chapter 9 sbvl

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Chng 9: Tnh Chuyn V Bng Phng Php Nng Lng

1 Cc Khi Nim

2 Th Nng Bin Dng n Hi

3 nh L Castigliano

4 Cng Thc Mohr

5 Nhn Biu Vrxaghin

1 Cc Khi Nim

z

L L

* Thanh chu ko_nn ng tm c bin dng di dc trc:

; ;z z n zL

N LL dzEF L

A

B

'B

O

* Thanh chu xon thun ty c gc xoay tng i gia hai mt ct:

;zL

M dzGJ G

y

z

y

y

* Thanh chu un phng:

y- Chuyn v thng ca trng tm mt ct ngangtheo phng vung gc vi trc thanh.

- Chuyn v xoay ca mt ct ngang quanh mt trc nm trongmt ct ngang.

1 Cc Khi Nim

1 2 3 42P 3P 4P

12

434

1P1M

* K hiu cho cc i lng lc (bao gm lc v ngu lc): P* k hiu cho lc ti v tr v theo phng kkP* K hiu cho cc i lng chuyn v (bao gm chuyn v thng vchuyn v xoay):

+ k hiu cho chuyn v ti v tr v theo phng kk

+ k hiu cho chuyn v ti v tr v theo phng k do nguynnhn m gy ra

km

* k hiu cho chuyn v n v ti v tr v theo phng k do lcgy ra

km1mP

1 Cc Khi Nim

2 Th Nng Bin Dng n Hi

222 2

1 1 1 1

2 2

1 1

2 2 2 2

. .2 2

n n n nyxz z

i i i ix yLi Li Li Li

n ny x

x yi iLi Li

MMN MU dz dz dz dzEF EJ EJ GJ

Q Qk dz k dzGF GF

F C

Cy

yy

F C

Cx

xx dFh

SJFkdF

bS

JFk 2

2

22

2

2 ;Vi

* i vi dm un phng, b qua nh hng lc ct:2

11 2

nx

i xLi

MU dz

EJ

* i vi thanh chu ko-nn ng tm:2

21 2

nz

i Li

NU dz

EF

* i vi thanh chu xon thun ty:2

31 2

nz

i Li

MU dzGJ

3 nh L Castigliano

kk P

U

=> Trong he an hoi tuyen tnh, chuyen v tai mt v tr va theomot phng nao o bang ao ham rieng cua the nang biendang an hoi tch luy trong he lay oi vi bien so la lc tai v trva theo phng can tnh chuyen v.

* i vi thanh chu ko-nn ng tm:

,,

1 1i

z iz in n

i kk

i ik i iL

NN

U P dzP E F

2

1 2

nz

i Li

NU dzEF

i

n

1i ii

k

i,zi,z

k LFEP

NN

* Nu ti v tr v theo phng cn tnh chuyn v khng c lc Pk ta tmt lc Pg ti v tr v theo phng cn tnh chuyn v. Sau khi o hm

ta cho Pg=0/z gN P

3 nh L Castigliano

* i vi h dn (h thanh-khp) ch chu ko hoc nn ng tm v cNz/(EF) = const trn sut chiu di Li

V D: Thanh ABC tuyt i cng c lin kt, chu lc v kch thc nhhnh v. Cc thanh BD v CE lm bng thp c m un n hi E v cdin tch mt ct ngang ln lt l F v 2F. Tnh chuyn v thng ngti A

2a 2a a

2aP

030

A B

C

D

E

A

C B

1P

060

030 060

030

D

Hnh 9.4

a

2P

V D: Cho h dn nh hnh v. Cc thanh trong dn lm bng thp cm un n hi E v c cng din tch mt ct ngang l F. Tnh chuyn vthng ng ti A

