bruce mayer, pe licensed electrical & mechanical engineer bmayer@chabotcollege

[email protected] • ENGR-36_Lec-26_Mass_Moment_of_Inertia.pptx1

Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Engineering 36

Chp10:Moment of

Interia



Mass Moments of Inertia The Previously Studied “Area Moment

of Inertia” does Not Actually have True Inertial Properties• The Area Version is More precisely Stated

as the SECOND Moment of Area Objects with Real mass DO have inertia

• i.e., an inertial Body will Resist Rotation by An Applied Torque Thru an F=ma Analog

Inertia ofMoment Mass IIαT



Mass Moment of Inertia

The Momentof Inertia is

the Resistanceto Spinning



Linear-Rotational Parallels



Mass Moment of Inertia The Angular acceleration, , about the axis

AA’ of the small mass m due to the application of a couple is proportional to r2m.• r2m moment of inertia of the mass m

with respect to the axis AA’

For a body of mass m the resistance to rotation about the axis AA’ is

inertiaofmomentmassdmr

mrmrmrmrI

2

223

22

21



Mass Radius of Gyration Imagine the entire Body Mass

Concentrated into a single Point Now place this mass a distance k from

the rotation axis so as to create the same resistance to rotation as the original body• This Condition Defines, Physically, the

Mass Radius of Gyration, k Mathematically

mIkormkdmrI 22



Ix, Iy, Iz

Mass Moment of inertia with respect to the y coordinate axis r is the ┴ distance to y-axis

dmxzdmrI y222

Similarly, for the moment of inertia with respect to the x and z axes

dmyxI

dmzyI

z

x22

22

Units Summary• SI 22 mkg dmrI

• US Customary Units

222

2 sftlbftftslbftslugI



Parallel Axis Theorem

Consider CENTRIODAL Axes (x’,y’,z’) Which are Translated Relative to the Original CoOrd Systems (x,y,z)

The Translation Relationships

zzz

yyy

xxx

'

'

'

In a Manner Similar to the Area Calculation• Two Middle Integrals are

1st-Moments Relative to the CG → 0

• The Last Integral is the Total Mass

Then Write Ix

dmzydmzzdmyydmzy

dmzzyydmzyI x2222

2222

220 0

m



Parallel Axis Theorem cont.

So Ix

Similarly for the Other two Axes

22

'

2222 00

zymII

mzydmzyI

xx

x

22 zymII xx

so

22

22

yxmII

xzmII

zz

yy

In General for any axis AA’ that is parallel to a centroidal axis BB’

2mdII Also the Radius of

Gyration222 dkk



Thin Plate Moment of Inertia For a thin plate of uniform thickness t and

homogeneous material of density , the mass moment of inertia with respect to axis AA’ contained in the plate

areaAAAA ItdArttdArdmrI ,222

Similarly, for perpendicular axis BB’ which is also

contained in the plateareaBBBB ItI ,

BBAA

areaBBareaAAareaCCC

II

IItJtI

,,,

For the axis CC’ which is PERPENDICULAR to the plate note that This is a POLAR Geometry



Polar Moment of Inertia The polar moment of inertia is an important parameter in

problems involving torsion of cylindrical shafts, Torsion in Welded Joints, and the rotation of slabs

In Torsion Problems, Define a Moment of Inertia Relative to the Pivot-Point, or “Pole”, at O

dArJO2

Relate JO to Ix & Iy Using The Pythagorean Theorem

xy

O

II

dAydAxdAyxdArJ

22222



Thin Plate Examples For the principal centroidal axes on a

rectangular plate 2

1212

1213

121

, maatabbatItI areaAAAA

21212

1213

121

, mbbtababtItI areaBBBB

22121

,, bamIII massBBmassAACC

For centroidal axes on a circular plate

2414

41

, mrrtItII areaAABBAA

221mrIII BBAACC

Area = ab

Area = πr2



3D Mass Moments by Integration• The Moment of inertia of a

homogeneous body is obtained from double or triple integrations of the form

dVrI 2• For bodies with two planes of symmetry,

the moment of inertia may be obtained from a single integration by choosing thin slabs perpendicular to the planes of symmetry for dm.

• The moment of inertia with respect to a particular axis for a COMPOSITE body may be obtained by ADDING the moments of inertia with respect to the same axis of the components.



Common Geometric Shapes



Example 1

Determine the moments of inertia of the steel forging with respect to the xyz coordinate axes, knowing that the specific weight of steel is 490 lb/ft3 (0.284 lb/in3)

SOLUTION PLAN• With the forging divided

into a Square-Bar and two Cylinders, compute the mass and moments of inertia of each component with respect to the xyz axes using the parallel axis theorem.

• Add the moments of inertia from the components to determine the total moments of inertia for the forging.



Example 1 cont.

For The Symmetrically Located Cylinders

ftslb0829.0

sft2.32ftin1728in31lb/ft490

2

233

323

m

gVm

23

21222

121

21

22212

sftlb1059.2

0829.00829.0

ymmaymII xx

Referring to the Geometric-Shape Table for the Cylinders• a = 1” (the radius)• L = 3”• xcentriod = 2.5”• ycentriod = 2”

Then the Axial (x) Moment of Inertia



Example 1 cont.2 Now the Transverse

(y & z) Moments of Inertia

23

212

5.221232

121

121

2221212

sftlb1017.4

0829.030829.0

3

xmLamxmII yy

23

21222

125.22

1232

121

121

2222121

sftlb1048.6

0829.030829.0

3

yxmLamI z

dz

2zd



Example 1 cont.3

For The Sq-Bar

ftslb211.0

sft2.32ftin1728in622lb/ft490

2

233

33

m

gVm

23

21222

126

121

22121

sftlb 1088.4

211.0

cbmII zx

Referring to the Geometric-Shape Table for the Block• a = 2”• b = 6”• c = 2”

Then the Transverse (x & z ) Moments of Inertia



Example 1 cont.4

And the Axial (y) Moment of Inertia

23

21222

122

121

22121

sftlb 10977.0

211.0

acmI y

33 1059.221088.4 xI23 sftlb1006.10

xI

33 1017.4210977.0 yI23 sftlb1032.9

yI

33 1048.621088.4 zI23 sftlb1084.17

zI

Add the moments of inertia from the components to determine the total moment of inertia.



T = Iα

When you take ME104 (Dynamics) at UCBerkeley you will learn that the Rotational Behavior of the CrankShaft depends on its Mass Moment of inertia



WhiteBoard Work

Some OtherMass

Moments

For theThick Ring 2

22innerouter

zRRmI



WhiteBoard Work

Find MASSMoment of

Inertiafor Prism

About the y-axis in this case



Bruce Mayer, PERegistered Electrical & Mechanical Engineer

[email protected]

Engineering 36

Appendix 00

sinhTµs

Tµx

dxdy



WhiteBoard Work

Find MASSMoment of

Inertiafor Roller

About axis AA’ in this case



Mass Moment of Inertia Last time we discussed the “Area

Moment of Intertia”• Since Areas do NOT have Inertial

properties, the Areal Moment is more properly called the “2nd Moment of Area”

Massive Objects DO physically have Inertial Properties• Finding the true “Moment of Inertia” is very

analogous to determination of the 2nd Moment of Area

bruce mayer, pe licensed electrical & mechanical engineer bmayer@chabotcollege

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