理論力學(xué)生教學(xué)課件dch6e

1§6-4、剛體的一般運(yùn)動(dòng)的運(yùn)動(dòng)學(xué)與動(dòng)力學(xué)2剛體一般運(yùn)動(dòng)的實(shí)例飛機(jī)的旋翼運(yùn)動(dòng)為剛體的一般運(yùn)動(dòng)3§6-4-1剛體一般運(yùn)動(dòng)的運(yùn)動(dòng)學(xué)問(wèn)題：如何確定自由剛體在空間的位置？4§6-4-1剛體一般運(yùn)動(dòng)的運(yùn)動(dòng)學(xué)一、剛體一般運(yùn)動(dòng)的運(yùn)動(dòng)方程x

y

z

定參考系x1

y1

z1平移參考系1xy1z1o'xzro

'y一般運(yùn)動(dòng)=平移運(yùn)動(dòng)+定點(diǎn)運(yùn)動(dòng)xO'

=

f1

(t)yO'

=

f2

(t)zO'

=

f3

(t)y

=

f1

(t)

q

=

f2

(t)

j

=

f3

(t)5§6-4-1剛體一般運(yùn)動(dòng)的運(yùn)動(dòng)學(xué)二、剛體一般運(yùn)動(dòng)時(shí)其上點(diǎn)的速度和加速度vM

=

ve

+

vrraM

=

ae

+

a=vo'

+w

·r'=

ao'

+a

·

r'+w

·(w

·

r')x11yz1o'r

'xyzro

'rM剛體的一般運(yùn)動(dòng)：隨基點(diǎn)的平移和繞基點(diǎn)的定點(diǎn)運(yùn)動(dòng)的合成動(dòng)點(diǎn)：M，動(dòng)系：ox1

y1

z（1

平移動(dòng)系）剛體上點(diǎn)的速度剛體上點(diǎn)的加速度6§6-4-1剛體一般運(yùn)動(dòng)的運(yùn)動(dòng)學(xué)例：半徑為R的保齡球在地面上純滾動(dòng)，已知該球繞鉛垂軸的角0，其大小均為常量ω。求保齡球的角速度，角加速度，球體最高點(diǎn)M的速度和加速度。1

1x速度是

w

z

，繞水平軸的角速度為

wx1y1o1w

zx1w11

1w

=

w

z

+w

x

=

w

0

k

-

w

0

i解：（1）求角速度和角加速度dta

=

dw=

0（2）求M點(diǎn)的速度1

1vM

=vO

+w

·rO

Mz1MPvO

=

rO

P

·w1

11vo0

=

v

=v?

+w

·rP

O1

O1Pw

z

，w

x

：常矢量1

11x=

rO

P

·(wz

+w

)1

1=

w

0

Rj11xOP=

r

·w

=-Rk

·(-w0

)i7§6-4-1剛體一般運(yùn)動(dòng)的運(yùn)動(dòng)學(xué)0O1Mx1z1+w

)·

r=

w

Rj

+(w0=

w

Rj+w

x

·

rO

M1

1=

2w

0

Rjx11z1yo11w

z1w

xMPvo1w

z

，w

x

：常矢量1

1vM

=vO

+w

·rO

M1

1vO

=

rO

P

·w1

1=

ro

P

·w=

w

0

Rj1

x11

1aM

=

aO'

+a

·rO

M

+w

·(w

·rO

M

)（3）求M點(diǎn)的加速度vo

=

w

0

Rj11aO

=

0=

w

0

Rj

-w

0

i

·

Rk=

w

0

Rj

+w

0

Rj1a

=

0,\

a

·

ro

M

=

0?