EF

2E F

A

B

Ca a

P030

060

030

060

PABN

ACN

A

* Tch nt ti A0 0

0 0

cos30 cos 60 0

sin 30 sin 60 0AB AC

AB AC

X N NY N N P

12

32

AB

AC

N P

N P

* Theo iu kin bn ng sut php

2

zz max

max

N PF F

2 2150 3,57

2 2.21PF cm cm

Chn 23,6F cm

* Chuyn v thng ng ti A: p dng nh l Castigliao

,,

1

z i ACABz in AB ACk

k i AB ACi i i AB AB AC AC

N NNN N NP P PL L LE F E F E F

1 3 34 4 3 3 4 3 3 150.2002 2 2 2 2 5,267

2 2100.3,63 4 3 4 3yA

P Pa Paa cm

EF E F EF

* Chuyn v theo phng ngang ti A:

t thm lc Pg theo phng ngang ti A

030

060gP

ABN

ACN

A

P1 32 2

3 12 2

AB g

AC g

N P P

N P P

* p dng nh l Castigliao

,,

1

z i ACABz i AB ACn

g g gk i AB AC

i i i AB AB AC AC

N NNN N NP P P

L L LE F E F E F

3 3 12 2 4 3 4 3 4 150.2002 2 2 2,25

2 4 4 2100.3,63xA

P Pa Paa cm

EF E F EF

a

2a

2a

q

P qa

A B

C DE

,E F

* Xt cn bng thanh ABCD


52

zz max

max

N qaF F

Chn 26,98F cm

a

2a

2a

q

P qa

AB

C DE

AXAY

B EN

045

00 .2 . sin 45 .2 .3 03 522 2

A BE

BE

m q a a N a P a

N qa P qa

25 5.15.2,5 6,978

2 2.19qaF cm

* Bin dng di dc trc ca thanh EB

2 25 .2 2

10 10.15.2,5 .1002 0,63921000.6,98

BE BEBE

qa aN L qaL cm

EF EF EF

* Chuyn v thng ng ti A: p dng nh l Castigliao

2 25 3

15 2 15 2.15.2,5 .1002 2 2 2 1,35721000.6,98

BEBE

D BEBE BE

qaNN qaP L a cmE F EF EF

4 Cng Thc Mohr

* Cng thc Mohr:

+ Trng thi m: l trng thi chu ti

+ Trng thi k: l trng thi n v bng cch b ti v t mt lcPk=1 ti v tr v theo phng cn tnh chuyn v

* To hai trng thi

n

1i L ii

m,yik,yix

n

1i L xii

m,xik,xin

1i L ii

m,zik,zikm

iii

dzFGQQ

kdzJEMM

dzFENN

* i vi h dn (h thanh-khp) ch chu ko hoc nn ng tm v cNz/(EF) =const trn sut chiu di Li

1

nzi zi

km ii i i

N N LE F

+ : ni lc trng thi mzN

+ : ni lc trng thi kzN

V D: Thanh ABCD tuyt i cng c lin kt, chu lc v kch thcnh hnh v. Thanh AE v BE lm bng thp c m un n hi E, ngsut cho php v c din tch mt ct ngang ln lt l 2F v F.

a

2a

2a

q

P qa

A B

C DE

,E F

, 2E F

060

+ Xc nh ng lc trong cc thanh AE v BE.

+ Xc nh din tch mt ct ngang F hai thanh AE v BE cng bn

+ Tnh chuyn vthng ng ti D

4 2

2

35 / ; 22.10 /

21 /

q kN m a mE kN cm

kN cm

V D: Thanh ABC tuyt i cng c lin kt, chu lc v kch thc nhhnh v. Thanh CD lm bng thp c m un n hi E v c din tchmt ct ngang F. Tnh chuyn v thng ng ti A

a a

P060

A B

C

D

045

a

2a

2a

q

P qa

A B

C DE

,E F

060

V D: Thanh ABCD tuyt i cng c lin kt, chu lc v kch thcnh hnh v. Thanh BE lm bng thp c m un n hi E v c dintch mt ct ngang F. Tnh chuyn v thng ng ti D

V d: Thanh AB tuyt i cng chu lin kt gi c nh ti A v c gibi thanh CD, h chu lc v c kch thc nh hnh v. Thanh CD c mtct ngang khng i din tch F v lm bng thp c m un n hi E,ng sut cho php

+ Xc nh din tch mt ct ngang F thanh CD bn.+ Tnh bin dng di dc trc ca thanh CD

2 4 221 / , 2,1.10 /kN cm E kN cm . Cho:

AB

2,5m

25 /q kN m

0,5m

030

C

D

+ Xc nh phn lc lin kt ti B v ng lc trong thanh CD.

