同濟(jì)大學(xué)理論力學(xué)孫杰習(xí)題解答練習(xí)冊(cè)2

理論力學(xué)習(xí)題解答練習(xí)冊(cè)動(dòng)力學(xué)

1P33

習(xí)題：1ORFx設(shè)F

=kx

當(dāng)

x=R

mg=kR

k

=RmgF

=

xRmg代入：–

F

=ma

并整理：x

+x=0Rg¨

二階常系數(shù)線性齊次微分方程通解：x=Csinw

t+

Dcosw

t

初始條件：x=

R

v=

0

運(yùn)動(dòng)方程：w

=√R√g˙v=x當(dāng)：x=

0

=

–

sinw

t

√gRsinw

t

=1v=

–√gR=–

7.9×103

m/sw

t

=p

/2t

=1266

sv其中：質(zhì)點(diǎn)運(yùn)動(dòng)微分方程直角坐標(biāo)法t=

0

x=

Rcosw

t

P33

習(xí)題：2AORMqatanmgFNt

方向：n方向：mgsinq

=

mat

其中：at

=dtdvdqdv=Rv∫

gRsinq

dq

=∫

v

dv

0

0

v

30°

v

2

=

gR(

2

–

)√3mgcosq

–

FN

=man

q其中：an

=Rv2FN

=

mg(3

–4

)21√3質(zhì)點(diǎn)運(yùn)動(dòng)微分方程自然坐標(biāo)法圖!

（分析）∴∴=0.598

mgP34

習(xí)題：1OCOO1O2OCevOvCvCwwwp

=p

=p

==

mw

2lp

=

mvC=

mvO

=

mew

0

p

=vCP34

習(xí)題：2jCAllxySFx

=

0

aCx

=

0

vCx

=

常

又

vC0

x

=

0

vCx

=

0

xC

=

常

=

0

PFN質(zhì)心運(yùn)動(dòng)守恒(水平)：當(dāng)桿處于j

角：xA

=l

cosjyA

=2l

sinjxA2+()2

=l

22yAsin2j

+cos2j

=1橢圓建立坐標(biāo)!xC=常求位移

!求軌跡方程P35

習(xí)題：3OCAqPllaCnaCtFOxFOywa質(zhì)心運(yùn)動(dòng)定理：S

Fx

=

maCx

FOx

=

–

m(

aCncosq

+

aCt

sinq)

=

–

(

w2cosq

+

a

sinq)

gPlS

Fy

=

maCy

FOy

–

P

=

m(

aCnsinq

–

aCt

cosq)

FOy

=

P

+

(

w2sinq

–

a

cosq)

gPl∴aCn

=

lw

2aCt

=

la

aCFO

（質(zhì)心運(yùn)動(dòng)定理）畫(huà)圖P36

習(xí)題：2ABCmmmJC桿121=

m(2l

)2JC盤(pán)JC環(huán)JC

=

JC桿

+

JC盤(pán)

+

JC環(huán)31=

ml

221=

mr

2

+

m(

l

+

r

)2=

mr

2

+

m(

l

+

r

)223=

ml

2

+

2

m(

l

+

r

)2

+mr

2

3127=

ml

2

+

4

m

lr

+

mr

2

37組合法

lr

r

l組合法軸!(記)P38

習(xí)題：3OBADCaCxaCyFOxFOyawmgmg剛體定軸轉(zhuǎn)動(dòng)微分方程：JOa

=SMO

(Fi

)=

ml

2

31731JO

=

m(2l

)

2

+[

m(2l

)2

+

m(2l

)2]

121617=kg·m2=2.83kg·m2SMO(Fi

)=

mgl

+

mg2l=

3mgla=20.75

rad/s2∴w=4

rad/s而質(zhì)心運(yùn)動(dòng)定理：23laCx

=

w

2aCy

=a

23l=6

m/s2=7.78

m/s2∴S

Fx

=

2maCx

FOx

=

–

2maCx

S

Fy

=

2maCy

2mg–

FOy

=

2maCy

=–

96N

FOy

=32.28N

FO

=101.3N

定軸轉(zhuǎn)動(dòng)a

w

aC

FO

aC?。篻=9.8m/s2P37

習(xí)題：1AOmm1m2aM動(dòng)量矩定理:wvvLO

=

JOw

+(

m1v)r+(m2v)r

21=(m

+m1

+m2)rv

SMO(Fi

)=

M–

m1gr

+

m2gr其中:v=

rw

JO

=

mr221dtdLO=SMO(Fi

)21(m

+m1

+m2)ra

=

M–

m1gr

+

m2gr∴M

=2.89kN·m對(duì)系統(tǒng)：?。篻=9.8m/s2dtdLO=SMO(Fi)研究對(duì)象m1gm2gP37

習(xí)題：2FⅠ

OGⅡ

OPGwFOxFOyFOxFOySMO(Fi

)=

Fr剛體定軸轉(zhuǎn)動(dòng)微分方程：JOa

=SMO

(Fi

)∴JOFra

=

JOPr=

質(zhì)心運(yùn)動(dòng)定理：SFx

=

0

FOx

=

0

SFy

=

0

FOy

-

G

-

F

=0∴FOy=

G

+

F=

G

+

Pa動(dòng)量矩定理：LO

=

JOw

+

vr

gP=

(JO

+

r2)w

gPSMO(Fi

)=

PrdtdLO=SMO(Fi

)(JO

+

r2)a

=

PrgPPra

=

JO

+

r2gPJOg

+

Pr2Pr=