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Page 1: 轉動慣量 Moments of Inertia - cct.me.ntut.edu.twcct.me.ntut.edu.tw/ccteducation/chchting/aiahtm/statics/lecture... · 轉動慣量或慣性矩 I. 轉動慣量也稱慣性矩(Moment

轉動慣量或慣性矩Moments of Inertia

Chen-Ching Ting(丁振卿)

Mechanical Engineering National Taipei University of Technology(國立台北科技大學機械系)

Homepage httpcctmentutedutwE-mail chchtingntutedutw

CCT Group

助教 吳穎彥

June 3 2011

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 1 54

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課程大綱

1 轉動慣量或慣性矩(Moment of Inertia)

2 面積慣性矩(Moment of Inertia for Area)面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)面積慣性矩之迴轉半徑(Radius of Gyration for Area)

3 複合面積之慣性矩(Moment of Inertia for Composite Area)

4 面積之慣性積(Product of Inertia for Area)平行軸定理(Parallel-Axis Theorem)

5 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)主軸慣性矩(Principal Moment of Inertia)

6 慣性矩之莫爾圓(Mohrrsquos Circle for Moment of Inertia)

7 質量慣性矩(Mass Moment of Inertia)平行軸定理(Parallel-Axis Theorem)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 2 54

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參考文獻

RC Hibbeler Statics Pearson Education Inc 歐亞書局有限公司ISBN=978-986-154-861-6 Chapter 10 2009

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 3 54

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轉轉轉動動動慣慣慣量量量或或或慣慣慣性性性矩矩矩Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 4 54

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慣性(Inertia)

慣性(Inertia)簡單來說乃物體持續維持不變的行為

伽利略(1632)提出一個不受任何外力(或者合外力為0)的物體將保持靜止或勻速直線運動

牛頓(1687)提出所有物體都將一直處於靜止或者勻速直線運動狀態直到出現施加其上的力改變它的運動狀態止

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 5 54

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轉動慣量或慣性矩 I

轉動慣量也稱慣性矩(Moment of Inertia)為物體對旋轉運動的慣性比較直線運動與旋轉運動得

F = ma = mdv

dt(1)

τ = Iα = Idω

dt(2)

其中F為力(N)m為質量(kg)a為加速度(ms2)v為速度(ms)τ為扭矩(N middotm)I為轉動慣量或慣性矩(kg middotm2)α為角加速度(rads2)ω為角速度(rads)

v = rω (3)

I = mr2 (4)

τ = rF = rmdv

dt= mr2

dt= Iα

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 6 54

轉動慣量或慣性矩 II

一般物件的動能K用轉動力學的定義取代

K =1

2mv2 (5)

=1

2m(rω)2 (6)

=1

2Iω2 (7)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 7 54

面面面積積積慣慣慣性性性矩矩矩Moment of Inertia for Area

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 8 54

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面積慣性矩

由左圖液面下y深度的壓力p為

p = γy (8)

其中γ = ρg為液體比重(SpecificWeight)液面下y深度之微小面積dA的受力dF為

dF = pdA = γydA (9)其扭矩dM為

dM = ydF = γy2dA (10)ˆdM = γ

ˆy2dA = γIx (11)

其中Ix =acutey2dA稱為x軸上之面積慣性矩

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 9 54

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極慣性矩(Polar Moment of Inertia)

極點

由左圖x軸上之面積慣性矩Ix及y軸上之面積慣性矩Iy如下

dIx = y2dA (12)

Ix =

ˆAy2dA (13)

dIy = x2dA (14)

Iy =

ˆAx2dA (15)

當微小面積dA對極點(O)或z軸旋轉時其面積極慣性矩JO如下

dJO = r2dA (16)

r2 = x2 + y2 (17)

JO =

ˆAr2dA =

ˆA(x2 + y2)dA = Ix + Iy (18)

其中r為微小面積dA到極點(O)的距離極點(O)簡單說乃z軸上的任意點由於x2 y2 r2及面

積(A)為正因此JO Ix Iy 均為正面積慣性矩的單位為m4mm4

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 10 54

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面面面積積積慣慣慣性性性矩矩矩之之之平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem for Area

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 11 54

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面積慣性矩之平行軸定理

極點

新極點

如左圖面積慣性矩之平行軸定理乃以面心(C)為新極點(O)設定新座標軸(xrsquoyrsquo)其中xrsquo軸平行於原座標x軸yrsquo軸平行於原座標y軸取微小面積dA其在新座標之點座標為(xrsquoyrsquo)則x軸上之面積慣性矩Ix及y軸上之面積慣性矩Iy可如下換算

Ix =

ˆA

(y prime + dy )2dA (19)

=

ˆAy prime2dA + 2dy

ˆAy primedA + d2

y

ˆAdA

Iy =

ˆA

(x prime + dx)2dA (20)

=

ˆAx prime2dA + 2dx

ˆAx primedA + d2

x

ˆAdA

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 12 54

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面積慣性矩之平行軸定理- Continued

由於面心在xrsquo軸上之等價力臂為y primeyrsquo軸上之等價力臂為x prime當以面心為轉軸點時即x prime = y prime = 0則x軸上之面積慣性矩Ix為ˆ

Ay prime2dA = macrIx prime (21)

ˆy primedA = y prime

ˆAdA = 0 (22)

d2y

ˆAdA = Ad2

y (23)

Ix =

ˆAy prime2dA + 2dy

ˆAy primedA + d2

y

ˆAdA (24)

= macrIx prime + Ad2y (25)

同理可得y軸上之面積慣性矩Iy

Iy =

ˆAx prime2dA + 2dx

ˆAx primedA + d2

x

ˆAdA (26)

= macrIy prime + Ad2x

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 13 54

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面積之極慣性矩

極點

新極點

由於通過面心之極慣性矩JC與極點(O)到面心(C)之距離d為

d2 = d2x + d2

y (27)

JC = macrIx prime + macrIy prime (28)

JO = Ix + Iy (29)

= macrIx prime + Ad2y + macrIy prime + Ad2

x (30)

= macrIx prime + macrIy prime + A(d2x + d2

y ) (31)

= JC + Ad2 (32)

也就是面積A之極慣性矩JO等於通過面積A之面心極慣性矩JC 與面積乘上極點(O)到面心(C)之距離d的平方和

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 14 54

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面面面積積積慣慣慣性性性矩矩矩之之之迴迴迴轉轉轉半半半徑徑徑Radius of Gyration for Area

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 15 54

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面積慣性矩之迴轉半徑

當x軸上之面積慣性矩Ix與y軸上之面積慣性矩Iy為已知時面積慣性矩之迴轉半徑可得知

dIx = y2dA (33)

Ix = kxA kx =

radicIxA

(34)

dIy = x2dA (35)

Iy = kyA ky =

radicIyA

(36)

dJO = r2dA (37)

JO = kOA kO =

radicJOA

(38)

其中(kx ky kO)分別為x軸y軸極點(O)之面積慣性矩迴轉半徑

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 16 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 17 54

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解題技巧 I

極點

新極點

繪圖並標出面積(A)之面心座標位置(C)接著以面心座標位置(C)為原點畫出新座標軸(xrsquoyrsquo)其中xrsquo軸平行於原(舊)座標x軸yrsquo軸平行於原(舊)座標y軸並寫出新舊座標軸的平移量dx與dy其中dx為yrsquo軸與y軸的平移量dy為xrsquo軸與x軸的平移量取微小面積dA其在原(舊)座標之點座標為(xy)在新座標之點座標為(xrsquoyrsquo)帶入x軸上之面積慣性矩Ix及y軸上之面積慣性矩Iy還有xrsquo軸上之面積慣性矩 macrIx prime及y軸上之面積慣性矩 macrIy prime可如下換算

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 18 54

解題技巧 II

Ix =

ˆAy2dA =

ˆAy prime2dA + 2dy

ˆAy primedA + d2

y

ˆAdA = macrIx prime + Ad2

y

macrIx prime =

ˆAy prime2dA (39)

Iy =

ˆAx2dA =

ˆAx prime2dA + 2dx

ˆAx primedA + d2

x

ˆAdA = macrIy prime + Ad2

x

macrIy prime =

ˆAx prime2dA (40)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 19 54

解題技巧 III

另外微小面積dA的面積慣性矩dIx dIy也可寫成

dIx = d macrIx prime + dAd2y (41)

Ix =

ˆdIx =

ˆd macrIx prime +

ˆdAd2

y (42)

dIy = d macrIy prime + dAd2x (43)

Iy =

ˆdIy =

ˆd macrIy prime +

ˆdAd2

x (44)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 20 54

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

面心慣性矩 慣性矩 極慣性矩

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 20 54

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 20 54

Example 10-1 解答

dA = bdy (45)

Ixb =

ˆAy2dA (46)

=

ˆ h

0y2bdy (47)

=1

3by3|h0 (48)

=1

3bh3 (49)

長方形之邊長b與h則面心慣性矩 macrIx prime macrIy prime與長方形之邊軸慣性矩Ix Iy為

macrIx prime =1

12bh3 (50)

macrIy prime =1

12hb3 (51)

Ix =1

3bh3 (52)

Iy =1

3hb3 (53)

JC = macrIx prime + macrIy prime (54)

JO = Ix + Iy (55)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 21 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 21 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 21 54

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Example 10-2 解答

取微小面積dA=ydx之長方形其邊長b=dx與h=ydy = y2所以

y2 = 400x (56)

y =radic

400x (57)

d macrIx prime = d(1

12bh3) =

1

12y3dx (58)

dIx = d macrIx prime + dAd2y =

1

12y3dx + ydx

(y2

)2=

1

3y3dx (59)

Ix =

ˆ 100

0

1

3y3dx =

ˆ 100

0

1

3

(radic400x

)3dx (60)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 22 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 22 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 22 54

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Example 10-3 解答

取微小面積dA=2ydx之長方形其邊長b=dx與h=2y所以

x2 + y2 = a2 (61)

y =radic

a2 minus x2 (62)

d macrIx prime = d(1

12bh3) =

1

12dx(2y)3 =

2

3y3dx (63)

macrIx prime =

ˆ a

minusa

2

3y3dx =

ˆ a

minusa

2

3

(radica2 minus x2

)3dx =

πa4

4(64)

圓之半徑a則面心慣性矩 macrIx prime macrIy prime為

macrIx prime =πa4

4(65)

macrIy prime =πa4

4(66)

JC = macrIx prime + macrIy prime =πa4

2(67)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 23 54

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複複複合合合面面面積積積之之之慣慣慣性性性矩矩矩Moment of Inertia for Composite Area

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 24 54

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (68)

k =

radicI

m(69)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (70)

= I1 + I2 + + In (71)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 25 54

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複合面積之慣性矩

複合面積之慣性矩(Ixc Iyc)為各局部面積之慣性矩(Ixi Iyi )合

Ixc =i=nsumi=1

Ixi (72)

= Ix1 + Ix2 + + Ixn (73)

Iyc =i=nsumi=1

Iyi (74)

= Iy1 + Iy2 + + Iyn (75)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 26 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

長方形與圓形複合

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

長方形複合

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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面面面積積積之之之慣慣慣性性性積積積Product of Inertia for Area

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 28 54

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面積之慣性積(Product of Inertia for Area)

為了下個章節求解慣性矩之極值(極大或極小值)方便數學運算在此先定義面積之慣性積(Ixy )為

dIxy = xydA (76)

Ixy =

ˆAxydA (77)

面積之慣性積的符號可為正負或零當慣性積之面積相對轉軸對稱時則面積之慣性積和為零

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 29 54

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 30 54

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平行軸定理

極點

新極點

如左圖面積慣性矩之平行軸定理乃以面心(C)為新極點(O)設定新座標軸(xrsquoyrsquo)其中xrsquo軸平行於原座標x軸yrsquo軸平行於原座標y軸取微小面積dA其在新座標之點座標為(xrsquoyrsquo)面積之慣性積(Ixy )為

dIxy = (x prime + dx)(y prime + dy )dA (78)

當以面心為轉軸點時即x prime = y prime = 0ˆAx primedA = x prime

ˆAdA = 0

ˆAy primedA = y prime

ˆAdA = 0 (79)

Ixy =

ˆA

(x prime + dx)(y prime + dy )dA (80)

=

ˆAx primey primedA + dx

ˆAy primedA + dy

ˆAx primedA + dxdy

ˆAdA (81)

= macrIx primey prime + Adxdy

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 31 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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傾傾傾斜斜斜軸軸軸面面面積積積之之之慣慣慣性性性矩矩矩

Moment of Inertia for Area about Inclined Axis

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 33 54

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傾斜軸面積之慣性矩

當轉動軸傾斜θ角時也就是座標軸旋轉θ角而Ix Iy Ixy為已知則新的座標軸(uv)可寫為

u = xcosθ + ysinθ (82)

v = ycosθ minus xsinθ (83)

由u軸上之面積慣性矩Iuv軸上之面積慣性矩Iv及慣性積Iuv得知

dIu = v2dA = (ycosθ minus xsinθ)2dA (84)

dIv = u2dA = (xcosθ + ysinθ)2dA (85)

dIuv = uvdA = (xcosθ + ysinθ)(ycosθ minus xsinθ)dA (86)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 34 54

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傾斜軸面積之慣性矩 I

由x軸上之面積慣性矩Ixy軸上之面積慣性矩Iy及慣性積Ixy得知

Ix =

ˆy2dA Iy =

ˆx2dA Ixy =

ˆxydA (87)

Iu =

ˆ(ycosθ minus xsinθ)2dA (88)

=

ˆ(y2cos2θ minus 2xycosθsinθ + x2sin2θ)dA (89)

= cos2θ

ˆy2dAminus 2cosθsinθ

ˆxydA + sin2θ

ˆx2dA (90)

= Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (91)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 35 54

傾斜軸面積之慣性矩 II

Iv =

ˆ(xcosθ + ysinθ)2dA (92)

=

ˆ(x2cos2θ + 2xycosθsinθ + y2sin2θ)dA (93)

= cos2θ

ˆx2dA + 2cosθsinθ

ˆxydA + sin2θ

ˆy2dA (94)

= Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (95)

Iuv =

ˆ(xcosθ + ysinθ)(ycosθ minus xsinθ)dA (96)

=

ˆ(xycos2θ + y2cosθsinθ minus x2cosθsinθ minus xysin2θ)dA (97)

= (cos2 minus sin2θ)

ˆxydA + cosθsinθ

ˆy2dAminus cosθsinθ

ˆx2dA

= Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 36 54

傾斜軸面積之慣性矩 III

再由

sin2θ + cos2θ = 1 (98)

sin(αplusmn β) = sinαcosβ plusmn cosαsinβ (99)

sin2θ = sin(θ + θ) = 2sinθcosθ (100)

cos(αplusmn β) = cosαcosβ ∓ sinαsinβ (101)

cos2θ = cos(θ + θ) = cos2θ minus sin2θ (102)

= 2cos2θ minus 1 = 1minus 2sin2θ (103)

所以

Iu = Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (104)

= Ix(cos2θ + 1

2) + Iy (

1minus cos2θ

2)minus Ixy sin2θ (105)

=Ix + Iy

2+

Ix minus Iy2

cos2θ minus Ixy sin2θ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 37 54

傾斜軸面積之慣性矩 IV

Iv = Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (106)

= Iy (cos2θ + 1

2) + Ix(

1minus cos2θ

2) + Ixy sin2θ (107)

=Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (108)

Iuv = Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ (109)

= Ixycos2θ +1

2Ixsin2θ minus 1

2Iy sin2θ (110)

=Ix minus Iy

2sin2θ + Ixycos2θ (111)

所以

JO = Iu + Iv = Ix + Iy (112)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 38 54

主主主軸軸軸慣慣慣性性性矩矩矩Principal Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 39 54

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主軸慣性矩 I

所謂主軸慣性矩(Principal Moment ofInertia)乃定義發生在慣性矩的極值(極大或極小值)上也就是慣性矩之一次微分式為零或斜率為零

dIu

dθ=

d(Ix+Iy

2+

IxminusIy2

cos2θ minus Ixy sin2θ)

dθ(113)

= minus2Ix minus Iy

2sin2θ minus 2Ixy cos2θ (114)

= 0 (115)

其中dsin2θ = cos2θ dcos2θ = minussin2θ當慣性矩之一次微分式為零時傾斜軸與主軸之夾角θ = θp

minus2Ix minus Iy

2sin2θp minus 2Ixycos2θp = 0 (116)

tan2θ =sin2θ

cos2θ=

minusIxy(Ix minus Iy )2

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 40 54

主軸慣性矩 II

cos2θp =IxminusIy2radic

(IxminusIy2 )2 + I 2xy

sin2θp =minusIxyradic

(IxminusIy2 )2 + I 2xy

(117)

(Iu)max =Ix + Iy

2+

Ix minus Iy2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

minus IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2+

radic(Ix minus Iy

2)2 + I 2xy (118)

(Iv )min =Ix + Iy

2minus Ix minus Iy

2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

+ IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2minusradic

(Ix minus Iy

2)2 + I 2xy (119)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 41 54

主軸慣性矩 III

由tan2θp = tan(π + 2θp)可得到θp1與θp2二個解即2θp2 = π + 2θp1也就是θp2 minus θp1 = π

2

tan2θp = tan(π + 2θp) (120)

2θp2 = π + 2θp1 (121)

θp2 minus θp1 =π

2(122)

所以

Imaxmin

=Ix + Iy

2plusmnradic

(Ix minus Iy

2)2 + I 2xy (123)

Iuv =Ix minus Iy

2

minusIxyradic(IxminusIy2 )2 + I 2xy

+ Ixy

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

(124)

= 0

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 42 54

課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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慣慣慣性性性矩矩矩之之之莫莫莫爾爾爾圓圓圓Mohrrsquos Circle for Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 44 54

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慣性矩之莫爾圓

由於Iu =

Ix + Iy2

+Ix minus Iy

2cos2θ minus Ixy sin2θ (125)

Iv =Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (126)

Iuv =Ix minus Iy

2sin2θ + Ixycos2θ (127)

將式(125)與式(127)各自分別平方並相加可得

(Iu minusIx + Iy

2)2 + I 2uv = (

Ix minus Iy2

)2 + I 2xy (128)

其中Ix Iy Ixy均為常數式(128)可進一步寫成

(Iu minus a)2 + I 2uv = R2 (129)

其中a =Ix+Iy2 R =

radic(IxminusIy2 )2 + I 2xycos2θp1 =

aRsin2θp1 =

minusIxyR 2θp1為OA轉向I軸方向即2θp1為

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 45 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 46 54

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解題技巧

首先確定轉動系統的Ix Iy Ixy接著算出莫爾圓之圓心座標a及半徑R並畫出莫爾圓再由莫爾圓上點出A(Ix Ixy )並由OA轉向I軸找出兩倍夾角2θp1及2θp2其中θp1及θp2為主軸(Principle Axis)與Ix軸的夾角由於主軸(Principle Axis)為極值之軸Imax = Iu及Imin = Iv決定θp1及θp2何者為Imax = Iu與Ix軸的夾角之方法乃比較Ix及Iy之大小Imax = Iu會靠近Ix及Iy之較大者也就是Imax = Iu與Ix及Iy之較大者的夾角比較小

a =Ix + Iy

2(130)

R =

radic(Ix minus Iy

2)2 + I 2xy (131)

2θp2 = π + 2θp1 θp2 minus θp1 =π

2(132)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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質質質量量量慣慣慣性性性矩矩矩Mass Moment of Inertia

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質量慣性矩

τ = Iα (133)

其中τ為旋轉扭矩I為質量慣性矩α為旋轉角速度

I =

ˆmr2dm (134)

=

ˆVr2ρdV (135)

= ρ

ˆVr2dV (136)

其中r為旋轉半徑或轉軸到旋轉體的距離

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

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平行軸定理

當通過物體質心(G)並在一轉軸上的質量慣性矩(IG )為已知時可透過座標軸轉換計算物體之質量慣性矩(I)

I =

ˆmr2dm (137)

=

ˆm

[(d + x prime)2 + y prime2]dm (138)

=

ˆm

(x prime2 + y prime2)dm + 2d

ˆmx primedm + d2

ˆmdm (139)

I = IG + md2 (140)

其中IG為在zrsquo軸上通過質心的質量慣性矩m為質量d為通過質心之軸zrsquo並平行原軸z之距離

acutem x primedm = x

acutem dm = 0 since x = 0

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (141)

k =

radicI

m(142)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (143)

= I1 + I2 + + In (144)

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課課課堂堂堂練練練習習習

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

兩圓盤複合體

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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結結結論論論

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結論

轉動慣量或慣性矩為物體對旋轉運動的慣性而所謂慣性乃物體持續維持不變的行為也就是定義慣性矩可表達物體對本身旋轉運動的維持或持續性好壞更簡單的說慣性矩越大之物體其維持旋轉運動趨勢的能力越大例如惰輪儲能裝置惰輪之慣性矩越大當其轉動以後其維持旋轉運動趨勢的能力越大

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  • 轉動慣量或慣性矩(Moment of Inertia)
  • 面積慣性矩(Moment of Inertia for Area)
    • 面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)
    • 面積慣性矩之迴轉半徑(Radius of Gyration for Area)
      • 複合面積之慣性矩(Moment of Inertia for Composite Area)
      • 面積之慣性積(Product of Inertia for Area)
        • 平行軸定理(Parallel-Axis Theorem)
          • 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)
            • 主軸慣性矩(Principal Moment of Inertia)
              • 慣性矩之莫爾圓(Mohrs Circle for Moment of Inertia)
              • 質量慣性矩(Mass Moment of Inertia)
                • 平行軸定理(Parallel-Axis Theorem)
Page 2: 轉動慣量 Moments of Inertia - cct.me.ntut.edu.twcct.me.ntut.edu.tw/ccteducation/chchting/aiahtm/statics/lecture... · 轉動慣量或慣性矩 I. 轉動慣量也稱慣性矩(Moment

課程大綱

1 轉動慣量或慣性矩(Moment of Inertia)

2 面積慣性矩(Moment of Inertia for Area)面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)面積慣性矩之迴轉半徑(Radius of Gyration for Area)

3 複合面積之慣性矩(Moment of Inertia for Composite Area)

4 面積之慣性積(Product of Inertia for Area)平行軸定理(Parallel-Axis Theorem)

5 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)主軸慣性矩(Principal Moment of Inertia)

6 慣性矩之莫爾圓(Mohrrsquos Circle for Moment of Inertia)

7 質量慣性矩(Mass Moment of Inertia)平行軸定理(Parallel-Axis Theorem)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 2 54

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參考文獻

RC Hibbeler Statics Pearson Education Inc 歐亞書局有限公司ISBN=978-986-154-861-6 Chapter 10 2009

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 3 54

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轉轉轉動動動慣慣慣量量量或或或慣慣慣性性性矩矩矩Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 4 54

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慣性(Inertia)

慣性(Inertia)簡單來說乃物體持續維持不變的行為

伽利略(1632)提出一個不受任何外力(或者合外力為0)的物體將保持靜止或勻速直線運動

牛頓(1687)提出所有物體都將一直處於靜止或者勻速直線運動狀態直到出現施加其上的力改變它的運動狀態止

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 5 54

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轉動慣量或慣性矩 I

轉動慣量也稱慣性矩(Moment of Inertia)為物體對旋轉運動的慣性比較直線運動與旋轉運動得

F = ma = mdv

dt(1)

τ = Iα = Idω

dt(2)

其中F為力(N)m為質量(kg)a為加速度(ms2)v為速度(ms)τ為扭矩(N middotm)I為轉動慣量或慣性矩(kg middotm2)α為角加速度(rads2)ω為角速度(rads)

v = rω (3)

I = mr2 (4)

τ = rF = rmdv

dt= mr2

dt= Iα

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轉動慣量或慣性矩 II

一般物件的動能K用轉動力學的定義取代

K =1

2mv2 (5)

=1

2m(rω)2 (6)

=1

2Iω2 (7)

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面面面積積積慣慣慣性性性矩矩矩Moment of Inertia for Area

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 8 54

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面積慣性矩

由左圖液面下y深度的壓力p為

p = γy (8)

其中γ = ρg為液體比重(SpecificWeight)液面下y深度之微小面積dA的受力dF為

dF = pdA = γydA (9)其扭矩dM為

dM = ydF = γy2dA (10)ˆdM = γ

ˆy2dA = γIx (11)

其中Ix =acutey2dA稱為x軸上之面積慣性矩

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極慣性矩(Polar Moment of Inertia)

極點

由左圖x軸上之面積慣性矩Ix及y軸上之面積慣性矩Iy如下

dIx = y2dA (12)

Ix =

ˆAy2dA (13)

dIy = x2dA (14)

Iy =

ˆAx2dA (15)

當微小面積dA對極點(O)或z軸旋轉時其面積極慣性矩JO如下

dJO = r2dA (16)

r2 = x2 + y2 (17)

JO =

ˆAr2dA =

ˆA(x2 + y2)dA = Ix + Iy (18)

其中r為微小面積dA到極點(O)的距離極點(O)簡單說乃z軸上的任意點由於x2 y2 r2及面

積(A)為正因此JO Ix Iy 均為正面積慣性矩的單位為m4mm4

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面面面積積積慣慣慣性性性矩矩矩之之之平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem for Area

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面積慣性矩之平行軸定理

極點

新極點

如左圖面積慣性矩之平行軸定理乃以面心(C)為新極點(O)設定新座標軸(xrsquoyrsquo)其中xrsquo軸平行於原座標x軸yrsquo軸平行於原座標y軸取微小面積dA其在新座標之點座標為(xrsquoyrsquo)則x軸上之面積慣性矩Ix及y軸上之面積慣性矩Iy可如下換算

Ix =

ˆA

(y prime + dy )2dA (19)

=

ˆAy prime2dA + 2dy

ˆAy primedA + d2

y

ˆAdA

Iy =

ˆA

(x prime + dx)2dA (20)

=

ˆAx prime2dA + 2dx

ˆAx primedA + d2

x

ˆAdA

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面積慣性矩之平行軸定理- Continued

由於面心在xrsquo軸上之等價力臂為y primeyrsquo軸上之等價力臂為x prime當以面心為轉軸點時即x prime = y prime = 0則x軸上之面積慣性矩Ix為ˆ

Ay prime2dA = macrIx prime (21)

ˆy primedA = y prime

ˆAdA = 0 (22)

d2y

ˆAdA = Ad2

y (23)

Ix =

ˆAy prime2dA + 2dy

ˆAy primedA + d2

y

ˆAdA (24)

= macrIx prime + Ad2y (25)

同理可得y軸上之面積慣性矩Iy

Iy =

ˆAx prime2dA + 2dx

ˆAx primedA + d2

x

ˆAdA (26)

= macrIy prime + Ad2x

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面積之極慣性矩

極點

新極點

由於通過面心之極慣性矩JC與極點(O)到面心(C)之距離d為

d2 = d2x + d2

y (27)

JC = macrIx prime + macrIy prime (28)

JO = Ix + Iy (29)

= macrIx prime + Ad2y + macrIy prime + Ad2

x (30)

= macrIx prime + macrIy prime + A(d2x + d2

y ) (31)

= JC + Ad2 (32)

也就是面積A之極慣性矩JO等於通過面積A之面心極慣性矩JC 與面積乘上極點(O)到面心(C)之距離d的平方和

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面面面積積積慣慣慣性性性矩矩矩之之之迴迴迴轉轉轉半半半徑徑徑Radius of Gyration for Area

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面積慣性矩之迴轉半徑

當x軸上之面積慣性矩Ix與y軸上之面積慣性矩Iy為已知時面積慣性矩之迴轉半徑可得知

dIx = y2dA (33)

Ix = kxA kx =

radicIxA

(34)

dIy = x2dA (35)

Iy = kyA ky =

radicIyA

(36)

dJO = r2dA (37)

JO = kOA kO =

radicJOA

(38)

其中(kx ky kO)分別為x軸y軸極點(O)之面積慣性矩迴轉半徑

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課課課堂堂堂練練練習習習

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解題技巧 I

極點

新極點

繪圖並標出面積(A)之面心座標位置(C)接著以面心座標位置(C)為原點畫出新座標軸(xrsquoyrsquo)其中xrsquo軸平行於原(舊)座標x軸yrsquo軸平行於原(舊)座標y軸並寫出新舊座標軸的平移量dx與dy其中dx為yrsquo軸與y軸的平移量dy為xrsquo軸與x軸的平移量取微小面積dA其在原(舊)座標之點座標為(xy)在新座標之點座標為(xrsquoyrsquo)帶入x軸上之面積慣性矩Ix及y軸上之面積慣性矩Iy還有xrsquo軸上之面積慣性矩 macrIx prime及y軸上之面積慣性矩 macrIy prime可如下換算

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解題技巧 II

Ix =

ˆAy2dA =

ˆAy prime2dA + 2dy

ˆAy primedA + d2

y

ˆAdA = macrIx prime + Ad2

y

macrIx prime =

ˆAy prime2dA (39)

Iy =

ˆAx2dA =

ˆAx prime2dA + 2dx

ˆAx primedA + d2

x

ˆAdA = macrIy prime + Ad2

x

macrIy prime =

ˆAx prime2dA (40)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 19 54

解題技巧 III

另外微小面積dA的面積慣性矩dIx dIy也可寫成

dIx = d macrIx prime + dAd2y (41)

Ix =

ˆdIx =

ˆd macrIx prime +

ˆdAd2

y (42)

dIy = d macrIy prime + dAd2x (43)

Iy =

ˆdIy =

ˆd macrIy prime +

ˆdAd2

x (44)

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

面心慣性矩 慣性矩 極慣性矩

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Example 10-1 解答

dA = bdy (45)

Ixb =

ˆAy2dA (46)

=

ˆ h

0y2bdy (47)

=1

3by3|h0 (48)

=1

3bh3 (49)

長方形之邊長b與h則面心慣性矩 macrIx prime macrIy prime與長方形之邊軸慣性矩Ix Iy為

macrIx prime =1

12bh3 (50)

macrIy prime =1

12hb3 (51)

Ix =1

3bh3 (52)

Iy =1

3hb3 (53)

JC = macrIx prime + macrIy prime (54)

JO = Ix + Iy (55)

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Example 10-2 解答

取微小面積dA=ydx之長方形其邊長b=dx與h=ydy = y2所以

y2 = 400x (56)

y =radic

400x (57)

d macrIx prime = d(1

12bh3) =

1

12y3dx (58)

dIx = d macrIx prime + dAd2y =

1

12y3dx + ydx

(y2

)2=

1

3y3dx (59)

Ix =

ˆ 100

0

1

3y3dx =

ˆ 100

0

1

3

(radic400x

)3dx (60)

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Example 10-3 解答

取微小面積dA=2ydx之長方形其邊長b=dx與h=2y所以

x2 + y2 = a2 (61)

y =radic

a2 minus x2 (62)

d macrIx prime = d(1

12bh3) =

1

12dx(2y)3 =

2

3y3dx (63)

macrIx prime =

ˆ a

minusa

2

3y3dx =

ˆ a

minusa

2

3

(radica2 minus x2

)3dx =

πa4

4(64)

圓之半徑a則面心慣性矩 macrIx prime macrIy prime為

macrIx prime =πa4

4(65)

macrIy prime =πa4

4(66)

JC = macrIx prime + macrIy prime =πa4

2(67)

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複複複合合合面面面積積積之之之慣慣慣性性性矩矩矩Moment of Inertia for Composite Area

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (68)

k =

radicI

m(69)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (70)

= I1 + I2 + + In (71)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 25 54

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複合面積之慣性矩

複合面積之慣性矩(Ixc Iyc)為各局部面積之慣性矩(Ixi Iyi )合

Ixc =i=nsumi=1

Ixi (72)

= Ix1 + Ix2 + + Ixn (73)

Iyc =i=nsumi=1

Iyi (74)

= Iy1 + Iy2 + + Iyn (75)

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課課課堂堂堂練練練習習習

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

長方形與圓形複合

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

長方形複合

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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面面面積積積之之之慣慣慣性性性積積積Product of Inertia for Area

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面積之慣性積(Product of Inertia for Area)

為了下個章節求解慣性矩之極值(極大或極小值)方便數學運算在此先定義面積之慣性積(Ixy )為

dIxy = xydA (76)

Ixy =

ˆAxydA (77)

面積之慣性積的符號可為正負或零當慣性積之面積相對轉軸對稱時則面積之慣性積和為零

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

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平行軸定理

極點

新極點

如左圖面積慣性矩之平行軸定理乃以面心(C)為新極點(O)設定新座標軸(xrsquoyrsquo)其中xrsquo軸平行於原座標x軸yrsquo軸平行於原座標y軸取微小面積dA其在新座標之點座標為(xrsquoyrsquo)面積之慣性積(Ixy )為

dIxy = (x prime + dx)(y prime + dy )dA (78)

當以面心為轉軸點時即x prime = y prime = 0ˆAx primedA = x prime

ˆAdA = 0

ˆAy primedA = y prime

ˆAdA = 0 (79)

Ixy =

ˆA

(x prime + dx)(y prime + dy )dA (80)

=

ˆAx primey primedA + dx

ˆAy primedA + dy

ˆAx primedA + dxdy

ˆAdA (81)

= macrIx primey prime + Adxdy

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課課課堂堂堂練練練習習習

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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傾傾傾斜斜斜軸軸軸面面面積積積之之之慣慣慣性性性矩矩矩

Moment of Inertia for Area about Inclined Axis

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傾斜軸面積之慣性矩

當轉動軸傾斜θ角時也就是座標軸旋轉θ角而Ix Iy Ixy為已知則新的座標軸(uv)可寫為

u = xcosθ + ysinθ (82)

v = ycosθ minus xsinθ (83)

由u軸上之面積慣性矩Iuv軸上之面積慣性矩Iv及慣性積Iuv得知

dIu = v2dA = (ycosθ minus xsinθ)2dA (84)

dIv = u2dA = (xcosθ + ysinθ)2dA (85)

dIuv = uvdA = (xcosθ + ysinθ)(ycosθ minus xsinθ)dA (86)

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傾斜軸面積之慣性矩 I

由x軸上之面積慣性矩Ixy軸上之面積慣性矩Iy及慣性積Ixy得知

Ix =

ˆy2dA Iy =

ˆx2dA Ixy =

ˆxydA (87)

Iu =

ˆ(ycosθ minus xsinθ)2dA (88)

=

ˆ(y2cos2θ minus 2xycosθsinθ + x2sin2θ)dA (89)

= cos2θ

ˆy2dAminus 2cosθsinθ

ˆxydA + sin2θ

ˆx2dA (90)

= Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (91)

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傾斜軸面積之慣性矩 II

Iv =

ˆ(xcosθ + ysinθ)2dA (92)

=

ˆ(x2cos2θ + 2xycosθsinθ + y2sin2θ)dA (93)

= cos2θ

ˆx2dA + 2cosθsinθ

ˆxydA + sin2θ

ˆy2dA (94)

= Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (95)

Iuv =

ˆ(xcosθ + ysinθ)(ycosθ minus xsinθ)dA (96)

=

ˆ(xycos2θ + y2cosθsinθ minus x2cosθsinθ minus xysin2θ)dA (97)

= (cos2 minus sin2θ)

ˆxydA + cosθsinθ

ˆy2dAminus cosθsinθ

ˆx2dA

= Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 36 54

傾斜軸面積之慣性矩 III

再由

sin2θ + cos2θ = 1 (98)

sin(αplusmn β) = sinαcosβ plusmn cosαsinβ (99)

sin2θ = sin(θ + θ) = 2sinθcosθ (100)

cos(αplusmn β) = cosαcosβ ∓ sinαsinβ (101)

cos2θ = cos(θ + θ) = cos2θ minus sin2θ (102)

= 2cos2θ minus 1 = 1minus 2sin2θ (103)

所以

Iu = Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (104)

= Ix(cos2θ + 1

2) + Iy (

1minus cos2θ

2)minus Ixy sin2θ (105)

=Ix + Iy

2+

Ix minus Iy2

cos2θ minus Ixy sin2θ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 37 54

傾斜軸面積之慣性矩 IV

Iv = Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (106)

= Iy (cos2θ + 1

2) + Ix(

1minus cos2θ

2) + Ixy sin2θ (107)

=Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (108)

Iuv = Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ (109)

= Ixycos2θ +1

2Ixsin2θ minus 1

2Iy sin2θ (110)

=Ix minus Iy

2sin2θ + Ixycos2θ (111)

所以

JO = Iu + Iv = Ix + Iy (112)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 38 54

主主主軸軸軸慣慣慣性性性矩矩矩Principal Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 39 54

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主軸慣性矩 I

所謂主軸慣性矩(Principal Moment ofInertia)乃定義發生在慣性矩的極值(極大或極小值)上也就是慣性矩之一次微分式為零或斜率為零

dIu

dθ=

d(Ix+Iy

2+

IxminusIy2

cos2θ minus Ixy sin2θ)

dθ(113)

= minus2Ix minus Iy

2sin2θ minus 2Ixy cos2θ (114)

= 0 (115)

其中dsin2θ = cos2θ dcos2θ = minussin2θ當慣性矩之一次微分式為零時傾斜軸與主軸之夾角θ = θp

minus2Ix minus Iy

2sin2θp minus 2Ixycos2θp = 0 (116)

tan2θ =sin2θ

cos2θ=

minusIxy(Ix minus Iy )2

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 40 54

主軸慣性矩 II

cos2θp =IxminusIy2radic

(IxminusIy2 )2 + I 2xy

sin2θp =minusIxyradic

(IxminusIy2 )2 + I 2xy

(117)

(Iu)max =Ix + Iy

2+

Ix minus Iy2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

minus IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2+

radic(Ix minus Iy

2)2 + I 2xy (118)

(Iv )min =Ix + Iy

2minus Ix minus Iy

2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

+ IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2minusradic

(Ix minus Iy

2)2 + I 2xy (119)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 41 54

主軸慣性矩 III

由tan2θp = tan(π + 2θp)可得到θp1與θp2二個解即2θp2 = π + 2θp1也就是θp2 minus θp1 = π

2

tan2θp = tan(π + 2θp) (120)

2θp2 = π + 2θp1 (121)

θp2 minus θp1 =π

2(122)

所以

Imaxmin

=Ix + Iy

2plusmnradic

(Ix minus Iy

2)2 + I 2xy (123)

Iuv =Ix minus Iy

2

minusIxyradic(IxminusIy2 )2 + I 2xy

+ Ixy

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

(124)

= 0

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 42 54

課課課堂堂堂練練練習習習

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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慣慣慣性性性矩矩矩之之之莫莫莫爾爾爾圓圓圓Mohrrsquos Circle for Moment of Inertia

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慣性矩之莫爾圓

由於Iu =

Ix + Iy2

+Ix minus Iy

2cos2θ minus Ixy sin2θ (125)

Iv =Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (126)

Iuv =Ix minus Iy

2sin2θ + Ixycos2θ (127)

將式(125)與式(127)各自分別平方並相加可得

(Iu minusIx + Iy

2)2 + I 2uv = (

Ix minus Iy2

)2 + I 2xy (128)

其中Ix Iy Ixy均為常數式(128)可進一步寫成

(Iu minus a)2 + I 2uv = R2 (129)

其中a =Ix+Iy2 R =

radic(IxminusIy2 )2 + I 2xycos2θp1 =

aRsin2θp1 =

minusIxyR 2θp1為OA轉向I軸方向即2θp1為

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 46 54

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解題技巧

首先確定轉動系統的Ix Iy Ixy接著算出莫爾圓之圓心座標a及半徑R並畫出莫爾圓再由莫爾圓上點出A(Ix Ixy )並由OA轉向I軸找出兩倍夾角2θp1及2θp2其中θp1及θp2為主軸(Principle Axis)與Ix軸的夾角由於主軸(Principle Axis)為極值之軸Imax = Iu及Imin = Iv決定θp1及θp2何者為Imax = Iu與Ix軸的夾角之方法乃比較Ix及Iy之大小Imax = Iu會靠近Ix及Iy之較大者也就是Imax = Iu與Ix及Iy之較大者的夾角比較小

a =Ix + Iy

2(130)

R =

radic(Ix minus Iy

2)2 + I 2xy (131)

2θp2 = π + 2θp1 θp2 minus θp1 =π

2(132)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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質質質量量量慣慣慣性性性矩矩矩Mass Moment of Inertia

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質量慣性矩

τ = Iα (133)

其中τ為旋轉扭矩I為質量慣性矩α為旋轉角速度

I =

ˆmr2dm (134)

=

ˆVr2ρdV (135)

= ρ

ˆVr2dV (136)

其中r為旋轉半徑或轉軸到旋轉體的距離

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 50 54

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平行軸定理

當通過物體質心(G)並在一轉軸上的質量慣性矩(IG )為已知時可透過座標軸轉換計算物體之質量慣性矩(I)

I =

ˆmr2dm (137)

=

ˆm

[(d + x prime)2 + y prime2]dm (138)

=

ˆm

(x prime2 + y prime2)dm + 2d

ˆmx primedm + d2

ˆmdm (139)

I = IG + md2 (140)

其中IG為在zrsquo軸上通過質心的質量慣性矩m為質量d為通過質心之軸zrsquo並平行原軸z之距離

acutem x primedm = x

acutem dm = 0 since x = 0

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (141)

k =

radicI

m(142)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (143)

= I1 + I2 + + In (144)

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課課課堂堂堂練練練習習習

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

兩圓盤複合體

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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結結結論論論

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結論

轉動慣量或慣性矩為物體對旋轉運動的慣性而所謂慣性乃物體持續維持不變的行為也就是定義慣性矩可表達物體對本身旋轉運動的維持或持續性好壞更簡單的說慣性矩越大之物體其維持旋轉運動趨勢的能力越大例如惰輪儲能裝置惰輪之慣性矩越大當其轉動以後其維持旋轉運動趨勢的能力越大

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 55 54

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  • 轉動慣量或慣性矩(Moment of Inertia)
  • 面積慣性矩(Moment of Inertia for Area)
    • 面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)
    • 面積慣性矩之迴轉半徑(Radius of Gyration for Area)
      • 複合面積之慣性矩(Moment of Inertia for Composite Area)
      • 面積之慣性積(Product of Inertia for Area)
        • 平行軸定理(Parallel-Axis Theorem)
          • 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)
            • 主軸慣性矩(Principal Moment of Inertia)
              • 慣性矩之莫爾圓(Mohrs Circle for Moment of Inertia)
              • 質量慣性矩(Mass Moment of Inertia)
                • 平行軸定理(Parallel-Axis Theorem)
Page 3: 轉動慣量 Moments of Inertia - cct.me.ntut.edu.twcct.me.ntut.edu.tw/ccteducation/chchting/aiahtm/statics/lecture... · 轉動慣量或慣性矩 I. 轉動慣量也稱慣性矩(Moment

參考文獻

RC Hibbeler Statics Pearson Education Inc 歐亞書局有限公司ISBN=978-986-154-861-6 Chapter 10 2009

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 3 54

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轉轉轉動動動慣慣慣量量量或或或慣慣慣性性性矩矩矩Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 4 54

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慣性(Inertia)

慣性(Inertia)簡單來說乃物體持續維持不變的行為

伽利略(1632)提出一個不受任何外力(或者合外力為0)的物體將保持靜止或勻速直線運動

牛頓(1687)提出所有物體都將一直處於靜止或者勻速直線運動狀態直到出現施加其上的力改變它的運動狀態止

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 5 54

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轉動慣量或慣性矩 I

轉動慣量也稱慣性矩(Moment of Inertia)為物體對旋轉運動的慣性比較直線運動與旋轉運動得

F = ma = mdv

dt(1)

τ = Iα = Idω

dt(2)

其中F為力(N)m為質量(kg)a為加速度(ms2)v為速度(ms)τ為扭矩(N middotm)I為轉動慣量或慣性矩(kg middotm2)α為角加速度(rads2)ω為角速度(rads)

v = rω (3)

I = mr2 (4)

τ = rF = rmdv

dt= mr2

dt= Iα

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 6 54

轉動慣量或慣性矩 II

一般物件的動能K用轉動力學的定義取代

K =1

2mv2 (5)

=1

2m(rω)2 (6)

=1

2Iω2 (7)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 7 54

面面面積積積慣慣慣性性性矩矩矩Moment of Inertia for Area

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 8 54

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面積慣性矩

由左圖液面下y深度的壓力p為

p = γy (8)

其中γ = ρg為液體比重(SpecificWeight)液面下y深度之微小面積dA的受力dF為

dF = pdA = γydA (9)其扭矩dM為

dM = ydF = γy2dA (10)ˆdM = γ

ˆy2dA = γIx (11)

其中Ix =acutey2dA稱為x軸上之面積慣性矩

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 9 54

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極慣性矩(Polar Moment of Inertia)

極點

由左圖x軸上之面積慣性矩Ix及y軸上之面積慣性矩Iy如下

dIx = y2dA (12)

Ix =

ˆAy2dA (13)

dIy = x2dA (14)

Iy =

ˆAx2dA (15)

當微小面積dA對極點(O)或z軸旋轉時其面積極慣性矩JO如下

dJO = r2dA (16)

r2 = x2 + y2 (17)

JO =

ˆAr2dA =

ˆA(x2 + y2)dA = Ix + Iy (18)

其中r為微小面積dA到極點(O)的距離極點(O)簡單說乃z軸上的任意點由於x2 y2 r2及面

積(A)為正因此JO Ix Iy 均為正面積慣性矩的單位為m4mm4

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 10 54

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面面面積積積慣慣慣性性性矩矩矩之之之平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem for Area

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 11 54

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面積慣性矩之平行軸定理

極點

新極點

如左圖面積慣性矩之平行軸定理乃以面心(C)為新極點(O)設定新座標軸(xrsquoyrsquo)其中xrsquo軸平行於原座標x軸yrsquo軸平行於原座標y軸取微小面積dA其在新座標之點座標為(xrsquoyrsquo)則x軸上之面積慣性矩Ix及y軸上之面積慣性矩Iy可如下換算

Ix =

ˆA

(y prime + dy )2dA (19)

=

ˆAy prime2dA + 2dy

ˆAy primedA + d2

y

ˆAdA

Iy =

ˆA

(x prime + dx)2dA (20)

=

ˆAx prime2dA + 2dx

ˆAx primedA + d2

x

ˆAdA

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面積慣性矩之平行軸定理- Continued

由於面心在xrsquo軸上之等價力臂為y primeyrsquo軸上之等價力臂為x prime當以面心為轉軸點時即x prime = y prime = 0則x軸上之面積慣性矩Ix為ˆ

Ay prime2dA = macrIx prime (21)

ˆy primedA = y prime

ˆAdA = 0 (22)

d2y

ˆAdA = Ad2

y (23)

Ix =

ˆAy prime2dA + 2dy

ˆAy primedA + d2

y

ˆAdA (24)

= macrIx prime + Ad2y (25)

同理可得y軸上之面積慣性矩Iy

Iy =

ˆAx prime2dA + 2dx

ˆAx primedA + d2

x

ˆAdA (26)

= macrIy prime + Ad2x

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面積之極慣性矩

極點

新極點

由於通過面心之極慣性矩JC與極點(O)到面心(C)之距離d為

d2 = d2x + d2

y (27)

JC = macrIx prime + macrIy prime (28)

JO = Ix + Iy (29)

= macrIx prime + Ad2y + macrIy prime + Ad2

x (30)

= macrIx prime + macrIy prime + A(d2x + d2

y ) (31)

= JC + Ad2 (32)

也就是面積A之極慣性矩JO等於通過面積A之面心極慣性矩JC 與面積乘上極點(O)到面心(C)之距離d的平方和

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面面面積積積慣慣慣性性性矩矩矩之之之迴迴迴轉轉轉半半半徑徑徑Radius of Gyration for Area

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面積慣性矩之迴轉半徑

當x軸上之面積慣性矩Ix與y軸上之面積慣性矩Iy為已知時面積慣性矩之迴轉半徑可得知

dIx = y2dA (33)

Ix = kxA kx =

radicIxA

(34)

dIy = x2dA (35)

Iy = kyA ky =

radicIyA

(36)

dJO = r2dA (37)

JO = kOA kO =

radicJOA

(38)

其中(kx ky kO)分別為x軸y軸極點(O)之面積慣性矩迴轉半徑

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課課課堂堂堂練練練習習習

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解題技巧 I

極點

新極點

繪圖並標出面積(A)之面心座標位置(C)接著以面心座標位置(C)為原點畫出新座標軸(xrsquoyrsquo)其中xrsquo軸平行於原(舊)座標x軸yrsquo軸平行於原(舊)座標y軸並寫出新舊座標軸的平移量dx與dy其中dx為yrsquo軸與y軸的平移量dy為xrsquo軸與x軸的平移量取微小面積dA其在原(舊)座標之點座標為(xy)在新座標之點座標為(xrsquoyrsquo)帶入x軸上之面積慣性矩Ix及y軸上之面積慣性矩Iy還有xrsquo軸上之面積慣性矩 macrIx prime及y軸上之面積慣性矩 macrIy prime可如下換算

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解題技巧 II

Ix =

ˆAy2dA =

ˆAy prime2dA + 2dy

ˆAy primedA + d2

y

ˆAdA = macrIx prime + Ad2

y

macrIx prime =

ˆAy prime2dA (39)

Iy =

ˆAx2dA =

ˆAx prime2dA + 2dx

ˆAx primedA + d2

x

ˆAdA = macrIy prime + Ad2

x

macrIy prime =

ˆAx prime2dA (40)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 19 54

解題技巧 III

另外微小面積dA的面積慣性矩dIx dIy也可寫成

dIx = d macrIx prime + dAd2y (41)

Ix =

ˆdIx =

ˆd macrIx prime +

ˆdAd2

y (42)

dIy = d macrIy prime + dAd2x (43)

Iy =

ˆdIy =

ˆd macrIy prime +

ˆdAd2

x (44)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 20 54

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

面心慣性矩 慣性矩 極慣性矩

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 20 54

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Example 10-1 解答

dA = bdy (45)

Ixb =

ˆAy2dA (46)

=

ˆ h

0y2bdy (47)

=1

3by3|h0 (48)

=1

3bh3 (49)

長方形之邊長b與h則面心慣性矩 macrIx prime macrIy prime與長方形之邊軸慣性矩Ix Iy為

macrIx prime =1

12bh3 (50)

macrIy prime =1

12hb3 (51)

Ix =1

3bh3 (52)

Iy =1

3hb3 (53)

JC = macrIx prime + macrIy prime (54)

JO = Ix + Iy (55)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 21 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 21 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 21 54

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Example 10-2 解答

取微小面積dA=ydx之長方形其邊長b=dx與h=ydy = y2所以

y2 = 400x (56)

y =radic

400x (57)

d macrIx prime = d(1

12bh3) =

1

12y3dx (58)

dIx = d macrIx prime + dAd2y =

1

12y3dx + ydx

(y2

)2=

1

3y3dx (59)

Ix =

ˆ 100

0

1

3y3dx =

ˆ 100

0

1

3

(radic400x

)3dx (60)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 22 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 22 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 22 54

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Example 10-3 解答

取微小面積dA=2ydx之長方形其邊長b=dx與h=2y所以

x2 + y2 = a2 (61)

y =radic

a2 minus x2 (62)

d macrIx prime = d(1

12bh3) =

1

12dx(2y)3 =

2

3y3dx (63)

macrIx prime =

ˆ a

minusa

2

3y3dx =

ˆ a

minusa

2

3

(radica2 minus x2

)3dx =

πa4

4(64)

圓之半徑a則面心慣性矩 macrIx prime macrIy prime為

macrIx prime =πa4

4(65)

macrIy prime =πa4

4(66)

JC = macrIx prime + macrIy prime =πa4

2(67)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 23 54

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複複複合合合面面面積積積之之之慣慣慣性性性矩矩矩Moment of Inertia for Composite Area

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (68)

k =

radicI

m(69)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (70)

= I1 + I2 + + In (71)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 25 54

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複合面積之慣性矩

複合面積之慣性矩(Ixc Iyc)為各局部面積之慣性矩(Ixi Iyi )合

Ixc =i=nsumi=1

Ixi (72)

= Ix1 + Ix2 + + Ixn (73)

Iyc =i=nsumi=1

Iyi (74)

= Iy1 + Iy2 + + Iyn (75)

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課課課堂堂堂練練練習習習

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

長方形與圓形複合

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

長方形複合

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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面面面積積積之之之慣慣慣性性性積積積Product of Inertia for Area

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 28 54

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面積之慣性積(Product of Inertia for Area)

為了下個章節求解慣性矩之極值(極大或極小值)方便數學運算在此先定義面積之慣性積(Ixy )為

dIxy = xydA (76)

Ixy =

ˆAxydA (77)

面積之慣性積的符號可為正負或零當慣性積之面積相對轉軸對稱時則面積之慣性積和為零

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 29 54

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 30 54

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平行軸定理

極點

新極點

如左圖面積慣性矩之平行軸定理乃以面心(C)為新極點(O)設定新座標軸(xrsquoyrsquo)其中xrsquo軸平行於原座標x軸yrsquo軸平行於原座標y軸取微小面積dA其在新座標之點座標為(xrsquoyrsquo)面積之慣性積(Ixy )為

dIxy = (x prime + dx)(y prime + dy )dA (78)

當以面心為轉軸點時即x prime = y prime = 0ˆAx primedA = x prime

ˆAdA = 0

ˆAy primedA = y prime

ˆAdA = 0 (79)

Ixy =

ˆA

(x prime + dx)(y prime + dy )dA (80)

=

ˆAx primey primedA + dx

ˆAy primedA + dy

ˆAx primedA + dxdy

ˆAdA (81)

= macrIx primey prime + Adxdy

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 31 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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傾傾傾斜斜斜軸軸軸面面面積積積之之之慣慣慣性性性矩矩矩

Moment of Inertia for Area about Inclined Axis

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 33 54

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傾斜軸面積之慣性矩

當轉動軸傾斜θ角時也就是座標軸旋轉θ角而Ix Iy Ixy為已知則新的座標軸(uv)可寫為

u = xcosθ + ysinθ (82)

v = ycosθ minus xsinθ (83)

由u軸上之面積慣性矩Iuv軸上之面積慣性矩Iv及慣性積Iuv得知

dIu = v2dA = (ycosθ minus xsinθ)2dA (84)

dIv = u2dA = (xcosθ + ysinθ)2dA (85)

dIuv = uvdA = (xcosθ + ysinθ)(ycosθ minus xsinθ)dA (86)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 34 54

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傾斜軸面積之慣性矩 I

由x軸上之面積慣性矩Ixy軸上之面積慣性矩Iy及慣性積Ixy得知

Ix =

ˆy2dA Iy =

ˆx2dA Ixy =

ˆxydA (87)

Iu =

ˆ(ycosθ minus xsinθ)2dA (88)

=

ˆ(y2cos2θ minus 2xycosθsinθ + x2sin2θ)dA (89)

= cos2θ

ˆy2dAminus 2cosθsinθ

ˆxydA + sin2θ

ˆx2dA (90)

= Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (91)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 35 54

傾斜軸面積之慣性矩 II

Iv =

ˆ(xcosθ + ysinθ)2dA (92)

=

ˆ(x2cos2θ + 2xycosθsinθ + y2sin2θ)dA (93)

= cos2θ

ˆx2dA + 2cosθsinθ

ˆxydA + sin2θ

ˆy2dA (94)

= Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (95)

Iuv =

ˆ(xcosθ + ysinθ)(ycosθ minus xsinθ)dA (96)

=

ˆ(xycos2θ + y2cosθsinθ minus x2cosθsinθ minus xysin2θ)dA (97)

= (cos2 minus sin2θ)

ˆxydA + cosθsinθ

ˆy2dAminus cosθsinθ

ˆx2dA

= Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 36 54

傾斜軸面積之慣性矩 III

再由

sin2θ + cos2θ = 1 (98)

sin(αplusmn β) = sinαcosβ plusmn cosαsinβ (99)

sin2θ = sin(θ + θ) = 2sinθcosθ (100)

cos(αplusmn β) = cosαcosβ ∓ sinαsinβ (101)

cos2θ = cos(θ + θ) = cos2θ minus sin2θ (102)

= 2cos2θ minus 1 = 1minus 2sin2θ (103)

所以

Iu = Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (104)

= Ix(cos2θ + 1

2) + Iy (

1minus cos2θ

2)minus Ixy sin2θ (105)

=Ix + Iy

2+

Ix minus Iy2

cos2θ minus Ixy sin2θ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 37 54

傾斜軸面積之慣性矩 IV

Iv = Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (106)

= Iy (cos2θ + 1

2) + Ix(

1minus cos2θ

2) + Ixy sin2θ (107)

=Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (108)

Iuv = Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ (109)

= Ixycos2θ +1

2Ixsin2θ minus 1

2Iy sin2θ (110)

=Ix minus Iy

2sin2θ + Ixycos2θ (111)

所以

JO = Iu + Iv = Ix + Iy (112)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 38 54

主主主軸軸軸慣慣慣性性性矩矩矩Principal Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 39 54

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主軸慣性矩 I

所謂主軸慣性矩(Principal Moment ofInertia)乃定義發生在慣性矩的極值(極大或極小值)上也就是慣性矩之一次微分式為零或斜率為零

dIu

dθ=

d(Ix+Iy

2+

IxminusIy2

cos2θ minus Ixy sin2θ)

dθ(113)

= minus2Ix minus Iy

2sin2θ minus 2Ixy cos2θ (114)

= 0 (115)

其中dsin2θ = cos2θ dcos2θ = minussin2θ當慣性矩之一次微分式為零時傾斜軸與主軸之夾角θ = θp

minus2Ix minus Iy

2sin2θp minus 2Ixycos2θp = 0 (116)

tan2θ =sin2θ

cos2θ=

minusIxy(Ix minus Iy )2

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 40 54

主軸慣性矩 II

cos2θp =IxminusIy2radic

(IxminusIy2 )2 + I 2xy

sin2θp =minusIxyradic

(IxminusIy2 )2 + I 2xy

(117)

(Iu)max =Ix + Iy

2+

Ix minus Iy2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

minus IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2+

radic(Ix minus Iy

2)2 + I 2xy (118)

(Iv )min =Ix + Iy

2minus Ix minus Iy

2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

+ IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2minusradic

(Ix minus Iy

2)2 + I 2xy (119)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 41 54

主軸慣性矩 III

由tan2θp = tan(π + 2θp)可得到θp1與θp2二個解即2θp2 = π + 2θp1也就是θp2 minus θp1 = π

2

tan2θp = tan(π + 2θp) (120)

2θp2 = π + 2θp1 (121)

θp2 minus θp1 =π

2(122)

所以

Imaxmin

=Ix + Iy

2plusmnradic

(Ix minus Iy

2)2 + I 2xy (123)

Iuv =Ix minus Iy

2

minusIxyradic(IxminusIy2 )2 + I 2xy

+ Ixy

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

(124)

= 0

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 42 54

課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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慣慣慣性性性矩矩矩之之之莫莫莫爾爾爾圓圓圓Mohrrsquos Circle for Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 44 54

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慣性矩之莫爾圓

由於Iu =

Ix + Iy2

+Ix minus Iy

2cos2θ minus Ixy sin2θ (125)

Iv =Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (126)

Iuv =Ix minus Iy

2sin2θ + Ixycos2θ (127)

將式(125)與式(127)各自分別平方並相加可得

(Iu minusIx + Iy

2)2 + I 2uv = (

Ix minus Iy2

)2 + I 2xy (128)

其中Ix Iy Ixy均為常數式(128)可進一步寫成

(Iu minus a)2 + I 2uv = R2 (129)

其中a =Ix+Iy2 R =

radic(IxminusIy2 )2 + I 2xycos2θp1 =

aRsin2θp1 =

minusIxyR 2θp1為OA轉向I軸方向即2θp1為

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 45 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 46 54

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解題技巧

首先確定轉動系統的Ix Iy Ixy接著算出莫爾圓之圓心座標a及半徑R並畫出莫爾圓再由莫爾圓上點出A(Ix Ixy )並由OA轉向I軸找出兩倍夾角2θp1及2θp2其中θp1及θp2為主軸(Principle Axis)與Ix軸的夾角由於主軸(Principle Axis)為極值之軸Imax = Iu及Imin = Iv決定θp1及θp2何者為Imax = Iu與Ix軸的夾角之方法乃比較Ix及Iy之大小Imax = Iu會靠近Ix及Iy之較大者也就是Imax = Iu與Ix及Iy之較大者的夾角比較小

a =Ix + Iy

2(130)

R =

radic(Ix minus Iy

2)2 + I 2xy (131)

2θp2 = π + 2θp1 θp2 minus θp1 =π

2(132)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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質質質量量量慣慣慣性性性矩矩矩Mass Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 48 54

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質量慣性矩

τ = Iα (133)

其中τ為旋轉扭矩I為質量慣性矩α為旋轉角速度

I =

ˆmr2dm (134)

=

ˆVr2ρdV (135)

= ρ

ˆVr2dV (136)

其中r為旋轉半徑或轉軸到旋轉體的距離

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 49 54

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 50 54

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平行軸定理

當通過物體質心(G)並在一轉軸上的質量慣性矩(IG )為已知時可透過座標軸轉換計算物體之質量慣性矩(I)

I =

ˆmr2dm (137)

=

ˆm

[(d + x prime)2 + y prime2]dm (138)

=

ˆm

(x prime2 + y prime2)dm + 2d

ˆmx primedm + d2

ˆmdm (139)

I = IG + md2 (140)

其中IG為在zrsquo軸上通過質心的質量慣性矩m為質量d為通過質心之軸zrsquo並平行原軸z之距離

acutem x primedm = x

acutem dm = 0 since x = 0

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 51 54

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (141)

k =

radicI

m(142)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (143)

= I1 + I2 + + In (144)

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課課課堂堂堂練練練習習習

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

兩圓盤複合體

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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結結結論論論

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結論

轉動慣量或慣性矩為物體對旋轉運動的慣性而所謂慣性乃物體持續維持不變的行為也就是定義慣性矩可表達物體對本身旋轉運動的維持或持續性好壞更簡單的說慣性矩越大之物體其維持旋轉運動趨勢的能力越大例如惰輪儲能裝置惰輪之慣性矩越大當其轉動以後其維持旋轉運動趨勢的能力越大

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  • 轉動慣量或慣性矩(Moment of Inertia)
  • 面積慣性矩(Moment of Inertia for Area)
    • 面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)
    • 面積慣性矩之迴轉半徑(Radius of Gyration for Area)
      • 複合面積之慣性矩(Moment of Inertia for Composite Area)
      • 面積之慣性積(Product of Inertia for Area)
        • 平行軸定理(Parallel-Axis Theorem)
          • 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)
            • 主軸慣性矩(Principal Moment of Inertia)
              • 慣性矩之莫爾圓(Mohrs Circle for Moment of Inertia)
              • 質量慣性矩(Mass Moment of Inertia)
                • 平行軸定理(Parallel-Axis Theorem)
Page 4: 轉動慣量 Moments of Inertia - cct.me.ntut.edu.twcct.me.ntut.edu.tw/ccteducation/chchting/aiahtm/statics/lecture... · 轉動慣量或慣性矩 I. 轉動慣量也稱慣性矩(Moment

轉轉轉動動動慣慣慣量量量或或或慣慣慣性性性矩矩矩Moment of Inertia

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慣性(Inertia)

慣性(Inertia)簡單來說乃物體持續維持不變的行為

伽利略(1632)提出一個不受任何外力(或者合外力為0)的物體將保持靜止或勻速直線運動

牛頓(1687)提出所有物體都將一直處於靜止或者勻速直線運動狀態直到出現施加其上的力改變它的運動狀態止

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轉動慣量或慣性矩 I

轉動慣量也稱慣性矩(Moment of Inertia)為物體對旋轉運動的慣性比較直線運動與旋轉運動得

F = ma = mdv

dt(1)

τ = Iα = Idω

dt(2)

其中F為力(N)m為質量(kg)a為加速度(ms2)v為速度(ms)τ為扭矩(N middotm)I為轉動慣量或慣性矩(kg middotm2)α為角加速度(rads2)ω為角速度(rads)

v = rω (3)

I = mr2 (4)

τ = rF = rmdv

dt= mr2

dt= Iα

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轉動慣量或慣性矩 II

一般物件的動能K用轉動力學的定義取代

K =1

2mv2 (5)

=1

2m(rω)2 (6)

=1

2Iω2 (7)

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面面面積積積慣慣慣性性性矩矩矩Moment of Inertia for Area

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面積慣性矩

由左圖液面下y深度的壓力p為

p = γy (8)

其中γ = ρg為液體比重(SpecificWeight)液面下y深度之微小面積dA的受力dF為

dF = pdA = γydA (9)其扭矩dM為

dM = ydF = γy2dA (10)ˆdM = γ

ˆy2dA = γIx (11)

其中Ix =acutey2dA稱為x軸上之面積慣性矩

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極慣性矩(Polar Moment of Inertia)

極點

由左圖x軸上之面積慣性矩Ix及y軸上之面積慣性矩Iy如下

dIx = y2dA (12)

Ix =

ˆAy2dA (13)

dIy = x2dA (14)

Iy =

ˆAx2dA (15)

當微小面積dA對極點(O)或z軸旋轉時其面積極慣性矩JO如下

dJO = r2dA (16)

r2 = x2 + y2 (17)

JO =

ˆAr2dA =

ˆA(x2 + y2)dA = Ix + Iy (18)

其中r為微小面積dA到極點(O)的距離極點(O)簡單說乃z軸上的任意點由於x2 y2 r2及面

積(A)為正因此JO Ix Iy 均為正面積慣性矩的單位為m4mm4

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面面面積積積慣慣慣性性性矩矩矩之之之平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem for Area

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面積慣性矩之平行軸定理

極點

新極點

如左圖面積慣性矩之平行軸定理乃以面心(C)為新極點(O)設定新座標軸(xrsquoyrsquo)其中xrsquo軸平行於原座標x軸yrsquo軸平行於原座標y軸取微小面積dA其在新座標之點座標為(xrsquoyrsquo)則x軸上之面積慣性矩Ix及y軸上之面積慣性矩Iy可如下換算

Ix =

ˆA

(y prime + dy )2dA (19)

=

ˆAy prime2dA + 2dy

ˆAy primedA + d2

y

ˆAdA

Iy =

ˆA

(x prime + dx)2dA (20)

=

ˆAx prime2dA + 2dx

ˆAx primedA + d2

x

ˆAdA

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面積慣性矩之平行軸定理- Continued

由於面心在xrsquo軸上之等價力臂為y primeyrsquo軸上之等價力臂為x prime當以面心為轉軸點時即x prime = y prime = 0則x軸上之面積慣性矩Ix為ˆ

Ay prime2dA = macrIx prime (21)

ˆy primedA = y prime

ˆAdA = 0 (22)

d2y

ˆAdA = Ad2

y (23)

Ix =

ˆAy prime2dA + 2dy

ˆAy primedA + d2

y

ˆAdA (24)

= macrIx prime + Ad2y (25)

同理可得y軸上之面積慣性矩Iy

Iy =

ˆAx prime2dA + 2dx

ˆAx primedA + d2

x

ˆAdA (26)

= macrIy prime + Ad2x

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面積之極慣性矩

極點

新極點

由於通過面心之極慣性矩JC與極點(O)到面心(C)之距離d為

d2 = d2x + d2

y (27)

JC = macrIx prime + macrIy prime (28)

JO = Ix + Iy (29)

= macrIx prime + Ad2y + macrIy prime + Ad2

x (30)

= macrIx prime + macrIy prime + A(d2x + d2

y ) (31)

= JC + Ad2 (32)

也就是面積A之極慣性矩JO等於通過面積A之面心極慣性矩JC 與面積乘上極點(O)到面心(C)之距離d的平方和

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面面面積積積慣慣慣性性性矩矩矩之之之迴迴迴轉轉轉半半半徑徑徑Radius of Gyration for Area

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面積慣性矩之迴轉半徑

當x軸上之面積慣性矩Ix與y軸上之面積慣性矩Iy為已知時面積慣性矩之迴轉半徑可得知

dIx = y2dA (33)

Ix = kxA kx =

radicIxA

(34)

dIy = x2dA (35)

Iy = kyA ky =

radicIyA

(36)

dJO = r2dA (37)

JO = kOA kO =

radicJOA

(38)

其中(kx ky kO)分別為x軸y軸極點(O)之面積慣性矩迴轉半徑

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課課課堂堂堂練練練習習習

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解題技巧 I

極點

新極點

繪圖並標出面積(A)之面心座標位置(C)接著以面心座標位置(C)為原點畫出新座標軸(xrsquoyrsquo)其中xrsquo軸平行於原(舊)座標x軸yrsquo軸平行於原(舊)座標y軸並寫出新舊座標軸的平移量dx與dy其中dx為yrsquo軸與y軸的平移量dy為xrsquo軸與x軸的平移量取微小面積dA其在原(舊)座標之點座標為(xy)在新座標之點座標為(xrsquoyrsquo)帶入x軸上之面積慣性矩Ix及y軸上之面積慣性矩Iy還有xrsquo軸上之面積慣性矩 macrIx prime及y軸上之面積慣性矩 macrIy prime可如下換算

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解題技巧 II

Ix =

ˆAy2dA =

ˆAy prime2dA + 2dy

ˆAy primedA + d2

y

ˆAdA = macrIx prime + Ad2

y

macrIx prime =

ˆAy prime2dA (39)

Iy =

ˆAx2dA =

ˆAx prime2dA + 2dx

ˆAx primedA + d2

x

ˆAdA = macrIy prime + Ad2

x

macrIy prime =

ˆAx prime2dA (40)

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解題技巧 III

另外微小面積dA的面積慣性矩dIx dIy也可寫成

dIx = d macrIx prime + dAd2y (41)

Ix =

ˆdIx =

ˆd macrIx prime +

ˆdAd2

y (42)

dIy = d macrIy prime + dAd2x (43)

Iy =

ˆdIy =

ˆd macrIy prime +

ˆdAd2

x (44)

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

面心慣性矩 慣性矩 極慣性矩

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Example 10-1 解答

dA = bdy (45)

Ixb =

ˆAy2dA (46)

=

ˆ h

0y2bdy (47)

=1

3by3|h0 (48)

=1

3bh3 (49)

長方形之邊長b與h則面心慣性矩 macrIx prime macrIy prime與長方形之邊軸慣性矩Ix Iy為

macrIx prime =1

12bh3 (50)

macrIy prime =1

12hb3 (51)

Ix =1

3bh3 (52)

Iy =1

3hb3 (53)

JC = macrIx prime + macrIy prime (54)

JO = Ix + Iy (55)

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Example 10-2 解答

取微小面積dA=ydx之長方形其邊長b=dx與h=ydy = y2所以

y2 = 400x (56)

y =radic

400x (57)

d macrIx prime = d(1

12bh3) =

1

12y3dx (58)

dIx = d macrIx prime + dAd2y =

1

12y3dx + ydx

(y2

)2=

1

3y3dx (59)

Ix =

ˆ 100

0

1

3y3dx =

ˆ 100

0

1

3

(radic400x

)3dx (60)

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Example 10-3 解答

取微小面積dA=2ydx之長方形其邊長b=dx與h=2y所以

x2 + y2 = a2 (61)

y =radic

a2 minus x2 (62)

d macrIx prime = d(1

12bh3) =

1

12dx(2y)3 =

2

3y3dx (63)

macrIx prime =

ˆ a

minusa

2

3y3dx =

ˆ a

minusa

2

3

(radica2 minus x2

)3dx =

πa4

4(64)

圓之半徑a則面心慣性矩 macrIx prime macrIy prime為

macrIx prime =πa4

4(65)

macrIy prime =πa4

4(66)

JC = macrIx prime + macrIy prime =πa4

2(67)

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複複複合合合面面面積積積之之之慣慣慣性性性矩矩矩Moment of Inertia for Composite Area

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (68)

k =

radicI

m(69)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (70)

= I1 + I2 + + In (71)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 25 54

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複合面積之慣性矩

複合面積之慣性矩(Ixc Iyc)為各局部面積之慣性矩(Ixi Iyi )合

Ixc =i=nsumi=1

Ixi (72)

= Ix1 + Ix2 + + Ixn (73)

Iyc =i=nsumi=1

Iyi (74)

= Iy1 + Iy2 + + Iyn (75)

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

長方形與圓形複合

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

長方形複合

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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面面面積積積之之之慣慣慣性性性積積積Product of Inertia for Area

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面積之慣性積(Product of Inertia for Area)

為了下個章節求解慣性矩之極值(極大或極小值)方便數學運算在此先定義面積之慣性積(Ixy )為

dIxy = xydA (76)

Ixy =

ˆAxydA (77)

面積之慣性積的符號可為正負或零當慣性積之面積相對轉軸對稱時則面積之慣性積和為零

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

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平行軸定理

極點

新極點

如左圖面積慣性矩之平行軸定理乃以面心(C)為新極點(O)設定新座標軸(xrsquoyrsquo)其中xrsquo軸平行於原座標x軸yrsquo軸平行於原座標y軸取微小面積dA其在新座標之點座標為(xrsquoyrsquo)面積之慣性積(Ixy )為

dIxy = (x prime + dx)(y prime + dy )dA (78)

當以面心為轉軸點時即x prime = y prime = 0ˆAx primedA = x prime

ˆAdA = 0

ˆAy primedA = y prime

ˆAdA = 0 (79)

Ixy =

ˆA

(x prime + dx)(y prime + dy )dA (80)

=

ˆAx primey primedA + dx

ˆAy primedA + dy

ˆAx primedA + dxdy

ˆAdA (81)

= macrIx primey prime + Adxdy

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 31 54

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課課課堂堂堂練練練習習習

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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傾傾傾斜斜斜軸軸軸面面面積積積之之之慣慣慣性性性矩矩矩

Moment of Inertia for Area about Inclined Axis

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傾斜軸面積之慣性矩

當轉動軸傾斜θ角時也就是座標軸旋轉θ角而Ix Iy Ixy為已知則新的座標軸(uv)可寫為

u = xcosθ + ysinθ (82)

v = ycosθ minus xsinθ (83)

由u軸上之面積慣性矩Iuv軸上之面積慣性矩Iv及慣性積Iuv得知

dIu = v2dA = (ycosθ minus xsinθ)2dA (84)

dIv = u2dA = (xcosθ + ysinθ)2dA (85)

dIuv = uvdA = (xcosθ + ysinθ)(ycosθ minus xsinθ)dA (86)

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傾斜軸面積之慣性矩 I

由x軸上之面積慣性矩Ixy軸上之面積慣性矩Iy及慣性積Ixy得知

Ix =

ˆy2dA Iy =

ˆx2dA Ixy =

ˆxydA (87)

Iu =

ˆ(ycosθ minus xsinθ)2dA (88)

=

ˆ(y2cos2θ minus 2xycosθsinθ + x2sin2θ)dA (89)

= cos2θ

ˆy2dAminus 2cosθsinθ

ˆxydA + sin2θ

ˆx2dA (90)

= Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (91)

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傾斜軸面積之慣性矩 II

Iv =

ˆ(xcosθ + ysinθ)2dA (92)

=

ˆ(x2cos2θ + 2xycosθsinθ + y2sin2θ)dA (93)

= cos2θ

ˆx2dA + 2cosθsinθ

ˆxydA + sin2θ

ˆy2dA (94)

= Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (95)

Iuv =

ˆ(xcosθ + ysinθ)(ycosθ minus xsinθ)dA (96)

=

ˆ(xycos2θ + y2cosθsinθ minus x2cosθsinθ minus xysin2θ)dA (97)

= (cos2 minus sin2θ)

ˆxydA + cosθsinθ

ˆy2dAminus cosθsinθ

ˆx2dA

= Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 36 54

傾斜軸面積之慣性矩 III

再由

sin2θ + cos2θ = 1 (98)

sin(αplusmn β) = sinαcosβ plusmn cosαsinβ (99)

sin2θ = sin(θ + θ) = 2sinθcosθ (100)

cos(αplusmn β) = cosαcosβ ∓ sinαsinβ (101)

cos2θ = cos(θ + θ) = cos2θ minus sin2θ (102)

= 2cos2θ minus 1 = 1minus 2sin2θ (103)

所以

Iu = Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (104)

= Ix(cos2θ + 1

2) + Iy (

1minus cos2θ

2)minus Ixy sin2θ (105)

=Ix + Iy

2+

Ix minus Iy2

cos2θ minus Ixy sin2θ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 37 54

傾斜軸面積之慣性矩 IV

Iv = Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (106)

= Iy (cos2θ + 1

2) + Ix(

1minus cos2θ

2) + Ixy sin2θ (107)

=Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (108)

Iuv = Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ (109)

= Ixycos2θ +1

2Ixsin2θ minus 1

2Iy sin2θ (110)

=Ix minus Iy

2sin2θ + Ixycos2θ (111)

所以

JO = Iu + Iv = Ix + Iy (112)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 38 54

主主主軸軸軸慣慣慣性性性矩矩矩Principal Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 39 54

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主軸慣性矩 I

所謂主軸慣性矩(Principal Moment ofInertia)乃定義發生在慣性矩的極值(極大或極小值)上也就是慣性矩之一次微分式為零或斜率為零

dIu

dθ=

d(Ix+Iy

2+

IxminusIy2

cos2θ minus Ixy sin2θ)

dθ(113)

= minus2Ix minus Iy

2sin2θ minus 2Ixy cos2θ (114)

= 0 (115)

其中dsin2θ = cos2θ dcos2θ = minussin2θ當慣性矩之一次微分式為零時傾斜軸與主軸之夾角θ = θp

minus2Ix minus Iy

2sin2θp minus 2Ixycos2θp = 0 (116)

tan2θ =sin2θ

cos2θ=

minusIxy(Ix minus Iy )2

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 40 54

主軸慣性矩 II

cos2θp =IxminusIy2radic

(IxminusIy2 )2 + I 2xy

sin2θp =minusIxyradic

(IxminusIy2 )2 + I 2xy

(117)

(Iu)max =Ix + Iy

2+

Ix minus Iy2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

minus IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2+

radic(Ix minus Iy

2)2 + I 2xy (118)

(Iv )min =Ix + Iy

2minus Ix minus Iy

2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

+ IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2minusradic

(Ix minus Iy

2)2 + I 2xy (119)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 41 54

主軸慣性矩 III

由tan2θp = tan(π + 2θp)可得到θp1與θp2二個解即2θp2 = π + 2θp1也就是θp2 minus θp1 = π

2

tan2θp = tan(π + 2θp) (120)

2θp2 = π + 2θp1 (121)

θp2 minus θp1 =π

2(122)

所以

Imaxmin

=Ix + Iy

2plusmnradic

(Ix minus Iy

2)2 + I 2xy (123)

Iuv =Ix minus Iy

2

minusIxyradic(IxminusIy2 )2 + I 2xy

+ Ixy

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

(124)

= 0

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課課課堂堂堂練練練習習習

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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慣慣慣性性性矩矩矩之之之莫莫莫爾爾爾圓圓圓Mohrrsquos Circle for Moment of Inertia

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慣性矩之莫爾圓

由於Iu =

Ix + Iy2

+Ix minus Iy

2cos2θ minus Ixy sin2θ (125)

Iv =Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (126)

Iuv =Ix minus Iy

2sin2θ + Ixycos2θ (127)

將式(125)與式(127)各自分別平方並相加可得

(Iu minusIx + Iy

2)2 + I 2uv = (

Ix minus Iy2

)2 + I 2xy (128)

其中Ix Iy Ixy均為常數式(128)可進一步寫成

(Iu minus a)2 + I 2uv = R2 (129)

其中a =Ix+Iy2 R =

radic(IxminusIy2 )2 + I 2xycos2θp1 =

aRsin2θp1 =

minusIxyR 2θp1為OA轉向I軸方向即2θp1為

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課課課堂堂堂練練練習習習

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解題技巧

首先確定轉動系統的Ix Iy Ixy接著算出莫爾圓之圓心座標a及半徑R並畫出莫爾圓再由莫爾圓上點出A(Ix Ixy )並由OA轉向I軸找出兩倍夾角2θp1及2θp2其中θp1及θp2為主軸(Principle Axis)與Ix軸的夾角由於主軸(Principle Axis)為極值之軸Imax = Iu及Imin = Iv決定θp1及θp2何者為Imax = Iu與Ix軸的夾角之方法乃比較Ix及Iy之大小Imax = Iu會靠近Ix及Iy之較大者也就是Imax = Iu與Ix及Iy之較大者的夾角比較小

a =Ix + Iy

2(130)

R =

radic(Ix minus Iy

2)2 + I 2xy (131)

2θp2 = π + 2θp1 θp2 minus θp1 =π

2(132)

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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質質質量量量慣慣慣性性性矩矩矩Mass Moment of Inertia

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質量慣性矩

τ = Iα (133)

其中τ為旋轉扭矩I為質量慣性矩α為旋轉角速度

I =

ˆmr2dm (134)

=

ˆVr2ρdV (135)

= ρ

ˆVr2dV (136)

其中r為旋轉半徑或轉軸到旋轉體的距離

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

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平行軸定理

當通過物體質心(G)並在一轉軸上的質量慣性矩(IG )為已知時可透過座標軸轉換計算物體之質量慣性矩(I)

I =

ˆmr2dm (137)

=

ˆm

[(d + x prime)2 + y prime2]dm (138)

=

ˆm

(x prime2 + y prime2)dm + 2d

ˆmx primedm + d2

ˆmdm (139)

I = IG + md2 (140)

其中IG為在zrsquo軸上通過質心的質量慣性矩m為質量d為通過質心之軸zrsquo並平行原軸z之距離

acutem x primedm = x

acutem dm = 0 since x = 0

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (141)

k =

radicI

m(142)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (143)

= I1 + I2 + + In (144)

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課課課堂堂堂練練練習習習

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

兩圓盤複合體

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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結結結論論論

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結論

轉動慣量或慣性矩為物體對旋轉運動的慣性而所謂慣性乃物體持續維持不變的行為也就是定義慣性矩可表達物體對本身旋轉運動的維持或持續性好壞更簡單的說慣性矩越大之物體其維持旋轉運動趨勢的能力越大例如惰輪儲能裝置惰輪之慣性矩越大當其轉動以後其維持旋轉運動趨勢的能力越大

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  • 轉動慣量或慣性矩(Moment of Inertia)
  • 面積慣性矩(Moment of Inertia for Area)
    • 面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)
    • 面積慣性矩之迴轉半徑(Radius of Gyration for Area)
      • 複合面積之慣性矩(Moment of Inertia for Composite Area)
      • 面積之慣性積(Product of Inertia for Area)
        • 平行軸定理(Parallel-Axis Theorem)
          • 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)
            • 主軸慣性矩(Principal Moment of Inertia)
              • 慣性矩之莫爾圓(Mohrs Circle for Moment of Inertia)
              • 質量慣性矩(Mass Moment of Inertia)
                • 平行軸定理(Parallel-Axis Theorem)
Page 5: 轉動慣量 Moments of Inertia - cct.me.ntut.edu.twcct.me.ntut.edu.tw/ccteducation/chchting/aiahtm/statics/lecture... · 轉動慣量或慣性矩 I. 轉動慣量也稱慣性矩(Moment

慣性(Inertia)

慣性(Inertia)簡單來說乃物體持續維持不變的行為

伽利略(1632)提出一個不受任何外力(或者合外力為0)的物體將保持靜止或勻速直線運動

牛頓(1687)提出所有物體都將一直處於靜止或者勻速直線運動狀態直到出現施加其上的力改變它的運動狀態止

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轉動慣量或慣性矩 I

轉動慣量也稱慣性矩(Moment of Inertia)為物體對旋轉運動的慣性比較直線運動與旋轉運動得

F = ma = mdv

dt(1)

τ = Iα = Idω

dt(2)

其中F為力(N)m為質量(kg)a為加速度(ms2)v為速度(ms)τ為扭矩(N middotm)I為轉動慣量或慣性矩(kg middotm2)α為角加速度(rads2)ω為角速度(rads)

v = rω (3)

I = mr2 (4)

τ = rF = rmdv

dt= mr2

dt= Iα

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轉動慣量或慣性矩 II

一般物件的動能K用轉動力學的定義取代

K =1

2mv2 (5)

=1

2m(rω)2 (6)

=1

2Iω2 (7)

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面面面積積積慣慣慣性性性矩矩矩Moment of Inertia for Area

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面積慣性矩

由左圖液面下y深度的壓力p為

p = γy (8)

其中γ = ρg為液體比重(SpecificWeight)液面下y深度之微小面積dA的受力dF為

dF = pdA = γydA (9)其扭矩dM為

dM = ydF = γy2dA (10)ˆdM = γ

ˆy2dA = γIx (11)

其中Ix =acutey2dA稱為x軸上之面積慣性矩

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極慣性矩(Polar Moment of Inertia)

極點

由左圖x軸上之面積慣性矩Ix及y軸上之面積慣性矩Iy如下

dIx = y2dA (12)

Ix =

ˆAy2dA (13)

dIy = x2dA (14)

Iy =

ˆAx2dA (15)

當微小面積dA對極點(O)或z軸旋轉時其面積極慣性矩JO如下

dJO = r2dA (16)

r2 = x2 + y2 (17)

JO =

ˆAr2dA =

ˆA(x2 + y2)dA = Ix + Iy (18)

其中r為微小面積dA到極點(O)的距離極點(O)簡單說乃z軸上的任意點由於x2 y2 r2及面

積(A)為正因此JO Ix Iy 均為正面積慣性矩的單位為m4mm4

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面面面積積積慣慣慣性性性矩矩矩之之之平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem for Area

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面積慣性矩之平行軸定理

極點

新極點

如左圖面積慣性矩之平行軸定理乃以面心(C)為新極點(O)設定新座標軸(xrsquoyrsquo)其中xrsquo軸平行於原座標x軸yrsquo軸平行於原座標y軸取微小面積dA其在新座標之點座標為(xrsquoyrsquo)則x軸上之面積慣性矩Ix及y軸上之面積慣性矩Iy可如下換算

Ix =

ˆA

(y prime + dy )2dA (19)

=

ˆAy prime2dA + 2dy

ˆAy primedA + d2

y

ˆAdA

Iy =

ˆA

(x prime + dx)2dA (20)

=

ˆAx prime2dA + 2dx

ˆAx primedA + d2

x

ˆAdA

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面積慣性矩之平行軸定理- Continued

由於面心在xrsquo軸上之等價力臂為y primeyrsquo軸上之等價力臂為x prime當以面心為轉軸點時即x prime = y prime = 0則x軸上之面積慣性矩Ix為ˆ

Ay prime2dA = macrIx prime (21)

ˆy primedA = y prime

ˆAdA = 0 (22)

d2y

ˆAdA = Ad2

y (23)

Ix =

ˆAy prime2dA + 2dy

ˆAy primedA + d2

y

ˆAdA (24)

= macrIx prime + Ad2y (25)

同理可得y軸上之面積慣性矩Iy

Iy =

ˆAx prime2dA + 2dx

ˆAx primedA + d2

x

ˆAdA (26)

= macrIy prime + Ad2x

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面積之極慣性矩

極點

新極點

由於通過面心之極慣性矩JC與極點(O)到面心(C)之距離d為

d2 = d2x + d2

y (27)

JC = macrIx prime + macrIy prime (28)

JO = Ix + Iy (29)

= macrIx prime + Ad2y + macrIy prime + Ad2

x (30)

= macrIx prime + macrIy prime + A(d2x + d2

y ) (31)

= JC + Ad2 (32)

也就是面積A之極慣性矩JO等於通過面積A之面心極慣性矩JC 與面積乘上極點(O)到面心(C)之距離d的平方和

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面面面積積積慣慣慣性性性矩矩矩之之之迴迴迴轉轉轉半半半徑徑徑Radius of Gyration for Area

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面積慣性矩之迴轉半徑

當x軸上之面積慣性矩Ix與y軸上之面積慣性矩Iy為已知時面積慣性矩之迴轉半徑可得知

dIx = y2dA (33)

Ix = kxA kx =

radicIxA

(34)

dIy = x2dA (35)

Iy = kyA ky =

radicIyA

(36)

dJO = r2dA (37)

JO = kOA kO =

radicJOA

(38)

其中(kx ky kO)分別為x軸y軸極點(O)之面積慣性矩迴轉半徑

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課課課堂堂堂練練練習習習

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解題技巧 I

極點

新極點

繪圖並標出面積(A)之面心座標位置(C)接著以面心座標位置(C)為原點畫出新座標軸(xrsquoyrsquo)其中xrsquo軸平行於原(舊)座標x軸yrsquo軸平行於原(舊)座標y軸並寫出新舊座標軸的平移量dx與dy其中dx為yrsquo軸與y軸的平移量dy為xrsquo軸與x軸的平移量取微小面積dA其在原(舊)座標之點座標為(xy)在新座標之點座標為(xrsquoyrsquo)帶入x軸上之面積慣性矩Ix及y軸上之面積慣性矩Iy還有xrsquo軸上之面積慣性矩 macrIx prime及y軸上之面積慣性矩 macrIy prime可如下換算

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解題技巧 II

Ix =

ˆAy2dA =

ˆAy prime2dA + 2dy

ˆAy primedA + d2

y

ˆAdA = macrIx prime + Ad2

y

macrIx prime =

ˆAy prime2dA (39)

Iy =

ˆAx2dA =

ˆAx prime2dA + 2dx

ˆAx primedA + d2

x

ˆAdA = macrIy prime + Ad2

x

macrIy prime =

ˆAx prime2dA (40)

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解題技巧 III

另外微小面積dA的面積慣性矩dIx dIy也可寫成

dIx = d macrIx prime + dAd2y (41)

Ix =

ˆdIx =

ˆd macrIx prime +

ˆdAd2

y (42)

dIy = d macrIy prime + dAd2x (43)

Iy =

ˆdIy =

ˆd macrIy prime +

ˆdAd2

x (44)

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

面心慣性矩 慣性矩 極慣性矩

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Example 10-1 解答

dA = bdy (45)

Ixb =

ˆAy2dA (46)

=

ˆ h

0y2bdy (47)

=1

3by3|h0 (48)

=1

3bh3 (49)

長方形之邊長b與h則面心慣性矩 macrIx prime macrIy prime與長方形之邊軸慣性矩Ix Iy為

macrIx prime =1

12bh3 (50)

macrIy prime =1

12hb3 (51)

Ix =1

3bh3 (52)

Iy =1

3hb3 (53)

JC = macrIx prime + macrIy prime (54)

JO = Ix + Iy (55)

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Example 10-2 解答

取微小面積dA=ydx之長方形其邊長b=dx與h=ydy = y2所以

y2 = 400x (56)

y =radic

400x (57)

d macrIx prime = d(1

12bh3) =

1

12y3dx (58)

dIx = d macrIx prime + dAd2y =

1

12y3dx + ydx

(y2

)2=

1

3y3dx (59)

Ix =

ˆ 100

0

1

3y3dx =

ˆ 100

0

1

3

(radic400x

)3dx (60)

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Example 10-3 解答

取微小面積dA=2ydx之長方形其邊長b=dx與h=2y所以

x2 + y2 = a2 (61)

y =radic

a2 minus x2 (62)

d macrIx prime = d(1

12bh3) =

1

12dx(2y)3 =

2

3y3dx (63)

macrIx prime =

ˆ a

minusa

2

3y3dx =

ˆ a

minusa

2

3

(radica2 minus x2

)3dx =

πa4

4(64)

圓之半徑a則面心慣性矩 macrIx prime macrIy prime為

macrIx prime =πa4

4(65)

macrIy prime =πa4

4(66)

JC = macrIx prime + macrIy prime =πa4

2(67)

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複複複合合合面面面積積積之之之慣慣慣性性性矩矩矩Moment of Inertia for Composite Area

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (68)

k =

radicI

m(69)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (70)

= I1 + I2 + + In (71)

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複合面積之慣性矩

複合面積之慣性矩(Ixc Iyc)為各局部面積之慣性矩(Ixi Iyi )合

Ixc =i=nsumi=1

Ixi (72)

= Ix1 + Ix2 + + Ixn (73)

Iyc =i=nsumi=1

Iyi (74)

= Iy1 + Iy2 + + Iyn (75)

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課課課堂堂堂練練練習習習

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

長方形與圓形複合

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

長方形複合

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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面面面積積積之之之慣慣慣性性性積積積Product of Inertia for Area

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面積之慣性積(Product of Inertia for Area)

為了下個章節求解慣性矩之極值(極大或極小值)方便數學運算在此先定義面積之慣性積(Ixy )為

dIxy = xydA (76)

Ixy =

ˆAxydA (77)

面積之慣性積的符號可為正負或零當慣性積之面積相對轉軸對稱時則面積之慣性積和為零

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

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平行軸定理

極點

新極點

如左圖面積慣性矩之平行軸定理乃以面心(C)為新極點(O)設定新座標軸(xrsquoyrsquo)其中xrsquo軸平行於原座標x軸yrsquo軸平行於原座標y軸取微小面積dA其在新座標之點座標為(xrsquoyrsquo)面積之慣性積(Ixy )為

dIxy = (x prime + dx)(y prime + dy )dA (78)

當以面心為轉軸點時即x prime = y prime = 0ˆAx primedA = x prime

ˆAdA = 0

ˆAy primedA = y prime

ˆAdA = 0 (79)

Ixy =

ˆA

(x prime + dx)(y prime + dy )dA (80)

=

ˆAx primey primedA + dx

ˆAy primedA + dy

ˆAx primedA + dxdy

ˆAdA (81)

= macrIx primey prime + Adxdy

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課課課堂堂堂練練練習習習

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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傾傾傾斜斜斜軸軸軸面面面積積積之之之慣慣慣性性性矩矩矩

Moment of Inertia for Area about Inclined Axis

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傾斜軸面積之慣性矩

當轉動軸傾斜θ角時也就是座標軸旋轉θ角而Ix Iy Ixy為已知則新的座標軸(uv)可寫為

u = xcosθ + ysinθ (82)

v = ycosθ minus xsinθ (83)

由u軸上之面積慣性矩Iuv軸上之面積慣性矩Iv及慣性積Iuv得知

dIu = v2dA = (ycosθ minus xsinθ)2dA (84)

dIv = u2dA = (xcosθ + ysinθ)2dA (85)

dIuv = uvdA = (xcosθ + ysinθ)(ycosθ minus xsinθ)dA (86)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 34 54

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傾斜軸面積之慣性矩 I

由x軸上之面積慣性矩Ixy軸上之面積慣性矩Iy及慣性積Ixy得知

Ix =

ˆy2dA Iy =

ˆx2dA Ixy =

ˆxydA (87)

Iu =

ˆ(ycosθ minus xsinθ)2dA (88)

=

ˆ(y2cos2θ minus 2xycosθsinθ + x2sin2θ)dA (89)

= cos2θ

ˆy2dAminus 2cosθsinθ

ˆxydA + sin2θ

ˆx2dA (90)

= Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (91)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 35 54

傾斜軸面積之慣性矩 II

Iv =

ˆ(xcosθ + ysinθ)2dA (92)

=

ˆ(x2cos2θ + 2xycosθsinθ + y2sin2θ)dA (93)

= cos2θ

ˆx2dA + 2cosθsinθ

ˆxydA + sin2θ

ˆy2dA (94)

= Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (95)

Iuv =

ˆ(xcosθ + ysinθ)(ycosθ minus xsinθ)dA (96)

=

ˆ(xycos2θ + y2cosθsinθ minus x2cosθsinθ minus xysin2θ)dA (97)

= (cos2 minus sin2θ)

ˆxydA + cosθsinθ

ˆy2dAminus cosθsinθ

ˆx2dA

= Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 36 54

傾斜軸面積之慣性矩 III

再由

sin2θ + cos2θ = 1 (98)

sin(αplusmn β) = sinαcosβ plusmn cosαsinβ (99)

sin2θ = sin(θ + θ) = 2sinθcosθ (100)

cos(αplusmn β) = cosαcosβ ∓ sinαsinβ (101)

cos2θ = cos(θ + θ) = cos2θ minus sin2θ (102)

= 2cos2θ minus 1 = 1minus 2sin2θ (103)

所以

Iu = Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (104)

= Ix(cos2θ + 1

2) + Iy (

1minus cos2θ

2)minus Ixy sin2θ (105)

=Ix + Iy

2+

Ix minus Iy2

cos2θ minus Ixy sin2θ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 37 54

傾斜軸面積之慣性矩 IV

Iv = Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (106)

= Iy (cos2θ + 1

2) + Ix(

1minus cos2θ

2) + Ixy sin2θ (107)

=Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (108)

Iuv = Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ (109)

= Ixycos2θ +1

2Ixsin2θ minus 1

2Iy sin2θ (110)

=Ix minus Iy

2sin2θ + Ixycos2θ (111)

所以

JO = Iu + Iv = Ix + Iy (112)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 38 54

主主主軸軸軸慣慣慣性性性矩矩矩Principal Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 39 54

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主軸慣性矩 I

所謂主軸慣性矩(Principal Moment ofInertia)乃定義發生在慣性矩的極值(極大或極小值)上也就是慣性矩之一次微分式為零或斜率為零

dIu

dθ=

d(Ix+Iy

2+

IxminusIy2

cos2θ minus Ixy sin2θ)

dθ(113)

= minus2Ix minus Iy

2sin2θ minus 2Ixy cos2θ (114)

= 0 (115)