+ Tnh chuyn v thng ng ti A.

m2 m2 m1

m2P060

A B

C

D

E

* Xt cn bng thanh ABC


32

zz max

max

N PF F

Chn 29,63F cm

23 3.200 9,62

2 2.18PF cm

* Theo iu kin cng

m2

m2P060

A B

C CEN

BDN

BY

0

0

30 sin 60 .2 .2 02

3 10 cos 60 02

B CE CE

x BD CE BD

m P N N P

F P N N N P

3

32 3,6.10CE CE

CE

PL N L

L EF EF L

24 3

3 3.200 2,42.2.10 .3,6.102 .

PF cmLE

L

* Tnh chuyn v thng ng ti C

+ Ta c: 3 3 1;2 2CE BD

N P N P

+ To trng thi k:

m2 m2 m1

m2

A B

C

D

E

m2

m21kP

A B

C CEN

BDN

BY

1kP

10 1.2 .2 02

10 02

B CE CE

x BD CE BD

m N N

F N N N

+ p dng cng thc Mohr, chuyn v thng ng ti C:

4 41

1 3 11 3 200. 200 2 22 2 3000 2000 1,732.10 .9,63 2.10 .9,63

nzi zi CE CE BD BD

km i CE BDi i i CE CE BD BD

N N N N N NL L L mmE F E F E F

EF

2EFEF

2EF

3EF EFA

P

BC

DE

a

a a

* Trng thi m h chu tc dng cati trng P. S dng phng php tchnt ta xc nh c ng lc trong ccthanh nh bng bn di

P

A

ABN

ADN045

BBCN

BDNBAN

045

DDEN DAN

DBNDCN

045

* Trng thi k nh hnh v. Tng t s dng phng php tch nt ta xcnh c ng lc trong cc thanh nh bng bn.

* Chuyn v thng ng ti A:

1

7,16y

nzi zi

A ii i i

N N PaLE F EF

EF

2EFEF

2EF

3EF EFA

1kP

BC

DE

a

a a

5 Nhn biu Vrxaghin

+ Trng thi m: l trng thi chu ti

+ Trng thi k: l trng thi n v bng cch b ti v t

* To hai trng thi

. Mt lc Pk=1 ti v tr cn tnh chuyn v thng

. Mt ngu lc Mk=1 ti v tr cn tnh chuyn v xoay

1 1

i in n

i c i ckm

i ii i i i

f fE F E J

NL

z

z

C

cf

NL

" "m

" "k

* Chuyn v ti mt v tr v theo mt phng

+ : din tch biu ni lc trng thi m+ : Cao ca biu ni lc trng thik ly ti trng tm biu ni lc trngthi m