其中dsin2θ = cos2θ dcos2θ = minussin2θ當慣性矩之一次微分式為零時傾斜軸與主軸之夾角θ = θp

minus2Ix minus Iy

2sin2θp minus 2Ixycos2θp = 0 (116)

tan2θ =sin2θ

cos2θ=

minusIxy(Ix minus Iy )2

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 40 54

主軸慣性矩 II

cos2θp =IxminusIy2radic

(IxminusIy2 )2 + I 2xy

sin2θp =minusIxyradic

(IxminusIy2 )2 + I 2xy

(117)

(Iu)max =Ix + Iy

2+

Ix minus Iy2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

minus IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2+

radic(Ix minus Iy

2)2 + I 2xy (118)

(Iv )min =Ix + Iy

2minus Ix minus Iy

2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

+ IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2minusradic

(Ix minus Iy

2)2 + I 2xy (119)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 41 54

主軸慣性矩 III

由tan2θp = tan(π + 2θp)可得到θp1與θp2二個解即2θp2 = π + 2θp1也就是θp2 minus θp1 = π

2

tan2θp = tan(π + 2θp) (120)

2θp2 = π + 2θp1 (121)

θp2 minus θp1 =π

2(122)

所以

Imaxmin

=Ix + Iy

2plusmnradic

(Ix minus Iy

2)2 + I 2xy (123)

Iuv =Ix minus Iy

2

minusIxyradic(IxminusIy2 )2 + I 2xy

+ Ixy

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

(124)

= 0

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 42 54

課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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慣慣慣性性性矩矩矩之之之莫莫莫爾爾爾圓圓圓Mohrrsquos Circle for Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 44 54

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慣性矩之莫爾圓

由於Iu =

Ix + Iy2

+Ix minus Iy

2cos2θ minus Ixy sin2θ (125)

Iv =Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (126)

Iuv =Ix minus Iy

2sin2θ + Ixycos2θ (127)

將式(125)與式(127)各自分別平方並相加可得

(Iu minusIx + Iy

2)2 + I 2uv = (

Ix minus Iy2

)2 + I 2xy (128)

其中Ix Iy Ixy均為常數式(128)可進一步寫成

(Iu minus a)2 + I 2uv = R2 (129)

其中a =Ix+Iy2 R =

radic(IxminusIy2 )2 + I 2xycos2θp1 =

aRsin2θp1 =

minusIxyR 2θp1為OA轉向I軸方向即2θp1為

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 45 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 46 54

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解題技巧

首先確定轉動系統的Ix Iy Ixy接著算出莫爾圓之圓心座標a及半徑R並畫出莫爾圓再由莫爾圓上點出A(Ix Ixy )並由OA轉向I軸找出兩倍夾角2θp1及2θp2其中θp1及θp2為主軸(Principle Axis)與Ix軸的夾角由於主軸(Principle Axis)為極值之軸Imax = Iu及Imin = Iv決定θp1及θp2何者為Imax = Iu與Ix軸的夾角之方法乃比較Ix及Iy之大小Imax = Iu會靠近Ix及Iy之較大者也就是Imax = Iu與Ix及Iy之較大者的夾角比較小

a =Ix + Iy

2(130)

R =

radic(Ix minus Iy

2)2 + I 2xy (131)

2θp2 = π + 2θp1 θp2 minus θp1 =π

2(132)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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質質質量量量慣慣慣性性性矩矩矩Mass Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 48 54

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質量慣性矩

τ = Iα (133)

其中τ為旋轉扭矩I為質量慣性矩α為旋轉角速度

I =

ˆmr2dm (134)

=

ˆVr2ρdV (135)

= ρ

ˆVr2dV (136)

其中r為旋轉半徑或轉軸到旋轉體的距離

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 49 54

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 50 54

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平行軸定理

當通過物體質心(G)並在一轉軸上的質量慣性矩(IG )為已知時可透過座標軸轉換計算物體之質量慣性矩(I)

I =

ˆmr2dm (137)

=

ˆm

[(d + x prime)2 + y prime2]dm (138)

=

ˆm

(x prime2 + y prime2)dm + 2d

ˆmx primedm + d2

ˆmdm (139)

I = IG + md2 (140)

其中IG為在zrsquo軸上通過質心的質量慣性矩m為質量d為通過質心之軸zrsquo並平行原軸z之距離

acutem x primedm = x

acutem dm = 0 since x = 0

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 51 54

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (141)

k =

radicI

m(142)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (143)

= I1 + I2 + + In (144)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 52 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

兩圓盤複合體

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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結結結論論論

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 54 54

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結論

轉動慣量或慣性矩為物體對旋轉運動的慣性而所謂慣性乃物體持續維持不變的行為也就是定義慣性矩可表達物體對本身旋轉運動的維持或持續性好壞更簡單的說慣性矩越大之物體其維持旋轉運動趨勢的能力越大例如惰輪儲能裝置惰輪之慣性矩越大當其轉動以後其維持旋轉運動趨勢的能力越大

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 55 54

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  • 轉動慣量或慣性矩(Moment of Inertia)
  • 面積慣性矩(Moment of Inertia for Area)
    • 面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)
    • 面積慣性矩之迴轉半徑(Radius of Gyration for Area)
      • 複合面積之慣性矩(Moment of Inertia for Composite Area)
      • 面積之慣性積(Product of Inertia for Area)
        • 平行軸定理(Parallel-Axis Theorem)
          • 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)
            • 主軸慣性矩(Principal Moment of Inertia)
              • 慣性矩之莫爾圓(Mohrs Circle for Moment of Inertia)
              • 質量慣性矩(Mass Moment of Inertia)
                • 平行軸定理(Parallel-Axis Theorem)
Page 6: 轉動慣量 Moments of Inertia - cct.me.ntut.edu.twcct.me.ntut.edu.tw/ccteducation/chchting/aiahtm/statics/lecture... · 轉動慣量或慣性矩 I. 轉動慣量也稱慣性矩(Moment

轉動慣量或慣性矩 I

轉動慣量也稱慣性矩(Moment of Inertia)為物體對旋轉運動的慣性比較直線運動與旋轉運動得

F = ma = mdv

dt(1)

τ = Iα = Idω

dt(2)

其中F為力(N)m為質量(kg)a為加速度(ms2)v為速度(ms)τ為扭矩(N middotm)I為轉動慣量或慣性矩(kg middotm2)α為角加速度(rads2)ω為角速度(rads)

v = rω (3)

I = mr2 (4)

τ = rF = rmdv

dt= mr2

dt= Iα

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轉動慣量或慣性矩 II

一般物件的動能K用轉動力學的定義取代

K =1

2mv2 (5)

=1

2m(rω)2 (6)

=1

2Iω2 (7)

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面面面積積積慣慣慣性性性矩矩矩Moment of Inertia for Area

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面積慣性矩

由左圖液面下y深度的壓力p為

p = γy (8)

其中γ = ρg為液體比重(SpecificWeight)液面下y深度之微小面積dA的受力dF為

dF = pdA = γydA (9)其扭矩dM為

dM = ydF = γy2dA (10)ˆdM = γ

ˆy2dA = γIx (11)

其中Ix =acutey2dA稱為x軸上之面積慣性矩

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極慣性矩(Polar Moment of Inertia)

極點

由左圖x軸上之面積慣性矩Ix及y軸上之面積慣性矩Iy如下

dIx = y2dA (12)

Ix =

ˆAy2dA (13)

dIy = x2dA (14)

Iy =

ˆAx2dA (15)

當微小面積dA對極點(O)或z軸旋轉時其面積極慣性矩JO如下

dJO = r2dA (16)

r2 = x2 + y2 (17)

JO =

ˆAr2dA =

ˆA(x2 + y2)dA = Ix + Iy (18)

其中r為微小面積dA到極點(O)的距離極點(O)簡單說乃z軸上的任意點由於x2 y2 r2及面

積(A)為正因此JO Ix Iy 均為正面積慣性矩的單位為m4mm4

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面面面積積積慣慣慣性性性矩矩矩之之之平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem for Area

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面積慣性矩之平行軸定理

極點

新極點

如左圖面積慣性矩之平行軸定理乃以面心(C)為新極點(O)設定新座標軸(xrsquoyrsquo)其中xrsquo軸平行於原座標x軸yrsquo軸平行於原座標y軸取微小面積dA其在新座標之點座標為(xrsquoyrsquo)則x軸上之面積慣性矩Ix及y軸上之面積慣性矩Iy可如下換算

Ix =

ˆA

(y prime + dy )2dA (19)

=

ˆAy prime2dA + 2dy

ˆAy primedA + d2

y

ˆAdA

Iy =

ˆA

(x prime + dx)2dA (20)

=

ˆAx prime2dA + 2dx

ˆAx primedA + d2

x

ˆAdA

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面積慣性矩之平行軸定理- Continued

由於面心在xrsquo軸上之等價力臂為y primeyrsquo軸上之等價力臂為x prime當以面心為轉軸點時即x prime = y prime = 0則x軸上之面積慣性矩Ix為ˆ

Ay prime2dA = macrIx prime (21)

ˆy primedA = y prime

ˆAdA = 0 (22)

d2y

ˆAdA = Ad2

y (23)

Ix =

ˆAy prime2dA + 2dy

ˆAy primedA + d2

y

ˆAdA (24)

= macrIx prime + Ad2y (25)

同理可得y軸上之面積慣性矩Iy

Iy =

ˆAx prime2dA + 2dx

ˆAx primedA + d2

x

ˆAdA (26)

= macrIy prime + Ad2x

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面積之極慣性矩

極點

新極點

由於通過面心之極慣性矩JC與極點(O)到面心(C)之距離d為

d2 = d2x + d2

y (27)

JC = macrIx prime + macrIy prime (28)

JO = Ix + Iy (29)

= macrIx prime + Ad2y + macrIy prime + Ad2

x (30)

= macrIx prime + macrIy prime + A(d2x + d2

y ) (31)

= JC + Ad2 (32)

也就是面積A之極慣性矩JO等於通過面積A之面心極慣性矩JC 與面積乘上極點(O)到面心(C)之距離d的平方和

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面面面積積積慣慣慣性性性矩矩矩之之之迴迴迴轉轉轉半半半徑徑徑Radius of Gyration for Area

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面積慣性矩之迴轉半徑

當x軸上之面積慣性矩Ix與y軸上之面積慣性矩Iy為已知時面積慣性矩之迴轉半徑可得知

dIx = y2dA (33)

Ix = kxA kx =

radicIxA

(34)

dIy = x2dA (35)

Iy = kyA ky =

radicIyA

(36)

dJO = r2dA (37)

JO = kOA kO =

radicJOA

(38)

其中(kx ky kO)分別為x軸y軸極點(O)之面積慣性矩迴轉半徑

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課課課堂堂堂練練練習習習

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解題技巧 I

極點

新極點

繪圖並標出面積(A)之面心座標位置(C)接著以面心座標位置(C)為原點畫出新座標軸(xrsquoyrsquo)其中xrsquo軸平行於原(舊)座標x軸yrsquo軸平行於原(舊)座標y軸並寫出新舊座標軸的平移量dx與dy其中dx為yrsquo軸與y軸的平移量dy為xrsquo軸與x軸的平移量取微小面積dA其在原(舊)座標之點座標為(xy)在新座標之點座標為(xrsquoyrsquo)帶入x軸上之面積慣性矩Ix及y軸上之面積慣性矩Iy還有xrsquo軸上之面積慣性矩 macrIx prime及y軸上之面積慣性矩 macrIy prime可如下換算

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解題技巧 II

Ix =

ˆAy2dA =

ˆAy prime2dA + 2dy

ˆAy primedA + d2

y

ˆAdA = macrIx prime + Ad2

y

macrIx prime =

ˆAy prime2dA (39)

Iy =

ˆAx2dA =

ˆAx prime2dA + 2dx

ˆAx primedA + d2

x

ˆAdA = macrIy prime + Ad2

x

macrIy prime =

ˆAx prime2dA (40)

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解題技巧 III

另外微小面積dA的面積慣性矩dIx dIy也可寫成

dIx = d macrIx prime + dAd2y (41)

Ix =

ˆdIx =

ˆd macrIx prime +

ˆdAd2

y (42)

dIy = d macrIy prime + dAd2x (43)

Iy =

ˆdIy =

ˆd macrIy prime +

ˆdAd2

x (44)

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面心慣性矩 慣性矩 極慣性矩

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Example 10-1 解答

dA = bdy (45)

Ixb =

ˆAy2dA (46)

=

ˆ h

0y2bdy (47)

=1

3by3|h0 (48)

=1

3bh3 (49)

長方形之邊長b與h則面心慣性矩 macrIx prime macrIy prime與長方形之邊軸慣性矩Ix Iy為

macrIx prime =1

12bh3 (50)

macrIy prime =1

12hb3 (51)

Ix =1

3bh3 (52)

Iy =1

3hb3 (53)

JC = macrIx prime + macrIy prime (54)

JO = Ix + Iy (55)

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Example 10-2 解答

取微小面積dA=ydx之長方形其邊長b=dx與h=ydy = y2所以

y2 = 400x (56)

y =radic

400x (57)

d macrIx prime = d(1

12bh3) =

1

12y3dx (58)

dIx = d macrIx prime + dAd2y =

1

12y3dx + ydx

(y2

)2=

1

3y3dx (59)

Ix =

ˆ 100

0

1

3y3dx =

ˆ 100

0

1

3

(radic400x

)3dx (60)

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Example 10-3 解答

取微小面積dA=2ydx之長方形其邊長b=dx與h=2y所以

x2 + y2 = a2 (61)

y =radic

a2 minus x2 (62)

d macrIx prime = d(1

12bh3) =

1

12dx(2y)3 =

2

3y3dx (63)

macrIx prime =

ˆ a

minusa

2

3y3dx =

ˆ a

minusa

2

3

(radica2 minus x2

)3dx =

πa4

4(64)

圓之半徑a則面心慣性矩 macrIx prime macrIy prime為

macrIx prime =πa4

4(65)

macrIy prime =πa4

4(66)

JC = macrIx prime + macrIy prime =πa4

2(67)

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複複複合合合面面面積積積之之之慣慣慣性性性矩矩矩Moment of Inertia for Composite Area

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (68)

k =

radicI

m(69)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (70)

= I1 + I2 + + In (71)

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複合面積之慣性矩

複合面積之慣性矩(Ixc Iyc)為各局部面積之慣性矩(Ixi Iyi )合

Ixc =i=nsumi=1

Ixi (72)

= Ix1 + Ix2 + + Ixn (73)

Iyc =i=nsumi=1

Iyi (74)

= Iy1 + Iy2 + + Iyn (75)

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課課課堂堂堂練練練習習習

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

長方形與圓形複合

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

長方形複合

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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面面面積積積之之之慣慣慣性性性積積積Product of Inertia for Area

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 28 54

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面積之慣性積(Product of Inertia for Area)

為了下個章節求解慣性矩之極值(極大或極小值)方便數學運算在此先定義面積之慣性積(Ixy )為

dIxy = xydA (76)

Ixy =

ˆAxydA (77)

面積之慣性積的符號可為正負或零當慣性積之面積相對轉軸對稱時則面積之慣性積和為零

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 29 54

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 30 54

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平行軸定理

極點

新極點

如左圖面積慣性矩之平行軸定理乃以面心(C)為新極點(O)設定新座標軸(xrsquoyrsquo)其中xrsquo軸平行於原座標x軸yrsquo軸平行於原座標y軸取微小面積dA其在新座標之點座標為(xrsquoyrsquo)面積之慣性積(Ixy )為

dIxy = (x prime + dx)(y prime + dy )dA (78)

當以面心為轉軸點時即x prime = y prime = 0ˆAx primedA = x prime

ˆAdA = 0

ˆAy primedA = y prime

ˆAdA = 0 (79)

Ixy =

ˆA

(x prime + dx)(y prime + dy )dA (80)

=

ˆAx primey primedA + dx

ˆAy primedA + dy

ˆAx primedA + dxdy

ˆAdA (81)

= macrIx primey prime + Adxdy

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 31 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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傾傾傾斜斜斜軸軸軸面面面積積積之之之慣慣慣性性性矩矩矩

Moment of Inertia for Area about Inclined Axis

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 33 54

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傾斜軸面積之慣性矩

當轉動軸傾斜θ角時也就是座標軸旋轉θ角而Ix Iy Ixy為已知則新的座標軸(uv)可寫為

u = xcosθ + ysinθ (82)

v = ycosθ minus xsinθ (83)

由u軸上之面積慣性矩Iuv軸上之面積慣性矩Iv及慣性積Iuv得知

dIu = v2dA = (ycosθ minus xsinθ)2dA (84)

dIv = u2dA = (xcosθ + ysinθ)2dA (85)

dIuv = uvdA = (xcosθ + ysinθ)(ycosθ minus xsinθ)dA (86)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 34 54

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傾斜軸面積之慣性矩 I

由x軸上之面積慣性矩Ixy軸上之面積慣性矩Iy及慣性積Ixy得知

Ix =

ˆy2dA Iy =

ˆx2dA Ixy =

ˆxydA (87)

Iu =

ˆ(ycosθ minus xsinθ)2dA (88)

=

ˆ(y2cos2θ minus 2xycosθsinθ + x2sin2θ)dA (89)

= cos2θ

ˆy2dAminus 2cosθsinθ

ˆxydA + sin2θ

ˆx2dA (90)

= Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (91)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 35 54

傾斜軸面積之慣性矩 II

Iv =

ˆ(xcosθ + ysinθ)2dA (92)

=

ˆ(x2cos2θ + 2xycosθsinθ + y2sin2θ)dA (93)

= cos2θ

ˆx2dA + 2cosθsinθ

ˆxydA + sin2θ

ˆy2dA (94)

= Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (95)

Iuv =

ˆ(xcosθ + ysinθ)(ycosθ minus xsinθ)dA (96)

=

ˆ(xycos2θ + y2cosθsinθ minus x2cosθsinθ minus xysin2θ)dA (97)

= (cos2 minus sin2θ)

ˆxydA + cosθsinθ

ˆy2dAminus cosθsinθ

ˆx2dA

= Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 36 54

傾斜軸面積之慣性矩 III

再由

sin2θ + cos2θ = 1 (98)

sin(αplusmn β) = sinαcosβ plusmn cosαsinβ (99)

sin2θ = sin(θ + θ) = 2sinθcosθ (100)

cos(αplusmn β) = cosαcosβ ∓ sinαsinβ (101)

cos2θ = cos(θ + θ) = cos2θ minus sin2θ (102)

= 2cos2θ minus 1 = 1minus 2sin2θ (103)

所以

Iu = Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (104)

= Ix(cos2θ + 1

2) + Iy (

1minus cos2θ

2)minus Ixy sin2θ (105)

=Ix + Iy

2+

Ix minus Iy2

cos2θ minus Ixy sin2θ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 37 54

傾斜軸面積之慣性矩 IV

Iv = Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (106)

= Iy (cos2θ + 1

2) + Ix(

1minus cos2θ

2) + Ixy sin2θ (107)

=Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (108)

Iuv = Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ (109)

= Ixycos2θ +1

2Ixsin2θ minus 1

2Iy sin2θ (110)

=Ix minus Iy

2sin2θ + Ixycos2θ (111)

所以

JO = Iu + Iv = Ix + Iy (112)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 38 54

主主主軸軸軸慣慣慣性性性矩矩矩Principal Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 39 54

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主軸慣性矩 I

所謂主軸慣性矩(Principal Moment ofInertia)乃定義發生在慣性矩的極值(極大或極小值)上也就是慣性矩之一次微分式為零或斜率為零

dIu

dθ=

d(Ix+Iy

2+

IxminusIy2

cos2θ minus Ixy sin2θ)

dθ(113)

= minus2Ix minus Iy

2sin2θ minus 2Ixy cos2θ (114)

= 0 (115)

其中dsin2θ = cos2θ dcos2θ = minussin2θ當慣性矩之一次微分式為零時傾斜軸與主軸之夾角θ = θp

minus2Ix minus Iy

2sin2θp minus 2Ixycos2θp = 0 (116)

tan2θ =sin2θ

cos2θ=

minusIxy(Ix minus Iy )2

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 40 54

主軸慣性矩 II

cos2θp =IxminusIy2radic

(IxminusIy2 )2 + I 2xy

sin2θp =minusIxyradic

(IxminusIy2 )2 + I 2xy

(117)

(Iu)max =Ix + Iy

2+

Ix minus Iy2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

minus IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2+

radic(Ix minus Iy

2)2 + I 2xy (118)

(Iv )min =Ix + Iy

2minus Ix minus Iy

2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

+ IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2minusradic

(Ix minus Iy

2)2 + I 2xy (119)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 41 54

主軸慣性矩 III

由tan2θp = tan(π + 2θp)可得到θp1與θp2二個解即2θp2 = π + 2θp1也就是θp2 minus θp1 = π

2

tan2θp = tan(π + 2θp) (120)

2θp2 = π + 2θp1 (121)

θp2 minus θp1 =π

2(122)

所以

Imaxmin

=Ix + Iy

2plusmnradic

(Ix minus Iy

2)2 + I 2xy (123)

Iuv =Ix minus Iy

2

minusIxyradic(IxminusIy2 )2 + I 2xy

+ Ixy

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

(124)

= 0

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 42 54

課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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慣慣慣性性性矩矩矩之之之莫莫莫爾爾爾圓圓圓Mohrrsquos Circle for Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 44 54

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慣性矩之莫爾圓

由於Iu =

Ix + Iy2

+Ix minus Iy

2cos2θ minus Ixy sin2θ (125)

Iv =Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (126)

Iuv =Ix minus Iy

2sin2θ + Ixycos2θ (127)

將式(125)與式(127)各自分別平方並相加可得

(Iu minusIx + Iy

2)2 + I 2uv = (

Ix minus Iy2

)2 + I 2xy (128)

其中Ix Iy Ixy均為常數式(128)可進一步寫成

(Iu minus a)2 + I 2uv = R2 (129)

其中a =Ix+Iy2 R =

radic(IxminusIy2 )2 + I 2xycos2θp1 =

aRsin2θp1 =

minusIxyR 2θp1為OA轉向I軸方向即2θp1為

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 45 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 46 54

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解題技巧

首先確定轉動系統的Ix Iy Ixy接著算出莫爾圓之圓心座標a及半徑R並畫出莫爾圓再由莫爾圓上點出A(Ix Ixy )並由OA轉向I軸找出兩倍夾角2θp1及2θp2其中θp1及θp2為主軸(Principle Axis)與Ix軸的夾角由於主軸(Principle Axis)為極值之軸Imax = Iu及Imin = Iv決定θp1及θp2何者為Imax = Iu與Ix軸的夾角之方法乃比較Ix及Iy之大小Imax = Iu會靠近Ix及Iy之較大者也就是Imax = Iu與Ix及Iy之較大者的夾角比較小

a =Ix + Iy

2(130)

R =

radic(Ix minus Iy

2)2 + I 2xy (131)

2θp2 = π + 2θp1 θp2 minus θp1 =π

2(132)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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質質質量量量慣慣慣性性性矩矩矩Mass Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 48 54

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質量慣性矩

τ = Iα (133)

其中τ為旋轉扭矩I為質量慣性矩α為旋轉角速度

I =

ˆmr2dm (134)

=

ˆVr2ρdV (135)

= ρ

ˆVr2dV (136)

其中r為旋轉半徑或轉軸到旋轉體的距離

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 49 54

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 50 54

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平行軸定理

當通過物體質心(G)並在一轉軸上的質量慣性矩(IG )為已知時可透過座標軸轉換計算物體之質量慣性矩(I)

I =

ˆmr2dm (137)

=

ˆm

[(d + x prime)2 + y prime2]dm (138)

=

ˆm

(x prime2 + y prime2)dm + 2d

ˆmx primedm + d2

ˆmdm (139)

I = IG + md2 (140)

其中IG為在zrsquo軸上通過質心的質量慣性矩m為質量d為通過質心之軸zrsquo並平行原軸z之距離

acutem x primedm = x

acutem dm = 0 since x = 0

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (141)

k =

radicI

m(142)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (143)

= I1 + I2 + + In (144)

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課課課堂堂堂練練練習習習

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

兩圓盤複合體

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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結結結論論論

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結論

轉動慣量或慣性矩為物體對旋轉運動的慣性而所謂慣性乃物體持續維持不變的行為也就是定義慣性矩可表達物體對本身旋轉運動的維持或持續性好壞更簡單的說慣性矩越大之物體其維持旋轉運動趨勢的能力越大例如惰輪儲能裝置惰輪之慣性矩越大當其轉動以後其維持旋轉運動趨勢的能力越大

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  • 轉動慣量或慣性矩(Moment of Inertia)
  • 面積慣性矩(Moment of Inertia for Area)
    • 面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)
    • 面積慣性矩之迴轉半徑(Radius of Gyration for Area)
      • 複合面積之慣性矩(Moment of Inertia for Composite Area)
      • 面積之慣性積(Product of Inertia for Area)
        • 平行軸定理(Parallel-Axis Theorem)
          • 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)
            • 主軸慣性矩(Principal Moment of Inertia)
              • 慣性矩之莫爾圓(Mohrs Circle for Moment of Inertia)
              • 質量慣性矩(Mass Moment of Inertia)
                • 平行軸定理(Parallel-Axis Theorem)
Page 7: 轉動慣量 Moments of Inertia - cct.me.ntut.edu.twcct.me.ntut.edu.tw/ccteducation/chchting/aiahtm/statics/lecture... · 轉動慣量或慣性矩 I. 轉動慣量也稱慣性矩(Moment

轉動慣量或慣性矩 II

一般物件的動能K用轉動力學的定義取代

K =1

2mv2 (5)

=1

2m(rω)2 (6)

=1

2Iω2 (7)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 7 54

面面面積積積慣慣慣性性性矩矩矩Moment of Inertia for Area

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面積慣性矩

由左圖液面下y深度的壓力p為

p = γy (8)

其中γ = ρg為液體比重(SpecificWeight)液面下y深度之微小面積dA的受力dF為

dF = pdA = γydA (9)其扭矩dM為

dM = ydF = γy2dA (10)ˆdM = γ

ˆy2dA = γIx (11)

其中Ix =acutey2dA稱為x軸上之面積慣性矩

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極慣性矩(Polar Moment of Inertia)

極點

由左圖x軸上之面積慣性矩Ix及y軸上之面積慣性矩Iy如下

dIx = y2dA (12)

Ix =

ˆAy2dA (13)

dIy = x2dA (14)

Iy =

ˆAx2dA (15)

當微小面積dA對極點(O)或z軸旋轉時其面積極慣性矩JO如下

dJO = r2dA (16)

r2 = x2 + y2 (17)

JO =

ˆAr2dA =

ˆA(x2 + y2)dA = Ix + Iy (18)

其中r為微小面積dA到極點(O)的距離極點(O)簡單說乃z軸上的任意點由於x2 y2 r2及面

積(A)為正因此JO Ix Iy 均為正面積慣性矩的單位為m4mm4

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面面面積積積慣慣慣性性性矩矩矩之之之平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem for Area

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面積慣性矩之平行軸定理

極點

新極點

如左圖面積慣性矩之平行軸定理乃以面心(C)為新極點(O)設定新座標軸(xrsquoyrsquo)其中xrsquo軸平行於原座標x軸yrsquo軸平行於原座標y軸取微小面積dA其在新座標之點座標為(xrsquoyrsquo)則x軸上之面積慣性矩Ix及y軸上之面積慣性矩Iy可如下換算

Ix =

ˆA

(y prime + dy )2dA (19)

=

ˆAy prime2dA + 2dy

ˆAy primedA + d2

y

ˆAdA

Iy =

ˆA

(x prime + dx)2dA (20)

=

ˆAx prime2dA + 2dx

ˆAx primedA + d2

x

ˆAdA

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面積慣性矩之平行軸定理- Continued

由於面心在xrsquo軸上之等價力臂為y primeyrsquo軸上之等價力臂為x prime當以面心為轉軸點時即x prime = y prime = 0則x軸上之面積慣性矩Ix為ˆ

Ay prime2dA = macrIx prime (21)

ˆy primedA = y prime

ˆAdA = 0 (22)

d2y

ˆAdA = Ad2

y (23)

Ix =

ˆAy prime2dA + 2dy

ˆAy primedA + d2

y

ˆAdA (24)

= macrIx prime + Ad2y (25)

同理可得y軸上之面積慣性矩Iy

Iy =

ˆAx prime2dA + 2dx

ˆAx primedA + d2

x

ˆAdA (26)

= macrIy prime + Ad2x

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面積之極慣性矩

極點

新極點

由於通過面心之極慣性矩JC與極點(O)到面心(C)之距離d為

d2 = d2x + d2

y (27)

JC = macrIx prime + macrIy prime (28)

JO = Ix + Iy (29)

= macrIx prime + Ad2y + macrIy prime + Ad2

x (30)

= macrIx prime + macrIy prime + A(d2x + d2

y ) (31)

= JC + Ad2 (32)

也就是面積A之極慣性矩JO等於通過面積A之面心極慣性矩JC 與面積乘上極點(O)到面心(C)之距離d的平方和

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面面面積積積慣慣慣性性性矩矩矩之之之迴迴迴轉轉轉半半半徑徑徑Radius of Gyration for Area

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面積慣性矩之迴轉半徑

當x軸上之面積慣性矩Ix與y軸上之面積慣性矩Iy為已知時面積慣性矩之迴轉半徑可得知

dIx = y2dA (33)

Ix = kxA kx =

radicIxA

(34)

dIy = x2dA (35)

Iy = kyA ky =

radicIyA

(36)

dJO = r2dA (37)

JO = kOA kO =

radicJOA

(38)

其中(kx ky kO)分別為x軸y軸極點(O)之面積慣性矩迴轉半徑

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課課課堂堂堂練練練習習習

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解題技巧 I

極點

新極點

繪圖並標出面積(A)之面心座標位置(C)接著以面心座標位置(C)為原點畫出新座標軸(xrsquoyrsquo)其中xrsquo軸平行於原(舊)座標x軸yrsquo軸平行於原(舊)座標y軸並寫出新舊座標軸的平移量dx與dy其中dx為yrsquo軸與y軸的平移量dy為xrsquo軸與x軸的平移量取微小面積dA其在原(舊)座標之點座標為(xy)在新座標之點座標為(xrsquoyrsquo)帶入x軸上之面積慣性矩Ix及y軸上之面積慣性矩Iy還有xrsquo軸上之面積慣性矩 macrIx prime及y軸上之面積慣性矩 macrIy prime可如下換算

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解題技巧 II

Ix =

ˆAy2dA =

ˆAy prime2dA + 2dy

ˆAy primedA + d2

y

ˆAdA = macrIx prime + Ad2

y

macrIx prime =

ˆAy prime2dA (39)

Iy =

ˆAx2dA =

ˆAx prime2dA + 2dx

ˆAx primedA + d2

x

ˆAdA = macrIy prime + Ad2

x

macrIy prime =

ˆAx prime2dA (40)

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解題技巧 III

另外微小面積dA的面積慣性矩dIx dIy也可寫成

dIx = d macrIx prime + dAd2y (41)

Ix =

ˆdIx =

ˆd macrIx prime +

ˆdAd2

y (42)

dIy = d macrIy prime + dAd2x (43)

Iy =

ˆdIy =

ˆd macrIy prime +

ˆdAd2

x (44)

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

面心慣性矩 慣性矩 極慣性矩

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Example 10-1 解答

dA = bdy (45)

Ixb =

ˆAy2dA (46)

=

ˆ h

0y2bdy (47)

=1

3by3|h0 (48)

=1

3bh3 (49)

長方形之邊長b與h則面心慣性矩 macrIx prime macrIy prime與長方形之邊軸慣性矩Ix Iy為

macrIx prime =1

12bh3 (50)

macrIy prime =1

12hb3 (51)

Ix =1

3bh3 (52)

Iy =1

3hb3 (53)

JC = macrIx prime + macrIy prime (54)

JO = Ix + Iy (55)

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Example 10-2 解答

取微小面積dA=ydx之長方形其邊長b=dx與h=ydy = y2所以

y2 = 400x (56)

y =radic

400x (57)

d macrIx prime = d(1

12bh3) =

1

12y3dx (58)

dIx = d macrIx prime + dAd2y =

1

12y3dx + ydx

(y2

)2=

1

3y3dx (59)

Ix =

ˆ 100

0

1

3y3dx =

ˆ 100

0

1

3

(radic400x

)3dx (60)

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Example 10-3 解答

取微小面積dA=2ydx之長方形其邊長b=dx與h=2y所以

x2 + y2 = a2 (61)

y =radic

a2 minus x2 (62)

d macrIx prime = d(1

12bh3) =

1

12dx(2y)3 =

2

3y3dx (63)

macrIx prime =

ˆ a

minusa

2

3y3dx =

ˆ a

minusa

2

3

(radica2 minus x2

)3dx =

πa4

4(64)

圓之半徑a則面心慣性矩 macrIx prime macrIy prime為

macrIx prime =πa4

4(65)

macrIy prime =πa4

4(66)

JC = macrIx prime + macrIy prime =πa4

2(67)

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複複複合合合面面面積積積之之之慣慣慣性性性矩矩矩Moment of Inertia for Composite Area

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (68)

k =

radicI

m(69)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (70)

= I1 + I2 + + In (71)

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複合面積之慣性矩

複合面積之慣性矩(Ixc Iyc)為各局部面積之慣性矩(Ixi Iyi )合

Ixc =i=nsumi=1

Ixi (72)

= Ix1 + Ix2 + + Ixn (73)

Iyc =i=nsumi=1

Iyi (74)

= Iy1 + Iy2 + + Iyn (75)

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課課課堂堂堂練練練習習習

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

長方形與圓形複合

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長方形複合

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面面面積積積之之之慣慣慣性性性積積積Product of Inertia for Area

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面積之慣性積(Product of Inertia for Area)

為了下個章節求解慣性矩之極值(極大或極小值)方便數學運算在此先定義面積之慣性積(Ixy )為

dIxy = xydA (76)

Ixy =

ˆAxydA (77)

面積之慣性積的符號可為正負或零當慣性積之面積相對轉軸對稱時則面積之慣性積和為零

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

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平行軸定理

極點

新極點

如左圖面積慣性矩之平行軸定理乃以面心(C)為新極點(O)設定新座標軸(xrsquoyrsquo)其中xrsquo軸平行於原座標x軸yrsquo軸平行於原座標y軸取微小面積dA其在新座標之點座標為(xrsquoyrsquo)面積之慣性積(Ixy )為

dIxy = (x prime + dx)(y prime + dy )dA (78)

當以面心為轉軸點時即x prime = y prime = 0ˆAx primedA = x prime

ˆAdA = 0

ˆAy primedA = y prime

ˆAdA = 0 (79)

Ixy =

ˆA

(x prime + dx)(y prime + dy )dA (80)

=

ˆAx primey primedA + dx

ˆAy primedA + dy

ˆAx primedA + dxdy

ˆAdA (81)

= macrIx primey prime + Adxdy

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課課課堂堂堂練練練習習習

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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傾傾傾斜斜斜軸軸軸面面面積積積之之之慣慣慣性性性矩矩矩

Moment of Inertia for Area about Inclined Axis

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傾斜軸面積之慣性矩

當轉動軸傾斜θ角時也就是座標軸旋轉θ角而Ix Iy Ixy為已知則新的座標軸(uv)可寫為

u = xcosθ + ysinθ (82)

v = ycosθ minus xsinθ (83)

由u軸上之面積慣性矩Iuv軸上之面積慣性矩Iv及慣性積Iuv得知

dIu = v2dA = (ycosθ minus xsinθ)2dA (84)

dIv = u2dA = (xcosθ + ysinθ)2dA (85)

dIuv = uvdA = (xcosθ + ysinθ)(ycosθ minus xsinθ)dA (86)

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傾斜軸面積之慣性矩 I

由x軸上之面積慣性矩Ixy軸上之面積慣性矩Iy及慣性積Ixy得知

Ix =

ˆy2dA Iy =

ˆx2dA Ixy =

ˆxydA (87)

Iu =

ˆ(ycosθ minus xsinθ)2dA (88)

=

ˆ(y2cos2θ minus 2xycosθsinθ + x2sin2θ)dA (89)

= cos2θ

ˆy2dAminus 2cosθsinθ

ˆxydA + sin2θ

ˆx2dA (90)

= Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (91)

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傾斜軸面積之慣性矩 II

Iv =

ˆ(xcosθ + ysinθ)2dA (92)

=

ˆ(x2cos2θ + 2xycosθsinθ + y2sin2θ)dA (93)

= cos2θ

ˆx2dA + 2cosθsinθ

ˆxydA + sin2θ

ˆy2dA (94)

= Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (95)

Iuv =

ˆ(xcosθ + ysinθ)(ycosθ minus xsinθ)dA (96)

=

ˆ(xycos2θ + y2cosθsinθ minus x2cosθsinθ minus xysin2θ)dA (97)

= (cos2 minus sin2θ)

ˆxydA + cosθsinθ

ˆy2dAminus cosθsinθ

ˆx2dA

= Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 36 54

傾斜軸面積之慣性矩 III

再由

sin2θ + cos2θ = 1 (98)

sin(αplusmn β) = sinαcosβ plusmn cosαsinβ (99)

sin2θ = sin(θ + θ) = 2sinθcosθ (100)

cos(αplusmn β) = cosαcosβ ∓ sinαsinβ (101)

cos2θ = cos(θ + θ) = cos2θ minus sin2θ (102)

= 2cos2θ minus 1 = 1minus 2sin2θ (103)

所以

Iu = Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (104)

= Ix(cos2θ + 1

2) + Iy (

1minus cos2θ

2)minus Ixy sin2θ (105)

=Ix + Iy

2+

Ix minus Iy2

cos2θ minus Ixy sin2θ

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傾斜軸面積之慣性矩 IV

Iv = Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (106)

= Iy (cos2θ + 1

2) + Ix(

1minus cos2θ

2) + Ixy sin2θ (107)

=Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (108)

Iuv = Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ (109)

= Ixycos2θ +1

2Ixsin2θ minus 1

2Iy sin2θ (110)

=Ix minus Iy

2sin2θ + Ixycos2θ (111)

所以

JO = Iu + Iv = Ix + Iy (112)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 38 54

主主主軸軸軸慣慣慣性性性矩矩矩Principal Moment of Inertia

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主軸慣性矩 I

所謂主軸慣性矩(Principal Moment ofInertia)乃定義發生在慣性矩的極值(極大或極小值)上也就是慣性矩之一次微分式為零或斜率為零

dIu

dθ=

d(Ix+Iy

2+

IxminusIy2

cos2θ minus Ixy sin2θ)

dθ(113)

= minus2Ix minus Iy

2sin2θ minus 2Ixy cos2θ (114)

= 0 (115)

其中dsin2θ = cos2θ dcos2θ = minussin2θ當慣性矩之一次微分式為零時傾斜軸與主軸之夾角θ = θp

minus2Ix minus Iy

2sin2θp minus 2Ixycos2θp = 0 (116)

tan2θ =sin2θ

cos2θ=

minusIxy(Ix minus Iy )2

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 40 54

主軸慣性矩 II

cos2θp =IxminusIy2radic

(IxminusIy2 )2 + I 2xy

sin2θp =minusIxyradic

(IxminusIy2 )2 + I 2xy

(117)

(Iu)max =Ix + Iy

2+

Ix minus Iy2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

minus IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2+

radic(Ix minus Iy

2)2 + I 2xy (118)

(Iv )min =Ix + Iy

2minus Ix minus Iy

2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

+ IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2minusradic

(Ix minus Iy

2)2 + I 2xy (119)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 41 54

主軸慣性矩 III

由tan2θp = tan(π + 2θp)可得到θp1與θp2二個解即2θp2 = π + 2θp1也就是θp2 minus θp1 = π

2

tan2θp = tan(π + 2θp) (120)

2θp2 = π + 2θp1 (121)

θp2 minus θp1 =π

2(122)

所以

Imaxmin

=Ix + Iy

2plusmnradic

(Ix minus Iy

2)2 + I 2xy (123)

Iuv =Ix minus Iy

2

minusIxyradic(IxminusIy2 )2 + I 2xy

+ Ixy

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

(124)

= 0

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 42 54

課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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慣慣慣性性性矩矩矩之之之莫莫莫爾爾爾圓圓圓Mohrrsquos Circle for Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 44 54

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慣性矩之莫爾圓

由於Iu =

Ix + Iy2

+Ix minus Iy

2cos2θ minus Ixy sin2θ (125)

Iv =Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (126)

Iuv =Ix minus Iy

2sin2θ + Ixycos2θ (127)

將式(125)與式(127)各自分別平方並相加可得

(Iu minusIx + Iy

2)2 + I 2uv = (

Ix minus Iy2

)2 + I 2xy (128)

其中Ix Iy Ixy均為常數式(128)可進一步寫成

(Iu minus a)2 + I 2uv = R2 (129)

其中a =Ix+Iy2 R =

radic(IxminusIy2 )2 + I 2xycos2θp1 =

aRsin2θp1 =

minusIxyR 2θp1為OA轉向I軸方向即2θp1為

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 45 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 46 54

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解題技巧

首先確定轉動系統的Ix Iy Ixy接著算出莫爾圓之圓心座標a及半徑R並畫出莫爾圓再由莫爾圓上點出A(Ix Ixy )並由OA轉向I軸找出兩倍夾角2θp1及2θp2其中θp1及θp2為主軸(Principle Axis)與Ix軸的夾角由於主軸(Principle Axis)為極值之軸Imax = Iu及Imin = Iv決定θp1及θp2何者為Imax = Iu與Ix軸的夾角之方法乃比較Ix及Iy之大小Imax = Iu會靠近Ix及Iy之較大者也就是Imax = Iu與Ix及Iy之較大者的夾角比較小

a =Ix + Iy

2(130)

R =

radic(Ix minus Iy

2)2 + I 2xy (131)

2θp2 = π + 2θp1 θp2 minus θp1 =π

2(132)

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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質質質量量量慣慣慣性性性矩矩矩Mass Moment of Inertia

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質量慣性矩

τ = Iα (133)

其中τ為旋轉扭矩I為質量慣性矩α為旋轉角速度

I =

ˆmr2dm (134)

=

ˆVr2ρdV (135)

= ρ

ˆVr2dV (136)

其中r為旋轉半徑或轉軸到旋轉體的距離

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 49 54

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 50 54

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平行軸定理

當通過物體質心(G)並在一轉軸上的質量慣性矩(IG )為已知時可透過座標軸轉換計算物體之質量慣性矩(I)

I =

ˆmr2dm (137)

=

ˆm

[(d + x prime)2 + y prime2]dm (138)

=

ˆm

(x prime2 + y prime2)dm + 2d

ˆmx primedm + d2

ˆmdm (139)

I = IG + md2 (140)

其中IG為在zrsquo軸上通過質心的質量慣性矩m為質量d為通過質心之軸zrsquo並平行原軸z之距離

acutem x primedm = x

acutem dm = 0 since x = 0

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (141)

k =

radicI

m(142)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (143)

= I1 + I2 + + In (144)

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

兩圓盤複合體

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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結結結論論論

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 54 54

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結論

轉動慣量或慣性矩為物體對旋轉運動的慣性而所謂慣性乃物體持續維持不變的行為也就是定義慣性矩可表達物體對本身旋轉運動的維持或持續性好壞更簡單的說慣性矩越大之物體其維持旋轉運動趨勢的能力越大例如惰輪儲能裝置惰輪之慣性矩越大當其轉動以後其維持旋轉運動趨勢的能力越大

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 55 54

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  • 轉動慣量或慣性矩(Moment of Inertia)
  • 面積慣性矩(Moment of Inertia for Area)
    • 面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)
    • 面積慣性矩之迴轉半徑(Radius of Gyration for Area)
      • 複合面積之慣性矩(Moment of Inertia for Composite Area)
      • 面積之慣性積(Product of Inertia for Area)
        • 平行軸定理(Parallel-Axis Theorem)
          • 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)
            • 主軸慣性矩(Principal Moment of Inertia)
              • 慣性矩之莫爾圓(Mohrs Circle for Moment of Inertia)
              • 質量慣性矩(Mass Moment of Inertia)
                • 平行軸定理(Parallel-Axis Theorem)
Page 8: 轉動慣量 Moments of Inertia - cct.me.ntut.edu.twcct.me.ntut.edu.tw/ccteducation/chchting/aiahtm/statics/lecture... · 轉動慣量或慣性矩 I. 轉動慣量也稱慣性矩(Moment

面面面積積積慣慣慣性性性矩矩矩Moment of Inertia for Area

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 8 54

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面積慣性矩

由左圖液面下y深度的壓力p為

p = γy (8)

其中γ = ρg為液體比重(SpecificWeight)液面下y深度之微小面積dA的受力dF為

dF = pdA = γydA (9)其扭矩dM為

dM = ydF = γy2dA (10)ˆdM = γ

ˆy2dA = γIx (11)

其中Ix =acutey2dA稱為x軸上之面積慣性矩

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 9 54

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極慣性矩(Polar Moment of Inertia)

極點

由左圖x軸上之面積慣性矩Ix及y軸上之面積慣性矩Iy如下

dIx = y2dA (12)

Ix =

ˆAy2dA (13)

dIy = x2dA (14)

Iy =

ˆAx2dA (15)

當微小面積dA對極點(O)或z軸旋轉時其面積極慣性矩JO如下

dJO = r2dA (16)

r2 = x2 + y2 (17)

JO =

ˆAr2dA =

ˆA(x2 + y2)dA = Ix + Iy (18)

其中r為微小面積dA到極點(O)的距離極點(O)簡單說乃z軸上的任意點由於x2 y2 r2及面

積(A)為正因此JO Ix Iy 均為正面積慣性矩的單位為m4mm4

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 10 54

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面面面積積積慣慣慣性性性矩矩矩之之之平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem for Area

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 11 54

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面積慣性矩之平行軸定理

極點

新極點

如左圖面積慣性矩之平行軸定理乃以面心(C)為新極點(O)設定新座標軸(xrsquoyrsquo)其中xrsquo軸平行於原座標x軸yrsquo軸平行於原座標y軸取微小面積dA其在新座標之點座標為(xrsquoyrsquo)則x軸上之面積慣性矩Ix及y軸上之面積慣性矩Iy可如下換算

Ix =

ˆA

(y prime + dy )2dA (19)

=

ˆAy prime2dA + 2dy

ˆAy primedA + d2

y

ˆAdA

Iy =

ˆA

(x prime + dx)2dA (20)

=

ˆAx prime2dA + 2dx

ˆAx primedA + d2

x

ˆAdA

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面積慣性矩之平行軸定理- Continued

由於面心在xrsquo軸上之等價力臂為y primeyrsquo軸上之等價力臂為x prime當以面心為轉軸點時即x prime = y prime = 0則x軸上之面積慣性矩Ix為ˆ

Ay prime2dA = macrIx prime (21)

ˆy primedA = y prime

ˆAdA = 0 (22)

d2y

ˆAdA = Ad2

y (23)

Ix =

ˆAy prime2dA + 2dy

ˆAy primedA + d2

y

ˆAdA (24)

= macrIx prime + Ad2y (25)

同理可得y軸上之面積慣性矩Iy

Iy =

ˆAx prime2dA + 2dx

ˆAx primedA + d2

x

ˆAdA (26)

= macrIy prime + Ad2x

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 13 54

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面積之極慣性矩

極點

新極點

由於通過面心之極慣性矩JC與極點(O)到面心(C)之距離d為

d2 = d2x + d2

y (27)

JC = macrIx prime + macrIy prime (28)

JO = Ix + Iy (29)

= macrIx prime + Ad2y + macrIy prime + Ad2

x (30)

= macrIx prime + macrIy prime + A(d2x + d2

y ) (31)

= JC + Ad2 (32)

也就是面積A之極慣性矩JO等於通過面積A之面心極慣性矩JC 與面積乘上極點(O)到面心(C)之距離d的平方和

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 14 54

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面面面積積積慣慣慣性性性矩矩矩之之之迴迴迴轉轉轉半半半徑徑徑Radius of Gyration for Area

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 15 54

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面積慣性矩之迴轉半徑

當x軸上之面積慣性矩Ix與y軸上之面積慣性矩Iy為已知時面積慣性矩之迴轉半徑可得知

dIx = y2dA (33)

Ix = kxA kx =

radicIxA

(34)

dIy = x2dA (35)

Iy = kyA ky =

radicIyA

(36)

dJO = r2dA (37)

JO = kOA kO =

radicJOA

(38)

其中(kx ky kO)分別為x軸y軸極點(O)之面積慣性矩迴轉半徑

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 17 54

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解題技巧 I

極點

新極點

繪圖並標出面積(A)之面心座標位置(C)接著以面心座標位置(C)為原點畫出新座標軸(xrsquoyrsquo)其中xrsquo軸平行於原(舊)座標x軸yrsquo軸平行於原(舊)座標y軸並寫出新舊座標軸的平移量dx與dy其中dx為yrsquo軸與y軸的平移量dy為xrsquo軸與x軸的平移量取微小面積dA其在原(舊)座標之點座標為(xy)在新座標之點座標為(xrsquoyrsquo)帶入x軸上之面積慣性矩Ix及y軸上之面積慣性矩Iy還有xrsquo軸上之面積慣性矩 macrIx prime及y軸上之面積慣性矩 macrIy prime可如下換算

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 18 54

解題技巧 II

Ix =

ˆAy2dA =

ˆAy prime2dA + 2dy

ˆAy primedA + d2

y

ˆAdA = macrIx prime + Ad2

y

macrIx prime =

ˆAy prime2dA (39)

Iy =

ˆAx2dA =

ˆAx prime2dA + 2dx

ˆAx primedA + d2

x

ˆAdA = macrIy prime + Ad2

x

macrIy prime =

ˆAx prime2dA (40)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 19 54

解題技巧 III

另外微小面積dA的面積慣性矩dIx dIy也可寫成

dIx = d macrIx prime + dAd2y (41)

Ix =

ˆdIx =

ˆd macrIx prime +

ˆdAd2

y (42)

dIy = d macrIy prime + dAd2x (43)

Iy =

ˆdIy =

ˆd macrIy prime +

ˆdAd2

x (44)

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

面心慣性矩 慣性矩 極慣性矩

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 20 54

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 20 54

Example 10-1 解答

dA = bdy (45)

Ixb =

ˆAy2dA (46)

=

ˆ h

0y2bdy (47)

=1

3by3|h0 (48)

=1

3bh3 (49)

長方形之邊長b與h則面心慣性矩 macrIx prime macrIy prime與長方形之邊軸慣性矩Ix Iy為

macrIx prime =1

12bh3 (50)

macrIy prime =1

12hb3 (51)

Ix =1

3bh3 (52)

Iy =1

3hb3 (53)

JC = macrIx prime + macrIy prime (54)

JO = Ix + Iy (55)

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 21 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Example 10-2 解答

取微小面積dA=ydx之長方形其邊長b=dx與h=ydy = y2所以

y2 = 400x (56)

y =radic

400x (57)

d macrIx prime = d(1

12bh3) =

1

12y3dx (58)

dIx = d macrIx prime + dAd2y =

1

12y3dx + ydx

(y2

)2=

1

3y3dx (59)

Ix =

ˆ 100

0

1

3y3dx =

ˆ 100

0

1

3

(radic400x

)3dx (60)

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 22 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 22 54

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Example 10-3 解答

取微小面積dA=2ydx之長方形其邊長b=dx與h=2y所以

x2 + y2 = a2 (61)

y =radic

a2 minus x2 (62)

d macrIx prime = d(1

12bh3) =

1

12dx(2y)3 =

2

3y3dx (63)

macrIx prime =

ˆ a

minusa

2

3y3dx =

ˆ a

minusa

2

3

(radica2 minus x2

)3dx =

πa4

4(64)

圓之半徑a則面心慣性矩 macrIx prime macrIy prime為

macrIx prime =πa4

4(65)

macrIy prime =πa4

4(66)

JC = macrIx prime + macrIy prime =πa4

2(67)

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複複複合合合面面面積積積之之之慣慣慣性性性矩矩矩Moment of Inertia for Composite Area

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (68)

k =

radicI

m(69)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (70)

= I1 + I2 + + In (71)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 25 54

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複合面積之慣性矩

複合面積之慣性矩(Ixc Iyc)為各局部面積之慣性矩(Ixi Iyi )合

Ixc =i=nsumi=1

Ixi (72)

= Ix1 + Ix2 + + Ixn (73)

Iyc =i=nsumi=1

Iyi (74)

= Iy1 + Iy2 + + Iyn (75)

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

長方形與圓形複合

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

長方形複合

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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面面面積積積之之之慣慣慣性性性積積積Product of Inertia for Area

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面積之慣性積(Product of Inertia for Area)

為了下個章節求解慣性矩之極值(極大或極小值)方便數學運算在此先定義面積之慣性積(Ixy )為

dIxy = xydA (76)

Ixy =

ˆAxydA (77)

面積之慣性積的符號可為正負或零當慣性積之面積相對轉軸對稱時則面積之慣性積和為零

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

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平行軸定理

極點

新極點

如左圖面積慣性矩之平行軸定理乃以面心(C)為新極點(O)設定新座標軸(xrsquoyrsquo)其中xrsquo軸平行於原座標x軸yrsquo軸平行於原座標y軸取微小面積dA其在新座標之點座標為(xrsquoyrsquo)面積之慣性積(Ixy )為

dIxy = (x prime + dx)(y prime + dy )dA (78)

當以面心為轉軸點時即x prime = y prime = 0ˆAx primedA = x prime

ˆAdA = 0

ˆAy primedA = y prime

ˆAdA = 0 (79)

Ixy =

ˆA

(x prime + dx)(y prime + dy )dA (80)

=

ˆAx primey primedA + dx

ˆAy primedA + dy

ˆAx primedA + dxdy

ˆAdA (81)

= macrIx primey prime + Adxdy

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課課課堂堂堂練練練習習習

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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傾傾傾斜斜斜軸軸軸面面面積積積之之之慣慣慣性性性矩矩矩

Moment of Inertia for Area about Inclined Axis

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傾斜軸面積之慣性矩

當轉動軸傾斜θ角時也就是座標軸旋轉θ角而Ix Iy Ixy為已知則新的座標軸(uv)可寫為

u = xcosθ + ysinθ (82)

v = ycosθ minus xsinθ (83)

由u軸上之面積慣性矩Iuv軸上之面積慣性矩Iv及慣性積Iuv得知

dIu = v2dA = (ycosθ minus xsinθ)2dA (84)

dIv = u2dA = (xcosθ + ysinθ)2dA (85)

dIuv = uvdA = (xcosθ + ysinθ)(ycosθ minus xsinθ)dA (86)

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傾斜軸面積之慣性矩 I

由x軸上之面積慣性矩Ixy軸上之面積慣性矩Iy及慣性積Ixy得知

Ix =

ˆy2dA Iy =

ˆx2dA Ixy =

ˆxydA (87)

Iu =

ˆ(ycosθ minus xsinθ)2dA (88)

=

ˆ(y2cos2θ minus 2xycosθsinθ + x2sin2θ)dA (89)

= cos2θ

ˆy2dAminus 2cosθsinθ

ˆxydA + sin2θ

ˆx2dA (90)

= Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (91)

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傾斜軸面積之慣性矩 II

Iv =

ˆ(xcosθ + ysinθ)2dA (92)

=

ˆ(x2cos2θ + 2xycosθsinθ + y2sin2θ)dA (93)

= cos2θ

ˆx2dA + 2cosθsinθ

ˆxydA + sin2θ

ˆy2dA (94)

= Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (95)

Iuv =

ˆ(xcosθ + ysinθ)(ycosθ minus xsinθ)dA (96)

=

ˆ(xycos2θ + y2cosθsinθ minus x2cosθsinθ minus xysin2θ)dA (97)

= (cos2 minus sin2θ)

ˆxydA + cosθsinθ

ˆy2dAminus cosθsinθ

ˆx2dA

= Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 36 54

傾斜軸面積之慣性矩 III

再由

sin2θ + cos2θ = 1 (98)

sin(αplusmn β) = sinαcosβ plusmn cosαsinβ (99)

sin2θ = sin(θ + θ) = 2sinθcosθ (100)

cos(αplusmn β) = cosαcosβ ∓ sinαsinβ (101)

cos2θ = cos(θ + θ) = cos2θ minus sin2θ (102)

= 2cos2θ minus 1 = 1minus 2sin2θ (103)

所以

Iu = Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (104)

= Ix(cos2θ + 1

2) + Iy (

1minus cos2θ

2)minus Ixy sin2θ (105)

=Ix + Iy

2+

Ix minus Iy2

cos2θ minus Ixy sin2θ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 37 54

傾斜軸面積之慣性矩 IV

Iv = Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (106)

= Iy (cos2θ + 1

2) + Ix(

1minus cos2θ

2) + Ixy sin2θ (107)

=Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (108)

Iuv = Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ (109)

= Ixycos2θ +1

2Ixsin2θ minus 1

2Iy sin2θ (110)

=Ix minus Iy

2sin2θ + Ixycos2θ (111)

所以

JO = Iu + Iv = Ix + Iy (112)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 38 54

主主主軸軸軸慣慣慣性性性矩矩矩Principal Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 39 54

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主軸慣性矩 I

所謂主軸慣性矩(Principal Moment ofInertia)乃定義發生在慣性矩的極值(極大或極小值)上也就是慣性矩之一次微分式為零或斜率為零

dIu

dθ=

d(Ix+Iy

2+

IxminusIy2

cos2θ minus Ixy sin2θ)

dθ(113)

= minus2Ix minus Iy

2sin2θ minus 2Ixy cos2θ (114)

= 0 (115)

其中dsin2θ = cos2θ dcos2θ = minussin2θ當慣性矩之一次微分式為零時傾斜軸與主軸之夾角θ = θp

minus2Ix minus Iy

2sin2θp minus 2Ixycos2θp = 0 (116)

tan2θ =sin2θ

cos2θ=

minusIxy(Ix minus Iy )2

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 40 54

主軸慣性矩 II

cos2θp =IxminusIy2radic

(IxminusIy2 )2 + I 2xy

sin2θp =minusIxyradic

(IxminusIy2 )2 + I 2xy

(117)

(Iu)max =Ix + Iy

2+

Ix minus Iy2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

minus IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2+

radic(Ix minus Iy

2)2 + I 2xy (118)

(Iv )min =Ix + Iy

2minus Ix minus Iy

2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

+ IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2minusradic

(Ix minus Iy

2)2 + I 2xy (119)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 41 54

主軸慣性矩 III

由tan2θp = tan(π + 2θp)可得到θp1與θp2二個解即2θp2 = π + 2θp1也就是θp2 minus θp1 = π

2

tan2θp = tan(π + 2θp) (120)

2θp2 = π + 2θp1 (121)

θp2 minus θp1 =π

2(122)

所以

Imaxmin

=Ix + Iy

2plusmnradic

(Ix minus Iy

2)2 + I 2xy (123)

Iuv =Ix minus Iy

2

minusIxyradic(IxminusIy2 )2 + I 2xy

+ Ixy

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

(124)

= 0

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 42 54

課課課堂堂堂練練練習習習

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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慣慣慣性性性矩矩矩之之之莫莫莫爾爾爾圓圓圓Mohrrsquos Circle for Moment of Inertia

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慣性矩之莫爾圓

由於Iu =

Ix + Iy2

+Ix minus Iy

2cos2θ minus Ixy sin2θ (125)

Iv =Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (126)

Iuv =Ix minus Iy

2sin2θ + Ixycos2θ (127)

將式(125)與式(127)各自分別平方並相加可得

(Iu minusIx + Iy

2)2 + I 2uv = (

Ix minus Iy2

)2 + I 2xy (128)

其中Ix Iy Ixy均為常數式(128)可進一步寫成

(Iu minus a)2 + I 2uv = R2 (129)

其中a =Ix+Iy2 R =

radic(IxminusIy2 )2 + I 2xycos2θp1 =

aRsin2θp1 =

minusIxyR 2θp1為OA轉向I軸方向即2θp1為

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 46 54

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解題技巧

首先確定轉動系統的Ix Iy Ixy接著算出莫爾圓之圓心座標a及半徑R並畫出莫爾圓再由莫爾圓上點出A(Ix Ixy )並由OA轉向I軸找出兩倍夾角2θp1及2θp2其中θp1及θp2為主軸(Principle Axis)與Ix軸的夾角由於主軸(Principle Axis)為極值之軸Imax = Iu及Imin = Iv決定θp1及θp2何者為Imax = Iu與Ix軸的夾角之方法乃比較Ix及Iy之大小Imax = Iu會靠近Ix及Iy之較大者也就是Imax = Iu與Ix及Iy之較大者的夾角比較小

a =Ix + Iy

2(130)

R =

radic(Ix minus Iy

2)2 + I 2xy (131)

2θp2 = π + 2θp1 θp2 minus θp1 =π

2(132)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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質質質量量量慣慣慣性性性矩矩矩Mass Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 48 54

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質量慣性矩

τ = Iα (133)

其中τ為旋轉扭矩I為質量慣性矩α為旋轉角速度

I =

ˆmr2dm (134)

=

ˆVr2ρdV (135)

= ρ

ˆVr2dV (136)

其中r為旋轉半徑或轉軸到旋轉體的距離

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

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平行軸定理

當通過物體質心(G)並在一轉軸上的質量慣性矩(IG )為已知時可透過座標軸轉換計算物體之質量慣性矩(I)

I =

ˆmr2dm (137)

=

ˆm

[(d + x prime)2 + y prime2]dm (138)

=

ˆm

(x prime2 + y prime2)dm + 2d

ˆmx primedm + d2

ˆmdm (139)

I = IG + md2 (140)

其中IG為在zrsquo軸上通過質心的質量慣性矩m為質量d為通過質心之軸zrsquo並平行原軸z之距離

acutem x primedm = x

acutem dm = 0 since x = 0

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (141)

k =

radicI

m(142)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (143)

= I1 + I2 + + In (144)

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

兩圓盤複合體

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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結結結論論論

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結論

轉動慣量或慣性矩為物體對旋轉運動的慣性而所謂慣性乃物體持續維持不變的行為也就是定義慣性矩可表達物體對本身旋轉運動的維持或持續性好壞更簡單的說慣性矩越大之物體其維持旋轉運動趨勢的能力越大例如惰輪儲能裝置惰輪之慣性矩越大當其轉動以後其維持旋轉運動趨勢的能力越大

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 55 54

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  • 轉動慣量或慣性矩(Moment of Inertia)
  • 面積慣性矩(Moment of Inertia for Area)
    • 面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)
    • 面積慣性矩之迴轉半徑(Radius of Gyration for Area)
      • 複合面積之慣性矩(Moment of Inertia for Composite Area)
      • 面積之慣性積(Product of Inertia for Area)
        • 平行軸定理(Parallel-Axis Theorem)
          • 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)
            • 主軸慣性矩(Principal Moment of Inertia)
              • 慣性矩之莫爾圓(Mohrs Circle for Moment of Inertia)
              • 質量慣性矩(Mass Moment of Inertia)
                • 平行軸定理(Parallel-Axis Theorem)
Page 9: 轉動慣量 Moments of Inertia - cct.me.ntut.edu.twcct.me.ntut.edu.tw/ccteducation/chchting/aiahtm/statics/lecture... · 轉動慣量或慣性矩 I. 轉動慣量也稱慣性矩(Moment

面積慣性矩

由左圖液面下y深度的壓力p為

p = γy (8)

其中γ = ρg為液體比重(SpecificWeight)液面下y深度之微小面積dA的受力dF為

dF = pdA = γydA (9)其扭矩dM為

dM = ydF = γy2dA (10)ˆdM = γ

ˆy2dA = γIx (11)

其中Ix =acutey2dA稱為x軸上之面積慣性矩

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 9 54

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極慣性矩(Polar Moment of Inertia)

極點

由左圖x軸上之面積慣性矩Ix及y軸上之面積慣性矩Iy如下

dIx = y2dA (12)

Ix =

ˆAy2dA (13)

dIy = x2dA (14)

Iy =

ˆAx2dA (15)

當微小面積dA對極點(O)或z軸旋轉時其面積極慣性矩JO如下

dJO = r2dA (16)

r2 = x2 + y2 (17)

JO =

ˆAr2dA =

ˆA(x2 + y2)dA = Ix + Iy (18)

其中r為微小面積dA到極點(O)的距離極點(O)簡單說乃z軸上的任意點由於x2 y2 r2及面

積(A)為正因此JO Ix Iy 均為正面積慣性矩的單位為m4mm4

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面面面積積積慣慣慣性性性矩矩矩之之之平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem for Area

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 11 54

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面積慣性矩之平行軸定理

極點

新極點

如左圖面積慣性矩之平行軸定理乃以面心(C)為新極點(O)設定新座標軸(xrsquoyrsquo)其中xrsquo軸平行於原座標x軸yrsquo軸平行於原座標y軸取微小面積dA其在新座標之點座標為(xrsquoyrsquo)則x軸上之面積慣性矩Ix及y軸上之面積慣性矩Iy可如下換算

Ix =

ˆA

(y prime + dy )2dA (19)

=

ˆAy prime2dA + 2dy

ˆAy primedA + d2

y

ˆAdA

Iy =

ˆA

(x prime + dx)2dA (20)

=

ˆAx prime2dA + 2dx

ˆAx primedA + d2

x

ˆAdA

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面積慣性矩之平行軸定理- Continued

由於面心在xrsquo軸上之等價力臂為y primeyrsquo軸上之等價力臂為x prime當以面心為轉軸點時即x prime = y prime = 0則x軸上之面積慣性矩Ix為ˆ

Ay prime2dA = macrIx prime (21)

ˆy primedA = y prime

ˆAdA = 0 (22)

d2y

ˆAdA = Ad2

y (23)

Ix =

ˆAy prime2dA + 2dy

ˆAy primedA + d2

y

ˆAdA (24)

= macrIx prime + Ad2y (25)

同理可得y軸上之面積慣性矩Iy

Iy =

ˆAx prime2dA + 2dx

ˆAx primedA + d2

x

ˆAdA (26)

= macrIy prime + Ad2x

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面積之極慣性矩

極點

新極點

由於通過面心之極慣性矩JC與極點(O)到面心(C)之距離d為

d2 = d2x + d2

y (27)

JC = macrIx prime + macrIy prime (28)

JO = Ix + Iy (29)

= macrIx prime + Ad2y + macrIy prime + Ad2

x (30)

= macrIx prime + macrIy prime + A(d2x + d2

y ) (31)

= JC + Ad2 (32)

也就是面積A之極慣性矩JO等於通過面積A之面心極慣性矩JC 與面積乘上極點(O)到面心(C)之距離d的平方和

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面面面積積積慣慣慣性性性矩矩矩之之之迴迴迴轉轉轉半半半徑徑徑Radius of Gyration for Area

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面積慣性矩之迴轉半徑

當x軸上之面積慣性矩Ix與y軸上之面積慣性矩Iy為已知時面積慣性矩之迴轉半徑可得知

dIx = y2dA (33)

Ix = kxA kx =

radicIxA

(34)

dIy = x2dA (35)

Iy = kyA ky =

radicIyA

(36)

dJO = r2dA (37)

JO = kOA kO =

radicJOA

(38)

其中(kx ky kO)分別為x軸y軸極點(O)之面積慣性矩迴轉半徑

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課課課堂堂堂練練練習習習

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解題技巧 I

極點

新極點

繪圖並標出面積(A)之面心座標位置(C)接著以面心座標位置(C)為原點畫出新座標軸(xrsquoyrsquo)其中xrsquo軸平行於原(舊)座標x軸yrsquo軸平行於原(舊)座標y軸並寫出新舊座標軸的平移量dx與dy其中dx為yrsquo軸與y軸的平移量dy為xrsquo軸與x軸的平移量取微小面積dA其在原(舊)座標之點座標為(xy)在新座標之點座標為(xrsquoyrsquo)帶入x軸上之面積慣性矩Ix及y軸上之面積慣性矩Iy還有xrsquo軸上之面積慣性矩 macrIx prime及y軸上之面積慣性矩 macrIy prime可如下換算

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解題技巧 II

Ix =

ˆAy2dA =

ˆAy prime2dA + 2dy

ˆAy primedA + d2

y

ˆAdA = macrIx prime + Ad2

y

macrIx prime =

ˆAy prime2dA (39)

Iy =

ˆAx2dA =

ˆAx prime2dA + 2dx

ˆAx primedA + d2

x

ˆAdA = macrIy prime + Ad2

x

macrIy prime =

ˆAx prime2dA (40)

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解題技巧 III

另外微小面積dA的面積慣性矩dIx dIy也可寫成

dIx = d macrIx prime + dAd2y (41)

Ix =

ˆdIx =

ˆd macrIx prime +

ˆdAd2

y (42)

dIy = d macrIy prime + dAd2x (43)

Iy =

ˆdIy =

ˆd macrIy prime +

ˆdAd2

x (44)

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

面心慣性矩 慣性矩 極慣性矩

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Example 10-1 解答

dA = bdy (45)

Ixb =

ˆAy2dA (46)

=

ˆ h

0y2bdy (47)

=1

3by3|h0 (48)

=1

3bh3 (49)

長方形之邊長b與h則面心慣性矩 macrIx prime macrIy prime與長方形之邊軸慣性矩Ix Iy為

macrIx prime =1

12bh3 (50)

macrIy prime =1

12hb3 (51)

Ix =1

3bh3 (52)

Iy =1

3hb3 (53)

JC = macrIx prime + macrIy prime (54)

JO = Ix + Iy (55)

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Example 10-2 解答

取微小面積dA=ydx之長方形其邊長b=dx與h=ydy = y2所以

y2 = 400x (56)

y =radic

400x (57)

d macrIx prime = d(1

12bh3) =

1

12y3dx (58)

dIx = d macrIx prime + dAd2y =

1

12y3dx + ydx

(y2

)2=

1

3y3dx (59)

Ix =

ˆ 100

0

1

3y3dx =

ˆ 100

0

1

3

(radic400x

)3dx (60)

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Example 10-3 解答

取微小面積dA=2ydx之長方形其邊長b=dx與h=2y所以

x2 + y2 = a2 (61)

y =radic

a2 minus x2 (62)

d macrIx prime = d(1

12bh3) =

1

12dx(2y)3 =

2

3y3dx (63)

macrIx prime =

ˆ a

minusa

2

3y3dx =

ˆ a

minusa

2

3

(radica2 minus x2

)3dx =

πa4

4(64)

圓之半徑a則面心慣性矩 macrIx prime macrIy prime為

macrIx prime =πa4

4(65)

macrIy prime =πa4

4(66)

JC = macrIx prime + macrIy prime =πa4

2(67)

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複複複合合合面面面積積積之之之慣慣慣性性性矩矩矩Moment of Inertia for Composite Area

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (68)

k =

radicI

m(69)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (70)

= I1 + I2 + + In (71)

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複合面積之慣性矩

複合面積之慣性矩(Ixc Iyc)為各局部面積之慣性矩(Ixi Iyi )合

Ixc =i=nsumi=1

Ixi (72)

= Ix1 + Ix2 + + Ixn (73)

Iyc =i=nsumi=1

Iyi (74)

= Iy1 + Iy2 + + Iyn (75)

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課課課堂堂堂練練練習習習

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

長方形與圓形複合

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

長方形複合

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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面面面積積積之之之慣慣慣性性性積積積Product of Inertia for Area

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面積之慣性積(Product of Inertia for Area)

為了下個章節求解慣性矩之極值(極大或極小值)方便數學運算在此先定義面積之慣性積(Ixy )為

dIxy = xydA (76)

Ixy =

ˆAxydA (77)

面積之慣性積的符號可為正負或零當慣性積之面積相對轉軸對稱時則面積之慣性積和為零

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

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平行軸定理

極點

新極點

如左圖面積慣性矩之平行軸定理乃以面心(C)為新極點(O)設定新座標軸(xrsquoyrsquo)其中xrsquo軸平行於原座標x軸yrsquo軸平行於原座標y軸取微小面積dA其在新座標之點座標為(xrsquoyrsquo)面積之慣性積(Ixy )為

dIxy = (x prime + dx)(y prime + dy )dA (78)

當以面心為轉軸點時即x prime = y prime = 0ˆAx primedA = x prime

ˆAdA = 0

ˆAy primedA = y prime

ˆAdA = 0 (79)

Ixy =

ˆA

(x prime + dx)(y prime + dy )dA (80)

=

ˆAx primey primedA + dx

ˆAy primedA + dy

ˆAx primedA + dxdy

ˆAdA (81)

= macrIx primey prime + Adxdy

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課課課堂堂堂練練練習習習

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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傾傾傾斜斜斜軸軸軸面面面積積積之之之慣慣慣性性性矩矩矩

Moment of Inertia for Area about Inclined Axis

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 33 54

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傾斜軸面積之慣性矩

當轉動軸傾斜θ角時也就是座標軸旋轉θ角而Ix Iy Ixy為已知則新的座標軸(uv)可寫為

u = xcosθ + ysinθ (82)

v = ycosθ minus xsinθ (83)

由u軸上之面積慣性矩Iuv軸上之面積慣性矩Iv及慣性積Iuv得知

dIu = v2dA = (ycosθ minus xsinθ)2dA (84)

dIv = u2dA = (xcosθ + ysinθ)2dA (85)

dIuv = uvdA = (xcosθ + ysinθ)(ycosθ minus xsinθ)dA (86)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 34 54

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傾斜軸面積之慣性矩 I

由x軸上之面積慣性矩Ixy軸上之面積慣性矩Iy及慣性積Ixy得知

Ix =

ˆy2dA Iy =

ˆx2dA Ixy =

ˆxydA (87)

Iu =

ˆ(ycosθ minus xsinθ)2dA (88)

=

ˆ(y2cos2θ minus 2xycosθsinθ + x2sin2θ)dA (89)

= cos2θ

ˆy2dAminus 2cosθsinθ

ˆxydA + sin2θ

ˆx2dA (90)

= Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (91)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 35 54

傾斜軸面積之慣性矩 II

Iv =

ˆ(xcosθ + ysinθ)2dA (92)

=

ˆ(x2cos2θ + 2xycosθsinθ + y2sin2θ)dA (93)

= cos2θ

ˆx2dA + 2cosθsinθ

ˆxydA + sin2θ

ˆy2dA (94)

= Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (95)

Iuv =

ˆ(xcosθ + ysinθ)(ycosθ minus xsinθ)dA (96)

=

ˆ(xycos2θ + y2cosθsinθ minus x2cosθsinθ minus xysin2θ)dA (97)

= (cos2 minus sin2θ)

ˆxydA + cosθsinθ

ˆy2dAminus cosθsinθ

ˆx2dA

= Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 36 54

傾斜軸面積之慣性矩 III

再由

sin2θ + cos2θ = 1 (98)

sin(αplusmn β) = sinαcosβ plusmn cosαsinβ (99)

sin2θ = sin(θ + θ) = 2sinθcosθ (100)

cos(αplusmn β) = cosαcosβ ∓ sinαsinβ (101)

cos2θ = cos(θ + θ) = cos2θ minus sin2θ (102)

= 2cos2θ minus 1 = 1minus 2sin2θ (103)

所以

Iu = Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (104)

= Ix(cos2θ + 1

2) + Iy (

1minus cos2θ

2)minus Ixy sin2θ (105)

=Ix + Iy

2+

Ix minus Iy2

cos2θ minus Ixy sin2θ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 37 54

傾斜軸面積之慣性矩 IV

Iv = Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (106)

= Iy (cos2θ + 1

2) + Ix(

1minus cos2θ

2) + Ixy sin2θ (107)

=Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (108)

Iuv = Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ (109)

= Ixycos2θ +1

2Ixsin2θ minus 1

2Iy sin2θ (110)

=Ix minus Iy

2sin2θ + Ixycos2θ (111)

所以

JO = Iu + Iv = Ix + Iy (112)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 38 54

主主主軸軸軸慣慣慣性性性矩矩矩Principal Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 39 54

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主軸慣性矩 I

所謂主軸慣性矩(Principal Moment ofInertia)乃定義發生在慣性矩的極值(極大或極小值)上也就是慣性矩之一次微分式為零或斜率為零

dIu

dθ=

d(Ix+Iy

2+

IxminusIy2

cos2θ minus Ixy sin2θ)

dθ(113)

= minus2Ix minus Iy

2sin2θ minus 2Ixy cos2θ (114)

= 0 (115)

其中dsin2θ = cos2θ dcos2θ = minussin2θ當慣性矩之一次微分式為零時傾斜軸與主軸之夾角θ = θp

minus2Ix minus Iy

2sin2θp minus 2Ixycos2θp = 0 (116)

tan2θ =sin2θ

cos2θ=

minusIxy(Ix minus Iy )2

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 40 54

主軸慣性矩 II

cos2θp =IxminusIy2radic

(IxminusIy2 )2 + I 2xy

sin2θp =minusIxyradic

(IxminusIy2 )2 + I 2xy

(117)

(Iu)max =Ix + Iy

2+

Ix minus Iy2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

minus IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2+

radic(Ix minus Iy

2)2 + I 2xy (118)

(Iv )min =Ix + Iy

2minus Ix minus Iy

2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

+ IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2minusradic

(Ix minus Iy

2)2 + I 2xy (119)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 41 54

主軸慣性矩 III

由tan2θp = tan(π + 2θp)可得到θp1與θp2二個解即2θp2 = π + 2θp1也就是θp2 minus θp1 = π

2

tan2θp = tan(π + 2θp) (120)

2θp2 = π + 2θp1 (121)

θp2 minus θp1 =π

2(122)

所以

Imaxmin

=Ix + Iy

2plusmnradic

(Ix minus Iy

2)2 + I 2xy (123)

Iuv =Ix minus Iy

2

minusIxyradic(IxminusIy2 )2 + I 2xy

+ Ixy

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

(124)

= 0

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 42 54

課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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慣慣慣性性性矩矩矩之之之莫莫莫爾爾爾圓圓圓Mohrrsquos Circle for Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 44 54

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慣性矩之莫爾圓

由於Iu =

Ix + Iy2

+Ix minus Iy

2cos2θ minus Ixy sin2θ (125)

Iv =Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (126)

Iuv =Ix minus Iy

2sin2θ + Ixycos2θ (127)

將式(125)與式(127)各自分別平方並相加可得

(Iu minusIx + Iy

2)2 + I 2uv = (

Ix minus Iy2

)2 + I 2xy (128)

其中Ix Iy Ixy均為常數式(128)可進一步寫成

(Iu minus a)2 + I 2uv = R2 (129)

其中a =Ix+Iy2 R =

radic(IxminusIy2 )2 + I 2xycos2θp1 =

aRsin2θp1 =

minusIxyR 2θp1為OA轉向I軸方向即2θp1為

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 45 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 46 54

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解題技巧

首先確定轉動系統的Ix Iy Ixy接著算出莫爾圓之圓心座標a及半徑R並畫出莫爾圓再由莫爾圓上點出A(Ix Ixy )並由OA轉向I軸找出兩倍夾角2θp1及2θp2其中θp1及θp2為主軸(Principle Axis)與Ix軸的夾角由於主軸(Principle Axis)為極值之軸Imax = Iu及Imin = Iv決定θp1及θp2何者為Imax = Iu與Ix軸的夾角之方法乃比較Ix及Iy之大小Imax = Iu會靠近Ix及Iy之較大者也就是Imax = Iu與Ix及Iy之較大者的夾角比較小

a =Ix + Iy

2(130)

R =

radic(Ix minus Iy

2)2 + I 2xy (131)

2θp2 = π + 2θp1 θp2 minus θp1 =π

2(132)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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質質質量量量慣慣慣性性性矩矩矩Mass Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 48 54

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質量慣性矩

τ = Iα (133)

其中τ為旋轉扭矩I為質量慣性矩α為旋轉角速度

I =

ˆmr2dm (134)

=

ˆVr2ρdV (135)

= ρ

ˆVr2dV (136)

其中r為旋轉半徑或轉軸到旋轉體的距離

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 49 54

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 50 54

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平行軸定理

當通過物體質心(G)並在一轉軸上的質量慣性矩(IG )為已知時可透過座標軸轉換計算物體之質量慣性矩(I)

I =

ˆmr2dm (137)

=

ˆm

[(d + x prime)2 + y prime2]dm (138)

=

ˆm

(x prime2 + y prime2)dm + 2d

ˆmx primedm + d2

ˆmdm (139)

I = IG + md2 (140)

其中IG為在zrsquo軸上通過質心的質量慣性矩m為質量d為通過質心之軸zrsquo並平行原軸z之距離

acutem x primedm = x

acutem dm = 0 since x = 0

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 51 54

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (141)

k =

radicI

m(142)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (143)

= I1 + I2 + + In (144)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 52 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

兩圓盤複合體

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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結結結論論論

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 54 54

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結論

轉動慣量或慣性矩為物體對旋轉運動的慣性而所謂慣性乃物體持續維持不變的行為也就是定義慣性矩可表達物體對本身旋轉運動的維持或持續性好壞更簡單的說慣性矩越大之物體其維持旋轉運動趨勢的能力越大例如惰輪儲能裝置惰輪之慣性矩越大當其轉動以後其維持旋轉運動趨勢的能力越大

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 55 54

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  • 轉動慣量或慣性矩(Moment of Inertia)
  • 面積慣性矩(Moment of Inertia for Area)
    • 面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)
    • 面積慣性矩之迴轉半徑(Radius of Gyration for Area)
      • 複合面積之慣性矩(Moment of Inertia for Composite Area)
      • 面積之慣性積(Product of Inertia for Area)
        • 平行軸定理(Parallel-Axis Theorem)
          • 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)
            • 主軸慣性矩(Principal Moment of Inertia)
              • 慣性矩之莫爾圓(Mohrs Circle for Moment of Inertia)
              • 質量慣性矩(Mass Moment of Inertia)
                • 平行軸定理(Parallel-Axis Theorem)
Page 10: 轉動慣量 Moments of Inertia - cct.me.ntut.edu.twcct.me.ntut.edu.tw/ccteducation/chchting/aiahtm/statics/lecture... · 轉動慣量或慣性矩 I. 轉動慣量也稱慣性矩(Moment

極慣性矩(Polar Moment of Inertia)

極點

由左圖x軸上之面積慣性矩Ix及y軸上之面積慣性矩Iy如下

dIx = y2dA (12)

Ix =

ˆAy2dA (13)

dIy = x2dA (14)

Iy =

ˆAx2dA (15)

當微小面積dA對極點(O)或z軸旋轉時其面積極慣性矩JO如下

dJO = r2dA (16)

r2 = x2 + y2 (17)

JO =

ˆAr2dA =

ˆA(x2 + y2)dA = Ix + Iy (18)

其中r為微小面積dA到極點(O)的距離極點(O)簡單說乃z軸上的任意點由於x2 y2 r2及面

積(A)為正因此JO Ix Iy 均為正面積慣性矩的單位為m4mm4

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面面面積積積慣慣慣性性性矩矩矩之之之平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem for Area

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面積慣性矩之平行軸定理

極點

新極點

如左圖面積慣性矩之平行軸定理乃以面心(C)為新極點(O)設定新座標軸(xrsquoyrsquo)其中xrsquo軸平行於原座標x軸yrsquo軸平行於原座標y軸取微小面積dA其在新座標之點座標為(xrsquoyrsquo)則x軸上之面積慣性矩Ix及y軸上之面積慣性矩Iy可如下換算

Ix =

ˆA

(y prime + dy )2dA (19)

=

ˆAy prime2dA + 2dy

ˆAy primedA + d2

y

ˆAdA

Iy =

ˆA

(x prime + dx)2dA (20)

=

ˆAx prime2dA + 2dx

ˆAx primedA + d2

x

ˆAdA

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面積慣性矩之平行軸定理- Continued

由於面心在xrsquo軸上之等價力臂為y primeyrsquo軸上之等價力臂為x prime當以面心為轉軸點時即x prime = y prime = 0則x軸上之面積慣性矩Ix為ˆ

Ay prime2dA = macrIx prime (21)

ˆy primedA = y prime

ˆAdA = 0 (22)

d2y

ˆAdA = Ad2

y (23)

Ix =

ˆAy prime2dA + 2dy

ˆAy primedA + d2

y

ˆAdA (24)

= macrIx prime + Ad2y (25)

同理可得y軸上之面積慣性矩Iy

Iy =

ˆAx prime2dA + 2dx

ˆAx primedA + d2

x

ˆAdA (26)

= macrIy prime + Ad2x

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面積之極慣性矩

極點

新極點

由於通過面心之極慣性矩JC與極點(O)到面心(C)之距離d為

d2 = d2x + d2

y (27)

JC = macrIx prime + macrIy prime (28)

JO = Ix + Iy (29)

= macrIx prime + Ad2y + macrIy prime + Ad2

x (30)

= macrIx prime + macrIy prime + A(d2x + d2

y ) (31)

= JC + Ad2 (32)

也就是面積A之極慣性矩JO等於通過面積A之面心極慣性矩JC 與面積乘上極點(O)到面心(C)之距離d的平方和

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面面面積積積慣慣慣性性性矩矩矩之之之迴迴迴轉轉轉半半半徑徑徑Radius of Gyration for Area

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面積慣性矩之迴轉半徑

當x軸上之面積慣性矩Ix與y軸上之面積慣性矩Iy為已知時面積慣性矩之迴轉半徑可得知

dIx = y2dA (33)

Ix = kxA kx =

radicIxA

(34)

dIy = x2dA (35)

Iy = kyA ky =

radicIyA

(36)

dJO = r2dA (37)

JO = kOA kO =

radicJOA

(38)

其中(kx ky kO)分別為x軸y軸極點(O)之面積慣性矩迴轉半徑

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課課課堂堂堂練練練習習習

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解題技巧 I

極點

新極點

繪圖並標出面積(A)之面心座標位置(C)接著以面心座標位置(C)為原點畫出新座標軸(xrsquoyrsquo)其中xrsquo軸平行於原(舊)座標x軸yrsquo軸平行於原(舊)座標y軸並寫出新舊座標軸的平移量dx與dy其中dx為yrsquo軸與y軸的平移量dy為xrsquo軸與x軸的平移量取微小面積dA其在原(舊)座標之點座標為(xy)在新座標之點座標為(xrsquoyrsquo)帶入x軸上之面積慣性矩Ix及y軸上之面積慣性矩Iy還有xrsquo軸上之面積慣性矩 macrIx prime及y軸上之面積慣性矩 macrIy prime可如下換算

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解題技巧 II

Ix =

ˆAy2dA =

ˆAy prime2dA + 2dy

ˆAy primedA + d2

y

ˆAdA = macrIx prime + Ad2

y

macrIx prime =

ˆAy prime2dA (39)

Iy =

ˆAx2dA =

ˆAx prime2dA + 2dx

ˆAx primedA + d2

x

ˆAdA = macrIy prime + Ad2

x

macrIy prime =

ˆAx prime2dA (40)

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解題技巧 III

另外微小面積dA的面積慣性矩dIx dIy也可寫成

dIx = d macrIx prime + dAd2y (41)

Ix =

ˆdIx =

ˆd macrIx prime +

ˆdAd2

y (42)

dIy = d macrIy prime + dAd2x (43)

Iy =

ˆdIy =

ˆd macrIy prime +

ˆdAd2

x (44)

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

面心慣性矩 慣性矩 極慣性矩

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Example 10-1 解答

dA = bdy (45)

Ixb =

ˆAy2dA (46)

=

ˆ h

0y2bdy (47)

=1

3by3|h0 (48)

=1

3bh3 (49)

長方形之邊長b與h則面心慣性矩 macrIx prime macrIy prime與長方形之邊軸慣性矩Ix Iy為

macrIx prime =1

12bh3 (50)

macrIy prime =1

12hb3 (51)

Ix =1

3bh3 (52)

Iy =1

3hb3 (53)

JC = macrIx prime + macrIy prime (54)

JO = Ix + Iy (55)

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Example 10-2 解答

取微小面積dA=ydx之長方形其邊長b=dx與h=ydy = y2所以

y2 = 400x (56)

y =radic

400x (57)

d macrIx prime = d(1

12bh3) =

1

12y3dx (58)

dIx = d macrIx prime + dAd2y =

1

12y3dx + ydx

(y2

)2=

1

3y3dx (59)

Ix =

ˆ 100

0

1

3y3dx =

ˆ 100

0

1

3

(radic400x

)3dx (60)

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Example 10-3 解答

取微小面積dA=2ydx之長方形其邊長b=dx與h=2y所以

x2 + y2 = a2 (61)

y =radic

a2 minus x2 (62)

d macrIx prime = d(1

12bh3) =

1

12dx(2y)3 =

2

3y3dx (63)

macrIx prime =

ˆ a

minusa

2

3y3dx =

ˆ a

minusa

2

3

(radica2 minus x2

)3dx =

πa4

4(64)

圓之半徑a則面心慣性矩 macrIx prime macrIy prime為

macrIx prime =πa4

4(65)

macrIy prime =πa4

4(66)

JC = macrIx prime + macrIy prime =πa4

2(67)

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複複複合合合面面面積積積之之之慣慣慣性性性矩矩矩Moment of Inertia for Composite Area

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (68)

k =

radicI

m(69)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (70)

= I1 + I2 + + In (71)

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複合面積之慣性矩

複合面積之慣性矩(Ixc Iyc)為各局部面積之慣性矩(Ixi Iyi )合

Ixc =i=nsumi=1

Ixi (72)

= Ix1 + Ix2 + + Ixn (73)

Iyc =i=nsumi=1

Iyi (74)

= Iy1 + Iy2 + + Iyn (75)

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課課課堂堂堂練練練習習習

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

長方形與圓形複合

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

長方形複合

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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面面面積積積之之之慣慣慣性性性積積積Product of Inertia for Area

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面積之慣性積(Product of Inertia for Area)

為了下個章節求解慣性矩之極值(極大或極小值)方便數學運算在此先定義面積之慣性積(Ixy )為

dIxy = xydA (76)

Ixy =

ˆAxydA (77)

面積之慣性積的符號可為正負或零當慣性積之面積相對轉軸對稱時則面積之慣性積和為零

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 29 54

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 30 54

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平行軸定理

極點

新極點

如左圖面積慣性矩之平行軸定理乃以面心(C)為新極點(O)設定新座標軸(xrsquoyrsquo)其中xrsquo軸平行於原座標x軸yrsquo軸平行於原座標y軸取微小面積dA其在新座標之點座標為(xrsquoyrsquo)面積之慣性積(Ixy )為

dIxy = (x prime + dx)(y prime + dy )dA (78)

當以面心為轉軸點時即x prime = y prime = 0ˆAx primedA = x prime

ˆAdA = 0

ˆAy primedA = y prime

ˆAdA = 0 (79)

Ixy =

ˆA

(x prime + dx)(y prime + dy )dA (80)

=

ˆAx primey primedA + dx

ˆAy primedA + dy

ˆAx primedA + dxdy

ˆAdA (81)

= macrIx primey prime + Adxdy

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 31 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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傾傾傾斜斜斜軸軸軸面面面積積積之之之慣慣慣性性性矩矩矩

Moment of Inertia for Area about Inclined Axis

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 33 54

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傾斜軸面積之慣性矩

當轉動軸傾斜θ角時也就是座標軸旋轉θ角而Ix Iy Ixy為已知則新的座標軸(uv)可寫為

u = xcosθ + ysinθ (82)

v = ycosθ minus xsinθ (83)

由u軸上之面積慣性矩Iuv軸上之面積慣性矩Iv及慣性積Iuv得知

dIu = v2dA = (ycosθ minus xsinθ)2dA (84)

dIv = u2dA = (xcosθ + ysinθ)2dA (85)

dIuv = uvdA = (xcosθ + ysinθ)(ycosθ minus xsinθ)dA (86)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 34 54

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傾斜軸面積之慣性矩 I

由x軸上之面積慣性矩Ixy軸上之面積慣性矩Iy及慣性積Ixy得知

Ix =

ˆy2dA Iy =

ˆx2dA Ixy =

ˆxydA (87)

Iu =

ˆ(ycosθ minus xsinθ)2dA (88)

=

ˆ(y2cos2θ minus 2xycosθsinθ + x2sin2θ)dA (89)

= cos2θ

ˆy2dAminus 2cosθsinθ

ˆxydA + sin2θ

ˆx2dA (90)

= Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (91)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 35 54

傾斜軸面積之慣性矩 II

Iv =

ˆ(xcosθ + ysinθ)2dA (92)

=

ˆ(x2cos2θ + 2xycosθsinθ + y2sin2θ)dA (93)

= cos2θ

ˆx2dA + 2cosθsinθ

ˆxydA + sin2θ

ˆy2dA (94)

= Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (95)

Iuv =

ˆ(xcosθ + ysinθ)(ycosθ minus xsinθ)dA (96)

=

ˆ(xycos2θ + y2cosθsinθ minus x2cosθsinθ minus xysin2θ)dA (97)

= (cos2 minus sin2θ)

ˆxydA + cosθsinθ

ˆy2dAminus cosθsinθ

ˆx2dA

= Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 36 54

傾斜軸面積之慣性矩 III

再由

sin2θ + cos2θ = 1 (98)

sin(αplusmn β) = sinαcosβ plusmn cosαsinβ (99)

sin2θ = sin(θ + θ) = 2sinθcosθ (100)

cos(αplusmn β) = cosαcosβ ∓ sinαsinβ (101)

cos2θ = cos(θ + θ) = cos2θ minus sin2θ (102)

= 2cos2θ minus 1 = 1minus 2sin2θ (103)

所以

Iu = Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (104)

= Ix(cos2θ + 1

2) + Iy (

1minus cos2θ

2)minus Ixy sin2θ (105)

=Ix + Iy

2+

Ix minus Iy2

cos2θ minus Ixy sin2θ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 37 54

傾斜軸面積之慣性矩 IV

Iv = Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (106)

= Iy (cos2θ + 1

2) + Ix(

1minus cos2θ

2) + Ixy sin2θ (107)

=Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (108)

Iuv = Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ (109)

= Ixycos2θ +1

2Ixsin2θ minus 1

2Iy sin2θ (110)

=Ix minus Iy

2sin2θ + Ixycos2θ (111)

所以

JO = Iu + Iv = Ix + Iy (112)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 38 54

主主主軸軸軸慣慣慣性性性矩矩矩Principal Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 39 54

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主軸慣性矩 I

所謂主軸慣性矩(Principal Moment ofInertia)乃定義發生在慣性矩的極值(極大或極小值)上也就是慣性矩之一次微分式為零或斜率為零

dIu

dθ=

d(Ix+Iy

2+

IxminusIy2

cos2θ minus Ixy sin2θ)

dθ(113)

= minus2Ix minus Iy

2sin2θ minus 2Ixy cos2θ (114)

= 0 (115)

其中dsin2θ = cos2θ dcos2θ = minussin2θ當慣性矩之一次微分式為零時傾斜軸與主軸之夾角θ = θp

minus2Ix minus Iy

2sin2θp minus 2Ixycos2θp = 0 (116)

tan2θ =sin2θ

cos2θ=

minusIxy(Ix minus Iy )2

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 40 54

主軸慣性矩 II

cos2θp =IxminusIy2radic

(IxminusIy2 )2 + I 2xy

sin2θp =minusIxyradic

(IxminusIy2 )2 + I 2xy

(117)

(Iu)max =Ix + Iy

2+

Ix minus Iy2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

minus IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2+

radic(Ix minus Iy

2)2 + I 2xy (118)

(Iv )min =Ix + Iy

2minus Ix minus Iy

2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

+ IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2minusradic

(Ix minus Iy

2)2 + I 2xy (119)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 41 54

主軸慣性矩 III

由tan2θp = tan(π + 2θp)可得到θp1與θp2二個解即2θp2 = π + 2θp1也就是θp2 minus θp1 = π

2

tan2θp = tan(π + 2θp) (120)

2θp2 = π + 2θp1 (121)

θp2 minus θp1 =π

2(122)

所以

Imaxmin

=Ix + Iy

2plusmnradic

(Ix minus Iy

2)2 + I 2xy (123)

Iuv =Ix minus Iy

2

minusIxyradic(IxminusIy2 )2 + I 2xy

+ Ixy

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

(124)

= 0

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 42 54

課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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慣慣慣性性性矩矩矩之之之莫莫莫爾爾爾圓圓圓Mohrrsquos Circle for Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 44 54

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慣性矩之莫爾圓

由於Iu =

Ix + Iy2

+Ix minus Iy

2cos2θ minus Ixy sin2θ (125)

Iv =Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (126)

Iuv =Ix minus Iy

2sin2θ + Ixycos2θ (127)

將式(125)與式(127)各自分別平方並相加可得

(Iu minusIx + Iy

2)2 + I 2uv = (

Ix minus Iy2

)2 + I 2xy (128)

其中Ix Iy Ixy均為常數式(128)可進一步寫成

(Iu minus a)2 + I 2uv = R2 (129)

其中a =Ix+Iy2 R =

radic(IxminusIy2 )2 + I 2xycos2θp1 =

aRsin2θp1 =

minusIxyR 2θp1為OA轉向I軸方向即2θp1為

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 45 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 46 54

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解題技巧

首先確定轉動系統的Ix Iy Ixy接著算出莫爾圓之圓心座標a及半徑R並畫出莫爾圓再由莫爾圓上點出A(Ix Ixy )並由OA轉向I軸找出兩倍夾角2θp1及2θp2其中θp1及θp2為主軸(Principle Axis)與Ix軸的夾角由於主軸(Principle Axis)為極值之軸Imax = Iu及Imin = Iv決定θp1及θp2何者為Imax = Iu與Ix軸的夾角之方法乃比較Ix及Iy之大小Imax = Iu會靠近Ix及Iy之較大者也就是Imax = Iu與Ix及Iy之較大者的夾角比較小

a =Ix + Iy

2(130)