cf

5 Nhn biu Vrxaghin

* Nhng lu khi thc hin php nhn biu

A B

Pl

Pl

xM

A B

q

l2

2ql

xM

A Bl

M

M

xM

A B

Pl

Pl

xM

A Bl

M

M

xM

A Bl

M

M

xM

A Bl

M

M

xM

A Bl

2 / 8ql

xM

q

A B

Ml

M

xM

A B

q

l2

2ql

xM

A Bl

M

M

xM

A B

1l

1 2 1 2/Pl l l l

xM

P

2l

* Biu ni lc ca mt s dng n gin

B

1l

1Pl

xM

P

2l

B

1l

2Pl

xM

P

2l

A CA C A B

1l

1 1 2/Ml l l

xM

2l

M

2 1 2/Ml l l

B

1l

xM

2l

B

1l

M

xM

2l

A CA C

MM

M

* Biu ni lc ca mt s dng n gin

A B

1l

1 2 1 2/Pl l l l

xM

P

2l

* Din tch, trng tm ca mt s hnh thng gp

C

dl

h

C

dl

h

C

dl

h

ld

C

1213

hl

d l

1334

hl

d l

2338

hl

d l

2312

hl

d l

* Cch chia din tch ca hnh phc tp

xM

1M

2M 2M 1 2M M

A B

PM

A B

M

A B

P

xM

1M

2M1M

2M

A B

1M

A B A B

2M 1M 2M

xM

1M

2M

1M

2M

A B

1M

A B A B

2M 1M 2M


1M

2M

1M 2M

2 / 8ql

lA B

1M

A

B2M

A Bl

q

A Bl

q1M

2M

1M

2M

l

1M

2M

A B

2M

A B

1M

2 / 8ql

A Bl

qA Bl

q1M 2M


M

l

2 / 8ql

M

A Bl

qM

A B

M

A Bl

q

1M

2M

l

1M

2M

A B2M

A B

1M 2 / 8ql

A Bl

qA Bl

q1M 2M

V d: Dm AD c cng chng un EJ=const.+ Tnh chuyn v thng ng ca mt ct ti B, C.+ Tnh chuyn v xoay ca mt ct ti A, B, C, D.

3P

a 2a

AB C

DP

3a

A B CD

3a2aa

P 2P

V d: Dm AD c cng chng un EJ=const.+ Tnh chuyn v thng ng ca mt ct ti A, C.+ Tnh chuyn v xoay ca mt ct ti A, B, C, D

A B CD

3a2aa

P 2P 2Pa

V d: Dm AC c cng chng un EJ=const.+ Tnh chuyn v thng ng ca mt ct ti C.+ Tnh chuyn v xoay ca mt ct ti A, B, C.