R =

radic(Ix minus Iy

2)2 + I 2xy (131)

2θp2 = π + 2θp1 θp2 minus θp1 =π

2(132)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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質質質量量量慣慣慣性性性矩矩矩Mass Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 48 54

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質量慣性矩

τ = Iα (133)

其中τ為旋轉扭矩I為質量慣性矩α為旋轉角速度

I =

ˆmr2dm (134)

=

ˆVr2ρdV (135)

= ρ

ˆVr2dV (136)

其中r為旋轉半徑或轉軸到旋轉體的距離

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 49 54

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 50 54

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平行軸定理

當通過物體質心(G)並在一轉軸上的質量慣性矩(IG )為已知時可透過座標軸轉換計算物體之質量慣性矩(I)

I =

ˆmr2dm (137)

=

ˆm

[(d + x prime)2 + y prime2]dm (138)

=

ˆm

(x prime2 + y prime2)dm + 2d

ˆmx primedm + d2

ˆmdm (139)

I = IG + md2 (140)

其中IG為在zrsquo軸上通過質心的質量慣性矩m為質量d為通過質心之軸zrsquo並平行原軸z之距離

acutem x primedm = x

acutem dm = 0 since x = 0

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 51 54

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (141)

k =

radicI

m(142)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (143)

= I1 + I2 + + In (144)

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課課課堂堂堂練練練習習習

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

兩圓盤複合體

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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結結結論論論

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結論

轉動慣量或慣性矩為物體對旋轉運動的慣性而所謂慣性乃物體持續維持不變的行為也就是定義慣性矩可表達物體對本身旋轉運動的維持或持續性好壞更簡單的說慣性矩越大之物體其維持旋轉運動趨勢的能力越大例如惰輪儲能裝置惰輪之慣性矩越大當其轉動以後其維持旋轉運動趨勢的能力越大

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 55 54

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  • 轉動慣量或慣性矩(Moment of Inertia)
  • 面積慣性矩(Moment of Inertia for Area)
    • 面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)
    • 面積慣性矩之迴轉半徑(Radius of Gyration for Area)
      • 複合面積之慣性矩(Moment of Inertia for Composite Area)
      • 面積之慣性積(Product of Inertia for Area)
        • 平行軸定理(Parallel-Axis Theorem)
          • 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)
            • 主軸慣性矩(Principal Moment of Inertia)
              • 慣性矩之莫爾圓(Mohrs Circle for Moment of Inertia)
              • 質量慣性矩(Mass Moment of Inertia)
                • 平行軸定理(Parallel-Axis Theorem)
Page 11: 轉動慣量 Moments of Inertia - cct.me.ntut.edu.twcct.me.ntut.edu.tw/ccteducation/chchting/aiahtm/statics/lecture... · 轉動慣量或慣性矩 I. 轉動慣量也稱慣性矩(Moment

面面面積積積慣慣慣性性性矩矩矩之之之平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem for Area

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面積慣性矩之平行軸定理

極點

新極點

如左圖面積慣性矩之平行軸定理乃以面心(C)為新極點(O)設定新座標軸(xrsquoyrsquo)其中xrsquo軸平行於原座標x軸yrsquo軸平行於原座標y軸取微小面積dA其在新座標之點座標為(xrsquoyrsquo)則x軸上之面積慣性矩Ix及y軸上之面積慣性矩Iy可如下換算

Ix =

ˆA

(y prime + dy )2dA (19)

=

ˆAy prime2dA + 2dy

ˆAy primedA + d2

y

ˆAdA

Iy =

ˆA

(x prime + dx)2dA (20)

=

ˆAx prime2dA + 2dx

ˆAx primedA + d2

x

ˆAdA

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面積慣性矩之平行軸定理- Continued

由於面心在xrsquo軸上之等價力臂為y primeyrsquo軸上之等價力臂為x prime當以面心為轉軸點時即x prime = y prime = 0則x軸上之面積慣性矩Ix為ˆ

Ay prime2dA = macrIx prime (21)

ˆy primedA = y prime

ˆAdA = 0 (22)

d2y

ˆAdA = Ad2

y (23)

Ix =

ˆAy prime2dA + 2dy

ˆAy primedA + d2

y

ˆAdA (24)

= macrIx prime + Ad2y (25)

同理可得y軸上之面積慣性矩Iy

Iy =

ˆAx prime2dA + 2dx

ˆAx primedA + d2

x

ˆAdA (26)

= macrIy prime + Ad2x

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面積之極慣性矩

極點

新極點

由於通過面心之極慣性矩JC與極點(O)到面心(C)之距離d為

d2 = d2x + d2

y (27)

JC = macrIx prime + macrIy prime (28)

JO = Ix + Iy (29)

= macrIx prime + Ad2y + macrIy prime + Ad2

x (30)

= macrIx prime + macrIy prime + A(d2x + d2

y ) (31)

= JC + Ad2 (32)

也就是面積A之極慣性矩JO等於通過面積A之面心極慣性矩JC 與面積乘上極點(O)到面心(C)之距離d的平方和

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面面面積積積慣慣慣性性性矩矩矩之之之迴迴迴轉轉轉半半半徑徑徑Radius of Gyration for Area

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面積慣性矩之迴轉半徑

當x軸上之面積慣性矩Ix與y軸上之面積慣性矩Iy為已知時面積慣性矩之迴轉半徑可得知

dIx = y2dA (33)

Ix = kxA kx =

radicIxA

(34)

dIy = x2dA (35)

Iy = kyA ky =

radicIyA

(36)

dJO = r2dA (37)

JO = kOA kO =

radicJOA

(38)

其中(kx ky kO)分別為x軸y軸極點(O)之面積慣性矩迴轉半徑

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課課課堂堂堂練練練習習習

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解題技巧 I

極點

新極點

繪圖並標出面積(A)之面心座標位置(C)接著以面心座標位置(C)為原點畫出新座標軸(xrsquoyrsquo)其中xrsquo軸平行於原(舊)座標x軸yrsquo軸平行於原(舊)座標y軸並寫出新舊座標軸的平移量dx與dy其中dx為yrsquo軸與y軸的平移量dy為xrsquo軸與x軸的平移量取微小面積dA其在原(舊)座標之點座標為(xy)在新座標之點座標為(xrsquoyrsquo)帶入x軸上之面積慣性矩Ix及y軸上之面積慣性矩Iy還有xrsquo軸上之面積慣性矩 macrIx prime及y軸上之面積慣性矩 macrIy prime可如下換算

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解題技巧 II

Ix =

ˆAy2dA =

ˆAy prime2dA + 2dy

ˆAy primedA + d2

y

ˆAdA = macrIx prime + Ad2

y

macrIx prime =

ˆAy prime2dA (39)

Iy =

ˆAx2dA =

ˆAx prime2dA + 2dx

ˆAx primedA + d2

x

ˆAdA = macrIy prime + Ad2

x

macrIy prime =

ˆAx prime2dA (40)

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解題技巧 III

另外微小面積dA的面積慣性矩dIx dIy也可寫成

dIx = d macrIx prime + dAd2y (41)

Ix =

ˆdIx =

ˆd macrIx prime +

ˆdAd2

y (42)

dIy = d macrIy prime + dAd2x (43)

Iy =

ˆdIy =

ˆd macrIy prime +

ˆdAd2

x (44)

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

面心慣性矩 慣性矩 極慣性矩

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Example 10-1 解答

dA = bdy (45)

Ixb =

ˆAy2dA (46)

=

ˆ h

0y2bdy (47)

=1

3by3|h0 (48)

=1

3bh3 (49)

長方形之邊長b與h則面心慣性矩 macrIx prime macrIy prime與長方形之邊軸慣性矩Ix Iy為

macrIx prime =1

12bh3 (50)

macrIy prime =1

12hb3 (51)

Ix =1

3bh3 (52)

Iy =1

3hb3 (53)

JC = macrIx prime + macrIy prime (54)

JO = Ix + Iy (55)

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Example 10-2 解答

取微小面積dA=ydx之長方形其邊長b=dx與h=ydy = y2所以

y2 = 400x (56)

y =radic

400x (57)

d macrIx prime = d(1

12bh3) =

1

12y3dx (58)

dIx = d macrIx prime + dAd2y =

1

12y3dx + ydx

(y2

)2=

1

3y3dx (59)

Ix =

ˆ 100

0

1

3y3dx =

ˆ 100

0

1

3

(radic400x

)3dx (60)

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Example 10-3 解答

取微小面積dA=2ydx之長方形其邊長b=dx與h=2y所以

x2 + y2 = a2 (61)

y =radic

a2 minus x2 (62)

d macrIx prime = d(1

12bh3) =

1

12dx(2y)3 =

2

3y3dx (63)

macrIx prime =

ˆ a

minusa

2

3y3dx =

ˆ a

minusa

2

3

(radica2 minus x2

)3dx =

πa4

4(64)

圓之半徑a則面心慣性矩 macrIx prime macrIy prime為

macrIx prime =πa4

4(65)

macrIy prime =πa4

4(66)

JC = macrIx prime + macrIy prime =πa4

2(67)

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複複複合合合面面面積積積之之之慣慣慣性性性矩矩矩Moment of Inertia for Composite Area

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (68)

k =

radicI

m(69)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (70)

= I1 + I2 + + In (71)

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複合面積之慣性矩

複合面積之慣性矩(Ixc Iyc)為各局部面積之慣性矩(Ixi Iyi )合

Ixc =i=nsumi=1

Ixi (72)

= Ix1 + Ix2 + + Ixn (73)

Iyc =i=nsumi=1

Iyi (74)

= Iy1 + Iy2 + + Iyn (75)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 26 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

長方形與圓形複合

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

長方形複合

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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面面面積積積之之之慣慣慣性性性積積積Product of Inertia for Area

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 28 54

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面積之慣性積(Product of Inertia for Area)

為了下個章節求解慣性矩之極值(極大或極小值)方便數學運算在此先定義面積之慣性積(Ixy )為

dIxy = xydA (76)

Ixy =

ˆAxydA (77)

面積之慣性積的符號可為正負或零當慣性積之面積相對轉軸對稱時則面積之慣性積和為零

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 29 54

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 30 54

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平行軸定理

極點

新極點

如左圖面積慣性矩之平行軸定理乃以面心(C)為新極點(O)設定新座標軸(xrsquoyrsquo)其中xrsquo軸平行於原座標x軸yrsquo軸平行於原座標y軸取微小面積dA其在新座標之點座標為(xrsquoyrsquo)面積之慣性積(Ixy )為

dIxy = (x prime + dx)(y prime + dy )dA (78)

當以面心為轉軸點時即x prime = y prime = 0ˆAx primedA = x prime

ˆAdA = 0

ˆAy primedA = y prime

ˆAdA = 0 (79)

Ixy =

ˆA

(x prime + dx)(y prime + dy )dA (80)

=

ˆAx primey primedA + dx

ˆAy primedA + dy

ˆAx primedA + dxdy

ˆAdA (81)

= macrIx primey prime + Adxdy

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 31 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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傾傾傾斜斜斜軸軸軸面面面積積積之之之慣慣慣性性性矩矩矩

Moment of Inertia for Area about Inclined Axis

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 33 54

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傾斜軸面積之慣性矩

當轉動軸傾斜θ角時也就是座標軸旋轉θ角而Ix Iy Ixy為已知則新的座標軸(uv)可寫為

u = xcosθ + ysinθ (82)

v = ycosθ minus xsinθ (83)

由u軸上之面積慣性矩Iuv軸上之面積慣性矩Iv及慣性積Iuv得知

dIu = v2dA = (ycosθ minus xsinθ)2dA (84)

dIv = u2dA = (xcosθ + ysinθ)2dA (85)

dIuv = uvdA = (xcosθ + ysinθ)(ycosθ minus xsinθ)dA (86)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 34 54

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傾斜軸面積之慣性矩 I

由x軸上之面積慣性矩Ixy軸上之面積慣性矩Iy及慣性積Ixy得知

Ix =

ˆy2dA Iy =

ˆx2dA Ixy =

ˆxydA (87)

Iu =

ˆ(ycosθ minus xsinθ)2dA (88)

=

ˆ(y2cos2θ minus 2xycosθsinθ + x2sin2θ)dA (89)

= cos2θ

ˆy2dAminus 2cosθsinθ

ˆxydA + sin2θ

ˆx2dA (90)

= Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (91)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 35 54

傾斜軸面積之慣性矩 II

Iv =

ˆ(xcosθ + ysinθ)2dA (92)

=

ˆ(x2cos2θ + 2xycosθsinθ + y2sin2θ)dA (93)

= cos2θ

ˆx2dA + 2cosθsinθ

ˆxydA + sin2θ

ˆy2dA (94)

= Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (95)

Iuv =

ˆ(xcosθ + ysinθ)(ycosθ minus xsinθ)dA (96)

=

ˆ(xycos2θ + y2cosθsinθ minus x2cosθsinθ minus xysin2θ)dA (97)

= (cos2 minus sin2θ)

ˆxydA + cosθsinθ

ˆy2dAminus cosθsinθ

ˆx2dA

= Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 36 54

傾斜軸面積之慣性矩 III

再由

sin2θ + cos2θ = 1 (98)

sin(αplusmn β) = sinαcosβ plusmn cosαsinβ (99)

sin2θ = sin(θ + θ) = 2sinθcosθ (100)

cos(αplusmn β) = cosαcosβ ∓ sinαsinβ (101)

cos2θ = cos(θ + θ) = cos2θ minus sin2θ (102)

= 2cos2θ minus 1 = 1minus 2sin2θ (103)

所以

Iu = Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (104)

= Ix(cos2θ + 1

2) + Iy (

1minus cos2θ

2)minus Ixy sin2θ (105)

=Ix + Iy

2+

Ix minus Iy2

cos2θ minus Ixy sin2θ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 37 54

傾斜軸面積之慣性矩 IV

Iv = Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (106)

= Iy (cos2θ + 1

2) + Ix(

1minus cos2θ

2) + Ixy sin2θ (107)

=Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (108)

Iuv = Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ (109)

= Ixycos2θ +1

2Ixsin2θ minus 1

2Iy sin2θ (110)

=Ix minus Iy

2sin2θ + Ixycos2θ (111)

所以

JO = Iu + Iv = Ix + Iy (112)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 38 54

主主主軸軸軸慣慣慣性性性矩矩矩Principal Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 39 54

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主軸慣性矩 I

所謂主軸慣性矩(Principal Moment ofInertia)乃定義發生在慣性矩的極值(極大或極小值)上也就是慣性矩之一次微分式為零或斜率為零

dIu

dθ=

d(Ix+Iy

2+

IxminusIy2

cos2θ minus Ixy sin2θ)

dθ(113)

= minus2Ix minus Iy

2sin2θ minus 2Ixy cos2θ (114)

= 0 (115)

其中dsin2θ = cos2θ dcos2θ = minussin2θ當慣性矩之一次微分式為零時傾斜軸與主軸之夾角θ = θp

minus2Ix minus Iy

2sin2θp minus 2Ixycos2θp = 0 (116)

tan2θ =sin2θ

cos2θ=

minusIxy(Ix minus Iy )2

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 40 54

主軸慣性矩 II

cos2θp =IxminusIy2radic

(IxminusIy2 )2 + I 2xy

sin2θp =minusIxyradic

(IxminusIy2 )2 + I 2xy

(117)

(Iu)max =Ix + Iy

2+

Ix minus Iy2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

minus IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2+

radic(Ix minus Iy

2)2 + I 2xy (118)

(Iv )min =Ix + Iy

2minus Ix minus Iy

2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

+ IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2minusradic

(Ix minus Iy

2)2 + I 2xy (119)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 41 54

主軸慣性矩 III

由tan2θp = tan(π + 2θp)可得到θp1與θp2二個解即2θp2 = π + 2θp1也就是θp2 minus θp1 = π

2

tan2θp = tan(π + 2θp) (120)

2θp2 = π + 2θp1 (121)

θp2 minus θp1 =π

2(122)

所以

Imaxmin

=Ix + Iy

2plusmnradic

(Ix minus Iy

2)2 + I 2xy (123)

Iuv =Ix minus Iy

2

minusIxyradic(IxminusIy2 )2 + I 2xy

+ Ixy

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

(124)

= 0

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 42 54

課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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慣慣慣性性性矩矩矩之之之莫莫莫爾爾爾圓圓圓Mohrrsquos Circle for Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 44 54

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慣性矩之莫爾圓

由於Iu =

Ix + Iy2

+Ix minus Iy

2cos2θ minus Ixy sin2θ (125)

Iv =Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (126)

Iuv =Ix minus Iy

2sin2θ + Ixycos2θ (127)

將式(125)與式(127)各自分別平方並相加可得

(Iu minusIx + Iy

2)2 + I 2uv = (

Ix minus Iy2

)2 + I 2xy (128)

其中Ix Iy Ixy均為常數式(128)可進一步寫成

(Iu minus a)2 + I 2uv = R2 (129)

其中a =Ix+Iy2 R =

radic(IxminusIy2 )2 + I 2xycos2θp1 =

aRsin2θp1 =

minusIxyR 2θp1為OA轉向I軸方向即2θp1為

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 45 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 46 54

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解題技巧

首先確定轉動系統的Ix Iy Ixy接著算出莫爾圓之圓心座標a及半徑R並畫出莫爾圓再由莫爾圓上點出A(Ix Ixy )並由OA轉向I軸找出兩倍夾角2θp1及2θp2其中θp1及θp2為主軸(Principle Axis)與Ix軸的夾角由於主軸(Principle Axis)為極值之軸Imax = Iu及Imin = Iv決定θp1及θp2何者為Imax = Iu與Ix軸的夾角之方法乃比較Ix及Iy之大小Imax = Iu會靠近Ix及Iy之較大者也就是Imax = Iu與Ix及Iy之較大者的夾角比較小

a =Ix + Iy

2(130)

R =

radic(Ix minus Iy

2)2 + I 2xy (131)

2θp2 = π + 2θp1 θp2 minus θp1 =π

2(132)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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質質質量量量慣慣慣性性性矩矩矩Mass Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 48 54

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質量慣性矩

τ = Iα (133)

其中τ為旋轉扭矩I為質量慣性矩α為旋轉角速度

I =

ˆmr2dm (134)

=

ˆVr2ρdV (135)

= ρ

ˆVr2dV (136)

其中r為旋轉半徑或轉軸到旋轉體的距離

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

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平行軸定理

當通過物體質心(G)並在一轉軸上的質量慣性矩(IG )為已知時可透過座標軸轉換計算物體之質量慣性矩(I)

I =

ˆmr2dm (137)

=

ˆm

[(d + x prime)2 + y prime2]dm (138)

=

ˆm

(x prime2 + y prime2)dm + 2d

ˆmx primedm + d2

ˆmdm (139)

I = IG + md2 (140)

其中IG為在zrsquo軸上通過質心的質量慣性矩m為質量d為通過質心之軸zrsquo並平行原軸z之距離

acutem x primedm = x

acutem dm = 0 since x = 0

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (141)

k =

radicI

m(142)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (143)

= I1 + I2 + + In (144)

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課課課堂堂堂練練練習習習

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

兩圓盤複合體

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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結結結論論論

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結論

轉動慣量或慣性矩為物體對旋轉運動的慣性而所謂慣性乃物體持續維持不變的行為也就是定義慣性矩可表達物體對本身旋轉運動的維持或持續性好壞更簡單的說慣性矩越大之物體其維持旋轉運動趨勢的能力越大例如惰輪儲能裝置惰輪之慣性矩越大當其轉動以後其維持旋轉運動趨勢的能力越大

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  • 轉動慣量或慣性矩(Moment of Inertia)
  • 面積慣性矩(Moment of Inertia for Area)
    • 面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)
    • 面積慣性矩之迴轉半徑(Radius of Gyration for Area)
      • 複合面積之慣性矩(Moment of Inertia for Composite Area)
      • 面積之慣性積(Product of Inertia for Area)
        • 平行軸定理(Parallel-Axis Theorem)
          • 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)
            • 主軸慣性矩(Principal Moment of Inertia)
              • 慣性矩之莫爾圓(Mohrs Circle for Moment of Inertia)
              • 質量慣性矩(Mass Moment of Inertia)
                • 平行軸定理(Parallel-Axis Theorem)
Page 12: 轉動慣量 Moments of Inertia - cct.me.ntut.edu.twcct.me.ntut.edu.tw/ccteducation/chchting/aiahtm/statics/lecture... · 轉動慣量或慣性矩 I. 轉動慣量也稱慣性矩(Moment

面積慣性矩之平行軸定理

極點

新極點

如左圖面積慣性矩之平行軸定理乃以面心(C)為新極點(O)設定新座標軸(xrsquoyrsquo)其中xrsquo軸平行於原座標x軸yrsquo軸平行於原座標y軸取微小面積dA其在新座標之點座標為(xrsquoyrsquo)則x軸上之面積慣性矩Ix及y軸上之面積慣性矩Iy可如下換算

Ix =

ˆA

(y prime + dy )2dA (19)

=

ˆAy prime2dA + 2dy

ˆAy primedA + d2

y

ˆAdA

Iy =

ˆA

(x prime + dx)2dA (20)

=

ˆAx prime2dA + 2dx

ˆAx primedA + d2

x

ˆAdA

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面積慣性矩之平行軸定理- Continued

由於面心在xrsquo軸上之等價力臂為y primeyrsquo軸上之等價力臂為x prime當以面心為轉軸點時即x prime = y prime = 0則x軸上之面積慣性矩Ix為ˆ

Ay prime2dA = macrIx prime (21)

ˆy primedA = y prime

ˆAdA = 0 (22)

d2y

ˆAdA = Ad2

y (23)

Ix =

ˆAy prime2dA + 2dy

ˆAy primedA + d2

y

ˆAdA (24)

= macrIx prime + Ad2y (25)

同理可得y軸上之面積慣性矩Iy

Iy =

ˆAx prime2dA + 2dx

ˆAx primedA + d2

x

ˆAdA (26)

= macrIy prime + Ad2x

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面積之極慣性矩

極點

新極點

由於通過面心之極慣性矩JC與極點(O)到面心(C)之距離d為

d2 = d2x + d2

y (27)

JC = macrIx prime + macrIy prime (28)

JO = Ix + Iy (29)

= macrIx prime + Ad2y + macrIy prime + Ad2

x (30)

= macrIx prime + macrIy prime + A(d2x + d2

y ) (31)

= JC + Ad2 (32)

也就是面積A之極慣性矩JO等於通過面積A之面心極慣性矩JC 與面積乘上極點(O)到面心(C)之距離d的平方和

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面面面積積積慣慣慣性性性矩矩矩之之之迴迴迴轉轉轉半半半徑徑徑Radius of Gyration for Area

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面積慣性矩之迴轉半徑

當x軸上之面積慣性矩Ix與y軸上之面積慣性矩Iy為已知時面積慣性矩之迴轉半徑可得知

dIx = y2dA (33)

Ix = kxA kx =

radicIxA

(34)

dIy = x2dA (35)

Iy = kyA ky =

radicIyA

(36)

dJO = r2dA (37)

JO = kOA kO =

radicJOA

(38)

其中(kx ky kO)分別為x軸y軸極點(O)之面積慣性矩迴轉半徑

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課課課堂堂堂練練練習習習

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解題技巧 I

極點

新極點

繪圖並標出面積(A)之面心座標位置(C)接著以面心座標位置(C)為原點畫出新座標軸(xrsquoyrsquo)其中xrsquo軸平行於原(舊)座標x軸yrsquo軸平行於原(舊)座標y軸並寫出新舊座標軸的平移量dx與dy其中dx為yrsquo軸與y軸的平移量dy為xrsquo軸與x軸的平移量取微小面積dA其在原(舊)座標之點座標為(xy)在新座標之點座標為(xrsquoyrsquo)帶入x軸上之面積慣性矩Ix及y軸上之面積慣性矩Iy還有xrsquo軸上之面積慣性矩 macrIx prime及y軸上之面積慣性矩 macrIy prime可如下換算

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解題技巧 II

Ix =

ˆAy2dA =

ˆAy prime2dA + 2dy

ˆAy primedA + d2

y

ˆAdA = macrIx prime + Ad2

y

macrIx prime =

ˆAy prime2dA (39)

Iy =

ˆAx2dA =

ˆAx prime2dA + 2dx

ˆAx primedA + d2

x

ˆAdA = macrIy prime + Ad2

x

macrIy prime =

ˆAx prime2dA (40)

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解題技巧 III

另外微小面積dA的面積慣性矩dIx dIy也可寫成

dIx = d macrIx prime + dAd2y (41)

Ix =

ˆdIx =

ˆd macrIx prime +

ˆdAd2

y (42)

dIy = d macrIy prime + dAd2x (43)

Iy =

ˆdIy =

ˆd macrIy prime +

ˆdAd2

x (44)

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

面心慣性矩 慣性矩 極慣性矩

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Example 10-1 解答

dA = bdy (45)

Ixb =

ˆAy2dA (46)

=

ˆ h

0y2bdy (47)

=1

3by3|h0 (48)

=1

3bh3 (49)

長方形之邊長b與h則面心慣性矩 macrIx prime macrIy prime與長方形之邊軸慣性矩Ix Iy為

macrIx prime =1

12bh3 (50)

macrIy prime =1

12hb3 (51)

Ix =1

3bh3 (52)

Iy =1

3hb3 (53)

JC = macrIx prime + macrIy prime (54)

JO = Ix + Iy (55)

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Example 10-2 解答

取微小面積dA=ydx之長方形其邊長b=dx與h=ydy = y2所以

y2 = 400x (56)

y =radic

400x (57)

d macrIx prime = d(1

12bh3) =

1

12y3dx (58)

dIx = d macrIx prime + dAd2y =

1

12y3dx + ydx

(y2

)2=

1

3y3dx (59)

Ix =

ˆ 100

0

1

3y3dx =

ˆ 100

0

1

3

(radic400x

)3dx (60)

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Example 10-3 解答

取微小面積dA=2ydx之長方形其邊長b=dx與h=2y所以

x2 + y2 = a2 (61)

y =radic

a2 minus x2 (62)

d macrIx prime = d(1

12bh3) =

1

12dx(2y)3 =

2

3y3dx (63)

macrIx prime =

ˆ a

minusa

2

3y3dx =

ˆ a

minusa

2

3

(radica2 minus x2

)3dx =

πa4

4(64)

圓之半徑a則面心慣性矩 macrIx prime macrIy prime為

macrIx prime =πa4

4(65)

macrIy prime =πa4

4(66)

JC = macrIx prime + macrIy prime =πa4

2(67)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 23 54

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複複複合合合面面面積積積之之之慣慣慣性性性矩矩矩Moment of Inertia for Composite Area

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 24 54

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (68)

k =

radicI

m(69)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (70)

= I1 + I2 + + In (71)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 25 54

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複合面積之慣性矩

複合面積之慣性矩(Ixc Iyc)為各局部面積之慣性矩(Ixi Iyi )合

Ixc =i=nsumi=1

Ixi (72)

= Ix1 + Ix2 + + Ixn (73)

Iyc =i=nsumi=1

Iyi (74)

= Iy1 + Iy2 + + Iyn (75)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 26 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

長方形與圓形複合

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

長方形複合

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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面面面積積積之之之慣慣慣性性性積積積Product of Inertia for Area

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 28 54

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面積之慣性積(Product of Inertia for Area)

為了下個章節求解慣性矩之極值(極大或極小值)方便數學運算在此先定義面積之慣性積(Ixy )為

dIxy = xydA (76)

Ixy =

ˆAxydA (77)

面積之慣性積的符號可為正負或零當慣性積之面積相對轉軸對稱時則面積之慣性積和為零

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 29 54

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 30 54

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平行軸定理

極點

新極點

如左圖面積慣性矩之平行軸定理乃以面心(C)為新極點(O)設定新座標軸(xrsquoyrsquo)其中xrsquo軸平行於原座標x軸yrsquo軸平行於原座標y軸取微小面積dA其在新座標之點座標為(xrsquoyrsquo)面積之慣性積(Ixy )為

dIxy = (x prime + dx)(y prime + dy )dA (78)

當以面心為轉軸點時即x prime = y prime = 0ˆAx primedA = x prime

ˆAdA = 0

ˆAy primedA = y prime

ˆAdA = 0 (79)

Ixy =

ˆA

(x prime + dx)(y prime + dy )dA (80)

=

ˆAx primey primedA + dx

ˆAy primedA + dy

ˆAx primedA + dxdy

ˆAdA (81)

= macrIx primey prime + Adxdy

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 31 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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傾傾傾斜斜斜軸軸軸面面面積積積之之之慣慣慣性性性矩矩矩

Moment of Inertia for Area about Inclined Axis

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 33 54

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傾斜軸面積之慣性矩

當轉動軸傾斜θ角時也就是座標軸旋轉θ角而Ix Iy Ixy為已知則新的座標軸(uv)可寫為

u = xcosθ + ysinθ (82)

v = ycosθ minus xsinθ (83)

由u軸上之面積慣性矩Iuv軸上之面積慣性矩Iv及慣性積Iuv得知

dIu = v2dA = (ycosθ minus xsinθ)2dA (84)

dIv = u2dA = (xcosθ + ysinθ)2dA (85)

dIuv = uvdA = (xcosθ + ysinθ)(ycosθ minus xsinθ)dA (86)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 34 54

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傾斜軸面積之慣性矩 I

由x軸上之面積慣性矩Ixy軸上之面積慣性矩Iy及慣性積Ixy得知

Ix =

ˆy2dA Iy =

ˆx2dA Ixy =

ˆxydA (87)

Iu =

ˆ(ycosθ minus xsinθ)2dA (88)

=

ˆ(y2cos2θ minus 2xycosθsinθ + x2sin2θ)dA (89)

= cos2θ

ˆy2dAminus 2cosθsinθ

ˆxydA + sin2θ

ˆx2dA (90)

= Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (91)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 35 54

傾斜軸面積之慣性矩 II

Iv =

ˆ(xcosθ + ysinθ)2dA (92)

=

ˆ(x2cos2θ + 2xycosθsinθ + y2sin2θ)dA (93)

= cos2θ

ˆx2dA + 2cosθsinθ

ˆxydA + sin2θ

ˆy2dA (94)

= Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (95)

Iuv =

ˆ(xcosθ + ysinθ)(ycosθ minus xsinθ)dA (96)

=

ˆ(xycos2θ + y2cosθsinθ minus x2cosθsinθ minus xysin2θ)dA (97)

= (cos2 minus sin2θ)

ˆxydA + cosθsinθ

ˆy2dAminus cosθsinθ

ˆx2dA

= Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 36 54

傾斜軸面積之慣性矩 III

再由

sin2θ + cos2θ = 1 (98)

sin(αplusmn β) = sinαcosβ plusmn cosαsinβ (99)

sin2θ = sin(θ + θ) = 2sinθcosθ (100)

cos(αplusmn β) = cosαcosβ ∓ sinαsinβ (101)

cos2θ = cos(θ + θ) = cos2θ minus sin2θ (102)

= 2cos2θ minus 1 = 1minus 2sin2θ (103)

所以

Iu = Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (104)

= Ix(cos2θ + 1

2) + Iy (

1minus cos2θ

2)minus Ixy sin2θ (105)

=Ix + Iy

2+

Ix minus Iy2

cos2θ minus Ixy sin2θ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 37 54

傾斜軸面積之慣性矩 IV

Iv = Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (106)

= Iy (cos2θ + 1

2) + Ix(

1minus cos2θ

2) + Ixy sin2θ (107)

=Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (108)

Iuv = Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ (109)

= Ixycos2θ +1

2Ixsin2θ minus 1

2Iy sin2θ (110)

=Ix minus Iy

2sin2θ + Ixycos2θ (111)

所以

JO = Iu + Iv = Ix + Iy (112)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 38 54

主主主軸軸軸慣慣慣性性性矩矩矩Principal Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 39 54

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主軸慣性矩 I

所謂主軸慣性矩(Principal Moment ofInertia)乃定義發生在慣性矩的極值(極大或極小值)上也就是慣性矩之一次微分式為零或斜率為零

dIu

dθ=

d(Ix+Iy

2+

IxminusIy2

cos2θ minus Ixy sin2θ)

dθ(113)

= minus2Ix minus Iy

2sin2θ minus 2Ixy cos2θ (114)

= 0 (115)

其中dsin2θ = cos2θ dcos2θ = minussin2θ當慣性矩之一次微分式為零時傾斜軸與主軸之夾角θ = θp

minus2Ix minus Iy

2sin2θp minus 2Ixycos2θp = 0 (116)

tan2θ =sin2θ

cos2θ=

minusIxy(Ix minus Iy )2

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 40 54

主軸慣性矩 II

cos2θp =IxminusIy2radic

(IxminusIy2 )2 + I 2xy

sin2θp =minusIxyradic

(IxminusIy2 )2 + I 2xy

(117)

(Iu)max =Ix + Iy

2+

Ix minus Iy2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

minus IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2+

radic(Ix minus Iy

2)2 + I 2xy (118)

(Iv )min =Ix + Iy

2minus Ix minus Iy

2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

+ IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2minusradic

(Ix minus Iy

2)2 + I 2xy (119)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 41 54

主軸慣性矩 III

由tan2θp = tan(π + 2θp)可得到θp1與θp2二個解即2θp2 = π + 2θp1也就是θp2 minus θp1 = π

2

tan2θp = tan(π + 2θp) (120)

2θp2 = π + 2θp1 (121)

θp2 minus θp1 =π

2(122)

所以

Imaxmin

=Ix + Iy

2plusmnradic

(Ix minus Iy

2)2 + I 2xy (123)

Iuv =Ix minus Iy

2

minusIxyradic(IxminusIy2 )2 + I 2xy

+ Ixy

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

(124)

= 0

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 42 54

課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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慣慣慣性性性矩矩矩之之之莫莫莫爾爾爾圓圓圓Mohrrsquos Circle for Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 44 54

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慣性矩之莫爾圓

由於Iu =

Ix + Iy2

+Ix minus Iy

2cos2θ minus Ixy sin2θ (125)

Iv =Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (126)

Iuv =Ix minus Iy

2sin2θ + Ixycos2θ (127)

將式(125)與式(127)各自分別平方並相加可得

(Iu minusIx + Iy

2)2 + I 2uv = (

Ix minus Iy2

)2 + I 2xy (128)

其中Ix Iy Ixy均為常數式(128)可進一步寫成

(Iu minus a)2 + I 2uv = R2 (129)

其中a =Ix+Iy2 R =

radic(IxminusIy2 )2 + I 2xycos2θp1 =

aRsin2θp1 =

minusIxyR 2θp1為OA轉向I軸方向即2θp1為

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 45 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 46 54

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解題技巧

首先確定轉動系統的Ix Iy Ixy接著算出莫爾圓之圓心座標a及半徑R並畫出莫爾圓再由莫爾圓上點出A(Ix Ixy )並由OA轉向I軸找出兩倍夾角2θp1及2θp2其中θp1及θp2為主軸(Principle Axis)與Ix軸的夾角由於主軸(Principle Axis)為極值之軸Imax = Iu及Imin = Iv決定θp1及θp2何者為Imax = Iu與Ix軸的夾角之方法乃比較Ix及Iy之大小Imax = Iu會靠近Ix及Iy之較大者也就是Imax = Iu與Ix及Iy之較大者的夾角比較小

a =Ix + Iy

2(130)

R =

radic(Ix minus Iy

2)2 + I 2xy (131)

2θp2 = π + 2θp1 θp2 minus θp1 =π

2(132)

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質質質量量量慣慣慣性性性矩矩矩Mass Moment of Inertia

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質量慣性矩

τ = Iα (133)

其中τ為旋轉扭矩I為質量慣性矩α為旋轉角速度

I =

ˆmr2dm (134)

=

ˆVr2ρdV (135)

= ρ

ˆVr2dV (136)

其中r為旋轉半徑或轉軸到旋轉體的距離

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

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平行軸定理

當通過物體質心(G)並在一轉軸上的質量慣性矩(IG )為已知時可透過座標軸轉換計算物體之質量慣性矩(I)

I =

ˆmr2dm (137)

=

ˆm

[(d + x prime)2 + y prime2]dm (138)

=

ˆm

(x prime2 + y prime2)dm + 2d

ˆmx primedm + d2

ˆmdm (139)

I = IG + md2 (140)

其中IG為在zrsquo軸上通過質心的質量慣性矩m為質量d為通過質心之軸zrsquo並平行原軸z之距離

acutem x primedm = x

acutem dm = 0 since x = 0

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (141)

k =

radicI

m(142)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (143)

= I1 + I2 + + In (144)

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課課課堂堂堂練練練習習習

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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兩圓盤複合體

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結結結論論論

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結論

轉動慣量或慣性矩為物體對旋轉運動的慣性而所謂慣性乃物體持續維持不變的行為也就是定義慣性矩可表達物體對本身旋轉運動的維持或持續性好壞更簡單的說慣性矩越大之物體其維持旋轉運動趨勢的能力越大例如惰輪儲能裝置惰輪之慣性矩越大當其轉動以後其維持旋轉運動趨勢的能力越大

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  • 轉動慣量或慣性矩(Moment of Inertia)
  • 面積慣性矩(Moment of Inertia for Area)
    • 面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)
    • 面積慣性矩之迴轉半徑(Radius of Gyration for Area)
      • 複合面積之慣性矩(Moment of Inertia for Composite Area)
      • 面積之慣性積(Product of Inertia for Area)
        • 平行軸定理(Parallel-Axis Theorem)
          • 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)
            • 主軸慣性矩(Principal Moment of Inertia)
              • 慣性矩之莫爾圓(Mohrs Circle for Moment of Inertia)
              • 質量慣性矩(Mass Moment of Inertia)
                • 平行軸定理(Parallel-Axis Theorem)
Page 13: 轉動慣量 Moments of Inertia - cct.me.ntut.edu.twcct.me.ntut.edu.tw/ccteducation/chchting/aiahtm/statics/lecture... · 轉動慣量或慣性矩 I. 轉動慣量也稱慣性矩(Moment

面積慣性矩之平行軸定理- Continued

由於面心在xrsquo軸上之等價力臂為y primeyrsquo軸上之等價力臂為x prime當以面心為轉軸點時即x prime = y prime = 0則x軸上之面積慣性矩Ix為ˆ

Ay prime2dA = macrIx prime (21)

ˆy primedA = y prime

ˆAdA = 0 (22)

d2y

ˆAdA = Ad2

y (23)

Ix =

ˆAy prime2dA + 2dy

ˆAy primedA + d2

y

ˆAdA (24)

= macrIx prime + Ad2y (25)

同理可得y軸上之面積慣性矩Iy

Iy =

ˆAx prime2dA + 2dx

ˆAx primedA + d2

x

ˆAdA (26)

= macrIy prime + Ad2x

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面積之極慣性矩

極點

新極點

由於通過面心之極慣性矩JC與極點(O)到面心(C)之距離d為

d2 = d2x + d2

y (27)

JC = macrIx prime + macrIy prime (28)

JO = Ix + Iy (29)

= macrIx prime + Ad2y + macrIy prime + Ad2

x (30)

= macrIx prime + macrIy prime + A(d2x + d2

y ) (31)

= JC + Ad2 (32)

也就是面積A之極慣性矩JO等於通過面積A之面心極慣性矩JC 與面積乘上極點(O)到面心(C)之距離d的平方和

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面面面積積積慣慣慣性性性矩矩矩之之之迴迴迴轉轉轉半半半徑徑徑Radius of Gyration for Area

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面積慣性矩之迴轉半徑

當x軸上之面積慣性矩Ix與y軸上之面積慣性矩Iy為已知時面積慣性矩之迴轉半徑可得知

dIx = y2dA (33)

Ix = kxA kx =

radicIxA

(34)

dIy = x2dA (35)

Iy = kyA ky =

radicIyA

(36)

dJO = r2dA (37)

JO = kOA kO =

radicJOA

(38)

其中(kx ky kO)分別為x軸y軸極點(O)之面積慣性矩迴轉半徑

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課課課堂堂堂練練練習習習

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解題技巧 I

極點

新極點

繪圖並標出面積(A)之面心座標位置(C)接著以面心座標位置(C)為原點畫出新座標軸(xrsquoyrsquo)其中xrsquo軸平行於原(舊)座標x軸yrsquo軸平行於原(舊)座標y軸並寫出新舊座標軸的平移量dx與dy其中dx為yrsquo軸與y軸的平移量dy為xrsquo軸與x軸的平移量取微小面積dA其在原(舊)座標之點座標為(xy)在新座標之點座標為(xrsquoyrsquo)帶入x軸上之面積慣性矩Ix及y軸上之面積慣性矩Iy還有xrsquo軸上之面積慣性矩 macrIx prime及y軸上之面積慣性矩 macrIy prime可如下換算

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解題技巧 II

Ix =

ˆAy2dA =

ˆAy prime2dA + 2dy

ˆAy primedA + d2

y

ˆAdA = macrIx prime + Ad2

y

macrIx prime =

ˆAy prime2dA (39)

Iy =

ˆAx2dA =

ˆAx prime2dA + 2dx

ˆAx primedA + d2

x

ˆAdA = macrIy prime + Ad2

x

macrIy prime =

ˆAx prime2dA (40)

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解題技巧 III

另外微小面積dA的面積慣性矩dIx dIy也可寫成

dIx = d macrIx prime + dAd2y (41)

Ix =

ˆdIx =

ˆd macrIx prime +

ˆdAd2

y (42)

dIy = d macrIy prime + dAd2x (43)

Iy =

ˆdIy =

ˆd macrIy prime +

ˆdAd2

x (44)

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

面心慣性矩 慣性矩 極慣性矩

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Example 10-1 解答

dA = bdy (45)

Ixb =

ˆAy2dA (46)

=

ˆ h

0y2bdy (47)

=1

3by3|h0 (48)

=1

3bh3 (49)

長方形之邊長b與h則面心慣性矩 macrIx prime macrIy prime與長方形之邊軸慣性矩Ix Iy為

macrIx prime =1

12bh3 (50)

macrIy prime =1

12hb3 (51)

Ix =1

3bh3 (52)

Iy =1

3hb3 (53)

JC = macrIx prime + macrIy prime (54)

JO = Ix + Iy (55)

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Example 10-2 解答

取微小面積dA=ydx之長方形其邊長b=dx與h=ydy = y2所以

y2 = 400x (56)

y =radic

400x (57)

d macrIx prime = d(1

12bh3) =

1

12y3dx (58)

dIx = d macrIx prime + dAd2y =

1

12y3dx + ydx

(y2

)2=

1

3y3dx (59)

Ix =

ˆ 100

0

1

3y3dx =

ˆ 100

0

1

3

(radic400x

)3dx (60)

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 22 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 22 54

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Example 10-3 解答

取微小面積dA=2ydx之長方形其邊長b=dx與h=2y所以

x2 + y2 = a2 (61)

y =radic

a2 minus x2 (62)

d macrIx prime = d(1

12bh3) =

1

12dx(2y)3 =

2

3y3dx (63)

macrIx prime =

ˆ a

minusa

2

3y3dx =

ˆ a

minusa

2

3

(radica2 minus x2

)3dx =

πa4

4(64)

圓之半徑a則面心慣性矩 macrIx prime macrIy prime為

macrIx prime =πa4

4(65)

macrIy prime =πa4

4(66)

JC = macrIx prime + macrIy prime =πa4

2(67)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 23 54

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複複複合合合面面面積積積之之之慣慣慣性性性矩矩矩Moment of Inertia for Composite Area

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 24 54

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (68)

k =

radicI

m(69)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (70)

= I1 + I2 + + In (71)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 25 54

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複合面積之慣性矩

複合面積之慣性矩(Ixc Iyc)為各局部面積之慣性矩(Ixi Iyi )合

Ixc =i=nsumi=1

Ixi (72)

= Ix1 + Ix2 + + Ixn (73)

Iyc =i=nsumi=1

Iyi (74)

= Iy1 + Iy2 + + Iyn (75)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 26 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

長方形與圓形複合

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

長方形複合

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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面面面積積積之之之慣慣慣性性性積積積Product of Inertia for Area

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 28 54

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面積之慣性積(Product of Inertia for Area)

為了下個章節求解慣性矩之極值(極大或極小值)方便數學運算在此先定義面積之慣性積(Ixy )為

dIxy = xydA (76)

Ixy =

ˆAxydA (77)

面積之慣性積的符號可為正負或零當慣性積之面積相對轉軸對稱時則面積之慣性積和為零

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

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平行軸定理

極點

新極點

如左圖面積慣性矩之平行軸定理乃以面心(C)為新極點(O)設定新座標軸(xrsquoyrsquo)其中xrsquo軸平行於原座標x軸yrsquo軸平行於原座標y軸取微小面積dA其在新座標之點座標為(xrsquoyrsquo)面積之慣性積(Ixy )為

dIxy = (x prime + dx)(y prime + dy )dA (78)

當以面心為轉軸點時即x prime = y prime = 0ˆAx primedA = x prime

ˆAdA = 0

ˆAy primedA = y prime

ˆAdA = 0 (79)

Ixy =

ˆA

(x prime + dx)(y prime + dy )dA (80)

=

ˆAx primey primedA + dx

ˆAy primedA + dy

ˆAx primedA + dxdy

ˆAdA (81)

= macrIx primey prime + Adxdy

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 31 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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傾傾傾斜斜斜軸軸軸面面面積積積之之之慣慣慣性性性矩矩矩

Moment of Inertia for Area about Inclined Axis

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 33 54

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傾斜軸面積之慣性矩

當轉動軸傾斜θ角時也就是座標軸旋轉θ角而Ix Iy Ixy為已知則新的座標軸(uv)可寫為

u = xcosθ + ysinθ (82)

v = ycosθ minus xsinθ (83)

由u軸上之面積慣性矩Iuv軸上之面積慣性矩Iv及慣性積Iuv得知

dIu = v2dA = (ycosθ minus xsinθ)2dA (84)

dIv = u2dA = (xcosθ + ysinθ)2dA (85)

dIuv = uvdA = (xcosθ + ysinθ)(ycosθ minus xsinθ)dA (86)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 34 54

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傾斜軸面積之慣性矩 I

由x軸上之面積慣性矩Ixy軸上之面積慣性矩Iy及慣性積Ixy得知

Ix =

ˆy2dA Iy =

ˆx2dA Ixy =

ˆxydA (87)

Iu =

ˆ(ycosθ minus xsinθ)2dA (88)

=

ˆ(y2cos2θ minus 2xycosθsinθ + x2sin2θ)dA (89)

= cos2θ

ˆy2dAminus 2cosθsinθ

ˆxydA + sin2θ

ˆx2dA (90)

= Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (91)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 35 54

傾斜軸面積之慣性矩 II

Iv =

ˆ(xcosθ + ysinθ)2dA (92)

=

ˆ(x2cos2θ + 2xycosθsinθ + y2sin2θ)dA (93)

= cos2θ

ˆx2dA + 2cosθsinθ

ˆxydA + sin2θ

ˆy2dA (94)

= Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (95)

Iuv =

ˆ(xcosθ + ysinθ)(ycosθ minus xsinθ)dA (96)

=

ˆ(xycos2θ + y2cosθsinθ minus x2cosθsinθ minus xysin2θ)dA (97)

= (cos2 minus sin2θ)

ˆxydA + cosθsinθ

ˆy2dAminus cosθsinθ

ˆx2dA

= Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 36 54

傾斜軸面積之慣性矩 III

再由

sin2θ + cos2θ = 1 (98)

sin(αplusmn β) = sinαcosβ plusmn cosαsinβ (99)

sin2θ = sin(θ + θ) = 2sinθcosθ (100)

cos(αplusmn β) = cosαcosβ ∓ sinαsinβ (101)

cos2θ = cos(θ + θ) = cos2θ minus sin2θ (102)

= 2cos2θ minus 1 = 1minus 2sin2θ (103)

所以

Iu = Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (104)

= Ix(cos2θ + 1

2) + Iy (

1minus cos2θ

2)minus Ixy sin2θ (105)

=Ix + Iy

2+

Ix minus Iy2

cos2θ minus Ixy sin2θ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 37 54

傾斜軸面積之慣性矩 IV

Iv = Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (106)

= Iy (cos2θ + 1

2) + Ix(

1minus cos2θ

2) + Ixy sin2θ (107)

=Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (108)

Iuv = Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ (109)

= Ixycos2θ +1

2Ixsin2θ minus 1

2Iy sin2θ (110)

=Ix minus Iy

2sin2θ + Ixycos2θ (111)

所以

JO = Iu + Iv = Ix + Iy (112)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 38 54

主主主軸軸軸慣慣慣性性性矩矩矩Principal Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 39 54

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主軸慣性矩 I

所謂主軸慣性矩(Principal Moment ofInertia)乃定義發生在慣性矩的極值(極大或極小值)上也就是慣性矩之一次微分式為零或斜率為零

dIu

dθ=

d(Ix+Iy

2+

IxminusIy2

cos2θ minus Ixy sin2θ)

dθ(113)

= minus2Ix minus Iy

2sin2θ minus 2Ixy cos2θ (114)

= 0 (115)

其中dsin2θ = cos2θ dcos2θ = minussin2θ當慣性矩之一次微分式為零時傾斜軸與主軸之夾角θ = θp

minus2Ix minus Iy

2sin2θp minus 2Ixycos2θp = 0 (116)

tan2θ =sin2θ

cos2θ=

minusIxy(Ix minus Iy )2

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 40 54

主軸慣性矩 II

cos2θp =IxminusIy2radic

(IxminusIy2 )2 + I 2xy

sin2θp =minusIxyradic

(IxminusIy2 )2 + I 2xy

(117)

(Iu)max =Ix + Iy

2+

Ix minus Iy2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

minus IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2+

radic(Ix minus Iy

2)2 + I 2xy (118)

(Iv )min =Ix + Iy

2minus Ix minus Iy

2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

+ IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2minusradic

(Ix minus Iy

2)2 + I 2xy (119)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 41 54

主軸慣性矩 III

由tan2θp = tan(π + 2θp)可得到θp1與θp2二個解即2θp2 = π + 2θp1也就是θp2 minus θp1 = π

2

tan2θp = tan(π + 2θp) (120)

2θp2 = π + 2θp1 (121)

θp2 minus θp1 =π

2(122)

所以

Imaxmin

=Ix + Iy

2plusmnradic

(Ix minus Iy

2)2 + I 2xy (123)

Iuv =Ix minus Iy

2

minusIxyradic(IxminusIy2 )2 + I 2xy

+ Ixy

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

(124)

= 0

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 42 54

課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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慣慣慣性性性矩矩矩之之之莫莫莫爾爾爾圓圓圓Mohrrsquos Circle for Moment of Inertia

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慣性矩之莫爾圓

由於Iu =

Ix + Iy2

+Ix minus Iy

2cos2θ minus Ixy sin2θ (125)

Iv =Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (126)

Iuv =Ix minus Iy

2sin2θ + Ixycos2θ (127)

將式(125)與式(127)各自分別平方並相加可得

(Iu minusIx + Iy

2)2 + I 2uv = (

Ix minus Iy2

)2 + I 2xy (128)

其中Ix Iy Ixy均為常數式(128)可進一步寫成

(Iu minus a)2 + I 2uv = R2 (129)

其中a =Ix+Iy2 R =

radic(IxminusIy2 )2 + I 2xycos2θp1 =

aRsin2θp1 =

minusIxyR 2θp1為OA轉向I軸方向即2θp1為

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課課課堂堂堂練練練習習習

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解題技巧

首先確定轉動系統的Ix Iy Ixy接著算出莫爾圓之圓心座標a及半徑R並畫出莫爾圓再由莫爾圓上點出A(Ix Ixy )並由OA轉向I軸找出兩倍夾角2θp1及2θp2其中θp1及θp2為主軸(Principle Axis)與Ix軸的夾角由於主軸(Principle Axis)為極值之軸Imax = Iu及Imin = Iv決定θp1及θp2何者為Imax = Iu與Ix軸的夾角之方法乃比較Ix及Iy之大小Imax = Iu會靠近Ix及Iy之較大者也就是Imax = Iu與Ix及Iy之較大者的夾角比較小

a =Ix + Iy

2(130)

R =

radic(Ix minus Iy

2)2 + I 2xy (131)

2θp2 = π + 2θp1 θp2 minus θp1 =π

2(132)

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質質質量量量慣慣慣性性性矩矩矩Mass Moment of Inertia

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質量慣性矩

τ = Iα (133)

其中τ為旋轉扭矩I為質量慣性矩α為旋轉角速度

I =

ˆmr2dm (134)

=

ˆVr2ρdV (135)

= ρ

ˆVr2dV (136)

其中r為旋轉半徑或轉軸到旋轉體的距離

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

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平行軸定理

當通過物體質心(G)並在一轉軸上的質量慣性矩(IG )為已知時可透過座標軸轉換計算物體之質量慣性矩(I)

I =

ˆmr2dm (137)

=

ˆm

[(d + x prime)2 + y prime2]dm (138)

=

ˆm

(x prime2 + y prime2)dm + 2d

ˆmx primedm + d2

ˆmdm (139)

I = IG + md2 (140)

其中IG為在zrsquo軸上通過質心的質量慣性矩m為質量d為通過質心之軸zrsquo並平行原軸z之距離

acutem x primedm = x

acutem dm = 0 since x = 0

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (141)

k =

radicI

m(142)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (143)

= I1 + I2 + + In (144)

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兩圓盤複合體

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結結結論論論

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結論

轉動慣量或慣性矩為物體對旋轉運動的慣性而所謂慣性乃物體持續維持不變的行為也就是定義慣性矩可表達物體對本身旋轉運動的維持或持續性好壞更簡單的說慣性矩越大之物體其維持旋轉運動趨勢的能力越大例如惰輪儲能裝置惰輪之慣性矩越大當其轉動以後其維持旋轉運動趨勢的能力越大

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  • 轉動慣量或慣性矩(Moment of Inertia)
  • 面積慣性矩(Moment of Inertia for Area)
    • 面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)
    • 面積慣性矩之迴轉半徑(Radius of Gyration for Area)
      • 複合面積之慣性矩(Moment of Inertia for Composite Area)
      • 面積之慣性積(Product of Inertia for Area)
        • 平行軸定理(Parallel-Axis Theorem)
          • 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)
            • 主軸慣性矩(Principal Moment of Inertia)
              • 慣性矩之莫爾圓(Mohrs Circle for Moment of Inertia)
              • 質量慣性矩(Mass Moment of Inertia)
                • 平行軸定理(Parallel-Axis Theorem)
Page 14: 轉動慣量 Moments of Inertia - cct.me.ntut.edu.twcct.me.ntut.edu.tw/ccteducation/chchting/aiahtm/statics/lecture... · 轉動慣量或慣性矩 I. 轉動慣量也稱慣性矩(Moment

面積之極慣性矩

極點

新極點

由於通過面心之極慣性矩JC與極點(O)到面心(C)之距離d為

d2 = d2x + d2

y (27)

JC = macrIx prime + macrIy prime (28)

JO = Ix + Iy (29)

= macrIx prime + Ad2y + macrIy prime + Ad2

x (30)

= macrIx prime + macrIy prime + A(d2x + d2

y ) (31)

= JC + Ad2 (32)

也就是面積A之極慣性矩JO等於通過面積A之面心極慣性矩JC 與面積乘上極點(O)到面心(C)之距離d的平方和

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面面面積積積慣慣慣性性性矩矩矩之之之迴迴迴轉轉轉半半半徑徑徑Radius of Gyration for Area

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面積慣性矩之迴轉半徑

當x軸上之面積慣性矩Ix與y軸上之面積慣性矩Iy為已知時面積慣性矩之迴轉半徑可得知

dIx = y2dA (33)

Ix = kxA kx =

radicIxA

(34)

dIy = x2dA (35)

Iy = kyA ky =

radicIyA

(36)

dJO = r2dA (37)

JO = kOA kO =

radicJOA

(38)

其中(kx ky kO)分別為x軸y軸極點(O)之面積慣性矩迴轉半徑

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課課課堂堂堂練練練習習習

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解題技巧 I

極點

新極點

繪圖並標出面積(A)之面心座標位置(C)接著以面心座標位置(C)為原點畫出新座標軸(xrsquoyrsquo)其中xrsquo軸平行於原(舊)座標x軸yrsquo軸平行於原(舊)座標y軸並寫出新舊座標軸的平移量dx與dy其中dx為yrsquo軸與y軸的平移量dy為xrsquo軸與x軸的平移量取微小面積dA其在原(舊)座標之點座標為(xy)在新座標之點座標為(xrsquoyrsquo)帶入x軸上之面積慣性矩Ix及y軸上之面積慣性矩Iy還有xrsquo軸上之面積慣性矩 macrIx prime及y軸上之面積慣性矩 macrIy prime可如下換算

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解題技巧 II

Ix =

ˆAy2dA =

ˆAy prime2dA + 2dy

ˆAy primedA + d2

y

ˆAdA = macrIx prime + Ad2

y

macrIx prime =

ˆAy prime2dA (39)

Iy =

ˆAx2dA =

ˆAx prime2dA + 2dx

ˆAx primedA + d2

x

ˆAdA = macrIy prime + Ad2

x

macrIy prime =

ˆAx prime2dA (40)

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解題技巧 III

另外微小面積dA的面積慣性矩dIx dIy也可寫成

dIx = d macrIx prime + dAd2y (41)

Ix =

ˆdIx =

ˆd macrIx prime +

ˆdAd2

y (42)

dIy = d macrIy prime + dAd2x (43)

Iy =

ˆdIy =

ˆd macrIy prime +

ˆdAd2

x (44)

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

面心慣性矩 慣性矩 極慣性矩

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Example 10-1 解答

dA = bdy (45)

Ixb =

ˆAy2dA (46)

=

ˆ h

0y2bdy (47)

=1

3by3|h0 (48)

=1

3bh3 (49)

長方形之邊長b與h則面心慣性矩 macrIx prime macrIy prime與長方形之邊軸慣性矩Ix Iy為

macrIx prime =1

12bh3 (50)

macrIy prime =1

12hb3 (51)

Ix =1

3bh3 (52)

Iy =1

3hb3 (53)

JC = macrIx prime + macrIy prime (54)

JO = Ix + Iy (55)

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Example 10-2 解答

取微小面積dA=ydx之長方形其邊長b=dx與h=ydy = y2所以

y2 = 400x (56)

y =radic

400x (57)

d macrIx prime = d(1

12bh3) =

1

12y3dx (58)

dIx = d macrIx prime + dAd2y =

1

12y3dx + ydx

(y2

)2=

1

3y3dx (59)

Ix =

ˆ 100

0

1

3y3dx =

ˆ 100

0

1

3

(radic400x

)3dx (60)

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 22 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 22 54

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Example 10-3 解答

取微小面積dA=2ydx之長方形其邊長b=dx與h=2y所以

x2 + y2 = a2 (61)

y =radic

a2 minus x2 (62)

d macrIx prime = d(1

12bh3) =

1

12dx(2y)3 =

2

3y3dx (63)

macrIx prime =

ˆ a

minusa

2

3y3dx =

ˆ a

minusa

2

3

(radica2 minus x2

)3dx =

πa4

4(64)

圓之半徑a則面心慣性矩 macrIx prime macrIy prime為

macrIx prime =πa4

4(65)

macrIy prime =πa4

4(66)

JC = macrIx prime + macrIy prime =πa4

2(67)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 23 54

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複複複合合合面面面積積積之之之慣慣慣性性性矩矩矩Moment of Inertia for Composite Area

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 24 54

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (68)

k =

radicI

m(69)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (70)

= I1 + I2 + + In (71)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 25 54

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複合面積之慣性矩

複合面積之慣性矩(Ixc Iyc)為各局部面積之慣性矩(Ixi Iyi )合

Ixc =i=nsumi=1

Ixi (72)

= Ix1 + Ix2 + + Ixn (73)

Iyc =i=nsumi=1

Iyi (74)

= Iy1 + Iy2 + + Iyn (75)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 26 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

長方形與圓形複合

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

長方形複合

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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面面面積積積之之之慣慣慣性性性積積積Product of Inertia for Area

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 28 54

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面積之慣性積(Product of Inertia for Area)

為了下個章節求解慣性矩之極值(極大或極小值)方便數學運算在此先定義面積之慣性積(Ixy )為

dIxy = xydA (76)

Ixy =

ˆAxydA (77)

面積之慣性積的符號可為正負或零當慣性積之面積相對轉軸對稱時則面積之慣性積和為零

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

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平行軸定理

極點

新極點

如左圖面積慣性矩之平行軸定理乃以面心(C)為新極點(O)設定新座標軸(xrsquoyrsquo)其中xrsquo軸平行於原座標x軸yrsquo軸平行於原座標y軸取微小面積dA其在新座標之點座標為(xrsquoyrsquo)面積之慣性積(Ixy )為

dIxy = (x prime + dx)(y prime + dy )dA (78)

當以面心為轉軸點時即x prime = y prime = 0ˆAx primedA = x prime

ˆAdA = 0

ˆAy primedA = y prime

ˆAdA = 0 (79)

Ixy =

ˆA

(x prime + dx)(y prime + dy )dA (80)

=

ˆAx primey primedA + dx

ˆAy primedA + dy

ˆAx primedA + dxdy

ˆAdA (81)

= macrIx primey prime + Adxdy

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 31 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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傾傾傾斜斜斜軸軸軸面面面積積積之之之慣慣慣性性性矩矩矩

Moment of Inertia for Area about Inclined Axis

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 33 54

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傾斜軸面積之慣性矩

當轉動軸傾斜θ角時也就是座標軸旋轉θ角而Ix Iy Ixy為已知則新的座標軸(uv)可寫為

u = xcosθ + ysinθ (82)

v = ycosθ minus xsinθ (83)

由u軸上之面積慣性矩Iuv軸上之面積慣性矩Iv及慣性積Iuv得知

dIu = v2dA = (ycosθ minus xsinθ)2dA (84)

dIv = u2dA = (xcosθ + ysinθ)2dA (85)

dIuv = uvdA = (xcosθ + ysinθ)(ycosθ minus xsinθ)dA (86)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 34 54

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傾斜軸面積之慣性矩 I

由x軸上之面積慣性矩Ixy軸上之面積慣性矩Iy及慣性積Ixy得知

Ix =

ˆy2dA Iy =

ˆx2dA Ixy =

ˆxydA (87)

Iu =

ˆ(ycosθ minus xsinθ)2dA (88)

=

ˆ(y2cos2θ minus 2xycosθsinθ + x2sin2θ)dA (89)

= cos2θ

ˆy2dAminus 2cosθsinθ

ˆxydA + sin2θ

ˆx2dA (90)

= Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (91)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 35 54

傾斜軸面積之慣性矩 II

Iv =

ˆ(xcosθ + ysinθ)2dA (92)

=

ˆ(x2cos2θ + 2xycosθsinθ + y2sin2θ)dA (93)

= cos2θ

ˆx2dA + 2cosθsinθ

ˆxydA + sin2θ

ˆy2dA (94)

= Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (95)

Iuv =

ˆ(xcosθ + ysinθ)(ycosθ minus xsinθ)dA (96)

=

ˆ(xycos2θ + y2cosθsinθ minus x2cosθsinθ minus xysin2θ)dA (97)

= (cos2 minus sin2θ)

ˆxydA + cosθsinθ

ˆy2dAminus cosθsinθ

ˆx2dA

= Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 36 54

傾斜軸面積之慣性矩 III

再由

sin2θ + cos2θ = 1 (98)

sin(αplusmn β) = sinαcosβ plusmn cosαsinβ (99)

sin2θ = sin(θ + θ) = 2sinθcosθ (100)

cos(αplusmn β) = cosαcosβ ∓ sinαsinβ (101)

cos2θ = cos(θ + θ) = cos2θ minus sin2θ (102)

= 2cos2θ minus 1 = 1minus 2sin2θ (103)

所以

Iu = Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (104)

= Ix(cos2θ + 1

2) + Iy (

1minus cos2θ

2)minus Ixy sin2θ (105)

=Ix + Iy

2+

Ix minus Iy2

cos2θ minus Ixy sin2θ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 37 54

傾斜軸面積之慣性矩 IV

Iv = Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (106)

= Iy (cos2θ + 1

2) + Ix(

1minus cos2θ

2) + Ixy sin2θ (107)

=Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (108)

Iuv = Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ (109)

= Ixycos2θ +1

2Ixsin2θ minus 1

2Iy sin2θ (110)

=Ix minus Iy

2sin2θ + Ixycos2θ (111)

所以

JO = Iu + Iv = Ix + Iy (112)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 38 54

主主主軸軸軸慣慣慣性性性矩矩矩Principal Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 39 54

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主軸慣性矩 I

所謂主軸慣性矩(Principal Moment ofInertia)乃定義發生在慣性矩的極值(極大或極小值)上也就是慣性矩之一次微分式為零或斜率為零

dIu

dθ=

d(Ix+Iy

2+

IxminusIy2

cos2θ minus Ixy sin2θ)

dθ(113)

= minus2Ix minus Iy

2sin2θ minus 2Ixy cos2θ (114)

= 0 (115)

其中dsin2θ = cos2θ dcos2θ = minussin2θ當慣性矩之一次微分式為零時傾斜軸與主軸之夾角θ = θp

minus2Ix minus Iy

2sin2θp minus 2Ixycos2θp = 0 (116)

tan2θ =sin2θ

cos2θ=

minusIxy(Ix minus Iy )2

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 40 54

主軸慣性矩 II

cos2θp =IxminusIy2radic

(IxminusIy2 )2 + I 2xy

sin2θp =minusIxyradic

(IxminusIy2 )2 + I 2xy

(117)

(Iu)max =Ix + Iy

2+

Ix minus Iy2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

minus IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2+

radic(Ix minus Iy

2)2 + I 2xy (118)

(Iv )min =Ix + Iy

2minus Ix minus Iy

2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

+ IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2minusradic

(Ix minus Iy

2)2 + I 2xy (119)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 41 54

主軸慣性矩 III

由tan2θp = tan(π + 2θp)可得到θp1與θp2二個解即2θp2 = π + 2θp1也就是θp2 minus θp1 = π

2

tan2θp = tan(π + 2θp) (120)

2θp2 = π + 2θp1 (121)

θp2 minus θp1 =π

2(122)

所以

Imaxmin

=Ix + Iy

2plusmnradic

(Ix minus Iy

2)2 + I 2xy (123)

Iuv =Ix minus Iy

2

minusIxyradic(IxminusIy2 )2 + I 2xy

+ Ixy

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

(124)

= 0

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 42 54

課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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慣慣慣性性性矩矩矩之之之莫莫莫爾爾爾圓圓圓Mohrrsquos Circle for Moment of Inertia

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慣性矩之莫爾圓

由於Iu =

Ix + Iy2

+Ix minus Iy

2cos2θ minus Ixy sin2θ (125)

Iv =Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (126)

Iuv =Ix minus Iy

2sin2θ + Ixycos2θ (127)

將式(125)與式(127)各自分別平方並相加可得

(Iu minusIx + Iy

2)2 + I 2uv = (

Ix minus Iy2

)2 + I 2xy (128)

其中Ix Iy Ixy均為常數式(128)可進一步寫成

(Iu minus a)2 + I 2uv = R2 (129)

其中a =Ix+Iy2 R =

radic(IxminusIy2 )2 + I 2xycos2θp1 =

aRsin2θp1 =

minusIxyR 2θp1為OA轉向I軸方向即2θp1為

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課課課堂堂堂練練練習習習

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解題技巧

首先確定轉動系統的Ix Iy Ixy接著算出莫爾圓之圓心座標a及半徑R並畫出莫爾圓再由莫爾圓上點出A(Ix Ixy )並由OA轉向I軸找出兩倍夾角2θp1及2θp2其中θp1及θp2為主軸(Principle Axis)與Ix軸的夾角由於主軸(Principle Axis)為極值之軸Imax = Iu及Imin = Iv決定θp1及θp2何者為Imax = Iu與Ix軸的夾角之方法乃比較Ix及Iy之大小Imax = Iu會靠近Ix及Iy之較大者也就是Imax = Iu與Ix及Iy之較大者的夾角比較小

a =Ix + Iy

2(130)

R =

radic(Ix minus Iy

2)2 + I 2xy (131)

2θp2 = π + 2θp1 θp2 minus θp1 =π

2(132)

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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質質質量量量慣慣慣性性性矩矩矩Mass Moment of Inertia

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質量慣性矩

τ = Iα (133)

其中τ為旋轉扭矩I為質量慣性矩α為旋轉角速度

I =

ˆmr2dm (134)

=

ˆVr2ρdV (135)

= ρ

ˆVr2dV (136)

其中r為旋轉半徑或轉軸到旋轉體的距離

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

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平行軸定理

當通過物體質心(G)並在一轉軸上的質量慣性矩(IG )為已知時可透過座標軸轉換計算物體之質量慣性矩(I)

I =

ˆmr2dm (137)

=

ˆm

[(d + x prime)2 + y prime2]dm (138)

=

ˆm

(x prime2 + y prime2)dm + 2d

ˆmx primedm + d2

ˆmdm (139)

I = IG + md2 (140)

其中IG為在zrsquo軸上通過質心的質量慣性矩m為質量d為通過質心之軸zrsquo並平行原軸z之距離

acutem x primedm = x

acutem dm = 0 since x = 0

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (141)

k =

radicI

m(142)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (143)

= I1 + I2 + + In (144)

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課課課堂堂堂練練練習習習

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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兩圓盤複合體

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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結結結論論論

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結論

轉動慣量或慣性矩為物體對旋轉運動的慣性而所謂慣性乃物體持續維持不變的行為也就是定義慣性矩可表達物體對本身旋轉運動的維持或持續性好壞更簡單的說慣性矩越大之物體其維持旋轉運動趨勢的能力越大例如惰輪儲能裝置惰輪之慣性矩越大當其轉動以後其維持旋轉運動趨勢的能力越大

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  • 轉動慣量或慣性矩(Moment of Inertia)
  • 面積慣性矩(Moment of Inertia for Area)
    • 面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)
    • 面積慣性矩之迴轉半徑(Radius of Gyration for Area)
      • 複合面積之慣性矩(Moment of Inertia for Composite Area)
      • 面積之慣性積(Product of Inertia for Area)
        • 平行軸定理(Parallel-Axis Theorem)
          • 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)
            • 主軸慣性矩(Principal Moment of Inertia)
              • 慣性矩之莫爾圓(Mohrs Circle for Moment of Inertia)
              • 質量慣性矩(Mass Moment of Inertia)
                • 平行軸定理(Parallel-Axis Theorem)
Page 15: 轉動慣量 Moments of Inertia - cct.me.ntut.edu.twcct.me.ntut.edu.tw/ccteducation/chchting/aiahtm/statics/lecture... · 轉動慣量或慣性矩 I. 轉動慣量也稱慣性矩(Moment

面面面積積積慣慣慣性性性矩矩矩之之之迴迴迴轉轉轉半半半徑徑徑Radius of Gyration for Area

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面積慣性矩之迴轉半徑

當x軸上之面積慣性矩Ix與y軸上之面積慣性矩Iy為已知時面積慣性矩之迴轉半徑可得知

dIx = y2dA (33)

Ix = kxA kx =

radicIxA

(34)

dIy = x2dA (35)

Iy = kyA ky =

radicIyA

(36)

dJO = r2dA (37)

JO = kOA kO =

radicJOA

(38)

其中(kx ky kO)分別為x軸y軸極點(O)之面積慣性矩迴轉半徑

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課課課堂堂堂練練練習習習

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解題技巧 I

極點

新極點

繪圖並標出面積(A)之面心座標位置(C)接著以面心座標位置(C)為原點畫出新座標軸(xrsquoyrsquo)其中xrsquo軸平行於原(舊)座標x軸yrsquo軸平行於原(舊)座標y軸並寫出新舊座標軸的平移量dx與dy其中dx為yrsquo軸與y軸的平移量dy為xrsquo軸與x軸的平移量取微小面積dA其在原(舊)座標之點座標為(xy)在新座標之點座標為(xrsquoyrsquo)帶入x軸上之面積慣性矩Ix及y軸上之面積慣性矩Iy還有xrsquo軸上之面積慣性矩 macrIx prime及y軸上之面積慣性矩 macrIy prime可如下換算

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解題技巧 II

Ix =

ˆAy2dA =

ˆAy prime2dA + 2dy

ˆAy primedA + d2

y

ˆAdA = macrIx prime + Ad2

y

macrIx prime =

ˆAy prime2dA (39)

Iy =

ˆAx2dA =

ˆAx prime2dA + 2dx

ˆAx primedA + d2

x

ˆAdA = macrIy prime + Ad2

x

macrIy prime =

ˆAx prime2dA (40)

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解題技巧 III

另外微小面積dA的面積慣性矩dIx dIy也可寫成

dIx = d macrIx prime + dAd2y (41)

Ix =

ˆdIx =

ˆd macrIx prime +

ˆdAd2

y (42)

dIy = d macrIy prime + dAd2x (43)

Iy =

ˆdIy =

ˆd macrIy prime +

ˆdAd2

x (44)

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

面心慣性矩 慣性矩 極慣性矩

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Example 10-1 解答

dA = bdy (45)

Ixb =

ˆAy2dA (46)

=

ˆ h

0y2bdy (47)

=1

3by3|h0 (48)

=1

3bh3 (49)

長方形之邊長b與h則面心慣性矩 macrIx prime macrIy prime與長方形之邊軸慣性矩Ix Iy為

macrIx prime =1

12bh3 (50)

macrIy prime =1

12hb3 (51)

Ix =1

3bh3 (52)

Iy =1

3hb3 (53)

JC = macrIx prime + macrIy prime (54)

JO = Ix + Iy (55)

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Example 10-2 解答

取微小面積dA=ydx之長方形其邊長b=dx與h=ydy = y2所以

y2 = 400x (56)

y =radic

400x (57)

d macrIx prime = d(1

12bh3) =

1

12y3dx (58)

dIx = d macrIx prime + dAd2y =

1

12y3dx + ydx

(y2

)2=

1

3y3dx (59)

Ix =

ˆ 100

0

1

3y3dx =

ˆ 100

0

1

3

(radic400x

)3dx (60)

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Example 10-3 解答

取微小面積dA=2ydx之長方形其邊長b=dx與h=2y所以

x2 + y2 = a2 (61)

y =radic

a2 minus x2 (62)

d macrIx prime = d(1

12bh3) =

1

12dx(2y)3 =

2

3y3dx (63)

macrIx prime =

ˆ a

minusa

2

3y3dx =

ˆ a

minusa

2

3

(radica2 minus x2

)3dx =

πa4

4(64)

圓之半徑a則面心慣性矩 macrIx prime macrIy prime為

macrIx prime =πa4

4(65)

macrIy prime =πa4

4(66)

JC = macrIx prime + macrIy prime =πa4

2(67)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 23 54

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複複複合合合面面面積積積之之之慣慣慣性性性矩矩矩Moment of Inertia for Composite Area

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 24 54

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (68)

k =

radicI

m(69)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (70)

= I1 + I2 + + In (71)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 25 54

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複合面積之慣性矩

複合面積之慣性矩(Ixc Iyc)為各局部面積之慣性矩(Ixi Iyi )合

Ixc =i=nsumi=1

Ixi (72)

= Ix1 + Ix2 + + Ixn (73)

Iyc =i=nsumi=1

Iyi (74)

= Iy1 + Iy2 + + Iyn (75)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 26 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

長方形與圓形複合

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

長方形複合

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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面面面積積積之之之慣慣慣性性性積積積Product of Inertia for Area

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 28 54

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面積之慣性積(Product of Inertia for Area)

為了下個章節求解慣性矩之極值(極大或極小值)方便數學運算在此先定義面積之慣性積(Ixy )為

dIxy = xydA (76)

Ixy =

ˆAxydA (77)

面積之慣性積的符號可為正負或零當慣性積之面積相對轉軸對稱時則面積之慣性積和為零

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 29 54

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 30 54

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平行軸定理

極點

新極點

如左圖面積慣性矩之平行軸定理乃以面心(C)為新極點(O)設定新座標軸(xrsquoyrsquo)其中xrsquo軸平行於原座標x軸yrsquo軸平行於原座標y軸取微小面積dA其在新座標之點座標為(xrsquoyrsquo)面積之慣性積(Ixy )為

dIxy = (x prime + dx)(y prime + dy )dA (78)

當以面心為轉軸點時即x prime = y prime = 0ˆAx primedA = x prime

ˆAdA = 0

ˆAy primedA = y prime

ˆAdA = 0 (79)

Ixy =

ˆA

(x prime + dx)(y prime + dy )dA (80)

=

ˆAx primey primedA + dx

ˆAy primedA + dy

ˆAx primedA + dxdy

ˆAdA (81)

= macrIx primey prime + Adxdy

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 31 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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傾傾傾斜斜斜軸軸軸面面面積積積之之之慣慣慣性性性矩矩矩

Moment of Inertia for Area about Inclined Axis

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 33 54

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傾斜軸面積之慣性矩

當轉動軸傾斜θ角時也就是座標軸旋轉θ角而Ix Iy Ixy為已知則新的座標軸(uv)可寫為

u = xcosθ + ysinθ (82)

v = ycosθ minus xsinθ (83)

由u軸上之面積慣性矩Iuv軸上之面積慣性矩Iv及慣性積Iuv得知

dIu = v2dA = (ycosθ minus xsinθ)2dA (84)

dIv = u2dA = (xcosθ + ysinθ)2dA (85)

dIuv = uvdA = (xcosθ + ysinθ)(ycosθ minus xsinθ)dA (86)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 34 54

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傾斜軸面積之慣性矩 I

由x軸上之面積慣性矩Ixy軸上之面積慣性矩Iy及慣性積Ixy得知

Ix =

ˆy2dA Iy =

ˆx2dA Ixy =

ˆxydA (87)

Iu =

ˆ(ycosθ minus xsinθ)2dA (88)

=

ˆ(y2cos2θ minus 2xycosθsinθ + x2sin2θ)dA (89)

= cos2θ

ˆy2dAminus 2cosθsinθ

ˆxydA + sin2θ

ˆx2dA (90)

= Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (91)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 35 54

傾斜軸面積之慣性矩 II

Iv =

ˆ(xcosθ + ysinθ)2dA (92)

=

ˆ(x2cos2θ + 2xycosθsinθ + y2sin2θ)dA (93)

= cos2θ

ˆx2dA + 2cosθsinθ

ˆxydA + sin2θ

ˆy2dA (94)

= Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (95)

Iuv =

ˆ(xcosθ + ysinθ)(ycosθ minus xsinθ)dA (96)

=

ˆ(xycos2θ + y2cosθsinθ minus x2cosθsinθ minus xysin2θ)dA (97)

= (cos2 minus sin2θ)

ˆxydA + cosθsinθ

ˆy2dAminus cosθsinθ

ˆx2dA

= Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 36 54

傾斜軸面積之慣性矩 III

再由

sin2θ + cos2θ = 1 (98)

sin(αplusmn β) = sinαcosβ plusmn cosαsinβ (99)

sin2θ = sin(θ + θ) = 2sinθcosθ (100)

cos(αplusmn β) = cosαcosβ ∓ sinαsinβ (101)

cos2θ = cos(θ + θ) = cos2θ minus sin2θ (102)

= 2cos2θ minus 1 = 1minus 2sin2θ (103)

所以

Iu = Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (104)

= Ix(cos2θ + 1

2) + Iy (

1minus cos2θ

2)minus Ixy sin2θ (105)

=Ix + Iy

2+

Ix minus Iy2

cos2θ minus Ixy sin2θ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 37 54

傾斜軸面積之慣性矩 IV

Iv = Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (106)

= Iy (cos2θ + 1

2) + Ix(

1minus cos2θ

2) + Ixy sin2θ (107)

=Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (108)

Iuv = Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ (109)

= Ixycos2θ +1

2Ixsin2θ minus 1

2Iy sin2θ (110)

=Ix minus Iy

2sin2θ + Ixycos2θ (111)

所以

JO = Iu + Iv = Ix + Iy (112)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 38 54

主主主軸軸軸慣慣慣性性性矩矩矩Principal Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 39 54

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主軸慣性矩 I

所謂主軸慣性矩(Principal Moment ofInertia)乃定義發生在慣性矩的極值(極大或極小值)上也就是慣性矩之一次微分式為零或斜率為零

dIu

dθ=

d(Ix+Iy

2+

IxminusIy2

cos2θ minus Ixy sin2θ)

dθ(113)

= minus2Ix minus Iy

2sin2θ minus 2Ixy cos2θ (114)

= 0 (115)

其中dsin2θ = cos2θ dcos2θ = minussin2θ當慣性矩之一次微分式為零時傾斜軸與主軸之夾角θ = θp

minus2Ix minus Iy

2sin2θp minus 2Ixycos2θp = 0 (116)

tan2θ =sin2θ

cos2θ=

minusIxy(Ix minus Iy )2

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 40 54

主軸慣性矩 II

cos2θp =IxminusIy2radic

(IxminusIy2 )2 + I 2xy

sin2θp =minusIxyradic

(IxminusIy2 )2 + I 2xy

(117)

(Iu)max =Ix + Iy

2+

Ix minus Iy2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

minus IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2+

radic(Ix minus Iy

2)2 + I 2xy (118)

(Iv )min =Ix + Iy

2minus Ix minus Iy

2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

+ IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2minusradic

(Ix minus Iy

2)2 + I 2xy (119)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 41 54

主軸慣性矩 III

由tan2θp = tan(π + 2θp)可得到θp1與θp2二個解即2θp2 = π + 2θp1也就是θp2 minus θp1 = π

2

tan2θp = tan(π + 2θp) (120)

2θp2 = π + 2θp1 (121)

θp2 minus θp1 =π

2(122)

所以

Imaxmin

=Ix + Iy

2plusmnradic

(Ix minus Iy

2)2 + I 2xy (123)

Iuv =Ix minus Iy

2

minusIxyradic(IxminusIy2 )2 + I 2xy

+ Ixy

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

(124)

= 0

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 42 54

課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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慣慣慣性性性矩矩矩之之之莫莫莫爾爾爾圓圓圓Mohrrsquos Circle for Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 44 54

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慣性矩之莫爾圓

由於Iu =

Ix + Iy2

+Ix minus Iy

2cos2θ minus Ixy sin2θ (125)

Iv =Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (126)

Iuv =Ix minus Iy

2sin2θ + Ixycos2θ (127)

將式(125)與式(127)各自分別平方並相加可得

(Iu minusIx + Iy

2)2 + I 2uv = (

Ix minus Iy2

)2 + I 2xy (128)

其中Ix Iy Ixy均為常數式(128)可進一步寫成

(Iu minus a)2 + I 2uv = R2 (129)

其中a =Ix+Iy2 R =

radic(IxminusIy2 )2 + I 2xycos2θp1 =

aRsin2θp1 =

minusIxyR 2θp1為OA轉向I軸方向即2θp1為

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 45 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 46 54

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解題技巧

首先確定轉動系統的Ix Iy Ixy接著算出莫爾圓之圓心座標a及半徑R並畫出莫爾圓再由莫爾圓上點出A(Ix Ixy )並由OA轉向I軸找出兩倍夾角2θp1及2θp2其中θp1及θp2為主軸(Principle Axis)與Ix軸的夾角由於主軸(Principle Axis)為極值之軸Imax = Iu及Imin = Iv決定θp1及θp2何者為Imax = Iu與Ix軸的夾角之方法乃比較Ix及Iy之大小Imax = Iu會靠近Ix及Iy之較大者也就是Imax = Iu與Ix及Iy之較大者的夾角比較小

a =Ix + Iy

2(130)

R =

radic(Ix minus Iy

2)2 + I 2xy (131)

2θp2 = π + 2θp1 θp2 minus θp1 =π

2(132)

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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質質質量量量慣慣慣性性性矩矩矩Mass Moment of Inertia

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質量慣性矩

τ = Iα (133)

其中τ為旋轉扭矩I為質量慣性矩α為旋轉角速度

I =

ˆmr2dm (134)

=

ˆVr2ρdV (135)

= ρ

ˆVr2dV (136)

其中r為旋轉半徑或轉軸到旋轉體的距離

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

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平行軸定理

當通過物體質心(G)並在一轉軸上的質量慣性矩(IG )為已知時可透過座標軸轉換計算物體之質量慣性矩(I)

I =

ˆmr2dm (137)

=

ˆm

[(d + x prime)2 + y prime2]dm (138)

=

ˆm

(x prime2 + y prime2)dm + 2d

ˆmx primedm + d2

ˆmdm (139)

I = IG + md2 (140)

其中IG為在zrsquo軸上通過質心的質量慣性矩m為質量d為通過質心之軸zrsquo並平行原軸z之距離

acutem x primedm = x

acutem dm = 0 since x = 0

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (141)

k =

radicI

m(142)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (143)

= I1 + I2 + + In (144)

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課課課堂堂堂練練練習習習

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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兩圓盤複合體

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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結結結論論論

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結論

轉動慣量或慣性矩為物體對旋轉運動的慣性而所謂慣性乃物體持續維持不變的行為也就是定義慣性矩可表達物體對本身旋轉運動的維持或持續性好壞更簡單的說慣性矩越大之物體其維持旋轉運動趨勢的能力越大例如惰輪儲能裝置惰輪之慣性矩越大當其轉動以後其維持旋轉運動趨勢的能力越大

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  • 轉動慣量或慣性矩(Moment of Inertia)
  • 面積慣性矩(Moment of Inertia for Area)
    • 面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)
    • 面積慣性矩之迴轉半徑(Radius of Gyration for Area)
      • 複合面積之慣性矩(Moment of Inertia for Composite Area)
      • 面積之慣性積(Product of Inertia for Area)
        • 平行軸定理(Parallel-Axis Theorem)
          • 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)
            • 主軸慣性矩(Principal Moment of Inertia)
              • 慣性矩之莫爾圓(Mohrs Circle for Moment of Inertia)
              • 質量慣性矩(Mass Moment of Inertia)
                • 平行軸定理(Parallel-Axis Theorem)
Page 16: 轉動慣量 Moments of Inertia - cct.me.ntut.edu.twcct.me.ntut.edu.tw/ccteducation/chchting/aiahtm/statics/lecture... · 轉動慣量或慣性矩 I. 轉動慣量也稱慣性矩(Moment

面積慣性矩之迴轉半徑

當x軸上之面積慣性矩Ix與y軸上之面積慣性矩Iy為已知時面積慣性矩之迴轉半徑可得知

dIx = y2dA (33)

Ix = kxA kx =

radicIxA

(34)

dIy = x2dA (35)

Iy = kyA ky =

radicIyA

(36)

dJO = r2dA (37)

JO = kOA kO =

radicJOA

(38)

其中(kx ky kO)分別為x軸y軸極點(O)之面積慣性矩迴轉半徑

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課課課堂堂堂練練練習習習

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解題技巧 I

極點

新極點

繪圖並標出面積(A)之面心座標位置(C)接著以面心座標位置(C)為原點畫出新座標軸(xrsquoyrsquo)其中xrsquo軸平行於原(舊)座標x軸yrsquo軸平行於原(舊)座標y軸並寫出新舊座標軸的平移量dx與dy其中dx為yrsquo軸與y軸的平移量dy為xrsquo軸與x軸的平移量取微小面積dA其在原(舊)座標之點座標為(xy)在新座標之點座標為(xrsquoyrsquo)帶入x軸上之面積慣性矩Ix及y軸上之面積慣性矩Iy還有xrsquo軸上之面積慣性矩 macrIx prime及y軸上之面積慣性矩 macrIy prime可如下換算

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解題技巧 II

Ix =

ˆAy2dA =

ˆAy prime2dA + 2dy

ˆAy primedA + d2

y

ˆAdA = macrIx prime + Ad2

y

macrIx prime =

ˆAy prime2dA (39)

Iy =

ˆAx2dA =

ˆAx prime2dA + 2dx

ˆAx primedA + d2

x

ˆAdA = macrIy prime + Ad2

x

macrIy prime =

ˆAx prime2dA (40)

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解題技巧 III

另外微小面積dA的面積慣性矩dIx dIy也可寫成

dIx = d macrIx prime + dAd2y (41)

Ix =

ˆdIx =

ˆd macrIx prime +

ˆdAd2

y (42)

dIy = d macrIy prime + dAd2x (43)

Iy =

ˆdIy =

ˆd macrIy prime +

ˆdAd2

x (44)

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

面心慣性矩 慣性矩 極慣性矩

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Example 10-1 解答

dA = bdy (45)

Ixb =

ˆAy2dA (46)

=

ˆ h

0y2bdy (47)

=1

3by3|h0 (48)

=1

3bh3 (49)

長方形之邊長b與h則面心慣性矩 macrIx prime macrIy prime與長方形之邊軸慣性矩Ix Iy為

macrIx prime =1

12bh3 (50)

macrIy prime =1

12hb3 (51)

Ix =1

3bh3 (52)

Iy =1

3hb3 (53)

JC = macrIx prime + macrIy prime (54)

JO = Ix + Iy (55)

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Example 10-2 解答

取微小面積dA=ydx之長方形其邊長b=dx與h=ydy = y2所以

y2 = 400x (56)

y =radic

400x (57)

d macrIx prime = d(1

12bh3) =

1

12y3dx (58)

dIx = d macrIx prime + dAd2y =

1

12y3dx + ydx

(y2

)2=

1

3y3dx (59)

Ix =

ˆ 100

0

1

3y3dx =

ˆ 100

0

1

3

(radic400x

)3dx (60)

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Example 10-3 解答

取微小面積dA=2ydx之長方形其邊長b=dx與h=2y所以

x2 + y2 = a2 (61)

y =radic

a2 minus x2 (62)

d macrIx prime = d(1

12bh3) =

1

12dx(2y)3 =

2

3y3dx (63)

macrIx prime =

ˆ a

minusa

2

3y3dx =

ˆ a

minusa

2

3

(radica2 minus x2

)3dx =

πa4

4(64)

圓之半徑a則面心慣性矩 macrIx prime macrIy prime為

macrIx prime =πa4

4(65)

macrIy prime =πa4

4(66)

JC = macrIx prime + macrIy prime =πa4

2(67)

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複複複合合合面面面積積積之之之慣慣慣性性性矩矩矩Moment of Inertia for Composite Area

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (68)

k =

radicI

m(69)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (70)

= I1 + I2 + + In (71)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 25 54

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複合面積之慣性矩

複合面積之慣性矩(Ixc Iyc)為各局部面積之慣性矩(Ixi Iyi )合

Ixc =i=nsumi=1

Ixi (72)

= Ix1 + Ix2 + + Ixn (73)

Iyc =i=nsumi=1

Iyi (74)

= Iy1 + Iy2 + + Iyn (75)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 26 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

長方形與圓形複合

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

長方形複合

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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面面面積積積之之之慣慣慣性性性積積積Product of Inertia for Area

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 28 54

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面積之慣性積(Product of Inertia for Area)

為了下個章節求解慣性矩之極值(極大或極小值)方便數學運算在此先定義面積之慣性積(Ixy )為

dIxy = xydA (76)

Ixy =

ˆAxydA (77)

面積之慣性積的符號可為正負或零當慣性積之面積相對轉軸對稱時則面積之慣性積和為零

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 29 54

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 30 54

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平行軸定理

極點

新極點

如左圖面積慣性矩之平行軸定理乃以面心(C)為新極點(O)設定新座標軸(xrsquoyrsquo)其中xrsquo軸平行於原座標x軸yrsquo軸平行於原座標y軸取微小面積dA其在新座標之點座標為(xrsquoyrsquo)面積之慣性積(Ixy )為

dIxy = (x prime + dx)(y prime + dy )dA (78)

當以面心為轉軸點時即x prime = y prime = 0ˆAx primedA = x prime

ˆAdA = 0

ˆAy primedA = y prime

ˆAdA = 0 (79)

Ixy =

ˆA

(x prime + dx)(y prime + dy )dA (80)

=

ˆAx primey primedA + dx

ˆAy primedA + dy

ˆAx primedA + dxdy

ˆAdA (81)

= macrIx primey prime + Adxdy

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 31 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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傾傾傾斜斜斜軸軸軸面面面積積積之之之慣慣慣性性性矩矩矩

Moment of Inertia for Area about Inclined Axis

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 33 54

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傾斜軸面積之慣性矩

當轉動軸傾斜θ角時也就是座標軸旋轉θ角而Ix Iy Ixy為已知則新的座標軸(uv)可寫為

u = xcosθ + ysinθ (82)

v = ycosθ minus xsinθ (83)

由u軸上之面積慣性矩Iuv軸上之面積慣性矩Iv及慣性積Iuv得知

dIu = v2dA = (ycosθ minus xsinθ)2dA (84)

dIv = u2dA = (xcosθ + ysinθ)2dA (85)

dIuv = uvdA = (xcosθ + ysinθ)(ycosθ minus xsinθ)dA (86)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 34 54

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傾斜軸面積之慣性矩 I

由x軸上之面積慣性矩Ixy軸上之面積慣性矩Iy及慣性積Ixy得知

Ix =

ˆy2dA Iy =

ˆx2dA Ixy =

ˆxydA (87)

Iu =

ˆ(ycosθ minus xsinθ)2dA (88)

=

ˆ(y2cos2θ minus 2xycosθsinθ + x2sin2θ)dA (89)

= cos2θ

ˆy2dAminus 2cosθsinθ

ˆxydA + sin2θ

ˆx2dA (90)

= Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (91)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 35 54

傾斜軸面積之慣性矩 II

Iv =

ˆ(xcosθ + ysinθ)2dA (92)

=

ˆ(x2cos2θ + 2xycosθsinθ + y2sin2θ)dA (93)

= cos2θ

ˆx2dA + 2cosθsinθ

ˆxydA + sin2θ

ˆy2dA (94)

= Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (95)

Iuv =

ˆ(xcosθ + ysinθ)(ycosθ minus xsinθ)dA (96)

=

ˆ(xycos2θ + y2cosθsinθ minus x2cosθsinθ minus xysin2θ)dA (97)

= (cos2 minus sin2θ)

ˆxydA + cosθsinθ

ˆy2dAminus cosθsinθ

ˆx2dA

= Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 36 54

傾斜軸面積之慣性矩 III

再由

sin2θ + cos2θ = 1 (98)

sin(αplusmn β) = sinαcosβ plusmn cosαsinβ (99)

sin2θ = sin(θ + θ) = 2sinθcosθ (100)

cos(αplusmn β) = cosαcosβ ∓ sinαsinβ (101)

cos2θ = cos(θ + θ) = cos2θ minus sin2θ (102)

= 2cos2θ minus 1 = 1minus 2sin2θ (103)

所以

Iu = Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (104)

= Ix(cos2θ + 1

2) + Iy (

1minus cos2θ

2)minus Ixy sin2θ (105)

=Ix + Iy

2+

Ix minus Iy2

cos2θ minus Ixy sin2θ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 37 54

傾斜軸面積之慣性矩 IV

Iv = Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (106)

= Iy (cos2θ + 1

2) + Ix(

1minus cos2θ

2) + Ixy sin2θ (107)

=Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (108)

Iuv = Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ (109)

= Ixycos2θ +1

2Ixsin2θ minus 1

2Iy sin2θ (110)

=Ix minus Iy

2sin2θ + Ixycos2θ (111)

所以

JO = Iu + Iv = Ix + Iy (112)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 38 54

主主主軸軸軸慣慣慣性性性矩矩矩Principal Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 39 54

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主軸慣性矩 I

所謂主軸慣性矩(Principal Moment ofInertia)乃定義發生在慣性矩的極值(極大或極小值)上也就是慣性矩之一次微分式為零或斜率為零

dIu

dθ=

d(Ix+Iy

2+

IxminusIy2

cos2θ minus Ixy sin2θ)

dθ(113)

= minus2Ix minus Iy

2sin2θ minus 2Ixy cos2θ (114)

= 0 (115)

其中dsin2θ = cos2θ dcos2θ = minussin2θ當慣性矩之一次微分式為零時傾斜軸與主軸之夾角θ = θp

minus2Ix minus Iy

2sin2θp minus 2Ixycos2θp = 0 (116)

tan2θ =sin2θ

cos2θ=

minusIxy(Ix minus Iy )2

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 40 54

主軸慣性矩 II

cos2θp =IxminusIy2radic

(IxminusIy2 )2 + I 2xy

sin2θp =minusIxyradic

(IxminusIy2 )2 + I 2xy

(117)

(Iu)max =Ix + Iy

2+

Ix minus Iy2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

minus IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2+

radic(Ix minus Iy

2)2 + I 2xy (118)

(Iv )min =Ix + Iy

2minus Ix minus Iy

2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

+ IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2minusradic

(Ix minus Iy

2)2 + I 2xy (119)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 41 54

主軸慣性矩 III

由tan2θp = tan(π + 2θp)可得到θp1與θp2二個解即2θp2 = π + 2θp1也就是θp2 minus θp1 = π

2

tan2θp = tan(π + 2θp) (120)

2θp2 = π + 2θp1 (121)

θp2 minus θp1 =π

2(122)

所以

Imaxmin

=Ix + Iy

2plusmnradic

(Ix minus Iy

2)2 + I 2xy (123)

Iuv =Ix minus Iy

2

minusIxyradic(IxminusIy2 )2 + I 2xy

+ Ixy

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

(124)

= 0

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 42 54

課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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慣慣慣性性性矩矩矩之之之莫莫莫爾爾爾圓圓圓Mohrrsquos Circle for Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 44 54

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慣性矩之莫爾圓

由於Iu =

Ix + Iy2

+Ix minus Iy

2cos2θ minus Ixy sin2θ (125)

Iv =Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (126)

Iuv =Ix minus Iy

2sin2θ + Ixycos2θ (127)

將式(125)與式(127)各自分別平方並相加可得

(Iu minusIx + Iy

2)2 + I 2uv = (

Ix minus Iy2

)2 + I 2xy (128)

其中Ix Iy Ixy均為常數式(128)可進一步寫成

(Iu minus a)2 + I 2uv = R2 (129)

其中a =Ix+Iy2 R =

radic(IxminusIy2 )2 + I 2xycos2θp1 =

aRsin2θp1 =

minusIxyR 2θp1為OA轉向I軸方向即2θp1為

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 45 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 46 54

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解題技巧

首先確定轉動系統的Ix Iy Ixy接著算出莫爾圓之圓心座標a及半徑R並畫出莫爾圓再由莫爾圓上點出A(Ix Ixy )並由OA轉向I軸找出兩倍夾角2θp1及2θp2其中θp1及θp2為主軸(Principle Axis)與Ix軸的夾角由於主軸(Principle Axis)為極值之軸Imax = Iu及Imin = Iv決定θp1及θp2何者為Imax = Iu與Ix軸的夾角之方法乃比較Ix及Iy之大小Imax = Iu會靠近Ix及Iy之較大者也就是Imax = Iu與Ix及Iy之較大者的夾角比較小

a =Ix + Iy

2(130)

R =

radic(Ix minus Iy

2)2 + I 2xy (131)

2θp2 = π + 2θp1 θp2 minus θp1 =π

2(132)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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質質質量量量慣慣慣性性性矩矩矩Mass Moment of Inertia

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質量慣性矩

τ = Iα (133)

其中τ為旋轉扭矩I為質量慣性矩α為旋轉角速度

I =

ˆmr2dm (134)

=

ˆVr2ρdV (135)

= ρ

ˆVr2dV (136)

其中r為旋轉半徑或轉軸到旋轉體的距離

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

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平行軸定理

當通過物體質心(G)並在一轉軸上的質量慣性矩(IG )為已知時可透過座標軸轉換計算物體之質量慣性矩(I)

I =

ˆmr2dm (137)

=

ˆm

[(d + x prime)2 + y prime2]dm (138)

=

ˆm

(x prime2 + y prime2)dm + 2d

ˆmx primedm + d2

ˆmdm (139)

I = IG + md2 (140)

其中IG為在zrsquo軸上通過質心的質量慣性矩m為質量d為通過質心之軸zrsquo並平行原軸z之距離

acutem x primedm = x

acutem dm = 0 since x = 0

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (141)

k =

radicI

m(142)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (143)

= I1 + I2 + + In (144)

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課課課堂堂堂練練練習習習

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

兩圓盤複合體

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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結結結論論論

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結論

轉動慣量或慣性矩為物體對旋轉運動的慣性而所謂慣性乃物體持續維持不變的行為也就是定義慣性矩可表達物體對本身旋轉運動的維持或持續性好壞更簡單的說慣性矩越大之物體其維持旋轉運動趨勢的能力越大例如惰輪儲能裝置惰輪之慣性矩越大當其轉動以後其維持旋轉運動趨勢的能力越大

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 55 54

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  • 轉動慣量或慣性矩(Moment of Inertia)
  • 面積慣性矩(Moment of Inertia for Area)
    • 面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)
    • 面積慣性矩之迴轉半徑(Radius of Gyration for Area)
      • 複合面積之慣性矩(Moment of Inertia for Composite Area)
      • 面積之慣性積(Product of Inertia for Area)
        • 平行軸定理(Parallel-Axis Theorem)
          • 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)
            • 主軸慣性矩(Principal Moment of Inertia)
              • 慣性矩之莫爾圓(Mohrs Circle for Moment of Inertia)
              • 質量慣性矩(Mass Moment of Inertia)
                • 平行軸定理(Parallel-Axis Theorem)
Page 17: 轉動慣量 Moments of Inertia - cct.me.ntut.edu.twcct.me.ntut.edu.tw/ccteducation/chchting/aiahtm/statics/lecture... · 轉動慣量或慣性矩 I. 轉動慣量也稱慣性矩(Moment

課課課堂堂堂練練練習習習

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解題技巧 I

極點

新極點

繪圖並標出面積(A)之面心座標位置(C)接著以面心座標位置(C)為原點畫出新座標軸(xrsquoyrsquo)其中xrsquo軸平行於原(舊)座標x軸yrsquo軸平行於原(舊)座標y軸並寫出新舊座標軸的平移量dx與dy其中dx為yrsquo軸與y軸的平移量dy為xrsquo軸與x軸的平移量取微小面積dA其在原(舊)座標之點座標為(xy)在新座標之點座標為(xrsquoyrsquo)帶入x軸上之面積慣性矩Ix及y軸上之面積慣性矩Iy還有xrsquo軸上之面積慣性矩 macrIx prime及y軸上之面積慣性矩 macrIy prime可如下換算

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解題技巧 II

Ix =

ˆAy2dA =

ˆAy prime2dA + 2dy

ˆAy primedA + d2

y

ˆAdA = macrIx prime + Ad2

y

macrIx prime =

ˆAy prime2dA (39)

Iy =

ˆAx2dA =

ˆAx prime2dA + 2dx

ˆAx primedA + d2

x

ˆAdA = macrIy prime + Ad2

x

macrIy prime =

ˆAx prime2dA (40)

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解題技巧 III

另外微小面積dA的面積慣性矩dIx dIy也可寫成

dIx = d macrIx prime + dAd2y (41)

Ix =

ˆdIx =

ˆd macrIx prime +

ˆdAd2

y (42)

dIy = d macrIy prime + dAd2x (43)

Iy =

ˆdIy =

ˆd macrIy prime +

ˆdAd2

x (44)

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

面心慣性矩 慣性矩 極慣性矩

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Example 10-1 解答

dA = bdy (45)

Ixb =

ˆAy2dA (46)

=

ˆ h

0y2bdy (47)

=1

3by3|h0 (48)

=1

3bh3 (49)

長方形之邊長b與h則面心慣性矩 macrIx prime macrIy prime與長方形之邊軸慣性矩Ix Iy為

macrIx prime =1

12bh3 (50)

macrIy prime =1

12hb3 (51)

Ix =1

3bh3 (52)

Iy =1

3hb3 (53)

JC = macrIx prime + macrIy prime (54)

JO = Ix + Iy (55)

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 21 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 21 54

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Example 10-2 解答

取微小面積dA=ydx之長方形其邊長b=dx與h=ydy = y2所以

y2 = 400x (56)

y =radic

400x (57)

d macrIx prime = d(1

12bh3) =

1

12y3dx (58)

dIx = d macrIx prime + dAd2y =

1

12y3dx + ydx

(y2

)2=

1

3y3dx (59)

Ix =

ˆ 100

0

1

3y3dx =

ˆ 100

0

1

3

(radic400x

)3dx (60)

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 22 54

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Example 10-3 解答

取微小面積dA=2ydx之長方形其邊長b=dx與h=2y所以

x2 + y2 = a2 (61)

y =radic

a2 minus x2 (62)

d macrIx prime = d(1

12bh3) =

1

12dx(2y)3 =

2

3y3dx (63)

macrIx prime =

ˆ a

minusa

2

3y3dx =

ˆ a

minusa

2

3

(radica2 minus x2

)3dx =

πa4

4(64)

圓之半徑a則面心慣性矩 macrIx prime macrIy prime為

macrIx prime =πa4

4(65)

macrIy prime =πa4

4(66)

JC = macrIx prime + macrIy prime =πa4

2(67)

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複複複合合合面面面積積積之之之慣慣慣性性性矩矩矩Moment of Inertia for Composite Area

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (68)

k =

radicI

m(69)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (70)

= I1 + I2 + + In (71)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 25 54

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複合面積之慣性矩

複合面積之慣性矩(Ixc Iyc)為各局部面積之慣性矩(Ixi Iyi )合

Ixc =i=nsumi=1

Ixi (72)

= Ix1 + Ix2 + + Ixn (73)

Iyc =i=nsumi=1

Iyi (74)

= Iy1 + Iy2 + + Iyn (75)

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課課課堂堂堂練練練習習習

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

長方形與圓形複合

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

長方形複合

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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面面面積積積之之之慣慣慣性性性積積積Product of Inertia for Area

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 28 54

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面積之慣性積(Product of Inertia for Area)

為了下個章節求解慣性矩之極值(極大或極小值)方便數學運算在此先定義面積之慣性積(Ixy )為

dIxy = xydA (76)

Ixy =

ˆAxydA (77)

面積之慣性積的符號可為正負或零當慣性積之面積相對轉軸對稱時則面積之慣性積和為零

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 29 54

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 30 54

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平行軸定理

極點

新極點

如左圖面積慣性矩之平行軸定理乃以面心(C)為新極點(O)設定新座標軸(xrsquoyrsquo)其中xrsquo軸平行於原座標x軸yrsquo軸平行於原座標y軸取微小面積dA其在新座標之點座標為(xrsquoyrsquo)面積之慣性積(Ixy )為

dIxy = (x prime + dx)(y prime + dy )dA (78)

當以面心為轉軸點時即x prime = y prime = 0ˆAx primedA = x prime

ˆAdA = 0

ˆAy primedA = y prime

ˆAdA = 0 (79)

Ixy =

ˆA

(x prime + dx)(y prime + dy )dA (80)

=

ˆAx primey primedA + dx

ˆAy primedA + dy

ˆAx primedA + dxdy

ˆAdA (81)

= macrIx primey prime + Adxdy

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 31 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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傾傾傾斜斜斜軸軸軸面面面積積積之之之慣慣慣性性性矩矩矩

Moment of Inertia for Area about Inclined Axis

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 33 54

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傾斜軸面積之慣性矩

當轉動軸傾斜θ角時也就是座標軸旋轉θ角而Ix Iy Ixy為已知則新的座標軸(uv)可寫為

u = xcosθ + ysinθ (82)

v = ycosθ minus xsinθ (83)

由u軸上之面積慣性矩Iuv軸上之面積慣性矩Iv及慣性積Iuv得知

dIu = v2dA = (ycosθ minus xsinθ)2dA (84)

dIv = u2dA = (xcosθ + ysinθ)2dA (85)

dIuv = uvdA = (xcosθ + ysinθ)(ycosθ minus xsinθ)dA (86)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 34 54

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傾斜軸面積之慣性矩 I

由x軸上之面積慣性矩Ixy軸上之面積慣性矩Iy及慣性積Ixy得知

Ix =

ˆy2dA Iy =

ˆx2dA Ixy =

ˆxydA (87)

Iu =

ˆ(ycosθ minus xsinθ)2dA (88)

=

ˆ(y2cos2θ minus 2xycosθsinθ + x2sin2θ)dA (89)

= cos2θ

ˆy2dAminus 2cosθsinθ

ˆxydA + sin2θ

ˆx2dA (90)

= Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (91)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 35 54

傾斜軸面積之慣性矩 II

Iv =

ˆ(xcosθ + ysinθ)2dA (92)

=

ˆ(x2cos2θ + 2xycosθsinθ + y2sin2θ)dA (93)

= cos2θ

ˆx2dA + 2cosθsinθ

ˆxydA + sin2θ

ˆy2dA (94)

= Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (95)

Iuv =

ˆ(xcosθ + ysinθ)(ycosθ minus xsinθ)dA (96)

=

ˆ(xycos2θ + y2cosθsinθ minus x2cosθsinθ minus xysin2θ)dA (97)

= (cos2 minus sin2θ)

ˆxydA + cosθsinθ

ˆy2dAminus cosθsinθ

ˆx2dA

= Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 36 54

傾斜軸面積之慣性矩 III

再由

sin2θ + cos2θ = 1 (98)

sin(αplusmn β) = sinαcosβ plusmn cosαsinβ (99)

sin2θ = sin(θ + θ) = 2sinθcosθ (100)

cos(αplusmn β) = cosαcosβ ∓ sinαsinβ (101)

cos2θ = cos(θ + θ) = cos2θ minus sin2θ (102)

= 2cos2θ minus 1 = 1minus 2sin2θ (103)

所以

Iu = Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (104)

= Ix(cos2θ + 1

2) + Iy (

1minus cos2θ

2)minus Ixy sin2θ (105)

=Ix + Iy

2+

Ix minus Iy2

cos2θ minus Ixy sin2θ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 37 54

傾斜軸面積之慣性矩 IV

Iv = Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (106)

= Iy (cos2θ + 1

2) + Ix(

1minus cos2θ

2) + Ixy sin2θ (107)

=Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (108)

Iuv = Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ (109)

= Ixycos2θ +1

2Ixsin2θ minus 1

2Iy sin2θ (110)

=Ix minus Iy

2sin2θ + Ixycos2θ (111)

所以

JO = Iu + Iv = Ix + Iy (112)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 38 54

主主主軸軸軸慣慣慣性性性矩矩矩Principal Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 39 54

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主軸慣性矩 I

所謂主軸慣性矩(Principal Moment ofInertia)乃定義發生在慣性矩的極值(極大或極小值)上也就是慣性矩之一次微分式為零或斜率為零

dIu

dθ=

d(Ix+Iy

2+

IxminusIy2

cos2θ minus Ixy sin2θ)

dθ(113)

= minus2Ix minus Iy

2sin2θ minus 2Ixy cos2θ (114)

= 0 (115)

其中dsin2θ = cos2θ dcos2θ = minussin2θ當慣性矩之一次微分式為零時傾斜軸與主軸之夾角θ = θp

minus2Ix minus Iy

2sin2θp minus 2Ixycos2θp = 0 (116)

tan2θ =sin2θ

cos2θ=

minusIxy(Ix minus Iy )2

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 40 54

主軸慣性矩 II

cos2θp =IxminusIy2radic

(IxminusIy2 )2 + I 2xy

sin2θp =minusIxyradic

(IxminusIy2 )2 + I 2xy

(117)

(Iu)max =Ix + Iy

2+

Ix minus Iy2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

minus IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2+

radic(Ix minus Iy

2)2 + I 2xy (118)

(Iv )min =Ix + Iy

2minus Ix minus Iy

2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

+ IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2minusradic

(Ix minus Iy

2)2 + I 2xy (119)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 41 54

主軸慣性矩 III

由tan2θp = tan(π + 2θp)可得到θp1與θp2二個解即2θp2 = π + 2θp1也就是θp2 minus θp1 = π

2

tan2θp = tan(π + 2θp) (120)

2θp2 = π + 2θp1 (121)

θp2 minus θp1 =π

2(122)

所以

Imaxmin

=Ix + Iy

2plusmnradic

(Ix minus Iy

2)2 + I 2xy (123)

Iuv =Ix minus Iy

2

minusIxyradic(IxminusIy2 )2 + I 2xy

+ Ixy

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

(124)

= 0

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 42 54

課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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慣慣慣性性性矩矩矩之之之莫莫莫爾爾爾圓圓圓Mohrrsquos Circle for Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 44 54

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慣性矩之莫爾圓

由於Iu =

Ix + Iy2

+Ix minus Iy

2cos2θ minus Ixy sin2θ (125)

Iv =Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (126)

Iuv =Ix minus Iy

2sin2θ + Ixycos2θ (127)

將式(125)與式(127)各自分別平方並相加可得

(Iu minusIx + Iy

2)2 + I 2uv = (

Ix minus Iy2

)2 + I 2xy (128)

其中Ix Iy Ixy均為常數式(128)可進一步寫成

(Iu minus a)2 + I 2uv = R2 (129)

其中a =Ix+Iy2 R =

radic(IxminusIy2 )2 + I 2xycos2θp1 =

aRsin2θp1 =

minusIxyR 2θp1為OA轉向I軸方向即2θp1為

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 45 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 46 54

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解題技巧

首先確定轉動系統的Ix Iy Ixy接著算出莫爾圓之圓心座標a及半徑R並畫出莫爾圓再由莫爾圓上點出A(Ix Ixy )並由OA轉向I軸找出兩倍夾角2θp1及2θp2其中θp1及θp2為主軸(Principle Axis)與Ix軸的夾角由於主軸(Principle Axis)為極值之軸Imax = Iu及Imin = Iv決定θp1及θp2何者為Imax = Iu與Ix軸的夾角之方法乃比較Ix及Iy之大小Imax = Iu會靠近Ix及Iy之較大者也就是Imax = Iu與Ix及Iy之較大者的夾角比較小

a =Ix + Iy

2(130)

R =

radic(Ix minus Iy

2)2 + I 2xy (131)

2θp2 = π + 2θp1 θp2 minus θp1 =π

2(132)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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質質質量量量慣慣慣性性性矩矩矩Mass Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 48 54

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質量慣性矩

τ = Iα (133)

其中τ為旋轉扭矩I為質量慣性矩α為旋轉角速度

I =

ˆmr2dm (134)

=

ˆVr2ρdV (135)

= ρ

ˆVr2dV (136)

其中r為旋轉半徑或轉軸到旋轉體的距離

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 49 54

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 50 54

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平行軸定理

當通過物體質心(G)並在一轉軸上的質量慣性矩(IG )為已知時可透過座標軸轉換計算物體之質量慣性矩(I)

I =

ˆmr2dm (137)

=

ˆm

[(d + x prime)2 + y prime2]dm (138)

=

ˆm

(x prime2 + y prime2)dm + 2d

ˆmx primedm + d2

ˆmdm (139)

I = IG + md2 (140)

其中IG為在zrsquo軸上通過質心的質量慣性矩m為質量d為通過質心之軸zrsquo並平行原軸z之距離

acutem x primedm = x

acutem dm = 0 since x = 0

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (141)

k =

radicI

m(142)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (143)

= I1 + I2 + + In (144)

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課課課堂堂堂練練練習習習

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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兩圓盤複合體

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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結結結論論論

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結論

轉動慣量或慣性矩為物體對旋轉運動的慣性而所謂慣性乃物體持續維持不變的行為也就是定義慣性矩可表達物體對本身旋轉運動的維持或持續性好壞更簡單的說慣性矩越大之物體其維持旋轉運動趨勢的能力越大例如惰輪儲能裝置惰輪之慣性矩越大當其轉動以後其維持旋轉運動趨勢的能力越大

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  • 轉動慣量或慣性矩(Moment of Inertia)
  • 面積慣性矩(Moment of Inertia for Area)
    • 面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)
    • 面積慣性矩之迴轉半徑(Radius of Gyration for Area)
      • 複合面積之慣性矩(Moment of Inertia for Composite Area)
      • 面積之慣性積(Product of Inertia for Area)
        • 平行軸定理(Parallel-Axis Theorem)
          • 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)
            • 主軸慣性矩(Principal Moment of Inertia)
              • 慣性矩之莫爾圓(Mohrs Circle for Moment of Inertia)
              • 質量慣性矩(Mass Moment of Inertia)
                • 平行軸定理(Parallel-Axis Theorem)
Page 18: 轉動慣量 Moments of Inertia - cct.me.ntut.edu.twcct.me.ntut.edu.tw/ccteducation/chchting/aiahtm/statics/lecture... · 轉動慣量或慣性矩 I. 轉動慣量也稱慣性矩(Moment

解題技巧 I

極點

新極點

繪圖並標出面積(A)之面心座標位置(C)接著以面心座標位置(C)為原點畫出新座標軸(xrsquoyrsquo)其中xrsquo軸平行於原(舊)座標x軸yrsquo軸平行於原(舊)座標y軸並寫出新舊座標軸的平移量dx與dy其中dx為yrsquo軸與y軸的平移量dy為xrsquo軸與x軸的平移量取微小面積dA其在原(舊)座標之點座標為(xy)在新座標之點座標為(xrsquoyrsquo)帶入x軸上之面積慣性矩Ix及y軸上之面積慣性矩Iy還有xrsquo軸上之面積慣性矩 macrIx prime及y軸上之面積慣性矩 macrIy prime可如下換算

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解題技巧 II

Ix =

ˆAy2dA =

ˆAy prime2dA + 2dy

ˆAy primedA + d2

y

ˆAdA = macrIx prime + Ad2

y

macrIx prime =

ˆAy prime2dA (39)

Iy =

ˆAx2dA =

ˆAx prime2dA + 2dx

ˆAx primedA + d2

x

ˆAdA = macrIy prime + Ad2

x

macrIy prime =

ˆAx prime2dA (40)

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解題技巧 III

另外微小面積dA的面積慣性矩dIx dIy也可寫成

dIx = d macrIx prime + dAd2y (41)

Ix =

ˆdIx =

ˆd macrIx prime +

ˆdAd2

y (42)

dIy = d macrIy prime + dAd2x (43)

Iy =

ˆdIy =

ˆd macrIy prime +

ˆdAd2

x (44)

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

面心慣性矩 慣性矩 極慣性矩

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Example 10-1 解答

dA = bdy (45)

Ixb =

ˆAy2dA (46)

=

ˆ h

0y2bdy (47)

=1

3by3|h0 (48)

=1

3bh3 (49)

長方形之邊長b與h則面心慣性矩 macrIx prime macrIy prime與長方形之邊軸慣性矩Ix Iy為

macrIx prime =1

12bh3 (50)

macrIy prime =1

12hb3 (51)

Ix =1

3bh3 (52)

Iy =1

3hb3 (53)

JC = macrIx prime + macrIy prime (54)

JO = Ix + Iy (55)

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Example 10-2 解答

取微小面積dA=ydx之長方形其邊長b=dx與h=ydy = y2所以

y2 = 400x (56)

y =radic

400x (57)

d macrIx prime = d(1

12bh3) =

1

12y3dx (58)

dIx = d macrIx prime + dAd2y =

1

12y3dx + ydx

(y2

)2=

1

3y3dx (59)

Ix =

ˆ 100

0

1

3y3dx =

ˆ 100

0

1

3

(radic400x

)3dx (60)

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Example 10-3 解答

取微小面積dA=2ydx之長方形其邊長b=dx與h=2y所以

x2 + y2 = a2 (61)

y =radic

a2 minus x2 (62)

d macrIx prime = d(1

12bh3) =

1

12dx(2y)3 =

2

3y3dx (63)

macrIx prime =

ˆ a

minusa

2

3y3dx =

ˆ a

minusa

2

3

(radica2 minus x2

)3dx =

πa4

4(64)

圓之半徑a則面心慣性矩 macrIx prime macrIy prime為

macrIx prime =πa4

4(65)

macrIy prime =πa4

4(66)

JC = macrIx prime + macrIy prime =πa4

2(67)

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複複複合合合面面面積積積之之之慣慣慣性性性矩矩矩Moment of Inertia for Composite Area

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (68)

k =

radicI

m(69)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (70)

= I1 + I2 + + In (71)

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複合面積之慣性矩

複合面積之慣性矩(Ixc Iyc)為各局部面積之慣性矩(Ixi Iyi )合

Ixc =i=nsumi=1

Ixi (72)

= Ix1 + Ix2 + + Ixn (73)

Iyc =i=nsumi=1

Iyi (74)

= Iy1 + Iy2 + + Iyn (75)

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課課課堂堂堂練練練習習習

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

長方形與圓形複合

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長方形複合

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面面面積積積之之之慣慣慣性性性積積積Product of Inertia for Area

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面積之慣性積(Product of Inertia for Area)

為了下個章節求解慣性矩之極值(極大或極小值)方便數學運算在此先定義面積之慣性積(Ixy )為

dIxy = xydA (76)

Ixy =

ˆAxydA (77)

面積之慣性積的符號可為正負或零當慣性積之面積相對轉軸對稱時則面積之慣性積和為零

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

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平行軸定理

極點

新極點

如左圖面積慣性矩之平行軸定理乃以面心(C)為新極點(O)設定新座標軸(xrsquoyrsquo)其中xrsquo軸平行於原座標x軸yrsquo軸平行於原座標y軸取微小面積dA其在新座標之點座標為(xrsquoyrsquo)面積之慣性積(Ixy )為

dIxy = (x prime + dx)(y prime + dy )dA (78)

當以面心為轉軸點時即x prime = y prime = 0ˆAx primedA = x prime

ˆAdA = 0

ˆAy primedA = y prime

ˆAdA = 0 (79)

Ixy =

ˆA

(x prime + dx)(y prime + dy )dA (80)

=

ˆAx primey primedA + dx

ˆAy primedA + dy

ˆAx primedA + dxdy

ˆAdA (81)

= macrIx primey prime + Adxdy

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 31 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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傾傾傾斜斜斜軸軸軸面面面積積積之之之慣慣慣性性性矩矩矩

Moment of Inertia for Area about Inclined Axis

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 33 54

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傾斜軸面積之慣性矩

當轉動軸傾斜θ角時也就是座標軸旋轉θ角而Ix Iy Ixy為已知則新的座標軸(uv)可寫為

u = xcosθ + ysinθ (82)

v = ycosθ minus xsinθ (83)

由u軸上之面積慣性矩Iuv軸上之面積慣性矩Iv及慣性積Iuv得知

dIu = v2dA = (ycosθ minus xsinθ)2dA (84)

dIv = u2dA = (xcosθ + ysinθ)2dA (85)

dIuv = uvdA = (xcosθ + ysinθ)(ycosθ minus xsinθ)dA (86)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 34 54

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傾斜軸面積之慣性矩 I

由x軸上之面積慣性矩Ixy軸上之面積慣性矩Iy及慣性積Ixy得知

Ix =

ˆy2dA Iy =

ˆx2dA Ixy =

ˆxydA (87)

Iu =

ˆ(ycosθ minus xsinθ)2dA (88)

=

ˆ(y2cos2θ minus 2xycosθsinθ + x2sin2θ)dA (89)

= cos2θ

ˆy2dAminus 2cosθsinθ

ˆxydA + sin2θ

ˆx2dA (90)

= Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (91)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 35 54

傾斜軸面積之慣性矩 II

Iv =

ˆ(xcosθ + ysinθ)2dA (92)

=

ˆ(x2cos2θ + 2xycosθsinθ + y2sin2θ)dA (93)

= cos2θ

ˆx2dA + 2cosθsinθ

ˆxydA + sin2θ

ˆy2dA (94)

= Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (95)

Iuv =

ˆ(xcosθ + ysinθ)(ycosθ minus xsinθ)dA (96)

=

ˆ(xycos2θ + y2cosθsinθ minus x2cosθsinθ minus xysin2θ)dA (97)

= (cos2 minus sin2θ)

ˆxydA + cosθsinθ

ˆy2dAminus cosθsinθ

ˆx2dA

= Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 36 54

傾斜軸面積之慣性矩 III

再由

sin2θ + cos2θ = 1 (98)

sin(αplusmn β) = sinαcosβ plusmn cosαsinβ (99)

sin2θ = sin(θ + θ) = 2sinθcosθ (100)

cos(αplusmn β) = cosαcosβ ∓ sinαsinβ (101)

cos2θ = cos(θ + θ) = cos2θ minus sin2θ (102)

= 2cos2θ minus 1 = 1minus 2sin2θ (103)

所以

Iu = Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (104)

= Ix(cos2θ + 1

2) + Iy (

1minus cos2θ

2)minus Ixy sin2θ (105)

=Ix + Iy

2+

Ix minus Iy2

cos2θ minus Ixy sin2θ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 37 54

傾斜軸面積之慣性矩 IV

Iv = Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (106)

= Iy (cos2θ + 1

2) + Ix(

1minus cos2θ

2) + Ixy sin2θ (107)

=Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (108)

Iuv = Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ (109)

= Ixycos2θ +1

2Ixsin2θ minus 1

2Iy sin2θ (110)

=Ix minus Iy

2sin2θ + Ixycos2θ (111)

所以

JO = Iu + Iv = Ix + Iy (112)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 38 54

主主主軸軸軸慣慣慣性性性矩矩矩Principal Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 39 54

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主軸慣性矩 I

所謂主軸慣性矩(Principal Moment ofInertia)乃定義發生在慣性矩的極值(極大或極小值)上也就是慣性矩之一次微分式為零或斜率為零

dIu

dθ=

d(Ix+Iy

2+

IxminusIy2

cos2θ minus Ixy sin2θ)

dθ(113)

= minus2Ix minus Iy

2sin2θ minus 2Ixy cos2θ (114)

= 0 (115)

其中dsin2θ = cos2θ dcos2θ = minussin2θ當慣性矩之一次微分式為零時傾斜軸與主軸之夾角θ = θp

minus2Ix minus Iy

2sin2θp minus 2Ixycos2θp = 0 (116)

tan2θ =sin2θ

cos2θ=

minusIxy(Ix minus Iy )2

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 40 54

主軸慣性矩 II

cos2θp =IxminusIy2radic

(IxminusIy2 )2 + I 2xy

sin2θp =minusIxyradic

(IxminusIy2 )2 + I 2xy

(117)

(Iu)max =Ix + Iy

2+

Ix minus Iy2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

minus IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2+

radic(Ix minus Iy

2)2 + I 2xy (118)

(Iv )min =Ix + Iy

2minus Ix minus Iy

2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

+ IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2minusradic

(Ix minus Iy

2)2 + I 2xy (119)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 41 54

主軸慣性矩 III

由tan2θp = tan(π + 2θp)可得到θp1與θp2二個解即2θp2 = π + 2θp1也就是θp2 minus θp1 = π

2

tan2θp = tan(π + 2θp) (120)

2θp2 = π + 2θp1 (121)

θp2 minus θp1 =π

2(122)

所以

Imaxmin

=Ix + Iy

2plusmnradic

(Ix minus Iy

2)2 + I 2xy (123)

Iuv =Ix minus Iy

2

minusIxyradic(IxminusIy2 )2 + I 2xy

+ Ixy

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

(124)

= 0

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 42 54

課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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慣慣慣性性性矩矩矩之之之莫莫莫爾爾爾圓圓圓Mohrrsquos Circle for Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 44 54

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慣性矩之莫爾圓

由於Iu =

Ix + Iy2

+Ix minus Iy

2cos2θ minus Ixy sin2θ (125)

Iv =Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (126)

Iuv =Ix minus Iy

2sin2θ + Ixycos2θ (127)

將式(125)與式(127)各自分別平方並相加可得

(Iu minusIx + Iy

2)2 + I 2uv = (

Ix minus Iy2

)2 + I 2xy (128)

其中Ix Iy Ixy均為常數式(128)可進一步寫成

(Iu minus a)2 + I 2uv = R2 (129)

其中a =Ix+Iy2 R =

radic(IxminusIy2 )2 + I 2xycos2θp1 =

aRsin2θp1 =

minusIxyR 2θp1為OA轉向I軸方向即2θp1為

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 45 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 46 54

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解題技巧

首先確定轉動系統的Ix Iy Ixy接著算出莫爾圓之圓心座標a及半徑R並畫出莫爾圓再由莫爾圓上點出A(Ix Ixy )並由OA轉向I軸找出兩倍夾角2θp1及2θp2其中θp1及θp2為主軸(Principle Axis)與Ix軸的夾角由於主軸(Principle Axis)為極值之軸Imax = Iu及Imin = Iv決定θp1及θp2何者為Imax = Iu與Ix軸的夾角之方法乃比較Ix及Iy之大小Imax = Iu會靠近Ix及Iy之較大者也就是Imax = Iu與Ix及Iy之較大者的夾角比較小

a =Ix + Iy

2(130)

R =

radic(Ix minus Iy

2)2 + I 2xy (131)

2θp2 = π + 2θp1 θp2 minus θp1 =π

2(132)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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質質質量量量慣慣慣性性性矩矩矩Mass Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 48 54

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質量慣性矩

τ = Iα (133)

其中τ為旋轉扭矩I為質量慣性矩α為旋轉角速度

I =

ˆmr2dm (134)

=

ˆVr2ρdV (135)

= ρ

ˆVr2dV (136)

其中r為旋轉半徑或轉軸到旋轉體的距離

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 49 54

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 50 54

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平行軸定理

當通過物體質心(G)並在一轉軸上的質量慣性矩(IG )為已知時可透過座標軸轉換計算物體之質量慣性矩(I)

I =

ˆmr2dm (137)

=

ˆm

[(d + x prime)2 + y prime2]dm (138)

=

ˆm

(x prime2 + y prime2)dm + 2d

ˆmx primedm + d2

ˆmdm (139)

I = IG + md2 (140)

其中IG為在zrsquo軸上通過質心的質量慣性矩m為質量d為通過質心之軸zrsquo並平行原軸z之距離

acutem x primedm = x

acutem dm = 0 since x = 0

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 51 54

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (141)

k =

radicI

m(142)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (143)

= I1 + I2 + + In (144)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 52 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

兩圓盤複合體

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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結結結論論論

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 54 54

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結論

轉動慣量或慣性矩為物體對旋轉運動的慣性而所謂慣性乃物體持續維持不變的行為也就是定義慣性矩可表達物體對本身旋轉運動的維持或持續性好壞更簡單的說慣性矩越大之物體其維持旋轉運動趨勢的能力越大例如惰輪儲能裝置惰輪之慣性矩越大當其轉動以後其維持旋轉運動趨勢的能力越大

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 55 54

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  • 轉動慣量或慣性矩(Moment of Inertia)
  • 面積慣性矩(Moment of Inertia for Area)
    • 面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)
    • 面積慣性矩之迴轉半徑(Radius of Gyration for Area)
      • 複合面積之慣性矩(Moment of Inertia for Composite Area)
      • 面積之慣性積(Product of Inertia for Area)
        • 平行軸定理(Parallel-Axis Theorem)
          • 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)
            • 主軸慣性矩(Principal Moment of Inertia)
              • 慣性矩之莫爾圓(Mohrs Circle for Moment of Inertia)
              • 質量慣性矩(Mass Moment of Inertia)
                • 平行軸定理(Parallel-Axis Theorem)
Page 19: 轉動慣量 Moments of Inertia - cct.me.ntut.edu.twcct.me.ntut.edu.tw/ccteducation/chchting/aiahtm/statics/lecture... · 轉動慣量或慣性矩 I. 轉動慣量也稱慣性矩(Moment

解題技巧 II

Ix =

ˆAy2dA =

ˆAy prime2dA + 2dy

ˆAy primedA + d2

y

ˆAdA = macrIx prime + Ad2

y

macrIx prime =

ˆAy prime2dA (39)

Iy =

ˆAx2dA =

ˆAx prime2dA + 2dx

ˆAx primedA + d2

x

ˆAdA = macrIy prime + Ad2

x

macrIy prime =

ˆAx prime2dA (40)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 19 54

解題技巧 III

另外微小面積dA的面積慣性矩dIx dIy也可寫成

dIx = d macrIx prime + dAd2y (41)

Ix =

ˆdIx =

ˆd macrIx prime +

ˆdAd2

y (42)

dIy = d macrIy prime + dAd2x (43)

Iy =

ˆdIy =

ˆd macrIy prime +

ˆdAd2

x (44)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 20 54

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

面心慣性矩 慣性矩 極慣性矩

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 20 54

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 20 54

Example 10-1 解答

dA = bdy (45)

Ixb =

ˆAy2dA (46)

=

ˆ h

0y2bdy (47)

=1

3by3|h0 (48)

=1

3bh3 (49)

長方形之邊長b與h則面心慣性矩 macrIx prime macrIy prime與長方形之邊軸慣性矩Ix Iy為

macrIx prime =1

12bh3 (50)

macrIy prime =1

12hb3 (51)

Ix =1

3bh3 (52)

Iy =1

3hb3 (53)

JC = macrIx prime + macrIy prime (54)

JO = Ix + Iy (55)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 21 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 21 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 21 54

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Example 10-2 解答

取微小面積dA=ydx之長方形其邊長b=dx與h=ydy = y2所以

y2 = 400x (56)

y =radic

400x (57)

d macrIx prime = d(1

12bh3) =

1

12y3dx (58)

dIx = d macrIx prime + dAd2y =

1

12y3dx + ydx

(y2

)2=

1

3y3dx (59)

Ix =

ˆ 100

0

1

3y3dx =

ˆ 100

0

1

3

(radic400x

)3dx (60)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 22 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 22 54

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Example 10-3 解答

取微小面積dA=2ydx之長方形其邊長b=dx與h=2y所以

x2 + y2 = a2 (61)

y =radic

a2 minus x2 (62)

d macrIx prime = d(1

12bh3) =

1

12dx(2y)3 =

2

3y3dx (63)

macrIx prime =

ˆ a

minusa

2

3y3dx =

ˆ a

minusa

2

3

(radica2 minus x2

)3dx =

πa4

4(64)

圓之半徑a則面心慣性矩 macrIx prime macrIy prime為

macrIx prime =πa4

4(65)

macrIy prime =πa4

4(66)

JC = macrIx prime + macrIy prime =πa4

2(67)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 23 54

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複複複合合合面面面積積積之之之慣慣慣性性性矩矩矩Moment of Inertia for Composite Area

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 24 54

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (68)

k =

radicI

m(69)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (70)

= I1 + I2 + + In (71)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 25 54

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複合面積之慣性矩

複合面積之慣性矩(Ixc Iyc)為各局部面積之慣性矩(Ixi Iyi )合

Ixc =i=nsumi=1

Ixi (72)

= Ix1 + Ix2 + + Ixn (73)

Iyc =i=nsumi=1

Iyi (74)

= Iy1 + Iy2 + + Iyn (75)

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課課課堂堂堂練練練習習習

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

長方形與圓形複合

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

長方形複合

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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面面面積積積之之之慣慣慣性性性積積積Product of Inertia for Area

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面積之慣性積(Product of Inertia for Area)

為了下個章節求解慣性矩之極值(極大或極小值)方便數學運算在此先定義面積之慣性積(Ixy )為

dIxy = xydA (76)

Ixy =

ˆAxydA (77)

面積之慣性積的符號可為正負或零當慣性積之面積相對轉軸對稱時則面積之慣性積和為零

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 29 54

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

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平行軸定理

極點

新極點

如左圖面積慣性矩之平行軸定理乃以面心(C)為新極點(O)設定新座標軸(xrsquoyrsquo)其中xrsquo軸平行於原座標x軸yrsquo軸平行於原座標y軸取微小面積dA其在新座標之點座標為(xrsquoyrsquo)面積之慣性積(Ixy )為

dIxy = (x prime + dx)(y prime + dy )dA (78)

當以面心為轉軸點時即x prime = y prime = 0ˆAx primedA = x prime

ˆAdA = 0

ˆAy primedA = y prime

ˆAdA = 0 (79)

Ixy =

ˆA

(x prime + dx)(y prime + dy )dA (80)

=

ˆAx primey primedA + dx

ˆAy primedA + dy

ˆAx primedA + dxdy

ˆAdA (81)

= macrIx primey prime + Adxdy

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 31 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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傾傾傾斜斜斜軸軸軸面面面積積積之之之慣慣慣性性性矩矩矩

Moment of Inertia for Area about Inclined Axis

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 33 54

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傾斜軸面積之慣性矩

當轉動軸傾斜θ角時也就是座標軸旋轉θ角而Ix Iy Ixy為已知則新的座標軸(uv)可寫為

u = xcosθ + ysinθ (82)

v = ycosθ minus xsinθ (83)

由u軸上之面積慣性矩Iuv軸上之面積慣性矩Iv及慣性積Iuv得知

dIu = v2dA = (ycosθ minus xsinθ)2dA (84)

dIv = u2dA = (xcosθ + ysinθ)2dA (85)

dIuv = uvdA = (xcosθ + ysinθ)(ycosθ minus xsinθ)dA (86)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 34 54

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傾斜軸面積之慣性矩 I

由x軸上之面積慣性矩Ixy軸上之面積慣性矩Iy及慣性積Ixy得知

Ix =

ˆy2dA Iy =

ˆx2dA Ixy =

ˆxydA (87)

Iu =

ˆ(ycosθ minus xsinθ)2dA (88)

=

ˆ(y2cos2θ minus 2xycosθsinθ + x2sin2θ)dA (89)

= cos2θ

ˆy2dAminus 2cosθsinθ

ˆxydA + sin2θ

ˆx2dA (90)

= Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (91)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 35 54

傾斜軸面積之慣性矩 II

Iv =

ˆ(xcosθ + ysinθ)2dA (92)

=

ˆ(x2cos2θ + 2xycosθsinθ + y2sin2θ)dA (93)

= cos2θ

ˆx2dA + 2cosθsinθ

ˆxydA + sin2θ

ˆy2dA (94)

= Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (95)

Iuv =

ˆ(xcosθ + ysinθ)(ycosθ minus xsinθ)dA (96)

=

ˆ(xycos2θ + y2cosθsinθ minus x2cosθsinθ minus xysin2θ)dA (97)

= (cos2 minus sin2θ)

ˆxydA + cosθsinθ

ˆy2dAminus cosθsinθ

ˆx2dA

= Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 36 54

傾斜軸面積之慣性矩 III

再由

sin2θ + cos2θ = 1 (98)

sin(αplusmn β) = sinαcosβ plusmn cosαsinβ (99)

sin2θ = sin(θ + θ) = 2sinθcosθ (100)

cos(αplusmn β) = cosαcosβ ∓ sinαsinβ (101)

cos2θ = cos(θ + θ) = cos2θ minus sin2θ (102)

= 2cos2θ minus 1 = 1minus 2sin2θ (103)

所以

Iu = Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (104)

= Ix(cos2θ + 1

2) + Iy (

1minus cos2θ

2)minus Ixy sin2θ (105)

=Ix + Iy

2+

Ix minus Iy2

cos2θ minus Ixy sin2θ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 37 54

傾斜軸面積之慣性矩 IV

Iv = Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (106)

= Iy (cos2θ + 1

2) + Ix(

1minus cos2θ

2) + Ixy sin2θ (107)

=Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (108)

Iuv = Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ (109)

= Ixycos2θ +1

2Ixsin2θ minus 1

2Iy sin2θ (110)

=Ix minus Iy

2sin2θ + Ixycos2θ (111)

所以

JO = Iu + Iv = Ix + Iy (112)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 38 54

主主主軸軸軸慣慣慣性性性矩矩矩Principal Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 39 54

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主軸慣性矩 I

所謂主軸慣性矩(Principal Moment ofInertia)乃定義發生在慣性矩的極值(極大或極小值)上也就是慣性矩之一次微分式為零或斜率為零

dIu

dθ=

d(Ix+Iy

2+

IxminusIy2

cos2θ minus Ixy sin2θ)

dθ(113)

= minus2Ix minus Iy

2sin2θ minus 2Ixy cos2θ (114)

= 0 (115)

其中dsin2θ = cos2θ dcos2θ = minussin2θ當慣性矩之一次微分式為零時傾斜軸與主軸之夾角θ = θp

minus2Ix minus Iy

2sin2θp minus 2Ixycos2θp = 0 (116)

tan2θ =sin2θ

cos2θ=

minusIxy(Ix minus Iy )2

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 40 54

主軸慣性矩 II

cos2θp =IxminusIy2radic

(IxminusIy2 )2 + I 2xy

sin2θp =minusIxyradic

(IxminusIy2 )2 + I 2xy

(117)

(Iu)max =Ix + Iy

2+

Ix minus Iy2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

minus IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2+

radic(Ix minus Iy

2)2 + I 2xy (118)

(Iv )min =Ix + Iy

2minus Ix minus Iy

2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

+ IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2minusradic

(Ix minus Iy

2)2 + I 2xy (119)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 41 54

主軸慣性矩 III

由tan2θp = tan(π + 2θp)可得到θp1與θp2二個解即2θp2 = π + 2θp1也就是θp2 minus θp1 = π

2

tan2θp = tan(π + 2θp) (120)

2θp2 = π + 2θp1 (121)

θp2 minus θp1 =π

2(122)

所以

Imaxmin

=Ix + Iy

2plusmnradic

(Ix minus Iy

2)2 + I 2xy (123)

Iuv =Ix minus Iy

2

minusIxyradic(IxminusIy2 )2 + I 2xy

+ Ixy

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

(124)

= 0

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 42 54

課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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慣慣慣性性性矩矩矩之之之莫莫莫爾爾爾圓圓圓Mohrrsquos Circle for Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 44 54

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慣性矩之莫爾圓

由於Iu =

Ix + Iy2

+Ix minus Iy

2cos2θ minus Ixy sin2θ (125)

Iv =Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (126)

Iuv =Ix minus Iy

2sin2θ + Ixycos2θ (127)

將式(125)與式(127)各自分別平方並相加可得

(Iu minusIx + Iy

2)2 + I 2uv = (

Ix minus Iy2

)2 + I 2xy (128)

其中Ix Iy Ixy均為常數式(128)可進一步寫成

(Iu minus a)2 + I 2uv = R2 (129)

其中a =Ix+Iy2 R =

radic(IxminusIy2 )2 + I 2xycos2θp1 =

aRsin2θp1 =

minusIxyR 2θp1為OA轉向I軸方向即2θp1為

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 45 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 46 54

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解題技巧

首先確定轉動系統的Ix Iy Ixy接著算出莫爾圓之圓心座標a及半徑R並畫出莫爾圓再由莫爾圓上點出A(Ix Ixy )並由OA轉向I軸找出兩倍夾角2θp1及2θp2其中θp1及θp2為主軸(Principle Axis)與Ix軸的夾角由於主軸(Principle Axis)為極值之軸Imax = Iu及Imin = Iv決定θp1及θp2何者為Imax = Iu與Ix軸的夾角之方法乃比較Ix及Iy之大小Imax = Iu會靠近Ix及Iy之較大者也就是Imax = Iu與Ix及Iy之較大者的夾角比較小

a =Ix + Iy

2(130)

R =

radic(Ix minus Iy

2)2 + I 2xy (131)

2θp2 = π + 2θp1 θp2 minus θp1 =π

2(132)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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質質質量量量慣慣慣性性性矩矩矩Mass Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 48 54

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質量慣性矩

τ = Iα (133)

其中τ為旋轉扭矩I為質量慣性矩α為旋轉角速度

I =

ˆmr2dm (134)

=

ˆVr2ρdV (135)

= ρ

ˆVr2dV (136)

其中r為旋轉半徑或轉軸到旋轉體的距離

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 49 54

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 50 54

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平行軸定理

當通過物體質心(G)並在一轉軸上的質量慣性矩(IG )為已知時可透過座標軸轉換計算物體之質量慣性矩(I)

I =

ˆmr2dm (137)

=

ˆm

[(d + x prime)2 + y prime2]dm (138)

=

ˆm

(x prime2 + y prime2)dm + 2d

ˆmx primedm + d2

ˆmdm (139)

I = IG + md2 (140)

其中IG為在zrsquo軸上通過質心的質量慣性矩m為質量d為通過質心之軸zrsquo並平行原軸z之距離

acutem x primedm = x

acutem dm = 0 since x = 0

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 51 54

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (141)

k =

radicI

m(142)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (143)

= I1 + I2 + + In (144)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 52 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

兩圓盤複合體

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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結結結論論論

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 54 54

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結論

轉動慣量或慣性矩為物體對旋轉運動的慣性而所謂慣性乃物體持續維持不變的行為也就是定義慣性矩可表達物體對本身旋轉運動的維持或持續性好壞更簡單的說慣性矩越大之物體其維持旋轉運動趨勢的能力越大例如惰輪儲能裝置惰輪之慣性矩越大當其轉動以後其維持旋轉運動趨勢的能力越大

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 55 54

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  • 轉動慣量或慣性矩(Moment of Inertia)
  • 面積慣性矩(Moment of Inertia for Area)
    • 面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)
    • 面積慣性矩之迴轉半徑(Radius of Gyration for Area)
      • 複合面積之慣性矩(Moment of Inertia for Composite Area)
      • 面積之慣性積(Product of Inertia for Area)
        • 平行軸定理(Parallel-Axis Theorem)
          • 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)
            • 主軸慣性矩(Principal Moment of Inertia)
              • 慣性矩之莫爾圓(Mohrs Circle for Moment of Inertia)
              • 質量慣性矩(Mass Moment of Inertia)
                • 平行軸定理(Parallel-Axis Theorem)
Page 20: 轉動慣量 Moments of Inertia - cct.me.ntut.edu.twcct.me.ntut.edu.tw/ccteducation/chchting/aiahtm/statics/lecture... · 轉動慣量或慣性矩 I. 轉動慣量也稱慣性矩(Moment

解題技巧 III

另外微小面積dA的面積慣性矩dIx dIy也可寫成

dIx = d macrIx prime + dAd2y (41)

Ix =

ˆdIx =

ˆd macrIx prime +

ˆdAd2

y (42)

dIy = d macrIy prime + dAd2x (43)

Iy =

ˆdIy =

ˆd macrIy prime +

ˆdAd2

x (44)

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面心慣性矩 慣性矩 極慣性矩

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Example 10-1 解答

dA = bdy (45)

Ixb =

ˆAy2dA (46)

=

ˆ h

0y2bdy (47)

=1

3by3|h0 (48)

=1

3bh3 (49)

長方形之邊長b與h則面心慣性矩 macrIx prime macrIy prime與長方形之邊軸慣性矩Ix Iy為

macrIx prime =1

12bh3 (50)

macrIy prime =1

12hb3 (51)

Ix =1

3bh3 (52)

Iy =1

3hb3 (53)

JC = macrIx prime + macrIy prime (54)

JO = Ix + Iy (55)

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Example 10-2 解答

取微小面積dA=ydx之長方形其邊長b=dx與h=ydy = y2所以

y2 = 400x (56)

y =radic

400x (57)

d macrIx prime = d(1

12bh3) =

1

12y3dx (58)

dIx = d macrIx prime + dAd2y =

1

12y3dx + ydx

(y2

)2=

1

3y3dx (59)

Ix =

ˆ 100

0

1

3y3dx =

ˆ 100

0

1

3

(radic400x

)3dx (60)

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Example 10-3 解答

取微小面積dA=2ydx之長方形其邊長b=dx與h=2y所以

x2 + y2 = a2 (61)

y =radic

a2 minus x2 (62)

d macrIx prime = d(1

12bh3) =

1

12dx(2y)3 =

2

3y3dx (63)

macrIx prime =

ˆ a

minusa

2

3y3dx =

ˆ a

minusa

2

3

(radica2 minus x2

)3dx =

πa4

4(64)

圓之半徑a則面心慣性矩 macrIx prime macrIy prime為

macrIx prime =πa4

4(65)

macrIy prime =πa4

4(66)

JC = macrIx prime + macrIy prime =πa4

2(67)

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複複複合合合面面面積積積之之之慣慣慣性性性矩矩矩Moment of Inertia for Composite Area

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (68)

k =

radicI

m(69)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (70)

= I1 + I2 + + In (71)

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複合面積之慣性矩

複合面積之慣性矩(Ixc Iyc)為各局部面積之慣性矩(Ixi Iyi )合

Ixc =i=nsumi=1

Ixi (72)

= Ix1 + Ix2 + + Ixn (73)

Iyc =i=nsumi=1

Iyi (74)

= Iy1 + Iy2 + + Iyn (75)

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課課課堂堂堂練練練習習習

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

長方形與圓形複合

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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長方形複合

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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面面面積積積之之之慣慣慣性性性積積積Product of Inertia for Area

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面積之慣性積(Product of Inertia for Area)

為了下個章節求解慣性矩之極值(極大或極小值)方便數學運算在此先定義面積之慣性積(Ixy )為

dIxy = xydA (76)

Ixy =

ˆAxydA (77)

面積之慣性積的符號可為正負或零當慣性積之面積相對轉軸對稱時則面積之慣性積和為零

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

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平行軸定理

極點

新極點

如左圖面積慣性矩之平行軸定理乃以面心(C)為新極點(O)設定新座標軸(xrsquoyrsquo)其中xrsquo軸平行於原座標x軸yrsquo軸平行於原座標y軸取微小面積dA其在新座標之點座標為(xrsquoyrsquo)面積之慣性積(Ixy )為

dIxy = (x prime + dx)(y prime + dy )dA (78)

當以面心為轉軸點時即x prime = y prime = 0ˆAx primedA = x prime

ˆAdA = 0

ˆAy primedA = y prime

ˆAdA = 0 (79)

Ixy =

ˆA

(x prime + dx)(y prime + dy )dA (80)

=

ˆAx primey primedA + dx

ˆAy primedA + dy

ˆAx primedA + dxdy

ˆAdA (81)

= macrIx primey prime + Adxdy

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課課課堂堂堂練練練習習習

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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傾傾傾斜斜斜軸軸軸面面面積積積之之之慣慣慣性性性矩矩矩

Moment of Inertia for Area about Inclined Axis

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傾斜軸面積之慣性矩

當轉動軸傾斜θ角時也就是座標軸旋轉θ角而Ix Iy Ixy為已知則新的座標軸(uv)可寫為

u = xcosθ + ysinθ (82)

v = ycosθ minus xsinθ (83)

由u軸上之面積慣性矩Iuv軸上之面積慣性矩Iv及慣性積Iuv得知

dIu = v2dA = (ycosθ minus xsinθ)2dA (84)

dIv = u2dA = (xcosθ + ysinθ)2dA (85)

dIuv = uvdA = (xcosθ + ysinθ)(ycosθ minus xsinθ)dA (86)

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傾斜軸面積之慣性矩 I

由x軸上之面積慣性矩Ixy軸上之面積慣性矩Iy及慣性積Ixy得知

Ix =

ˆy2dA Iy =

ˆx2dA Ixy =

ˆxydA (87)

Iu =

ˆ(ycosθ minus xsinθ)2dA (88)

=

ˆ(y2cos2θ minus 2xycosθsinθ + x2sin2θ)dA (89)

= cos2θ

ˆy2dAminus 2cosθsinθ

ˆxydA + sin2θ

ˆx2dA (90)

= Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (91)

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傾斜軸面積之慣性矩 II

Iv =

ˆ(xcosθ + ysinθ)2dA (92)

=

ˆ(x2cos2θ + 2xycosθsinθ + y2sin2θ)dA (93)

= cos2θ

ˆx2dA + 2cosθsinθ

ˆxydA + sin2θ

ˆy2dA (94)

= Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (95)

Iuv =

ˆ(xcosθ + ysinθ)(ycosθ minus xsinθ)dA (96)

=

ˆ(xycos2θ + y2cosθsinθ minus x2cosθsinθ minus xysin2θ)dA (97)

= (cos2 minus sin2θ)

ˆxydA + cosθsinθ

ˆy2dAminus cosθsinθ

ˆx2dA

= Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ

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傾斜軸面積之慣性矩 III

再由

sin2θ + cos2θ = 1 (98)

sin(αplusmn β) = sinαcosβ plusmn cosαsinβ (99)

sin2θ = sin(θ + θ) = 2sinθcosθ (100)

cos(αplusmn β) = cosαcosβ ∓ sinαsinβ (101)

cos2θ = cos(θ + θ) = cos2θ minus sin2θ (102)

= 2cos2θ minus 1 = 1minus 2sin2θ (103)

所以

Iu = Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (104)

= Ix(cos2θ + 1

2) + Iy (

1minus cos2θ

2)minus Ixy sin2θ (105)

=Ix + Iy

2+

Ix minus Iy2

cos2θ minus Ixy sin2θ

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傾斜軸面積之慣性矩 IV

Iv = Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (106)

= Iy (cos2θ + 1

2) + Ix(

1minus cos2θ

2) + Ixy sin2θ (107)

=Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (108)

Iuv = Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ (109)

= Ixycos2θ +1

2Ixsin2θ minus 1

2Iy sin2θ (110)

=Ix minus Iy

2sin2θ + Ixycos2θ (111)

所以

JO = Iu + Iv = Ix + Iy (112)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 38 54

主主主軸軸軸慣慣慣性性性矩矩矩Principal Moment of Inertia

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主軸慣性矩 I

所謂主軸慣性矩(Principal Moment ofInertia)乃定義發生在慣性矩的極值(極大或極小值)上也就是慣性矩之一次微分式為零或斜率為零

dIu

dθ=

d(Ix+Iy

2+

IxminusIy2

cos2θ minus Ixy sin2θ)

dθ(113)

= minus2Ix minus Iy

2sin2θ minus 2Ixy cos2θ (114)

= 0 (115)

其中dsin2θ = cos2θ dcos2θ = minussin2θ當慣性矩之一次微分式為零時傾斜軸與主軸之夾角θ = θp

minus2Ix minus Iy

2sin2θp minus 2Ixycos2θp = 0 (116)

tan2θ =sin2θ

cos2θ=

minusIxy(Ix minus Iy )2

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 40 54

主軸慣性矩 II

cos2θp =IxminusIy2radic

(IxminusIy2 )2 + I 2xy

sin2θp =minusIxyradic

(IxminusIy2 )2 + I 2xy

(117)

(Iu)max =Ix + Iy

2+

Ix minus Iy2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

minus IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2+

radic(Ix minus Iy

2)2 + I 2xy (118)

(Iv )min =Ix + Iy

2minus Ix minus Iy

2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

+ IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2minusradic

(Ix minus Iy

2)2 + I 2xy (119)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 41 54

主軸慣性矩 III

由tan2θp = tan(π + 2θp)可得到θp1與θp2二個解即2θp2 = π + 2θp1也就是θp2 minus θp1 = π

2

tan2θp = tan(π + 2θp) (120)

2θp2 = π + 2θp1 (121)

θp2 minus θp1 =π

2(122)

所以

Imaxmin

=Ix + Iy

2plusmnradic

(Ix minus Iy

2)2 + I 2xy (123)

Iuv =Ix minus Iy

2

minusIxyradic(IxminusIy2 )2 + I 2xy

+ Ixy

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

(124)

= 0

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 42 54

課課課堂堂堂練練練習習習

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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慣慣慣性性性矩矩矩之之之莫莫莫爾爾爾圓圓圓Mohrrsquos Circle for Moment of Inertia

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慣性矩之莫爾圓

由於Iu =

Ix + Iy2

+Ix minus Iy

2cos2θ minus Ixy sin2θ (125)

Iv =Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (126)

Iuv =Ix minus Iy

2sin2θ + Ixycos2θ (127)

將式(125)與式(127)各自分別平方並相加可得

(Iu minusIx + Iy

2)2 + I 2uv = (

Ix minus Iy2

)2 + I 2xy (128)

其中Ix Iy Ixy均為常數式(128)可進一步寫成

(Iu minus a)2 + I 2uv = R2 (129)

其中a =Ix+Iy2 R =

radic(IxminusIy2 )2 + I 2xycos2θp1 =

aRsin2θp1 =

minusIxyR 2θp1為OA轉向I軸方向即2θp1為

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 46 54

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解題技巧

首先確定轉動系統的Ix Iy Ixy接著算出莫爾圓之圓心座標a及半徑R並畫出莫爾圓再由莫爾圓上點出A(Ix Ixy )並由OA轉向I軸找出兩倍夾角2θp1及2θp2其中θp1及θp2為主軸(Principle Axis)與Ix軸的夾角由於主軸(Principle Axis)為極值之軸Imax = Iu及Imin = Iv決定θp1及θp2何者為Imax = Iu與Ix軸的夾角之方法乃比較Ix及Iy之大小Imax = Iu會靠近Ix及Iy之較大者也就是Imax = Iu與Ix及Iy之較大者的夾角比較小

a =Ix + Iy

2(130)

R =

radic(Ix minus Iy

2)2 + I 2xy (131)

2θp2 = π + 2θp1 θp2 minus θp1 =π

2(132)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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質質質量量量慣慣慣性性性矩矩矩Mass Moment of Inertia

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質量慣性矩

τ = Iα (133)

其中τ為旋轉扭矩I為質量慣性矩α為旋轉角速度

I =

ˆmr2dm (134)

=

ˆVr2ρdV (135)

= ρ

ˆVr2dV (136)

其中r為旋轉半徑或轉軸到旋轉體的距離

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

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平行軸定理

當通過物體質心(G)並在一轉軸上的質量慣性矩(IG )為已知時可透過座標軸轉換計算物體之質量慣性矩(I)

I =

ˆmr2dm (137)

=

ˆm

[(d + x prime)2 + y prime2]dm (138)

=

ˆm

(x prime2 + y prime2)dm + 2d

ˆmx primedm + d2

ˆmdm (139)

I = IG + md2 (140)

其中IG為在zrsquo軸上通過質心的質量慣性矩m為質量d為通過質心之軸zrsquo並平行原軸z之距離

acutem x primedm = x

acutem dm = 0 since x = 0

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (141)

k =

radicI

m(142)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (143)

= I1 + I2 + + In (144)

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課課課堂堂堂練練練習習習

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

兩圓盤複合體

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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結結結論論論

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 54 54

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結論

轉動慣量或慣性矩為物體對旋轉運動的慣性而所謂慣性乃物體持續維持不變的行為也就是定義慣性矩可表達物體對本身旋轉運動的維持或持續性好壞更簡單的說慣性矩越大之物體其維持旋轉運動趨勢的能力越大例如惰輪儲能裝置惰輪之慣性矩越大當其轉動以後其維持旋轉運動趨勢的能力越大

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 55 54

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  • 轉動慣量或慣性矩(Moment of Inertia)
  • 面積慣性矩(Moment of Inertia for Area)
    • 面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)
    • 面積慣性矩之迴轉半徑(Radius of Gyration for Area)
      • 複合面積之慣性矩(Moment of Inertia for Composite Area)
      • 面積之慣性積(Product of Inertia for Area)
        • 平行軸定理(Parallel-Axis Theorem)
          • 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)
            • 主軸慣性矩(Principal Moment of Inertia)
              • 慣性矩之莫爾圓(Mohrs Circle for Moment of Inertia)
              • 質量慣性矩(Mass Moment of Inertia)
                • 平行軸定理(Parallel-Axis Theorem)
Page 21: 轉動慣量 Moments of Inertia - cct.me.ntut.edu.twcct.me.ntut.edu.tw/ccteducation/chchting/aiahtm/statics/lecture... · 轉動慣量或慣性矩 I. 轉動慣量也稱慣性矩(Moment

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

面心慣性矩 慣性矩 極慣性矩

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Example 10-1 解答

dA = bdy (45)

Ixb =

ˆAy2dA (46)

=

ˆ h

0y2bdy (47)

=1

3by3|h0 (48)

=1

3bh3 (49)

長方形之邊長b與h則面心慣性矩 macrIx prime macrIy prime與長方形之邊軸慣性矩Ix Iy為

macrIx prime =1

12bh3 (50)

macrIy prime =1

12hb3 (51)

Ix =1

3bh3 (52)

Iy =1

3hb3 (53)

JC = macrIx prime + macrIy prime (54)

JO = Ix + Iy (55)

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Example 10-2 解答

取微小面積dA=ydx之長方形其邊長b=dx與h=ydy = y2所以

y2 = 400x (56)

y =radic

400x (57)

d macrIx prime = d(1

12bh3) =

1

12y3dx (58)

dIx = d macrIx prime + dAd2y =

1

12y3dx + ydx

(y2

)2=

1

3y3dx (59)

Ix =

ˆ 100

0

1

3y3dx =

ˆ 100

0

1

3

(radic400x

)3dx (60)

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 22 54

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Example 10-3 解答

取微小面積dA=2ydx之長方形其邊長b=dx與h=2y所以

x2 + y2 = a2 (61)

y =radic

a2 minus x2 (62)

d macrIx prime = d(1

12bh3) =

1

12dx(2y)3 =

2

3y3dx (63)

macrIx prime =

ˆ a

minusa

2

3y3dx =

ˆ a

minusa

2

3

(radica2 minus x2

)3dx =

πa4

4(64)

圓之半徑a則面心慣性矩 macrIx prime macrIy prime為

macrIx prime =πa4

4(65)

macrIy prime =πa4

4(66)

JC = macrIx prime + macrIy prime =πa4

2(67)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 23 54

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複複複合合合面面面積積積之之之慣慣慣性性性矩矩矩Moment of Inertia for Composite Area

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 24 54

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (68)

k =

radicI

m(69)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (70)

= I1 + I2 + + In (71)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 25 54

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複合面積之慣性矩

複合面積之慣性矩(Ixc Iyc)為各局部面積之慣性矩(Ixi Iyi )合

Ixc =i=nsumi=1

Ixi (72)

= Ix1 + Ix2 + + Ixn (73)

Iyc =i=nsumi=1

Iyi (74)

= Iy1 + Iy2 + + Iyn (75)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 26 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

長方形與圓形複合

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

長方形複合

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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面面面積積積之之之慣慣慣性性性積積積Product of Inertia for Area

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 28 54

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面積之慣性積(Product of Inertia for Area)

為了下個章節求解慣性矩之極值(極大或極小值)方便數學運算在此先定義面積之慣性積(Ixy )為

dIxy = xydA (76)

Ixy =

ˆAxydA (77)

面積之慣性積的符號可為正負或零當慣性積之面積相對轉軸對稱時則面積之慣性積和為零

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 29 54

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

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平行軸定理

極點

新極點

如左圖面積慣性矩之平行軸定理乃以面心(C)為新極點(O)設定新座標軸(xrsquoyrsquo)其中xrsquo軸平行於原座標x軸yrsquo軸平行於原座標y軸取微小面積dA其在新座標之點座標為(xrsquoyrsquo)面積之慣性積(Ixy )為

dIxy = (x prime + dx)(y prime + dy )dA (78)

當以面心為轉軸點時即x prime = y prime = 0ˆAx primedA = x prime

ˆAdA = 0

ˆAy primedA = y prime

ˆAdA = 0 (79)

Ixy =

ˆA

(x prime + dx)(y prime + dy )dA (80)

=

ˆAx primey primedA + dx

ˆAy primedA + dy

ˆAx primedA + dxdy

ˆAdA (81)

= macrIx primey prime + Adxdy

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 31 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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傾傾傾斜斜斜軸軸軸面面面積積積之之之慣慣慣性性性矩矩矩

Moment of Inertia for Area about Inclined Axis

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 33 54

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傾斜軸面積之慣性矩

當轉動軸傾斜θ角時也就是座標軸旋轉θ角而Ix Iy Ixy為已知則新的座標軸(uv)可寫為

u = xcosθ + ysinθ (82)

v = ycosθ minus xsinθ (83)

由u軸上之面積慣性矩Iuv軸上之面積慣性矩Iv及慣性積Iuv得知

dIu = v2dA = (ycosθ minus xsinθ)2dA (84)

dIv = u2dA = (xcosθ + ysinθ)2dA (85)

dIuv = uvdA = (xcosθ + ysinθ)(ycosθ minus xsinθ)dA (86)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 34 54

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傾斜軸面積之慣性矩 I

由x軸上之面積慣性矩Ixy軸上之面積慣性矩Iy及慣性積Ixy得知

Ix =

ˆy2dA Iy =

ˆx2dA Ixy =

ˆxydA (87)

Iu =

ˆ(ycosθ minus xsinθ)2dA (88)

=

ˆ(y2cos2θ minus 2xycosθsinθ + x2sin2θ)dA (89)

= cos2θ

ˆy2dAminus 2cosθsinθ

ˆxydA + sin2θ

ˆx2dA (90)

= Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (91)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 35 54

傾斜軸面積之慣性矩 II

Iv =

ˆ(xcosθ + ysinθ)2dA (92)

=

ˆ(x2cos2θ + 2xycosθsinθ + y2sin2θ)dA (93)

= cos2θ

ˆx2dA + 2cosθsinθ

ˆxydA + sin2θ

ˆy2dA (94)

= Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (95)

Iuv =

ˆ(xcosθ + ysinθ)(ycosθ minus xsinθ)dA (96)

=

ˆ(xycos2θ + y2cosθsinθ minus x2cosθsinθ minus xysin2θ)dA (97)

= (cos2 minus sin2θ)

ˆxydA + cosθsinθ

ˆy2dAminus cosθsinθ

ˆx2dA

= Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 36 54

傾斜軸面積之慣性矩 III

再由

sin2θ + cos2θ = 1 (98)

sin(αplusmn β) = sinαcosβ plusmn cosαsinβ (99)

sin2θ = sin(θ + θ) = 2sinθcosθ (100)

cos(αplusmn β) = cosαcosβ ∓ sinαsinβ (101)

cos2θ = cos(θ + θ) = cos2θ minus sin2θ (102)

= 2cos2θ minus 1 = 1minus 2sin2θ (103)

所以

Iu = Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (104)

= Ix(cos2θ + 1

2) + Iy (

1minus cos2θ

2)minus Ixy sin2θ (105)

=Ix + Iy

2+

Ix minus Iy2

cos2θ minus Ixy sin2θ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 37 54

傾斜軸面積之慣性矩 IV

Iv = Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (106)

= Iy (cos2θ + 1

2) + Ix(

1minus cos2θ

2) + Ixy sin2θ (107)

=Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (108)

Iuv = Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ (109)

= Ixycos2θ +1

2Ixsin2θ minus 1

2Iy sin2θ (110)

=Ix minus Iy

2sin2θ + Ixycos2θ (111)

所以

JO = Iu + Iv = Ix + Iy (112)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 38 54

主主主軸軸軸慣慣慣性性性矩矩矩Principal Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 39 54

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主軸慣性矩 I

所謂主軸慣性矩(Principal Moment ofInertia)乃定義發生在慣性矩的極值(極大或極小值)上也就是慣性矩之一次微分式為零或斜率為零

dIu

dθ=

d(Ix+Iy

2+

IxminusIy2

cos2θ minus Ixy sin2θ)

dθ(113)

= minus2Ix minus Iy

2sin2θ minus 2Ixy cos2θ (114)

= 0 (115)

其中dsin2θ = cos2θ dcos2θ = minussin2θ當慣性矩之一次微分式為零時傾斜軸與主軸之夾角θ = θp

minus2Ix minus Iy

2sin2θp minus 2Ixycos2θp = 0 (116)

tan2θ =sin2θ

cos2θ=

minusIxy(Ix minus Iy )2

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 40 54

主軸慣性矩 II

cos2θp =IxminusIy2radic

(IxminusIy2 )2 + I 2xy

sin2θp =minusIxyradic

(IxminusIy2 )2 + I 2xy

(117)

(Iu)max =Ix + Iy

2+

Ix minus Iy2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

minus IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2+

radic(Ix minus Iy

2)2 + I 2xy (118)

(Iv )min =Ix + Iy

2minus Ix minus Iy

2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

+ IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2minusradic

(Ix minus Iy

2)2 + I 2xy (119)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 41 54

主軸慣性矩 III

由tan2θp = tan(π + 2θp)可得到θp1與θp2二個解即2θp2 = π + 2θp1也就是θp2 minus θp1 = π

2

tan2θp = tan(π + 2θp) (120)

2θp2 = π + 2θp1 (121)

θp2 minus θp1 =π

2(122)

所以

Imaxmin

=Ix + Iy

2plusmnradic

(Ix minus Iy

2)2 + I 2xy (123)

Iuv =Ix minus Iy

2

minusIxyradic(IxminusIy2 )2 + I 2xy

+ Ixy

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

(124)

= 0

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 42 54

課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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慣慣慣性性性矩矩矩之之之莫莫莫爾爾爾圓圓圓Mohrrsquos Circle for Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 44 54

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慣性矩之莫爾圓

由於Iu =

Ix + Iy2

+Ix minus Iy

2cos2θ minus Ixy sin2θ (125)

Iv =Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (126)

Iuv =Ix minus Iy

2sin2θ + Ixycos2θ (127)

將式(125)與式(127)各自分別平方並相加可得

(Iu minusIx + Iy

2)2 + I 2uv = (

Ix minus Iy2

)2 + I 2xy (128)

其中Ix Iy Ixy均為常數式(128)可進一步寫成

(Iu minus a)2 + I 2uv = R2 (129)

其中a =Ix+Iy2 R =

radic(IxminusIy2 )2 + I 2xycos2θp1 =

aRsin2θp1 =

minusIxyR 2θp1為OA轉向I軸方向即2θp1為

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課課課堂堂堂練練練習習習

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解題技巧

首先確定轉動系統的Ix Iy Ixy接著算出莫爾圓之圓心座標a及半徑R並畫出莫爾圓再由莫爾圓上點出A(Ix Ixy )並由OA轉向I軸找出兩倍夾角2θp1及2θp2其中θp1及θp2為主軸(Principle Axis)與Ix軸的夾角由於主軸(Principle Axis)為極值之軸Imax = Iu及Imin = Iv決定θp1及θp2何者為Imax = Iu與Ix軸的夾角之方法乃比較Ix及Iy之大小Imax = Iu會靠近Ix及Iy之較大者也就是Imax = Iu與Ix及Iy之較大者的夾角比較小

a =Ix + Iy

2(130)

R =

radic(Ix minus Iy

2)2 + I 2xy (131)

2θp2 = π + 2θp1 θp2 minus θp1 =π

2(132)

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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質質質量量量慣慣慣性性性矩矩矩Mass Moment of Inertia

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質量慣性矩

τ = Iα (133)

其中τ為旋轉扭矩I為質量慣性矩α為旋轉角速度

I =

ˆmr2dm (134)

=

ˆVr2ρdV (135)

= ρ

ˆVr2dV (136)

其中r為旋轉半徑或轉軸到旋轉體的距離

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

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平行軸定理

當通過物體質心(G)並在一轉軸上的質量慣性矩(IG )為已知時可透過座標軸轉換計算物體之質量慣性矩(I)

I =

ˆmr2dm (137)

=

ˆm

[(d + x prime)2 + y prime2]dm (138)

=

ˆm

(x prime2 + y prime2)dm + 2d

ˆmx primedm + d2

ˆmdm (139)

I = IG + md2 (140)

其中IG為在zrsquo軸上通過質心的質量慣性矩m為質量d為通過質心之軸zrsquo並平行原軸z之距離

acutem x primedm = x

acutem dm = 0 since x = 0

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (141)

k =

radicI

m(142)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (143)

= I1 + I2 + + In (144)

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課課課堂堂堂練練練習習習

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

兩圓盤複合體

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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結結結論論論

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結論

轉動慣量或慣性矩為物體對旋轉運動的慣性而所謂慣性乃物體持續維持不變的行為也就是定義慣性矩可表達物體對本身旋轉運動的維持或持續性好壞更簡單的說慣性矩越大之物體其維持旋轉運動趨勢的能力越大例如惰輪儲能裝置惰輪之慣性矩越大當其轉動以後其維持旋轉運動趨勢的能力越大

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  • 轉動慣量或慣性矩(Moment of Inertia)
  • 面積慣性矩(Moment of Inertia for Area)
    • 面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)
    • 面積慣性矩之迴轉半徑(Radius of Gyration for Area)
      • 複合面積之慣性矩(Moment of Inertia for Composite Area)
      • 面積之慣性積(Product of Inertia for Area)
        • 平行軸定理(Parallel-Axis Theorem)
          • 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)
            • 主軸慣性矩(Principal Moment of Inertia)
              • 慣性矩之莫爾圓(Mohrs Circle for Moment of Inertia)
              • 質量慣性矩(Mass Moment of Inertia)
                • 平行軸定理(Parallel-Axis Theorem)
Page 22: 轉動慣量 Moments of Inertia - cct.me.ntut.edu.twcct.me.ntut.edu.tw/ccteducation/chchting/aiahtm/statics/lecture... · 轉動慣量或慣性矩 I. 轉動慣量也稱慣性矩(Moment

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Example 10-1 解答

dA = bdy (45)

Ixb =

ˆAy2dA (46)

=

ˆ h

0y2bdy (47)

=1

3by3|h0 (48)

=1

3bh3 (49)

長方形之邊長b與h則面心慣性矩 macrIx prime macrIy prime與長方形之邊軸慣性矩Ix Iy為

macrIx prime =1

12bh3 (50)

macrIy prime =1

12hb3 (51)

Ix =1

3bh3 (52)

Iy =1

3hb3 (53)

JC = macrIx prime + macrIy prime (54)

JO = Ix + Iy (55)

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Example 10-2 解答

取微小面積dA=ydx之長方形其邊長b=dx與h=ydy = y2所以

y2 = 400x (56)

y =radic

400x (57)

d macrIx prime = d(1

12bh3) =

1

12y3dx (58)

dIx = d macrIx prime + dAd2y =

1

12y3dx + ydx

(y2

)2=

1

3y3dx (59)

Ix =

ˆ 100

0

1

3y3dx =

ˆ 100

0

1

3

(radic400x

)3dx (60)

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Example 10-3 解答

取微小面積dA=2ydx之長方形其邊長b=dx與h=2y所以

x2 + y2 = a2 (61)

y =radic

a2 minus x2 (62)

d macrIx prime = d(1

12bh3) =

1

12dx(2y)3 =

2

3y3dx (63)

macrIx prime =

ˆ a

minusa

2

3y3dx =

ˆ a

minusa

2

3

(radica2 minus x2

)3dx =

πa4

4(64)

圓之半徑a則面心慣性矩 macrIx prime macrIy prime為

macrIx prime =πa4

4(65)

macrIy prime =πa4

4(66)

JC = macrIx prime + macrIy prime =πa4

2(67)

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複複複合合合面面面積積積之之之慣慣慣性性性矩矩矩Moment of Inertia for Composite Area

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (68)

k =

radicI

m(69)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (70)

= I1 + I2 + + In (71)

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複合面積之慣性矩

複合面積之慣性矩(Ixc Iyc)為各局部面積之慣性矩(Ixi Iyi )合

Ixc =i=nsumi=1

Ixi (72)

= Ix1 + Ix2 + + Ixn (73)

Iyc =i=nsumi=1

Iyi (74)

= Iy1 + Iy2 + + Iyn (75)

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課課課堂堂堂練練練習習習

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

長方形與圓形複合

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

長方形複合

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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面面面積積積之之之慣慣慣性性性積積積Product of Inertia for Area

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 28 54

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面積之慣性積(Product of Inertia for Area)

為了下個章節求解慣性矩之極值(極大或極小值)方便數學運算在此先定義面積之慣性積(Ixy )為

dIxy = xydA (76)

Ixy =

ˆAxydA (77)

面積之慣性積的符號可為正負或零當慣性積之面積相對轉軸對稱時則面積之慣性積和為零

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 29 54

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 30 54

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平行軸定理

極點

新極點

如左圖面積慣性矩之平行軸定理乃以面心(C)為新極點(O)設定新座標軸(xrsquoyrsquo)其中xrsquo軸平行於原座標x軸yrsquo軸平行於原座標y軸取微小面積dA其在新座標之點座標為(xrsquoyrsquo)面積之慣性積(Ixy )為

dIxy = (x prime + dx)(y prime + dy )dA (78)

當以面心為轉軸點時即x prime = y prime = 0ˆAx primedA = x prime

ˆAdA = 0

ˆAy primedA = y prime

ˆAdA = 0 (79)

Ixy =

ˆA

(x prime + dx)(y prime + dy )dA (80)

=

ˆAx primey primedA + dx

ˆAy primedA + dy

ˆAx primedA + dxdy

ˆAdA (81)

= macrIx primey prime + Adxdy

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 31 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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傾傾傾斜斜斜軸軸軸面面面積積積之之之慣慣慣性性性矩矩矩

Moment of Inertia for Area about Inclined Axis

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 33 54

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傾斜軸面積之慣性矩

當轉動軸傾斜θ角時也就是座標軸旋轉θ角而Ix Iy Ixy為已知則新的座標軸(uv)可寫為

u = xcosθ + ysinθ (82)

v = ycosθ minus xsinθ (83)

由u軸上之面積慣性矩Iuv軸上之面積慣性矩Iv及慣性積Iuv得知

dIu = v2dA = (ycosθ minus xsinθ)2dA (84)

dIv = u2dA = (xcosθ + ysinθ)2dA (85)

dIuv = uvdA = (xcosθ + ysinθ)(ycosθ minus xsinθ)dA (86)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 34 54

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傾斜軸面積之慣性矩 I

由x軸上之面積慣性矩Ixy軸上之面積慣性矩Iy及慣性積Ixy得知

Ix =

ˆy2dA Iy =

ˆx2dA Ixy =

ˆxydA (87)

Iu =

ˆ(ycosθ minus xsinθ)2dA (88)

=

ˆ(y2cos2θ minus 2xycosθsinθ + x2sin2θ)dA (89)

= cos2θ

ˆy2dAminus 2cosθsinθ

ˆxydA + sin2θ

ˆx2dA (90)

= Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (91)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 35 54

傾斜軸面積之慣性矩 II

Iv =

ˆ(xcosθ + ysinθ)2dA (92)

=

ˆ(x2cos2θ + 2xycosθsinθ + y2sin2θ)dA (93)

= cos2θ

ˆx2dA + 2cosθsinθ

ˆxydA + sin2θ

ˆy2dA (94)

= Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (95)

Iuv =

ˆ(xcosθ + ysinθ)(ycosθ minus xsinθ)dA (96)

=

ˆ(xycos2θ + y2cosθsinθ minus x2cosθsinθ minus xysin2θ)dA (97)

= (cos2 minus sin2θ)

ˆxydA + cosθsinθ

ˆy2dAminus cosθsinθ

ˆx2dA

= Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 36 54

傾斜軸面積之慣性矩 III

再由

sin2θ + cos2θ = 1 (98)

sin(αplusmn β) = sinαcosβ plusmn cosαsinβ (99)

sin2θ = sin(θ + θ) = 2sinθcosθ (100)

cos(αplusmn β) = cosαcosβ ∓ sinαsinβ (101)

cos2θ = cos(θ + θ) = cos2θ minus sin2θ (102)

= 2cos2θ minus 1 = 1minus 2sin2θ (103)

所以

Iu = Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (104)

= Ix(cos2θ + 1

2) + Iy (

1minus cos2θ

2)minus Ixy sin2θ (105)

=Ix + Iy

2+

Ix minus Iy2

cos2θ minus Ixy sin2θ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 37 54

傾斜軸面積之慣性矩 IV

Iv = Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (106)

= Iy (cos2θ + 1

2) + Ix(

1minus cos2θ

2) + Ixy sin2θ (107)

=Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (108)

Iuv = Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ (109)

= Ixycos2θ +1

2Ixsin2θ minus 1

2Iy sin2θ (110)

=Ix minus Iy

2sin2θ + Ixycos2θ (111)

所以

JO = Iu + Iv = Ix + Iy (112)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 38 54

主主主軸軸軸慣慣慣性性性矩矩矩Principal Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 39 54

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主軸慣性矩 I

所謂主軸慣性矩(Principal Moment ofInertia)乃定義發生在慣性矩的極值(極大或極小值)上也就是慣性矩之一次微分式為零或斜率為零

dIu

dθ=

d(Ix+Iy

2+

IxminusIy2

cos2θ minus Ixy sin2θ)

dθ(113)

= minus2Ix minus Iy

2sin2θ minus 2Ixy cos2θ (114)

= 0 (115)

其中dsin2θ = cos2θ dcos2θ = minussin2θ當慣性矩之一次微分式為零時傾斜軸與主軸之夾角θ = θp

minus2Ix minus Iy

2sin2θp minus 2Ixycos2θp = 0 (116)

tan2θ =sin2θ

cos2θ=

minusIxy(Ix minus Iy )2

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 40 54

主軸慣性矩 II

cos2θp =IxminusIy2radic

(IxminusIy2 )2 + I 2xy

sin2θp =minusIxyradic

(IxminusIy2 )2 + I 2xy

(117)

(Iu)max =Ix + Iy

2+

Ix minus Iy2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

minus IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2+

radic(Ix minus Iy

2)2 + I 2xy (118)

(Iv )min =Ix + Iy

2minus Ix minus Iy

2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

+ IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2minusradic

(Ix minus Iy

2)2 + I 2xy (119)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 41 54

主軸慣性矩 III

由tan2θp = tan(π + 2θp)可得到θp1與θp2二個解即2θp2 = π + 2θp1也就是θp2 minus θp1 = π

2

tan2θp = tan(π + 2θp) (120)

2θp2 = π + 2θp1 (121)

θp2 minus θp1 =π

2(122)

所以

Imaxmin

=Ix + Iy

2plusmnradic

(Ix minus Iy

2)2 + I 2xy (123)

Iuv =Ix minus Iy

2

minusIxyradic(IxminusIy2 )2 + I 2xy

+ Ixy

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

(124)

= 0

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 42 54

課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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慣慣慣性性性矩矩矩之之之莫莫莫爾爾爾圓圓圓Mohrrsquos Circle for Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 44 54

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慣性矩之莫爾圓

由於Iu =

Ix + Iy2

+Ix minus Iy

2cos2θ minus Ixy sin2θ (125)

Iv =Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (126)

Iuv =Ix minus Iy

2sin2θ + Ixycos2θ (127)

將式(125)與式(127)各自分別平方並相加可得

(Iu minusIx + Iy

2)2 + I 2uv = (

Ix minus Iy2

)2 + I 2xy (128)

其中Ix Iy Ixy均為常數式(128)可進一步寫成

(Iu minus a)2 + I 2uv = R2 (129)

其中a =Ix+Iy2 R =

radic(IxminusIy2 )2 + I 2xycos2θp1 =

aRsin2θp1 =

minusIxyR 2θp1為OA轉向I軸方向即2θp1為

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 45 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 46 54

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解題技巧

首先確定轉動系統的Ix Iy Ixy接著算出莫爾圓之圓心座標a及半徑R並畫出莫爾圓再由莫爾圓上點出A(Ix Ixy )並由OA轉向I軸找出兩倍夾角2θp1及2θp2其中θp1及θp2為主軸(Principle Axis)與Ix軸的夾角由於主軸(Principle Axis)為極值之軸Imax = Iu及Imin = Iv決定θp1及θp2何者為Imax = Iu與Ix軸的夾角之方法乃比較Ix及Iy之大小Imax = Iu會靠近Ix及Iy之較大者也就是Imax = Iu與Ix及Iy之較大者的夾角比較小

a =Ix + Iy

2(130)

R =

radic(Ix minus Iy

2)2 + I 2xy (131)

2θp2 = π + 2θp1 θp2 minus θp1 =π

2(132)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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質質質量量量慣慣慣性性性矩矩矩Mass Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 48 54

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質量慣性矩

τ = Iα (133)

其中τ為旋轉扭矩I為質量慣性矩α為旋轉角速度

I =

ˆmr2dm (134)

=

ˆVr2ρdV (135)

= ρ

ˆVr2dV (136)

其中r為旋轉半徑或轉軸到旋轉體的距離

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 49 54

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 50 54

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平行軸定理

當通過物體質心(G)並在一轉軸上的質量慣性矩(IG )為已知時可透過座標軸轉換計算物體之質量慣性矩(I)

I =

ˆmr2dm (137)

=

ˆm

[(d + x prime)2 + y prime2]dm (138)

=

ˆm

(x prime2 + y prime2)dm + 2d

ˆmx primedm + d2

ˆmdm (139)

I = IG + md2 (140)

其中IG為在zrsquo軸上通過質心的質量慣性矩m為質量d為通過質心之軸zrsquo並平行原軸z之距離

acutem x primedm = x

acutem dm = 0 since x = 0

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 51 54

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (141)

k =

radicI

m(142)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (143)

= I1 + I2 + + In (144)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 52 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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兩圓盤複合體

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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結結結論論論

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 54 54

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結論

轉動慣量或慣性矩為物體對旋轉運動的慣性而所謂慣性乃物體持續維持不變的行為也就是定義慣性矩可表達物體對本身旋轉運動的維持或持續性好壞更簡單的說慣性矩越大之物體其維持旋轉運動趨勢的能力越大例如惰輪儲能裝置惰輪之慣性矩越大當其轉動以後其維持旋轉運動趨勢的能力越大

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 55 54

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  • 轉動慣量或慣性矩(Moment of Inertia)
  • 面積慣性矩(Moment of Inertia for Area)
    • 面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)
    • 面積慣性矩之迴轉半徑(Radius of Gyration for Area)
      • 複合面積之慣性矩(Moment of Inertia for Composite Area)
      • 面積之慣性積(Product of Inertia for Area)
        • 平行軸定理(Parallel-Axis Theorem)
          • 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)
            • 主軸慣性矩(Principal Moment of Inertia)
              • 慣性矩之莫爾圓(Mohrs Circle for Moment of Inertia)
              • 質量慣性矩(Mass Moment of Inertia)
                • 平行軸定理(Parallel-Axis Theorem)
Page 23: 轉動慣量 Moments of Inertia - cct.me.ntut.edu.twcct.me.ntut.edu.tw/ccteducation/chchting/aiahtm/statics/lecture... · 轉動慣量或慣性矩 I. 轉動慣量也稱慣性矩(Moment

Example 10-1 解答

dA = bdy (45)

Ixb =

ˆAy2dA (46)

=

ˆ h

0y2bdy (47)

=1

3by3|h0 (48)

=1

3bh3 (49)

長方形之邊長b與h則面心慣性矩 macrIx prime macrIy prime與長方形之邊軸慣性矩Ix Iy為

macrIx prime =1

12bh3 (50)

macrIy prime =1

12hb3 (51)

Ix =1

3bh3 (52)

Iy =1

3hb3 (53)

JC = macrIx prime + macrIy prime (54)

JO = Ix + Iy (55)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 21 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 21 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Example 10-2 解答

取微小面積dA=ydx之長方形其邊長b=dx與h=ydy = y2所以

y2 = 400x (56)

y =radic

400x (57)

d macrIx prime = d(1

12bh3) =

1

12y3dx (58)

dIx = d macrIx prime + dAd2y =

1

12y3dx + ydx

(y2

)2=

1

3y3dx (59)

Ix =

ˆ 100

0

1

3y3dx =

ˆ 100

0

1

3

(radic400x

)3dx (60)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 22 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 22 54

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Example 10-3 解答

取微小面積dA=2ydx之長方形其邊長b=dx與h=2y所以

x2 + y2 = a2 (61)

y =radic

a2 minus x2 (62)

d macrIx prime = d(1

12bh3) =

1

12dx(2y)3 =

2

3y3dx (63)

macrIx prime =

ˆ a

minusa

2

3y3dx =

ˆ a

minusa

2

3

(radica2 minus x2

)3dx =

πa4

4(64)

圓之半徑a則面心慣性矩 macrIx prime macrIy prime為

macrIx prime =πa4

4(65)

macrIy prime =πa4

4(66)

JC = macrIx prime + macrIy prime =πa4

2(67)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 23 54

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複複複合合合面面面積積積之之之慣慣慣性性性矩矩矩Moment of Inertia for Composite Area

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 24 54

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (68)

k =

radicI

m(69)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (70)

= I1 + I2 + + In (71)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 25 54

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複合面積之慣性矩

複合面積之慣性矩(Ixc Iyc)為各局部面積之慣性矩(Ixi Iyi )合

Ixc =i=nsumi=1

Ixi (72)

= Ix1 + Ix2 + + Ixn (73)

Iyc =i=nsumi=1

Iyi (74)

= Iy1 + Iy2 + + Iyn (75)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 26 54

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課課課堂堂堂練練練習習習

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

長方形與圓形複合

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長方形複合

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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面面面積積積之之之慣慣慣性性性積積積Product of Inertia for Area

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 28 54

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面積之慣性積(Product of Inertia for Area)

為了下個章節求解慣性矩之極值(極大或極小值)方便數學運算在此先定義面積之慣性積(Ixy )為

dIxy = xydA (76)

Ixy =

ˆAxydA (77)

面積之慣性積的符號可為正負或零當慣性積之面積相對轉軸對稱時則面積之慣性積和為零

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 29 54

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

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平行軸定理

極點

新極點

如左圖面積慣性矩之平行軸定理乃以面心(C)為新極點(O)設定新座標軸(xrsquoyrsquo)其中xrsquo軸平行於原座標x軸yrsquo軸平行於原座標y軸取微小面積dA其在新座標之點座標為(xrsquoyrsquo)面積之慣性積(Ixy )為

dIxy = (x prime + dx)(y prime + dy )dA (78)

當以面心為轉軸點時即x prime = y prime = 0ˆAx primedA = x prime

ˆAdA = 0

ˆAy primedA = y prime

ˆAdA = 0 (79)

Ixy =

ˆA

(x prime + dx)(y prime + dy )dA (80)

=

ˆAx primey primedA + dx

ˆAy primedA + dy

ˆAx primedA + dxdy

ˆAdA (81)

= macrIx primey prime + Adxdy

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課課課堂堂堂練練練習習習

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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傾傾傾斜斜斜軸軸軸面面面積積積之之之慣慣慣性性性矩矩矩

Moment of Inertia for Area about Inclined Axis

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傾斜軸面積之慣性矩

當轉動軸傾斜θ角時也就是座標軸旋轉θ角而Ix Iy Ixy為已知則新的座標軸(uv)可寫為

u = xcosθ + ysinθ (82)

v = ycosθ minus xsinθ (83)

由u軸上之面積慣性矩Iuv軸上之面積慣性矩Iv及慣性積Iuv得知

dIu = v2dA = (ycosθ minus xsinθ)2dA (84)

dIv = u2dA = (xcosθ + ysinθ)2dA (85)

dIuv = uvdA = (xcosθ + ysinθ)(ycosθ minus xsinθ)dA (86)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 34 54

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傾斜軸面積之慣性矩 I

由x軸上之面積慣性矩Ixy軸上之面積慣性矩Iy及慣性積Ixy得知

Ix =

ˆy2dA Iy =

ˆx2dA Ixy =

ˆxydA (87)

Iu =

ˆ(ycosθ minus xsinθ)2dA (88)

=

ˆ(y2cos2θ minus 2xycosθsinθ + x2sin2θ)dA (89)

= cos2θ

ˆy2dAminus 2cosθsinθ

ˆxydA + sin2θ

ˆx2dA (90)

= Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (91)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 35 54

傾斜軸面積之慣性矩 II

Iv =

ˆ(xcosθ + ysinθ)2dA (92)

=

ˆ(x2cos2θ + 2xycosθsinθ + y2sin2θ)dA (93)

= cos2θ

ˆx2dA + 2cosθsinθ

ˆxydA + sin2θ

ˆy2dA (94)

= Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (95)

Iuv =

ˆ(xcosθ + ysinθ)(ycosθ minus xsinθ)dA (96)

=

ˆ(xycos2θ + y2cosθsinθ minus x2cosθsinθ minus xysin2θ)dA (97)

= (cos2 minus sin2θ)

ˆxydA + cosθsinθ

ˆy2dAminus cosθsinθ

ˆx2dA

= Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 36 54

傾斜軸面積之慣性矩 III

再由

sin2θ + cos2θ = 1 (98)

sin(αplusmn β) = sinαcosβ plusmn cosαsinβ (99)

sin2θ = sin(θ + θ) = 2sinθcosθ (100)

cos(αplusmn β) = cosαcosβ ∓ sinαsinβ (101)

cos2θ = cos(θ + θ) = cos2θ minus sin2θ (102)

= 2cos2θ minus 1 = 1minus 2sin2θ (103)

所以

Iu = Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (104)

= Ix(cos2θ + 1

2) + Iy (

1minus cos2θ

2)minus Ixy sin2θ (105)

=Ix + Iy

2+

Ix minus Iy2

cos2θ minus Ixy sin2θ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 37 54

傾斜軸面積之慣性矩 IV

Iv = Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (106)

= Iy (cos2θ + 1

2) + Ix(

1minus cos2θ

2) + Ixy sin2θ (107)

=Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (108)

Iuv = Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ (109)

= Ixycos2θ +1

2Ixsin2θ minus 1

2Iy sin2θ (110)

=Ix minus Iy

2sin2θ + Ixycos2θ (111)

所以

JO = Iu + Iv = Ix + Iy (112)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 38 54

主主主軸軸軸慣慣慣性性性矩矩矩Principal Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 39 54

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主軸慣性矩 I

所謂主軸慣性矩(Principal Moment ofInertia)乃定義發生在慣性矩的極值(極大或極小值)上也就是慣性矩之一次微分式為零或斜率為零

dIu

dθ=

d(Ix+Iy

2+

IxminusIy2

cos2θ minus Ixy sin2θ)

dθ(113)

= minus2Ix minus Iy

2sin2θ minus 2Ixy cos2θ (114)

= 0 (115)

其中dsin2θ = cos2θ dcos2θ = minussin2θ當慣性矩之一次微分式為零時傾斜軸與主軸之夾角θ = θp

minus2Ix minus Iy

2sin2θp minus 2Ixycos2θp = 0 (116)

tan2θ =sin2θ

cos2θ=

minusIxy(Ix minus Iy )2

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 40 54

主軸慣性矩 II

cos2θp =IxminusIy2radic

(IxminusIy2 )2 + I 2xy

sin2θp =minusIxyradic

(IxminusIy2 )2 + I 2xy

(117)

(Iu)max =Ix + Iy

2+

Ix minus Iy2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

minus IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2+

radic(Ix minus Iy

2)2 + I 2xy (118)

(Iv )min =Ix + Iy

2minus Ix minus Iy

2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

+ IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2minusradic

(Ix minus Iy

2)2 + I 2xy (119)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 41 54

主軸慣性矩 III

由tan2θp = tan(π + 2θp)可得到θp1與θp2二個解即2θp2 = π + 2θp1也就是θp2 minus θp1 = π

2

tan2θp = tan(π + 2θp) (120)

2θp2 = π + 2θp1 (121)

θp2 minus θp1 =π

2(122)

所以

Imaxmin

=Ix + Iy

2plusmnradic

(Ix minus Iy

2)2 + I 2xy (123)

Iuv =Ix minus Iy

2

minusIxyradic(IxminusIy2 )2 + I 2xy

+ Ixy

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

(124)

= 0

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 42 54

課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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慣慣慣性性性矩矩矩之之之莫莫莫爾爾爾圓圓圓Mohrrsquos Circle for Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 44 54

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慣性矩之莫爾圓

由於Iu =

Ix + Iy2

+Ix minus Iy

2cos2θ minus Ixy sin2θ (125)

Iv =Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (126)

Iuv =Ix minus Iy

2sin2θ + Ixycos2θ (127)

將式(125)與式(127)各自分別平方並相加可得

(Iu minusIx + Iy

2)2 + I 2uv = (

Ix minus Iy2

)2 + I 2xy (128)

其中Ix Iy Ixy均為常數式(128)可進一步寫成

(Iu minus a)2 + I 2uv = R2 (129)