P qaq

3a a

A B C

3P

a 2a

AB C

DP

3a

* Phn lc lin kt ti A, D

50 . 3 .3 6 0370 6 .5 3 .3 03

A D D

D A A

m P a P a Y a Y P

m Y a P a P a Y P

3P

a 2a

AB C

DP

3a DYAY

* Biu lc ct, mmen un trng thi m nh hnh b, c

3P

a 2a

AB C

D

P

3aAY DY73P

43P

53P

yQ

xM

73Pa

5Pa

3a

A CD

1kP

3a

xM

1,5a

12

3

4

1cf

2cf

3cf 4cf

)a

)b

)c

)d

)e

3a

A CD

1kM

3a

xM

1

1cf 2cf 3cf 4cf

)f

)g

* Trng thi k v biu mmen un khi tnh chuyn vthng ng ca mt ct ti Cnh hnh d, e

* Trng thi k v biu mmen un khi tnh chuyn vxoay ca mt ct ti A nhhnh f, g

* Chuyn v thng ng ca mtct ti C

3P

a 2a

AB C

D

P

3aAY DY73P

43P

53P

yQ

xM

73Pa

5Pa

3a

A CD

1kP

3a

xM

1,5a

12

3

4

1cf

2cf

3cf 4cf

)a

)b

)c

)d

)e

3a

A CD

1kM

3a

xM

1

1cf 2cf 3cf 4cf

)f

)g

4

1

2 2 2 2

3

1

1 7 14 8 7 15. . . .6 3 3 3 6 2

473

iC i ci

fEJ

a aPa Pa a Pa Pa aEJ

PaEJ

* Chuyn v xoay ca mt ct tiA

4

1

2 2 2 2

2

1

1 7 8 14 2 8 11 15 1. . . .6 9 3 3 3 18 2 3

14918

iA i ci

fEJ

Pa Pa Pa PaEJ

PaEJ

P

a 2a

A B C

2a

D

M Pa

* Biu mmen un trng thi m nh hnh b

* Trng thi k v biu mmen un khi tnh chuyn v thng ngca mt ct ti A nh hnh c, d

* Trng thi k v biu mmen un khi tnh chuyn v xoay camt ct ti C nh hnh e, f

P

a 2a

A B C

2a

D

M Pa

0,5Pa

Pa

xM

)a

)b

1kP

a

A B D)c

xM

a)d

1cf

2cf

3cf

4cf

1kP

a

A B D)e

xM

0,5)f

2cf

3cf

4cf

2a 2a

C

1kM

0,5

12

3 4

* Chuyn v thng ng ca mtct ti A

* Chuyn v xoay ca mt ct tiC

4

1

2 2 2 2

3

1

1 5 2. . 0,5 . 0,5 .2 6 3 3

56

iA i ci

fEJ

a a aPa Pa a Pa PaEJ

PaEJ

4

1

2 2 2

2

1

1 1 1 1. 0,5 . 0,5 .6 3 3

16

iC i ci

fEJ

Pa Pa PaEJ

PaEJ


* Trng thi k v biu mmen un khi tnh chuyn v thng ngca mt ct ti C nh hnh d, e

* Trng thi k v biu mmen un khi tnh chuyn v xoay camt ct ti C nh hnh f, g

P qa2M qaq

3a a

A B C

* Chuyn v thng ng ca mt ctti C

* Chuyn v xoay ca mt ct ti C

P qa2M qa q

3a a

A B CAY BY

56

qa

136

qa

qa

2qa

297 /72qa

yQ

xM2qa 1kP

3a a

A B C

xM

a

1cf

2cf

3cf

4cf

12

3 4

a

A B C

xM

1

1cf

2cf

3cf

4cf

1kM

)a

)b

)c

)d

)e

)f

)g

4

1

3 3 3 3

4

1

1 3 9 3 2 2. . . 0,5 .2 3 4 2 2 3 3

724

iC i ci

fEJ

a a a aqa qa qa qaEJ

qaEJ

4

1

3 3 3 3

3

1

1 3 1 9 1 3 2. . . 0,5 .12 3 4 2 2 3

18

iC i ci

fEJ

qa qa qa qaEJ

qaEJ


* Trng thi k v biu mmen un khi tnh chuyn v thng ngca mt ct ti D nh hnh d, e

* Trng thi k v biu mmen un khi tnh chuyn v xoay camt ct ti B nh hnh f, g

2a a a

A B C D

P qaq

22M qa 2M qa

* Chuyn v thng ng ca mt ctti D

* Chuyn v thng ng ca mtct ti B

2a a a

A B C D

P qaq

22M qa 2M qa

AYCY

qa16

qa

176

qa

22qa253

qa

yQ

213

qa21

72qa

xM

2qa

a

A C D

1kP a

a

A C D

a2a

1kP

2 / 3a

xM

1cf

2cf

B

3cf 4cf 5cf 6cf 7cf

1cf 2cf 3cf 4cf 5cf

)a

)b

)c

)d

)e

)f

xM

12

3

45

6

7

7

1

3 3 3 3

3 3 3

4

1

2 5 4 1 7 1 5. . . .1 3 3 3 9 6 9 12 6

8 1 2. . .9 2 3 2

4924

iD i ci

fEJ

a a a aqa qa qa qa

a a aEJ qa qa qa

qaEJ

7

1

3 3 3

3 3

4

1

2 5 4 1 4. . .1 3 3 3 9 6 91 2. .

12 3 92336

iB i ci

fEJ

a a aqa qa qa

a aEJ qa qa

qaEJ

* Biu mmen un, lc dc trng thi m nh hnh b, c

* Trng thi k v biu mmen un, lc dc khi tnh chuyn vthng ng ca mt ct ti C nh hnh d, e, f

P qaq

3a

aA

B C

P qaq

3a

aA

B C

20,5qa

23,5qa

20,5qa

xM

qa

zN

3a

aA

B C

1kP a

a

xM

1

zN

1

1cf

2cf

2

3

3cf

4

4cf

)a )b )c

)d )e )f

* Chuyn v thngng ca mt ctti C

4

2 4 234 3 3 3

1

1 1 1 3 3 9 3 .1 3. . . 6,1256 4 2 2i

cC i c

i

f a qa qa qaf qa qa a qa aEJ EF EJ EF EJ EF

chapter 9 sbvl

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