其中a =Ix+Iy2 R =

radic(IxminusIy2 )2 + I 2xycos2θp1 =

aRsin2θp1 =

minusIxyR 2θp1為OA轉向I軸方向即2θp1為

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 45 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 46 54

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解題技巧

首先確定轉動系統的Ix Iy Ixy接著算出莫爾圓之圓心座標a及半徑R並畫出莫爾圓再由莫爾圓上點出A(Ix Ixy )並由OA轉向I軸找出兩倍夾角2θp1及2θp2其中θp1及θp2為主軸(Principle Axis)與Ix軸的夾角由於主軸(Principle Axis)為極值之軸Imax = Iu及Imin = Iv決定θp1及θp2何者為Imax = Iu與Ix軸的夾角之方法乃比較Ix及Iy之大小Imax = Iu會靠近Ix及Iy之較大者也就是Imax = Iu與Ix及Iy之較大者的夾角比較小

a =Ix + Iy

2(130)

R =

radic(Ix minus Iy

2)2 + I 2xy (131)

2θp2 = π + 2θp1 θp2 minus θp1 =π

2(132)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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質質質量量量慣慣慣性性性矩矩矩Mass Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 48 54

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質量慣性矩

τ = Iα (133)

其中τ為旋轉扭矩I為質量慣性矩α為旋轉角速度

I =

ˆmr2dm (134)

=

ˆVr2ρdV (135)

= ρ

ˆVr2dV (136)

其中r為旋轉半徑或轉軸到旋轉體的距離

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 49 54

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 50 54

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平行軸定理

當通過物體質心(G)並在一轉軸上的質量慣性矩(IG )為已知時可透過座標軸轉換計算物體之質量慣性矩(I)

I =

ˆmr2dm (137)

=

ˆm

[(d + x prime)2 + y prime2]dm (138)

=

ˆm

(x prime2 + y prime2)dm + 2d

ˆmx primedm + d2

ˆmdm (139)

I = IG + md2 (140)

其中IG為在zrsquo軸上通過質心的質量慣性矩m為質量d為通過質心之軸zrsquo並平行原軸z之距離

acutem x primedm = x

acutem dm = 0 since x = 0

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 51 54

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (141)

k =

radicI

m(142)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (143)

= I1 + I2 + + In (144)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 52 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

兩圓盤複合體

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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結結結論論論

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 54 54

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結論

轉動慣量或慣性矩為物體對旋轉運動的慣性而所謂慣性乃物體持續維持不變的行為也就是定義慣性矩可表達物體對本身旋轉運動的維持或持續性好壞更簡單的說慣性矩越大之物體其維持旋轉運動趨勢的能力越大例如惰輪儲能裝置惰輪之慣性矩越大當其轉動以後其維持旋轉運動趨勢的能力越大

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 55 54

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  • 轉動慣量或慣性矩(Moment of Inertia)
  • 面積慣性矩(Moment of Inertia for Area)
    • 面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)
    • 面積慣性矩之迴轉半徑(Radius of Gyration for Area)
      • 複合面積之慣性矩(Moment of Inertia for Composite Area)
      • 面積之慣性積(Product of Inertia for Area)
        • 平行軸定理(Parallel-Axis Theorem)
          • 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)
            • 主軸慣性矩(Principal Moment of Inertia)
              • 慣性矩之莫爾圓(Mohrs Circle for Moment of Inertia)
              • 質量慣性矩(Mass Moment of Inertia)
                • 平行軸定理(Parallel-Axis Theorem)
Page 24: 轉動慣量 Moments of Inertia - cct.me.ntut.edu.twcct.me.ntut.edu.tw/ccteducation/chchting/aiahtm/statics/lecture... · 轉動慣量或慣性矩 I. 轉動慣量也稱慣性矩(Moment

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 21 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Example 10-2 解答

取微小面積dA=ydx之長方形其邊長b=dx與h=ydy = y2所以

y2 = 400x (56)

y =radic

400x (57)

d macrIx prime = d(1

12bh3) =

1

12y3dx (58)

dIx = d macrIx prime + dAd2y =

1

12y3dx + ydx

(y2

)2=

1

3y3dx (59)

Ix =

ˆ 100

0

1

3y3dx =

ˆ 100

0

1

3

(radic400x

)3dx (60)

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 22 54

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Example 10-3 解答

取微小面積dA=2ydx之長方形其邊長b=dx與h=2y所以

x2 + y2 = a2 (61)

y =radic

a2 minus x2 (62)

d macrIx prime = d(1

12bh3) =

1

12dx(2y)3 =

2

3y3dx (63)

macrIx prime =

ˆ a

minusa

2

3y3dx =

ˆ a

minusa

2

3

(radica2 minus x2

)3dx =

πa4

4(64)

圓之半徑a則面心慣性矩 macrIx prime macrIy prime為

macrIx prime =πa4

4(65)

macrIy prime =πa4

4(66)

JC = macrIx prime + macrIy prime =πa4

2(67)

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複複複合合合面面面積積積之之之慣慣慣性性性矩矩矩Moment of Inertia for Composite Area

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (68)

k =

radicI

m(69)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (70)

= I1 + I2 + + In (71)

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複合面積之慣性矩

複合面積之慣性矩(Ixc Iyc)為各局部面積之慣性矩(Ixi Iyi )合

Ixc =i=nsumi=1

Ixi (72)

= Ix1 + Ix2 + + Ixn (73)

Iyc =i=nsumi=1

Iyi (74)

= Iy1 + Iy2 + + Iyn (75)

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

長方形與圓形複合

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

長方形複合

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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面面面積積積之之之慣慣慣性性性積積積Product of Inertia for Area

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 28 54

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面積之慣性積(Product of Inertia for Area)

為了下個章節求解慣性矩之極值(極大或極小值)方便數學運算在此先定義面積之慣性積(Ixy )為

dIxy = xydA (76)

Ixy =

ˆAxydA (77)

面積之慣性積的符號可為正負或零當慣性積之面積相對轉軸對稱時則面積之慣性積和為零

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

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平行軸定理

極點

新極點

如左圖面積慣性矩之平行軸定理乃以面心(C)為新極點(O)設定新座標軸(xrsquoyrsquo)其中xrsquo軸平行於原座標x軸yrsquo軸平行於原座標y軸取微小面積dA其在新座標之點座標為(xrsquoyrsquo)面積之慣性積(Ixy )為

dIxy = (x prime + dx)(y prime + dy )dA (78)

當以面心為轉軸點時即x prime = y prime = 0ˆAx primedA = x prime

ˆAdA = 0

ˆAy primedA = y prime

ˆAdA = 0 (79)

Ixy =

ˆA

(x prime + dx)(y prime + dy )dA (80)

=

ˆAx primey primedA + dx

ˆAy primedA + dy

ˆAx primedA + dxdy

ˆAdA (81)

= macrIx primey prime + Adxdy

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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傾傾傾斜斜斜軸軸軸面面面積積積之之之慣慣慣性性性矩矩矩

Moment of Inertia for Area about Inclined Axis

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傾斜軸面積之慣性矩

當轉動軸傾斜θ角時也就是座標軸旋轉θ角而Ix Iy Ixy為已知則新的座標軸(uv)可寫為

u = xcosθ + ysinθ (82)

v = ycosθ minus xsinθ (83)

由u軸上之面積慣性矩Iuv軸上之面積慣性矩Iv及慣性積Iuv得知

dIu = v2dA = (ycosθ minus xsinθ)2dA (84)

dIv = u2dA = (xcosθ + ysinθ)2dA (85)

dIuv = uvdA = (xcosθ + ysinθ)(ycosθ minus xsinθ)dA (86)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 34 54

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傾斜軸面積之慣性矩 I

由x軸上之面積慣性矩Ixy軸上之面積慣性矩Iy及慣性積Ixy得知

Ix =

ˆy2dA Iy =

ˆx2dA Ixy =

ˆxydA (87)

Iu =

ˆ(ycosθ minus xsinθ)2dA (88)

=

ˆ(y2cos2θ minus 2xycosθsinθ + x2sin2θ)dA (89)

= cos2θ

ˆy2dAminus 2cosθsinθ

ˆxydA + sin2θ

ˆx2dA (90)

= Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (91)

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傾斜軸面積之慣性矩 II

Iv =

ˆ(xcosθ + ysinθ)2dA (92)

=

ˆ(x2cos2θ + 2xycosθsinθ + y2sin2θ)dA (93)

= cos2θ

ˆx2dA + 2cosθsinθ

ˆxydA + sin2θ

ˆy2dA (94)

= Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (95)

Iuv =

ˆ(xcosθ + ysinθ)(ycosθ minus xsinθ)dA (96)

=

ˆ(xycos2θ + y2cosθsinθ minus x2cosθsinθ minus xysin2θ)dA (97)

= (cos2 minus sin2θ)

ˆxydA + cosθsinθ

ˆy2dAminus cosθsinθ

ˆx2dA

= Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 36 54

傾斜軸面積之慣性矩 III

再由

sin2θ + cos2θ = 1 (98)

sin(αplusmn β) = sinαcosβ plusmn cosαsinβ (99)

sin2θ = sin(θ + θ) = 2sinθcosθ (100)

cos(αplusmn β) = cosαcosβ ∓ sinαsinβ (101)

cos2θ = cos(θ + θ) = cos2θ minus sin2θ (102)

= 2cos2θ minus 1 = 1minus 2sin2θ (103)

所以

Iu = Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (104)

= Ix(cos2θ + 1

2) + Iy (

1minus cos2θ

2)minus Ixy sin2θ (105)

=Ix + Iy

2+

Ix minus Iy2

cos2θ minus Ixy sin2θ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 37 54

傾斜軸面積之慣性矩 IV

Iv = Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (106)

= Iy (cos2θ + 1

2) + Ix(

1minus cos2θ

2) + Ixy sin2θ (107)

=Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (108)

Iuv = Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ (109)

= Ixycos2θ +1

2Ixsin2θ minus 1

2Iy sin2θ (110)

=Ix minus Iy

2sin2θ + Ixycos2θ (111)

所以

JO = Iu + Iv = Ix + Iy (112)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 38 54

主主主軸軸軸慣慣慣性性性矩矩矩Principal Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 39 54

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主軸慣性矩 I

所謂主軸慣性矩(Principal Moment ofInertia)乃定義發生在慣性矩的極值(極大或極小值)上也就是慣性矩之一次微分式為零或斜率為零

dIu

dθ=

d(Ix+Iy

2+

IxminusIy2

cos2θ minus Ixy sin2θ)

dθ(113)

= minus2Ix minus Iy

2sin2θ minus 2Ixy cos2θ (114)

= 0 (115)

其中dsin2θ = cos2θ dcos2θ = minussin2θ當慣性矩之一次微分式為零時傾斜軸與主軸之夾角θ = θp

minus2Ix minus Iy

2sin2θp minus 2Ixycos2θp = 0 (116)

tan2θ =sin2θ

cos2θ=

minusIxy(Ix minus Iy )2

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 40 54

主軸慣性矩 II

cos2θp =IxminusIy2radic

(IxminusIy2 )2 + I 2xy

sin2θp =minusIxyradic

(IxminusIy2 )2 + I 2xy

(117)

(Iu)max =Ix + Iy

2+

Ix minus Iy2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

minus IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2+

radic(Ix minus Iy

2)2 + I 2xy (118)

(Iv )min =Ix + Iy

2minus Ix minus Iy

2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

+ IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2minusradic

(Ix minus Iy

2)2 + I 2xy (119)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 41 54

主軸慣性矩 III

由tan2θp = tan(π + 2θp)可得到θp1與θp2二個解即2θp2 = π + 2θp1也就是θp2 minus θp1 = π

2

tan2θp = tan(π + 2θp) (120)

2θp2 = π + 2θp1 (121)

θp2 minus θp1 =π

2(122)

所以

Imaxmin

=Ix + Iy

2plusmnradic

(Ix minus Iy

2)2 + I 2xy (123)

Iuv =Ix minus Iy

2

minusIxyradic(IxminusIy2 )2 + I 2xy

+ Ixy

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

(124)

= 0

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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慣慣慣性性性矩矩矩之之之莫莫莫爾爾爾圓圓圓Mohrrsquos Circle for Moment of Inertia

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慣性矩之莫爾圓

由於Iu =

Ix + Iy2

+Ix minus Iy

2cos2θ minus Ixy sin2θ (125)

Iv =Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (126)

Iuv =Ix minus Iy

2sin2θ + Ixycos2θ (127)

將式(125)與式(127)各自分別平方並相加可得

(Iu minusIx + Iy

2)2 + I 2uv = (

Ix minus Iy2

)2 + I 2xy (128)

其中Ix Iy Ixy均為常數式(128)可進一步寫成

(Iu minus a)2 + I 2uv = R2 (129)

其中a =Ix+Iy2 R =

radic(IxminusIy2 )2 + I 2xycos2θp1 =

aRsin2θp1 =

minusIxyR 2θp1為OA轉向I軸方向即2θp1為

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課課課堂堂堂練練練習習習

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解題技巧

首先確定轉動系統的Ix Iy Ixy接著算出莫爾圓之圓心座標a及半徑R並畫出莫爾圓再由莫爾圓上點出A(Ix Ixy )並由OA轉向I軸找出兩倍夾角2θp1及2θp2其中θp1及θp2為主軸(Principle Axis)與Ix軸的夾角由於主軸(Principle Axis)為極值之軸Imax = Iu及Imin = Iv決定θp1及θp2何者為Imax = Iu與Ix軸的夾角之方法乃比較Ix及Iy之大小Imax = Iu會靠近Ix及Iy之較大者也就是Imax = Iu與Ix及Iy之較大者的夾角比較小

a =Ix + Iy

2(130)

R =

radic(Ix minus Iy

2)2 + I 2xy (131)

2θp2 = π + 2θp1 θp2 minus θp1 =π

2(132)

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質質質量量量慣慣慣性性性矩矩矩Mass Moment of Inertia

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質量慣性矩

τ = Iα (133)

其中τ為旋轉扭矩I為質量慣性矩α為旋轉角速度

I =

ˆmr2dm (134)

=

ˆVr2ρdV (135)

= ρ

ˆVr2dV (136)

其中r為旋轉半徑或轉軸到旋轉體的距離

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

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平行軸定理

當通過物體質心(G)並在一轉軸上的質量慣性矩(IG )為已知時可透過座標軸轉換計算物體之質量慣性矩(I)

I =

ˆmr2dm (137)

=

ˆm

[(d + x prime)2 + y prime2]dm (138)

=

ˆm

(x prime2 + y prime2)dm + 2d

ˆmx primedm + d2

ˆmdm (139)

I = IG + md2 (140)

其中IG為在zrsquo軸上通過質心的質量慣性矩m為質量d為通過質心之軸zrsquo並平行原軸z之距離

acutem x primedm = x

acutem dm = 0 since x = 0

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (141)

k =

radicI

m(142)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (143)

= I1 + I2 + + In (144)

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課課課堂堂堂練練練習習習

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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兩圓盤複合體

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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結結結論論論

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結論

轉動慣量或慣性矩為物體對旋轉運動的慣性而所謂慣性乃物體持續維持不變的行為也就是定義慣性矩可表達物體對本身旋轉運動的維持或持續性好壞更簡單的說慣性矩越大之物體其維持旋轉運動趨勢的能力越大例如惰輪儲能裝置惰輪之慣性矩越大當其轉動以後其維持旋轉運動趨勢的能力越大

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  • 轉動慣量或慣性矩(Moment of Inertia)
  • 面積慣性矩(Moment of Inertia for Area)
    • 面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)
    • 面積慣性矩之迴轉半徑(Radius of Gyration for Area)
      • 複合面積之慣性矩(Moment of Inertia for Composite Area)
      • 面積之慣性積(Product of Inertia for Area)
        • 平行軸定理(Parallel-Axis Theorem)
          • 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)
            • 主軸慣性矩(Principal Moment of Inertia)
              • 慣性矩之莫爾圓(Mohrs Circle for Moment of Inertia)
              • 質量慣性矩(Mass Moment of Inertia)
                • 平行軸定理(Parallel-Axis Theorem)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Example 10-2 解答

取微小面積dA=ydx之長方形其邊長b=dx與h=ydy = y2所以

y2 = 400x (56)

y =radic

400x (57)

d macrIx prime = d(1

12bh3) =

1

12y3dx (58)

dIx = d macrIx prime + dAd2y =

1

12y3dx + ydx

(y2

)2=

1

3y3dx (59)

Ix =

ˆ 100

0

1

3y3dx =

ˆ 100

0

1

3

(radic400x

)3dx (60)

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Example 10-3 解答

取微小面積dA=2ydx之長方形其邊長b=dx與h=2y所以

x2 + y2 = a2 (61)

y =radic

a2 minus x2 (62)

d macrIx prime = d(1

12bh3) =

1

12dx(2y)3 =

2

3y3dx (63)

macrIx prime =

ˆ a

minusa

2

3y3dx =

ˆ a

minusa

2

3

(radica2 minus x2

)3dx =

πa4

4(64)

圓之半徑a則面心慣性矩 macrIx prime macrIy prime為

macrIx prime =πa4

4(65)

macrIy prime =πa4

4(66)

JC = macrIx prime + macrIy prime =πa4

2(67)

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複複複合合合面面面積積積之之之慣慣慣性性性矩矩矩Moment of Inertia for Composite Area

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (68)

k =

radicI

m(69)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (70)

= I1 + I2 + + In (71)

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複合面積之慣性矩

複合面積之慣性矩(Ixc Iyc)為各局部面積之慣性矩(Ixi Iyi )合

Ixc =i=nsumi=1

Ixi (72)

= Ix1 + Ix2 + + Ixn (73)

Iyc =i=nsumi=1

Iyi (74)

= Iy1 + Iy2 + + Iyn (75)

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課課課堂堂堂練練練習習習

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

長方形與圓形複合

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

長方形複合

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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面面面積積積之之之慣慣慣性性性積積積Product of Inertia for Area

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 28 54

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面積之慣性積(Product of Inertia for Area)

為了下個章節求解慣性矩之極值(極大或極小值)方便數學運算在此先定義面積之慣性積(Ixy )為

dIxy = xydA (76)

Ixy =

ˆAxydA (77)

面積之慣性積的符號可為正負或零當慣性積之面積相對轉軸對稱時則面積之慣性積和為零

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 29 54

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 30 54

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平行軸定理

極點

新極點

如左圖面積慣性矩之平行軸定理乃以面心(C)為新極點(O)設定新座標軸(xrsquoyrsquo)其中xrsquo軸平行於原座標x軸yrsquo軸平行於原座標y軸取微小面積dA其在新座標之點座標為(xrsquoyrsquo)面積之慣性積(Ixy )為

dIxy = (x prime + dx)(y prime + dy )dA (78)

當以面心為轉軸點時即x prime = y prime = 0ˆAx primedA = x prime

ˆAdA = 0

ˆAy primedA = y prime

ˆAdA = 0 (79)

Ixy =

ˆA

(x prime + dx)(y prime + dy )dA (80)

=

ˆAx primey primedA + dx

ˆAy primedA + dy

ˆAx primedA + dxdy

ˆAdA (81)

= macrIx primey prime + Adxdy

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 31 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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傾傾傾斜斜斜軸軸軸面面面積積積之之之慣慣慣性性性矩矩矩

Moment of Inertia for Area about Inclined Axis

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 33 54

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傾斜軸面積之慣性矩

當轉動軸傾斜θ角時也就是座標軸旋轉θ角而Ix Iy Ixy為已知則新的座標軸(uv)可寫為

u = xcosθ + ysinθ (82)

v = ycosθ minus xsinθ (83)

由u軸上之面積慣性矩Iuv軸上之面積慣性矩Iv及慣性積Iuv得知

dIu = v2dA = (ycosθ minus xsinθ)2dA (84)

dIv = u2dA = (xcosθ + ysinθ)2dA (85)

dIuv = uvdA = (xcosθ + ysinθ)(ycosθ minus xsinθ)dA (86)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 34 54

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傾斜軸面積之慣性矩 I

由x軸上之面積慣性矩Ixy軸上之面積慣性矩Iy及慣性積Ixy得知

Ix =

ˆy2dA Iy =

ˆx2dA Ixy =

ˆxydA (87)

Iu =

ˆ(ycosθ minus xsinθ)2dA (88)

=

ˆ(y2cos2θ minus 2xycosθsinθ + x2sin2θ)dA (89)

= cos2θ

ˆy2dAminus 2cosθsinθ

ˆxydA + sin2θ

ˆx2dA (90)

= Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (91)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 35 54

傾斜軸面積之慣性矩 II

Iv =

ˆ(xcosθ + ysinθ)2dA (92)

=

ˆ(x2cos2θ + 2xycosθsinθ + y2sin2θ)dA (93)

= cos2θ

ˆx2dA + 2cosθsinθ

ˆxydA + sin2θ

ˆy2dA (94)

= Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (95)

Iuv =

ˆ(xcosθ + ysinθ)(ycosθ minus xsinθ)dA (96)

=

ˆ(xycos2θ + y2cosθsinθ minus x2cosθsinθ minus xysin2θ)dA (97)

= (cos2 minus sin2θ)

ˆxydA + cosθsinθ

ˆy2dAminus cosθsinθ

ˆx2dA

= Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 36 54

傾斜軸面積之慣性矩 III

再由

sin2θ + cos2θ = 1 (98)

sin(αplusmn β) = sinαcosβ plusmn cosαsinβ (99)

sin2θ = sin(θ + θ) = 2sinθcosθ (100)

cos(αplusmn β) = cosαcosβ ∓ sinαsinβ (101)

cos2θ = cos(θ + θ) = cos2θ minus sin2θ (102)

= 2cos2θ minus 1 = 1minus 2sin2θ (103)

所以

Iu = Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (104)

= Ix(cos2θ + 1

2) + Iy (

1minus cos2θ

2)minus Ixy sin2θ (105)

=Ix + Iy

2+

Ix minus Iy2

cos2θ minus Ixy sin2θ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 37 54

傾斜軸面積之慣性矩 IV

Iv = Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (106)

= Iy (cos2θ + 1

2) + Ix(

1minus cos2θ

2) + Ixy sin2θ (107)

=Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (108)

Iuv = Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ (109)

= Ixycos2θ +1

2Ixsin2θ minus 1

2Iy sin2θ (110)

=Ix minus Iy

2sin2θ + Ixycos2θ (111)

所以

JO = Iu + Iv = Ix + Iy (112)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 38 54

主主主軸軸軸慣慣慣性性性矩矩矩Principal Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 39 54

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主軸慣性矩 I

所謂主軸慣性矩(Principal Moment ofInertia)乃定義發生在慣性矩的極值(極大或極小值)上也就是慣性矩之一次微分式為零或斜率為零

dIu

dθ=

d(Ix+Iy

2+

IxminusIy2

cos2θ minus Ixy sin2θ)

dθ(113)

= minus2Ix minus Iy

2sin2θ minus 2Ixy cos2θ (114)

= 0 (115)

其中dsin2θ = cos2θ dcos2θ = minussin2θ當慣性矩之一次微分式為零時傾斜軸與主軸之夾角θ = θp

minus2Ix minus Iy

2sin2θp minus 2Ixycos2θp = 0 (116)

tan2θ =sin2θ

cos2θ=

minusIxy(Ix minus Iy )2

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 40 54

主軸慣性矩 II

cos2θp =IxminusIy2radic

(IxminusIy2 )2 + I 2xy

sin2θp =minusIxyradic

(IxminusIy2 )2 + I 2xy

(117)

(Iu)max =Ix + Iy

2+

Ix minus Iy2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

minus IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2+

radic(Ix minus Iy

2)2 + I 2xy (118)

(Iv )min =Ix + Iy

2minus Ix minus Iy

2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

+ IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2minusradic

(Ix minus Iy

2)2 + I 2xy (119)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 41 54

主軸慣性矩 III

由tan2θp = tan(π + 2θp)可得到θp1與θp2二個解即2θp2 = π + 2θp1也就是θp2 minus θp1 = π

2

tan2θp = tan(π + 2θp) (120)

2θp2 = π + 2θp1 (121)

θp2 minus θp1 =π

2(122)

所以

Imaxmin

=Ix + Iy

2plusmnradic

(Ix minus Iy

2)2 + I 2xy (123)

Iuv =Ix minus Iy

2

minusIxyradic(IxminusIy2 )2 + I 2xy

+ Ixy

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

(124)

= 0

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 42 54

課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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慣慣慣性性性矩矩矩之之之莫莫莫爾爾爾圓圓圓Mohrrsquos Circle for Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 44 54

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慣性矩之莫爾圓

由於Iu =

Ix + Iy2

+Ix minus Iy

2cos2θ minus Ixy sin2θ (125)

Iv =Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (126)

Iuv =Ix minus Iy

2sin2θ + Ixycos2θ (127)

將式(125)與式(127)各自分別平方並相加可得

(Iu minusIx + Iy

2)2 + I 2uv = (

Ix minus Iy2

)2 + I 2xy (128)

其中Ix Iy Ixy均為常數式(128)可進一步寫成

(Iu minus a)2 + I 2uv = R2 (129)

其中a =Ix+Iy2 R =

radic(IxminusIy2 )2 + I 2xycos2θp1 =

aRsin2θp1 =

minusIxyR 2θp1為OA轉向I軸方向即2θp1為

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 45 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 46 54

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解題技巧

首先確定轉動系統的Ix Iy Ixy接著算出莫爾圓之圓心座標a及半徑R並畫出莫爾圓再由莫爾圓上點出A(Ix Ixy )並由OA轉向I軸找出兩倍夾角2θp1及2θp2其中θp1及θp2為主軸(Principle Axis)與Ix軸的夾角由於主軸(Principle Axis)為極值之軸Imax = Iu及Imin = Iv決定θp1及θp2何者為Imax = Iu與Ix軸的夾角之方法乃比較Ix及Iy之大小Imax = Iu會靠近Ix及Iy之較大者也就是Imax = Iu與Ix及Iy之較大者的夾角比較小

a =Ix + Iy

2(130)

R =

radic(Ix minus Iy

2)2 + I 2xy (131)

2θp2 = π + 2θp1 θp2 minus θp1 =π

2(132)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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質質質量量量慣慣慣性性性矩矩矩Mass Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 48 54

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質量慣性矩

τ = Iα (133)

其中τ為旋轉扭矩I為質量慣性矩α為旋轉角速度

I =

ˆmr2dm (134)

=

ˆVr2ρdV (135)

= ρ

ˆVr2dV (136)

其中r為旋轉半徑或轉軸到旋轉體的距離

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 49 54

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 50 54

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平行軸定理

當通過物體質心(G)並在一轉軸上的質量慣性矩(IG )為已知時可透過座標軸轉換計算物體之質量慣性矩(I)

I =

ˆmr2dm (137)

=

ˆm

[(d + x prime)2 + y prime2]dm (138)

=

ˆm

(x prime2 + y prime2)dm + 2d

ˆmx primedm + d2

ˆmdm (139)

I = IG + md2 (140)

其中IG為在zrsquo軸上通過質心的質量慣性矩m為質量d為通過質心之軸zrsquo並平行原軸z之距離

acutem x primedm = x

acutem dm = 0 since x = 0

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 51 54

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (141)

k =

radicI

m(142)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (143)

= I1 + I2 + + In (144)

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課課課堂堂堂練練練習習習

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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兩圓盤複合體

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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結結結論論論

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結論

轉動慣量或慣性矩為物體對旋轉運動的慣性而所謂慣性乃物體持續維持不變的行為也就是定義慣性矩可表達物體對本身旋轉運動的維持或持續性好壞更簡單的說慣性矩越大之物體其維持旋轉運動趨勢的能力越大例如惰輪儲能裝置惰輪之慣性矩越大當其轉動以後其維持旋轉運動趨勢的能力越大

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 55 54

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  • 轉動慣量或慣性矩(Moment of Inertia)
  • 面積慣性矩(Moment of Inertia for Area)
    • 面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)
    • 面積慣性矩之迴轉半徑(Radius of Gyration for Area)
      • 複合面積之慣性矩(Moment of Inertia for Composite Area)
      • 面積之慣性積(Product of Inertia for Area)
        • 平行軸定理(Parallel-Axis Theorem)
          • 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)
            • 主軸慣性矩(Principal Moment of Inertia)
              • 慣性矩之莫爾圓(Mohrs Circle for Moment of Inertia)
              • 質量慣性矩(Mass Moment of Inertia)
                • 平行軸定理(Parallel-Axis Theorem)
Page 26: 轉動慣量 Moments of Inertia - cct.me.ntut.edu.twcct.me.ntut.edu.tw/ccteducation/chchting/aiahtm/statics/lecture... · 轉動慣量或慣性矩 I. 轉動慣量也稱慣性矩(Moment

Example 10-2 解答

取微小面積dA=ydx之長方形其邊長b=dx與h=ydy = y2所以

y2 = 400x (56)

y =radic

400x (57)

d macrIx prime = d(1

12bh3) =

1

12y3dx (58)

dIx = d macrIx prime + dAd2y =

1

12y3dx + ydx

(y2

)2=

1

3y3dx (59)

Ix =

ˆ 100

0

1

3y3dx =

ˆ 100

0

1

3

(radic400x

)3dx (60)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 22 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Example 10-3 解答

取微小面積dA=2ydx之長方形其邊長b=dx與h=2y所以

x2 + y2 = a2 (61)

y =radic

a2 minus x2 (62)

d macrIx prime = d(1

12bh3) =

1

12dx(2y)3 =

2

3y3dx (63)

macrIx prime =

ˆ a

minusa

2

3y3dx =

ˆ a

minusa

2

3

(radica2 minus x2

)3dx =

πa4

4(64)

圓之半徑a則面心慣性矩 macrIx prime macrIy prime為

macrIx prime =πa4

4(65)

macrIy prime =πa4

4(66)

JC = macrIx prime + macrIy prime =πa4

2(67)

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複複複合合合面面面積積積之之之慣慣慣性性性矩矩矩Moment of Inertia for Composite Area

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (68)

k =

radicI

m(69)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (70)

= I1 + I2 + + In (71)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 25 54

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複合面積之慣性矩

複合面積之慣性矩(Ixc Iyc)為各局部面積之慣性矩(Ixi Iyi )合

Ixc =i=nsumi=1

Ixi (72)

= Ix1 + Ix2 + + Ixn (73)

Iyc =i=nsumi=1

Iyi (74)

= Iy1 + Iy2 + + Iyn (75)

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課課課堂堂堂練練練習習習

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

長方形與圓形複合

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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長方形複合

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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面面面積積積之之之慣慣慣性性性積積積Product of Inertia for Area

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面積之慣性積(Product of Inertia for Area)

為了下個章節求解慣性矩之極值(極大或極小值)方便數學運算在此先定義面積之慣性積(Ixy )為

dIxy = xydA (76)

Ixy =

ˆAxydA (77)

面積之慣性積的符號可為正負或零當慣性積之面積相對轉軸對稱時則面積之慣性積和為零

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

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平行軸定理

極點

新極點

如左圖面積慣性矩之平行軸定理乃以面心(C)為新極點(O)設定新座標軸(xrsquoyrsquo)其中xrsquo軸平行於原座標x軸yrsquo軸平行於原座標y軸取微小面積dA其在新座標之點座標為(xrsquoyrsquo)面積之慣性積(Ixy )為

dIxy = (x prime + dx)(y prime + dy )dA (78)

當以面心為轉軸點時即x prime = y prime = 0ˆAx primedA = x prime

ˆAdA = 0

ˆAy primedA = y prime

ˆAdA = 0 (79)

Ixy =

ˆA

(x prime + dx)(y prime + dy )dA (80)

=

ˆAx primey primedA + dx

ˆAy primedA + dy

ˆAx primedA + dxdy

ˆAdA (81)

= macrIx primey prime + Adxdy

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課課課堂堂堂練練練習習習

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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傾傾傾斜斜斜軸軸軸面面面積積積之之之慣慣慣性性性矩矩矩

Moment of Inertia for Area about Inclined Axis

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傾斜軸面積之慣性矩

當轉動軸傾斜θ角時也就是座標軸旋轉θ角而Ix Iy Ixy為已知則新的座標軸(uv)可寫為

u = xcosθ + ysinθ (82)

v = ycosθ minus xsinθ (83)

由u軸上之面積慣性矩Iuv軸上之面積慣性矩Iv及慣性積Iuv得知

dIu = v2dA = (ycosθ minus xsinθ)2dA (84)

dIv = u2dA = (xcosθ + ysinθ)2dA (85)

dIuv = uvdA = (xcosθ + ysinθ)(ycosθ minus xsinθ)dA (86)

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傾斜軸面積之慣性矩 I

由x軸上之面積慣性矩Ixy軸上之面積慣性矩Iy及慣性積Ixy得知

Ix =

ˆy2dA Iy =

ˆx2dA Ixy =

ˆxydA (87)

Iu =

ˆ(ycosθ minus xsinθ)2dA (88)

=

ˆ(y2cos2θ minus 2xycosθsinθ + x2sin2θ)dA (89)

= cos2θ

ˆy2dAminus 2cosθsinθ

ˆxydA + sin2θ

ˆx2dA (90)

= Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (91)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 35 54

傾斜軸面積之慣性矩 II

Iv =

ˆ(xcosθ + ysinθ)2dA (92)

=

ˆ(x2cos2θ + 2xycosθsinθ + y2sin2θ)dA (93)

= cos2θ

ˆx2dA + 2cosθsinθ

ˆxydA + sin2θ

ˆy2dA (94)

= Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (95)

Iuv =

ˆ(xcosθ + ysinθ)(ycosθ minus xsinθ)dA (96)

=

ˆ(xycos2θ + y2cosθsinθ minus x2cosθsinθ minus xysin2θ)dA (97)

= (cos2 minus sin2θ)

ˆxydA + cosθsinθ

ˆy2dAminus cosθsinθ

ˆx2dA

= Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 36 54

傾斜軸面積之慣性矩 III

再由

sin2θ + cos2θ = 1 (98)

sin(αplusmn β) = sinαcosβ plusmn cosαsinβ (99)

sin2θ = sin(θ + θ) = 2sinθcosθ (100)

cos(αplusmn β) = cosαcosβ ∓ sinαsinβ (101)

cos2θ = cos(θ + θ) = cos2θ minus sin2θ (102)

= 2cos2θ minus 1 = 1minus 2sin2θ (103)

所以

Iu = Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (104)

= Ix(cos2θ + 1

2) + Iy (

1minus cos2θ

2)minus Ixy sin2θ (105)

=Ix + Iy

2+

Ix minus Iy2

cos2θ minus Ixy sin2θ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 37 54

傾斜軸面積之慣性矩 IV

Iv = Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (106)

= Iy (cos2θ + 1

2) + Ix(

1minus cos2θ

2) + Ixy sin2θ (107)

=Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (108)

Iuv = Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ (109)

= Ixycos2θ +1

2Ixsin2θ minus 1

2Iy sin2θ (110)

=Ix minus Iy

2sin2θ + Ixycos2θ (111)

所以

JO = Iu + Iv = Ix + Iy (112)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 38 54

主主主軸軸軸慣慣慣性性性矩矩矩Principal Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 39 54

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主軸慣性矩 I

所謂主軸慣性矩(Principal Moment ofInertia)乃定義發生在慣性矩的極值(極大或極小值)上也就是慣性矩之一次微分式為零或斜率為零

dIu

dθ=

d(Ix+Iy

2+

IxminusIy2

cos2θ minus Ixy sin2θ)

dθ(113)

= minus2Ix minus Iy

2sin2θ minus 2Ixy cos2θ (114)

= 0 (115)

其中dsin2θ = cos2θ dcos2θ = minussin2θ當慣性矩之一次微分式為零時傾斜軸與主軸之夾角θ = θp

minus2Ix minus Iy

2sin2θp minus 2Ixycos2θp = 0 (116)

tan2θ =sin2θ

cos2θ=

minusIxy(Ix minus Iy )2

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 40 54

主軸慣性矩 II

cos2θp =IxminusIy2radic

(IxminusIy2 )2 + I 2xy

sin2θp =minusIxyradic

(IxminusIy2 )2 + I 2xy

(117)

(Iu)max =Ix + Iy

2+

Ix minus Iy2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

minus IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2+

radic(Ix minus Iy

2)2 + I 2xy (118)

(Iv )min =Ix + Iy

2minus Ix minus Iy

2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

+ IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2minusradic

(Ix minus Iy

2)2 + I 2xy (119)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 41 54

主軸慣性矩 III

由tan2θp = tan(π + 2θp)可得到θp1與θp2二個解即2θp2 = π + 2θp1也就是θp2 minus θp1 = π

2

tan2θp = tan(π + 2θp) (120)

2θp2 = π + 2θp1 (121)

θp2 minus θp1 =π

2(122)

所以

Imaxmin

=Ix + Iy

2plusmnradic

(Ix minus Iy

2)2 + I 2xy (123)

Iuv =Ix minus Iy

2

minusIxyradic(IxminusIy2 )2 + I 2xy

+ Ixy

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

(124)

= 0

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 42 54

課課課堂堂堂練練練習習習

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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慣慣慣性性性矩矩矩之之之莫莫莫爾爾爾圓圓圓Mohrrsquos Circle for Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 44 54

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慣性矩之莫爾圓

由於Iu =

Ix + Iy2

+Ix minus Iy

2cos2θ minus Ixy sin2θ (125)

Iv =Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (126)

Iuv =Ix minus Iy

2sin2θ + Ixycos2θ (127)

將式(125)與式(127)各自分別平方並相加可得

(Iu minusIx + Iy

2)2 + I 2uv = (

Ix minus Iy2

)2 + I 2xy (128)

其中Ix Iy Ixy均為常數式(128)可進一步寫成

(Iu minus a)2 + I 2uv = R2 (129)

其中a =Ix+Iy2 R =

radic(IxminusIy2 )2 + I 2xycos2θp1 =

aRsin2θp1 =

minusIxyR 2θp1為OA轉向I軸方向即2θp1為

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 45 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 46 54

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解題技巧

首先確定轉動系統的Ix Iy Ixy接著算出莫爾圓之圓心座標a及半徑R並畫出莫爾圓再由莫爾圓上點出A(Ix Ixy )並由OA轉向I軸找出兩倍夾角2θp1及2θp2其中θp1及θp2為主軸(Principle Axis)與Ix軸的夾角由於主軸(Principle Axis)為極值之軸Imax = Iu及Imin = Iv決定θp1及θp2何者為Imax = Iu與Ix軸的夾角之方法乃比較Ix及Iy之大小Imax = Iu會靠近Ix及Iy之較大者也就是Imax = Iu與Ix及Iy之較大者的夾角比較小

a =Ix + Iy

2(130)

R =

radic(Ix minus Iy

2)2 + I 2xy (131)

2θp2 = π + 2θp1 θp2 minus θp1 =π

2(132)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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質質質量量量慣慣慣性性性矩矩矩Mass Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 48 54

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質量慣性矩

τ = Iα (133)

其中τ為旋轉扭矩I為質量慣性矩α為旋轉角速度

I =

ˆmr2dm (134)

=

ˆVr2ρdV (135)

= ρ

ˆVr2dV (136)

其中r為旋轉半徑或轉軸到旋轉體的距離

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 49 54

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 50 54

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平行軸定理

當通過物體質心(G)並在一轉軸上的質量慣性矩(IG )為已知時可透過座標軸轉換計算物體之質量慣性矩(I)

I =

ˆmr2dm (137)

=

ˆm

[(d + x prime)2 + y prime2]dm (138)

=

ˆm

(x prime2 + y prime2)dm + 2d

ˆmx primedm + d2

ˆmdm (139)

I = IG + md2 (140)

其中IG為在zrsquo軸上通過質心的質量慣性矩m為質量d為通過質心之軸zrsquo並平行原軸z之距離

acutem x primedm = x

acutem dm = 0 since x = 0

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (141)

k =

radicI

m(142)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (143)

= I1 + I2 + + In (144)

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

兩圓盤複合體

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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結結結論論論

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 54 54

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結論

轉動慣量或慣性矩為物體對旋轉運動的慣性而所謂慣性乃物體持續維持不變的行為也就是定義慣性矩可表達物體對本身旋轉運動的維持或持續性好壞更簡單的說慣性矩越大之物體其維持旋轉運動趨勢的能力越大例如惰輪儲能裝置惰輪之慣性矩越大當其轉動以後其維持旋轉運動趨勢的能力越大

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 55 54

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  • 轉動慣量或慣性矩(Moment of Inertia)
  • 面積慣性矩(Moment of Inertia for Area)
    • 面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)
    • 面積慣性矩之迴轉半徑(Radius of Gyration for Area)
      • 複合面積之慣性矩(Moment of Inertia for Composite Area)
      • 面積之慣性積(Product of Inertia for Area)
        • 平行軸定理(Parallel-Axis Theorem)
          • 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)
            • 主軸慣性矩(Principal Moment of Inertia)
              • 慣性矩之莫爾圓(Mohrs Circle for Moment of Inertia)
              • 質量慣性矩(Mass Moment of Inertia)
                • 平行軸定理(Parallel-Axis Theorem)
Page 27: 轉動慣量 Moments of Inertia - cct.me.ntut.edu.twcct.me.ntut.edu.tw/ccteducation/chchting/aiahtm/statics/lecture... · 轉動慣量或慣性矩 I. 轉動慣量也稱慣性矩(Moment

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 22 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 22 54

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Example 10-3 解答

取微小面積dA=2ydx之長方形其邊長b=dx與h=2y所以

x2 + y2 = a2 (61)

y =radic

a2 minus x2 (62)

d macrIx prime = d(1

12bh3) =

1

12dx(2y)3 =

2

3y3dx (63)

macrIx prime =

ˆ a

minusa

2

3y3dx =

ˆ a

minusa

2

3

(radica2 minus x2

)3dx =

πa4

4(64)

圓之半徑a則面心慣性矩 macrIx prime macrIy prime為

macrIx prime =πa4

4(65)

macrIy prime =πa4

4(66)

JC = macrIx prime + macrIy prime =πa4

2(67)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 23 54

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複複複合合合面面面積積積之之之慣慣慣性性性矩矩矩Moment of Inertia for Composite Area

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 24 54

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (68)

k =

radicI

m(69)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (70)

= I1 + I2 + + In (71)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 25 54

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複合面積之慣性矩

複合面積之慣性矩(Ixc Iyc)為各局部面積之慣性矩(Ixi Iyi )合

Ixc =i=nsumi=1

Ixi (72)

= Ix1 + Ix2 + + Ixn (73)

Iyc =i=nsumi=1

Iyi (74)

= Iy1 + Iy2 + + Iyn (75)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 26 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

長方形與圓形複合

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

長方形複合

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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面面面積積積之之之慣慣慣性性性積積積Product of Inertia for Area

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 28 54

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面積之慣性積(Product of Inertia for Area)

為了下個章節求解慣性矩之極值(極大或極小值)方便數學運算在此先定義面積之慣性積(Ixy )為

dIxy = xydA (76)

Ixy =

ˆAxydA (77)

面積之慣性積的符號可為正負或零當慣性積之面積相對轉軸對稱時則面積之慣性積和為零

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 29 54

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 30 54

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平行軸定理

極點

新極點

如左圖面積慣性矩之平行軸定理乃以面心(C)為新極點(O)設定新座標軸(xrsquoyrsquo)其中xrsquo軸平行於原座標x軸yrsquo軸平行於原座標y軸取微小面積dA其在新座標之點座標為(xrsquoyrsquo)面積之慣性積(Ixy )為

dIxy = (x prime + dx)(y prime + dy )dA (78)

當以面心為轉軸點時即x prime = y prime = 0ˆAx primedA = x prime

ˆAdA = 0

ˆAy primedA = y prime

ˆAdA = 0 (79)

Ixy =

ˆA

(x prime + dx)(y prime + dy )dA (80)

=

ˆAx primey primedA + dx

ˆAy primedA + dy

ˆAx primedA + dxdy

ˆAdA (81)

= macrIx primey prime + Adxdy

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 31 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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傾傾傾斜斜斜軸軸軸面面面積積積之之之慣慣慣性性性矩矩矩

Moment of Inertia for Area about Inclined Axis

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 33 54

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傾斜軸面積之慣性矩

當轉動軸傾斜θ角時也就是座標軸旋轉θ角而Ix Iy Ixy為已知則新的座標軸(uv)可寫為

u = xcosθ + ysinθ (82)

v = ycosθ minus xsinθ (83)

由u軸上之面積慣性矩Iuv軸上之面積慣性矩Iv及慣性積Iuv得知

dIu = v2dA = (ycosθ minus xsinθ)2dA (84)

dIv = u2dA = (xcosθ + ysinθ)2dA (85)

dIuv = uvdA = (xcosθ + ysinθ)(ycosθ minus xsinθ)dA (86)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 34 54

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傾斜軸面積之慣性矩 I

由x軸上之面積慣性矩Ixy軸上之面積慣性矩Iy及慣性積Ixy得知

Ix =

ˆy2dA Iy =

ˆx2dA Ixy =

ˆxydA (87)

Iu =

ˆ(ycosθ minus xsinθ)2dA (88)

=

ˆ(y2cos2θ minus 2xycosθsinθ + x2sin2θ)dA (89)

= cos2θ

ˆy2dAminus 2cosθsinθ

ˆxydA + sin2θ

ˆx2dA (90)

= Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (91)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 35 54

傾斜軸面積之慣性矩 II

Iv =

ˆ(xcosθ + ysinθ)2dA (92)

=

ˆ(x2cos2θ + 2xycosθsinθ + y2sin2θ)dA (93)

= cos2θ

ˆx2dA + 2cosθsinθ

ˆxydA + sin2θ

ˆy2dA (94)

= Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (95)

Iuv =

ˆ(xcosθ + ysinθ)(ycosθ minus xsinθ)dA (96)

=

ˆ(xycos2θ + y2cosθsinθ minus x2cosθsinθ minus xysin2θ)dA (97)

= (cos2 minus sin2θ)

ˆxydA + cosθsinθ

ˆy2dAminus cosθsinθ

ˆx2dA

= Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 36 54

傾斜軸面積之慣性矩 III

再由

sin2θ + cos2θ = 1 (98)

sin(αplusmn β) = sinαcosβ plusmn cosαsinβ (99)

sin2θ = sin(θ + θ) = 2sinθcosθ (100)

cos(αplusmn β) = cosαcosβ ∓ sinαsinβ (101)

cos2θ = cos(θ + θ) = cos2θ minus sin2θ (102)

= 2cos2θ minus 1 = 1minus 2sin2θ (103)

所以

Iu = Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (104)

= Ix(cos2θ + 1

2) + Iy (

1minus cos2θ

2)minus Ixy sin2θ (105)

=Ix + Iy

2+

Ix minus Iy2

cos2θ minus Ixy sin2θ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 37 54

傾斜軸面積之慣性矩 IV

Iv = Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (106)

= Iy (cos2θ + 1

2) + Ix(

1minus cos2θ

2) + Ixy sin2θ (107)

=Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (108)

Iuv = Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ (109)

= Ixycos2θ +1

2Ixsin2θ minus 1

2Iy sin2θ (110)

=Ix minus Iy

2sin2θ + Ixycos2θ (111)

所以

JO = Iu + Iv = Ix + Iy (112)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 38 54

主主主軸軸軸慣慣慣性性性矩矩矩Principal Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 39 54

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主軸慣性矩 I

所謂主軸慣性矩(Principal Moment ofInertia)乃定義發生在慣性矩的極值(極大或極小值)上也就是慣性矩之一次微分式為零或斜率為零

dIu

dθ=

d(Ix+Iy

2+

IxminusIy2

cos2θ minus Ixy sin2θ)

dθ(113)

= minus2Ix minus Iy

2sin2θ minus 2Ixy cos2θ (114)

= 0 (115)

其中dsin2θ = cos2θ dcos2θ = minussin2θ當慣性矩之一次微分式為零時傾斜軸與主軸之夾角θ = θp

minus2Ix minus Iy

2sin2θp minus 2Ixycos2θp = 0 (116)

tan2θ =sin2θ

cos2θ=

minusIxy(Ix minus Iy )2

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 40 54

主軸慣性矩 II

cos2θp =IxminusIy2radic

(IxminusIy2 )2 + I 2xy

sin2θp =minusIxyradic

(IxminusIy2 )2 + I 2xy

(117)

(Iu)max =Ix + Iy

2+

Ix minus Iy2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

minus IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2+

radic(Ix minus Iy

2)2 + I 2xy (118)

(Iv )min =Ix + Iy

2minus Ix minus Iy

2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

+ IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2minusradic

(Ix minus Iy

2)2 + I 2xy (119)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 41 54

主軸慣性矩 III

由tan2θp = tan(π + 2θp)可得到θp1與θp2二個解即2θp2 = π + 2θp1也就是θp2 minus θp1 = π

2

tan2θp = tan(π + 2θp) (120)

2θp2 = π + 2θp1 (121)

θp2 minus θp1 =π

2(122)

所以

Imaxmin

=Ix + Iy

2plusmnradic

(Ix minus Iy

2)2 + I 2xy (123)

Iuv =Ix minus Iy

2

minusIxyradic(IxminusIy2 )2 + I 2xy

+ Ixy

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

(124)

= 0

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 42 54

課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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慣慣慣性性性矩矩矩之之之莫莫莫爾爾爾圓圓圓Mohrrsquos Circle for Moment of Inertia

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慣性矩之莫爾圓

由於Iu =

Ix + Iy2

+Ix minus Iy

2cos2θ minus Ixy sin2θ (125)

Iv =Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (126)

Iuv =Ix minus Iy

2sin2θ + Ixycos2θ (127)

將式(125)與式(127)各自分別平方並相加可得

(Iu minusIx + Iy

2)2 + I 2uv = (

Ix minus Iy2

)2 + I 2xy (128)

其中Ix Iy Ixy均為常數式(128)可進一步寫成

(Iu minus a)2 + I 2uv = R2 (129)

其中a =Ix+Iy2 R =

radic(IxminusIy2 )2 + I 2xycos2θp1 =

aRsin2θp1 =

minusIxyR 2θp1為OA轉向I軸方向即2θp1為

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課課課堂堂堂練練練習習習

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解題技巧

首先確定轉動系統的Ix Iy Ixy接著算出莫爾圓之圓心座標a及半徑R並畫出莫爾圓再由莫爾圓上點出A(Ix Ixy )並由OA轉向I軸找出兩倍夾角2θp1及2θp2其中θp1及θp2為主軸(Principle Axis)與Ix軸的夾角由於主軸(Principle Axis)為極值之軸Imax = Iu及Imin = Iv決定θp1及θp2何者為Imax = Iu與Ix軸的夾角之方法乃比較Ix及Iy之大小Imax = Iu會靠近Ix及Iy之較大者也就是Imax = Iu與Ix及Iy之較大者的夾角比較小

a =Ix + Iy

2(130)

R =

radic(Ix minus Iy

2)2 + I 2xy (131)

2θp2 = π + 2θp1 θp2 minus θp1 =π

2(132)

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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質質質量量量慣慣慣性性性矩矩矩Mass Moment of Inertia

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質量慣性矩

τ = Iα (133)

其中τ為旋轉扭矩I為質量慣性矩α為旋轉角速度

I =

ˆmr2dm (134)

=

ˆVr2ρdV (135)

= ρ

ˆVr2dV (136)

其中r為旋轉半徑或轉軸到旋轉體的距離

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

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平行軸定理

當通過物體質心(G)並在一轉軸上的質量慣性矩(IG )為已知時可透過座標軸轉換計算物體之質量慣性矩(I)

I =

ˆmr2dm (137)

=

ˆm

[(d + x prime)2 + y prime2]dm (138)

=

ˆm

(x prime2 + y prime2)dm + 2d

ˆmx primedm + d2

ˆmdm (139)

I = IG + md2 (140)

其中IG為在zrsquo軸上通過質心的質量慣性矩m為質量d為通過質心之軸zrsquo並平行原軸z之距離

acutem x primedm = x

acutem dm = 0 since x = 0

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (141)

k =

radicI

m(142)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (143)

= I1 + I2 + + In (144)

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課課課堂堂堂練練練習習習

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

兩圓盤複合體

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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結結結論論論

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結論

轉動慣量或慣性矩為物體對旋轉運動的慣性而所謂慣性乃物體持續維持不變的行為也就是定義慣性矩可表達物體對本身旋轉運動的維持或持續性好壞更簡單的說慣性矩越大之物體其維持旋轉運動趨勢的能力越大例如惰輪儲能裝置惰輪之慣性矩越大當其轉動以後其維持旋轉運動趨勢的能力越大

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 55 54

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  • 轉動慣量或慣性矩(Moment of Inertia)
  • 面積慣性矩(Moment of Inertia for Area)
    • 面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)
    • 面積慣性矩之迴轉半徑(Radius of Gyration for Area)
      • 複合面積之慣性矩(Moment of Inertia for Composite Area)
      • 面積之慣性積(Product of Inertia for Area)
        • 平行軸定理(Parallel-Axis Theorem)
          • 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)
            • 主軸慣性矩(Principal Moment of Inertia)
              • 慣性矩之莫爾圓(Mohrs Circle for Moment of Inertia)
              • 質量慣性矩(Mass Moment of Inertia)
                • 平行軸定理(Parallel-Axis Theorem)
Page 28: 轉動慣量 Moments of Inertia - cct.me.ntut.edu.twcct.me.ntut.edu.tw/ccteducation/chchting/aiahtm/statics/lecture... · 轉動慣量或慣性矩 I. 轉動慣量也稱慣性矩(Moment

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 22 54

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Example 10-3 解答

取微小面積dA=2ydx之長方形其邊長b=dx與h=2y所以

x2 + y2 = a2 (61)

y =radic

a2 minus x2 (62)

d macrIx prime = d(1

12bh3) =

1

12dx(2y)3 =

2

3y3dx (63)

macrIx prime =

ˆ a

minusa

2

3y3dx =

ˆ a

minusa

2

3

(radica2 minus x2

)3dx =

πa4

4(64)

圓之半徑a則面心慣性矩 macrIx prime macrIy prime為

macrIx prime =πa4

4(65)

macrIy prime =πa4

4(66)

JC = macrIx prime + macrIy prime =πa4

2(67)

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複複複合合合面面面積積積之之之慣慣慣性性性矩矩矩Moment of Inertia for Composite Area

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (68)

k =

radicI

m(69)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (70)

= I1 + I2 + + In (71)

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複合面積之慣性矩

複合面積之慣性矩(Ixc Iyc)為各局部面積之慣性矩(Ixi Iyi )合

Ixc =i=nsumi=1

Ixi (72)

= Ix1 + Ix2 + + Ixn (73)

Iyc =i=nsumi=1

Iyi (74)

= Iy1 + Iy2 + + Iyn (75)

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課課課堂堂堂練練練習習習

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

長方形與圓形複合

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

長方形複合

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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面面面積積積之之之慣慣慣性性性積積積Product of Inertia for Area

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面積之慣性積(Product of Inertia for Area)

為了下個章節求解慣性矩之極值(極大或極小值)方便數學運算在此先定義面積之慣性積(Ixy )為

dIxy = xydA (76)

Ixy =

ˆAxydA (77)

面積之慣性積的符號可為正負或零當慣性積之面積相對轉軸對稱時則面積之慣性積和為零

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

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平行軸定理

極點

新極點

如左圖面積慣性矩之平行軸定理乃以面心(C)為新極點(O)設定新座標軸(xrsquoyrsquo)其中xrsquo軸平行於原座標x軸yrsquo軸平行於原座標y軸取微小面積dA其在新座標之點座標為(xrsquoyrsquo)面積之慣性積(Ixy )為

dIxy = (x prime + dx)(y prime + dy )dA (78)

當以面心為轉軸點時即x prime = y prime = 0ˆAx primedA = x prime

ˆAdA = 0

ˆAy primedA = y prime

ˆAdA = 0 (79)

Ixy =

ˆA

(x prime + dx)(y prime + dy )dA (80)

=

ˆAx primey primedA + dx

ˆAy primedA + dy

ˆAx primedA + dxdy

ˆAdA (81)

= macrIx primey prime + Adxdy

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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傾傾傾斜斜斜軸軸軸面面面積積積之之之慣慣慣性性性矩矩矩

Moment of Inertia for Area about Inclined Axis

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 33 54

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傾斜軸面積之慣性矩

當轉動軸傾斜θ角時也就是座標軸旋轉θ角而Ix Iy Ixy為已知則新的座標軸(uv)可寫為

u = xcosθ + ysinθ (82)

v = ycosθ minus xsinθ (83)

由u軸上之面積慣性矩Iuv軸上之面積慣性矩Iv及慣性積Iuv得知

dIu = v2dA = (ycosθ minus xsinθ)2dA (84)

dIv = u2dA = (xcosθ + ysinθ)2dA (85)

dIuv = uvdA = (xcosθ + ysinθ)(ycosθ minus xsinθ)dA (86)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 34 54

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傾斜軸面積之慣性矩 I

由x軸上之面積慣性矩Ixy軸上之面積慣性矩Iy及慣性積Ixy得知

Ix =

ˆy2dA Iy =

ˆx2dA Ixy =

ˆxydA (87)

Iu =

ˆ(ycosθ minus xsinθ)2dA (88)

=

ˆ(y2cos2θ minus 2xycosθsinθ + x2sin2θ)dA (89)

= cos2θ

ˆy2dAminus 2cosθsinθ

ˆxydA + sin2θ

ˆx2dA (90)

= Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (91)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 35 54

傾斜軸面積之慣性矩 II

Iv =

ˆ(xcosθ + ysinθ)2dA (92)

=

ˆ(x2cos2θ + 2xycosθsinθ + y2sin2θ)dA (93)

= cos2θ

ˆx2dA + 2cosθsinθ

ˆxydA + sin2θ

ˆy2dA (94)

= Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (95)

Iuv =

ˆ(xcosθ + ysinθ)(ycosθ minus xsinθ)dA (96)

=

ˆ(xycos2θ + y2cosθsinθ minus x2cosθsinθ minus xysin2θ)dA (97)

= (cos2 minus sin2θ)

ˆxydA + cosθsinθ

ˆy2dAminus cosθsinθ

ˆx2dA

= Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 36 54

傾斜軸面積之慣性矩 III

再由

sin2θ + cos2θ = 1 (98)

sin(αplusmn β) = sinαcosβ plusmn cosαsinβ (99)

sin2θ = sin(θ + θ) = 2sinθcosθ (100)

cos(αplusmn β) = cosαcosβ ∓ sinαsinβ (101)

cos2θ = cos(θ + θ) = cos2θ minus sin2θ (102)

= 2cos2θ minus 1 = 1minus 2sin2θ (103)

所以

Iu = Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (104)

= Ix(cos2θ + 1

2) + Iy (

1minus cos2θ

2)minus Ixy sin2θ (105)

=Ix + Iy

2+

Ix minus Iy2

cos2θ minus Ixy sin2θ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 37 54

傾斜軸面積之慣性矩 IV

Iv = Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (106)

= Iy (cos2θ + 1

2) + Ix(

1minus cos2θ

2) + Ixy sin2θ (107)

=Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (108)

Iuv = Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ (109)

= Ixycos2θ +1

2Ixsin2θ minus 1

2Iy sin2θ (110)

=Ix minus Iy

2sin2θ + Ixycos2θ (111)

所以

JO = Iu + Iv = Ix + Iy (112)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 38 54

主主主軸軸軸慣慣慣性性性矩矩矩Principal Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 39 54

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主軸慣性矩 I

所謂主軸慣性矩(Principal Moment ofInertia)乃定義發生在慣性矩的極值(極大或極小值)上也就是慣性矩之一次微分式為零或斜率為零

dIu

dθ=

d(Ix+Iy

2+

IxminusIy2

cos2θ minus Ixy sin2θ)

dθ(113)

= minus2Ix minus Iy

2sin2θ minus 2Ixy cos2θ (114)

= 0 (115)

其中dsin2θ = cos2θ dcos2θ = minussin2θ當慣性矩之一次微分式為零時傾斜軸與主軸之夾角θ = θp

minus2Ix minus Iy

2sin2θp minus 2Ixycos2θp = 0 (116)

tan2θ =sin2θ

cos2θ=

minusIxy(Ix minus Iy )2

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 40 54

主軸慣性矩 II

cos2θp =IxminusIy2radic

(IxminusIy2 )2 + I 2xy

sin2θp =minusIxyradic

(IxminusIy2 )2 + I 2xy

(117)

(Iu)max =Ix + Iy

2+

Ix minus Iy2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

minus IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2+

radic(Ix minus Iy

2)2 + I 2xy (118)

(Iv )min =Ix + Iy

2minus Ix minus Iy

2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

+ IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2minusradic

(Ix minus Iy

2)2 + I 2xy (119)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 41 54

主軸慣性矩 III

由tan2θp = tan(π + 2θp)可得到θp1與θp2二個解即2θp2 = π + 2θp1也就是θp2 minus θp1 = π

2

tan2θp = tan(π + 2θp) (120)

2θp2 = π + 2θp1 (121)

θp2 minus θp1 =π

2(122)

所以

Imaxmin

=Ix + Iy

2plusmnradic

(Ix minus Iy

2)2 + I 2xy (123)

Iuv =Ix minus Iy

2

minusIxyradic(IxminusIy2 )2 + I 2xy

+ Ixy

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

(124)

= 0

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 42 54

課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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慣慣慣性性性矩矩矩之之之莫莫莫爾爾爾圓圓圓Mohrrsquos Circle for Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 44 54

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慣性矩之莫爾圓

由於Iu =

Ix + Iy2

+Ix minus Iy

2cos2θ minus Ixy sin2θ (125)

Iv =Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (126)

Iuv =Ix minus Iy

2sin2θ + Ixycos2θ (127)

將式(125)與式(127)各自分別平方並相加可得

(Iu minusIx + Iy

2)2 + I 2uv = (

Ix minus Iy2

)2 + I 2xy (128)

其中Ix Iy Ixy均為常數式(128)可進一步寫成

(Iu minus a)2 + I 2uv = R2 (129)

其中a =Ix+Iy2 R =

radic(IxminusIy2 )2 + I 2xycos2θp1 =

aRsin2θp1 =

minusIxyR 2θp1為OA轉向I軸方向即2θp1為

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 45 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 46 54

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解題技巧

首先確定轉動系統的Ix Iy Ixy接著算出莫爾圓之圓心座標a及半徑R並畫出莫爾圓再由莫爾圓上點出A(Ix Ixy )並由OA轉向I軸找出兩倍夾角2θp1及2θp2其中θp1及θp2為主軸(Principle Axis)與Ix軸的夾角由於主軸(Principle Axis)為極值之軸Imax = Iu及Imin = Iv決定θp1及θp2何者為Imax = Iu與Ix軸的夾角之方法乃比較Ix及Iy之大小Imax = Iu會靠近Ix及Iy之較大者也就是Imax = Iu與Ix及Iy之較大者的夾角比較小

a =Ix + Iy

2(130)

R =

radic(Ix minus Iy

2)2 + I 2xy (131)

2θp2 = π + 2θp1 θp2 minus θp1 =π

2(132)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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質質質量量量慣慣慣性性性矩矩矩Mass Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 48 54

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質量慣性矩

τ = Iα (133)

其中τ為旋轉扭矩I為質量慣性矩α為旋轉角速度

I =

ˆmr2dm (134)

=

ˆVr2ρdV (135)

= ρ

ˆVr2dV (136)

其中r為旋轉半徑或轉軸到旋轉體的距離

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 49 54

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 50 54

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平行軸定理

當通過物體質心(G)並在一轉軸上的質量慣性矩(IG )為已知時可透過座標軸轉換計算物體之質量慣性矩(I)

I =

ˆmr2dm (137)

=

ˆm

[(d + x prime)2 + y prime2]dm (138)

=

ˆm

(x prime2 + y prime2)dm + 2d

ˆmx primedm + d2

ˆmdm (139)

I = IG + md2 (140)

其中IG為在zrsquo軸上通過質心的質量慣性矩m為質量d為通過質心之軸zrsquo並平行原軸z之距離

acutem x primedm = x

acutem dm = 0 since x = 0

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 51 54

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (141)

k =

radicI

m(142)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (143)

= I1 + I2 + + In (144)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 52 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

兩圓盤複合體

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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結結結論論論

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 54 54

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結論

轉動慣量或慣性矩為物體對旋轉運動的慣性而所謂慣性乃物體持續維持不變的行為也就是定義慣性矩可表達物體對本身旋轉運動的維持或持續性好壞更簡單的說慣性矩越大之物體其維持旋轉運動趨勢的能力越大例如惰輪儲能裝置惰輪之慣性矩越大當其轉動以後其維持旋轉運動趨勢的能力越大

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 55 54

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  • 轉動慣量或慣性矩(Moment of Inertia)
  • 面積慣性矩(Moment of Inertia for Area)
    • 面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)
    • 面積慣性矩之迴轉半徑(Radius of Gyration for Area)
      • 複合面積之慣性矩(Moment of Inertia for Composite Area)
      • 面積之慣性積(Product of Inertia for Area)
        • 平行軸定理(Parallel-Axis Theorem)
          • 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)
            • 主軸慣性矩(Principal Moment of Inertia)
              • 慣性矩之莫爾圓(Mohrs Circle for Moment of Inertia)
              • 質量慣性矩(Mass Moment of Inertia)
                • 平行軸定理(Parallel-Axis Theorem)
Page 29: 轉動慣量 Moments of Inertia - cct.me.ntut.edu.twcct.me.ntut.edu.tw/ccteducation/chchting/aiahtm/statics/lecture... · 轉動慣量或慣性矩 I. 轉動慣量也稱慣性矩(Moment

Example 10-3 解答

取微小面積dA=2ydx之長方形其邊長b=dx與h=2y所以

x2 + y2 = a2 (61)

y =radic

a2 minus x2 (62)

d macrIx prime = d(1

12bh3) =

1

12dx(2y)3 =

2

3y3dx (63)

macrIx prime =

ˆ a

minusa

2

3y3dx =

ˆ a

minusa

2

3

(radica2 minus x2

)3dx =

πa4

4(64)

圓之半徑a則面心慣性矩 macrIx prime macrIy prime為

macrIx prime =πa4

4(65)

macrIy prime =πa4

4(66)

JC = macrIx prime + macrIy prime =πa4

2(67)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 23 54

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複複複合合合面面面積積積之之之慣慣慣性性性矩矩矩Moment of Inertia for Composite Area

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 24 54

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (68)

k =

radicI

m(69)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (70)

= I1 + I2 + + In (71)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 25 54

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複合面積之慣性矩

複合面積之慣性矩(Ixc Iyc)為各局部面積之慣性矩(Ixi Iyi )合

Ixc =i=nsumi=1

Ixi (72)

= Ix1 + Ix2 + + Ixn (73)

Iyc =i=nsumi=1

Iyi (74)

= Iy1 + Iy2 + + Iyn (75)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 26 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

長方形與圓形複合

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

長方形複合

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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面面面積積積之之之慣慣慣性性性積積積Product of Inertia for Area

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 28 54

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面積之慣性積(Product of Inertia for Area)

為了下個章節求解慣性矩之極值(極大或極小值)方便數學運算在此先定義面積之慣性積(Ixy )為

dIxy = xydA (76)

Ixy =

ˆAxydA (77)

面積之慣性積的符號可為正負或零當慣性積之面積相對轉軸對稱時則面積之慣性積和為零

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 29 54

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 30 54

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平行軸定理

極點

新極點

如左圖面積慣性矩之平行軸定理乃以面心(C)為新極點(O)設定新座標軸(xrsquoyrsquo)其中xrsquo軸平行於原座標x軸yrsquo軸平行於原座標y軸取微小面積dA其在新座標之點座標為(xrsquoyrsquo)面積之慣性積(Ixy )為

dIxy = (x prime + dx)(y prime + dy )dA (78)

當以面心為轉軸點時即x prime = y prime = 0ˆAx primedA = x prime

ˆAdA = 0

ˆAy primedA = y prime

ˆAdA = 0 (79)

Ixy =

ˆA

(x prime + dx)(y prime + dy )dA (80)

=

ˆAx primey primedA + dx

ˆAy primedA + dy

ˆAx primedA + dxdy

ˆAdA (81)

= macrIx primey prime + Adxdy

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課課課堂堂堂練練練習習習

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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傾傾傾斜斜斜軸軸軸面面面積積積之之之慣慣慣性性性矩矩矩

Moment of Inertia for Area about Inclined Axis

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 33 54

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傾斜軸面積之慣性矩

當轉動軸傾斜θ角時也就是座標軸旋轉θ角而Ix Iy Ixy為已知則新的座標軸(uv)可寫為

u = xcosθ + ysinθ (82)

v = ycosθ minus xsinθ (83)

由u軸上之面積慣性矩Iuv軸上之面積慣性矩Iv及慣性積Iuv得知

dIu = v2dA = (ycosθ minus xsinθ)2dA (84)

dIv = u2dA = (xcosθ + ysinθ)2dA (85)

dIuv = uvdA = (xcosθ + ysinθ)(ycosθ minus xsinθ)dA (86)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 34 54

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傾斜軸面積之慣性矩 I

由x軸上之面積慣性矩Ixy軸上之面積慣性矩Iy及慣性積Ixy得知

Ix =

ˆy2dA Iy =

ˆx2dA Ixy =

ˆxydA (87)

Iu =

ˆ(ycosθ minus xsinθ)2dA (88)

=

ˆ(y2cos2θ minus 2xycosθsinθ + x2sin2θ)dA (89)

= cos2θ

ˆy2dAminus 2cosθsinθ

ˆxydA + sin2θ

ˆx2dA (90)

= Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (91)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 35 54

傾斜軸面積之慣性矩 II

Iv =

ˆ(xcosθ + ysinθ)2dA (92)

=

ˆ(x2cos2θ + 2xycosθsinθ + y2sin2θ)dA (93)

= cos2θ

ˆx2dA + 2cosθsinθ

ˆxydA + sin2θ

ˆy2dA (94)

= Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (95)

Iuv =

ˆ(xcosθ + ysinθ)(ycosθ minus xsinθ)dA (96)

=

ˆ(xycos2θ + y2cosθsinθ minus x2cosθsinθ minus xysin2θ)dA (97)

= (cos2 minus sin2θ)

ˆxydA + cosθsinθ

ˆy2dAminus cosθsinθ

ˆx2dA

= Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 36 54

傾斜軸面積之慣性矩 III

再由

sin2θ + cos2θ = 1 (98)

sin(αplusmn β) = sinαcosβ plusmn cosαsinβ (99)

sin2θ = sin(θ + θ) = 2sinθcosθ (100)

cos(αplusmn β) = cosαcosβ ∓ sinαsinβ (101)

cos2θ = cos(θ + θ) = cos2θ minus sin2θ (102)

= 2cos2θ minus 1 = 1minus 2sin2θ (103)

所以

Iu = Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (104)

= Ix(cos2θ + 1

2) + Iy (

1minus cos2θ

2)minus Ixy sin2θ (105)

=Ix + Iy

2+

Ix minus Iy2

cos2θ minus Ixy sin2θ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 37 54

傾斜軸面積之慣性矩 IV

Iv = Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (106)

= Iy (cos2θ + 1

2) + Ix(

1minus cos2θ

2) + Ixy sin2θ (107)

=Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (108)

Iuv = Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ (109)

= Ixycos2θ +1

2Ixsin2θ minus 1

2Iy sin2θ (110)

=Ix minus Iy

2sin2θ + Ixycos2θ (111)

所以

JO = Iu + Iv = Ix + Iy (112)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 38 54

主主主軸軸軸慣慣慣性性性矩矩矩Principal Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 39 54

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主軸慣性矩 I

所謂主軸慣性矩(Principal Moment ofInertia)乃定義發生在慣性矩的極值(極大或極小值)上也就是慣性矩之一次微分式為零或斜率為零

dIu

dθ=

d(Ix+Iy

2+

IxminusIy2

cos2θ minus Ixy sin2θ)

dθ(113)

= minus2Ix minus Iy

2sin2θ minus 2Ixy cos2θ (114)

= 0 (115)

其中dsin2θ = cos2θ dcos2θ = minussin2θ當慣性矩之一次微分式為零時傾斜軸與主軸之夾角θ = θp

minus2Ix minus Iy

2sin2θp minus 2Ixycos2θp = 0 (116)

tan2θ =sin2θ

cos2θ=

minusIxy(Ix minus Iy )2

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 40 54

主軸慣性矩 II

cos2θp =IxminusIy2radic

(IxminusIy2 )2 + I 2xy

sin2θp =minusIxyradic

(IxminusIy2 )2 + I 2xy

(117)

(Iu)max =Ix + Iy

2+

Ix minus Iy2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

minus IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2+

radic(Ix minus Iy

2)2 + I 2xy (118)

(Iv )min =Ix + Iy

2minus Ix minus Iy

2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

+ IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2minusradic

(Ix minus Iy

2)2 + I 2xy (119)

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主軸慣性矩 III

由tan2θp = tan(π + 2θp)可得到θp1與θp2二個解即2θp2 = π + 2θp1也就是θp2 minus θp1 = π

2

tan2θp = tan(π + 2θp) (120)

2θp2 = π + 2θp1 (121)

θp2 minus θp1 =π

2(122)

所以

Imaxmin

=Ix + Iy

2plusmnradic

(Ix minus Iy

2)2 + I 2xy (123)

Iuv =Ix minus Iy

2

minusIxyradic(IxminusIy2 )2 + I 2xy

+ Ixy

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

(124)

= 0

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課課課堂堂堂練練練習習習

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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慣慣慣性性性矩矩矩之之之莫莫莫爾爾爾圓圓圓Mohrrsquos Circle for Moment of Inertia

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慣性矩之莫爾圓

由於Iu =

Ix + Iy2

+Ix minus Iy

2cos2θ minus Ixy sin2θ (125)

Iv =Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (126)

Iuv =Ix minus Iy

2sin2θ + Ixycos2θ (127)

將式(125)與式(127)各自分別平方並相加可得

(Iu minusIx + Iy

2)2 + I 2uv = (

Ix minus Iy2

)2 + I 2xy (128)

其中Ix Iy Ixy均為常數式(128)可進一步寫成

(Iu minus a)2 + I 2uv = R2 (129)

其中a =Ix+Iy2 R =

radic(IxminusIy2 )2 + I 2xycos2θp1 =

aRsin2θp1 =

minusIxyR 2θp1為OA轉向I軸方向即2θp1為

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課課課堂堂堂練練練習習習

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解題技巧

首先確定轉動系統的Ix Iy Ixy接著算出莫爾圓之圓心座標a及半徑R並畫出莫爾圓再由莫爾圓上點出A(Ix Ixy )並由OA轉向I軸找出兩倍夾角2θp1及2θp2其中θp1及θp2為主軸(Principle Axis)與Ix軸的夾角由於主軸(Principle Axis)為極值之軸Imax = Iu及Imin = Iv決定θp1及θp2何者為Imax = Iu與Ix軸的夾角之方法乃比較Ix及Iy之大小Imax = Iu會靠近Ix及Iy之較大者也就是Imax = Iu與Ix及Iy之較大者的夾角比較小

a =Ix + Iy

2(130)

R =

radic(Ix minus Iy

2)2 + I 2xy (131)

2θp2 = π + 2θp1 θp2 minus θp1 =π

2(132)

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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質質質量量量慣慣慣性性性矩矩矩Mass Moment of Inertia

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質量慣性矩

τ = Iα (133)

其中τ為旋轉扭矩I為質量慣性矩α為旋轉角速度

I =

ˆmr2dm (134)

=

ˆVr2ρdV (135)

= ρ

ˆVr2dV (136)

其中r為旋轉半徑或轉軸到旋轉體的距離

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

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平行軸定理

當通過物體質心(G)並在一轉軸上的質量慣性矩(IG )為已知時可透過座標軸轉換計算物體之質量慣性矩(I)

I =

ˆmr2dm (137)

=

ˆm

[(d + x prime)2 + y prime2]dm (138)

=

ˆm

(x prime2 + y prime2)dm + 2d

ˆmx primedm + d2

ˆmdm (139)

I = IG + md2 (140)

其中IG為在zrsquo軸上通過質心的質量慣性矩m為質量d為通過質心之軸zrsquo並平行原軸z之距離

acutem x primedm = x

acutem dm = 0 since x = 0

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (141)

k =

radicI

m(142)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (143)

= I1 + I2 + + In (144)

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課課課堂堂堂練練練習習習

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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兩圓盤複合體

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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結結結論論論

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結論

轉動慣量或慣性矩為物體對旋轉運動的慣性而所謂慣性乃物體持續維持不變的行為也就是定義慣性矩可表達物體對本身旋轉運動的維持或持續性好壞更簡單的說慣性矩越大之物體其維持旋轉運動趨勢的能力越大例如惰輪儲能裝置惰輪之慣性矩越大當其轉動以後其維持旋轉運動趨勢的能力越大

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  • 轉動慣量或慣性矩(Moment of Inertia)
  • 面積慣性矩(Moment of Inertia for Area)
    • 面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)
    • 面積慣性矩之迴轉半徑(Radius of Gyration for Area)
      • 複合面積之慣性矩(Moment of Inertia for Composite Area)
      • 面積之慣性積(Product of Inertia for Area)
        • 平行軸定理(Parallel-Axis Theorem)
          • 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)
            • 主軸慣性矩(Principal Moment of Inertia)
              • 慣性矩之莫爾圓(Mohrs Circle for Moment of Inertia)
              • 質量慣性矩(Mass Moment of Inertia)
                • 平行軸定理(Parallel-Axis Theorem)
Page 30: 轉動慣量 Moments of Inertia - cct.me.ntut.edu.twcct.me.ntut.edu.tw/ccteducation/chchting/aiahtm/statics/lecture... · 轉動慣量或慣性矩 I. 轉動慣量也稱慣性矩(Moment

複複複合合合面面面積積積之之之慣慣慣性性性矩矩矩Moment of Inertia for Composite Area

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 24 54

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (68)

k =

radicI

m(69)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (70)

= I1 + I2 + + In (71)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 25 54

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複合面積之慣性矩

複合面積之慣性矩(Ixc Iyc)為各局部面積之慣性矩(Ixi Iyi )合

Ixc =i=nsumi=1

Ixi (72)

= Ix1 + Ix2 + + Ixn (73)

Iyc =i=nsumi=1

Iyi (74)

= Iy1 + Iy2 + + Iyn (75)

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課課課堂堂堂練練練習習習

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

長方形與圓形複合

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

長方形複合

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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面面面積積積之之之慣慣慣性性性積積積Product of Inertia for Area

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面積之慣性積(Product of Inertia for Area)

為了下個章節求解慣性矩之極值(極大或極小值)方便數學運算在此先定義面積之慣性積(Ixy )為

dIxy = xydA (76)

Ixy =

ˆAxydA (77)

面積之慣性積的符號可為正負或零當慣性積之面積相對轉軸對稱時則面積之慣性積和為零

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 30 54

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平行軸定理

極點

新極點

如左圖面積慣性矩之平行軸定理乃以面心(C)為新極點(O)設定新座標軸(xrsquoyrsquo)其中xrsquo軸平行於原座標x軸yrsquo軸平行於原座標y軸取微小面積dA其在新座標之點座標為(xrsquoyrsquo)面積之慣性積(Ixy )為

dIxy = (x prime + dx)(y prime + dy )dA (78)

當以面心為轉軸點時即x prime = y prime = 0ˆAx primedA = x prime

ˆAdA = 0

ˆAy primedA = y prime

ˆAdA = 0 (79)

Ixy =

ˆA

(x prime + dx)(y prime + dy )dA (80)

=

ˆAx primey primedA + dx

ˆAy primedA + dy

ˆAx primedA + dxdy

ˆAdA (81)

= macrIx primey prime + Adxdy

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 31 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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傾傾傾斜斜斜軸軸軸面面面積積積之之之慣慣慣性性性矩矩矩

Moment of Inertia for Area about Inclined Axis

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 33 54

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傾斜軸面積之慣性矩

當轉動軸傾斜θ角時也就是座標軸旋轉θ角而Ix Iy Ixy為已知則新的座標軸(uv)可寫為

u = xcosθ + ysinθ (82)

v = ycosθ minus xsinθ (83)

由u軸上之面積慣性矩Iuv軸上之面積慣性矩Iv及慣性積Iuv得知

dIu = v2dA = (ycosθ minus xsinθ)2dA (84)

dIv = u2dA = (xcosθ + ysinθ)2dA (85)

dIuv = uvdA = (xcosθ + ysinθ)(ycosθ minus xsinθ)dA (86)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 34 54

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傾斜軸面積之慣性矩 I

由x軸上之面積慣性矩Ixy軸上之面積慣性矩Iy及慣性積Ixy得知

Ix =

ˆy2dA Iy =

ˆx2dA Ixy =

ˆxydA (87)

Iu =

ˆ(ycosθ minus xsinθ)2dA (88)

=

ˆ(y2cos2θ minus 2xycosθsinθ + x2sin2θ)dA (89)

= cos2θ

ˆy2dAminus 2cosθsinθ

ˆxydA + sin2θ

ˆx2dA (90)

= Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (91)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 35 54

傾斜軸面積之慣性矩 II

Iv =

ˆ(xcosθ + ysinθ)2dA (92)

=

ˆ(x2cos2θ + 2xycosθsinθ + y2sin2θ)dA (93)

= cos2θ

ˆx2dA + 2cosθsinθ

ˆxydA + sin2θ

ˆy2dA (94)

= Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (95)

Iuv =

ˆ(xcosθ + ysinθ)(ycosθ minus xsinθ)dA (96)

=

ˆ(xycos2θ + y2cosθsinθ minus x2cosθsinθ minus xysin2θ)dA (97)

= (cos2 minus sin2θ)

ˆxydA + cosθsinθ

ˆy2dAminus cosθsinθ

ˆx2dA

= Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 36 54

傾斜軸面積之慣性矩 III

再由

sin2θ + cos2θ = 1 (98)

sin(αplusmn β) = sinαcosβ plusmn cosαsinβ (99)

sin2θ = sin(θ + θ) = 2sinθcosθ (100)

cos(αplusmn β) = cosαcosβ ∓ sinαsinβ (101)

cos2θ = cos(θ + θ) = cos2θ minus sin2θ (102)

= 2cos2θ minus 1 = 1minus 2sin2θ (103)

所以

Iu = Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (104)

= Ix(cos2θ + 1

2) + Iy (

1minus cos2θ

2)minus Ixy sin2θ (105)

=Ix + Iy

2+

Ix minus Iy2

cos2θ minus Ixy sin2θ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 37 54

傾斜軸面積之慣性矩 IV

Iv = Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (106)

= Iy (cos2θ + 1

2) + Ix(

1minus cos2θ

2) + Ixy sin2θ (107)

=Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (108)

Iuv = Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ (109)

= Ixycos2θ +1

2Ixsin2θ minus 1

2Iy sin2θ (110)

=Ix minus Iy

2sin2θ + Ixycos2θ (111)

所以

JO = Iu + Iv = Ix + Iy (112)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 38 54

主主主軸軸軸慣慣慣性性性矩矩矩Principal Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 39 54

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主軸慣性矩 I

所謂主軸慣性矩(Principal Moment ofInertia)乃定義發生在慣性矩的極值(極大或極小值)上也就是慣性矩之一次微分式為零或斜率為零

dIu

dθ=

d(Ix+Iy

2+

IxminusIy2

cos2θ minus Ixy sin2θ)

dθ(113)

= minus2Ix minus Iy

2sin2θ minus 2Ixy cos2θ (114)

= 0 (115)

其中dsin2θ = cos2θ dcos2θ = minussin2θ當慣性矩之一次微分式為零時傾斜軸與主軸之夾角θ = θp

minus2Ix minus Iy

2sin2θp minus 2Ixycos2θp = 0 (116)

tan2θ =sin2θ

cos2θ=

minusIxy(Ix minus Iy )2

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 40 54

主軸慣性矩 II

cos2θp =IxminusIy2radic

(IxminusIy2 )2 + I 2xy

sin2θp =minusIxyradic

(IxminusIy2 )2 + I 2xy

(117)

(Iu)max =Ix + Iy

2+

Ix minus Iy2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

minus IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2+

radic(Ix minus Iy

2)2 + I 2xy (118)

(Iv )min =Ix + Iy

2minus Ix minus Iy

2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

+ IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2minusradic

(Ix minus Iy

2)2 + I 2xy (119)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 41 54

主軸慣性矩 III

由tan2θp = tan(π + 2θp)可得到θp1與θp2二個解即2θp2 = π + 2θp1也就是θp2 minus θp1 = π

2

tan2θp = tan(π + 2θp) (120)

2θp2 = π + 2θp1 (121)

θp2 minus θp1 =π

2(122)

所以

Imaxmin

=Ix + Iy

2plusmnradic

(Ix minus Iy

2)2 + I 2xy (123)

Iuv =Ix minus Iy

2

minusIxyradic(IxminusIy2 )2 + I 2xy

+ Ixy

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

(124)

= 0

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 42 54

課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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慣慣慣性性性矩矩矩之之之莫莫莫爾爾爾圓圓圓Mohrrsquos Circle for Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 44 54

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慣性矩之莫爾圓

由於Iu =

Ix + Iy2

+Ix minus Iy

2cos2θ minus Ixy sin2θ (125)

Iv =Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (126)

Iuv =Ix minus Iy

2sin2θ + Ixycos2θ (127)

將式(125)與式(127)各自分別平方並相加可得

(Iu minusIx + Iy

2)2 + I 2uv = (

Ix minus Iy2

)2 + I 2xy (128)

其中Ix Iy Ixy均為常數式(128)可進一步寫成

(Iu minus a)2 + I 2uv = R2 (129)

其中a =Ix+Iy2 R =

radic(IxminusIy2 )2 + I 2xycos2θp1 =

aRsin2θp1 =

minusIxyR 2θp1為OA轉向I軸方向即2θp1為

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 45 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 46 54

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解題技巧

首先確定轉動系統的Ix Iy Ixy接著算出莫爾圓之圓心座標a及半徑R並畫出莫爾圓再由莫爾圓上點出A(Ix Ixy )並由OA轉向I軸找出兩倍夾角2θp1及2θp2其中θp1及θp2為主軸(Principle Axis)與Ix軸的夾角由於主軸(Principle Axis)為極值之軸Imax = Iu及Imin = Iv決定θp1及θp2何者為Imax = Iu與Ix軸的夾角之方法乃比較Ix及Iy之大小Imax = Iu會靠近Ix及Iy之較大者也就是Imax = Iu與Ix及Iy之較大者的夾角比較小

a =Ix + Iy

2(130)

R =

radic(Ix minus Iy

2)2 + I 2xy (131)

2θp2 = π + 2θp1 θp2 minus θp1 =π

2(132)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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質質質量量量慣慣慣性性性矩矩矩Mass Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 48 54

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質量慣性矩

τ = Iα (133)

其中τ為旋轉扭矩I為質量慣性矩α為旋轉角速度

I =

ˆmr2dm (134)

=

ˆVr2ρdV (135)

= ρ

ˆVr2dV (136)

其中r為旋轉半徑或轉軸到旋轉體的距離

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 49 54

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 50 54

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平行軸定理

當通過物體質心(G)並在一轉軸上的質量慣性矩(IG )為已知時可透過座標軸轉換計算物體之質量慣性矩(I)

I =

ˆmr2dm (137)

=

ˆm

[(d + x prime)2 + y prime2]dm (138)

=

ˆm

(x prime2 + y prime2)dm + 2d

ˆmx primedm + d2

ˆmdm (139)

I = IG + md2 (140)

其中IG為在zrsquo軸上通過質心的質量慣性矩m為質量d為通過質心之軸zrsquo並平行原軸z之距離

acutem x primedm = x

acutem dm = 0 since x = 0

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 51 54

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (141)

k =

radicI

m(142)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (143)

= I1 + I2 + + In (144)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 52 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

兩圓盤複合體

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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結結結論論論

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 54 54

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結論

轉動慣量或慣性矩為物體對旋轉運動的慣性而所謂慣性乃物體持續維持不變的行為也就是定義慣性矩可表達物體對本身旋轉運動的維持或持續性好壞更簡單的說慣性矩越大之物體其維持旋轉運動趨勢的能力越大例如惰輪儲能裝置惰輪之慣性矩越大當其轉動以後其維持旋轉運動趨勢的能力越大

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 55 54

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  • 轉動慣量或慣性矩(Moment of Inertia)
  • 面積慣性矩(Moment of Inertia for Area)
    • 面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)
    • 面積慣性矩之迴轉半徑(Radius of Gyration for Area)
      • 複合面積之慣性矩(Moment of Inertia for Composite Area)
      • 面積之慣性積(Product of Inertia for Area)
        • 平行軸定理(Parallel-Axis Theorem)
          • 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)
            • 主軸慣性矩(Principal Moment of Inertia)
              • 慣性矩之莫爾圓(Mohrs Circle for Moment of Inertia)
              • 質量慣性矩(Mass Moment of Inertia)
                • 平行軸定理(Parallel-Axis Theorem)
Page 31: 轉動慣量 Moments of Inertia - cct.me.ntut.edu.twcct.me.ntut.edu.tw/ccteducation/chchting/aiahtm/statics/lecture... · 轉動慣量或慣性矩 I. 轉動慣量也稱慣性矩(Moment

迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (68)

k =

radicI

m(69)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (70)

= I1 + I2 + + In (71)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 25 54

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複合面積之慣性矩

複合面積之慣性矩(Ixc Iyc)為各局部面積之慣性矩(Ixi Iyi )合

Ixc =i=nsumi=1

Ixi (72)

= Ix1 + Ix2 + + Ixn (73)

Iyc =i=nsumi=1

Iyi (74)

= Iy1 + Iy2 + + Iyn (75)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 26 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

長方形與圓形複合

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

長方形複合

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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面面面積積積之之之慣慣慣性性性積積積Product of Inertia for Area

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 28 54

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面積之慣性積(Product of Inertia for Area)

為了下個章節求解慣性矩之極值(極大或極小值)方便數學運算在此先定義面積之慣性積(Ixy )為

dIxy = xydA (76)

Ixy =

ˆAxydA (77)

面積之慣性積的符號可為正負或零當慣性積之面積相對轉軸對稱時則面積之慣性積和為零

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 29 54

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 30 54

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平行軸定理

極點

新極點

如左圖面積慣性矩之平行軸定理乃以面心(C)為新極點(O)設定新座標軸(xrsquoyrsquo)其中xrsquo軸平行於原座標x軸yrsquo軸平行於原座標y軸取微小面積dA其在新座標之點座標為(xrsquoyrsquo)面積之慣性積(Ixy )為

dIxy = (x prime + dx)(y prime + dy )dA (78)

當以面心為轉軸點時即x prime = y prime = 0ˆAx primedA = x prime

ˆAdA = 0

ˆAy primedA = y prime

ˆAdA = 0 (79)

Ixy =

ˆA

(x prime + dx)(y prime + dy )dA (80)

=

ˆAx primey primedA + dx

ˆAy primedA + dy

ˆAx primedA + dxdy

ˆAdA (81)

= macrIx primey prime + Adxdy

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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傾傾傾斜斜斜軸軸軸面面面積積積之之之慣慣慣性性性矩矩矩

Moment of Inertia for Area about Inclined Axis

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 33 54

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傾斜軸面積之慣性矩

當轉動軸傾斜θ角時也就是座標軸旋轉θ角而Ix Iy Ixy為已知則新的座標軸(uv)可寫為

u = xcosθ + ysinθ (82)

v = ycosθ minus xsinθ (83)

由u軸上之面積慣性矩Iuv軸上之面積慣性矩Iv及慣性積Iuv得知

dIu = v2dA = (ycosθ minus xsinθ)2dA (84)

dIv = u2dA = (xcosθ + ysinθ)2dA (85)

dIuv = uvdA = (xcosθ + ysinθ)(ycosθ minus xsinθ)dA (86)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 34 54

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傾斜軸面積之慣性矩 I

由x軸上之面積慣性矩Ixy軸上之面積慣性矩Iy及慣性積Ixy得知

Ix =

ˆy2dA Iy =

ˆx2dA Ixy =

ˆxydA (87)

Iu =

ˆ(ycosθ minus xsinθ)2dA (88)

=

ˆ(y2cos2θ minus 2xycosθsinθ + x2sin2θ)dA (89)

= cos2θ

ˆy2dAminus 2cosθsinθ

ˆxydA + sin2θ

ˆx2dA (90)

= Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (91)

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傾斜軸面積之慣性矩 II

Iv =

ˆ(xcosθ + ysinθ)2dA (92)

=

ˆ(x2cos2θ + 2xycosθsinθ + y2sin2θ)dA (93)

= cos2θ

ˆx2dA + 2cosθsinθ

ˆxydA + sin2θ

ˆy2dA (94)

= Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (95)

Iuv =

ˆ(xcosθ + ysinθ)(ycosθ minus xsinθ)dA (96)

=

ˆ(xycos2θ + y2cosθsinθ minus x2cosθsinθ minus xysin2θ)dA (97)

= (cos2 minus sin2θ)

ˆxydA + cosθsinθ

ˆy2dAminus cosθsinθ

ˆx2dA

= Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 36 54

傾斜軸面積之慣性矩 III

再由

sin2θ + cos2θ = 1 (98)

sin(αplusmn β) = sinαcosβ plusmn cosαsinβ (99)

sin2θ = sin(θ + θ) = 2sinθcosθ (100)

cos(αplusmn β) = cosαcosβ ∓ sinαsinβ (101)

cos2θ = cos(θ + θ) = cos2θ minus sin2θ (102)

= 2cos2θ minus 1 = 1minus 2sin2θ (103)

所以

Iu = Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (104)

= Ix(cos2θ + 1

2) + Iy (

1minus cos2θ

2)minus Ixy sin2θ (105)

=Ix + Iy

2+

Ix minus Iy2

cos2θ minus Ixy sin2θ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 37 54

傾斜軸面積之慣性矩 IV

Iv = Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (106)

= Iy (cos2θ + 1

2) + Ix(

1minus cos2θ

2) + Ixy sin2θ (107)

=Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (108)

Iuv = Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ (109)

= Ixycos2θ +1

2Ixsin2θ minus 1

2Iy sin2θ (110)

=Ix minus Iy

2sin2θ + Ixycos2θ (111)

所以

JO = Iu + Iv = Ix + Iy (112)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 38 54

主主主軸軸軸慣慣慣性性性矩矩矩Principal Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 39 54

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主軸慣性矩 I

所謂主軸慣性矩(Principal Moment ofInertia)乃定義發生在慣性矩的極值(極大或極小值)上也就是慣性矩之一次微分式為零或斜率為零

dIu

dθ=

d(Ix+Iy

2+

IxminusIy2

cos2θ minus Ixy sin2θ)

dθ(113)

= minus2Ix minus Iy

2sin2θ minus 2Ixy cos2θ (114)

= 0 (115)

其中dsin2θ = cos2θ dcos2θ = minussin2θ當慣性矩之一次微分式為零時傾斜軸與主軸之夾角θ = θp

minus2Ix minus Iy

2sin2θp minus 2Ixycos2θp = 0 (116)

tan2θ =sin2θ

cos2θ=

minusIxy(Ix minus Iy )2

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 40 54

主軸慣性矩 II

cos2θp =IxminusIy2radic

(IxminusIy2 )2 + I 2xy

sin2θp =minusIxyradic

(IxminusIy2 )2 + I 2xy

(117)

(Iu)max =Ix + Iy

2+

Ix minus Iy2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

minus IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2+

radic(Ix minus Iy

2)2 + I 2xy (118)

(Iv )min =Ix + Iy

2minus Ix minus Iy

2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

+ IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2minusradic

(Ix minus Iy

2)2 + I 2xy (119)

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主軸慣性矩 III

由tan2θp = tan(π + 2θp)可得到θp1與θp2二個解即2θp2 = π + 2θp1也就是θp2 minus θp1 = π

2

tan2θp = tan(π + 2θp) (120)

2θp2 = π + 2θp1 (121)

θp2 minus θp1 =π

2(122)

所以

Imaxmin

=Ix + Iy

2plusmnradic

(Ix minus Iy

2)2 + I 2xy (123)

Iuv =Ix minus Iy

2

minusIxyradic(IxminusIy2 )2 + I 2xy

+ Ixy

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

(124)

= 0

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課課課堂堂堂練練練習習習

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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慣慣慣性性性矩矩矩之之之莫莫莫爾爾爾圓圓圓Mohrrsquos Circle for Moment of Inertia

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慣性矩之莫爾圓

由於Iu =

Ix + Iy2

+Ix minus Iy

2cos2θ minus Ixy sin2θ (125)

Iv =Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (126)

Iuv =Ix minus Iy

2sin2θ + Ixycos2θ (127)

將式(125)與式(127)各自分別平方並相加可得

(Iu minusIx + Iy

2)2 + I 2uv = (

Ix minus Iy2

)2 + I 2xy (128)

其中Ix Iy Ixy均為常數式(128)可進一步寫成

(Iu minus a)2 + I 2uv = R2 (129)

其中a =Ix+Iy2 R =

radic(IxminusIy2 )2 + I 2xycos2θp1 =

aRsin2θp1 =

minusIxyR 2θp1為OA轉向I軸方向即2θp1為

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課課課堂堂堂練練練習習習

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解題技巧

首先確定轉動系統的Ix Iy Ixy接著算出莫爾圓之圓心座標a及半徑R並畫出莫爾圓再由莫爾圓上點出A(Ix Ixy )並由OA轉向I軸找出兩倍夾角2θp1及2θp2其中θp1及θp2為主軸(Principle Axis)與Ix軸的夾角由於主軸(Principle Axis)為極值之軸Imax = Iu及Imin = Iv決定θp1及θp2何者為Imax = Iu與Ix軸的夾角之方法乃比較Ix及Iy之大小Imax = Iu會靠近Ix及Iy之較大者也就是Imax = Iu與Ix及Iy之較大者的夾角比較小

a =Ix + Iy

2(130)

R =

radic(Ix minus Iy

2)2 + I 2xy (131)

2θp2 = π + 2θp1 θp2 minus θp1 =π

2(132)

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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質質質量量量慣慣慣性性性矩矩矩Mass Moment of Inertia

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質量慣性矩

τ = Iα (133)

其中τ為旋轉扭矩I為質量慣性矩α為旋轉角速度

I =

ˆmr2dm (134)

=

ˆVr2ρdV (135)

= ρ

ˆVr2dV (136)

其中r為旋轉半徑或轉軸到旋轉體的距離

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

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平行軸定理

當通過物體質心(G)並在一轉軸上的質量慣性矩(IG )為已知時可透過座標軸轉換計算物體之質量慣性矩(I)

I =

ˆmr2dm (137)

=

ˆm

[(d + x prime)2 + y prime2]dm (138)

=

ˆm

(x prime2 + y prime2)dm + 2d

ˆmx primedm + d2

ˆmdm (139)

I = IG + md2 (140)

其中IG為在zrsquo軸上通過質心的質量慣性矩m為質量d為通過質心之軸zrsquo並平行原軸z之距離

acutem x primedm = x

acutem dm = 0 since x = 0

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (141)

k =

radicI

m(142)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (143)

= I1 + I2 + + In (144)

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

兩圓盤複合體

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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結結結論論論

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結論

轉動慣量或慣性矩為物體對旋轉運動的慣性而所謂慣性乃物體持續維持不變的行為也就是定義慣性矩可表達物體對本身旋轉運動的維持或持續性好壞更簡單的說慣性矩越大之物體其維持旋轉運動趨勢的能力越大例如惰輪儲能裝置惰輪之慣性矩越大當其轉動以後其維持旋轉運動趨勢的能力越大

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  • 轉動慣量或慣性矩(Moment of Inertia)
  • 面積慣性矩(Moment of Inertia for Area)
    • 面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)
    • 面積慣性矩之迴轉半徑(Radius of Gyration for Area)
      • 複合面積之慣性矩(Moment of Inertia for Composite Area)
      • 面積之慣性積(Product of Inertia for Area)
        • 平行軸定理(Parallel-Axis Theorem)
          • 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)
            • 主軸慣性矩(Principal Moment of Inertia)
              • 慣性矩之莫爾圓(Mohrs Circle for Moment of Inertia)
              • 質量慣性矩(Mass Moment of Inertia)
                • 平行軸定理(Parallel-Axis Theorem)
Page 32: 轉動慣量 Moments of Inertia - cct.me.ntut.edu.twcct.me.ntut.edu.tw/ccteducation/chchting/aiahtm/statics/lecture... · 轉動慣量或慣性矩 I. 轉動慣量也稱慣性矩(Moment

複合面積之慣性矩

複合面積之慣性矩(Ixc Iyc)為各局部面積之慣性矩(Ixi Iyi )合

Ixc =i=nsumi=1

Ixi (72)

= Ix1 + Ix2 + + Ixn (73)

Iyc =i=nsumi=1

Iyi (74)

= Iy1 + Iy2 + + Iyn (75)

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課課課堂堂堂練練練習習習

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

長方形與圓形複合

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長方形複合

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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面面面積積積之之之慣慣慣性性性積積積Product of Inertia for Area

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面積之慣性積(Product of Inertia for Area)

為了下個章節求解慣性矩之極值(極大或極小值)方便數學運算在此先定義面積之慣性積(Ixy )為

dIxy = xydA (76)

Ixy =

ˆAxydA (77)

面積之慣性積的符號可為正負或零當慣性積之面積相對轉軸對稱時則面積之慣性積和為零

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

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平行軸定理

極點

新極點

如左圖面積慣性矩之平行軸定理乃以面心(C)為新極點(O)設定新座標軸(xrsquoyrsquo)其中xrsquo軸平行於原座標x軸yrsquo軸平行於原座標y軸取微小面積dA其在新座標之點座標為(xrsquoyrsquo)面積之慣性積(Ixy )為

dIxy = (x prime + dx)(y prime + dy )dA (78)

當以面心為轉軸點時即x prime = y prime = 0ˆAx primedA = x prime

ˆAdA = 0

ˆAy primedA = y prime

ˆAdA = 0 (79)

Ixy =

ˆA

(x prime + dx)(y prime + dy )dA (80)

=

ˆAx primey primedA + dx

ˆAy primedA + dy

ˆAx primedA + dxdy

ˆAdA (81)

= macrIx primey prime + Adxdy

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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傾傾傾斜斜斜軸軸軸面面面積積積之之之慣慣慣性性性矩矩矩

Moment of Inertia for Area about Inclined Axis

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 33 54

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傾斜軸面積之慣性矩

當轉動軸傾斜θ角時也就是座標軸旋轉θ角而Ix Iy Ixy為已知則新的座標軸(uv)可寫為

u = xcosθ + ysinθ (82)

v = ycosθ minus xsinθ (83)

由u軸上之面積慣性矩Iuv軸上之面積慣性矩Iv及慣性積Iuv得知

dIu = v2dA = (ycosθ minus xsinθ)2dA (84)

dIv = u2dA = (xcosθ + ysinθ)2dA (85)

dIuv = uvdA = (xcosθ + ysinθ)(ycosθ minus xsinθ)dA (86)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 34 54

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傾斜軸面積之慣性矩 I

由x軸上之面積慣性矩Ixy軸上之面積慣性矩Iy及慣性積Ixy得知

Ix =

ˆy2dA Iy =

ˆx2dA Ixy =

ˆxydA (87)

Iu =

ˆ(ycosθ minus xsinθ)2dA (88)

=

ˆ(y2cos2θ minus 2xycosθsinθ + x2sin2θ)dA (89)

= cos2θ

ˆy2dAminus 2cosθsinθ

ˆxydA + sin2θ

ˆx2dA (90)

= Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (91)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 35 54

傾斜軸面積之慣性矩 II

Iv =

ˆ(xcosθ + ysinθ)2dA (92)

=

ˆ(x2cos2θ + 2xycosθsinθ + y2sin2θ)dA (93)

= cos2θ

ˆx2dA + 2cosθsinθ

ˆxydA + sin2θ

ˆy2dA (94)

= Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (95)

Iuv =

ˆ(xcosθ + ysinθ)(ycosθ minus xsinθ)dA (96)

=

ˆ(xycos2θ + y2cosθsinθ minus x2cosθsinθ minus xysin2θ)dA (97)

= (cos2 minus sin2θ)

ˆxydA + cosθsinθ

ˆy2dAminus cosθsinθ

ˆx2dA

= Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 36 54

傾斜軸面積之慣性矩 III

再由

sin2θ + cos2θ = 1 (98)

sin(αplusmn β) = sinαcosβ plusmn cosαsinβ (99)

sin2θ = sin(θ + θ) = 2sinθcosθ (100)

cos(αplusmn β) = cosαcosβ ∓ sinαsinβ (101)

cos2θ = cos(θ + θ) = cos2θ minus sin2θ (102)

= 2cos2θ minus 1 = 1minus 2sin2θ (103)

所以

Iu = Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (104)

= Ix(cos2θ + 1

2) + Iy (

1minus cos2θ

2)minus Ixy sin2θ (105)

=Ix + Iy

2+

Ix minus Iy2

cos2θ minus Ixy sin2θ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 37 54

傾斜軸面積之慣性矩 IV

Iv = Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (106)

= Iy (cos2θ + 1

2) + Ix(

1minus cos2θ

2) + Ixy sin2θ (107)

=Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (108)

Iuv = Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ (109)

= Ixycos2θ +1

2Ixsin2θ minus 1

2Iy sin2θ (110)

=Ix minus Iy

2sin2θ + Ixycos2θ (111)

所以

JO = Iu + Iv = Ix + Iy (112)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 38 54

主主主軸軸軸慣慣慣性性性矩矩矩Principal Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 39 54

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主軸慣性矩 I

所謂主軸慣性矩(Principal Moment ofInertia)乃定義發生在慣性矩的極值(極大或極小值)上也就是慣性矩之一次微分式為零或斜率為零

dIu

dθ=

d(Ix+Iy

2+

IxminusIy2

cos2θ minus Ixy sin2θ)

dθ(113)

= minus2Ix minus Iy

2sin2θ minus 2Ixy cos2θ (114)

= 0 (115)

其中dsin2θ = cos2θ dcos2θ = minussin2θ當慣性矩之一次微分式為零時傾斜軸與主軸之夾角θ = θp

minus2Ix minus Iy

2sin2θp minus 2Ixycos2θp = 0 (116)

tan2θ =sin2θ

cos2θ=

minusIxy(Ix minus Iy )2

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 40 54

主軸慣性矩 II

cos2θp =IxminusIy2radic

(IxminusIy2 )2 + I 2xy

sin2θp =minusIxyradic

(IxminusIy2 )2 + I 2xy

(117)

(Iu)max =Ix + Iy

2+

Ix minus Iy2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

minus IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2+

radic(Ix minus Iy

2)2 + I 2xy (118)

(Iv )min =Ix + Iy

2minus Ix minus Iy

2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

+ IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2minusradic

(Ix minus Iy

2)2 + I 2xy (119)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 41 54

主軸慣性矩 III

由tan2θp = tan(π + 2θp)可得到θp1與θp2二個解即2θp2 = π + 2θp1也就是θp2 minus θp1 = π

2

tan2θp = tan(π + 2θp) (120)

2θp2 = π + 2θp1 (121)

θp2 minus θp1 =π

2(122)

所以

Imaxmin

=Ix + Iy

2plusmnradic

(Ix minus Iy

2)2 + I 2xy (123)

Iuv =Ix minus Iy

2

minusIxyradic(IxminusIy2 )2 + I 2xy

+ Ixy

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

(124)

= 0

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 42 54

課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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慣慣慣性性性矩矩矩之之之莫莫莫爾爾爾圓圓圓Mohrrsquos Circle for Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 44 54

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慣性矩之莫爾圓

由於Iu =

Ix + Iy2

+Ix minus Iy

2cos2θ minus Ixy sin2θ (125)

Iv =Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (126)

Iuv =Ix minus Iy

2sin2θ + Ixycos2θ (127)

將式(125)與式(127)各自分別平方並相加可得

(Iu minusIx + Iy

2)2 + I 2uv = (

Ix minus Iy2

)2 + I 2xy (128)

其中Ix Iy Ixy均為常數式(128)可進一步寫成

(Iu minus a)2 + I 2uv = R2 (129)

其中a =Ix+Iy2 R =

radic(IxminusIy2 )2 + I 2xycos2θp1 =

aRsin2θp1 =

minusIxyR 2θp1為OA轉向I軸方向即2θp1為

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 45 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 46 54

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解題技巧

首先確定轉動系統的Ix Iy Ixy接著算出莫爾圓之圓心座標a及半徑R並畫出莫爾圓再由莫爾圓上點出A(Ix Ixy )並由OA轉向I軸找出兩倍夾角2θp1及2θp2其中θp1及θp2為主軸(Principle Axis)與Ix軸的夾角由於主軸(Principle Axis)為極值之軸Imax = Iu及Imin = Iv決定θp1及θp2何者為Imax = Iu與Ix軸的夾角之方法乃比較Ix及Iy之大小Imax = Iu會靠近Ix及Iy之較大者也就是Imax = Iu與Ix及Iy之較大者的夾角比較小

a =Ix + Iy

2(130)

R =

radic(Ix minus Iy

2)2 + I 2xy (131)

2θp2 = π + 2θp1 θp2 minus θp1 =π

2(132)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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質質質量量量慣慣慣性性性矩矩矩Mass Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 48 54

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質量慣性矩

τ = Iα (133)

其中τ為旋轉扭矩I為質量慣性矩α為旋轉角速度

I =

ˆmr2dm (134)

=

ˆVr2ρdV (135)

= ρ

ˆVr2dV (136)

其中r為旋轉半徑或轉軸到旋轉體的距離

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 49 54

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 50 54

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平行軸定理

當通過物體質心(G)並在一轉軸上的質量慣性矩(IG )為已知時可透過座標軸轉換計算物體之質量慣性矩(I)

I =

ˆmr2dm (137)

=

ˆm

[(d + x prime)2 + y prime2]dm (138)

=

ˆm

(x prime2 + y prime2)dm + 2d

ˆmx primedm + d2

ˆmdm (139)

I = IG + md2 (140)

其中IG為在zrsquo軸上通過質心的質量慣性矩m為質量d為通過質心之軸zrsquo並平行原軸z之距離

acutem x primedm = x

acutem dm = 0 since x = 0

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 51 54

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (141)

k =

radicI

m(142)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (143)

= I1 + I2 + + In (144)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 52 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

兩圓盤複合體

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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結結結論論論

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 54 54

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結論

轉動慣量或慣性矩為物體對旋轉運動的慣性而所謂慣性乃物體持續維持不變的行為也就是定義慣性矩可表達物體對本身旋轉運動的維持或持續性好壞更簡單的說慣性矩越大之物體其維持旋轉運動趨勢的能力越大例如惰輪儲能裝置惰輪之慣性矩越大當其轉動以後其維持旋轉運動趨勢的能力越大

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 55 54

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  • 轉動慣量或慣性矩(Moment of Inertia)
  • 面積慣性矩(Moment of Inertia for Area)
    • 面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)
    • 面積慣性矩之迴轉半徑(Radius of Gyration for Area)
      • 複合面積之慣性矩(Moment of Inertia for Composite Area)
      • 面積之慣性積(Product of Inertia for Area)
        • 平行軸定理(Parallel-Axis Theorem)
          • 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)
            • 主軸慣性矩(Principal Moment of Inertia)
              • 慣性矩之莫爾圓(Mohrs Circle for Moment of Inertia)
              • 質量慣性矩(Mass Moment of Inertia)
                • 平行軸定理(Parallel-Axis Theorem)
Page 33: 轉動慣量 Moments of Inertia - cct.me.ntut.edu.twcct.me.ntut.edu.tw/ccteducation/chchting/aiahtm/statics/lecture... · 轉動慣量或慣性矩 I. 轉動慣量也稱慣性矩(Moment

課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

長方形與圓形複合

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

長方形複合

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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面面面積積積之之之慣慣慣性性性積積積Product of Inertia for Area

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 28 54

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面積之慣性積(Product of Inertia for Area)

為了下個章節求解慣性矩之極值(極大或極小值)方便數學運算在此先定義面積之慣性積(Ixy )為

dIxy = xydA (76)

Ixy =

ˆAxydA (77)

面積之慣性積的符號可為正負或零當慣性積之面積相對轉軸對稱時則面積之慣性積和為零

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 29 54

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 30 54

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平行軸定理

極點

新極點

如左圖面積慣性矩之平行軸定理乃以面心(C)為新極點(O)設定新座標軸(xrsquoyrsquo)其中xrsquo軸平行於原座標x軸yrsquo軸平行於原座標y軸取微小面積dA其在新座標之點座標為(xrsquoyrsquo)面積之慣性積(Ixy )為

dIxy = (x prime + dx)(y prime + dy )dA (78)

當以面心為轉軸點時即x prime = y prime = 0ˆAx primedA = x prime

ˆAdA = 0

ˆAy primedA = y prime

ˆAdA = 0 (79)

Ixy =

ˆA

(x prime + dx)(y prime + dy )dA (80)

=

ˆAx primey primedA + dx

ˆAy primedA + dy

ˆAx primedA + dxdy

ˆAdA (81)

= macrIx primey prime + Adxdy

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 31 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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傾傾傾斜斜斜軸軸軸面面面積積積之之之慣慣慣性性性矩矩矩

Moment of Inertia for Area about Inclined Axis

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 33 54

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傾斜軸面積之慣性矩

當轉動軸傾斜θ角時也就是座標軸旋轉θ角而Ix Iy Ixy為已知則新的座標軸(uv)可寫為

u = xcosθ + ysinθ (82)

v = ycosθ minus xsinθ (83)

由u軸上之面積慣性矩Iuv軸上之面積慣性矩Iv及慣性積Iuv得知

dIu = v2dA = (ycosθ minus xsinθ)2dA (84)

dIv = u2dA = (xcosθ + ysinθ)2dA (85)

dIuv = uvdA = (xcosθ + ysinθ)(ycosθ minus xsinθ)dA (86)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 34 54

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傾斜軸面積之慣性矩 I

由x軸上之面積慣性矩Ixy軸上之面積慣性矩Iy及慣性積Ixy得知

Ix =

ˆy2dA Iy =

ˆx2dA Ixy =

ˆxydA (87)

Iu =

ˆ(ycosθ minus xsinθ)2dA (88)

=

ˆ(y2cos2θ minus 2xycosθsinθ + x2sin2θ)dA (89)

= cos2θ

ˆy2dAminus 2cosθsinθ

ˆxydA + sin2θ

ˆx2dA (90)

= Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (91)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 35 54

傾斜軸面積之慣性矩 II

Iv =

ˆ(xcosθ + ysinθ)2dA (92)

=

ˆ(x2cos2θ + 2xycosθsinθ + y2sin2θ)dA (93)

= cos2θ

ˆx2dA + 2cosθsinθ

ˆxydA + sin2θ

ˆy2dA (94)

= Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (95)

Iuv =

ˆ(xcosθ + ysinθ)(ycosθ minus xsinθ)dA (96)

=

ˆ(xycos2θ + y2cosθsinθ minus x2cosθsinθ minus xysin2θ)dA (97)

= (cos2 minus sin2θ)

ˆxydA + cosθsinθ

ˆy2dAminus cosθsinθ

ˆx2dA

= Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 36 54

傾斜軸面積之慣性矩 III

再由

sin2θ + cos2θ = 1 (98)

sin(αplusmn β) = sinαcosβ plusmn cosαsinβ (99)

sin2θ = sin(θ + θ) = 2sinθcosθ (100)

cos(αplusmn β) = cosαcosβ ∓ sinαsinβ (101)

cos2θ = cos(θ + θ) = cos2θ minus sin2θ (102)

= 2cos2θ minus 1 = 1minus 2sin2θ (103)

所以

Iu = Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (104)

= Ix(cos2θ + 1

2) + Iy (

1minus cos2θ

2)minus Ixy sin2θ (105)

=Ix + Iy

2+

Ix minus Iy2

cos2θ minus Ixy sin2θ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 37 54

傾斜軸面積之慣性矩 IV

Iv = Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (106)

= Iy (cos2θ + 1

2) + Ix(

1minus cos2θ

2) + Ixy sin2θ (107)

=Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (108)

Iuv = Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ (109)

= Ixycos2θ +1

2Ixsin2θ minus 1

2Iy sin2θ (110)

=Ix minus Iy

2sin2θ + Ixycos2θ (111)

所以

JO = Iu + Iv = Ix + Iy (112)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 38 54

主主主軸軸軸慣慣慣性性性矩矩矩Principal Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 39 54

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主軸慣性矩 I

所謂主軸慣性矩(Principal Moment ofInertia)乃定義發生在慣性矩的極值(極大或極小值)上也就是慣性矩之一次微分式為零或斜率為零

dIu

dθ=

d(Ix+Iy

2+

IxminusIy2

cos2θ minus Ixy sin2θ)

dθ(113)

= minus2Ix minus Iy

2sin2θ minus 2Ixy cos2θ (114)

= 0 (115)

其中dsin2θ = cos2θ dcos2θ = minussin2θ當慣性矩之一次微分式為零時傾斜軸與主軸之夾角θ = θp

minus2Ix minus Iy

2sin2θp minus 2Ixycos2θp = 0 (116)

tan2θ =sin2θ

cos2θ=

minusIxy(Ix minus Iy )2

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 40 54

主軸慣性矩 II

cos2θp =IxminusIy2radic

(IxminusIy2 )2 + I 2xy

sin2θp =minusIxyradic

(IxminusIy2 )2 + I 2xy

(117)

(Iu)max =Ix + Iy

2+

Ix minus Iy2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

minus IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2+

radic(Ix minus Iy

2)2 + I 2xy (118)

(Iv )min =Ix + Iy

2minus Ix minus Iy

2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

+ IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2minusradic

(Ix minus Iy

2)2 + I 2xy (119)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 41 54

主軸慣性矩 III

由tan2θp = tan(π + 2θp)可得到θp1與θp2二個解即2θp2 = π + 2θp1也就是θp2 minus θp1 = π

2

tan2θp = tan(π + 2θp) (120)

2θp2 = π + 2θp1 (121)

θp2 minus θp1 =π

2(122)

所以

Imaxmin

=Ix + Iy

2plusmnradic

(Ix minus Iy

2)2 + I 2xy (123)

Iuv =Ix minus Iy

2

minusIxyradic(IxminusIy2 )2 + I 2xy

+ Ixy

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

(124)

= 0

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 42 54

課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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慣慣慣性性性矩矩矩之之之莫莫莫爾爾爾圓圓圓Mohrrsquos Circle for Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 44 54

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慣性矩之莫爾圓

由於Iu =

Ix + Iy2

+Ix minus Iy

2cos2θ minus Ixy sin2θ (125)

Iv =Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (126)

Iuv =Ix minus Iy

2sin2θ + Ixycos2θ (127)

將式(125)與式(127)各自分別平方並相加可得

(Iu minusIx + Iy

2)2 + I 2uv = (

Ix minus Iy2

)2 + I 2xy (128)

其中Ix Iy Ixy均為常數式(128)可進一步寫成

(Iu minus a)2 + I 2uv = R2 (129)

其中a =Ix+Iy2 R =

radic(IxminusIy2 )2 + I 2xycos2θp1 =

aRsin2θp1 =

minusIxyR 2θp1為OA轉向I軸方向即2θp1為

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課課課堂堂堂練練練習習習

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解題技巧

首先確定轉動系統的Ix Iy Ixy接著算出莫爾圓之圓心座標a及半徑R並畫出莫爾圓再由莫爾圓上點出A(Ix Ixy )並由OA轉向I軸找出兩倍夾角2θp1及2θp2其中θp1及θp2為主軸(Principle Axis)與Ix軸的夾角由於主軸(Principle Axis)為極值之軸Imax = Iu及Imin = Iv決定θp1及θp2何者為Imax = Iu與Ix軸的夾角之方法乃比較Ix及Iy之大小Imax = Iu會靠近Ix及Iy之較大者也就是Imax = Iu與Ix及Iy之較大者的夾角比較小

a =Ix + Iy

2(130)

R =

radic(Ix minus Iy

2)2 + I 2xy (131)

2θp2 = π + 2θp1 θp2 minus θp1 =π

2(132)

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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質質質量量量慣慣慣性性性矩矩矩Mass Moment of Inertia

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質量慣性矩

τ = Iα (133)

其中τ為旋轉扭矩I為質量慣性矩α為旋轉角速度

I =

ˆmr2dm (134)

=

ˆVr2ρdV (135)

= ρ

ˆVr2dV (136)

其中r為旋轉半徑或轉軸到旋轉體的距離

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

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平行軸定理

當通過物體質心(G)並在一轉軸上的質量慣性矩(IG )為已知時可透過座標軸轉換計算物體之質量慣性矩(I)

I =

ˆmr2dm (137)

=

ˆm

[(d + x prime)2 + y prime2]dm (138)

=

ˆm

(x prime2 + y prime2)dm + 2d

ˆmx primedm + d2

ˆmdm (139)

I = IG + md2 (140)

其中IG為在zrsquo軸上通過質心的質量慣性矩m為質量d為通過質心之軸zrsquo並平行原軸z之距離

acutem x primedm = x

acutem dm = 0 since x = 0

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (141)

k =

radicI

m(142)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (143)

= I1 + I2 + + In (144)

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課課課堂堂堂練練練習習習

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

兩圓盤複合體

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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結結結論論論

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結論

轉動慣量或慣性矩為物體對旋轉運動的慣性而所謂慣性乃物體持續維持不變的行為也就是定義慣性矩可表達物體對本身旋轉運動的維持或持續性好壞更簡單的說慣性矩越大之物體其維持旋轉運動趨勢的能力越大例如惰輪儲能裝置惰輪之慣性矩越大當其轉動以後其維持旋轉運動趨勢的能力越大

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 55 54

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  • 轉動慣量或慣性矩(Moment of Inertia)
  • 面積慣性矩(Moment of Inertia for Area)
    • 面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)
    • 面積慣性矩之迴轉半徑(Radius of Gyration for Area)
      • 複合面積之慣性矩(Moment of Inertia for Composite Area)
      • 面積之慣性積(Product of Inertia for Area)
        • 平行軸定理(Parallel-Axis Theorem)
          • 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)
            • 主軸慣性矩(Principal Moment of Inertia)
              • 慣性矩之莫爾圓(Mohrs Circle for Moment of Inertia)
              • 質量慣性矩(Mass Moment of Inertia)
                • 平行軸定理(Parallel-Axis Theorem)
Page 34: 轉動慣量 Moments of Inertia - cct.me.ntut.edu.twcct.me.ntut.edu.tw/ccteducation/chchting/aiahtm/statics/lecture... · 轉動慣量或慣性矩 I. 轉動慣量也稱慣性矩(Moment

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

長方形與圓形複合

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

長方形複合

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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面面面積積積之之之慣慣慣性性性積積積Product of Inertia for Area

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面積之慣性積(Product of Inertia for Area)

為了下個章節求解慣性矩之極值(極大或極小值)方便數學運算在此先定義面積之慣性積(Ixy )為

dIxy = xydA (76)

Ixy =

ˆAxydA (77)

面積之慣性積的符號可為正負或零當慣性積之面積相對轉軸對稱時則面積之慣性積和為零

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

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平行軸定理

極點

新極點

如左圖面積慣性矩之平行軸定理乃以面心(C)為新極點(O)設定新座標軸(xrsquoyrsquo)其中xrsquo軸平行於原座標x軸yrsquo軸平行於原座標y軸取微小面積dA其在新座標之點座標為(xrsquoyrsquo)面積之慣性積(Ixy )為

dIxy = (x prime + dx)(y prime + dy )dA (78)

當以面心為轉軸點時即x prime = y prime = 0ˆAx primedA = x prime

ˆAdA = 0

ˆAy primedA = y prime

ˆAdA = 0 (79)

Ixy =

ˆA

(x prime + dx)(y prime + dy )dA (80)

=

ˆAx primey primedA + dx

ˆAy primedA + dy

ˆAx primedA + dxdy

ˆAdA (81)

= macrIx primey prime + Adxdy

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課課課堂堂堂練練練習習習

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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傾傾傾斜斜斜軸軸軸面面面積積積之之之慣慣慣性性性矩矩矩

Moment of Inertia for Area about Inclined Axis

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傾斜軸面積之慣性矩

當轉動軸傾斜θ角時也就是座標軸旋轉θ角而Ix Iy Ixy為已知則新的座標軸(uv)可寫為

u = xcosθ + ysinθ (82)

v = ycosθ minus xsinθ (83)

由u軸上之面積慣性矩Iuv軸上之面積慣性矩Iv及慣性積Iuv得知

dIu = v2dA = (ycosθ minus xsinθ)2dA (84)

dIv = u2dA = (xcosθ + ysinθ)2dA (85)

dIuv = uvdA = (xcosθ + ysinθ)(ycosθ minus xsinθ)dA (86)

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傾斜軸面積之慣性矩 I

由x軸上之面積慣性矩Ixy軸上之面積慣性矩Iy及慣性積Ixy得知

Ix =

ˆy2dA Iy =

ˆx2dA Ixy =

ˆxydA (87)

Iu =

ˆ(ycosθ minus xsinθ)2dA (88)

=

ˆ(y2cos2θ minus 2xycosθsinθ + x2sin2θ)dA (89)

= cos2θ

ˆy2dAminus 2cosθsinθ

ˆxydA + sin2θ

ˆx2dA (90)

= Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (91)

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傾斜軸面積之慣性矩 II

Iv =

ˆ(xcosθ + ysinθ)2dA (92)

=

ˆ(x2cos2θ + 2xycosθsinθ + y2sin2θ)dA (93)

= cos2θ

ˆx2dA + 2cosθsinθ

ˆxydA + sin2θ

ˆy2dA (94)

= Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (95)

Iuv =

ˆ(xcosθ + ysinθ)(ycosθ minus xsinθ)dA (96)

=

ˆ(xycos2θ + y2cosθsinθ minus x2cosθsinθ minus xysin2θ)dA (97)

= (cos2 minus sin2θ)

ˆxydA + cosθsinθ

ˆy2dAminus cosθsinθ

ˆx2dA

= Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 36 54

傾斜軸面積之慣性矩 III

再由

sin2θ + cos2θ = 1 (98)

sin(αplusmn β) = sinαcosβ plusmn cosαsinβ (99)

sin2θ = sin(θ + θ) = 2sinθcosθ (100)

cos(αplusmn β) = cosαcosβ ∓ sinαsinβ (101)

cos2θ = cos(θ + θ) = cos2θ minus sin2θ (102)

= 2cos2θ minus 1 = 1minus 2sin2θ (103)

所以

Iu = Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (104)

= Ix(cos2θ + 1

2) + Iy (

1minus cos2θ

2)minus Ixy sin2θ (105)

=Ix + Iy

2+

Ix minus Iy2

cos2θ minus Ixy sin2θ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 37 54

傾斜軸面積之慣性矩 IV

Iv = Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (106)

= Iy (cos2θ + 1

2) + Ix(

1minus cos2θ

2) + Ixy sin2θ (107)

=Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (108)

Iuv = Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ (109)

= Ixycos2θ +1

2Ixsin2θ minus 1

2Iy sin2θ (110)

=Ix minus Iy

2sin2θ + Ixycos2θ (111)

所以

JO = Iu + Iv = Ix + Iy (112)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 38 54

主主主軸軸軸慣慣慣性性性矩矩矩Principal Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 39 54

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主軸慣性矩 I

所謂主軸慣性矩(Principal Moment ofInertia)乃定義發生在慣性矩的極值(極大或極小值)上也就是慣性矩之一次微分式為零或斜率為零

dIu

dθ=

d(Ix+Iy

2+

IxminusIy2

cos2θ minus Ixy sin2θ)

dθ(113)

= minus2Ix minus Iy

2sin2θ minus 2Ixy cos2θ (114)

= 0 (115)

其中dsin2θ = cos2θ dcos2θ = minussin2θ當慣性矩之一次微分式為零時傾斜軸與主軸之夾角θ = θp

minus2Ix minus Iy

2sin2θp minus 2Ixycos2θp = 0 (116)

tan2θ =sin2θ

cos2θ=

minusIxy(Ix minus Iy )2

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 40 54

主軸慣性矩 II

cos2θp =IxminusIy2radic

(IxminusIy2 )2 + I 2xy

sin2θp =minusIxyradic

(IxminusIy2 )2 + I 2xy

(117)

(Iu)max =Ix + Iy

2+

Ix minus Iy2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

minus IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2+

radic(Ix minus Iy

2)2 + I 2xy (118)

(Iv )min =Ix + Iy

2minus Ix minus Iy

2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

+ IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2minusradic

(Ix minus Iy

2)2 + I 2xy (119)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 41 54

主軸慣性矩 III

由tan2θp = tan(π + 2θp)可得到θp1與θp2二個解即2θp2 = π + 2θp1也就是θp2 minus θp1 = π

2

tan2θp = tan(π + 2θp) (120)

2θp2 = π + 2θp1 (121)

θp2 minus θp1 =π

2(122)

所以

Imaxmin

=Ix + Iy

2plusmnradic

(Ix minus Iy

2)2 + I 2xy (123)

Iuv =Ix minus Iy

2

minusIxyradic(IxminusIy2 )2 + I 2xy

+ Ixy

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

(124)

= 0

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課課課堂堂堂練練練習習習

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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慣慣慣性性性矩矩矩之之之莫莫莫爾爾爾圓圓圓Mohrrsquos Circle for Moment of Inertia

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慣性矩之莫爾圓

由於Iu =

Ix + Iy2

+Ix minus Iy

2cos2θ minus Ixy sin2θ (125)

Iv =Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (126)

Iuv =Ix minus Iy

2sin2θ + Ixycos2θ (127)

將式(125)與式(127)各自分別平方並相加可得

(Iu minusIx + Iy

2)2 + I 2uv = (

Ix minus Iy2

)2 + I 2xy (128)

其中Ix Iy Ixy均為常數式(128)可進一步寫成

(Iu minus a)2 + I 2uv = R2 (129)

其中a =Ix+Iy2 R =

radic(IxminusIy2 )2 + I 2xycos2θp1 =

aRsin2θp1 =

minusIxyR 2θp1為OA轉向I軸方向即2θp1為

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 46 54

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解題技巧

首先確定轉動系統的Ix Iy Ixy接著算出莫爾圓之圓心座標a及半徑R並畫出莫爾圓再由莫爾圓上點出A(Ix Ixy )並由OA轉向I軸找出兩倍夾角2θp1及2θp2其中θp1及θp2為主軸(Principle Axis)與Ix軸的夾角由於主軸(Principle Axis)為極值之軸Imax = Iu及Imin = Iv決定θp1及θp2何者為Imax = Iu與Ix軸的夾角之方法乃比較Ix及Iy之大小Imax = Iu會靠近Ix及Iy之較大者也就是Imax = Iu與Ix及Iy之較大者的夾角比較小

a =Ix + Iy

2(130)

R =

radic(Ix minus Iy

2)2 + I 2xy (131)

2θp2 = π + 2θp1 θp2 minus θp1 =π

2(132)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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質質質量量量慣慣慣性性性矩矩矩Mass Moment of Inertia

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質量慣性矩

τ = Iα (133)

其中τ為旋轉扭矩I為質量慣性矩α為旋轉角速度

I =

ˆmr2dm (134)

=

ˆVr2ρdV (135)

= ρ

ˆVr2dV (136)

其中r為旋轉半徑或轉軸到旋轉體的距離

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

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平行軸定理

當通過物體質心(G)並在一轉軸上的質量慣性矩(IG )為已知時可透過座標軸轉換計算物體之質量慣性矩(I)

I =

ˆmr2dm (137)

=

ˆm

[(d + x prime)2 + y prime2]dm (138)

=

ˆm

(x prime2 + y prime2)dm + 2d

ˆmx primedm + d2

ˆmdm (139)

I = IG + md2 (140)

其中IG為在zrsquo軸上通過質心的質量慣性矩m為質量d為通過質心之軸zrsquo並平行原軸z之距離

acutem x primedm = x

acutem dm = 0 since x = 0

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (141)

k =

radicI

m(142)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (143)

= I1 + I2 + + In (144)

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課課課堂堂堂練練練習習習

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

兩圓盤複合體

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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結結結論論論

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結論

轉動慣量或慣性矩為物體對旋轉運動的慣性而所謂慣性乃物體持續維持不變的行為也就是定義慣性矩可表達物體對本身旋轉運動的維持或持續性好壞更簡單的說慣性矩越大之物體其維持旋轉運動趨勢的能力越大例如惰輪儲能裝置惰輪之慣性矩越大當其轉動以後其維持旋轉運動趨勢的能力越大

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 55 54

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  • 轉動慣量或慣性矩(Moment of Inertia)
  • 面積慣性矩(Moment of Inertia for Area)
    • 面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)
    • 面積慣性矩之迴轉半徑(Radius of Gyration for Area)
      • 複合面積之慣性矩(Moment of Inertia for Composite Area)
      • 面積之慣性積(Product of Inertia for Area)
        • 平行軸定理(Parallel-Axis Theorem)
          • 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)
            • 主軸慣性矩(Principal Moment of Inertia)
              • 慣性矩之莫爾圓(Mohrs Circle for Moment of Inertia)
              • 質量慣性矩(Mass Moment of Inertia)
                • 平行軸定理(Parallel-Axis Theorem)
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

長方形複合

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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面面面積積積之之之慣慣慣性性性積積積Product of Inertia for Area

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 28 54

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面積之慣性積(Product of Inertia for Area)

為了下個章節求解慣性矩之極值(極大或極小值)方便數學運算在此先定義面積之慣性積(Ixy )為

dIxy = xydA (76)

Ixy =

ˆAxydA (77)

面積之慣性積的符號可為正負或零當慣性積之面積相對轉軸對稱時則面積之慣性積和為零

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 29 54

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 30 54

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平行軸定理

極點

新極點

如左圖面積慣性矩之平行軸定理乃以面心(C)為新極點(O)設定新座標軸(xrsquoyrsquo)其中xrsquo軸平行於原座標x軸yrsquo軸平行於原座標y軸取微小面積dA其在新座標之點座標為(xrsquoyrsquo)面積之慣性積(Ixy )為

dIxy = (x prime + dx)(y prime + dy )dA (78)

當以面心為轉軸點時即x prime = y prime = 0ˆAx primedA = x prime

ˆAdA = 0

ˆAy primedA = y prime

ˆAdA = 0 (79)

Ixy =

ˆA

(x prime + dx)(y prime + dy )dA (80)

=

ˆAx primey primedA + dx

ˆAy primedA + dy

ˆAx primedA + dxdy

ˆAdA (81)

= macrIx primey prime + Adxdy

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 31 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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傾傾傾斜斜斜軸軸軸面面面積積積之之之慣慣慣性性性矩矩矩

Moment of Inertia for Area about Inclined Axis

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 33 54

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傾斜軸面積之慣性矩

當轉動軸傾斜θ角時也就是座標軸旋轉θ角而Ix Iy Ixy為已知則新的座標軸(uv)可寫為

u = xcosθ + ysinθ (82)

v = ycosθ minus xsinθ (83)

由u軸上之面積慣性矩Iuv軸上之面積慣性矩Iv及慣性積Iuv得知

dIu = v2dA = (ycosθ minus xsinθ)2dA (84)

dIv = u2dA = (xcosθ + ysinθ)2dA (85)

dIuv = uvdA = (xcosθ + ysinθ)(ycosθ minus xsinθ)dA (86)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 34 54

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傾斜軸面積之慣性矩 I

由x軸上之面積慣性矩Ixy軸上之面積慣性矩Iy及慣性積Ixy得知

Ix =

ˆy2dA Iy =

ˆx2dA Ixy =

ˆxydA (87)

Iu =

ˆ(ycosθ minus xsinθ)2dA (88)

=

ˆ(y2cos2θ minus 2xycosθsinθ + x2sin2θ)dA (89)

= cos2θ

ˆy2dAminus 2cosθsinθ

ˆxydA + sin2θ

ˆx2dA (90)

= Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (91)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 35 54

傾斜軸面積之慣性矩 II

Iv =

ˆ(xcosθ + ysinθ)2dA (92)

=

ˆ(x2cos2θ + 2xycosθsinθ + y2sin2θ)dA (93)

= cos2θ

ˆx2dA + 2cosθsinθ

ˆxydA + sin2θ

ˆy2dA (94)

= Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (95)

Iuv =

ˆ(xcosθ + ysinθ)(ycosθ minus xsinθ)dA (96)

=

ˆ(xycos2θ + y2cosθsinθ minus x2cosθsinθ minus xysin2θ)dA (97)

= (cos2 minus sin2θ)

ˆxydA + cosθsinθ

ˆy2dAminus cosθsinθ

ˆx2dA

= Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 36 54

傾斜軸面積之慣性矩 III

再由

sin2θ + cos2θ = 1 (98)

sin(αplusmn β) = sinαcosβ plusmn cosαsinβ (99)

sin2θ = sin(θ + θ) = 2sinθcosθ (100)

cos(αplusmn β) = cosαcosβ ∓ sinαsinβ (101)

cos2θ = cos(θ + θ) = cos2θ minus sin2θ (102)

= 2cos2θ minus 1 = 1minus 2sin2θ (103)

所以

Iu = Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (104)

= Ix(cos2θ + 1

2) + Iy (

1minus cos2θ

2)minus Ixy sin2θ (105)

=Ix + Iy

2+

Ix minus Iy2

cos2θ minus Ixy sin2θ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 37 54

傾斜軸面積之慣性矩 IV

Iv = Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (106)

= Iy (cos2θ + 1

2) + Ix(

1minus cos2θ

2) + Ixy sin2θ (107)

=Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (108)

Iuv = Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ (109)

= Ixycos2θ +1

2Ixsin2θ minus 1

2Iy sin2θ (110)

=Ix minus Iy

2sin2θ + Ixycos2θ (111)

所以

JO = Iu + Iv = Ix + Iy (112)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 38 54

主主主軸軸軸慣慣慣性性性矩矩矩Principal Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 39 54

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主軸慣性矩 I

所謂主軸慣性矩(Principal Moment ofInertia)乃定義發生在慣性矩的極值(極大或極小值)上也就是慣性矩之一次微分式為零或斜率為零

dIu

dθ=

d(Ix+Iy

2+

IxminusIy2

cos2θ minus Ixy sin2θ)

dθ(113)

= minus2Ix minus Iy

2sin2θ minus 2Ixy cos2θ (114)

= 0 (115)

其中dsin2θ = cos2θ dcos2θ = minussin2θ當慣性矩之一次微分式為零時傾斜軸與主軸之夾角θ = θp

minus2Ix minus Iy

2sin2θp minus 2Ixycos2θp = 0 (116)

tan2θ =sin2θ

cos2θ=

minusIxy(Ix minus Iy )2

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 40 54

主軸慣性矩 II

cos2θp =IxminusIy2radic

(IxminusIy2 )2 + I 2xy

sin2θp =minusIxyradic

(IxminusIy2 )2 + I 2xy

(117)

(Iu)max =Ix + Iy

2+

Ix minus Iy2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

minus IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2+

radic(Ix minus Iy

2)2 + I 2xy (118)

(Iv )min =Ix + Iy

2minus Ix minus Iy

2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

+ IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2minusradic

(Ix minus Iy

2)2 + I 2xy (119)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 41 54

主軸慣性矩 III

由tan2θp = tan(π + 2θp)可得到θp1與θp2二個解即2θp2 = π + 2θp1也就是θp2 minus θp1 = π

2

tan2θp = tan(π + 2θp) (120)

2θp2 = π + 2θp1 (121)

θp2 minus θp1 =π

2(122)

所以

Imaxmin

=Ix + Iy

2plusmnradic

(Ix minus Iy

2)2 + I 2xy (123)

Iuv =Ix minus Iy

2

minusIxyradic(IxminusIy2 )2 + I 2xy

+ Ixy

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

(124)

= 0

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 42 54

課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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慣慣慣性性性矩矩矩之之之莫莫莫爾爾爾圓圓圓Mohrrsquos Circle for Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 44 54

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慣性矩之莫爾圓

由於Iu =

Ix + Iy2

+Ix minus Iy

2cos2θ minus Ixy sin2θ (125)

Iv =Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (126)

Iuv =Ix minus Iy

2sin2θ + Ixycos2θ (127)

將式(125)與式(127)各自分別平方並相加可得

(Iu minusIx + Iy

2)2 + I 2uv = (

Ix minus Iy2

)2 + I 2xy (128)

其中Ix Iy Ixy均為常數式(128)可進一步寫成

(Iu minus a)2 + I 2uv = R2 (129)

其中a =Ix+Iy2 R =

radic(IxminusIy2 )2 + I 2xycos2θp1 =

aRsin2θp1 =

minusIxyR 2θp1為OA轉向I軸方向即2θp1為

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 45 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 46 54

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解題技巧

首先確定轉動系統的Ix Iy Ixy接著算出莫爾圓之圓心座標a及半徑R並畫出莫爾圓再由莫爾圓上點出A(Ix Ixy )並由OA轉向I軸找出兩倍夾角2θp1及2θp2其中θp1及θp2為主軸(Principle Axis)與Ix軸的夾角由於主軸(Principle Axis)為極值之軸Imax = Iu及Imin = Iv決定θp1及θp2何者為Imax = Iu與Ix軸的夾角之方法乃比較Ix及Iy之大小Imax = Iu會靠近Ix及Iy之較大者也就是Imax = Iu與Ix及Iy之較大者的夾角比較小

a =Ix + Iy

2(130)

R =

radic(Ix minus Iy

2)2 + I 2xy (131)

2θp2 = π + 2θp1 θp2 minus θp1 =π

2(132)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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質質質量量量慣慣慣性性性矩矩矩Mass Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 48 54

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質量慣性矩

τ = Iα (133)

其中τ為旋轉扭矩I為質量慣性矩α為旋轉角速度

I =

ˆmr2dm (134)

=

ˆVr2ρdV (135)

= ρ

ˆVr2dV (136)

其中r為旋轉半徑或轉軸到旋轉體的距離

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 49 54

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 50 54

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平行軸定理

當通過物體質心(G)並在一轉軸上的質量慣性矩(IG )為已知時可透過座標軸轉換計算物體之質量慣性矩(I)

I =

ˆmr2dm (137)

=

ˆm

[(d + x prime)2 + y prime2]dm (138)

=

ˆm

(x prime2 + y prime2)dm + 2d

ˆmx primedm + d2

ˆmdm (139)

I = IG + md2 (140)

其中IG為在zrsquo軸上通過質心的質量慣性矩m為質量d為通過質心之軸zrsquo並平行原軸z之距離

acutem x primedm = x

acutem dm = 0 since x = 0

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 51 54

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (141)

k =

radicI

m(142)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (143)

= I1 + I2 + + In (144)

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

兩圓盤複合體

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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結結結論論論

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 54 54

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結論

轉動慣量或慣性矩為物體對旋轉運動的慣性而所謂慣性乃物體持續維持不變的行為也就是定義慣性矩可表達物體對本身旋轉運動的維持或持續性好壞更簡單的說慣性矩越大之物體其維持旋轉運動趨勢的能力越大例如惰輪儲能裝置惰輪之慣性矩越大當其轉動以後其維持旋轉運動趨勢的能力越大

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 55 54

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  • 轉動慣量或慣性矩(Moment of Inertia)
  • 面積慣性矩(Moment of Inertia for Area)
    • 面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)
    • 面積慣性矩之迴轉半徑(Radius of Gyration for Area)
      • 複合面積之慣性矩(Moment of Inertia for Composite Area)
      • 面積之慣性積(Product of Inertia for Area)
        • 平行軸定理(Parallel-Axis Theorem)
          • 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)
            • 主軸慣性矩(Principal Moment of Inertia)
              • 慣性矩之莫爾圓(Mohrs Circle for Moment of Inertia)
              • 質量慣性矩(Mass Moment of Inertia)
                • 平行軸定理(Parallel-Axis Theorem)
Page 36: 轉動慣量 Moments of Inertia - cct.me.ntut.edu.twcct.me.ntut.edu.tw/ccteducation/chchting/aiahtm/statics/lecture... · 轉動慣量或慣性矩 I. 轉動慣量也稱慣性矩(Moment

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

長方形複合

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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面面面積積積之之之慣慣慣性性性積積積Product of Inertia for Area

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 28 54

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面積之慣性積(Product of Inertia for Area)

為了下個章節求解慣性矩之極值(極大或極小值)方便數學運算在此先定義面積之慣性積(Ixy )為

dIxy = xydA (76)

Ixy =

ˆAxydA (77)

面積之慣性積的符號可為正負或零當慣性積之面積相對轉軸對稱時則面積之慣性積和為零

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 29 54

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 30 54

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平行軸定理

極點

新極點

如左圖面積慣性矩之平行軸定理乃以面心(C)為新極點(O)設定新座標軸(xrsquoyrsquo)其中xrsquo軸平行於原座標x軸yrsquo軸平行於原座標y軸取微小面積dA其在新座標之點座標為(xrsquoyrsquo)面積之慣性積(Ixy )為

dIxy = (x prime + dx)(y prime + dy )dA (78)

當以面心為轉軸點時即x prime = y prime = 0ˆAx primedA = x prime

ˆAdA = 0

ˆAy primedA = y prime

ˆAdA = 0 (79)

Ixy =

ˆA

(x prime + dx)(y prime + dy )dA (80)

=

ˆAx primey primedA + dx

ˆAy primedA + dy

ˆAx primedA + dxdy

ˆAdA (81)

= macrIx primey prime + Adxdy

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 31 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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傾傾傾斜斜斜軸軸軸面面面積積積之之之慣慣慣性性性矩矩矩

Moment of Inertia for Area about Inclined Axis

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 33 54

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傾斜軸面積之慣性矩

當轉動軸傾斜θ角時也就是座標軸旋轉θ角而Ix Iy Ixy為已知則新的座標軸(uv)可寫為

u = xcosθ + ysinθ (82)

v = ycosθ minus xsinθ (83)

由u軸上之面積慣性矩Iuv軸上之面積慣性矩Iv及慣性積Iuv得知

dIu = v2dA = (ycosθ minus xsinθ)2dA (84)

dIv = u2dA = (xcosθ + ysinθ)2dA (85)

dIuv = uvdA = (xcosθ + ysinθ)(ycosθ minus xsinθ)dA (86)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 34 54

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傾斜軸面積之慣性矩 I

由x軸上之面積慣性矩Ixy軸上之面積慣性矩Iy及慣性積Ixy得知

Ix =

ˆy2dA Iy =

ˆx2dA Ixy =

ˆxydA (87)

Iu =

ˆ(ycosθ minus xsinθ)2dA (88)

=

ˆ(y2cos2θ minus 2xycosθsinθ + x2sin2θ)dA (89)

= cos2θ

ˆy2dAminus 2cosθsinθ

ˆxydA + sin2θ

ˆx2dA (90)

= Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (91)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 35 54

傾斜軸面積之慣性矩 II

Iv =

ˆ(xcosθ + ysinθ)2dA (92)

=

ˆ(x2cos2θ + 2xycosθsinθ + y2sin2θ)dA (93)

= cos2θ

ˆx2dA + 2cosθsinθ

ˆxydA + sin2θ

ˆy2dA (94)

= Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (95)

Iuv =

ˆ(xcosθ + ysinθ)(ycosθ minus xsinθ)dA (96)

=

ˆ(xycos2θ + y2cosθsinθ minus x2cosθsinθ minus xysin2θ)dA (97)

= (cos2 minus sin2θ)

ˆxydA + cosθsinθ

ˆy2dAminus cosθsinθ

ˆx2dA

= Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 36 54

傾斜軸面積之慣性矩 III

再由

sin2θ + cos2θ = 1 (98)

sin(αplusmn β) = sinαcosβ plusmn cosαsinβ (99)

sin2θ = sin(θ + θ) = 2sinθcosθ (100)

cos(αplusmn β) = cosαcosβ ∓ sinαsinβ (101)

cos2θ = cos(θ + θ) = cos2θ minus sin2θ (102)

= 2cos2θ minus 1 = 1minus 2sin2θ (103)

所以

Iu = Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (104)

= Ix(cos2θ + 1

2) + Iy (

1minus cos2θ

2)minus Ixy sin2θ (105)

=Ix + Iy

2+

Ix minus Iy2

cos2θ minus Ixy sin2θ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 37 54

傾斜軸面積之慣性矩 IV

Iv = Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (106)

= Iy (cos2θ + 1

2) + Ix(

1minus cos2θ

2) + Ixy sin2θ (107)

=Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (108)

Iuv = Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ (109)

= Ixycos2θ +1

2Ixsin2θ minus 1

2Iy sin2θ (110)

=Ix minus Iy

2sin2θ + Ixycos2θ (111)

所以

JO = Iu + Iv = Ix + Iy (112)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 38 54

主主主軸軸軸慣慣慣性性性矩矩矩Principal Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 39 54

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主軸慣性矩 I

所謂主軸慣性矩(Principal Moment ofInertia)乃定義發生在慣性矩的極值(極大或極小值)上也就是慣性矩之一次微分式為零或斜率為零

dIu

dθ=

d(Ix+Iy

2+

IxminusIy2

cos2θ minus Ixy sin2θ)

dθ(113)

= minus2Ix minus Iy

2sin2θ minus 2Ixy cos2θ (114)

= 0 (115)

其中dsin2θ = cos2θ dcos2θ = minussin2θ當慣性矩之一次微分式為零時傾斜軸與主軸之夾角θ = θp

minus2Ix minus Iy

2sin2θp minus 2Ixycos2θp = 0 (116)

tan2θ =sin2θ

cos2θ=

minusIxy(Ix minus Iy )2

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 40 54

主軸慣性矩 II

cos2θp =IxminusIy2radic

(IxminusIy2 )2 + I 2xy

sin2θp =minusIxyradic

(IxminusIy2 )2 + I 2xy

(117)

(Iu)max =Ix + Iy

2+

Ix minus Iy2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

minus IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2+

radic(Ix minus Iy

2)2 + I 2xy (118)

(Iv )min =Ix + Iy

2minus Ix minus Iy

2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

+ IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2minusradic

(Ix minus Iy

2)2 + I 2xy (119)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 41 54

主軸慣性矩 III

由tan2θp = tan(π + 2θp)可得到θp1與θp2二個解即2θp2 = π + 2θp1也就是θp2 minus θp1 = π

2

tan2θp = tan(π + 2θp) (120)

2θp2 = π + 2θp1 (121)

θp2 minus θp1 =π

2(122)

所以

Imaxmin

=Ix + Iy

2plusmnradic

(Ix minus Iy

2)2 + I 2xy (123)

Iuv =Ix minus Iy

2

minusIxyradic(IxminusIy2 )2 + I 2xy

+ Ixy

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

(124)

= 0

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 42 54

課課課堂堂堂練練練習習習

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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慣慣慣性性性矩矩矩之之之莫莫莫爾爾爾圓圓圓Mohrrsquos Circle for Moment of Inertia

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慣性矩之莫爾圓

由於Iu =

Ix + Iy2

+Ix minus Iy

2cos2θ minus Ixy sin2θ (125)

Iv =Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (126)

Iuv =Ix minus Iy

2sin2θ + Ixycos2θ (127)

將式(125)與式(127)各自分別平方並相加可得

(Iu minusIx + Iy

2)2 + I 2uv = (

Ix minus Iy2

)2 + I 2xy (128)

其中Ix Iy Ixy均為常數式(128)可進一步寫成

(Iu minus a)2 + I 2uv = R2 (129)

其中a =Ix+Iy2 R =

radic(IxminusIy2 )2 + I 2xycos2θp1 =

aRsin2θp1 =

minusIxyR 2θp1為OA轉向I軸方向即2θp1為

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課課課堂堂堂練練練習習習

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解題技巧

首先確定轉動系統的Ix Iy Ixy接著算出莫爾圓之圓心座標a及半徑R並畫出莫爾圓再由莫爾圓上點出A(Ix Ixy )並由OA轉向I軸找出兩倍夾角2θp1及2θp2其中θp1及θp2為主軸(Principle Axis)與Ix軸的夾角由於主軸(Principle Axis)為極值之軸Imax = Iu及Imin = Iv決定θp1及θp2何者為Imax = Iu與Ix軸的夾角之方法乃比較Ix及Iy之大小Imax = Iu會靠近Ix及Iy之較大者也就是Imax = Iu與Ix及Iy之較大者的夾角比較小

a =Ix + Iy

2(130)

R =

radic(Ix minus Iy

2)2 + I 2xy (131)

2θp2 = π + 2θp1 θp2 minus θp1 =π

2(132)

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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質質質量量量慣慣慣性性性矩矩矩Mass Moment of Inertia

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質量慣性矩

τ = Iα (133)

其中τ為旋轉扭矩I為質量慣性矩α為旋轉角速度

I =

ˆmr2dm (134)

=

ˆVr2ρdV (135)

= ρ

ˆVr2dV (136)

其中r為旋轉半徑或轉軸到旋轉體的距離

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

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平行軸定理

當通過物體質心(G)並在一轉軸上的質量慣性矩(IG )為已知時可透過座標軸轉換計算物體之質量慣性矩(I)

I =

ˆmr2dm (137)

=

ˆm

[(d + x prime)2 + y prime2]dm (138)

=

ˆm

(x prime2 + y prime2)dm + 2d

ˆmx primedm + d2

ˆmdm (139)

I = IG + md2 (140)

其中IG為在zrsquo軸上通過質心的質量慣性矩m為質量d為通過質心之軸zrsquo並平行原軸z之距離

acutem x primedm = x

acutem dm = 0 since x = 0

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (141)

k =

radicI

m(142)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (143)

= I1 + I2 + + In (144)

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課課課堂堂堂練練練習習習

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

兩圓盤複合體

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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結結結論論論

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結論

轉動慣量或慣性矩為物體對旋轉運動的慣性而所謂慣性乃物體持續維持不變的行為也就是定義慣性矩可表達物體對本身旋轉運動的維持或持續性好壞更簡單的說慣性矩越大之物體其維持旋轉運動趨勢的能力越大例如惰輪儲能裝置惰輪之慣性矩越大當其轉動以後其維持旋轉運動趨勢的能力越大

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 55 54

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  • 轉動慣量或慣性矩(Moment of Inertia)
  • 面積慣性矩(Moment of Inertia for Area)
    • 面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)
    • 面積慣性矩之迴轉半徑(Radius of Gyration for Area)
      • 複合面積之慣性矩(Moment of Inertia for Composite Area)
      • 面積之慣性積(Product of Inertia for Area)
        • 平行軸定理(Parallel-Axis Theorem)
          • 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)
            • 主軸慣性矩(Principal Moment of Inertia)
              • 慣性矩之莫爾圓(Mohrs Circle for Moment of Inertia)
              • 質量慣性矩(Mass Moment of Inertia)
                • 平行軸定理(Parallel-Axis Theorem)
Page 37: 轉動慣量 Moments of Inertia - cct.me.ntut.edu.twcct.me.ntut.edu.tw/ccteducation/chchting/aiahtm/statics/lecture... · 轉動慣量或慣性矩 I. 轉動慣量也稱慣性矩(Moment

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 27 54

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面面面積積積之之之慣慣慣性性性積積積Product of Inertia for Area

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 28 54

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面積之慣性積(Product of Inertia for Area)

為了下個章節求解慣性矩之極值(極大或極小值)方便數學運算在此先定義面積之慣性積(Ixy )為

dIxy = xydA (76)

Ixy =

ˆAxydA (77)

面積之慣性積的符號可為正負或零當慣性積之面積相對轉軸對稱時則面積之慣性積和為零

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 30 54

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平行軸定理

極點

新極點

如左圖面積慣性矩之平行軸定理乃以面心(C)為新極點(O)設定新座標軸(xrsquoyrsquo)其中xrsquo軸平行於原座標x軸yrsquo軸平行於原座標y軸取微小面積dA其在新座標之點座標為(xrsquoyrsquo)面積之慣性積(Ixy )為

dIxy = (x prime + dx)(y prime + dy )dA (78)

當以面心為轉軸點時即x prime = y prime = 0ˆAx primedA = x prime

ˆAdA = 0

ˆAy primedA = y prime

ˆAdA = 0 (79)

Ixy =

ˆA

(x prime + dx)(y prime + dy )dA (80)

=

ˆAx primey primedA + dx

ˆAy primedA + dy

ˆAx primedA + dxdy

ˆAdA (81)

= macrIx primey prime + Adxdy

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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傾傾傾斜斜斜軸軸軸面面面積積積之之之慣慣慣性性性矩矩矩

Moment of Inertia for Area about Inclined Axis

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傾斜軸面積之慣性矩

當轉動軸傾斜θ角時也就是座標軸旋轉θ角而Ix Iy Ixy為已知則新的座標軸(uv)可寫為

u = xcosθ + ysinθ (82)

v = ycosθ minus xsinθ (83)

由u軸上之面積慣性矩Iuv軸上之面積慣性矩Iv及慣性積Iuv得知

dIu = v2dA = (ycosθ minus xsinθ)2dA (84)

dIv = u2dA = (xcosθ + ysinθ)2dA (85)

dIuv = uvdA = (xcosθ + ysinθ)(ycosθ minus xsinθ)dA (86)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 34 54

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傾斜軸面積之慣性矩 I

由x軸上之面積慣性矩Ixy軸上之面積慣性矩Iy及慣性積Ixy得知

Ix =

ˆy2dA Iy =

ˆx2dA Ixy =

ˆxydA (87)

Iu =

ˆ(ycosθ minus xsinθ)2dA (88)

=

ˆ(y2cos2θ minus 2xycosθsinθ + x2sin2θ)dA (89)

= cos2θ

ˆy2dAminus 2cosθsinθ

ˆxydA + sin2θ

ˆx2dA (90)

= Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (91)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 35 54

傾斜軸面積之慣性矩 II

Iv =

ˆ(xcosθ + ysinθ)2dA (92)

=

ˆ(x2cos2θ + 2xycosθsinθ + y2sin2θ)dA (93)

= cos2θ

ˆx2dA + 2cosθsinθ

ˆxydA + sin2θ

ˆy2dA (94)

= Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (95)

Iuv =

ˆ(xcosθ + ysinθ)(ycosθ minus xsinθ)dA (96)

=

ˆ(xycos2θ + y2cosθsinθ minus x2cosθsinθ minus xysin2θ)dA (97)

= (cos2 minus sin2θ)

ˆxydA + cosθsinθ

ˆy2dAminus cosθsinθ

ˆx2dA

= Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 36 54

傾斜軸面積之慣性矩 III

再由

sin2θ + cos2θ = 1 (98)

sin(αplusmn β) = sinαcosβ plusmn cosαsinβ (99)

sin2θ = sin(θ + θ) = 2sinθcosθ (100)

cos(αplusmn β) = cosαcosβ ∓ sinαsinβ (101)

cos2θ = cos(θ + θ) = cos2θ minus sin2θ (102)

= 2cos2θ minus 1 = 1minus 2sin2θ (103)

所以

Iu = Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (104)

= Ix(cos2θ + 1

2) + Iy (

1minus cos2θ

2)minus Ixy sin2θ (105)

=Ix + Iy

2+

Ix minus Iy2

cos2θ minus Ixy sin2θ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 37 54

傾斜軸面積之慣性矩 IV

Iv = Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (106)

= Iy (cos2θ + 1

2) + Ix(

1minus cos2θ

2) + Ixy sin2θ (107)

=Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (108)

Iuv = Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ (109)

= Ixycos2θ +1

2Ixsin2θ minus 1

2Iy sin2θ (110)

=Ix minus Iy

2sin2θ + Ixycos2θ (111)

所以

JO = Iu + Iv = Ix + Iy (112)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 38 54

主主主軸軸軸慣慣慣性性性矩矩矩Principal Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 39 54

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主軸慣性矩 I

所謂主軸慣性矩(Principal Moment ofInertia)乃定義發生在慣性矩的極值(極大或極小值)上也就是慣性矩之一次微分式為零或斜率為零

dIu

dθ=

d(Ix+Iy

2+

IxminusIy2

cos2θ minus Ixy sin2θ)

dθ(113)

= minus2Ix minus Iy

2sin2θ minus 2Ixy cos2θ (114)

= 0 (115)

其中dsin2θ = cos2θ dcos2θ = minussin2θ當慣性矩之一次微分式為零時傾斜軸與主軸之夾角θ = θp

minus2Ix minus Iy

2sin2θp minus 2Ixycos2θp = 0 (116)

tan2θ =sin2θ

cos2θ=

minusIxy(Ix minus Iy )2

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 40 54

主軸慣性矩 II

cos2θp =IxminusIy2radic

(IxminusIy2 )2 + I 2xy

sin2θp =minusIxyradic

(IxminusIy2 )2 + I 2xy

(117)

(Iu)max =Ix + Iy

2+

Ix minus Iy2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

minus IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2+

radic(Ix minus Iy

2)2 + I 2xy (118)

(Iv )min =Ix + Iy

2minus Ix minus Iy

2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

+ IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2minusradic

(Ix minus Iy

2)2 + I 2xy (119)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 41 54

主軸慣性矩 III

由tan2θp = tan(π + 2θp)可得到θp1與θp2二個解即2θp2 = π + 2θp1也就是θp2 minus θp1 = π

2

tan2θp = tan(π + 2θp) (120)

2θp2 = π + 2θp1 (121)

θp2 minus θp1 =π

2(122)

所以

Imaxmin

=Ix + Iy

2plusmnradic

(Ix minus Iy

2)2 + I 2xy (123)

Iuv =Ix minus Iy

2

minusIxyradic(IxminusIy2 )2 + I 2xy

+ Ixy

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

(124)

= 0

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 42 54

課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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慣慣慣性性性矩矩矩之之之莫莫莫爾爾爾圓圓圓Mohrrsquos Circle for Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 44 54

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慣性矩之莫爾圓

由於Iu =

Ix + Iy2

+Ix minus Iy

2cos2θ minus Ixy sin2θ (125)

Iv =Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (126)

Iuv =Ix minus Iy

2sin2θ + Ixycos2θ (127)

將式(125)與式(127)各自分別平方並相加可得

(Iu minusIx + Iy

2)2 + I 2uv = (

Ix minus Iy2

)2 + I 2xy (128)

其中Ix Iy Ixy均為常數式(128)可進一步寫成

(Iu minus a)2 + I 2uv = R2 (129)

其中a =Ix+Iy2 R =

radic(IxminusIy2 )2 + I 2xycos2θp1 =

aRsin2θp1 =

minusIxyR 2θp1為OA轉向I軸方向即2θp1為

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 45 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 46 54

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解題技巧

首先確定轉動系統的Ix Iy Ixy接著算出莫爾圓之圓心座標a及半徑R並畫出莫爾圓再由莫爾圓上點出A(Ix Ixy )並由OA轉向I軸找出兩倍夾角2θp1及2θp2其中θp1及θp2為主軸(Principle Axis)與Ix軸的夾角由於主軸(Principle Axis)為極值之軸Imax = Iu及Imin = Iv決定θp1及θp2何者為Imax = Iu與Ix軸的夾角之方法乃比較Ix及Iy之大小Imax = Iu會靠近Ix及Iy之較大者也就是Imax = Iu與Ix及Iy之較大者的夾角比較小

a =Ix + Iy

2(130)

R =

radic(Ix minus Iy

2)2 + I 2xy (131)

2θp2 = π + 2θp1 θp2 minus θp1 =π

2(132)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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質質質量量量慣慣慣性性性矩矩矩Mass Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 48 54

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質量慣性矩

τ = Iα (133)

其中τ為旋轉扭矩I為質量慣性矩α為旋轉角速度

I =

ˆmr2dm (134)

=

ˆVr2ρdV (135)

= ρ

ˆVr2dV (136)

其中r為旋轉半徑或轉軸到旋轉體的距離

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 49 54

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 50 54

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平行軸定理

當通過物體質心(G)並在一轉軸上的質量慣性矩(IG )為已知時可透過座標軸轉換計算物體之質量慣性矩(I)

I =

ˆmr2dm (137)

=

ˆm

[(d + x prime)2 + y prime2]dm (138)

=

ˆm

(x prime2 + y prime2)dm + 2d

ˆmx primedm + d2

ˆmdm (139)

I = IG + md2 (140)

其中IG為在zrsquo軸上通過質心的質量慣性矩m為質量d為通過質心之軸zrsquo並平行原軸z之距離

acutem x primedm = x

acutem dm = 0 since x = 0

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 51 54

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (141)

k =

radicI

m(142)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (143)

= I1 + I2 + + In (144)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 52 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

兩圓盤複合體

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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結結結論論論

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 54 54

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結論

轉動慣量或慣性矩為物體對旋轉運動的慣性而所謂慣性乃物體持續維持不變的行為也就是定義慣性矩可表達物體對本身旋轉運動的維持或持續性好壞更簡單的說慣性矩越大之物體其維持旋轉運動趨勢的能力越大例如惰輪儲能裝置惰輪之慣性矩越大當其轉動以後其維持旋轉運動趨勢的能力越大

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 55 54

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  • 轉動慣量或慣性矩(Moment of Inertia)
  • 面積慣性矩(Moment of Inertia for Area)
    • 面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)
    • 面積慣性矩之迴轉半徑(Radius of Gyration for Area)
      • 複合面積之慣性矩(Moment of Inertia for Composite Area)
      • 面積之慣性積(Product of Inertia for Area)
        • 平行軸定理(Parallel-Axis Theorem)
          • 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)
            • 主軸慣性矩(Principal Moment of Inertia)
              • 慣性矩之莫爾圓(Mohrs Circle for Moment of Inertia)
              • 質量慣性矩(Mass Moment of Inertia)
                • 平行軸定理(Parallel-Axis Theorem)
Page 38: 轉動慣量 Moments of Inertia - cct.me.ntut.edu.twcct.me.ntut.edu.tw/ccteducation/chchting/aiahtm/statics/lecture... · 轉動慣量或慣性矩 I. 轉動慣量也稱慣性矩(Moment

面面面積積積之之之慣慣慣性性性積積積Product of Inertia for Area

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 28 54

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面積之慣性積(Product of Inertia for Area)

為了下個章節求解慣性矩之極值(極大或極小值)方便數學運算在此先定義面積之慣性積(Ixy )為

dIxy = xydA (76)

Ixy =

ˆAxydA (77)

面積之慣性積的符號可為正負或零當慣性積之面積相對轉軸對稱時則面積之慣性積和為零

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 29 54

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 30 54

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平行軸定理

極點

新極點

如左圖面積慣性矩之平行軸定理乃以面心(C)為新極點(O)設定新座標軸(xrsquoyrsquo)其中xrsquo軸平行於原座標x軸yrsquo軸平行於原座標y軸取微小面積dA其在新座標之點座標為(xrsquoyrsquo)面積之慣性積(Ixy )為

dIxy = (x prime + dx)(y prime + dy )dA (78)

當以面心為轉軸點時即x prime = y prime = 0ˆAx primedA = x prime

ˆAdA = 0

ˆAy primedA = y prime

ˆAdA = 0 (79)

Ixy =

ˆA

(x prime + dx)(y prime + dy )dA (80)

=

ˆAx primey primedA + dx

ˆAy primedA + dy

ˆAx primedA + dxdy

ˆAdA (81)

= macrIx primey prime + Adxdy

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 31 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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傾傾傾斜斜斜軸軸軸面面面積積積之之之慣慣慣性性性矩矩矩

Moment of Inertia for Area about Inclined Axis

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 33 54

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傾斜軸面積之慣性矩

當轉動軸傾斜θ角時也就是座標軸旋轉θ角而Ix Iy Ixy為已知則新的座標軸(uv)可寫為

u = xcosθ + ysinθ (82)

v = ycosθ minus xsinθ (83)

由u軸上之面積慣性矩Iuv軸上之面積慣性矩Iv及慣性積Iuv得知

dIu = v2dA = (ycosθ minus xsinθ)2dA (84)

dIv = u2dA = (xcosθ + ysinθ)2dA (85)

dIuv = uvdA = (xcosθ + ysinθ)(ycosθ minus xsinθ)dA (86)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 34 54

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傾斜軸面積之慣性矩 I

由x軸上之面積慣性矩Ixy軸上之面積慣性矩Iy及慣性積Ixy得知

Ix =

ˆy2dA Iy =

ˆx2dA Ixy =

ˆxydA (87)

Iu =

ˆ(ycosθ minus xsinθ)2dA (88)

=

ˆ(y2cos2θ minus 2xycosθsinθ + x2sin2θ)dA (89)

= cos2θ

ˆy2dAminus 2cosθsinθ

ˆxydA + sin2θ

ˆx2dA (90)

= Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (91)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 35 54

傾斜軸面積之慣性矩 II

Iv =

ˆ(xcosθ + ysinθ)2dA (92)

=

ˆ(x2cos2θ + 2xycosθsinθ + y2sin2θ)dA (93)

= cos2θ

ˆx2dA + 2cosθsinθ

ˆxydA + sin2θ

ˆy2dA (94)

= Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (95)

Iuv =

ˆ(xcosθ + ysinθ)(ycosθ minus xsinθ)dA (96)

=

ˆ(xycos2θ + y2cosθsinθ minus x2cosθsinθ minus xysin2θ)dA (97)

= (cos2 minus sin2θ)

ˆxydA + cosθsinθ

ˆy2dAminus cosθsinθ

ˆx2dA

= Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 36 54

傾斜軸面積之慣性矩 III

再由

sin2θ + cos2θ = 1 (98)

sin(αplusmn β) = sinαcosβ plusmn cosαsinβ (99)

sin2θ = sin(θ + θ) = 2sinθcosθ (100)

cos(αplusmn β) = cosαcosβ ∓ sinαsinβ (101)

cos2θ = cos(θ + θ) = cos2θ minus sin2θ (102)

= 2cos2θ minus 1 = 1minus 2sin2θ (103)

所以

Iu = Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (104)

= Ix(cos2θ + 1

2) + Iy (

1minus cos2θ

2)minus Ixy sin2θ (105)

=Ix + Iy

2+

Ix minus Iy2

cos2θ minus Ixy sin2θ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 37 54

傾斜軸面積之慣性矩 IV

Iv = Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (106)

= Iy (cos2θ + 1

2) + Ix(

1minus cos2θ

2) + Ixy sin2θ (107)

=Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (108)

Iuv = Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ (109)

= Ixycos2θ +1

2Ixsin2θ minus 1

2Iy sin2θ (110)

=Ix minus Iy

2sin2θ + Ixycos2θ (111)

所以

JO = Iu + Iv = Ix + Iy (112)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 38 54

主主主軸軸軸慣慣慣性性性矩矩矩Principal Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 39 54

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主軸慣性矩 I

所謂主軸慣性矩(Principal Moment ofInertia)乃定義發生在慣性矩的極值(極大或極小值)上也就是慣性矩之一次微分式為零或斜率為零

dIu

dθ=

d(Ix+Iy

2+

IxminusIy2

cos2θ minus Ixy sin2θ)

dθ(113)

= minus2Ix minus Iy

2sin2θ minus 2Ixy cos2θ (114)

= 0 (115)

其中dsin2θ = cos2θ dcos2θ = minussin2θ當慣性矩之一次微分式為零時傾斜軸與主軸之夾角θ = θp

minus2Ix minus Iy

2sin2θp minus 2Ixycos2θp = 0 (116)

tan2θ =sin2θ

cos2θ=

minusIxy(Ix minus Iy )2

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 40 54

主軸慣性矩 II

cos2θp =IxminusIy2radic

(IxminusIy2 )2 + I 2xy

sin2θp =minusIxyradic

(IxminusIy2 )2 + I 2xy

(117)

(Iu)max =Ix + Iy

2+

Ix minus Iy2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

minus IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2+

radic(Ix minus Iy

2)2 + I 2xy (118)

(Iv )min =Ix + Iy

2minus Ix minus Iy

2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

+ IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2minusradic

(Ix minus Iy

2)2 + I 2xy (119)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 41 54

主軸慣性矩 III

由tan2θp = tan(π + 2θp)可得到θp1與θp2二個解即2θp2 = π + 2θp1也就是θp2 minus θp1 = π

2

tan2θp = tan(π + 2θp) (120)

2θp2 = π + 2θp1 (121)

θp2 minus θp1 =π

2(122)

所以

Imaxmin

=Ix + Iy

2plusmnradic

(Ix minus Iy

2)2 + I 2xy (123)

Iuv =Ix minus Iy

2

minusIxyradic(IxminusIy2 )2 + I 2xy

+ Ixy

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

(124)

= 0

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 42 54

課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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慣慣慣性性性矩矩矩之之之莫莫莫爾爾爾圓圓圓Mohrrsquos Circle for Moment of Inertia

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慣性矩之莫爾圓

由於Iu =

Ix + Iy2

+Ix minus Iy

2cos2θ minus Ixy sin2θ (125)

Iv =Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (126)

Iuv =Ix minus Iy

2sin2θ + Ixycos2θ (127)

將式(125)與式(127)各自分別平方並相加可得

(Iu minusIx + Iy

2)2 + I 2uv = (

Ix minus Iy2

)2 + I 2xy (128)

其中Ix Iy Ixy均為常數式(128)可進一步寫成

(Iu minus a)2 + I 2uv = R2 (129)

其中a =Ix+Iy2 R =

radic(IxminusIy2 )2 + I 2xycos2θp1 =

aRsin2θp1 =

minusIxyR 2θp1為OA轉向I軸方向即2θp1為

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 45 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 46 54

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解題技巧

首先確定轉動系統的Ix Iy Ixy接著算出莫爾圓之圓心座標a及半徑R並畫出莫爾圓再由莫爾圓上點出A(Ix Ixy )並由OA轉向I軸找出兩倍夾角2θp1及2θp2其中θp1及θp2為主軸(Principle Axis)與Ix軸的夾角由於主軸(Principle Axis)為極值之軸Imax = Iu及Imin = Iv決定θp1及θp2何者為Imax = Iu與Ix軸的夾角之方法乃比較Ix及Iy之大小Imax = Iu會靠近Ix及Iy之較大者也就是Imax = Iu與Ix及Iy之較大者的夾角比較小

a =Ix + Iy

2(130)

R =

radic(Ix minus Iy

2)2 + I 2xy (131)

2θp2 = π + 2θp1 θp2 minus θp1 =π

2(132)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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質質質量量量慣慣慣性性性矩矩矩Mass Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 48 54

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質量慣性矩

τ = Iα (133)

其中τ為旋轉扭矩I為質量慣性矩α為旋轉角速度

I =

ˆmr2dm (134)

=

ˆVr2ρdV (135)

= ρ

ˆVr2dV (136)

其中r為旋轉半徑或轉軸到旋轉體的距離

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 49 54

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 50 54

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平行軸定理

當通過物體質心(G)並在一轉軸上的質量慣性矩(IG )為已知時可透過座標軸轉換計算物體之質量慣性矩(I)

I =

ˆmr2dm (137)

=

ˆm

[(d + x prime)2 + y prime2]dm (138)

=

ˆm

(x prime2 + y prime2)dm + 2d

ˆmx primedm + d2

ˆmdm (139)

I = IG + md2 (140)

其中IG為在zrsquo軸上通過質心的質量慣性矩m為質量d為通過質心之軸zrsquo並平行原軸z之距離

acutem x primedm = x

acutem dm = 0 since x = 0

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (141)

k =

radicI

m(142)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (143)

= I1 + I2 + + In (144)

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課課課堂堂堂練練練習習習

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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兩圓盤複合體

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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結結結論論論

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結論

轉動慣量或慣性矩為物體對旋轉運動的慣性而所謂慣性乃物體持續維持不變的行為也就是定義慣性矩可表達物體對本身旋轉運動的維持或持續性好壞更簡單的說慣性矩越大之物體其維持旋轉運動趨勢的能力越大例如惰輪儲能裝置惰輪之慣性矩越大當其轉動以後其維持旋轉運動趨勢的能力越大

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 55 54

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  • 轉動慣量或慣性矩(Moment of Inertia)
  • 面積慣性矩(Moment of Inertia for Area)
    • 面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)
    • 面積慣性矩之迴轉半徑(Radius of Gyration for Area)
      • 複合面積之慣性矩(Moment of Inertia for Composite Area)
      • 面積之慣性積(Product of Inertia for Area)
        • 平行軸定理(Parallel-Axis Theorem)
          • 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)
            • 主軸慣性矩(Principal Moment of Inertia)
              • 慣性矩之莫爾圓(Mohrs Circle for Moment of Inertia)
              • 質量慣性矩(Mass Moment of Inertia)
                • 平行軸定理(Parallel-Axis Theorem)
Page 39: 轉動慣量 Moments of Inertia - cct.me.ntut.edu.twcct.me.ntut.edu.tw/ccteducation/chchting/aiahtm/statics/lecture... · 轉動慣量或慣性矩 I. 轉動慣量也稱慣性矩(Moment

面積之慣性積(Product of Inertia for Area)

為了下個章節求解慣性矩之極值(極大或極小值)方便數學運算在此先定義面積之慣性積(Ixy )為

dIxy = xydA (76)

Ixy =

ˆAxydA (77)

面積之慣性積的符號可為正負或零當慣性積之面積相對轉軸對稱時則面積之慣性積和為零

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 30 54

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平行軸定理

極點

新極點

如左圖面積慣性矩之平行軸定理乃以面心(C)為新極點(O)設定新座標軸(xrsquoyrsquo)其中xrsquo軸平行於原座標x軸yrsquo軸平行於原座標y軸取微小面積dA其在新座標之點座標為(xrsquoyrsquo)面積之慣性積(Ixy )為

dIxy = (x prime + dx)(y prime + dy )dA (78)

當以面心為轉軸點時即x prime = y prime = 0ˆAx primedA = x prime

ˆAdA = 0

ˆAy primedA = y prime

ˆAdA = 0 (79)

Ixy =

ˆA

(x prime + dx)(y prime + dy )dA (80)

=

ˆAx primey primedA + dx

ˆAy primedA + dy

ˆAx primedA + dxdy

ˆAdA (81)

= macrIx primey prime + Adxdy

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 31 54

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課課課堂堂堂練練練習習習

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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傾傾傾斜斜斜軸軸軸面面面積積積之之之慣慣慣性性性矩矩矩

Moment of Inertia for Area about Inclined Axis

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傾斜軸面積之慣性矩

當轉動軸傾斜θ角時也就是座標軸旋轉θ角而Ix Iy Ixy為已知則新的座標軸(uv)可寫為

u = xcosθ + ysinθ (82)

v = ycosθ minus xsinθ (83)

由u軸上之面積慣性矩Iuv軸上之面積慣性矩Iv及慣性積Iuv得知

dIu = v2dA = (ycosθ minus xsinθ)2dA (84)

dIv = u2dA = (xcosθ + ysinθ)2dA (85)

dIuv = uvdA = (xcosθ + ysinθ)(ycosθ minus xsinθ)dA (86)

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傾斜軸面積之慣性矩 I

由x軸上之面積慣性矩Ixy軸上之面積慣性矩Iy及慣性積Ixy得知

Ix =

ˆy2dA Iy =

ˆx2dA Ixy =

ˆxydA (87)

Iu =

ˆ(ycosθ minus xsinθ)2dA (88)

=

ˆ(y2cos2θ minus 2xycosθsinθ + x2sin2θ)dA (89)

= cos2θ

ˆy2dAminus 2cosθsinθ

ˆxydA + sin2θ

ˆx2dA (90)

= Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (91)

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傾斜軸面積之慣性矩 II

Iv =

ˆ(xcosθ + ysinθ)2dA (92)

=

ˆ(x2cos2θ + 2xycosθsinθ + y2sin2θ)dA (93)

= cos2θ

ˆx2dA + 2cosθsinθ

ˆxydA + sin2θ

ˆy2dA (94)

= Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (95)

Iuv =

ˆ(xcosθ + ysinθ)(ycosθ minus xsinθ)dA (96)

=

ˆ(xycos2θ + y2cosθsinθ minus x2cosθsinθ minus xysin2θ)dA (97)

= (cos2 minus sin2θ)

ˆxydA + cosθsinθ

ˆy2dAminus cosθsinθ

ˆx2dA

= Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 36 54

傾斜軸面積之慣性矩 III

再由

sin2θ + cos2θ = 1 (98)

sin(αplusmn β) = sinαcosβ plusmn cosαsinβ (99)

sin2θ = sin(θ + θ) = 2sinθcosθ (100)

cos(αplusmn β) = cosαcosβ ∓ sinαsinβ (101)

cos2θ = cos(θ + θ) = cos2θ minus sin2θ (102)

= 2cos2θ minus 1 = 1minus 2sin2θ (103)

所以

Iu = Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (104)

= Ix(cos2θ + 1

2) + Iy (

1minus cos2θ

2)minus Ixy sin2θ (105)

=Ix + Iy

2+

Ix minus Iy2

cos2θ minus Ixy sin2θ

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傾斜軸面積之慣性矩 IV

Iv = Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (106)

= Iy (cos2θ + 1

2) + Ix(

1minus cos2θ

2) + Ixy sin2θ (107)

=Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (108)

Iuv = Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ (109)

= Ixycos2θ +1

2Ixsin2θ minus 1

2Iy sin2θ (110)

=Ix minus Iy

2sin2θ + Ixycos2θ (111)

所以

JO = Iu + Iv = Ix + Iy (112)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 38 54

主主主軸軸軸慣慣慣性性性矩矩矩Principal Moment of Inertia

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主軸慣性矩 I

所謂主軸慣性矩(Principal Moment ofInertia)乃定義發生在慣性矩的極值(極大或極小值)上也就是慣性矩之一次微分式為零或斜率為零

dIu

dθ=

d(Ix+Iy

2+

IxminusIy2

cos2θ minus Ixy sin2θ)

dθ(113)

= minus2Ix minus Iy

2sin2θ minus 2Ixy cos2θ (114)

= 0 (115)

其中dsin2θ = cos2θ dcos2θ = minussin2θ當慣性矩之一次微分式為零時傾斜軸與主軸之夾角θ = θp

minus2Ix minus Iy

2sin2θp minus 2Ixycos2θp = 0 (116)

tan2θ =sin2θ

cos2θ=

minusIxy(Ix minus Iy )2

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主軸慣性矩 II

cos2θp =IxminusIy2radic

(IxminusIy2 )2 + I 2xy

sin2θp =minusIxyradic

(IxminusIy2 )2 + I 2xy

(117)

(Iu)max =Ix + Iy

2+

Ix minus Iy2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

minus IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2+

radic(Ix minus Iy

2)2 + I 2xy (118)

(Iv )min =Ix + Iy

2minus Ix minus Iy

2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

+ IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2minusradic

(Ix minus Iy

2)2 + I 2xy (119)

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主軸慣性矩 III

由tan2θp = tan(π + 2θp)可得到θp1與θp2二個解即2θp2 = π + 2θp1也就是θp2 minus θp1 = π

2

tan2θp = tan(π + 2θp) (120)

2θp2 = π + 2θp1 (121)

θp2 minus θp1 =π

2(122)

所以

Imaxmin

=Ix + Iy

2plusmnradic

(Ix minus Iy

2)2 + I 2xy (123)

Iuv =Ix minus Iy

2

minusIxyradic(IxminusIy2 )2 + I 2xy

+ Ixy

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

(124)

= 0

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 42 54

課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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慣慣慣性性性矩矩矩之之之莫莫莫爾爾爾圓圓圓Mohrrsquos Circle for Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 44 54

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慣性矩之莫爾圓

由於Iu =

Ix + Iy2

+Ix minus Iy

2cos2θ minus Ixy sin2θ (125)

Iv =Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (126)

Iuv =Ix minus Iy

2sin2θ + Ixycos2θ (127)

將式(125)與式(127)各自分別平方並相加可得

(Iu minusIx + Iy

2)2 + I 2uv = (

Ix minus Iy2

)2 + I 2xy (128)

其中Ix Iy Ixy均為常數式(128)可進一步寫成

(Iu minus a)2 + I 2uv = R2 (129)

其中a =Ix+Iy2 R =

radic(IxminusIy2 )2 + I 2xycos2θp1 =

aRsin2θp1 =

minusIxyR 2θp1為OA轉向I軸方向即2θp1為

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 45 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 46 54

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解題技巧

首先確定轉動系統的Ix Iy Ixy接著算出莫爾圓之圓心座標a及半徑R並畫出莫爾圓再由莫爾圓上點出A(Ix Ixy )並由OA轉向I軸找出兩倍夾角2θp1及2θp2其中θp1及θp2為主軸(Principle Axis)與Ix軸的夾角由於主軸(Principle Axis)為極值之軸Imax = Iu及Imin = Iv決定θp1及θp2何者為Imax = Iu與Ix軸的夾角之方法乃比較Ix及Iy之大小Imax = Iu會靠近Ix及Iy之較大者也就是Imax = Iu與Ix及Iy之較大者的夾角比較小

a =Ix + Iy

2(130)

R =

radic(Ix minus Iy

2)2 + I 2xy (131)

2θp2 = π + 2θp1 θp2 minus θp1 =π

2(132)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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質質質量量量慣慣慣性性性矩矩矩Mass Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 48 54

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質量慣性矩

τ = Iα (133)

其中τ為旋轉扭矩I為質量慣性矩α為旋轉角速度

I =

ˆmr2dm (134)

=

ˆVr2ρdV (135)

= ρ

ˆVr2dV (136)

其中r為旋轉半徑或轉軸到旋轉體的距離

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 49 54

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 50 54

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平行軸定理

當通過物體質心(G)並在一轉軸上的質量慣性矩(IG )為已知時可透過座標軸轉換計算物體之質量慣性矩(I)

I =

ˆmr2dm (137)

=

ˆm

[(d + x prime)2 + y prime2]dm (138)

=

ˆm

(x prime2 + y prime2)dm + 2d

ˆmx primedm + d2

ˆmdm (139)

I = IG + md2 (140)

其中IG為在zrsquo軸上通過質心的質量慣性矩m為質量d為通過質心之軸zrsquo並平行原軸z之距離

acutem x primedm = x

acutem dm = 0 since x = 0

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (141)

k =

radicI

m(142)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (143)

= I1 + I2 + + In (144)

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

兩圓盤複合體

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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結結結論論論

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 54 54

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結論

轉動慣量或慣性矩為物體對旋轉運動的慣性而所謂慣性乃物體持續維持不變的行為也就是定義慣性矩可表達物體對本身旋轉運動的維持或持續性好壞更簡單的說慣性矩越大之物體其維持旋轉運動趨勢的能力越大例如惰輪儲能裝置惰輪之慣性矩越大當其轉動以後其維持旋轉運動趨勢的能力越大

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 55 54

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  • 轉動慣量或慣性矩(Moment of Inertia)
  • 面積慣性矩(Moment of Inertia for Area)
    • 面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)
    • 面積慣性矩之迴轉半徑(Radius of Gyration for Area)
      • 複合面積之慣性矩(Moment of Inertia for Composite Area)
      • 面積之慣性積(Product of Inertia for Area)
        • 平行軸定理(Parallel-Axis Theorem)
          • 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)
            • 主軸慣性矩(Principal Moment of Inertia)
              • 慣性矩之莫爾圓(Mohrs Circle for Moment of Inertia)
              • 質量慣性矩(Mass Moment of Inertia)
                • 平行軸定理(Parallel-Axis Theorem)
Page 40: 轉動慣量 Moments of Inertia - cct.me.ntut.edu.twcct.me.ntut.edu.tw/ccteducation/chchting/aiahtm/statics/lecture... · 轉動慣量或慣性矩 I. 轉動慣量也稱慣性矩(Moment

平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 30 54

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平行軸定理

極點

新極點

如左圖面積慣性矩之平行軸定理乃以面心(C)為新極點(O)設定新座標軸(xrsquoyrsquo)其中xrsquo軸平行於原座標x軸yrsquo軸平行於原座標y軸取微小面積dA其在新座標之點座標為(xrsquoyrsquo)面積之慣性積(Ixy )為

dIxy = (x prime + dx)(y prime + dy )dA (78)

當以面心為轉軸點時即x prime = y prime = 0ˆAx primedA = x prime

ˆAdA = 0

ˆAy primedA = y prime

ˆAdA = 0 (79)

Ixy =

ˆA

(x prime + dx)(y prime + dy )dA (80)

=

ˆAx primey primedA + dx

ˆAy primedA + dy

ˆAx primedA + dxdy

ˆAdA (81)

= macrIx primey prime + Adxdy

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 31 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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傾傾傾斜斜斜軸軸軸面面面積積積之之之慣慣慣性性性矩矩矩

Moment of Inertia for Area about Inclined Axis

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 33 54

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傾斜軸面積之慣性矩

當轉動軸傾斜θ角時也就是座標軸旋轉θ角而Ix Iy Ixy為已知則新的座標軸(uv)可寫為

u = xcosθ + ysinθ (82)

v = ycosθ minus xsinθ (83)

由u軸上之面積慣性矩Iuv軸上之面積慣性矩Iv及慣性積Iuv得知

dIu = v2dA = (ycosθ minus xsinθ)2dA (84)

dIv = u2dA = (xcosθ + ysinθ)2dA (85)

dIuv = uvdA = (xcosθ + ysinθ)(ycosθ minus xsinθ)dA (86)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 34 54

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傾斜軸面積之慣性矩 I

由x軸上之面積慣性矩Ixy軸上之面積慣性矩Iy及慣性積Ixy得知

Ix =

ˆy2dA Iy =

ˆx2dA Ixy =

ˆxydA (87)

Iu =

ˆ(ycosθ minus xsinθ)2dA (88)

=

ˆ(y2cos2θ minus 2xycosθsinθ + x2sin2θ)dA (89)

= cos2θ

ˆy2dAminus 2cosθsinθ

ˆxydA + sin2θ

ˆx2dA (90)

= Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (91)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 35 54

傾斜軸面積之慣性矩 II

Iv =

ˆ(xcosθ + ysinθ)2dA (92)

=

ˆ(x2cos2θ + 2xycosθsinθ + y2sin2θ)dA (93)

= cos2θ

ˆx2dA + 2cosθsinθ

ˆxydA + sin2θ

ˆy2dA (94)

= Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (95)

Iuv =

ˆ(xcosθ + ysinθ)(ycosθ minus xsinθ)dA (96)

=

ˆ(xycos2θ + y2cosθsinθ minus x2cosθsinθ minus xysin2θ)dA (97)

= (cos2 minus sin2θ)

ˆxydA + cosθsinθ

ˆy2dAminus cosθsinθ

ˆx2dA

= Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 36 54

傾斜軸面積之慣性矩 III

再由

sin2θ + cos2θ = 1 (98)

sin(αplusmn β) = sinαcosβ plusmn cosαsinβ (99)

sin2θ = sin(θ + θ) = 2sinθcosθ (100)

cos(αplusmn β) = cosαcosβ ∓ sinαsinβ (101)

cos2θ = cos(θ + θ) = cos2θ minus sin2θ (102)

= 2cos2θ minus 1 = 1minus 2sin2θ (103)

所以

Iu = Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (104)

= Ix(cos2θ + 1

2) + Iy (

1minus cos2θ

2)minus Ixy sin2θ (105)

=Ix + Iy

2+

Ix minus Iy2

cos2θ minus Ixy sin2θ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 37 54

傾斜軸面積之慣性矩 IV

Iv = Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (106)

= Iy (cos2θ + 1

2) + Ix(

1minus cos2θ

2) + Ixy sin2θ (107)

=Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (108)

Iuv = Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ (109)

= Ixycos2θ +1

2Ixsin2θ minus 1

2Iy sin2θ (110)

=Ix minus Iy

2sin2θ + Ixycos2θ (111)

所以

JO = Iu + Iv = Ix + Iy (112)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 38 54

主主主軸軸軸慣慣慣性性性矩矩矩Principal Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 39 54

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主軸慣性矩 I

所謂主軸慣性矩(Principal Moment ofInertia)乃定義發生在慣性矩的極值(極大或極小值)上也就是慣性矩之一次微分式為零或斜率為零

dIu

dθ=

d(Ix+Iy

2+

IxminusIy2

cos2θ minus Ixy sin2θ)

dθ(113)

= minus2Ix minus Iy

2sin2θ minus 2Ixy cos2θ (114)

= 0 (115)

其中dsin2θ = cos2θ dcos2θ = minussin2θ當慣性矩之一次微分式為零時傾斜軸與主軸之夾角θ = θp

minus2Ix minus Iy

2sin2θp minus 2Ixycos2θp = 0 (116)

tan2θ =sin2θ

cos2θ=

minusIxy(Ix minus Iy )2

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 40 54

主軸慣性矩 II

cos2θp =IxminusIy2radic

(IxminusIy2 )2 + I 2xy

sin2θp =minusIxyradic

(IxminusIy2 )2 + I 2xy

(117)

(Iu)max =Ix + Iy

2+

Ix minus Iy2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

minus IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2+

radic(Ix minus Iy

2)2 + I 2xy (118)

(Iv )min =Ix + Iy

2minus Ix minus Iy

2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

+ IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2minusradic

(Ix minus Iy

2)2 + I 2xy (119)

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主軸慣性矩 III

由tan2θp = tan(π + 2θp)可得到θp1與θp2二個解即2θp2 = π + 2θp1也就是θp2 minus θp1 = π

2

tan2θp = tan(π + 2θp) (120)

2θp2 = π + 2θp1 (121)

θp2 minus θp1 =π

2(122)

所以

Imaxmin

=Ix + Iy

2plusmnradic

(Ix minus Iy

2)2 + I 2xy (123)

Iuv =Ix minus Iy

2

minusIxyradic(IxminusIy2 )2 + I 2xy

+ Ixy

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

(124)

= 0

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課課課堂堂堂練練練習習習

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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慣慣慣性性性矩矩矩之之之莫莫莫爾爾爾圓圓圓Mohrrsquos Circle for Moment of Inertia

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慣性矩之莫爾圓

由於Iu =

Ix + Iy2

+Ix minus Iy

2cos2θ minus Ixy sin2θ (125)

Iv =Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (126)

Iuv =Ix minus Iy

2sin2θ + Ixycos2θ (127)

將式(125)與式(127)各自分別平方並相加可得

(Iu minusIx + Iy

2)2 + I 2uv = (

Ix minus Iy2

)2 + I 2xy (128)

其中Ix Iy Ixy均為常數式(128)可進一步寫成

(Iu minus a)2 + I 2uv = R2 (129)

其中a =Ix+Iy2 R =

radic(IxminusIy2 )2 + I 2xycos2θp1 =

aRsin2θp1 =

minusIxyR 2θp1為OA轉向I軸方向即2θp1為

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課課課堂堂堂練練練習習習

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解題技巧

首先確定轉動系統的Ix Iy Ixy接著算出莫爾圓之圓心座標a及半徑R並畫出莫爾圓再由莫爾圓上點出A(Ix Ixy )並由OA轉向I軸找出兩倍夾角2θp1及2θp2其中θp1及θp2為主軸(Principle Axis)與Ix軸的夾角由於主軸(Principle Axis)為極值之軸Imax = Iu及Imin = Iv決定θp1及θp2何者為Imax = Iu與Ix軸的夾角之方法乃比較Ix及Iy之大小Imax = Iu會靠近Ix及Iy之較大者也就是Imax = Iu與Ix及Iy之較大者的夾角比較小

a =Ix + Iy

2(130)

R =

radic(Ix minus Iy

2)2 + I 2xy (131)

2θp2 = π + 2θp1 θp2 minus θp1 =π

2(132)

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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質質質量量量慣慣慣性性性矩矩矩Mass Moment of Inertia

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質量慣性矩

τ = Iα (133)

其中τ為旋轉扭矩I為質量慣性矩α為旋轉角速度

I =

ˆmr2dm (134)

=

ˆVr2ρdV (135)

= ρ

ˆVr2dV (136)

其中r為旋轉半徑或轉軸到旋轉體的距離

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

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平行軸定理

當通過物體質心(G)並在一轉軸上的質量慣性矩(IG )為已知時可透過座標軸轉換計算物體之質量慣性矩(I)

I =

ˆmr2dm (137)

=

ˆm

[(d + x prime)2 + y prime2]dm (138)

=

ˆm

(x prime2 + y prime2)dm + 2d

ˆmx primedm + d2

ˆmdm (139)

I = IG + md2 (140)

其中IG為在zrsquo軸上通過質心的質量慣性矩m為質量d為通過質心之軸zrsquo並平行原軸z之距離

acutem x primedm = x

acutem dm = 0 since x = 0

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (141)

k =

radicI

m(142)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (143)

= I1 + I2 + + In (144)

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課課課堂堂堂練練練習習習

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

兩圓盤複合體

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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結結結論論論

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結論

轉動慣量或慣性矩為物體對旋轉運動的慣性而所謂慣性乃物體持續維持不變的行為也就是定義慣性矩可表達物體對本身旋轉運動的維持或持續性好壞更簡單的說慣性矩越大之物體其維持旋轉運動趨勢的能力越大例如惰輪儲能裝置惰輪之慣性矩越大當其轉動以後其維持旋轉運動趨勢的能力越大

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 55 54

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  • 轉動慣量或慣性矩(Moment of Inertia)
  • 面積慣性矩(Moment of Inertia for Area)
    • 面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)
    • 面積慣性矩之迴轉半徑(Radius of Gyration for Area)
      • 複合面積之慣性矩(Moment of Inertia for Composite Area)
      • 面積之慣性積(Product of Inertia for Area)
        • 平行軸定理(Parallel-Axis Theorem)
          • 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)
            • 主軸慣性矩(Principal Moment of Inertia)
              • 慣性矩之莫爾圓(Mohrs Circle for Moment of Inertia)
              • 質量慣性矩(Mass Moment of Inertia)
                • 平行軸定理(Parallel-Axis Theorem)
Page 41: 轉動慣量 Moments of Inertia - cct.me.ntut.edu.twcct.me.ntut.edu.tw/ccteducation/chchting/aiahtm/statics/lecture... · 轉動慣量或慣性矩 I. 轉動慣量也稱慣性矩(Moment

平行軸定理

極點

新極點

如左圖面積慣性矩之平行軸定理乃以面心(C)為新極點(O)設定新座標軸(xrsquoyrsquo)其中xrsquo軸平行於原座標x軸yrsquo軸平行於原座標y軸取微小面積dA其在新座標之點座標為(xrsquoyrsquo)面積之慣性積(Ixy )為

dIxy = (x prime + dx)(y prime + dy )dA (78)

當以面心為轉軸點時即x prime = y prime = 0ˆAx primedA = x prime

ˆAdA = 0

ˆAy primedA = y prime

ˆAdA = 0 (79)

Ixy =

ˆA

(x prime + dx)(y prime + dy )dA (80)

=

ˆAx primey primedA + dx

ˆAy primedA + dy

ˆAx primedA + dxdy

ˆAdA (81)

= macrIx primey prime + Adxdy

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課課課堂堂堂練練練習習習

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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傾傾傾斜斜斜軸軸軸面面面積積積之之之慣慣慣性性性矩矩矩

Moment of Inertia for Area about Inclined Axis

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 33 54

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傾斜軸面積之慣性矩

當轉動軸傾斜θ角時也就是座標軸旋轉θ角而Ix Iy Ixy為已知則新的座標軸(uv)可寫為

u = xcosθ + ysinθ (82)

v = ycosθ minus xsinθ (83)

由u軸上之面積慣性矩Iuv軸上之面積慣性矩Iv及慣性積Iuv得知

dIu = v2dA = (ycosθ minus xsinθ)2dA (84)

dIv = u2dA = (xcosθ + ysinθ)2dA (85)

dIuv = uvdA = (xcosθ + ysinθ)(ycosθ minus xsinθ)dA (86)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 34 54

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傾斜軸面積之慣性矩 I

由x軸上之面積慣性矩Ixy軸上之面積慣性矩Iy及慣性積Ixy得知

Ix =

ˆy2dA Iy =

ˆx2dA Ixy =

ˆxydA (87)

Iu =

ˆ(ycosθ minus xsinθ)2dA (88)

=

ˆ(y2cos2θ minus 2xycosθsinθ + x2sin2θ)dA (89)

= cos2θ

ˆy2dAminus 2cosθsinθ

ˆxydA + sin2θ

ˆx2dA (90)

= Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (91)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 35 54

傾斜軸面積之慣性矩 II

Iv =

ˆ(xcosθ + ysinθ)2dA (92)

=

ˆ(x2cos2θ + 2xycosθsinθ + y2sin2θ)dA (93)

= cos2θ

ˆx2dA + 2cosθsinθ

ˆxydA + sin2θ

ˆy2dA (94)

= Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (95)

Iuv =

ˆ(xcosθ + ysinθ)(ycosθ minus xsinθ)dA (96)

=

ˆ(xycos2θ + y2cosθsinθ minus x2cosθsinθ minus xysin2θ)dA (97)

= (cos2 minus sin2θ)

ˆxydA + cosθsinθ

ˆy2dAminus cosθsinθ

ˆx2dA

= Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 36 54

傾斜軸面積之慣性矩 III

再由

sin2θ + cos2θ = 1 (98)

sin(αplusmn β) = sinαcosβ plusmn cosαsinβ (99)

sin2θ = sin(θ + θ) = 2sinθcosθ (100)

cos(αplusmn β) = cosαcosβ ∓ sinαsinβ (101)

cos2θ = cos(θ + θ) = cos2θ minus sin2θ (102)

= 2cos2θ minus 1 = 1minus 2sin2θ (103)

所以

Iu = Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (104)

= Ix(cos2θ + 1

2) + Iy (

1minus cos2θ

2)minus Ixy sin2θ (105)

=Ix + Iy

2+

Ix minus Iy2

cos2θ minus Ixy sin2θ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 37 54

傾斜軸面積之慣性矩 IV

Iv = Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (106)

= Iy (cos2θ + 1

2) + Ix(

1minus cos2θ

2) + Ixy sin2θ (107)

=Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (108)

Iuv = Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ (109)

= Ixycos2θ +1

2Ixsin2θ minus 1

2Iy sin2θ (110)

=Ix minus Iy

2sin2θ + Ixycos2θ (111)

所以

JO = Iu + Iv = Ix + Iy (112)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 38 54

主主主軸軸軸慣慣慣性性性矩矩矩Principal Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 39 54

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主軸慣性矩 I

所謂主軸慣性矩(Principal Moment ofInertia)乃定義發生在慣性矩的極值(極大或極小值)上也就是慣性矩之一次微分式為零或斜率為零

dIu

dθ=

d(Ix+Iy

2+

IxminusIy2

cos2θ minus Ixy sin2θ)

dθ(113)

= minus2Ix minus Iy

2sin2θ minus 2Ixy cos2θ (114)

= 0 (115)

其中dsin2θ = cos2θ dcos2θ = minussin2θ當慣性矩之一次微分式為零時傾斜軸與主軸之夾角θ = θp

minus2Ix minus Iy

2sin2θp minus 2Ixycos2θp = 0 (116)

tan2θ =sin2θ

cos2θ=

minusIxy(Ix minus Iy )2

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 40 54

主軸慣性矩 II

cos2θp =IxminusIy2radic

(IxminusIy2 )2 + I 2xy

sin2θp =minusIxyradic

(IxminusIy2 )2 + I 2xy

(117)

(Iu)max =Ix + Iy

2+

Ix minus Iy2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

minus IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2+

radic(Ix minus Iy

2)2 + I 2xy (118)

(Iv )min =Ix + Iy

2minus Ix minus Iy

2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

+ IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2minusradic

(Ix minus Iy

2)2 + I 2xy (119)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 41 54

主軸慣性矩 III

由tan2θp = tan(π + 2θp)可得到θp1與θp2二個解即2θp2 = π + 2θp1也就是θp2 minus θp1 = π

2

tan2θp = tan(π + 2θp) (120)

2θp2 = π + 2θp1 (121)

θp2 minus θp1 =π

2(122)

所以

Imaxmin

=Ix + Iy

2plusmnradic

(Ix minus Iy

2)2 + I 2xy (123)

Iuv =Ix minus Iy

2

minusIxyradic(IxminusIy2 )2 + I 2xy

+ Ixy

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

(124)

= 0

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 42 54

課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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慣慣慣性性性矩矩矩之之之莫莫莫爾爾爾圓圓圓Mohrrsquos Circle for Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 44 54

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慣性矩之莫爾圓

由於Iu =

Ix + Iy2

+Ix minus Iy

2cos2θ minus Ixy sin2θ (125)

Iv =Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (126)

Iuv =Ix minus Iy

2sin2θ + Ixycos2θ (127)

將式(125)與式(127)各自分別平方並相加可得

(Iu minusIx + Iy

2)2 + I 2uv = (

Ix minus Iy2

)2 + I 2xy (128)

其中Ix Iy Ixy均為常數式(128)可進一步寫成

(Iu minus a)2 + I 2uv = R2 (129)

其中a =Ix+Iy2 R =

radic(IxminusIy2 )2 + I 2xycos2θp1 =

aRsin2θp1 =

minusIxyR 2θp1為OA轉向I軸方向即2θp1為

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 45 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 46 54

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解題技巧

首先確定轉動系統的Ix Iy Ixy接著算出莫爾圓之圓心座標a及半徑R並畫出莫爾圓再由莫爾圓上點出A(Ix Ixy )並由OA轉向I軸找出兩倍夾角2θp1及2θp2其中θp1及θp2為主軸(Principle Axis)與Ix軸的夾角由於主軸(Principle Axis)為極值之軸Imax = Iu及Imin = Iv決定θp1及θp2何者為Imax = Iu與Ix軸的夾角之方法乃比較Ix及Iy之大小Imax = Iu會靠近Ix及Iy之較大者也就是Imax = Iu與Ix及Iy之較大者的夾角比較小

a =Ix + Iy

2(130)

R =

radic(Ix minus Iy

2)2 + I 2xy (131)

2θp2 = π + 2θp1 θp2 minus θp1 =π

2(132)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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質質質量量量慣慣慣性性性矩矩矩Mass Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 48 54

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質量慣性矩

τ = Iα (133)

其中τ為旋轉扭矩I為質量慣性矩α為旋轉角速度

I =

ˆmr2dm (134)

=

ˆVr2ρdV (135)

= ρ

ˆVr2dV (136)

其中r為旋轉半徑或轉軸到旋轉體的距離

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 49 54

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 50 54

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平行軸定理

當通過物體質心(G)並在一轉軸上的質量慣性矩(IG )為已知時可透過座標軸轉換計算物體之質量慣性矩(I)

I =

ˆmr2dm (137)

=

ˆm

[(d + x prime)2 + y prime2]dm (138)

=

ˆm

(x prime2 + y prime2)dm + 2d

ˆmx primedm + d2

ˆmdm (139)

I = IG + md2 (140)

其中IG為在zrsquo軸上通過質心的質量慣性矩m為質量d為通過質心之軸zrsquo並平行原軸z之距離

acutem x primedm = x

acutem dm = 0 since x = 0

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 51 54

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (141)

k =

radicI

m(142)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (143)

= I1 + I2 + + In (144)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 52 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

兩圓盤複合體

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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結結結論論論

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 54 54

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結論

轉動慣量或慣性矩為物體對旋轉運動的慣性而所謂慣性乃物體持續維持不變的行為也就是定義慣性矩可表達物體對本身旋轉運動的維持或持續性好壞更簡單的說慣性矩越大之物體其維持旋轉運動趨勢的能力越大例如惰輪儲能裝置惰輪之慣性矩越大當其轉動以後其維持旋轉運動趨勢的能力越大

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 55 54

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  • 轉動慣量或慣性矩(Moment of Inertia)
  • 面積慣性矩(Moment of Inertia for Area)
    • 面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)
    • 面積慣性矩之迴轉半徑(Radius of Gyration for Area)
      • 複合面積之慣性矩(Moment of Inertia for Composite Area)
      • 面積之慣性積(Product of Inertia for Area)
        • 平行軸定理(Parallel-Axis Theorem)
          • 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)
            • 主軸慣性矩(Principal Moment of Inertia)
              • 慣性矩之莫爾圓(Mohrs Circle for Moment of Inertia)
              • 質量慣性矩(Mass Moment of Inertia)
                • 平行軸定理(Parallel-Axis Theorem)
Page 42: 轉動慣量 Moments of Inertia - cct.me.ntut.edu.twcct.me.ntut.edu.tw/ccteducation/chchting/aiahtm/statics/lecture... · 轉動慣量或慣性矩 I. 轉動慣量也稱慣性矩(Moment

課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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傾傾傾斜斜斜軸軸軸面面面積積積之之之慣慣慣性性性矩矩矩

Moment of Inertia for Area about Inclined Axis

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 33 54

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傾斜軸面積之慣性矩

當轉動軸傾斜θ角時也就是座標軸旋轉θ角而Ix Iy Ixy為已知則新的座標軸(uv)可寫為

u = xcosθ + ysinθ (82)

v = ycosθ minus xsinθ (83)

由u軸上之面積慣性矩Iuv軸上之面積慣性矩Iv及慣性積Iuv得知

dIu = v2dA = (ycosθ minus xsinθ)2dA (84)

dIv = u2dA = (xcosθ + ysinθ)2dA (85)

dIuv = uvdA = (xcosθ + ysinθ)(ycosθ minus xsinθ)dA (86)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 34 54

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傾斜軸面積之慣性矩 I

由x軸上之面積慣性矩Ixy軸上之面積慣性矩Iy及慣性積Ixy得知

Ix =

ˆy2dA Iy =

ˆx2dA Ixy =

ˆxydA (87)

Iu =

ˆ(ycosθ minus xsinθ)2dA (88)

=

ˆ(y2cos2θ minus 2xycosθsinθ + x2sin2θ)dA (89)

= cos2θ

ˆy2dAminus 2cosθsinθ

ˆxydA + sin2θ

ˆx2dA (90)

= Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (91)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 35 54

傾斜軸面積之慣性矩 II

Iv =

ˆ(xcosθ + ysinθ)2dA (92)

=

ˆ(x2cos2θ + 2xycosθsinθ + y2sin2θ)dA (93)

= cos2θ

ˆx2dA + 2cosθsinθ

ˆxydA + sin2θ

ˆy2dA (94)

= Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (95)

Iuv =

ˆ(xcosθ + ysinθ)(ycosθ minus xsinθ)dA (96)

=

ˆ(xycos2θ + y2cosθsinθ minus x2cosθsinθ minus xysin2θ)dA (97)

= (cos2 minus sin2θ)

ˆxydA + cosθsinθ

ˆy2dAminus cosθsinθ

ˆx2dA

= Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 36 54

傾斜軸面積之慣性矩 III

再由

sin2θ + cos2θ = 1 (98)

sin(αplusmn β) = sinαcosβ plusmn cosαsinβ (99)

sin2θ = sin(θ + θ) = 2sinθcosθ (100)

cos(αplusmn β) = cosαcosβ ∓ sinαsinβ (101)

cos2θ = cos(θ + θ) = cos2θ minus sin2θ (102)

= 2cos2θ minus 1 = 1minus 2sin2θ (103)

所以

Iu = Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (104)

= Ix(cos2θ + 1

2) + Iy (

1minus cos2θ

2)minus Ixy sin2θ (105)

=Ix + Iy

2+

Ix minus Iy2

cos2θ minus Ixy sin2θ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 37 54

傾斜軸面積之慣性矩 IV

Iv = Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (106)

= Iy (cos2θ + 1

2) + Ix(

1minus cos2θ

2) + Ixy sin2θ (107)

=Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (108)

Iuv = Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ (109)

= Ixycos2θ +1

2Ixsin2θ minus 1

2Iy sin2θ (110)

=Ix minus Iy

2sin2θ + Ixycos2θ (111)

所以

JO = Iu + Iv = Ix + Iy (112)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 38 54

主主主軸軸軸慣慣慣性性性矩矩矩Principal Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 39 54

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主軸慣性矩 I

所謂主軸慣性矩(Principal Moment ofInertia)乃定義發生在慣性矩的極值(極大或極小值)上也就是慣性矩之一次微分式為零或斜率為零

dIu

dθ=

d(Ix+Iy

2+

IxminusIy2

cos2θ minus Ixy sin2θ)

dθ(113)

= minus2Ix minus Iy

2sin2θ minus 2Ixy cos2θ (114)

= 0 (115)

其中dsin2θ = cos2θ dcos2θ = minussin2θ當慣性矩之一次微分式為零時傾斜軸與主軸之夾角θ = θp

minus2Ix minus Iy

2sin2θp minus 2Ixycos2θp = 0 (116)

tan2θ =sin2θ

cos2θ=

minusIxy(Ix minus Iy )2

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 40 54

主軸慣性矩 II

cos2θp =IxminusIy2radic

(IxminusIy2 )2 + I 2xy

sin2θp =minusIxyradic

(IxminusIy2 )2 + I 2xy

(117)

(Iu)max =Ix + Iy

2+

Ix minus Iy2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

minus IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2+

radic(Ix minus Iy

2)2 + I 2xy (118)

(Iv )min =Ix + Iy

2minus Ix minus Iy

2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

+ IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2minusradic

(Ix minus Iy

2)2 + I 2xy (119)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 41 54

主軸慣性矩 III

由tan2θp = tan(π + 2θp)可得到θp1與θp2二個解即2θp2 = π + 2θp1也就是θp2 minus θp1 = π

2

tan2θp = tan(π + 2θp) (120)

2θp2 = π + 2θp1 (121)

θp2 minus θp1 =π

2(122)

所以

Imaxmin

=Ix + Iy

2plusmnradic

(Ix minus Iy

2)2 + I 2xy (123)

Iuv =Ix minus Iy

2

minusIxyradic(IxminusIy2 )2 + I 2xy

+ Ixy

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

(124)

= 0

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 42 54

課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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慣慣慣性性性矩矩矩之之之莫莫莫爾爾爾圓圓圓Mohrrsquos Circle for Moment of Inertia

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慣性矩之莫爾圓

由於Iu =

Ix + Iy2

+Ix minus Iy

2cos2θ minus Ixy sin2θ (125)

Iv =Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (126)

Iuv =Ix minus Iy

2sin2θ + Ixycos2θ (127)

將式(125)與式(127)各自分別平方並相加可得

(Iu minusIx + Iy

2)2 + I 2uv = (

Ix minus Iy2

)2 + I 2xy (128)

其中Ix Iy Ixy均為常數式(128)可進一步寫成

(Iu minus a)2 + I 2uv = R2 (129)

其中a =Ix+Iy2 R =

radic(IxminusIy2 )2 + I 2xycos2θp1 =

aRsin2θp1 =

minusIxyR 2θp1為OA轉向I軸方向即2θp1為

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 45 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 46 54

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解題技巧

首先確定轉動系統的Ix Iy Ixy接著算出莫爾圓之圓心座標a及半徑R並畫出莫爾圓再由莫爾圓上點出A(Ix Ixy )並由OA轉向I軸找出兩倍夾角2θp1及2θp2其中θp1及θp2為主軸(Principle Axis)與Ix軸的夾角由於主軸(Principle Axis)為極值之軸Imax = Iu及Imin = Iv決定θp1及θp2何者為Imax = Iu與Ix軸的夾角之方法乃比較Ix及Iy之大小Imax = Iu會靠近Ix及Iy之較大者也就是Imax = Iu與Ix及Iy之較大者的夾角比較小

a =Ix + Iy

2(130)

R =

radic(Ix minus Iy

2)2 + I 2xy (131)

2θp2 = π + 2θp1 θp2 minus θp1 =π

2(132)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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質質質量量量慣慣慣性性性矩矩矩Mass Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 48 54

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質量慣性矩

τ = Iα (133)

其中τ為旋轉扭矩I為質量慣性矩α為旋轉角速度

I =

ˆmr2dm (134)

=

ˆVr2ρdV (135)

= ρ

ˆVr2dV (136)

其中r為旋轉半徑或轉軸到旋轉體的距離

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 49 54

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 50 54

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平行軸定理

當通過物體質心(G)並在一轉軸上的質量慣性矩(IG )為已知時可透過座標軸轉換計算物體之質量慣性矩(I)

I =

ˆmr2dm (137)

=

ˆm

[(d + x prime)2 + y prime2]dm (138)

=

ˆm

(x prime2 + y prime2)dm + 2d

ˆmx primedm + d2

ˆmdm (139)

I = IG + md2 (140)

其中IG為在zrsquo軸上通過質心的質量慣性矩m為質量d為通過質心之軸zrsquo並平行原軸z之距離

acutem x primedm = x

acutem dm = 0 since x = 0

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (141)

k =

radicI

m(142)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (143)

= I1 + I2 + + In (144)

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課課課堂堂堂練練練習習習

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

兩圓盤複合體

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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結結結論論論

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結論

轉動慣量或慣性矩為物體對旋轉運動的慣性而所謂慣性乃物體持續維持不變的行為也就是定義慣性矩可表達物體對本身旋轉運動的維持或持續性好壞更簡單的說慣性矩越大之物體其維持旋轉運動趨勢的能力越大例如惰輪儲能裝置惰輪之慣性矩越大當其轉動以後其維持旋轉運動趨勢的能力越大

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 55 54

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  • 轉動慣量或慣性矩(Moment of Inertia)
  • 面積慣性矩(Moment of Inertia for Area)
    • 面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)
    • 面積慣性矩之迴轉半徑(Radius of Gyration for Area)
      • 複合面積之慣性矩(Moment of Inertia for Composite Area)
      • 面積之慣性積(Product of Inertia for Area)
        • 平行軸定理(Parallel-Axis Theorem)
          • 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)
            • 主軸慣性矩(Principal Moment of Inertia)
              • 慣性矩之莫爾圓(Mohrs Circle for Moment of Inertia)
              • 質量慣性矩(Mass Moment of Inertia)
                • 平行軸定理(Parallel-Axis Theorem)
Page 43: 轉動慣量 Moments of Inertia - cct.me.ntut.edu.twcct.me.ntut.edu.tw/ccteducation/chchting/aiahtm/statics/lecture... · 轉動慣量或慣性矩 I. 轉動慣量也稱慣性矩(Moment

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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傾傾傾斜斜斜軸軸軸面面面積積積之之之慣慣慣性性性矩矩矩

Moment of Inertia for Area about Inclined Axis

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 33 54

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傾斜軸面積之慣性矩

當轉動軸傾斜θ角時也就是座標軸旋轉θ角而Ix Iy Ixy為已知則新的座標軸(uv)可寫為

u = xcosθ + ysinθ (82)

v = ycosθ minus xsinθ (83)

由u軸上之面積慣性矩Iuv軸上之面積慣性矩Iv及慣性積Iuv得知

dIu = v2dA = (ycosθ minus xsinθ)2dA (84)

dIv = u2dA = (xcosθ + ysinθ)2dA (85)

dIuv = uvdA = (xcosθ + ysinθ)(ycosθ minus xsinθ)dA (86)

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傾斜軸面積之慣性矩 I

由x軸上之面積慣性矩Ixy軸上之面積慣性矩Iy及慣性積Ixy得知

Ix =

ˆy2dA Iy =

ˆx2dA Ixy =

ˆxydA (87)

Iu =

ˆ(ycosθ minus xsinθ)2dA (88)

=

ˆ(y2cos2θ minus 2xycosθsinθ + x2sin2θ)dA (89)

= cos2θ

ˆy2dAminus 2cosθsinθ

ˆxydA + sin2θ

ˆx2dA (90)

= Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (91)

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傾斜軸面積之慣性矩 II

Iv =

ˆ(xcosθ + ysinθ)2dA (92)

=

ˆ(x2cos2θ + 2xycosθsinθ + y2sin2θ)dA (93)

= cos2θ

ˆx2dA + 2cosθsinθ

ˆxydA + sin2θ

ˆy2dA (94)

= Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (95)

Iuv =

ˆ(xcosθ + ysinθ)(ycosθ minus xsinθ)dA (96)

=

ˆ(xycos2θ + y2cosθsinθ minus x2cosθsinθ minus xysin2θ)dA (97)

= (cos2 minus sin2θ)

ˆxydA + cosθsinθ

ˆy2dAminus cosθsinθ

ˆx2dA

= Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ

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傾斜軸面積之慣性矩 III

再由

sin2θ + cos2θ = 1 (98)

sin(αplusmn β) = sinαcosβ plusmn cosαsinβ (99)

sin2θ = sin(θ + θ) = 2sinθcosθ (100)

cos(αplusmn β) = cosαcosβ ∓ sinαsinβ (101)

cos2θ = cos(θ + θ) = cos2θ minus sin2θ (102)

= 2cos2θ minus 1 = 1minus 2sin2θ (103)

所以

Iu = Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (104)

= Ix(cos2θ + 1

2) + Iy (

1minus cos2θ

2)minus Ixy sin2θ (105)

=Ix + Iy

2+

Ix minus Iy2

cos2θ minus Ixy sin2θ

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傾斜軸面積之慣性矩 IV

Iv = Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (106)

= Iy (cos2θ + 1

2) + Ix(

1minus cos2θ

2) + Ixy sin2θ (107)

=Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (108)

Iuv = Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ (109)

= Ixycos2θ +1

2Ixsin2θ minus 1

2Iy sin2θ (110)

=Ix minus Iy

2sin2θ + Ixycos2θ (111)

所以

JO = Iu + Iv = Ix + Iy (112)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 38 54

主主主軸軸軸慣慣慣性性性矩矩矩Principal Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 39 54

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主軸慣性矩 I

所謂主軸慣性矩(Principal Moment ofInertia)乃定義發生在慣性矩的極值(極大或極小值)上也就是慣性矩之一次微分式為零或斜率為零

dIu

dθ=

d(Ix+Iy

2+

IxminusIy2

cos2θ minus Ixy sin2θ)

dθ(113)

= minus2Ix minus Iy

2sin2θ minus 2Ixy cos2θ (114)

= 0 (115)

其中dsin2θ = cos2θ dcos2θ = minussin2θ當慣性矩之一次微分式為零時傾斜軸與主軸之夾角θ = θp

minus2Ix minus Iy

2sin2θp minus 2Ixycos2θp = 0 (116)

tan2θ =sin2θ

cos2θ=

minusIxy(Ix minus Iy )2

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 40 54

主軸慣性矩 II

cos2θp =IxminusIy2radic

(IxminusIy2 )2 + I 2xy

sin2θp =minusIxyradic

(IxminusIy2 )2 + I 2xy

(117)

(Iu)max =Ix + Iy

2+

Ix minus Iy2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

minus IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2+

radic(Ix minus Iy

2)2 + I 2xy (118)

(Iv )min =Ix + Iy

2minus Ix minus Iy

2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

+ IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2minusradic

(Ix minus Iy

2)2 + I 2xy (119)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 41 54

主軸慣性矩 III

由tan2θp = tan(π + 2θp)可得到θp1與θp2二個解即2θp2 = π + 2θp1也就是θp2 minus θp1 = π

2

tan2θp = tan(π + 2θp) (120)

2θp2 = π + 2θp1 (121)

θp2 minus θp1 =π

2(122)

所以

Imaxmin

=Ix + Iy

2plusmnradic

(Ix minus Iy

2)2 + I 2xy (123)

Iuv =Ix minus Iy

2

minusIxyradic(IxminusIy2 )2 + I 2xy

+ Ixy

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

(124)

= 0

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 42 54

課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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慣慣慣性性性矩矩矩之之之莫莫莫爾爾爾圓圓圓Mohrrsquos Circle for Moment of Inertia

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慣性矩之莫爾圓

由於Iu =

Ix + Iy2

+Ix minus Iy

2cos2θ minus Ixy sin2θ (125)

Iv =Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (126)

Iuv =Ix minus Iy

2sin2θ + Ixycos2θ (127)

將式(125)與式(127)各自分別平方並相加可得

(Iu minusIx + Iy

2)2 + I 2uv = (

Ix minus Iy2

)2 + I 2xy (128)

其中Ix Iy Ixy均為常數式(128)可進一步寫成

(Iu minus a)2 + I 2uv = R2 (129)

其中a =Ix+Iy2 R =

radic(IxminusIy2 )2 + I 2xycos2θp1 =

aRsin2θp1 =

minusIxyR 2θp1為OA轉向I軸方向即2θp1為

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課課課堂堂堂練練練習習習

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解題技巧

首先確定轉動系統的Ix Iy Ixy接著算出莫爾圓之圓心座標a及半徑R並畫出莫爾圓再由莫爾圓上點出A(Ix Ixy )並由OA轉向I軸找出兩倍夾角2θp1及2θp2其中θp1及θp2為主軸(Principle Axis)與Ix軸的夾角由於主軸(Principle Axis)為極值之軸Imax = Iu及Imin = Iv決定θp1及θp2何者為Imax = Iu與Ix軸的夾角之方法乃比較Ix及Iy之大小Imax = Iu會靠近Ix及Iy之較大者也就是Imax = Iu與Ix及Iy之較大者的夾角比較小

a =Ix + Iy

2(130)

R =

radic(Ix minus Iy

2)2 + I 2xy (131)

2θp2 = π + 2θp1 θp2 minus θp1 =π

2(132)

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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質質質量量量慣慣慣性性性矩矩矩Mass Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 48 54

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質量慣性矩

τ = Iα (133)

其中τ為旋轉扭矩I為質量慣性矩α為旋轉角速度

I =

ˆmr2dm (134)

=

ˆVr2ρdV (135)

= ρ

ˆVr2dV (136)

其中r為旋轉半徑或轉軸到旋轉體的距離

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 49 54

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 50 54

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平行軸定理

當通過物體質心(G)並在一轉軸上的質量慣性矩(IG )為已知時可透過座標軸轉換計算物體之質量慣性矩(I)

I =

ˆmr2dm (137)

=

ˆm

[(d + x prime)2 + y prime2]dm (138)

=

ˆm

(x prime2 + y prime2)dm + 2d

ˆmx primedm + d2

ˆmdm (139)

I = IG + md2 (140)

其中IG為在zrsquo軸上通過質心的質量慣性矩m為質量d為通過質心之軸zrsquo並平行原軸z之距離

acutem x primedm = x

acutem dm = 0 since x = 0

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 51 54

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (141)

k =

radicI

m(142)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (143)

= I1 + I2 + + In (144)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 52 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

兩圓盤複合體

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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結結結論論論

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 54 54

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結論

轉動慣量或慣性矩為物體對旋轉運動的慣性而所謂慣性乃物體持續維持不變的行為也就是定義慣性矩可表達物體對本身旋轉運動的維持或持續性好壞更簡單的說慣性矩越大之物體其維持旋轉運動趨勢的能力越大例如惰輪儲能裝置惰輪之慣性矩越大當其轉動以後其維持旋轉運動趨勢的能力越大

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 55 54

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  • 轉動慣量或慣性矩(Moment of Inertia)
  • 面積慣性矩(Moment of Inertia for Area)
    • 面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)
    • 面積慣性矩之迴轉半徑(Radius of Gyration for Area)
      • 複合面積之慣性矩(Moment of Inertia for Composite Area)
      • 面積之慣性積(Product of Inertia for Area)
        • 平行軸定理(Parallel-Axis Theorem)
          • 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)
            • 主軸慣性矩(Principal Moment of Inertia)
              • 慣性矩之莫爾圓(Mohrs Circle for Moment of Inertia)
              • 質量慣性矩(Mass Moment of Inertia)
                • 平行軸定理(Parallel-Axis Theorem)
Page 44: 轉動慣量 Moments of Inertia - cct.me.ntut.edu.twcct.me.ntut.edu.tw/ccteducation/chchting/aiahtm/statics/lecture... · 轉動慣量或慣性矩 I. 轉動慣量也稱慣性矩(Moment

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 32 54

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傾傾傾斜斜斜軸軸軸面面面積積積之之之慣慣慣性性性矩矩矩

Moment of Inertia for Area about Inclined Axis

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 33 54

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傾斜軸面積之慣性矩

當轉動軸傾斜θ角時也就是座標軸旋轉θ角而Ix Iy Ixy為已知則新的座標軸(uv)可寫為

u = xcosθ + ysinθ (82)

v = ycosθ minus xsinθ (83)

由u軸上之面積慣性矩Iuv軸上之面積慣性矩Iv及慣性積Iuv得知

dIu = v2dA = (ycosθ minus xsinθ)2dA (84)

dIv = u2dA = (xcosθ + ysinθ)2dA (85)

dIuv = uvdA = (xcosθ + ysinθ)(ycosθ minus xsinθ)dA (86)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 34 54

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傾斜軸面積之慣性矩 I

由x軸上之面積慣性矩Ixy軸上之面積慣性矩Iy及慣性積Ixy得知

Ix =

ˆy2dA Iy =

ˆx2dA Ixy =

ˆxydA (87)

Iu =

ˆ(ycosθ minus xsinθ)2dA (88)

=

ˆ(y2cos2θ minus 2xycosθsinθ + x2sin2θ)dA (89)

= cos2θ

ˆy2dAminus 2cosθsinθ

ˆxydA + sin2θ

ˆx2dA (90)

= Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (91)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 35 54

傾斜軸面積之慣性矩 II

Iv =

ˆ(xcosθ + ysinθ)2dA (92)

=

ˆ(x2cos2θ + 2xycosθsinθ + y2sin2θ)dA (93)

= cos2θ

ˆx2dA + 2cosθsinθ

ˆxydA + sin2θ

ˆy2dA (94)

= Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (95)

Iuv =

ˆ(xcosθ + ysinθ)(ycosθ minus xsinθ)dA (96)

=

ˆ(xycos2θ + y2cosθsinθ minus x2cosθsinθ minus xysin2θ)dA (97)

= (cos2 minus sin2θ)

ˆxydA + cosθsinθ

ˆy2dAminus cosθsinθ

ˆx2dA

= Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 36 54

傾斜軸面積之慣性矩 III

再由

sin2θ + cos2θ = 1 (98)

sin(αplusmn β) = sinαcosβ plusmn cosαsinβ (99)

sin2θ = sin(θ + θ) = 2sinθcosθ (100)

cos(αplusmn β) = cosαcosβ ∓ sinαsinβ (101)

cos2θ = cos(θ + θ) = cos2θ minus sin2θ (102)

= 2cos2θ minus 1 = 1minus 2sin2θ (103)

所以

Iu = Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (104)

= Ix(cos2θ + 1

2) + Iy (

1minus cos2θ

2)minus Ixy sin2θ (105)

=Ix + Iy

2+

Ix minus Iy2

cos2θ minus Ixy sin2θ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 37 54

傾斜軸面積之慣性矩 IV

Iv = Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (106)

= Iy (cos2θ + 1

2) + Ix(

1minus cos2θ

2) + Ixy sin2θ (107)

=Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (108)

Iuv = Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ (109)

= Ixycos2θ +1

2Ixsin2θ minus 1

2Iy sin2θ (110)

=Ix minus Iy

2sin2θ + Ixycos2θ (111)

所以

JO = Iu + Iv = Ix + Iy (112)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 38 54

主主主軸軸軸慣慣慣性性性矩矩矩Principal Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 39 54

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主軸慣性矩 I

所謂主軸慣性矩(Principal Moment ofInertia)乃定義發生在慣性矩的極值(極大或極小值)上也就是慣性矩之一次微分式為零或斜率為零

dIu

dθ=

d(Ix+Iy

2+

IxminusIy2

cos2θ minus Ixy sin2θ)

dθ(113)

= minus2Ix minus Iy

2sin2θ minus 2Ixy cos2θ (114)

= 0 (115)

其中dsin2θ = cos2θ dcos2θ = minussin2θ當慣性矩之一次微分式為零時傾斜軸與主軸之夾角θ = θp

minus2Ix minus Iy

2sin2θp minus 2Ixycos2θp = 0 (116)

tan2θ =sin2θ

cos2θ=

minusIxy(Ix minus Iy )2

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 40 54

主軸慣性矩 II

cos2θp =IxminusIy2radic

(IxminusIy2 )2 + I 2xy

sin2θp =minusIxyradic

(IxminusIy2 )2 + I 2xy

(117)

(Iu)max =Ix + Iy

2+

Ix minus Iy2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

minus IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2+

radic(Ix minus Iy

2)2 + I 2xy (118)

(Iv )min =Ix + Iy

2minus Ix minus Iy

2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

+ IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2minusradic

(Ix minus Iy

2)2 + I 2xy (119)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 41 54

主軸慣性矩 III

由tan2θp = tan(π + 2θp)可得到θp1與θp2二個解即2θp2 = π + 2θp1也就是θp2 minus θp1 = π

2

tan2θp = tan(π + 2θp) (120)

2θp2 = π + 2θp1 (121)

θp2 minus θp1 =π

2(122)

所以

Imaxmin

=Ix + Iy

2plusmnradic

(Ix minus Iy

2)2 + I 2xy (123)

Iuv =Ix minus Iy

2

minusIxyradic(IxminusIy2 )2 + I 2xy

+ Ixy

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

(124)

= 0

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 42 54

課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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慣慣慣性性性矩矩矩之之之莫莫莫爾爾爾圓圓圓Mohrrsquos Circle for Moment of Inertia

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慣性矩之莫爾圓

由於Iu =

Ix + Iy2

+Ix minus Iy

2cos2θ minus Ixy sin2θ (125)

Iv =Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (126)

Iuv =Ix minus Iy

2sin2θ + Ixycos2θ (127)

將式(125)與式(127)各自分別平方並相加可得

(Iu minusIx + Iy

2)2 + I 2uv = (

Ix minus Iy2

)2 + I 2xy (128)

其中Ix Iy Ixy均為常數式(128)可進一步寫成

(Iu minus a)2 + I 2uv = R2 (129)

其中a =Ix+Iy2 R =

radic(IxminusIy2 )2 + I 2xycos2θp1 =

aRsin2θp1 =

minusIxyR 2θp1為OA轉向I軸方向即2θp1為

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課課課堂堂堂練練練習習習

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解題技巧

首先確定轉動系統的Ix Iy Ixy接著算出莫爾圓之圓心座標a及半徑R並畫出莫爾圓再由莫爾圓上點出A(Ix Ixy )並由OA轉向I軸找出兩倍夾角2θp1及2θp2其中θp1及θp2為主軸(Principle Axis)與Ix軸的夾角由於主軸(Principle Axis)為極值之軸Imax = Iu及Imin = Iv決定θp1及θp2何者為Imax = Iu與Ix軸的夾角之方法乃比較Ix及Iy之大小Imax = Iu會靠近Ix及Iy之較大者也就是Imax = Iu與Ix及Iy之較大者的夾角比較小

a =Ix + Iy

2(130)

R =

radic(Ix minus Iy

2)2 + I 2xy (131)

2θp2 = π + 2θp1 θp2 minus θp1 =π

2(132)

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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質質質量量量慣慣慣性性性矩矩矩Mass Moment of Inertia

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質量慣性矩

τ = Iα (133)

其中τ為旋轉扭矩I為質量慣性矩α為旋轉角速度

I =

ˆmr2dm (134)

=

ˆVr2ρdV (135)

= ρ

ˆVr2dV (136)

其中r為旋轉半徑或轉軸到旋轉體的距離

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

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平行軸定理

當通過物體質心(G)並在一轉軸上的質量慣性矩(IG )為已知時可透過座標軸轉換計算物體之質量慣性矩(I)

I =

ˆmr2dm (137)

=

ˆm

[(d + x prime)2 + y prime2]dm (138)

=

ˆm

(x prime2 + y prime2)dm + 2d

ˆmx primedm + d2

ˆmdm (139)

I = IG + md2 (140)

其中IG為在zrsquo軸上通過質心的質量慣性矩m為質量d為通過質心之軸zrsquo並平行原軸z之距離

acutem x primedm = x

acutem dm = 0 since x = 0

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (141)

k =

radicI

m(142)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (143)

= I1 + I2 + + In (144)

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課課課堂堂堂練練練習習習

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

兩圓盤複合體

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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結結結論論論

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結論

轉動慣量或慣性矩為物體對旋轉運動的慣性而所謂慣性乃物體持續維持不變的行為也就是定義慣性矩可表達物體對本身旋轉運動的維持或持續性好壞更簡單的說慣性矩越大之物體其維持旋轉運動趨勢的能力越大例如惰輪儲能裝置惰輪之慣性矩越大當其轉動以後其維持旋轉運動趨勢的能力越大

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  • 轉動慣量或慣性矩(Moment of Inertia)
  • 面積慣性矩(Moment of Inertia for Area)
    • 面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)
    • 面積慣性矩之迴轉半徑(Radius of Gyration for Area)
      • 複合面積之慣性矩(Moment of Inertia for Composite Area)
      • 面積之慣性積(Product of Inertia for Area)
        • 平行軸定理(Parallel-Axis Theorem)
          • 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)
            • 主軸慣性矩(Principal Moment of Inertia)
              • 慣性矩之莫爾圓(Mohrs Circle for Moment of Inertia)
              • 質量慣性矩(Mass Moment of Inertia)
                • 平行軸定理(Parallel-Axis Theorem)
Page 45: 轉動慣量 Moments of Inertia - cct.me.ntut.edu.twcct.me.ntut.edu.tw/ccteducation/chchting/aiahtm/statics/lecture... · 轉動慣量或慣性矩 I. 轉動慣量也稱慣性矩(Moment

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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傾傾傾斜斜斜軸軸軸面面面積積積之之之慣慣慣性性性矩矩矩

Moment of Inertia for Area about Inclined Axis

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 33 54

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傾斜軸面積之慣性矩

當轉動軸傾斜θ角時也就是座標軸旋轉θ角而Ix Iy Ixy為已知則新的座標軸(uv)可寫為

u = xcosθ + ysinθ (82)

v = ycosθ minus xsinθ (83)

由u軸上之面積慣性矩Iuv軸上之面積慣性矩Iv及慣性積Iuv得知

dIu = v2dA = (ycosθ minus xsinθ)2dA (84)

dIv = u2dA = (xcosθ + ysinθ)2dA (85)

dIuv = uvdA = (xcosθ + ysinθ)(ycosθ minus xsinθ)dA (86)

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傾斜軸面積之慣性矩 I

由x軸上之面積慣性矩Ixy軸上之面積慣性矩Iy及慣性積Ixy得知

Ix =

ˆy2dA Iy =

ˆx2dA Ixy =

ˆxydA (87)

Iu =

ˆ(ycosθ minus xsinθ)2dA (88)

=

ˆ(y2cos2θ minus 2xycosθsinθ + x2sin2θ)dA (89)

= cos2θ

ˆy2dAminus 2cosθsinθ

ˆxydA + sin2θ

ˆx2dA (90)

= Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (91)

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傾斜軸面積之慣性矩 II

Iv =

ˆ(xcosθ + ysinθ)2dA (92)

=

ˆ(x2cos2θ + 2xycosθsinθ + y2sin2θ)dA (93)

= cos2θ

ˆx2dA + 2cosθsinθ

ˆxydA + sin2θ

ˆy2dA (94)

= Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (95)

Iuv =

ˆ(xcosθ + ysinθ)(ycosθ minus xsinθ)dA (96)

=

ˆ(xycos2θ + y2cosθsinθ minus x2cosθsinθ minus xysin2θ)dA (97)

= (cos2 minus sin2θ)

ˆxydA + cosθsinθ

ˆy2dAminus cosθsinθ

ˆx2dA

= Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ

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傾斜軸面積之慣性矩 III

再由

sin2θ + cos2θ = 1 (98)

sin(αplusmn β) = sinαcosβ plusmn cosαsinβ (99)

sin2θ = sin(θ + θ) = 2sinθcosθ (100)

cos(αplusmn β) = cosαcosβ ∓ sinαsinβ (101)

cos2θ = cos(θ + θ) = cos2θ minus sin2θ (102)

= 2cos2θ minus 1 = 1minus 2sin2θ (103)

所以

Iu = Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (104)

= Ix(cos2θ + 1

2) + Iy (

1minus cos2θ

2)minus Ixy sin2θ (105)

=Ix + Iy

2+

Ix minus Iy2

cos2θ minus Ixy sin2θ

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傾斜軸面積之慣性矩 IV

Iv = Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (106)

= Iy (cos2θ + 1

2) + Ix(

1minus cos2θ

2) + Ixy sin2θ (107)

=Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (108)

Iuv = Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ (109)

= Ixycos2θ +1

2Ixsin2θ minus 1

2Iy sin2θ (110)

=Ix minus Iy

2sin2θ + Ixycos2θ (111)

所以

JO = Iu + Iv = Ix + Iy (112)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 38 54

主主主軸軸軸慣慣慣性性性矩矩矩Principal Moment of Inertia

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主軸慣性矩 I

所謂主軸慣性矩(Principal Moment ofInertia)乃定義發生在慣性矩的極值(極大或極小值)上也就是慣性矩之一次微分式為零或斜率為零

dIu

dθ=

d(Ix+Iy

2+

IxminusIy2

cos2θ minus Ixy sin2θ)

dθ(113)

= minus2Ix minus Iy

2sin2θ minus 2Ixy cos2θ (114)

= 0 (115)

其中dsin2θ = cos2θ dcos2θ = minussin2θ當慣性矩之一次微分式為零時傾斜軸與主軸之夾角θ = θp

minus2Ix minus Iy

2sin2θp minus 2Ixycos2θp = 0 (116)

tan2θ =sin2θ

cos2θ=

minusIxy(Ix minus Iy )2

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 40 54

主軸慣性矩 II

cos2θp =IxminusIy2radic

(IxminusIy2 )2 + I 2xy

sin2θp =minusIxyradic

(IxminusIy2 )2 + I 2xy

(117)

(Iu)max =Ix + Iy

2+

Ix minus Iy2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

minus IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2+

radic(Ix minus Iy

2)2 + I 2xy (118)

(Iv )min =Ix + Iy

2minus Ix minus Iy

2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

+ IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2minusradic

(Ix minus Iy

2)2 + I 2xy (119)

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主軸慣性矩 III

由tan2θp = tan(π + 2θp)可得到θp1與θp2二個解即2θp2 = π + 2θp1也就是θp2 minus θp1 = π

2

tan2θp = tan(π + 2θp) (120)

2θp2 = π + 2θp1 (121)

θp2 minus θp1 =π

2(122)

所以

Imaxmin

=Ix + Iy

2plusmnradic

(Ix minus Iy

2)2 + I 2xy (123)

Iuv =Ix minus Iy

2

minusIxyradic(IxminusIy2 )2 + I 2xy

+ Ixy

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

(124)

= 0

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 42 54

課課課堂堂堂練練練習習習

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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慣慣慣性性性矩矩矩之之之莫莫莫爾爾爾圓圓圓Mohrrsquos Circle for Moment of Inertia

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慣性矩之莫爾圓

由於Iu =

Ix + Iy2

+Ix minus Iy

2cos2θ minus Ixy sin2θ (125)

Iv =Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (126)

Iuv =Ix minus Iy

2sin2θ + Ixycos2θ (127)

將式(125)與式(127)各自分別平方並相加可得

(Iu minusIx + Iy

2)2 + I 2uv = (

Ix minus Iy2

)2 + I 2xy (128)

其中Ix Iy Ixy均為常數式(128)可進一步寫成

(Iu minus a)2 + I 2uv = R2 (129)

其中a =Ix+Iy2 R =

radic(IxminusIy2 )2 + I 2xycos2θp1 =

aRsin2θp1 =

minusIxyR 2θp1為OA轉向I軸方向即2θp1為

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課課課堂堂堂練練練習習習

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解題技巧

首先確定轉動系統的Ix Iy Ixy接著算出莫爾圓之圓心座標a及半徑R並畫出莫爾圓再由莫爾圓上點出A(Ix Ixy )並由OA轉向I軸找出兩倍夾角2θp1及2θp2其中θp1及θp2為主軸(Principle Axis)與Ix軸的夾角由於主軸(Principle Axis)為極值之軸Imax = Iu及Imin = Iv決定θp1及θp2何者為Imax = Iu與Ix軸的夾角之方法乃比較Ix及Iy之大小Imax = Iu會靠近Ix及Iy之較大者也就是Imax = Iu與Ix及Iy之較大者的夾角比較小

a =Ix + Iy

2(130)

R =

radic(Ix minus Iy

2)2 + I 2xy (131)

2θp2 = π + 2θp1 θp2 minus θp1 =π

2(132)

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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質質質量量量慣慣慣性性性矩矩矩Mass Moment of Inertia

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質量慣性矩

τ = Iα (133)

其中τ為旋轉扭矩I為質量慣性矩α為旋轉角速度

I =

ˆmr2dm (134)

=

ˆVr2ρdV (135)

= ρ

ˆVr2dV (136)

其中r為旋轉半徑或轉軸到旋轉體的距離

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

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平行軸定理

當通過物體質心(G)並在一轉軸上的質量慣性矩(IG )為已知時可透過座標軸轉換計算物體之質量慣性矩(I)

I =

ˆmr2dm (137)

=

ˆm

[(d + x prime)2 + y prime2]dm (138)

=

ˆm

(x prime2 + y prime2)dm + 2d

ˆmx primedm + d2

ˆmdm (139)

I = IG + md2 (140)

其中IG為在zrsquo軸上通過質心的質量慣性矩m為質量d為通過質心之軸zrsquo並平行原軸z之距離

acutem x primedm = x

acutem dm = 0 since x = 0

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (141)

k =

radicI

m(142)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (143)

= I1 + I2 + + In (144)

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課課課堂堂堂練練練習習習

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

兩圓盤複合體

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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結結結論論論

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結論

轉動慣量或慣性矩為物體對旋轉運動的慣性而所謂慣性乃物體持續維持不變的行為也就是定義慣性矩可表達物體對本身旋轉運動的維持或持續性好壞更簡單的說慣性矩越大之物體其維持旋轉運動趨勢的能力越大例如惰輪儲能裝置惰輪之慣性矩越大當其轉動以後其維持旋轉運動趨勢的能力越大

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 55 54

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  • 轉動慣量或慣性矩(Moment of Inertia)
  • 面積慣性矩(Moment of Inertia for Area)
    • 面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)
    • 面積慣性矩之迴轉半徑(Radius of Gyration for Area)
      • 複合面積之慣性矩(Moment of Inertia for Composite Area)
      • 面積之慣性積(Product of Inertia for Area)
        • 平行軸定理(Parallel-Axis Theorem)
          • 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)
            • 主軸慣性矩(Principal Moment of Inertia)
              • 慣性矩之莫爾圓(Mohrs Circle for Moment of Inertia)
              • 質量慣性矩(Mass Moment of Inertia)
                • 平行軸定理(Parallel-Axis Theorem)
Page 46: 轉動慣量 Moments of Inertia - cct.me.ntut.edu.twcct.me.ntut.edu.tw/ccteducation/chchting/aiahtm/statics/lecture... · 轉動慣量或慣性矩 I. 轉動慣量也稱慣性矩(Moment

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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傾傾傾斜斜斜軸軸軸面面面積積積之之之慣慣慣性性性矩矩矩

Moment of Inertia for Area about Inclined Axis

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 33 54

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傾斜軸面積之慣性矩

當轉動軸傾斜θ角時也就是座標軸旋轉θ角而Ix Iy Ixy為已知則新的座標軸(uv)可寫為

u = xcosθ + ysinθ (82)

v = ycosθ minus xsinθ (83)

由u軸上之面積慣性矩Iuv軸上之面積慣性矩Iv及慣性積Iuv得知

dIu = v2dA = (ycosθ minus xsinθ)2dA (84)

dIv = u2dA = (xcosθ + ysinθ)2dA (85)

dIuv = uvdA = (xcosθ + ysinθ)(ycosθ minus xsinθ)dA (86)

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傾斜軸面積之慣性矩 I

由x軸上之面積慣性矩Ixy軸上之面積慣性矩Iy及慣性積Ixy得知

Ix =

ˆy2dA Iy =

ˆx2dA Ixy =

ˆxydA (87)

Iu =

ˆ(ycosθ minus xsinθ)2dA (88)

=

ˆ(y2cos2θ minus 2xycosθsinθ + x2sin2θ)dA (89)

= cos2θ

ˆy2dAminus 2cosθsinθ

ˆxydA + sin2θ

ˆx2dA (90)

= Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (91)

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傾斜軸面積之慣性矩 II

Iv =

ˆ(xcosθ + ysinθ)2dA (92)

=

ˆ(x2cos2θ + 2xycosθsinθ + y2sin2θ)dA (93)

= cos2θ

ˆx2dA + 2cosθsinθ

ˆxydA + sin2θ

ˆy2dA (94)

= Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (95)

Iuv =

ˆ(xcosθ + ysinθ)(ycosθ minus xsinθ)dA (96)

=

ˆ(xycos2θ + y2cosθsinθ minus x2cosθsinθ minus xysin2θ)dA (97)

= (cos2 minus sin2θ)

ˆxydA + cosθsinθ

ˆy2dAminus cosθsinθ

ˆx2dA

= Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 36 54

傾斜軸面積之慣性矩 III

再由

sin2θ + cos2θ = 1 (98)

sin(αplusmn β) = sinαcosβ plusmn cosαsinβ (99)

sin2θ = sin(θ + θ) = 2sinθcosθ (100)

cos(αplusmn β) = cosαcosβ ∓ sinαsinβ (101)

cos2θ = cos(θ + θ) = cos2θ minus sin2θ (102)

= 2cos2θ minus 1 = 1minus 2sin2θ (103)

所以

Iu = Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (104)

= Ix(cos2θ + 1

2) + Iy (

1minus cos2θ

2)minus Ixy sin2θ (105)

=Ix + Iy

2+

Ix minus Iy2

cos2θ minus Ixy sin2θ

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傾斜軸面積之慣性矩 IV

Iv = Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (106)

= Iy (cos2θ + 1

2) + Ix(

1minus cos2θ

2) + Ixy sin2θ (107)

=Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (108)

Iuv = Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ (109)

= Ixycos2θ +1

2Ixsin2θ minus 1

2Iy sin2θ (110)

=Ix minus Iy

2sin2θ + Ixycos2θ (111)

所以

JO = Iu + Iv = Ix + Iy (112)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 38 54

主主主軸軸軸慣慣慣性性性矩矩矩Principal Moment of Inertia

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主軸慣性矩 I

所謂主軸慣性矩(Principal Moment ofInertia)乃定義發生在慣性矩的極值(極大或極小值)上也就是慣性矩之一次微分式為零或斜率為零

dIu

dθ=

d(Ix+Iy

2+

IxminusIy2

cos2θ minus Ixy sin2θ)

dθ(113)

= minus2Ix minus Iy

2sin2θ minus 2Ixy cos2θ (114)

= 0 (115)

其中dsin2θ = cos2θ dcos2θ = minussin2θ當慣性矩之一次微分式為零時傾斜軸與主軸之夾角θ = θp

minus2Ix minus Iy

2sin2θp minus 2Ixycos2θp = 0 (116)

tan2θ =sin2θ

cos2θ=

minusIxy(Ix minus Iy )2

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 40 54

主軸慣性矩 II

cos2θp =IxminusIy2radic

(IxminusIy2 )2 + I 2xy

sin2θp =minusIxyradic

(IxminusIy2 )2 + I 2xy

(117)

(Iu)max =Ix + Iy

2+

Ix minus Iy2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

minus IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2+

radic(Ix minus Iy

2)2 + I 2xy (118)

(Iv )min =Ix + Iy

2minus Ix minus Iy

2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

+ IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2minusradic

(Ix minus Iy

2)2 + I 2xy (119)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 41 54

主軸慣性矩 III

由tan2θp = tan(π + 2θp)可得到θp1與θp2二個解即2θp2 = π + 2θp1也就是θp2 minus θp1 = π

2

tan2θp = tan(π + 2θp) (120)

2θp2 = π + 2θp1 (121)

θp2 minus θp1 =π

2(122)

所以

Imaxmin

=Ix + Iy

2plusmnradic

(Ix minus Iy

2)2 + I 2xy (123)

Iuv =Ix minus Iy

2

minusIxyradic(IxminusIy2 )2 + I 2xy

+ Ixy

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

(124)

= 0

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 42 54

課課課堂堂堂練練練習習習

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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慣慣慣性性性矩矩矩之之之莫莫莫爾爾爾圓圓圓Mohrrsquos Circle for Moment of Inertia

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慣性矩之莫爾圓

由於Iu =

Ix + Iy2

+Ix minus Iy

2cos2θ minus Ixy sin2θ (125)

Iv =Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (126)

Iuv =Ix minus Iy

2sin2θ + Ixycos2θ (127)

將式(125)與式(127)各自分別平方並相加可得

(Iu minusIx + Iy

2)2 + I 2uv = (

Ix minus Iy2

)2 + I 2xy (128)

其中Ix Iy Ixy均為常數式(128)可進一步寫成

(Iu minus a)2 + I 2uv = R2 (129)

其中a =Ix+Iy2 R =

radic(IxminusIy2 )2 + I 2xycos2θp1 =

aRsin2θp1 =

minusIxyR 2θp1為OA轉向I軸方向即2θp1為

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課課課堂堂堂練練練習習習

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解題技巧

首先確定轉動系統的Ix Iy Ixy接著算出莫爾圓之圓心座標a及半徑R並畫出莫爾圓再由莫爾圓上點出A(Ix Ixy )並由OA轉向I軸找出兩倍夾角2θp1及2θp2其中θp1及θp2為主軸(Principle Axis)與Ix軸的夾角由於主軸(Principle Axis)為極值之軸Imax = Iu及Imin = Iv決定θp1及θp2何者為Imax = Iu與Ix軸的夾角之方法乃比較Ix及Iy之大小Imax = Iu會靠近Ix及Iy之較大者也就是Imax = Iu與Ix及Iy之較大者的夾角比較小

a =Ix + Iy

2(130)

R =

radic(Ix minus Iy

2)2 + I 2xy (131)

2θp2 = π + 2θp1 θp2 minus θp1 =π

2(132)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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質質質量量量慣慣慣性性性矩矩矩Mass Moment of Inertia

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質量慣性矩

τ = Iα (133)

其中τ為旋轉扭矩I為質量慣性矩α為旋轉角速度

I =

ˆmr2dm (134)

=

ˆVr2ρdV (135)

= ρ

ˆVr2dV (136)

其中r為旋轉半徑或轉軸到旋轉體的距離

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

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平行軸定理

當通過物體質心(G)並在一轉軸上的質量慣性矩(IG )為已知時可透過座標軸轉換計算物體之質量慣性矩(I)

I =

ˆmr2dm (137)

=

ˆm

[(d + x prime)2 + y prime2]dm (138)

=

ˆm

(x prime2 + y prime2)dm + 2d

ˆmx primedm + d2

ˆmdm (139)

I = IG + md2 (140)

其中IG為在zrsquo軸上通過質心的質量慣性矩m為質量d為通過質心之軸zrsquo並平行原軸z之距離

acutem x primedm = x

acutem dm = 0 since x = 0

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (141)

k =

radicI

m(142)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (143)

= I1 + I2 + + In (144)

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

兩圓盤複合體

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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結結結論論論

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結論

轉動慣量或慣性矩為物體對旋轉運動的慣性而所謂慣性乃物體持續維持不變的行為也就是定義慣性矩可表達物體對本身旋轉運動的維持或持續性好壞更簡單的說慣性矩越大之物體其維持旋轉運動趨勢的能力越大例如惰輪儲能裝置惰輪之慣性矩越大當其轉動以後其維持旋轉運動趨勢的能力越大

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 55 54

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  • 轉動慣量或慣性矩(Moment of Inertia)
  • 面積慣性矩(Moment of Inertia for Area)
    • 面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)
    • 面積慣性矩之迴轉半徑(Radius of Gyration for Area)
      • 複合面積之慣性矩(Moment of Inertia for Composite Area)
      • 面積之慣性積(Product of Inertia for Area)
        • 平行軸定理(Parallel-Axis Theorem)
          • 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)
            • 主軸慣性矩(Principal Moment of Inertia)
              • 慣性矩之莫爾圓(Mohrs Circle for Moment of Inertia)
              • 質量慣性矩(Mass Moment of Inertia)
                • 平行軸定理(Parallel-Axis Theorem)
Page 47: 轉動慣量 Moments of Inertia - cct.me.ntut.edu.twcct.me.ntut.edu.tw/ccteducation/chchting/aiahtm/statics/lecture... · 轉動慣量或慣性矩 I. 轉動慣量也稱慣性矩(Moment

傾傾傾斜斜斜軸軸軸面面面積積積之之之慣慣慣性性性矩矩矩

Moment of Inertia for Area about Inclined Axis

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 33 54

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傾斜軸面積之慣性矩

當轉動軸傾斜θ角時也就是座標軸旋轉θ角而Ix Iy Ixy為已知則新的座標軸(uv)可寫為

u = xcosθ + ysinθ (82)

v = ycosθ minus xsinθ (83)

由u軸上之面積慣性矩Iuv軸上之面積慣性矩Iv及慣性積Iuv得知

dIu = v2dA = (ycosθ minus xsinθ)2dA (84)

dIv = u2dA = (xcosθ + ysinθ)2dA (85)

dIuv = uvdA = (xcosθ + ysinθ)(ycosθ minus xsinθ)dA (86)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 34 54

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傾斜軸面積之慣性矩 I

由x軸上之面積慣性矩Ixy軸上之面積慣性矩Iy及慣性積Ixy得知

Ix =

ˆy2dA Iy =

ˆx2dA Ixy =

ˆxydA (87)

Iu =

ˆ(ycosθ minus xsinθ)2dA (88)

=

ˆ(y2cos2θ minus 2xycosθsinθ + x2sin2θ)dA (89)

= cos2θ

ˆy2dAminus 2cosθsinθ

ˆxydA + sin2θ

ˆx2dA (90)

= Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (91)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 35 54

傾斜軸面積之慣性矩 II

Iv =

ˆ(xcosθ + ysinθ)2dA (92)

=

ˆ(x2cos2θ + 2xycosθsinθ + y2sin2θ)dA (93)

= cos2θ

ˆx2dA + 2cosθsinθ

ˆxydA + sin2θ

ˆy2dA (94)

= Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (95)

Iuv =

ˆ(xcosθ + ysinθ)(ycosθ minus xsinθ)dA (96)

=

ˆ(xycos2θ + y2cosθsinθ minus x2cosθsinθ minus xysin2θ)dA (97)

= (cos2 minus sin2θ)

ˆxydA + cosθsinθ

ˆy2dAminus cosθsinθ

ˆx2dA

= Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ

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傾斜軸面積之慣性矩 III

再由

sin2θ + cos2θ = 1 (98)

sin(αplusmn β) = sinαcosβ plusmn cosαsinβ (99)

sin2θ = sin(θ + θ) = 2sinθcosθ (100)

cos(αplusmn β) = cosαcosβ ∓ sinαsinβ (101)

cos2θ = cos(θ + θ) = cos2θ minus sin2θ (102)

= 2cos2θ minus 1 = 1minus 2sin2θ (103)

所以

Iu = Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (104)

= Ix(cos2θ + 1

2) + Iy (

1minus cos2θ

2)minus Ixy sin2θ (105)

=Ix + Iy

2+

Ix minus Iy2

cos2θ minus Ixy sin2θ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 37 54

傾斜軸面積之慣性矩 IV

Iv = Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (106)

= Iy (cos2θ + 1

2) + Ix(

1minus cos2θ

2) + Ixy sin2θ (107)

=Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (108)

Iuv = Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ (109)

= Ixycos2θ +1

2Ixsin2θ minus 1

2Iy sin2θ (110)

=Ix minus Iy

2sin2θ + Ixycos2θ (111)

所以

JO = Iu + Iv = Ix + Iy (112)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 38 54

主主主軸軸軸慣慣慣性性性矩矩矩Principal Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 39 54

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主軸慣性矩 I

所謂主軸慣性矩(Principal Moment ofInertia)乃定義發生在慣性矩的極值(極大或極小值)上也就是慣性矩之一次微分式為零或斜率為零

dIu

dθ=

d(Ix+Iy

2+

IxminusIy2

cos2θ minus Ixy sin2θ)

dθ(113)

= minus2Ix minus Iy

2sin2θ minus 2Ixy cos2θ (114)

= 0 (115)

其中dsin2θ = cos2θ dcos2θ = minussin2θ當慣性矩之一次微分式為零時傾斜軸與主軸之夾角θ = θp

minus2Ix minus Iy

2sin2θp minus 2Ixycos2θp = 0 (116)

tan2θ =sin2θ

cos2θ=

minusIxy(Ix minus Iy )2

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 40 54

主軸慣性矩 II

cos2θp =IxminusIy2radic

(IxminusIy2 )2 + I 2xy

sin2θp =minusIxyradic

(IxminusIy2 )2 + I 2xy

(117)

(Iu)max =Ix + Iy

2+

Ix minus Iy2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

minus IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2+

radic(Ix minus Iy

2)2 + I 2xy (118)

(Iv )min =Ix + Iy

2minus Ix minus Iy

2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

+ IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2minusradic

(Ix minus Iy

2)2 + I 2xy (119)

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主軸慣性矩 III

由tan2θp = tan(π + 2θp)可得到θp1與θp2二個解即2θp2 = π + 2θp1也就是θp2 minus θp1 = π

2

tan2θp = tan(π + 2θp) (120)

2θp2 = π + 2θp1 (121)

θp2 minus θp1 =π

2(122)

所以

Imaxmin

=Ix + Iy

2plusmnradic

(Ix minus Iy

2)2 + I 2xy (123)

Iuv =Ix minus Iy

2

minusIxyradic(IxminusIy2 )2 + I 2xy

+ Ixy

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

(124)

= 0

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課課課堂堂堂練練練習習習

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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慣慣慣性性性矩矩矩之之之莫莫莫爾爾爾圓圓圓Mohrrsquos Circle for Moment of Inertia

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慣性矩之莫爾圓

由於Iu =

Ix + Iy2

+Ix minus Iy

2cos2θ minus Ixy sin2θ (125)

Iv =Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (126)

Iuv =Ix minus Iy

2sin2θ + Ixycos2θ (127)

將式(125)與式(127)各自分別平方並相加可得

(Iu minusIx + Iy

2)2 + I 2uv = (

Ix minus Iy2

)2 + I 2xy (128)

其中Ix Iy Ixy均為常數式(128)可進一步寫成

(Iu minus a)2 + I 2uv = R2 (129)

其中a =Ix+Iy2 R =

radic(IxminusIy2 )2 + I 2xycos2θp1 =

aRsin2θp1 =

minusIxyR 2θp1為OA轉向I軸方向即2θp1為

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課課課堂堂堂練練練習習習

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解題技巧

首先確定轉動系統的Ix Iy Ixy接著算出莫爾圓之圓心座標a及半徑R並畫出莫爾圓再由莫爾圓上點出A(Ix Ixy )並由OA轉向I軸找出兩倍夾角2θp1及2θp2其中θp1及θp2為主軸(Principle Axis)與Ix軸的夾角由於主軸(Principle Axis)為極值之軸Imax = Iu及Imin = Iv決定θp1及θp2何者為Imax = Iu與Ix軸的夾角之方法乃比較Ix及Iy之大小Imax = Iu會靠近Ix及Iy之較大者也就是Imax = Iu與Ix及Iy之較大者的夾角比較小

a =Ix + Iy

2(130)

R =

radic(Ix minus Iy

2)2 + I 2xy (131)

2θp2 = π + 2θp1 θp2 minus θp1 =π

2(132)

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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質質質量量量慣慣慣性性性矩矩矩Mass Moment of Inertia

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質量慣性矩

τ = Iα (133)

其中τ為旋轉扭矩I為質量慣性矩α為旋轉角速度

I =

ˆmr2dm (134)

=

ˆVr2ρdV (135)

= ρ

ˆVr2dV (136)

其中r為旋轉半徑或轉軸到旋轉體的距離

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

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平行軸定理

當通過物體質心(G)並在一轉軸上的質量慣性矩(IG )為已知時可透過座標軸轉換計算物體之質量慣性矩(I)

I =

ˆmr2dm (137)

=

ˆm

[(d + x prime)2 + y prime2]dm (138)

=

ˆm

(x prime2 + y prime2)dm + 2d

ˆmx primedm + d2

ˆmdm (139)

I = IG + md2 (140)

其中IG為在zrsquo軸上通過質心的質量慣性矩m為質量d為通過質心之軸zrsquo並平行原軸z之距離

acutem x primedm = x

acutem dm = 0 since x = 0

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (141)

k =

radicI

m(142)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (143)

= I1 + I2 + + In (144)

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課課課堂堂堂練練練習習習

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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兩圓盤複合體

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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結結結論論論

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結論

轉動慣量或慣性矩為物體對旋轉運動的慣性而所謂慣性乃物體持續維持不變的行為也就是定義慣性矩可表達物體對本身旋轉運動的維持或持續性好壞更簡單的說慣性矩越大之物體其維持旋轉運動趨勢的能力越大例如惰輪儲能裝置惰輪之慣性矩越大當其轉動以後其維持旋轉運動趨勢的能力越大

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 55 54

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  • 轉動慣量或慣性矩(Moment of Inertia)
  • 面積慣性矩(Moment of Inertia for Area)
    • 面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)
    • 面積慣性矩之迴轉半徑(Radius of Gyration for Area)
      • 複合面積之慣性矩(Moment of Inertia for Composite Area)
      • 面積之慣性積(Product of Inertia for Area)
        • 平行軸定理(Parallel-Axis Theorem)
          • 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)
            • 主軸慣性矩(Principal Moment of Inertia)
              • 慣性矩之莫爾圓(Mohrs Circle for Moment of Inertia)
              • 質量慣性矩(Mass Moment of Inertia)
                • 平行軸定理(Parallel-Axis Theorem)
Page 48: 轉動慣量 Moments of Inertia - cct.me.ntut.edu.twcct.me.ntut.edu.tw/ccteducation/chchting/aiahtm/statics/lecture... · 轉動慣量或慣性矩 I. 轉動慣量也稱慣性矩(Moment

傾斜軸面積之慣性矩

當轉動軸傾斜θ角時也就是座標軸旋轉θ角而Ix Iy Ixy為已知則新的座標軸(uv)可寫為

u = xcosθ + ysinθ (82)

v = ycosθ minus xsinθ (83)

由u軸上之面積慣性矩Iuv軸上之面積慣性矩Iv及慣性積Iuv得知

dIu = v2dA = (ycosθ minus xsinθ)2dA (84)

dIv = u2dA = (xcosθ + ysinθ)2dA (85)

dIuv = uvdA = (xcosθ + ysinθ)(ycosθ minus xsinθ)dA (86)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 34 54

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傾斜軸面積之慣性矩 I

由x軸上之面積慣性矩Ixy軸上之面積慣性矩Iy及慣性積Ixy得知

Ix =

ˆy2dA Iy =

ˆx2dA Ixy =

ˆxydA (87)

Iu =

ˆ(ycosθ minus xsinθ)2dA (88)

=

ˆ(y2cos2θ minus 2xycosθsinθ + x2sin2θ)dA (89)

= cos2θ

ˆy2dAminus 2cosθsinθ

ˆxydA + sin2θ

ˆx2dA (90)

= Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (91)

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傾斜軸面積之慣性矩 II

Iv =

ˆ(xcosθ + ysinθ)2dA (92)

=

ˆ(x2cos2θ + 2xycosθsinθ + y2sin2θ)dA (93)

= cos2θ

ˆx2dA + 2cosθsinθ

ˆxydA + sin2θ

ˆy2dA (94)

= Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (95)

Iuv =

ˆ(xcosθ + ysinθ)(ycosθ minus xsinθ)dA (96)

=

ˆ(xycos2θ + y2cosθsinθ minus x2cosθsinθ minus xysin2θ)dA (97)

= (cos2 minus sin2θ)

ˆxydA + cosθsinθ

ˆy2dAminus cosθsinθ

ˆx2dA

= Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 36 54

傾斜軸面積之慣性矩 III

再由

sin2θ + cos2θ = 1 (98)

sin(αplusmn β) = sinαcosβ plusmn cosαsinβ (99)

sin2θ = sin(θ + θ) = 2sinθcosθ (100)

cos(αplusmn β) = cosαcosβ ∓ sinαsinβ (101)

cos2θ = cos(θ + θ) = cos2θ minus sin2θ (102)

= 2cos2θ minus 1 = 1minus 2sin2θ (103)

所以

Iu = Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (104)

= Ix(cos2θ + 1

2) + Iy (

1minus cos2θ

2)minus Ixy sin2θ (105)

=Ix + Iy

2+

Ix minus Iy2

cos2θ minus Ixy sin2θ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 37 54

傾斜軸面積之慣性矩 IV

Iv = Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (106)

= Iy (cos2θ + 1

2) + Ix(

1minus cos2θ

2) + Ixy sin2θ (107)

=Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (108)

Iuv = Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ (109)

= Ixycos2θ +1

2Ixsin2θ minus 1

2Iy sin2θ (110)

=Ix minus Iy

2sin2θ + Ixycos2θ (111)

所以

JO = Iu + Iv = Ix + Iy (112)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 38 54

主主主軸軸軸慣慣慣性性性矩矩矩Principal Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 39 54

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主軸慣性矩 I

所謂主軸慣性矩(Principal Moment ofInertia)乃定義發生在慣性矩的極值(極大或極小值)上也就是慣性矩之一次微分式為零或斜率為零

dIu

dθ=

d(Ix+Iy

2+

IxminusIy2

cos2θ minus Ixy sin2θ)

dθ(113)

= minus2Ix minus Iy

2sin2θ minus 2Ixy cos2θ (114)

= 0 (115)

其中dsin2θ = cos2θ dcos2θ = minussin2θ當慣性矩之一次微分式為零時傾斜軸與主軸之夾角θ = θp

minus2Ix minus Iy

2sin2θp minus 2Ixycos2θp = 0 (116)

tan2θ =sin2θ

cos2θ=

minusIxy(Ix minus Iy )2

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 40 54

主軸慣性矩 II

cos2θp =IxminusIy2radic

(IxminusIy2 )2 + I 2xy

sin2θp =minusIxyradic

(IxminusIy2 )2 + I 2xy

(117)

(Iu)max =Ix + Iy

2+

Ix minus Iy2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

minus IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2+

radic(Ix minus Iy

2)2 + I 2xy (118)

(Iv )min =Ix + Iy

2minus Ix minus Iy

2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

+ IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2minusradic

(Ix minus Iy

2)2 + I 2xy (119)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 41 54

主軸慣性矩 III

由tan2θp = tan(π + 2θp)可得到θp1與θp2二個解即2θp2 = π + 2θp1也就是θp2 minus θp1 = π

2

tan2θp = tan(π + 2θp) (120)

2θp2 = π + 2θp1 (121)

θp2 minus θp1 =π

2(122)

所以

Imaxmin

=Ix + Iy

2plusmnradic

(Ix minus Iy

2)2 + I 2xy (123)

Iuv =Ix minus Iy

2

minusIxyradic(IxminusIy2 )2 + I 2xy

+ Ixy

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

(124)

= 0

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課課課堂堂堂練練練習習習

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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慣慣慣性性性矩矩矩之之之莫莫莫爾爾爾圓圓圓Mohrrsquos Circle for Moment of Inertia

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慣性矩之莫爾圓

由於Iu =

Ix + Iy2

+Ix minus Iy

2cos2θ minus Ixy sin2θ (125)

Iv =Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (126)

Iuv =Ix minus Iy

2sin2θ + Ixycos2θ (127)

將式(125)與式(127)各自分別平方並相加可得

(Iu minusIx + Iy

2)2 + I 2uv = (

Ix minus Iy2

)2 + I 2xy (128)

其中Ix Iy Ixy均為常數式(128)可進一步寫成

(Iu minus a)2 + I 2uv = R2 (129)

其中a =Ix+Iy2 R =

radic(IxminusIy2 )2 + I 2xycos2θp1 =

aRsin2θp1 =

minusIxyR 2θp1為OA轉向I軸方向即2θp1為

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 46 54

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解題技巧

首先確定轉動系統的Ix Iy Ixy接著算出莫爾圓之圓心座標a及半徑R並畫出莫爾圓再由莫爾圓上點出A(Ix Ixy )並由OA轉向I軸找出兩倍夾角2θp1及2θp2其中θp1及θp2為主軸(Principle Axis)與Ix軸的夾角由於主軸(Principle Axis)為極值之軸Imax = Iu及Imin = Iv決定θp1及θp2何者為Imax = Iu與Ix軸的夾角之方法乃比較Ix及Iy之大小Imax = Iu會靠近Ix及Iy之較大者也就是Imax = Iu與Ix及Iy之較大者的夾角比較小

a =Ix + Iy

2(130)

R =

radic(Ix minus Iy

2)2 + I 2xy (131)

2θp2 = π + 2θp1 θp2 minus θp1 =π

2(132)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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質質質量量量慣慣慣性性性矩矩矩Mass Moment of Inertia

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質量慣性矩

τ = Iα (133)

其中τ為旋轉扭矩I為質量慣性矩α為旋轉角速度

I =

ˆmr2dm (134)

=

ˆVr2ρdV (135)

= ρ

ˆVr2dV (136)

其中r為旋轉半徑或轉軸到旋轉體的距離

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

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平行軸定理

當通過物體質心(G)並在一轉軸上的質量慣性矩(IG )為已知時可透過座標軸轉換計算物體之質量慣性矩(I)

I =

ˆmr2dm (137)

=

ˆm

[(d + x prime)2 + y prime2]dm (138)

=

ˆm

(x prime2 + y prime2)dm + 2d

ˆmx primedm + d2

ˆmdm (139)

I = IG + md2 (140)

其中IG為在zrsquo軸上通過質心的質量慣性矩m為質量d為通過質心之軸zrsquo並平行原軸z之距離

acutem x primedm = x

acutem dm = 0 since x = 0

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (141)

k =

radicI

m(142)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (143)

= I1 + I2 + + In (144)

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課課課堂堂堂練練練習習習

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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兩圓盤複合體

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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結結結論論論

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結論

轉動慣量或慣性矩為物體對旋轉運動的慣性而所謂慣性乃物體持續維持不變的行為也就是定義慣性矩可表達物體對本身旋轉運動的維持或持續性好壞更簡單的說慣性矩越大之物體其維持旋轉運動趨勢的能力越大例如惰輪儲能裝置惰輪之慣性矩越大當其轉動以後其維持旋轉運動趨勢的能力越大

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 55 54

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  • 轉動慣量或慣性矩(Moment of Inertia)
  • 面積慣性矩(Moment of Inertia for Area)
    • 面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)
    • 面積慣性矩之迴轉半徑(Radius of Gyration for Area)
      • 複合面積之慣性矩(Moment of Inertia for Composite Area)
      • 面積之慣性積(Product of Inertia for Area)
        • 平行軸定理(Parallel-Axis Theorem)
          • 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)
            • 主軸慣性矩(Principal Moment of Inertia)
              • 慣性矩之莫爾圓(Mohrs Circle for Moment of Inertia)
              • 質量慣性矩(Mass Moment of Inertia)
                • 平行軸定理(Parallel-Axis Theorem)
Page 49: 轉動慣量 Moments of Inertia - cct.me.ntut.edu.twcct.me.ntut.edu.tw/ccteducation/chchting/aiahtm/statics/lecture... · 轉動慣量或慣性矩 I. 轉動慣量也稱慣性矩(Moment

傾斜軸面積之慣性矩 I

由x軸上之面積慣性矩Ixy軸上之面積慣性矩Iy及慣性積Ixy得知

Ix =

ˆy2dA Iy =

ˆx2dA Ixy =

ˆxydA (87)

Iu =

ˆ(ycosθ minus xsinθ)2dA (88)

=

ˆ(y2cos2θ minus 2xycosθsinθ + x2sin2θ)dA (89)

= cos2θ

ˆy2dAminus 2cosθsinθ

ˆxydA + sin2θ

ˆx2dA (90)

= Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (91)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 35 54

傾斜軸面積之慣性矩 II

Iv =

ˆ(xcosθ + ysinθ)2dA (92)

=

ˆ(x2cos2θ + 2xycosθsinθ + y2sin2θ)dA (93)

= cos2θ

ˆx2dA + 2cosθsinθ

ˆxydA + sin2θ

ˆy2dA (94)

= Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (95)

Iuv =

ˆ(xcosθ + ysinθ)(ycosθ minus xsinθ)dA (96)

=

ˆ(xycos2θ + y2cosθsinθ minus x2cosθsinθ minus xysin2θ)dA (97)

= (cos2 minus sin2θ)

ˆxydA + cosθsinθ

ˆy2dAminus cosθsinθ

ˆx2dA

= Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 36 54

傾斜軸面積之慣性矩 III

再由

sin2θ + cos2θ = 1 (98)

sin(αplusmn β) = sinαcosβ plusmn cosαsinβ (99)

sin2θ = sin(θ + θ) = 2sinθcosθ (100)

cos(αplusmn β) = cosαcosβ ∓ sinαsinβ (101)

cos2θ = cos(θ + θ) = cos2θ minus sin2θ (102)

= 2cos2θ minus 1 = 1minus 2sin2θ (103)

所以

Iu = Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (104)

= Ix(cos2θ + 1

2) + Iy (

1minus cos2θ

2)minus Ixy sin2θ (105)

=Ix + Iy

2+

Ix minus Iy2

cos2θ minus Ixy sin2θ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 37 54

傾斜軸面積之慣性矩 IV

Iv = Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (106)

= Iy (cos2θ + 1

2) + Ix(

1minus cos2θ

2) + Ixy sin2θ (107)

=Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (108)

Iuv = Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ (109)

= Ixycos2θ +1

2Ixsin2θ minus 1

2Iy sin2θ (110)

=Ix minus Iy

2sin2θ + Ixycos2θ (111)

所以

JO = Iu + Iv = Ix + Iy (112)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 38 54

主主主軸軸軸慣慣慣性性性矩矩矩Principal Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 39 54

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主軸慣性矩 I

所謂主軸慣性矩(Principal Moment ofInertia)乃定義發生在慣性矩的極值(極大或極小值)上也就是慣性矩之一次微分式為零或斜率為零

dIu

dθ=

d(Ix+Iy

2+

IxminusIy2

cos2θ minus Ixy sin2θ)

dθ(113)

= minus2Ix minus Iy

2sin2θ minus 2Ixy cos2θ (114)

= 0 (115)

其中dsin2θ = cos2θ dcos2θ = minussin2θ當慣性矩之一次微分式為零時傾斜軸與主軸之夾角θ = θp

minus2Ix minus Iy

2sin2θp minus 2Ixycos2θp = 0 (116)

tan2θ =sin2θ

cos2θ=

minusIxy(Ix minus Iy )2

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 40 54

主軸慣性矩 II

cos2θp =IxminusIy2radic

(IxminusIy2 )2 + I 2xy

sin2θp =minusIxyradic

(IxminusIy2 )2 + I 2xy

(117)

(Iu)max =Ix + Iy

2+

Ix minus Iy2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

minus IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2+

radic(Ix minus Iy

2)2 + I 2xy (118)

(Iv )min =Ix + Iy

2minus Ix minus Iy

2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

+ IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2minusradic

(Ix minus Iy

2)2 + I 2xy (119)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 41 54

主軸慣性矩 III

由tan2θp = tan(π + 2θp)可得到θp1與θp2二個解即2θp2 = π + 2θp1也就是θp2 minus θp1 = π

2

tan2θp = tan(π + 2θp) (120)

2θp2 = π + 2θp1 (121)

θp2 minus θp1 =π

2(122)

所以

Imaxmin

=Ix + Iy

2plusmnradic

(Ix minus Iy

2)2 + I 2xy (123)

Iuv =Ix minus Iy

2

minusIxyradic(IxminusIy2 )2 + I 2xy

+ Ixy

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

(124)

= 0

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 42 54

課課課堂堂堂練練練習習習

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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慣慣慣性性性矩矩矩之之之莫莫莫爾爾爾圓圓圓Mohrrsquos Circle for Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 44 54

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慣性矩之莫爾圓

由於Iu =

Ix + Iy2

+Ix minus Iy

2cos2θ minus Ixy sin2θ (125)

Iv =Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (126)

Iuv =Ix minus Iy

2sin2θ + Ixycos2θ (127)

將式(125)與式(127)各自分別平方並相加可得

(Iu minusIx + Iy

2)2 + I 2uv = (

Ix minus Iy2

)2 + I 2xy (128)

其中Ix Iy Ixy均為常數式(128)可進一步寫成

(Iu minus a)2 + I 2uv = R2 (129)

其中a =Ix+Iy2 R =

radic(IxminusIy2 )2 + I 2xycos2θp1 =

aRsin2θp1 =

minusIxyR 2θp1為OA轉向I軸方向即2θp1為

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 45 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 46 54

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解題技巧

首先確定轉動系統的Ix Iy Ixy接著算出莫爾圓之圓心座標a及半徑R並畫出莫爾圓再由莫爾圓上點出A(Ix Ixy )並由OA轉向I軸找出兩倍夾角2θp1及2θp2其中θp1及θp2為主軸(Principle Axis)與Ix軸的夾角由於主軸(Principle Axis)為極值之軸Imax = Iu及Imin = Iv決定θp1及θp2何者為Imax = Iu與Ix軸的夾角之方法乃比較Ix及Iy之大小Imax = Iu會靠近Ix及Iy之較大者也就是Imax = Iu與Ix及Iy之較大者的夾角比較小

a =Ix + Iy

2(130)

R =

radic(Ix minus Iy

2)2 + I 2xy (131)

2θp2 = π + 2θp1 θp2 minus θp1 =π

2(132)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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質質質量量量慣慣慣性性性矩矩矩Mass Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 48 54

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質量慣性矩

τ = Iα (133)

其中τ為旋轉扭矩I為質量慣性矩α為旋轉角速度

I =

ˆmr2dm (134)

=

ˆVr2ρdV (135)

= ρ

ˆVr2dV (136)

其中r為旋轉半徑或轉軸到旋轉體的距離

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 49 54

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 50 54

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平行軸定理

當通過物體質心(G)並在一轉軸上的質量慣性矩(IG )為已知時可透過座標軸轉換計算物體之質量慣性矩(I)

I =

ˆmr2dm (137)

=

ˆm

[(d + x prime)2 + y prime2]dm (138)

=

ˆm

(x prime2 + y prime2)dm + 2d

ˆmx primedm + d2

ˆmdm (139)

I = IG + md2 (140)

其中IG為在zrsquo軸上通過質心的質量慣性矩m為質量d為通過質心之軸zrsquo並平行原軸z之距離

acutem x primedm = x

acutem dm = 0 since x = 0

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 51 54

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (141)

k =

radicI

m(142)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (143)

= I1 + I2 + + In (144)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 52 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

兩圓盤複合體

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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結結結論論論

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 54 54

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結論

轉動慣量或慣性矩為物體對旋轉運動的慣性而所謂慣性乃物體持續維持不變的行為也就是定義慣性矩可表達物體對本身旋轉運動的維持或持續性好壞更簡單的說慣性矩越大之物體其維持旋轉運動趨勢的能力越大例如惰輪儲能裝置惰輪之慣性矩越大當其轉動以後其維持旋轉運動趨勢的能力越大

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 55 54

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  • 轉動慣量或慣性矩(Moment of Inertia)
  • 面積慣性矩(Moment of Inertia for Area)
    • 面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)
    • 面積慣性矩之迴轉半徑(Radius of Gyration for Area)
      • 複合面積之慣性矩(Moment of Inertia for Composite Area)
      • 面積之慣性積(Product of Inertia for Area)
        • 平行軸定理(Parallel-Axis Theorem)
          • 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)
            • 主軸慣性矩(Principal Moment of Inertia)
              • 慣性矩之莫爾圓(Mohrs Circle for Moment of Inertia)
              • 質量慣性矩(Mass Moment of Inertia)
                • 平行軸定理(Parallel-Axis Theorem)
Page 50: 轉動慣量 Moments of Inertia - cct.me.ntut.edu.twcct.me.ntut.edu.tw/ccteducation/chchting/aiahtm/statics/lecture... · 轉動慣量或慣性矩 I. 轉動慣量也稱慣性矩(Moment

傾斜軸面積之慣性矩 II

Iv =

ˆ(xcosθ + ysinθ)2dA (92)

=

ˆ(x2cos2θ + 2xycosθsinθ + y2sin2θ)dA (93)

= cos2θ

ˆx2dA + 2cosθsinθ

ˆxydA + sin2θ

ˆy2dA (94)

= Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (95)

Iuv =

ˆ(xcosθ + ysinθ)(ycosθ minus xsinθ)dA (96)

=

ˆ(xycos2θ + y2cosθsinθ minus x2cosθsinθ minus xysin2θ)dA (97)

= (cos2 minus sin2θ)

ˆxydA + cosθsinθ

ˆy2dAminus cosθsinθ

ˆx2dA

= Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 36 54

傾斜軸面積之慣性矩 III

再由

sin2θ + cos2θ = 1 (98)

sin(αplusmn β) = sinαcosβ plusmn cosαsinβ (99)

sin2θ = sin(θ + θ) = 2sinθcosθ (100)

cos(αplusmn β) = cosαcosβ ∓ sinαsinβ (101)

cos2θ = cos(θ + θ) = cos2θ minus sin2θ (102)

= 2cos2θ minus 1 = 1minus 2sin2θ (103)

所以

Iu = Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (104)

= Ix(cos2θ + 1

2) + Iy (

1minus cos2θ

2)minus Ixy sin2θ (105)

=Ix + Iy

2+

Ix minus Iy2

cos2θ minus Ixy sin2θ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 37 54

傾斜軸面積之慣性矩 IV

Iv = Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (106)

= Iy (cos2θ + 1

2) + Ix(

1minus cos2θ

2) + Ixy sin2θ (107)

=Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (108)

Iuv = Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ (109)

= Ixycos2θ +1

2Ixsin2θ minus 1

2Iy sin2θ (110)

=Ix minus Iy

2sin2θ + Ixycos2θ (111)

所以

JO = Iu + Iv = Ix + Iy (112)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 38 54

主主主軸軸軸慣慣慣性性性矩矩矩Principal Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 39 54

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主軸慣性矩 I

所謂主軸慣性矩(Principal Moment ofInertia)乃定義發生在慣性矩的極值(極大或極小值)上也就是慣性矩之一次微分式為零或斜率為零

dIu

dθ=

d(Ix+Iy

2+

IxminusIy2

cos2θ minus Ixy sin2θ)

dθ(113)

= minus2Ix minus Iy

2sin2θ minus 2Ixy cos2θ (114)

= 0 (115)

其中dsin2θ = cos2θ dcos2θ = minussin2θ當慣性矩之一次微分式為零時傾斜軸與主軸之夾角θ = θp

minus2Ix minus Iy

2sin2θp minus 2Ixycos2θp = 0 (116)

tan2θ =sin2θ

cos2θ=

minusIxy(Ix minus Iy )2

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 40 54

主軸慣性矩 II

cos2θp =IxminusIy2radic

(IxminusIy2 )2 + I 2xy

sin2θp =minusIxyradic

(IxminusIy2 )2 + I 2xy

(117)

(Iu)max =Ix + Iy

2+

Ix minus Iy2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

minus IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2+

radic(Ix minus Iy

2)2 + I 2xy (118)

(Iv )min =Ix + Iy

2minus Ix minus Iy

2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

+ IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2minusradic

(Ix minus Iy

2)2 + I 2xy (119)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 41 54

主軸慣性矩 III

由tan2θp = tan(π + 2θp)可得到θp1與θp2二個解即2θp2 = π + 2θp1也就是θp2 minus θp1 = π

2

tan2θp = tan(π + 2θp) (120)

2θp2 = π + 2θp1 (121)

θp2 minus θp1 =π

2(122)

所以

Imaxmin

=Ix + Iy

2plusmnradic

(Ix minus Iy

2)2 + I 2xy (123)

Iuv =Ix minus Iy

2

minusIxyradic(IxminusIy2 )2 + I 2xy

+ Ixy

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

(124)

= 0

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 42 54

課課課堂堂堂練練練習習習

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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慣慣慣性性性矩矩矩之之之莫莫莫爾爾爾圓圓圓Mohrrsquos Circle for Moment of Inertia

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慣性矩之莫爾圓

由於Iu =

Ix + Iy2

+Ix minus Iy

2cos2θ minus Ixy sin2θ (125)

Iv =Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (126)

Iuv =Ix minus Iy

2sin2θ + Ixycos2θ (127)

將式(125)與式(127)各自分別平方並相加可得

(Iu minusIx + Iy

2)2 + I 2uv = (

Ix minus Iy2

)2 + I 2xy (128)

其中Ix Iy Ixy均為常數式(128)可進一步寫成

(Iu minus a)2 + I 2uv = R2 (129)

其中a =Ix+Iy2 R =

radic(IxminusIy2 )2 + I 2xycos2θp1 =

aRsin2θp1 =

minusIxyR 2θp1為OA轉向I軸方向即2θp1為

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 45 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 46 54

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解題技巧

首先確定轉動系統的Ix Iy Ixy接著算出莫爾圓之圓心座標a及半徑R並畫出莫爾圓再由莫爾圓上點出A(Ix Ixy )並由OA轉向I軸找出兩倍夾角2θp1及2θp2其中θp1及θp2為主軸(Principle Axis)與Ix軸的夾角由於主軸(Principle Axis)為極值之軸Imax = Iu及Imin = Iv決定θp1及θp2何者為Imax = Iu與Ix軸的夾角之方法乃比較Ix及Iy之大小Imax = Iu會靠近Ix及Iy之較大者也就是Imax = Iu與Ix及Iy之較大者的夾角比較小

a =Ix + Iy

2(130)

R =

radic(Ix minus Iy

2)2 + I 2xy (131)

2θp2 = π + 2θp1 θp2 minus θp1 =π

2(132)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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質質質量量量慣慣慣性性性矩矩矩Mass Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 48 54

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質量慣性矩

τ = Iα (133)

其中τ為旋轉扭矩I為質量慣性矩α為旋轉角速度

I =

ˆmr2dm (134)

=

ˆVr2ρdV (135)

= ρ

ˆVr2dV (136)

其中r為旋轉半徑或轉軸到旋轉體的距離

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 50 54

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平行軸定理

當通過物體質心(G)並在一轉軸上的質量慣性矩(IG )為已知時可透過座標軸轉換計算物體之質量慣性矩(I)

I =

ˆmr2dm (137)

=

ˆm

[(d + x prime)2 + y prime2]dm (138)

=

ˆm

(x prime2 + y prime2)dm + 2d

ˆmx primedm + d2

ˆmdm (139)

I = IG + md2 (140)

其中IG為在zrsquo軸上通過質心的質量慣性矩m為質量d為通過質心之軸zrsquo並平行原軸z之距離

acutem x primedm = x

acutem dm = 0 since x = 0

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (141)

k =

radicI

m(142)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (143)

= I1 + I2 + + In (144)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 52 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

兩圓盤複合體

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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結結結論論論

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結論

轉動慣量或慣性矩為物體對旋轉運動的慣性而所謂慣性乃物體持續維持不變的行為也就是定義慣性矩可表達物體對本身旋轉運動的維持或持續性好壞更簡單的說慣性矩越大之物體其維持旋轉運動趨勢的能力越大例如惰輪儲能裝置惰輪之慣性矩越大當其轉動以後其維持旋轉運動趨勢的能力越大

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 55 54

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  • 轉動慣量或慣性矩(Moment of Inertia)
  • 面積慣性矩(Moment of Inertia for Area)
    • 面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)
    • 面積慣性矩之迴轉半徑(Radius of Gyration for Area)
      • 複合面積之慣性矩(Moment of Inertia for Composite Area)
      • 面積之慣性積(Product of Inertia for Area)
        • 平行軸定理(Parallel-Axis Theorem)
          • 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)
            • 主軸慣性矩(Principal Moment of Inertia)
              • 慣性矩之莫爾圓(Mohrs Circle for Moment of Inertia)
              • 質量慣性矩(Mass Moment of Inertia)
                • 平行軸定理(Parallel-Axis Theorem)
Page 51: 轉動慣量 Moments of Inertia - cct.me.ntut.edu.twcct.me.ntut.edu.tw/ccteducation/chchting/aiahtm/statics/lecture... · 轉動慣量或慣性矩 I. 轉動慣量也稱慣性矩(Moment

傾斜軸面積之慣性矩 III

再由

sin2θ + cos2θ = 1 (98)

sin(αplusmn β) = sinαcosβ plusmn cosαsinβ (99)

sin2θ = sin(θ + θ) = 2sinθcosθ (100)

cos(αplusmn β) = cosαcosβ ∓ sinαsinβ (101)

cos2θ = cos(θ + θ) = cos2θ minus sin2θ (102)

= 2cos2θ minus 1 = 1minus 2sin2θ (103)

所以

Iu = Ixcos2θ minus 2Ixycosθsinθ + Iy sin

2θ (104)

= Ix(cos2θ + 1

2) + Iy (

1minus cos2θ

2)minus Ixy sin2θ (105)

=Ix + Iy

2+

Ix minus Iy2

cos2θ minus Ixy sin2θ

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 37 54

傾斜軸面積之慣性矩 IV

Iv = Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (106)

= Iy (cos2θ + 1

2) + Ix(

1minus cos2θ

2) + Ixy sin2θ (107)

=Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (108)

Iuv = Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ (109)

= Ixycos2θ +1

2Ixsin2θ minus 1

2Iy sin2θ (110)

=Ix minus Iy

2sin2θ + Ixycos2θ (111)

所以

JO = Iu + Iv = Ix + Iy (112)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 38 54

主主主軸軸軸慣慣慣性性性矩矩矩Principal Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 39 54

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主軸慣性矩 I

所謂主軸慣性矩(Principal Moment ofInertia)乃定義發生在慣性矩的極值(極大或極小值)上也就是慣性矩之一次微分式為零或斜率為零

dIu

dθ=

d(Ix+Iy

2+

IxminusIy2

cos2θ minus Ixy sin2θ)

dθ(113)

= minus2Ix minus Iy

2sin2θ minus 2Ixy cos2θ (114)

= 0 (115)

其中dsin2θ = cos2θ dcos2θ = minussin2θ當慣性矩之一次微分式為零時傾斜軸與主軸之夾角θ = θp

minus2Ix minus Iy

2sin2θp minus 2Ixycos2θp = 0 (116)

tan2θ =sin2θ

cos2θ=

minusIxy(Ix minus Iy )2

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 40 54

主軸慣性矩 II

cos2θp =IxminusIy2radic

(IxminusIy2 )2 + I 2xy

sin2θp =minusIxyradic

(IxminusIy2 )2 + I 2xy

(117)

(Iu)max =Ix + Iy

2+

Ix minus Iy2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

minus IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2+

radic(Ix minus Iy

2)2 + I 2xy (118)

(Iv )min =Ix + Iy

2minus Ix minus Iy

2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

+ IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2minusradic

(Ix minus Iy

2)2 + I 2xy (119)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 41 54

主軸慣性矩 III

由tan2θp = tan(π + 2θp)可得到θp1與θp2二個解即2θp2 = π + 2θp1也就是θp2 minus θp1 = π

2

tan2θp = tan(π + 2θp) (120)

2θp2 = π + 2θp1 (121)

θp2 minus θp1 =π

2(122)

所以

Imaxmin

=Ix + Iy

2plusmnradic

(Ix minus Iy

2)2 + I 2xy (123)

Iuv =Ix minus Iy

2

minusIxyradic(IxminusIy2 )2 + I 2xy

+ Ixy

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

(124)

= 0

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課課課堂堂堂練練練習習習

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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慣慣慣性性性矩矩矩之之之莫莫莫爾爾爾圓圓圓Mohrrsquos Circle for Moment of Inertia

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慣性矩之莫爾圓

由於Iu =

Ix + Iy2

+Ix minus Iy

2cos2θ minus Ixy sin2θ (125)

Iv =Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (126)

Iuv =Ix minus Iy

2sin2θ + Ixycos2θ (127)

將式(125)與式(127)各自分別平方並相加可得

(Iu minusIx + Iy

2)2 + I 2uv = (

Ix minus Iy2

)2 + I 2xy (128)

其中Ix Iy Ixy均為常數式(128)可進一步寫成

(Iu minus a)2 + I 2uv = R2 (129)

其中a =Ix+Iy2 R =

radic(IxminusIy2 )2 + I 2xycos2θp1 =

aRsin2θp1 =

minusIxyR 2θp1為OA轉向I軸方向即2θp1為

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課課課堂堂堂練練練習習習

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解題技巧

首先確定轉動系統的Ix Iy Ixy接著算出莫爾圓之圓心座標a及半徑R並畫出莫爾圓再由莫爾圓上點出A(Ix Ixy )並由OA轉向I軸找出兩倍夾角2θp1及2θp2其中θp1及θp2為主軸(Principle Axis)與Ix軸的夾角由於主軸(Principle Axis)為極值之軸Imax = Iu及Imin = Iv決定θp1及θp2何者為Imax = Iu與Ix軸的夾角之方法乃比較Ix及Iy之大小Imax = Iu會靠近Ix及Iy之較大者也就是Imax = Iu與Ix及Iy之較大者的夾角比較小

a =Ix + Iy

2(130)

R =

radic(Ix minus Iy

2)2 + I 2xy (131)

2θp2 = π + 2θp1 θp2 minus θp1 =π

2(132)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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質質質量量量慣慣慣性性性矩矩矩Mass Moment of Inertia

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質量慣性矩

τ = Iα (133)

其中τ為旋轉扭矩I為質量慣性矩α為旋轉角速度

I =

ˆmr2dm (134)

=

ˆVr2ρdV (135)

= ρ

ˆVr2dV (136)

其中r為旋轉半徑或轉軸到旋轉體的距離

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

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平行軸定理

當通過物體質心(G)並在一轉軸上的質量慣性矩(IG )為已知時可透過座標軸轉換計算物體之質量慣性矩(I)

I =

ˆmr2dm (137)

=

ˆm

[(d + x prime)2 + y prime2]dm (138)

=

ˆm

(x prime2 + y prime2)dm + 2d

ˆmx primedm + d2

ˆmdm (139)

I = IG + md2 (140)

其中IG為在zrsquo軸上通過質心的質量慣性矩m為質量d為通過質心之軸zrsquo並平行原軸z之距離

acutem x primedm = x

acutem dm = 0 since x = 0

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (141)

k =

radicI

m(142)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (143)

= I1 + I2 + + In (144)

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

兩圓盤複合體

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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結結結論論論

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結論

轉動慣量或慣性矩為物體對旋轉運動的慣性而所謂慣性乃物體持續維持不變的行為也就是定義慣性矩可表達物體對本身旋轉運動的維持或持續性好壞更簡單的說慣性矩越大之物體其維持旋轉運動趨勢的能力越大例如惰輪儲能裝置惰輪之慣性矩越大當其轉動以後其維持旋轉運動趨勢的能力越大

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 55 54

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  • 轉動慣量或慣性矩(Moment of Inertia)
  • 面積慣性矩(Moment of Inertia for Area)
    • 面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)
    • 面積慣性矩之迴轉半徑(Radius of Gyration for Area)
      • 複合面積之慣性矩(Moment of Inertia for Composite Area)
      • 面積之慣性積(Product of Inertia for Area)
        • 平行軸定理(Parallel-Axis Theorem)
          • 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)
            • 主軸慣性矩(Principal Moment of Inertia)
              • 慣性矩之莫爾圓(Mohrs Circle for Moment of Inertia)
              • 質量慣性矩(Mass Moment of Inertia)
                • 平行軸定理(Parallel-Axis Theorem)
Page 52: 轉動慣量 Moments of Inertia - cct.me.ntut.edu.twcct.me.ntut.edu.tw/ccteducation/chchting/aiahtm/statics/lecture... · 轉動慣量或慣性矩 I. 轉動慣量也稱慣性矩(Moment

傾斜軸面積之慣性矩 IV

Iv = Iycos2θ + 2Ixycosθsinθ + Ixsin

2θ (106)

= Iy (cos2θ + 1

2) + Ix(

1minus cos2θ

2) + Ixy sin2θ (107)

=Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (108)

Iuv = Ixy (cos2 minus sin2θ) + Ixcosθsinθ minus Iycosθsinθ (109)

= Ixycos2θ +1

2Ixsin2θ minus 1

2Iy sin2θ (110)

=Ix minus Iy

2sin2θ + Ixycos2θ (111)

所以

JO = Iu + Iv = Ix + Iy (112)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 38 54

主主主軸軸軸慣慣慣性性性矩矩矩Principal Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 39 54

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主軸慣性矩 I

所謂主軸慣性矩(Principal Moment ofInertia)乃定義發生在慣性矩的極值(極大或極小值)上也就是慣性矩之一次微分式為零或斜率為零

dIu

dθ=

d(Ix+Iy

2+

IxminusIy2

cos2θ minus Ixy sin2θ)

dθ(113)

= minus2Ix minus Iy

2sin2θ minus 2Ixy cos2θ (114)

= 0 (115)

其中dsin2θ = cos2θ dcos2θ = minussin2θ當慣性矩之一次微分式為零時傾斜軸與主軸之夾角θ = θp

minus2Ix minus Iy

2sin2θp minus 2Ixycos2θp = 0 (116)

tan2θ =sin2θ

cos2θ=

minusIxy(Ix minus Iy )2

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 40 54

主軸慣性矩 II

cos2θp =IxminusIy2radic

(IxminusIy2 )2 + I 2xy

sin2θp =minusIxyradic

(IxminusIy2 )2 + I 2xy

(117)

(Iu)max =Ix + Iy

2+

Ix minus Iy2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

minus IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2+

radic(Ix minus Iy

2)2 + I 2xy (118)

(Iv )min =Ix + Iy

2minus Ix minus Iy

2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

+ IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2minusradic

(Ix minus Iy

2)2 + I 2xy (119)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 41 54

主軸慣性矩 III

由tan2θp = tan(π + 2θp)可得到θp1與θp2二個解即2θp2 = π + 2θp1也就是θp2 minus θp1 = π

2

tan2θp = tan(π + 2θp) (120)

2θp2 = π + 2θp1 (121)

θp2 minus θp1 =π

2(122)

所以

Imaxmin

=Ix + Iy

2plusmnradic

(Ix minus Iy

2)2 + I 2xy (123)

Iuv =Ix minus Iy

2

minusIxyradic(IxminusIy2 )2 + I 2xy

+ Ixy

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

(124)

= 0

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課課課堂堂堂練練練習習習

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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慣慣慣性性性矩矩矩之之之莫莫莫爾爾爾圓圓圓Mohrrsquos Circle for Moment of Inertia

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慣性矩之莫爾圓

由於Iu =

Ix + Iy2

+Ix minus Iy

2cos2θ minus Ixy sin2θ (125)

Iv =Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (126)

Iuv =Ix minus Iy

2sin2θ + Ixycos2θ (127)

將式(125)與式(127)各自分別平方並相加可得

(Iu minusIx + Iy

2)2 + I 2uv = (

Ix minus Iy2

)2 + I 2xy (128)

其中Ix Iy Ixy均為常數式(128)可進一步寫成

(Iu minus a)2 + I 2uv = R2 (129)

其中a =Ix+Iy2 R =

radic(IxminusIy2 )2 + I 2xycos2θp1 =

aRsin2θp1 =

minusIxyR 2θp1為OA轉向I軸方向即2θp1為

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課課課堂堂堂練練練習習習

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解題技巧

首先確定轉動系統的Ix Iy Ixy接著算出莫爾圓之圓心座標a及半徑R並畫出莫爾圓再由莫爾圓上點出A(Ix Ixy )並由OA轉向I軸找出兩倍夾角2θp1及2θp2其中θp1及θp2為主軸(Principle Axis)與Ix軸的夾角由於主軸(Principle Axis)為極值之軸Imax = Iu及Imin = Iv決定θp1及θp2何者為Imax = Iu與Ix軸的夾角之方法乃比較Ix及Iy之大小Imax = Iu會靠近Ix及Iy之較大者也就是Imax = Iu與Ix及Iy之較大者的夾角比較小

a =Ix + Iy

2(130)

R =

radic(Ix minus Iy

2)2 + I 2xy (131)

2θp2 = π + 2θp1 θp2 minus θp1 =π

2(132)

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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質質質量量量慣慣慣性性性矩矩矩Mass Moment of Inertia

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質量慣性矩

τ = Iα (133)

其中τ為旋轉扭矩I為質量慣性矩α為旋轉角速度

I =

ˆmr2dm (134)

=

ˆVr2ρdV (135)

= ρ

ˆVr2dV (136)

其中r為旋轉半徑或轉軸到旋轉體的距離

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

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平行軸定理

當通過物體質心(G)並在一轉軸上的質量慣性矩(IG )為已知時可透過座標軸轉換計算物體之質量慣性矩(I)

I =

ˆmr2dm (137)

=

ˆm

[(d + x prime)2 + y prime2]dm (138)

=

ˆm

(x prime2 + y prime2)dm + 2d

ˆmx primedm + d2

ˆmdm (139)

I = IG + md2 (140)

其中IG為在zrsquo軸上通過質心的質量慣性矩m為質量d為通過質心之軸zrsquo並平行原軸z之距離

acutem x primedm = x

acutem dm = 0 since x = 0

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (141)

k =

radicI

m(142)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (143)

= I1 + I2 + + In (144)

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課課課堂堂堂練練練習習習

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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兩圓盤複合體

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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結結結論論論

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結論

轉動慣量或慣性矩為物體對旋轉運動的慣性而所謂慣性乃物體持續維持不變的行為也就是定義慣性矩可表達物體對本身旋轉運動的維持或持續性好壞更簡單的說慣性矩越大之物體其維持旋轉運動趨勢的能力越大例如惰輪儲能裝置惰輪之慣性矩越大當其轉動以後其維持旋轉運動趨勢的能力越大

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 55 54

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  • 轉動慣量或慣性矩(Moment of Inertia)
  • 面積慣性矩(Moment of Inertia for Area)
    • 面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)
    • 面積慣性矩之迴轉半徑(Radius of Gyration for Area)
      • 複合面積之慣性矩(Moment of Inertia for Composite Area)
      • 面積之慣性積(Product of Inertia for Area)
        • 平行軸定理(Parallel-Axis Theorem)
          • 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)
            • 主軸慣性矩(Principal Moment of Inertia)
              • 慣性矩之莫爾圓(Mohrs Circle for Moment of Inertia)
              • 質量慣性矩(Mass Moment of Inertia)
                • 平行軸定理(Parallel-Axis Theorem)
Page 53: 轉動慣量 Moments of Inertia - cct.me.ntut.edu.twcct.me.ntut.edu.tw/ccteducation/chchting/aiahtm/statics/lecture... · 轉動慣量或慣性矩 I. 轉動慣量也稱慣性矩(Moment

主主主軸軸軸慣慣慣性性性矩矩矩Principal Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 39 54

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主軸慣性矩 I

所謂主軸慣性矩(Principal Moment ofInertia)乃定義發生在慣性矩的極值(極大或極小值)上也就是慣性矩之一次微分式為零或斜率為零

dIu

dθ=

d(Ix+Iy

2+

IxminusIy2

cos2θ minus Ixy sin2θ)

dθ(113)

= minus2Ix minus Iy

2sin2θ minus 2Ixy cos2θ (114)

= 0 (115)

其中dsin2θ = cos2θ dcos2θ = minussin2θ當慣性矩之一次微分式為零時傾斜軸與主軸之夾角θ = θp

minus2Ix minus Iy

2sin2θp minus 2Ixycos2θp = 0 (116)

tan2θ =sin2θ

cos2θ=

minusIxy(Ix minus Iy )2

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主軸慣性矩 II

cos2θp =IxminusIy2radic

(IxminusIy2 )2 + I 2xy

sin2θp =minusIxyradic

(IxminusIy2 )2 + I 2xy

(117)

(Iu)max =Ix + Iy

2+

Ix minus Iy2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

minus IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2+

radic(Ix minus Iy

2)2 + I 2xy (118)

(Iv )min =Ix + Iy

2minus Ix minus Iy

2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

+ IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2minusradic

(Ix minus Iy

2)2 + I 2xy (119)

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主軸慣性矩 III

由tan2θp = tan(π + 2θp)可得到θp1與θp2二個解即2θp2 = π + 2θp1也就是θp2 minus θp1 = π

2

tan2θp = tan(π + 2θp) (120)

2θp2 = π + 2θp1 (121)

θp2 minus θp1 =π

2(122)

所以

Imaxmin

=Ix + Iy

2plusmnradic

(Ix minus Iy

2)2 + I 2xy (123)

Iuv =Ix minus Iy

2

minusIxyradic(IxminusIy2 )2 + I 2xy

+ Ixy

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

(124)

= 0

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課課課堂堂堂練練練習習習

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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慣慣慣性性性矩矩矩之之之莫莫莫爾爾爾圓圓圓Mohrrsquos Circle for Moment of Inertia

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慣性矩之莫爾圓

由於Iu =

Ix + Iy2

+Ix minus Iy

2cos2θ minus Ixy sin2θ (125)

Iv =Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (126)

Iuv =Ix minus Iy

2sin2θ + Ixycos2θ (127)

將式(125)與式(127)各自分別平方並相加可得

(Iu minusIx + Iy

2)2 + I 2uv = (

Ix minus Iy2

)2 + I 2xy (128)

其中Ix Iy Ixy均為常數式(128)可進一步寫成

(Iu minus a)2 + I 2uv = R2 (129)

其中a =Ix+Iy2 R =

radic(IxminusIy2 )2 + I 2xycos2θp1 =

aRsin2θp1 =

minusIxyR 2θp1為OA轉向I軸方向即2θp1為

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課課課堂堂堂練練練習習習

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解題技巧

首先確定轉動系統的Ix Iy Ixy接著算出莫爾圓之圓心座標a及半徑R並畫出莫爾圓再由莫爾圓上點出A(Ix Ixy )並由OA轉向I軸找出兩倍夾角2θp1及2θp2其中θp1及θp2為主軸(Principle Axis)與Ix軸的夾角由於主軸(Principle Axis)為極值之軸Imax = Iu及Imin = Iv決定θp1及θp2何者為Imax = Iu與Ix軸的夾角之方法乃比較Ix及Iy之大小Imax = Iu會靠近Ix及Iy之較大者也就是Imax = Iu與Ix及Iy之較大者的夾角比較小

a =Ix + Iy

2(130)

R =

radic(Ix minus Iy

2)2 + I 2xy (131)

2θp2 = π + 2θp1 θp2 minus θp1 =π

2(132)

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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質質質量量量慣慣慣性性性矩矩矩Mass Moment of Inertia

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質量慣性矩

τ = Iα (133)

其中τ為旋轉扭矩I為質量慣性矩α為旋轉角速度

I =

ˆmr2dm (134)

=

ˆVr2ρdV (135)

= ρ

ˆVr2dV (136)

其中r為旋轉半徑或轉軸到旋轉體的距離

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

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平行軸定理

當通過物體質心(G)並在一轉軸上的質量慣性矩(IG )為已知時可透過座標軸轉換計算物體之質量慣性矩(I)

I =

ˆmr2dm (137)

=

ˆm

[(d + x prime)2 + y prime2]dm (138)

=

ˆm

(x prime2 + y prime2)dm + 2d

ˆmx primedm + d2

ˆmdm (139)

I = IG + md2 (140)

其中IG為在zrsquo軸上通過質心的質量慣性矩m為質量d為通過質心之軸zrsquo並平行原軸z之距離

acutem x primedm = x

acutem dm = 0 since x = 0

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (141)

k =

radicI

m(142)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (143)

= I1 + I2 + + In (144)

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課課課堂堂堂練練練習習習

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

兩圓盤複合體

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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結結結論論論

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結論

轉動慣量或慣性矩為物體對旋轉運動的慣性而所謂慣性乃物體持續維持不變的行為也就是定義慣性矩可表達物體對本身旋轉運動的維持或持續性好壞更簡單的說慣性矩越大之物體其維持旋轉運動趨勢的能力越大例如惰輪儲能裝置惰輪之慣性矩越大當其轉動以後其維持旋轉運動趨勢的能力越大

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  • 轉動慣量或慣性矩(Moment of Inertia)
  • 面積慣性矩(Moment of Inertia for Area)
    • 面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)
    • 面積慣性矩之迴轉半徑(Radius of Gyration for Area)
      • 複合面積之慣性矩(Moment of Inertia for Composite Area)
      • 面積之慣性積(Product of Inertia for Area)
        • 平行軸定理(Parallel-Axis Theorem)
          • 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)
            • 主軸慣性矩(Principal Moment of Inertia)
              • 慣性矩之莫爾圓(Mohrs Circle for Moment of Inertia)
              • 質量慣性矩(Mass Moment of Inertia)
                • 平行軸定理(Parallel-Axis Theorem)
Page 54: 轉動慣量 Moments of Inertia - cct.me.ntut.edu.twcct.me.ntut.edu.tw/ccteducation/chchting/aiahtm/statics/lecture... · 轉動慣量或慣性矩 I. 轉動慣量也稱慣性矩(Moment

主軸慣性矩 I

所謂主軸慣性矩(Principal Moment ofInertia)乃定義發生在慣性矩的極值(極大或極小值)上也就是慣性矩之一次微分式為零或斜率為零

dIu

dθ=

d(Ix+Iy

2+

IxminusIy2

cos2θ minus Ixy sin2θ)

dθ(113)

= minus2Ix minus Iy

2sin2θ minus 2Ixy cos2θ (114)

= 0 (115)

其中dsin2θ = cos2θ dcos2θ = minussin2θ當慣性矩之一次微分式為零時傾斜軸與主軸之夾角θ = θp

minus2Ix minus Iy

2sin2θp minus 2Ixycos2θp = 0 (116)

tan2θ =sin2θ

cos2θ=

minusIxy(Ix minus Iy )2

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主軸慣性矩 II

cos2θp =IxminusIy2radic

(IxminusIy2 )2 + I 2xy

sin2θp =minusIxyradic

(IxminusIy2 )2 + I 2xy

(117)

(Iu)max =Ix + Iy

2+

Ix minus Iy2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

minus IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2+

radic(Ix minus Iy

2)2 + I 2xy (118)

(Iv )min =Ix + Iy

2minus Ix minus Iy

2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

+ IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2minusradic

(Ix minus Iy

2)2 + I 2xy (119)

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主軸慣性矩 III

由tan2θp = tan(π + 2θp)可得到θp1與θp2二個解即2θp2 = π + 2θp1也就是θp2 minus θp1 = π

2

tan2θp = tan(π + 2θp) (120)

2θp2 = π + 2θp1 (121)

θp2 minus θp1 =π

2(122)

所以

Imaxmin

=Ix + Iy

2plusmnradic

(Ix minus Iy

2)2 + I 2xy (123)

Iuv =Ix minus Iy

2

minusIxyradic(IxminusIy2 )2 + I 2xy

+ Ixy

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

(124)

= 0

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課課課堂堂堂練練練習習習

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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慣慣慣性性性矩矩矩之之之莫莫莫爾爾爾圓圓圓Mohrrsquos Circle for Moment of Inertia

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慣性矩之莫爾圓

由於Iu =

Ix + Iy2

+Ix minus Iy

2cos2θ minus Ixy sin2θ (125)

Iv =Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (126)

Iuv =Ix minus Iy

2sin2θ + Ixycos2θ (127)

將式(125)與式(127)各自分別平方並相加可得

(Iu minusIx + Iy

2)2 + I 2uv = (

Ix minus Iy2

)2 + I 2xy (128)

其中Ix Iy Ixy均為常數式(128)可進一步寫成

(Iu minus a)2 + I 2uv = R2 (129)

其中a =Ix+Iy2 R =

radic(IxminusIy2 )2 + I 2xycos2θp1 =

aRsin2θp1 =

minusIxyR 2θp1為OA轉向I軸方向即2θp1為

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課課課堂堂堂練練練習習習

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解題技巧

首先確定轉動系統的Ix Iy Ixy接著算出莫爾圓之圓心座標a及半徑R並畫出莫爾圓再由莫爾圓上點出A(Ix Ixy )並由OA轉向I軸找出兩倍夾角2θp1及2θp2其中θp1及θp2為主軸(Principle Axis)與Ix軸的夾角由於主軸(Principle Axis)為極值之軸Imax = Iu及Imin = Iv決定θp1及θp2何者為Imax = Iu與Ix軸的夾角之方法乃比較Ix及Iy之大小Imax = Iu會靠近Ix及Iy之較大者也就是Imax = Iu與Ix及Iy之較大者的夾角比較小

a =Ix + Iy

2(130)

R =

radic(Ix minus Iy

2)2 + I 2xy (131)

2θp2 = π + 2θp1 θp2 minus θp1 =π

2(132)

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

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質質質量量量慣慣慣性性性矩矩矩Mass Moment of Inertia

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質量慣性矩

τ = Iα (133)

其中τ為旋轉扭矩I為質量慣性矩α為旋轉角速度

I =

ˆmr2dm (134)

=

ˆVr2ρdV (135)

= ρ

ˆVr2dV (136)

其中r為旋轉半徑或轉軸到旋轉體的距離

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 49 54

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 50 54

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平行軸定理

當通過物體質心(G)並在一轉軸上的質量慣性矩(IG )為已知時可透過座標軸轉換計算物體之質量慣性矩(I)

I =

ˆmr2dm (137)

=

ˆm

[(d + x prime)2 + y prime2]dm (138)

=

ˆm

(x prime2 + y prime2)dm + 2d

ˆmx primedm + d2

ˆmdm (139)

I = IG + md2 (140)

其中IG為在zrsquo軸上通過質心的質量慣性矩m為質量d為通過質心之軸zrsquo並平行原軸z之距離

acutem x primedm = x

acutem dm = 0 since x = 0

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 51 54

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (141)

k =

radicI

m(142)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (143)

= I1 + I2 + + In (144)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 52 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

兩圓盤複合體

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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結結結論論論

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 54 54

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結論

轉動慣量或慣性矩為物體對旋轉運動的慣性而所謂慣性乃物體持續維持不變的行為也就是定義慣性矩可表達物體對本身旋轉運動的維持或持續性好壞更簡單的說慣性矩越大之物體其維持旋轉運動趨勢的能力越大例如惰輪儲能裝置惰輪之慣性矩越大當其轉動以後其維持旋轉運動趨勢的能力越大

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 55 54

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  • 轉動慣量或慣性矩(Moment of Inertia)
  • 面積慣性矩(Moment of Inertia for Area)
    • 面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)
    • 面積慣性矩之迴轉半徑(Radius of Gyration for Area)
      • 複合面積之慣性矩(Moment of Inertia for Composite Area)
      • 面積之慣性積(Product of Inertia for Area)
        • 平行軸定理(Parallel-Axis Theorem)
          • 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)
            • 主軸慣性矩(Principal Moment of Inertia)
              • 慣性矩之莫爾圓(Mohrs Circle for Moment of Inertia)
              • 質量慣性矩(Mass Moment of Inertia)
                • 平行軸定理(Parallel-Axis Theorem)
Page 55: 轉動慣量 Moments of Inertia - cct.me.ntut.edu.twcct.me.ntut.edu.tw/ccteducation/chchting/aiahtm/statics/lecture... · 轉動慣量或慣性矩 I. 轉動慣量也稱慣性矩(Moment

主軸慣性矩 II

cos2θp =IxminusIy2radic

(IxminusIy2 )2 + I 2xy

sin2θp =minusIxyradic

(IxminusIy2 )2 + I 2xy

(117)

(Iu)max =Ix + Iy

2+

Ix minus Iy2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

minus IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2+

radic(Ix minus Iy

2)2 + I 2xy (118)

(Iv )min =Ix + Iy

2minus Ix minus Iy

2

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

+ IxyminusIxyradic

(IxminusIy2 )2 + I 2xy

=Ix + Iy

2minusradic

(Ix minus Iy

2)2 + I 2xy (119)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 41 54

主軸慣性矩 III

由tan2θp = tan(π + 2θp)可得到θp1與θp2二個解即2θp2 = π + 2θp1也就是θp2 minus θp1 = π

2

tan2θp = tan(π + 2θp) (120)

2θp2 = π + 2θp1 (121)

θp2 minus θp1 =π

2(122)

所以

Imaxmin

=Ix + Iy

2plusmnradic

(Ix minus Iy

2)2 + I 2xy (123)

Iuv =Ix minus Iy

2

minusIxyradic(IxminusIy2 )2 + I 2xy

+ Ixy

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

(124)

= 0

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 42 54

課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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慣慣慣性性性矩矩矩之之之莫莫莫爾爾爾圓圓圓Mohrrsquos Circle for Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 44 54

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慣性矩之莫爾圓

由於Iu =

Ix + Iy2

+Ix minus Iy

2cos2θ minus Ixy sin2θ (125)

Iv =Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (126)

Iuv =Ix minus Iy

2sin2θ + Ixycos2θ (127)

將式(125)與式(127)各自分別平方並相加可得

(Iu minusIx + Iy

2)2 + I 2uv = (

Ix minus Iy2

)2 + I 2xy (128)

其中Ix Iy Ixy均為常數式(128)可進一步寫成

(Iu minus a)2 + I 2uv = R2 (129)

其中a =Ix+Iy2 R =

radic(IxminusIy2 )2 + I 2xycos2θp1 =

aRsin2θp1 =

minusIxyR 2θp1為OA轉向I軸方向即2θp1為

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 45 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 46 54

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解題技巧

首先確定轉動系統的Ix Iy Ixy接著算出莫爾圓之圓心座標a及半徑R並畫出莫爾圓再由莫爾圓上點出A(Ix Ixy )並由OA轉向I軸找出兩倍夾角2θp1及2θp2其中θp1及θp2為主軸(Principle Axis)與Ix軸的夾角由於主軸(Principle Axis)為極值之軸Imax = Iu及Imin = Iv決定θp1及θp2何者為Imax = Iu與Ix軸的夾角之方法乃比較Ix及Iy之大小Imax = Iu會靠近Ix及Iy之較大者也就是Imax = Iu與Ix及Iy之較大者的夾角比較小

a =Ix + Iy

2(130)

R =

radic(Ix minus Iy

2)2 + I 2xy (131)

2θp2 = π + 2θp1 θp2 minus θp1 =π

2(132)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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質質質量量量慣慣慣性性性矩矩矩Mass Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 48 54

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質量慣性矩

τ = Iα (133)

其中τ為旋轉扭矩I為質量慣性矩α為旋轉角速度

I =

ˆmr2dm (134)

=

ˆVr2ρdV (135)

= ρ

ˆVr2dV (136)

其中r為旋轉半徑或轉軸到旋轉體的距離

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 49 54

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 50 54

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平行軸定理

當通過物體質心(G)並在一轉軸上的質量慣性矩(IG )為已知時可透過座標軸轉換計算物體之質量慣性矩(I)

I =

ˆmr2dm (137)

=

ˆm

[(d + x prime)2 + y prime2]dm (138)

=

ˆm

(x prime2 + y prime2)dm + 2d

ˆmx primedm + d2

ˆmdm (139)

I = IG + md2 (140)

其中IG為在zrsquo軸上通過質心的質量慣性矩m為質量d為通過質心之軸zrsquo並平行原軸z之距離

acutem x primedm = x

acutem dm = 0 since x = 0

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 51 54

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (141)

k =

radicI

m(142)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (143)

= I1 + I2 + + In (144)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 52 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

兩圓盤複合體

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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結結結論論論

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 54 54

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結論

轉動慣量或慣性矩為物體對旋轉運動的慣性而所謂慣性乃物體持續維持不變的行為也就是定義慣性矩可表達物體對本身旋轉運動的維持或持續性好壞更簡單的說慣性矩越大之物體其維持旋轉運動趨勢的能力越大例如惰輪儲能裝置惰輪之慣性矩越大當其轉動以後其維持旋轉運動趨勢的能力越大

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 55 54

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  • 轉動慣量或慣性矩(Moment of Inertia)
  • 面積慣性矩(Moment of Inertia for Area)
    • 面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)
    • 面積慣性矩之迴轉半徑(Radius of Gyration for Area)
      • 複合面積之慣性矩(Moment of Inertia for Composite Area)
      • 面積之慣性積(Product of Inertia for Area)
        • 平行軸定理(Parallel-Axis Theorem)
          • 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)
            • 主軸慣性矩(Principal Moment of Inertia)
              • 慣性矩之莫爾圓(Mohrs Circle for Moment of Inertia)
              • 質量慣性矩(Mass Moment of Inertia)
                • 平行軸定理(Parallel-Axis Theorem)
Page 56: 轉動慣量 Moments of Inertia - cct.me.ntut.edu.twcct.me.ntut.edu.tw/ccteducation/chchting/aiahtm/statics/lecture... · 轉動慣量或慣性矩 I. 轉動慣量也稱慣性矩(Moment

主軸慣性矩 III

由tan2θp = tan(π + 2θp)可得到θp1與θp2二個解即2θp2 = π + 2θp1也就是θp2 minus θp1 = π

2

tan2θp = tan(π + 2θp) (120)

2θp2 = π + 2θp1 (121)

θp2 minus θp1 =π

2(122)

所以

Imaxmin

=Ix + Iy

2plusmnradic

(Ix minus Iy

2)2 + I 2xy (123)

Iuv =Ix minus Iy

2

minusIxyradic(IxminusIy2 )2 + I 2xy

+ Ixy

IxminusIy2radic

(IxminusIy2 )2 + I 2xy

(124)

= 0

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 42 54

課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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慣慣慣性性性矩矩矩之之之莫莫莫爾爾爾圓圓圓Mohrrsquos Circle for Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 44 54

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慣性矩之莫爾圓

由於Iu =

Ix + Iy2

+Ix minus Iy

2cos2θ minus Ixy sin2θ (125)

Iv =Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (126)

Iuv =Ix minus Iy

2sin2θ + Ixycos2θ (127)

將式(125)與式(127)各自分別平方並相加可得

(Iu minusIx + Iy

2)2 + I 2uv = (

Ix minus Iy2

)2 + I 2xy (128)

其中Ix Iy Ixy均為常數式(128)可進一步寫成

(Iu minus a)2 + I 2uv = R2 (129)

其中a =Ix+Iy2 R =

radic(IxminusIy2 )2 + I 2xycos2θp1 =

aRsin2θp1 =

minusIxyR 2θp1為OA轉向I軸方向即2θp1為

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 45 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 46 54

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解題技巧

首先確定轉動系統的Ix Iy Ixy接著算出莫爾圓之圓心座標a及半徑R並畫出莫爾圓再由莫爾圓上點出A(Ix Ixy )並由OA轉向I軸找出兩倍夾角2θp1及2θp2其中θp1及θp2為主軸(Principle Axis)與Ix軸的夾角由於主軸(Principle Axis)為極值之軸Imax = Iu及Imin = Iv決定θp1及θp2何者為Imax = Iu與Ix軸的夾角之方法乃比較Ix及Iy之大小Imax = Iu會靠近Ix及Iy之較大者也就是Imax = Iu與Ix及Iy之較大者的夾角比較小

a =Ix + Iy

2(130)

R =

radic(Ix minus Iy

2)2 + I 2xy (131)

2θp2 = π + 2θp1 θp2 minus θp1 =π

2(132)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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質質質量量量慣慣慣性性性矩矩矩Mass Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 48 54

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質量慣性矩

τ = Iα (133)

其中τ為旋轉扭矩I為質量慣性矩α為旋轉角速度

I =

ˆmr2dm (134)

=

ˆVr2ρdV (135)

= ρ

ˆVr2dV (136)

其中r為旋轉半徑或轉軸到旋轉體的距離

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 49 54

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 50 54

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平行軸定理

當通過物體質心(G)並在一轉軸上的質量慣性矩(IG )為已知時可透過座標軸轉換計算物體之質量慣性矩(I)

I =

ˆmr2dm (137)

=

ˆm

[(d + x prime)2 + y prime2]dm (138)

=

ˆm

(x prime2 + y prime2)dm + 2d

ˆmx primedm + d2

ˆmdm (139)

I = IG + md2 (140)

其中IG為在zrsquo軸上通過質心的質量慣性矩m為質量d為通過質心之軸zrsquo並平行原軸z之距離

acutem x primedm = x

acutem dm = 0 since x = 0

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 51 54

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (141)

k =

radicI

m(142)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (143)

= I1 + I2 + + In (144)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 52 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

兩圓盤複合體

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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結結結論論論

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 54 54

flip_clock_black_12swf
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結論

轉動慣量或慣性矩為物體對旋轉運動的慣性而所謂慣性乃物體持續維持不變的行為也就是定義慣性矩可表達物體對本身旋轉運動的維持或持續性好壞更簡單的說慣性矩越大之物體其維持旋轉運動趨勢的能力越大例如惰輪儲能裝置惰輪之慣性矩越大當其轉動以後其維持旋轉運動趨勢的能力越大

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 55 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
  • 轉動慣量或慣性矩(Moment of Inertia)
  • 面積慣性矩(Moment of Inertia for Area)
    • 面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)
    • 面積慣性矩之迴轉半徑(Radius of Gyration for Area)
      • 複合面積之慣性矩(Moment of Inertia for Composite Area)
      • 面積之慣性積(Product of Inertia for Area)
        • 平行軸定理(Parallel-Axis Theorem)
          • 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)
            • 主軸慣性矩(Principal Moment of Inertia)
              • 慣性矩之莫爾圓(Mohrs Circle for Moment of Inertia)
              • 質量慣性矩(Mass Moment of Inertia)
                • 平行軸定理(Parallel-Axis Theorem)
Page 57: 轉動慣量 Moments of Inertia - cct.me.ntut.edu.twcct.me.ntut.edu.tw/ccteducation/chchting/aiahtm/statics/lecture... · 轉動慣量或慣性矩 I. 轉動慣量也稱慣性矩(Moment

課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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慣慣慣性性性矩矩矩之之之莫莫莫爾爾爾圓圓圓Mohrrsquos Circle for Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 44 54

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慣性矩之莫爾圓

由於Iu =

Ix + Iy2

+Ix minus Iy

2cos2θ minus Ixy sin2θ (125)

Iv =Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (126)

Iuv =Ix minus Iy

2sin2θ + Ixycos2θ (127)

將式(125)與式(127)各自分別平方並相加可得

(Iu minusIx + Iy

2)2 + I 2uv = (

Ix minus Iy2

)2 + I 2xy (128)

其中Ix Iy Ixy均為常數式(128)可進一步寫成

(Iu minus a)2 + I 2uv = R2 (129)

其中a =Ix+Iy2 R =

radic(IxminusIy2 )2 + I 2xycos2θp1 =

aRsin2θp1 =

minusIxyR 2θp1為OA轉向I軸方向即2θp1為

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 45 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 46 54

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解題技巧

首先確定轉動系統的Ix Iy Ixy接著算出莫爾圓之圓心座標a及半徑R並畫出莫爾圓再由莫爾圓上點出A(Ix Ixy )並由OA轉向I軸找出兩倍夾角2θp1及2θp2其中θp1及θp2為主軸(Principle Axis)與Ix軸的夾角由於主軸(Principle Axis)為極值之軸Imax = Iu及Imin = Iv決定θp1及θp2何者為Imax = Iu與Ix軸的夾角之方法乃比較Ix及Iy之大小Imax = Iu會靠近Ix及Iy之較大者也就是Imax = Iu與Ix及Iy之較大者的夾角比較小

a =Ix + Iy

2(130)

R =

radic(Ix minus Iy

2)2 + I 2xy (131)

2θp2 = π + 2θp1 θp2 minus θp1 =π

2(132)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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質質質量量量慣慣慣性性性矩矩矩Mass Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 48 54

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質量慣性矩

τ = Iα (133)

其中τ為旋轉扭矩I為質量慣性矩α為旋轉角速度

I =

ˆmr2dm (134)

=

ˆVr2ρdV (135)

= ρ

ˆVr2dV (136)

其中r為旋轉半徑或轉軸到旋轉體的距離

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 49 54

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 50 54

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平行軸定理

當通過物體質心(G)並在一轉軸上的質量慣性矩(IG )為已知時可透過座標軸轉換計算物體之質量慣性矩(I)

I =

ˆmr2dm (137)

=

ˆm

[(d + x prime)2 + y prime2]dm (138)

=

ˆm

(x prime2 + y prime2)dm + 2d

ˆmx primedm + d2

ˆmdm (139)

I = IG + md2 (140)

其中IG為在zrsquo軸上通過質心的質量慣性矩m為質量d為通過質心之軸zrsquo並平行原軸z之距離

acutem x primedm = x

acutem dm = 0 since x = 0

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 51 54

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (141)

k =

radicI

m(142)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (143)

= I1 + I2 + + In (144)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 52 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

兩圓盤複合體

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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結結結論論論

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 54 54

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結論

轉動慣量或慣性矩為物體對旋轉運動的慣性而所謂慣性乃物體持續維持不變的行為也就是定義慣性矩可表達物體對本身旋轉運動的維持或持續性好壞更簡單的說慣性矩越大之物體其維持旋轉運動趨勢的能力越大例如惰輪儲能裝置惰輪之慣性矩越大當其轉動以後其維持旋轉運動趨勢的能力越大

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 55 54

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  • 轉動慣量或慣性矩(Moment of Inertia)
  • 面積慣性矩(Moment of Inertia for Area)
    • 面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)
    • 面積慣性矩之迴轉半徑(Radius of Gyration for Area)
      • 複合面積之慣性矩(Moment of Inertia for Composite Area)
      • 面積之慣性積(Product of Inertia for Area)
        • 平行軸定理(Parallel-Axis Theorem)
          • 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)
            • 主軸慣性矩(Principal Moment of Inertia)
              • 慣性矩之莫爾圓(Mohrs Circle for Moment of Inertia)
              • 質量慣性矩(Mass Moment of Inertia)
                • 平行軸定理(Parallel-Axis Theorem)
Page 58: 轉動慣量 Moments of Inertia - cct.me.ntut.edu.twcct.me.ntut.edu.tw/ccteducation/chchting/aiahtm/statics/lecture... · 轉動慣量或慣性矩 I. 轉動慣量也稱慣性矩(Moment

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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慣慣慣性性性矩矩矩之之之莫莫莫爾爾爾圓圓圓Mohrrsquos Circle for Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 44 54

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慣性矩之莫爾圓

由於Iu =

Ix + Iy2

+Ix minus Iy

2cos2θ minus Ixy sin2θ (125)

Iv =Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (126)

Iuv =Ix minus Iy

2sin2θ + Ixycos2θ (127)

將式(125)與式(127)各自分別平方並相加可得

(Iu minusIx + Iy

2)2 + I 2uv = (

Ix minus Iy2

)2 + I 2xy (128)

其中Ix Iy Ixy均為常數式(128)可進一步寫成

(Iu minus a)2 + I 2uv = R2 (129)

其中a =Ix+Iy2 R =

radic(IxminusIy2 )2 + I 2xycos2θp1 =

aRsin2θp1 =

minusIxyR 2θp1為OA轉向I軸方向即2θp1為

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 45 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 46 54

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解題技巧

首先確定轉動系統的Ix Iy Ixy接著算出莫爾圓之圓心座標a及半徑R並畫出莫爾圓再由莫爾圓上點出A(Ix Ixy )並由OA轉向I軸找出兩倍夾角2θp1及2θp2其中θp1及θp2為主軸(Principle Axis)與Ix軸的夾角由於主軸(Principle Axis)為極值之軸Imax = Iu及Imin = Iv決定θp1及θp2何者為Imax = Iu與Ix軸的夾角之方法乃比較Ix及Iy之大小Imax = Iu會靠近Ix及Iy之較大者也就是Imax = Iu與Ix及Iy之較大者的夾角比較小

a =Ix + Iy

2(130)

R =

radic(Ix minus Iy

2)2 + I 2xy (131)

2θp2 = π + 2θp1 θp2 minus θp1 =π

2(132)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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質質質量量量慣慣慣性性性矩矩矩Mass Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 48 54

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質量慣性矩

τ = Iα (133)

其中τ為旋轉扭矩I為質量慣性矩α為旋轉角速度

I =

ˆmr2dm (134)

=

ˆVr2ρdV (135)

= ρ

ˆVr2dV (136)

其中r為旋轉半徑或轉軸到旋轉體的距離

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 49 54

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 50 54

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平行軸定理

當通過物體質心(G)並在一轉軸上的質量慣性矩(IG )為已知時可透過座標軸轉換計算物體之質量慣性矩(I)

I =

ˆmr2dm (137)

=

ˆm

[(d + x prime)2 + y prime2]dm (138)

=

ˆm

(x prime2 + y prime2)dm + 2d

ˆmx primedm + d2

ˆmdm (139)

I = IG + md2 (140)

其中IG為在zrsquo軸上通過質心的質量慣性矩m為質量d為通過質心之軸zrsquo並平行原軸z之距離

acutem x primedm = x

acutem dm = 0 since x = 0

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 51 54

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (141)

k =

radicI

m(142)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (143)

= I1 + I2 + + In (144)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 52 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

兩圓盤複合體

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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結結結論論論

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 54 54

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結論

轉動慣量或慣性矩為物體對旋轉運動的慣性而所謂慣性乃物體持續維持不變的行為也就是定義慣性矩可表達物體對本身旋轉運動的維持或持續性好壞更簡單的說慣性矩越大之物體其維持旋轉運動趨勢的能力越大例如惰輪儲能裝置惰輪之慣性矩越大當其轉動以後其維持旋轉運動趨勢的能力越大

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 55 54

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  • 轉動慣量或慣性矩(Moment of Inertia)
  • 面積慣性矩(Moment of Inertia for Area)
    • 面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)
    • 面積慣性矩之迴轉半徑(Radius of Gyration for Area)
      • 複合面積之慣性矩(Moment of Inertia for Composite Area)
      • 面積之慣性積(Product of Inertia for Area)
        • 平行軸定理(Parallel-Axis Theorem)
          • 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)
            • 主軸慣性矩(Principal Moment of Inertia)
              • 慣性矩之莫爾圓(Mohrs Circle for Moment of Inertia)
              • 質量慣性矩(Mass Moment of Inertia)
                • 平行軸定理(Parallel-Axis Theorem)
Page 59: 轉動慣量 Moments of Inertia - cct.me.ntut.edu.twcct.me.ntut.edu.tw/ccteducation/chchting/aiahtm/statics/lecture... · 轉動慣量或慣性矩 I. 轉動慣量也稱慣性矩(Moment

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 43 54

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慣慣慣性性性矩矩矩之之之莫莫莫爾爾爾圓圓圓Mohrrsquos Circle for Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 44 54

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慣性矩之莫爾圓

由於Iu =

Ix + Iy2

+Ix minus Iy

2cos2θ minus Ixy sin2θ (125)

Iv =Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (126)

Iuv =Ix minus Iy

2sin2θ + Ixycos2θ (127)

將式(125)與式(127)各自分別平方並相加可得

(Iu minusIx + Iy

2)2 + I 2uv = (

Ix minus Iy2

)2 + I 2xy (128)

其中Ix Iy Ixy均為常數式(128)可進一步寫成

(Iu minus a)2 + I 2uv = R2 (129)

其中a =Ix+Iy2 R =

radic(IxminusIy2 )2 + I 2xycos2θp1 =

aRsin2θp1 =

minusIxyR 2θp1為OA轉向I軸方向即2θp1為

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 45 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 46 54

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解題技巧

首先確定轉動系統的Ix Iy Ixy接著算出莫爾圓之圓心座標a及半徑R並畫出莫爾圓再由莫爾圓上點出A(Ix Ixy )並由OA轉向I軸找出兩倍夾角2θp1及2θp2其中θp1及θp2為主軸(Principle Axis)與Ix軸的夾角由於主軸(Principle Axis)為極值之軸Imax = Iu及Imin = Iv決定θp1及θp2何者為Imax = Iu與Ix軸的夾角之方法乃比較Ix及Iy之大小Imax = Iu會靠近Ix及Iy之較大者也就是Imax = Iu與Ix及Iy之較大者的夾角比較小

a =Ix + Iy

2(130)

R =

radic(Ix minus Iy

2)2 + I 2xy (131)

2θp2 = π + 2θp1 θp2 minus θp1 =π

2(132)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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質質質量量量慣慣慣性性性矩矩矩Mass Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 48 54

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質量慣性矩

τ = Iα (133)

其中τ為旋轉扭矩I為質量慣性矩α為旋轉角速度

I =

ˆmr2dm (134)

=

ˆVr2ρdV (135)

= ρ

ˆVr2dV (136)

其中r為旋轉半徑或轉軸到旋轉體的距離

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 49 54

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 50 54

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平行軸定理

當通過物體質心(G)並在一轉軸上的質量慣性矩(IG )為已知時可透過座標軸轉換計算物體之質量慣性矩(I)

I =

ˆmr2dm (137)

=

ˆm

[(d + x prime)2 + y prime2]dm (138)

=

ˆm

(x prime2 + y prime2)dm + 2d

ˆmx primedm + d2

ˆmdm (139)

I = IG + md2 (140)

其中IG為在zrsquo軸上通過質心的質量慣性矩m為質量d為通過質心之軸zrsquo並平行原軸z之距離

acutem x primedm = x

acutem dm = 0 since x = 0

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 51 54

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (141)

k =

radicI

m(142)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (143)

= I1 + I2 + + In (144)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 52 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

兩圓盤複合體

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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結結結論論論

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 54 54

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結論

轉動慣量或慣性矩為物體對旋轉運動的慣性而所謂慣性乃物體持續維持不變的行為也就是定義慣性矩可表達物體對本身旋轉運動的維持或持續性好壞更簡單的說慣性矩越大之物體其維持旋轉運動趨勢的能力越大例如惰輪儲能裝置惰輪之慣性矩越大當其轉動以後其維持旋轉運動趨勢的能力越大

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 55 54

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  • 轉動慣量或慣性矩(Moment of Inertia)
  • 面積慣性矩(Moment of Inertia for Area)
    • 面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)
    • 面積慣性矩之迴轉半徑(Radius of Gyration for Area)
      • 複合面積之慣性矩(Moment of Inertia for Composite Area)
      • 面積之慣性積(Product of Inertia for Area)
        • 平行軸定理(Parallel-Axis Theorem)
          • 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)
            • 主軸慣性矩(Principal Moment of Inertia)
              • 慣性矩之莫爾圓(Mohrs Circle for Moment of Inertia)
              • 質量慣性矩(Mass Moment of Inertia)
                • 平行軸定理(Parallel-Axis Theorem)
Page 60: 轉動慣量 Moments of Inertia - cct.me.ntut.edu.twcct.me.ntut.edu.tw/ccteducation/chchting/aiahtm/statics/lecture... · 轉動慣量或慣性矩 I. 轉動慣量也稱慣性矩(Moment

慣慣慣性性性矩矩矩之之之莫莫莫爾爾爾圓圓圓Mohrrsquos Circle for Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 44 54

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慣性矩之莫爾圓

由於Iu =

Ix + Iy2

+Ix minus Iy

2cos2θ minus Ixy sin2θ (125)

Iv =Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (126)

Iuv =Ix minus Iy

2sin2θ + Ixycos2θ (127)

將式(125)與式(127)各自分別平方並相加可得

(Iu minusIx + Iy

2)2 + I 2uv = (

Ix minus Iy2

)2 + I 2xy (128)

其中Ix Iy Ixy均為常數式(128)可進一步寫成

(Iu minus a)2 + I 2uv = R2 (129)

其中a =Ix+Iy2 R =

radic(IxminusIy2 )2 + I 2xycos2θp1 =

aRsin2θp1 =

minusIxyR 2θp1為OA轉向I軸方向即2θp1為

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 45 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 46 54

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解題技巧

首先確定轉動系統的Ix Iy Ixy接著算出莫爾圓之圓心座標a及半徑R並畫出莫爾圓再由莫爾圓上點出A(Ix Ixy )並由OA轉向I軸找出兩倍夾角2θp1及2θp2其中θp1及θp2為主軸(Principle Axis)與Ix軸的夾角由於主軸(Principle Axis)為極值之軸Imax = Iu及Imin = Iv決定θp1及θp2何者為Imax = Iu與Ix軸的夾角之方法乃比較Ix及Iy之大小Imax = Iu會靠近Ix及Iy之較大者也就是Imax = Iu與Ix及Iy之較大者的夾角比較小

a =Ix + Iy

2(130)

R =

radic(Ix minus Iy

2)2 + I 2xy (131)

2θp2 = π + 2θp1 θp2 minus θp1 =π

2(132)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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質質質量量量慣慣慣性性性矩矩矩Mass Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 48 54

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質量慣性矩

τ = Iα (133)

其中τ為旋轉扭矩I為質量慣性矩α為旋轉角速度

I =

ˆmr2dm (134)

=

ˆVr2ρdV (135)

= ρ

ˆVr2dV (136)

其中r為旋轉半徑或轉軸到旋轉體的距離

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 49 54

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 50 54

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平行軸定理

當通過物體質心(G)並在一轉軸上的質量慣性矩(IG )為已知時可透過座標軸轉換計算物體之質量慣性矩(I)

I =

ˆmr2dm (137)

=

ˆm

[(d + x prime)2 + y prime2]dm (138)

=

ˆm

(x prime2 + y prime2)dm + 2d

ˆmx primedm + d2

ˆmdm (139)

I = IG + md2 (140)

其中IG為在zrsquo軸上通過質心的質量慣性矩m為質量d為通過質心之軸zrsquo並平行原軸z之距離

acutem x primedm = x

acutem dm = 0 since x = 0

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 51 54

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (141)

k =

radicI

m(142)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (143)

= I1 + I2 + + In (144)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 52 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

兩圓盤複合體

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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結結結論論論

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 54 54

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結論

轉動慣量或慣性矩為物體對旋轉運動的慣性而所謂慣性乃物體持續維持不變的行為也就是定義慣性矩可表達物體對本身旋轉運動的維持或持續性好壞更簡單的說慣性矩越大之物體其維持旋轉運動趨勢的能力越大例如惰輪儲能裝置惰輪之慣性矩越大當其轉動以後其維持旋轉運動趨勢的能力越大

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 55 54

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  • 轉動慣量或慣性矩(Moment of Inertia)
  • 面積慣性矩(Moment of Inertia for Area)
    • 面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)
    • 面積慣性矩之迴轉半徑(Radius of Gyration for Area)
      • 複合面積之慣性矩(Moment of Inertia for Composite Area)
      • 面積之慣性積(Product of Inertia for Area)
        • 平行軸定理(Parallel-Axis Theorem)
          • 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)
            • 主軸慣性矩(Principal Moment of Inertia)
              • 慣性矩之莫爾圓(Mohrs Circle for Moment of Inertia)
              • 質量慣性矩(Mass Moment of Inertia)
                • 平行軸定理(Parallel-Axis Theorem)
Page 61: 轉動慣量 Moments of Inertia - cct.me.ntut.edu.twcct.me.ntut.edu.tw/ccteducation/chchting/aiahtm/statics/lecture... · 轉動慣量或慣性矩 I. 轉動慣量也稱慣性矩(Moment

慣性矩之莫爾圓

由於Iu =

Ix + Iy2

+Ix minus Iy

2cos2θ minus Ixy sin2θ (125)

Iv =Ix + Iy

2minus Ix minus Iy

2cos2θ + Ixy sin2θ (126)

Iuv =Ix minus Iy

2sin2θ + Ixycos2θ (127)

將式(125)與式(127)各自分別平方並相加可得

(Iu minusIx + Iy

2)2 + I 2uv = (

Ix minus Iy2

)2 + I 2xy (128)

其中Ix Iy Ixy均為常數式(128)可進一步寫成

(Iu minus a)2 + I 2uv = R2 (129)

其中a =Ix+Iy2 R =

radic(IxminusIy2 )2 + I 2xycos2θp1 =

aRsin2θp1 =

minusIxyR 2θp1為OA轉向I軸方向即2θp1為

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 45 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 46 54

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解題技巧

首先確定轉動系統的Ix Iy Ixy接著算出莫爾圓之圓心座標a及半徑R並畫出莫爾圓再由莫爾圓上點出A(Ix Ixy )並由OA轉向I軸找出兩倍夾角2θp1及2θp2其中θp1及θp2為主軸(Principle Axis)與Ix軸的夾角由於主軸(Principle Axis)為極值之軸Imax = Iu及Imin = Iv決定θp1及θp2何者為Imax = Iu與Ix軸的夾角之方法乃比較Ix及Iy之大小Imax = Iu會靠近Ix及Iy之較大者也就是Imax = Iu與Ix及Iy之較大者的夾角比較小

a =Ix + Iy

2(130)

R =

radic(Ix minus Iy

2)2 + I 2xy (131)

2θp2 = π + 2θp1 θp2 minus θp1 =π

2(132)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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質質質量量量慣慣慣性性性矩矩矩Mass Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 48 54

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質量慣性矩

τ = Iα (133)

其中τ為旋轉扭矩I為質量慣性矩α為旋轉角速度

I =

ˆmr2dm (134)

=

ˆVr2ρdV (135)

= ρ

ˆVr2dV (136)

其中r為旋轉半徑或轉軸到旋轉體的距離

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 49 54

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 50 54

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平行軸定理

當通過物體質心(G)並在一轉軸上的質量慣性矩(IG )為已知時可透過座標軸轉換計算物體之質量慣性矩(I)

I =

ˆmr2dm (137)

=

ˆm

[(d + x prime)2 + y prime2]dm (138)

=

ˆm

(x prime2 + y prime2)dm + 2d

ˆmx primedm + d2

ˆmdm (139)

I = IG + md2 (140)

其中IG為在zrsquo軸上通過質心的質量慣性矩m為質量d為通過質心之軸zrsquo並平行原軸z之距離

acutem x primedm = x

acutem dm = 0 since x = 0

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 51 54

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (141)

k =

radicI

m(142)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (143)

= I1 + I2 + + In (144)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 52 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

兩圓盤複合體

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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結結結論論論

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 54 54

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結論

轉動慣量或慣性矩為物體對旋轉運動的慣性而所謂慣性乃物體持續維持不變的行為也就是定義慣性矩可表達物體對本身旋轉運動的維持或持續性好壞更簡單的說慣性矩越大之物體其維持旋轉運動趨勢的能力越大例如惰輪儲能裝置惰輪之慣性矩越大當其轉動以後其維持旋轉運動趨勢的能力越大

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 55 54

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Media File (applicationx-shockwave-flash)
  • 轉動慣量或慣性矩(Moment of Inertia)
  • 面積慣性矩(Moment of Inertia for Area)
    • 面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)
    • 面積慣性矩之迴轉半徑(Radius of Gyration for Area)
      • 複合面積之慣性矩(Moment of Inertia for Composite Area)
      • 面積之慣性積(Product of Inertia for Area)
        • 平行軸定理(Parallel-Axis Theorem)
          • 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)
            • 主軸慣性矩(Principal Moment of Inertia)
              • 慣性矩之莫爾圓(Mohrs Circle for Moment of Inertia)
              • 質量慣性矩(Mass Moment of Inertia)
                • 平行軸定理(Parallel-Axis Theorem)
Page 62: 轉動慣量 Moments of Inertia - cct.me.ntut.edu.twcct.me.ntut.edu.tw/ccteducation/chchting/aiahtm/statics/lecture... · 轉動慣量或慣性矩 I. 轉動慣量也稱慣性矩(Moment

課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 46 54

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Media File (applicationx-shockwave-flash)

解題技巧

首先確定轉動系統的Ix Iy Ixy接著算出莫爾圓之圓心座標a及半徑R並畫出莫爾圓再由莫爾圓上點出A(Ix Ixy )並由OA轉向I軸找出兩倍夾角2θp1及2θp2其中θp1及θp2為主軸(Principle Axis)與Ix軸的夾角由於主軸(Principle Axis)為極值之軸Imax = Iu及Imin = Iv決定θp1及θp2何者為Imax = Iu與Ix軸的夾角之方法乃比較Ix及Iy之大小Imax = Iu會靠近Ix及Iy之較大者也就是Imax = Iu與Ix及Iy之較大者的夾角比較小

a =Ix + Iy

2(130)

R =

radic(Ix minus Iy

2)2 + I 2xy (131)

2θp2 = π + 2θp1 θp2 minus θp1 =π

2(132)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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Media File (applicationx-shockwave-flash)

質質質量量量慣慣慣性性性矩矩矩Mass Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 48 54

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質量慣性矩

τ = Iα (133)

其中τ為旋轉扭矩I為質量慣性矩α為旋轉角速度

I =

ˆmr2dm (134)

=

ˆVr2ρdV (135)

= ρ

ˆVr2dV (136)

其中r為旋轉半徑或轉軸到旋轉體的距離

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 49 54

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 50 54

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平行軸定理

當通過物體質心(G)並在一轉軸上的質量慣性矩(IG )為已知時可透過座標軸轉換計算物體之質量慣性矩(I)

I =

ˆmr2dm (137)

=

ˆm

[(d + x prime)2 + y prime2]dm (138)

=

ˆm

(x prime2 + y prime2)dm + 2d

ˆmx primedm + d2

ˆmdm (139)

I = IG + md2 (140)

其中IG為在zrsquo軸上通過質心的質量慣性矩m為質量d為通過質心之軸zrsquo並平行原軸z之距離

acutem x primedm = x

acutem dm = 0 since x = 0

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 51 54

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (141)

k =

radicI

m(142)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (143)

= I1 + I2 + + In (144)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 52 54

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Media File (applicationx-shockwave-flash)

課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

兩圓盤複合體

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

結結結論論論

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 54 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

結論

轉動慣量或慣性矩為物體對旋轉運動的慣性而所謂慣性乃物體持續維持不變的行為也就是定義慣性矩可表達物體對本身旋轉運動的維持或持續性好壞更簡單的說慣性矩越大之物體其維持旋轉運動趨勢的能力越大例如惰輪儲能裝置惰輪之慣性矩越大當其轉動以後其維持旋轉運動趨勢的能力越大

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 55 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
  • 轉動慣量或慣性矩(Moment of Inertia)
  • 面積慣性矩(Moment of Inertia for Area)
    • 面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)
    • 面積慣性矩之迴轉半徑(Radius of Gyration for Area)
      • 複合面積之慣性矩(Moment of Inertia for Composite Area)
      • 面積之慣性積(Product of Inertia for Area)
        • 平行軸定理(Parallel-Axis Theorem)
          • 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)
            • 主軸慣性矩(Principal Moment of Inertia)
              • 慣性矩之莫爾圓(Mohrs Circle for Moment of Inertia)
              • 質量慣性矩(Mass Moment of Inertia)
                • 平行軸定理(Parallel-Axis Theorem)
Page 63: 轉動慣量 Moments of Inertia - cct.me.ntut.edu.twcct.me.ntut.edu.tw/ccteducation/chchting/aiahtm/statics/lecture... · 轉動慣量或慣性矩 I. 轉動慣量也稱慣性矩(Moment

解題技巧

首先確定轉動系統的Ix Iy Ixy接著算出莫爾圓之圓心座標a及半徑R並畫出莫爾圓再由莫爾圓上點出A(Ix Ixy )並由OA轉向I軸找出兩倍夾角2θp1及2θp2其中θp1及θp2為主軸(Principle Axis)與Ix軸的夾角由於主軸(Principle Axis)為極值之軸Imax = Iu及Imin = Iv決定θp1及θp2何者為Imax = Iu與Ix軸的夾角之方法乃比較Ix及Iy之大小Imax = Iu會靠近Ix及Iy之較大者也就是Imax = Iu與Ix及Iy之較大者的夾角比較小

a =Ix + Iy

2(130)

R =

radic(Ix minus Iy

2)2 + I 2xy (131)

2θp2 = π + 2θp1 θp2 minus θp1 =π

2(132)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

質質質量量量慣慣慣性性性矩矩矩Mass Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 48 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

質量慣性矩

τ = Iα (133)

其中τ為旋轉扭矩I為質量慣性矩α為旋轉角速度

I =

ˆmr2dm (134)

=

ˆVr2ρdV (135)

= ρ

ˆVr2dV (136)

其中r為旋轉半徑或轉軸到旋轉體的距離

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 49 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 50 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

平行軸定理

當通過物體質心(G)並在一轉軸上的質量慣性矩(IG )為已知時可透過座標軸轉換計算物體之質量慣性矩(I)

I =

ˆmr2dm (137)

=

ˆm

[(d + x prime)2 + y prime2]dm (138)

=

ˆm

(x prime2 + y prime2)dm + 2d

ˆmx primedm + d2

ˆmdm (139)

I = IG + md2 (140)

其中IG為在zrsquo軸上通過質心的質量慣性矩m為質量d為通過質心之軸zrsquo並平行原軸z之距離

acutem x primedm = x

acutem dm = 0 since x = 0

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 51 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (141)

k =

radicI

m(142)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (143)

= I1 + I2 + + In (144)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 52 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

兩圓盤複合體

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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結結結論論論

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 54 54

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結論

轉動慣量或慣性矩為物體對旋轉運動的慣性而所謂慣性乃物體持續維持不變的行為也就是定義慣性矩可表達物體對本身旋轉運動的維持或持續性好壞更簡單的說慣性矩越大之物體其維持旋轉運動趨勢的能力越大例如惰輪儲能裝置惰輪之慣性矩越大當其轉動以後其維持旋轉運動趨勢的能力越大

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 55 54

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  • 轉動慣量或慣性矩(Moment of Inertia)
  • 面積慣性矩(Moment of Inertia for Area)
    • 面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)
    • 面積慣性矩之迴轉半徑(Radius of Gyration for Area)
      • 複合面積之慣性矩(Moment of Inertia for Composite Area)
      • 面積之慣性積(Product of Inertia for Area)
        • 平行軸定理(Parallel-Axis Theorem)
          • 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)
            • 主軸慣性矩(Principal Moment of Inertia)
              • 慣性矩之莫爾圓(Mohrs Circle for Moment of Inertia)
              • 質量慣性矩(Mass Moment of Inertia)
                • 平行軸定理(Parallel-Axis Theorem)
Page 64: 轉動慣量 Moments of Inertia - cct.me.ntut.edu.twcct.me.ntut.edu.tw/ccteducation/chchting/aiahtm/statics/lecture... · 轉動慣量或慣性矩 I. 轉動慣量也稱慣性矩(Moment

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

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質質質量量量慣慣慣性性性矩矩矩Mass Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 48 54

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質量慣性矩

τ = Iα (133)

其中τ為旋轉扭矩I為質量慣性矩α為旋轉角速度

I =

ˆmr2dm (134)

=

ˆVr2ρdV (135)

= ρ

ˆVr2dV (136)

其中r為旋轉半徑或轉軸到旋轉體的距離

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 49 54

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 50 54

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平行軸定理

當通過物體質心(G)並在一轉軸上的質量慣性矩(IG )為已知時可透過座標軸轉換計算物體之質量慣性矩(I)

I =

ˆmr2dm (137)

=

ˆm

[(d + x prime)2 + y prime2]dm (138)

=

ˆm

(x prime2 + y prime2)dm + 2d

ˆmx primedm + d2

ˆmdm (139)

I = IG + md2 (140)

其中IG為在zrsquo軸上通過質心的質量慣性矩m為質量d為通過質心之軸zrsquo並平行原軸z之距離

acutem x primedm = x

acutem dm = 0 since x = 0

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 51 54

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (141)

k =

radicI

m(142)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (143)

= I1 + I2 + + In (144)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 52 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

兩圓盤複合體

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

結結結論論論

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 54 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

結論

轉動慣量或慣性矩為物體對旋轉運動的慣性而所謂慣性乃物體持續維持不變的行為也就是定義慣性矩可表達物體對本身旋轉運動的維持或持續性好壞更簡單的說慣性矩越大之物體其維持旋轉運動趨勢的能力越大例如惰輪儲能裝置惰輪之慣性矩越大當其轉動以後其維持旋轉運動趨勢的能力越大

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 55 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
  • 轉動慣量或慣性矩(Moment of Inertia)
  • 面積慣性矩(Moment of Inertia for Area)
    • 面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)
    • 面積慣性矩之迴轉半徑(Radius of Gyration for Area)
      • 複合面積之慣性矩(Moment of Inertia for Composite Area)
      • 面積之慣性積(Product of Inertia for Area)
        • 平行軸定理(Parallel-Axis Theorem)
          • 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)
            • 主軸慣性矩(Principal Moment of Inertia)
              • 慣性矩之莫爾圓(Mohrs Circle for Moment of Inertia)
              • 質量慣性矩(Mass Moment of Inertia)
                • 平行軸定理(Parallel-Axis Theorem)
Page 65: 轉動慣量 Moments of Inertia - cct.me.ntut.edu.twcct.me.ntut.edu.tw/ccteducation/chchting/aiahtm/statics/lecture... · 轉動慣量或慣性矩 I. 轉動慣量也稱慣性矩(Moment

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 47 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

質質質量量量慣慣慣性性性矩矩矩Mass Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 48 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

質量慣性矩

τ = Iα (133)

其中τ為旋轉扭矩I為質量慣性矩α為旋轉角速度

I =

ˆmr2dm (134)

=

ˆVr2ρdV (135)

= ρ

ˆVr2dV (136)

其中r為旋轉半徑或轉軸到旋轉體的距離

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 49 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 50 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

平行軸定理

當通過物體質心(G)並在一轉軸上的質量慣性矩(IG )為已知時可透過座標軸轉換計算物體之質量慣性矩(I)

I =

ˆmr2dm (137)

=

ˆm

[(d + x prime)2 + y prime2]dm (138)

=

ˆm

(x prime2 + y prime2)dm + 2d

ˆmx primedm + d2

ˆmdm (139)

I = IG + md2 (140)

其中IG為在zrsquo軸上通過質心的質量慣性矩m為質量d為通過質心之軸zrsquo並平行原軸z之距離

acutem x primedm = x

acutem dm = 0 since x = 0

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 51 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (141)

k =

radicI

m(142)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (143)

= I1 + I2 + + In (144)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 52 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

兩圓盤複合體

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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結結結論論論

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 54 54

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結論

轉動慣量或慣性矩為物體對旋轉運動的慣性而所謂慣性乃物體持續維持不變的行為也就是定義慣性矩可表達物體對本身旋轉運動的維持或持續性好壞更簡單的說慣性矩越大之物體其維持旋轉運動趨勢的能力越大例如惰輪儲能裝置惰輪之慣性矩越大當其轉動以後其維持旋轉運動趨勢的能力越大

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 55 54

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  • 轉動慣量或慣性矩(Moment of Inertia)
  • 面積慣性矩(Moment of Inertia for Area)
    • 面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)
    • 面積慣性矩之迴轉半徑(Radius of Gyration for Area)
      • 複合面積之慣性矩(Moment of Inertia for Composite Area)
      • 面積之慣性積(Product of Inertia for Area)
        • 平行軸定理(Parallel-Axis Theorem)
          • 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)
            • 主軸慣性矩(Principal Moment of Inertia)
              • 慣性矩之莫爾圓(Mohrs Circle for Moment of Inertia)
              • 質量慣性矩(Mass Moment of Inertia)
                • 平行軸定理(Parallel-Axis Theorem)
Page 66: 轉動慣量 Moments of Inertia - cct.me.ntut.edu.twcct.me.ntut.edu.tw/ccteducation/chchting/aiahtm/statics/lecture... · 轉動慣量或慣性矩 I. 轉動慣量也稱慣性矩(Moment

質質質量量量慣慣慣性性性矩矩矩Mass Moment of Inertia

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 48 54

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質量慣性矩

τ = Iα (133)

其中τ為旋轉扭矩I為質量慣性矩α為旋轉角速度

I =

ˆmr2dm (134)

=

ˆVr2ρdV (135)

= ρ

ˆVr2dV (136)

其中r為旋轉半徑或轉軸到旋轉體的距離

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 49 54

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平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 50 54

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平行軸定理

當通過物體質心(G)並在一轉軸上的質量慣性矩(IG )為已知時可透過座標軸轉換計算物體之質量慣性矩(I)

I =

ˆmr2dm (137)

=

ˆm

[(d + x prime)2 + y prime2]dm (138)

=

ˆm

(x prime2 + y prime2)dm + 2d

ˆmx primedm + d2

ˆmdm (139)

I = IG + md2 (140)

其中IG為在zrsquo軸上通過質心的質量慣性矩m為質量d為通過質心之軸zrsquo並平行原軸z之距離

acutem x primedm = x

acutem dm = 0 since x = 0

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 51 54

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迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (141)

k =

radicI

m(142)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (143)

= I1 + I2 + + In (144)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 52 54

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課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

兩圓盤複合體

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

結結結論論論

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 54 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

結論

轉動慣量或慣性矩為物體對旋轉運動的慣性而所謂慣性乃物體持續維持不變的行為也就是定義慣性矩可表達物體對本身旋轉運動的維持或持續性好壞更簡單的說慣性矩越大之物體其維持旋轉運動趨勢的能力越大例如惰輪儲能裝置惰輪之慣性矩越大當其轉動以後其維持旋轉運動趨勢的能力越大

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 55 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
  • 轉動慣量或慣性矩(Moment of Inertia)
  • 面積慣性矩(Moment of Inertia for Area)
    • 面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)
    • 面積慣性矩之迴轉半徑(Radius of Gyration for Area)
      • 複合面積之慣性矩(Moment of Inertia for Composite Area)
      • 面積之慣性積(Product of Inertia for Area)
        • 平行軸定理(Parallel-Axis Theorem)
          • 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)
            • 主軸慣性矩(Principal Moment of Inertia)
              • 慣性矩之莫爾圓(Mohrs Circle for Moment of Inertia)
              • 質量慣性矩(Mass Moment of Inertia)
                • 平行軸定理(Parallel-Axis Theorem)
Page 67: 轉動慣量 Moments of Inertia - cct.me.ntut.edu.twcct.me.ntut.edu.tw/ccteducation/chchting/aiahtm/statics/lecture... · 轉動慣量或慣性矩 I. 轉動慣量也稱慣性矩(Moment

質量慣性矩

τ = Iα (133)

其中τ為旋轉扭矩I為質量慣性矩α為旋轉角速度

I =

ˆmr2dm (134)

=

ˆVr2ρdV (135)

= ρ

ˆVr2dV (136)

其中r為旋轉半徑或轉軸到旋轉體的距離

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 49 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 50 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

平行軸定理

當通過物體質心(G)並在一轉軸上的質量慣性矩(IG )為已知時可透過座標軸轉換計算物體之質量慣性矩(I)

I =

ˆmr2dm (137)

=

ˆm

[(d + x prime)2 + y prime2]dm (138)

=

ˆm

(x prime2 + y prime2)dm + 2d

ˆmx primedm + d2

ˆmdm (139)

I = IG + md2 (140)

其中IG為在zrsquo軸上通過質心的質量慣性矩m為質量d為通過質心之軸zrsquo並平行原軸z之距離

acutem x primedm = x

acutem dm = 0 since x = 0

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 51 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (141)

k =

radicI

m(142)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (143)

= I1 + I2 + + In (144)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 52 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

兩圓盤複合體

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

結結結論論論

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 54 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

結論

轉動慣量或慣性矩為物體對旋轉運動的慣性而所謂慣性乃物體持續維持不變的行為也就是定義慣性矩可表達物體對本身旋轉運動的維持或持續性好壞更簡單的說慣性矩越大之物體其維持旋轉運動趨勢的能力越大例如惰輪儲能裝置惰輪之慣性矩越大當其轉動以後其維持旋轉運動趨勢的能力越大

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 55 54

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Media File (applicationx-shockwave-flash)
  • 轉動慣量或慣性矩(Moment of Inertia)
  • 面積慣性矩(Moment of Inertia for Area)
    • 面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)
    • 面積慣性矩之迴轉半徑(Radius of Gyration for Area)
      • 複合面積之慣性矩(Moment of Inertia for Composite Area)
      • 面積之慣性積(Product of Inertia for Area)
        • 平行軸定理(Parallel-Axis Theorem)
          • 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)
            • 主軸慣性矩(Principal Moment of Inertia)
              • 慣性矩之莫爾圓(Mohrs Circle for Moment of Inertia)
              • 質量慣性矩(Mass Moment of Inertia)
                • 平行軸定理(Parallel-Axis Theorem)
Page 68: 轉動慣量 Moments of Inertia - cct.me.ntut.edu.twcct.me.ntut.edu.tw/ccteducation/chchting/aiahtm/statics/lecture... · 轉動慣量或慣性矩 I. 轉動慣量也稱慣性矩(Moment

平平平行行行軸軸軸定定定理理理Parallel-Axis Theorem

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 50 54

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Media File (applicationx-shockwave-flash)

平行軸定理

當通過物體質心(G)並在一轉軸上的質量慣性矩(IG )為已知時可透過座標軸轉換計算物體之質量慣性矩(I)

I =

ˆmr2dm (137)

=

ˆm

[(d + x prime)2 + y prime2]dm (138)

=

ˆm

(x prime2 + y prime2)dm + 2d

ˆmx primedm + d2

ˆmdm (139)

I = IG + md2 (140)

其中IG為在zrsquo軸上通過質心的質量慣性矩m為質量d為通過質心之軸zrsquo並平行原軸z之距離

acutem x primedm = x

acutem dm = 0 since x = 0

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 51 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (141)

k =

radicI

m(142)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (143)

= I1 + I2 + + In (144)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 52 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

兩圓盤複合體

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Media File (applicationx-shockwave-flash)

結結結論論論

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 54 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

結論

轉動慣量或慣性矩為物體對旋轉運動的慣性而所謂慣性乃物體持續維持不變的行為也就是定義慣性矩可表達物體對本身旋轉運動的維持或持續性好壞更簡單的說慣性矩越大之物體其維持旋轉運動趨勢的能力越大例如惰輪儲能裝置惰輪之慣性矩越大當其轉動以後其維持旋轉運動趨勢的能力越大

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 55 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
  • 轉動慣量或慣性矩(Moment of Inertia)
  • 面積慣性矩(Moment of Inertia for Area)
    • 面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)
    • 面積慣性矩之迴轉半徑(Radius of Gyration for Area)
      • 複合面積之慣性矩(Moment of Inertia for Composite Area)
      • 面積之慣性積(Product of Inertia for Area)
        • 平行軸定理(Parallel-Axis Theorem)
          • 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)
            • 主軸慣性矩(Principal Moment of Inertia)
              • 慣性矩之莫爾圓(Mohrs Circle for Moment of Inertia)
              • 質量慣性矩(Mass Moment of Inertia)
                • 平行軸定理(Parallel-Axis Theorem)
Page 69: 轉動慣量 Moments of Inertia - cct.me.ntut.edu.twcct.me.ntut.edu.tw/ccteducation/chchting/aiahtm/statics/lecture... · 轉動慣量或慣性矩 I. 轉動慣量也稱慣性矩(Moment

平行軸定理

當通過物體質心(G)並在一轉軸上的質量慣性矩(IG )為已知時可透過座標軸轉換計算物體之質量慣性矩(I)

I =

ˆmr2dm (137)

=

ˆm

[(d + x prime)2 + y prime2]dm (138)

=

ˆm

(x prime2 + y prime2)dm + 2d

ˆmx primedm + d2

ˆmdm (139)

I = IG + md2 (140)

其中IG為在zrsquo軸上通過質心的質量慣性矩m為質量d為通過質心之軸zrsquo並平行原軸z之距離

acutem x primedm = x

acutem dm = 0 since x = 0

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 51 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (141)

k =

radicI

m(142)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (143)

= I1 + I2 + + In (144)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 52 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

兩圓盤複合體

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

結結結論論論

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 54 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

結論

轉動慣量或慣性矩為物體對旋轉運動的慣性而所謂慣性乃物體持續維持不變的行為也就是定義慣性矩可表達物體對本身旋轉運動的維持或持續性好壞更簡單的說慣性矩越大之物體其維持旋轉運動趨勢的能力越大例如惰輪儲能裝置惰輪之慣性矩越大當其轉動以後其維持旋轉運動趨勢的能力越大

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 55 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
  • 轉動慣量或慣性矩(Moment of Inertia)
  • 面積慣性矩(Moment of Inertia for Area)
    • 面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)
    • 面積慣性矩之迴轉半徑(Radius of Gyration for Area)
      • 複合面積之慣性矩(Moment of Inertia for Composite Area)
      • 面積之慣性積(Product of Inertia for Area)
        • 平行軸定理(Parallel-Axis Theorem)
          • 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)
            • 主軸慣性矩(Principal Moment of Inertia)
              • 慣性矩之莫爾圓(Mohrs Circle for Moment of Inertia)
              • 質量慣性矩(Mass Moment of Inertia)
                • 平行軸定理(Parallel-Axis Theorem)
Page 70: 轉動慣量 Moments of Inertia - cct.me.ntut.edu.twcct.me.ntut.edu.tw/ccteducation/chchting/aiahtm/statics/lecture... · 轉動慣量或慣性矩 I. 轉動慣量也稱慣性矩(Moment

迴旋半徑及複合體

迴轉半徑(k)是一個可以用來計算轉動慣量的物理量當一力矩作用於一個物體上時物體會依照轉動慣量(I) 呈現應有的旋轉運動物體對於一直軸或質心的迴轉半徑(k)是此物體所有粒子對於此直軸或質心的均方根距離

I = mk2 (141)

k =

radicI

m(142)

複合體之質量慣性矩(Ic)為各局部物體之質量慣性矩(Ii )合

Ic =i=nsumi=1

Ii (143)

= I1 + I2 + + In (144)

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 52 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

兩圓盤複合體

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Media File (applicationx-shockwave-flash)

結結結論論論

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 54 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

結論

轉動慣量或慣性矩為物體對旋轉運動的慣性而所謂慣性乃物體持續維持不變的行為也就是定義慣性矩可表達物體對本身旋轉運動的維持或持續性好壞更簡單的說慣性矩越大之物體其維持旋轉運動趨勢的能力越大例如惰輪儲能裝置惰輪之慣性矩越大當其轉動以後其維持旋轉運動趨勢的能力越大

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 55 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
  • 轉動慣量或慣性矩(Moment of Inertia)
  • 面積慣性矩(Moment of Inertia for Area)
    • 面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)
    • 面積慣性矩之迴轉半徑(Radius of Gyration for Area)
      • 複合面積之慣性矩(Moment of Inertia for Composite Area)
      • 面積之慣性積(Product of Inertia for Area)
        • 平行軸定理(Parallel-Axis Theorem)
          • 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)
            • 主軸慣性矩(Principal Moment of Inertia)
              • 慣性矩之莫爾圓(Mohrs Circle for Moment of Inertia)
              • 質量慣性矩(Mass Moment of Inertia)
                • 平行軸定理(Parallel-Axis Theorem)
Page 71: 轉動慣量 Moments of Inertia - cct.me.ntut.edu.twcct.me.ntut.edu.tw/ccteducation/chchting/aiahtm/statics/lecture... · 轉動慣量或慣性矩 I. 轉動慣量也稱慣性矩(Moment

課課課堂堂堂練練練習習習

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

兩圓盤複合體

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

結結結論論論

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 54 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

結論

轉動慣量或慣性矩為物體對旋轉運動的慣性而所謂慣性乃物體持續維持不變的行為也就是定義慣性矩可表達物體對本身旋轉運動的維持或持續性好壞更簡單的說慣性矩越大之物體其維持旋轉運動趨勢的能力越大例如惰輪儲能裝置惰輪之慣性矩越大當其轉動以後其維持旋轉運動趨勢的能力越大

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 55 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
  • 轉動慣量或慣性矩(Moment of Inertia)
  • 面積慣性矩(Moment of Inertia for Area)
    • 面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)
    • 面積慣性矩之迴轉半徑(Radius of Gyration for Area)
      • 複合面積之慣性矩(Moment of Inertia for Composite Area)
      • 面積之慣性積(Product of Inertia for Area)
        • 平行軸定理(Parallel-Axis Theorem)
          • 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)
            • 主軸慣性矩(Principal Moment of Inertia)
              • 慣性矩之莫爾圓(Mohrs Circle for Moment of Inertia)
              • 質量慣性矩(Mass Moment of Inertia)
                • 平行軸定理(Parallel-Axis Theorem)
Page 72: 轉動慣量 Moments of Inertia - cct.me.ntut.edu.twcct.me.ntut.edu.tw/ccteducation/chchting/aiahtm/statics/lecture... · 轉動慣量或慣性矩 I. 轉動慣量也稱慣性矩(Moment

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

兩圓盤複合體

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

結結結論論論

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 54 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

結論

轉動慣量或慣性矩為物體對旋轉運動的慣性而所謂慣性乃物體持續維持不變的行為也就是定義慣性矩可表達物體對本身旋轉運動的維持或持續性好壞更簡單的說慣性矩越大之物體其維持旋轉運動趨勢的能力越大例如惰輪儲能裝置惰輪之慣性矩越大當其轉動以後其維持旋轉運動趨勢的能力越大

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 55 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
  • 轉動慣量或慣性矩(Moment of Inertia)
  • 面積慣性矩(Moment of Inertia for Area)
    • 面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)
    • 面積慣性矩之迴轉半徑(Radius of Gyration for Area)
      • 複合面積之慣性矩(Moment of Inertia for Composite Area)
      • 面積之慣性積(Product of Inertia for Area)
        • 平行軸定理(Parallel-Axis Theorem)
          • 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)
            • 主軸慣性矩(Principal Moment of Inertia)
              • 慣性矩之莫爾圓(Mohrs Circle for Moment of Inertia)
              • 質量慣性矩(Mass Moment of Inertia)
                • 平行軸定理(Parallel-Axis Theorem)
Page 73: 轉動慣量 Moments of Inertia - cct.me.ntut.edu.twcct.me.ntut.edu.tw/ccteducation/chchting/aiahtm/statics/lecture... · 轉動慣量或慣性矩 I. 轉動慣量也稱慣性矩(Moment

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

兩圓盤複合體

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

結結結論論論

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 54 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

結論

轉動慣量或慣性矩為物體對旋轉運動的慣性而所謂慣性乃物體持續維持不變的行為也就是定義慣性矩可表達物體對本身旋轉運動的維持或持續性好壞更簡單的說慣性矩越大之物體其維持旋轉運動趨勢的能力越大例如惰輪儲能裝置惰輪之慣性矩越大當其轉動以後其維持旋轉運動趨勢的能力越大

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 55 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
  • 轉動慣量或慣性矩(Moment of Inertia)
  • 面積慣性矩(Moment of Inertia for Area)
    • 面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)
    • 面積慣性矩之迴轉半徑(Radius of Gyration for Area)
      • 複合面積之慣性矩(Moment of Inertia for Composite Area)
      • 面積之慣性積(Product of Inertia for Area)
        • 平行軸定理(Parallel-Axis Theorem)
          • 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)
            • 主軸慣性矩(Principal Moment of Inertia)
              • 慣性矩之莫爾圓(Mohrs Circle for Moment of Inertia)
              • 質量慣性矩(Mass Moment of Inertia)
                • 平行軸定理(Parallel-Axis Theorem)
Page 74: 轉動慣量 Moments of Inertia - cct.me.ntut.edu.twcct.me.ntut.edu.tw/ccteducation/chchting/aiahtm/statics/lecture... · 轉動慣量或慣性矩 I. 轉動慣量也稱慣性矩(Moment

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

兩圓盤複合體

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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結結結論論論

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 54 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

結論

轉動慣量或慣性矩為物體對旋轉運動的慣性而所謂慣性乃物體持續維持不變的行為也就是定義慣性矩可表達物體對本身旋轉運動的維持或持續性好壞更簡單的說慣性矩越大之物體其維持旋轉運動趨勢的能力越大例如惰輪儲能裝置惰輪之慣性矩越大當其轉動以後其維持旋轉運動趨勢的能力越大

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 55 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
  • 轉動慣量或慣性矩(Moment of Inertia)
  • 面積慣性矩(Moment of Inertia for Area)
    • 面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)
    • 面積慣性矩之迴轉半徑(Radius of Gyration for Area)
      • 複合面積之慣性矩(Moment of Inertia for Composite Area)
      • 面積之慣性積(Product of Inertia for Area)
        • 平行軸定理(Parallel-Axis Theorem)
          • 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)
            • 主軸慣性矩(Principal Moment of Inertia)
              • 慣性矩之莫爾圓(Mohrs Circle for Moment of Inertia)
              • 質量慣性矩(Mass Moment of Inertia)
                • 平行軸定理(Parallel-Axis Theorem)
Page 75: 轉動慣量 Moments of Inertia - cct.me.ntut.edu.twcct.me.ntut.edu.tw/ccteducation/chchting/aiahtm/statics/lecture... · 轉動慣量或慣性矩 I. 轉動慣量也稱慣性矩(Moment

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

兩圓盤複合體

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

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Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

結結結論論論

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 54 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

結論

轉動慣量或慣性矩為物體對旋轉運動的慣性而所謂慣性乃物體持續維持不變的行為也就是定義慣性矩可表達物體對本身旋轉運動的維持或持續性好壞更簡單的說慣性矩越大之物體其維持旋轉運動趨勢的能力越大例如惰輪儲能裝置惰輪之慣性矩越大當其轉動以後其維持旋轉運動趨勢的能力越大

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 55 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
  • 轉動慣量或慣性矩(Moment of Inertia)
  • 面積慣性矩(Moment of Inertia for Area)
    • 面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)
    • 面積慣性矩之迴轉半徑(Radius of Gyration for Area)
      • 複合面積之慣性矩(Moment of Inertia for Composite Area)
      • 面積之慣性積(Product of Inertia for Area)
        • 平行軸定理(Parallel-Axis Theorem)
          • 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)
            • 主軸慣性矩(Principal Moment of Inertia)
              • 慣性矩之莫爾圓(Mohrs Circle for Moment of Inertia)
              • 質量慣性矩(Mass Moment of Inertia)
                • 平行軸定理(Parallel-Axis Theorem)
Page 76: 轉動慣量 Moments of Inertia - cct.me.ntut.edu.twcct.me.ntut.edu.tw/ccteducation/chchting/aiahtm/statics/lecture... · 轉動慣量或慣性矩 I. 轉動慣量也稱慣性矩(Moment

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

兩圓盤複合體

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

結結結論論論

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 54 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

結論

轉動慣量或慣性矩為物體對旋轉運動的慣性而所謂慣性乃物體持續維持不變的行為也就是定義慣性矩可表達物體對本身旋轉運動的維持或持續性好壞更簡單的說慣性矩越大之物體其維持旋轉運動趨勢的能力越大例如惰輪儲能裝置惰輪之慣性矩越大當其轉動以後其維持旋轉運動趨勢的能力越大

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 55 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
  • 轉動慣量或慣性矩(Moment of Inertia)
  • 面積慣性矩(Moment of Inertia for Area)
    • 面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)
    • 面積慣性矩之迴轉半徑(Radius of Gyration for Area)
      • 複合面積之慣性矩(Moment of Inertia for Composite Area)
      • 面積之慣性積(Product of Inertia for Area)
        • 平行軸定理(Parallel-Axis Theorem)
          • 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)
            • 主軸慣性矩(Principal Moment of Inertia)
              • 慣性矩之莫爾圓(Mohrs Circle for Moment of Inertia)
              • 質量慣性矩(Mass Moment of Inertia)
                • 平行軸定理(Parallel-Axis Theorem)
Page 77: 轉動慣量 Moments of Inertia - cct.me.ntut.edu.twcct.me.ntut.edu.tw/ccteducation/chchting/aiahtm/statics/lecture... · 轉動慣量或慣性矩 I. 轉動慣量也稱慣性矩(Moment

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

結結結論論論

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 54 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

結論

轉動慣量或慣性矩為物體對旋轉運動的慣性而所謂慣性乃物體持續維持不變的行為也就是定義慣性矩可表達物體對本身旋轉運動的維持或持續性好壞更簡單的說慣性矩越大之物體其維持旋轉運動趨勢的能力越大例如惰輪儲能裝置惰輪之慣性矩越大當其轉動以後其維持旋轉運動趨勢的能力越大

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 55 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
  • 轉動慣量或慣性矩(Moment of Inertia)
  • 面積慣性矩(Moment of Inertia for Area)
    • 面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)
    • 面積慣性矩之迴轉半徑(Radius of Gyration for Area)
      • 複合面積之慣性矩(Moment of Inertia for Composite Area)
      • 面積之慣性積(Product of Inertia for Area)
        • 平行軸定理(Parallel-Axis Theorem)
          • 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)
            • 主軸慣性矩(Principal Moment of Inertia)
              • 慣性矩之莫爾圓(Mohrs Circle for Moment of Inertia)
              • 質量慣性矩(Mass Moment of Inertia)
                • 平行軸定理(Parallel-Axis Theorem)
Page 78: 轉動慣量 Moments of Inertia - cct.me.ntut.edu.twcct.me.ntut.edu.tw/ccteducation/chchting/aiahtm/statics/lecture... · 轉動慣量或慣性矩 I. 轉動慣量也稱慣性矩(Moment

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

結結結論論論

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 54 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

結論

轉動慣量或慣性矩為物體對旋轉運動的慣性而所謂慣性乃物體持續維持不變的行為也就是定義慣性矩可表達物體對本身旋轉運動的維持或持續性好壞更簡單的說慣性矩越大之物體其維持旋轉運動趨勢的能力越大例如惰輪儲能裝置惰輪之慣性矩越大當其轉動以後其維持旋轉運動趨勢的能力越大

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 55 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
  • 轉動慣量或慣性矩(Moment of Inertia)
  • 面積慣性矩(Moment of Inertia for Area)
    • 面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)
    • 面積慣性矩之迴轉半徑(Radius of Gyration for Area)
      • 複合面積之慣性矩(Moment of Inertia for Composite Area)
      • 面積之慣性積(Product of Inertia for Area)
        • 平行軸定理(Parallel-Axis Theorem)
          • 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)
            • 主軸慣性矩(Principal Moment of Inertia)
              • 慣性矩之莫爾圓(Mohrs Circle for Moment of Inertia)
              • 質量慣性矩(Mass Moment of Inertia)
                • 平行軸定理(Parallel-Axis Theorem)
Page 79: 轉動慣量 Moments of Inertia - cct.me.ntut.edu.twcct.me.ntut.edu.tw/ccteducation/chchting/aiahtm/statics/lecture... · 轉動慣量或慣性矩 I. 轉動慣量也稱慣性矩(Moment

Engineering Mechanics Statics Twelfth EditionRussell C Hibbeler

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 53 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

結結結論論論

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 54 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

結論

轉動慣量或慣性矩為物體對旋轉運動的慣性而所謂慣性乃物體持續維持不變的行為也就是定義慣性矩可表達物體對本身旋轉運動的維持或持續性好壞更簡單的說慣性矩越大之物體其維持旋轉運動趨勢的能力越大例如惰輪儲能裝置惰輪之慣性矩越大當其轉動以後其維持旋轉運動趨勢的能力越大

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 55 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
  • 轉動慣量或慣性矩(Moment of Inertia)
  • 面積慣性矩(Moment of Inertia for Area)
    • 面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)
    • 面積慣性矩之迴轉半徑(Radius of Gyration for Area)
      • 複合面積之慣性矩(Moment of Inertia for Composite Area)
      • 面積之慣性積(Product of Inertia for Area)
        • 平行軸定理(Parallel-Axis Theorem)
          • 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)
            • 主軸慣性矩(Principal Moment of Inertia)
              • 慣性矩之莫爾圓(Mohrs Circle for Moment of Inertia)
              • 質量慣性矩(Mass Moment of Inertia)
                • 平行軸定理(Parallel-Axis Theorem)
Page 80: 轉動慣量 Moments of Inertia - cct.me.ntut.edu.twcct.me.ntut.edu.tw/ccteducation/chchting/aiahtm/statics/lecture... · 轉動慣量或慣性矩 I. 轉動慣量也稱慣性矩(Moment

結結結論論論

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 54 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)

結論

轉動慣量或慣性矩為物體對旋轉運動的慣性而所謂慣性乃物體持續維持不變的行為也就是定義慣性矩可表達物體對本身旋轉運動的維持或持續性好壞更簡單的說慣性矩越大之物體其維持旋轉運動趨勢的能力越大例如惰輪儲能裝置惰輪之慣性矩越大當其轉動以後其維持旋轉運動趨勢的能力越大

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 55 54

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  • 轉動慣量或慣性矩(Moment of Inertia)
  • 面積慣性矩(Moment of Inertia for Area)
    • 面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)
    • 面積慣性矩之迴轉半徑(Radius of Gyration for Area)
      • 複合面積之慣性矩(Moment of Inertia for Composite Area)
      • 面積之慣性積(Product of Inertia for Area)
        • 平行軸定理(Parallel-Axis Theorem)
          • 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)
            • 主軸慣性矩(Principal Moment of Inertia)
              • 慣性矩之莫爾圓(Mohrs Circle for Moment of Inertia)
              • 質量慣性矩(Mass Moment of Inertia)
                • 平行軸定理(Parallel-Axis Theorem)
Page 81: 轉動慣量 Moments of Inertia - cct.me.ntut.edu.twcct.me.ntut.edu.tw/ccteducation/chchting/aiahtm/statics/lecture... · 轉動慣量或慣性矩 I. 轉動慣量也稱慣性矩(Moment

結論

轉動慣量或慣性矩為物體對旋轉運動的慣性而所謂慣性乃物體持續維持不變的行為也就是定義慣性矩可表達物體對本身旋轉運動的維持或持續性好壞更簡單的說慣性矩越大之物體其維持旋轉運動趨勢的能力越大例如惰輪儲能裝置惰輪之慣性矩越大當其轉動以後其維持旋轉運動趨勢的能力越大

Chen-Ching Ting (丁振卿) Mechanical Engineering National Taipei University of Technology (國立台北科技大學機械系) Homepage httpcctmentutedutw E-mail chchtingntutedutw CCT Group[2mm] 助教 吳穎彥 ()轉動慣量或慣性矩Moments of Inertia June 3 2011 55 54

flip_clock_black_12swf
Media File (applicationx-shockwave-flash)
  • 轉動慣量或慣性矩(Moment of Inertia)
  • 面積慣性矩(Moment of Inertia for Area)
    • 面積慣性矩之平行軸定理(Parallel-Axis Theorem for Area)
    • 面積慣性矩之迴轉半徑(Radius of Gyration for Area)
      • 複合面積之慣性矩(Moment of Inertia for Composite Area)
      • 面積之慣性積(Product of Inertia for Area)
        • 平行軸定理(Parallel-Axis Theorem)
          • 傾斜軸面積之慣性矩(Moment of Inertia for Area about Inclined Axis)
            • 主軸慣性矩(Principal Moment of Inertia)
              • 慣性矩之莫爾圓(Mohrs Circle for Moment of Inertia)
              • 質量慣性矩(Mass Moment of Inertia)
                • 平行軸定理(Parallel-Axis Theorem)