?8§6-4-1剛體一般運(yùn)動(dòng)的運(yùn)動(dòng)學(xué)1z1y1o1w

z1w

xMP1vo

1

aM

=

w

·(w

·

rO

M

)11vMO=

(w

·

rOM

)=

(w

z

+w

x

)·

rO

M1

1

1=

w

x1·

rO

M=

w

0

Rj1MMO1a

=

w

·

v·vMO

+w

x

·vMO1

1

1=

(w

z

+w

x

)·vMO

=

w

z1

1

1

10x1a

M0

0

0

0

0=

w

k

·w

Rj

+(-w

)i

·w

Rj

=

-w

2

Ri

-w

2

Rk思考題：如何求該瞬時(shí)M點(diǎn)運(yùn)動(dòng)的曲率半徑?9§6-4-1剛體一般運(yùn)動(dòng)的運(yùn)動(dòng)學(xué)例:已知半徑為R鋼球在地面上純滾動(dòng)。O為球心，A、B、C、O共面，圖示瞬時(shí)A、B兩點(diǎn)的速度水平向右，大小均為u。求此瞬時(shí)球的角速度。ABvABCOvM

=vo'

+w

·r'取C為基點(diǎn)vA

=w

·rCA

(1)vB

=w

·rCB

(2)w

·rCA

-w

·rCB

=0w

·(rCA

-rCB)

=0w

·rBA

=0u2Rw

//rBA由（2）得：w

=vwvA

=

vB10§6-4-1剛體一般運(yùn)動(dòng)的運(yùn)動(dòng)學(xué)三、剛體一般運(yùn)動(dòng)的運(yùn)動(dòng)微分方程eFC

ima

==eC

i

C

dtM

(F

)dLrJCz

'w

z

'

+(JCy

'

-

JCx

'

)w

x'w

y

'

=

MCz

'JCx

'w

x'

+(JCz

'

-

JCy

'

)w

y

'w

z

'

=

MCx

'C

z=

FmyC

=

FymzmxC

=

Fx質(zhì)心運(yùn)動(dòng)定理：相對(duì)質(zhì)心的動(dòng)量矩定理：JCy

'w

y

'+(JCx

'-JCz

')w

x'w

z

'=

MCy

'1x1yz1xzro

'yrMr

'c§6-4-2剛體一般運(yùn)動(dòng)的動(dòng)力學(xué)剛體一般運(yùn)動(dòng)基本物理量的計(jì)算

（1）動(dòng)量：

p

=

mvCO（2）對(duì)固定點(diǎn)O的動(dòng)量矩：L=

r

·(mv

)

+

LrOC

C

CCz

'

z

'C=

JCx

'w

x

'i'+JCy

'w

y

'

j'+J

w

k

'其中：Lr（3）動(dòng)能：212C

Cmv

+

T

rT

=22y'

Cy'z'y'x'Cx

Cy Cz

JCx'vCx

CymvCz

mT

=1[v

v

v

]JCz'

wz'

11

w

,

wx'

m

v

+1[w

w

w

]

J222Cz

'

z

'Cy

'

y

'Cx'

x

'Cw

2w

2w

2+

1

J+

1

J=

1

J其中：T

r其中：Cx',Cy',Cz'為中心慣量主軸§6-4-2剛體一般運(yùn)動(dòng)的動(dòng)力學(xué)（4）角速度在慣量主軸上的投影：xyzx'z'yy'qyqN歐拉角節(jié)線w

zjw

z

'w

nj=

y

cos

q

+

jw

z

'

1

j

12cosq0

q0

y

sinqsinj

cosjsinj0wz'

w

=sinqcosjy'wx'

w

=y

k

+q

n

+j

k

'w

=

w

z

+w

n

+w

z

'w

x'

=y

sinq

sin

j

+q

cosjw

y

'

=y

sinq

cosj

-q

sinjB(j,q)13§6-4-2剛體一般運(yùn)動(dòng)的動(dòng)力學(xué)CCCCzCyCxv

v

v=

x

y

z

]

j

y

wz'

w

=

B(q,j)

qy'wx'

y

qj]BT

JB

q

C

CCC

xC

22T

=

1

[x

y

z

]m

y

+

1

[y

2j

1=

qT

Mq

00BT

JBCCC

zC

qy

j],

M

=

mzqT

=

[x

y22y'

Cy'z'y'x'Cx

Cy Cz

JCx'vCx

CymvCz

mT

=1[v

v

v

]JCz'

wz'

w

,

wx'

m

v

+1[w

w

w

]

J14§6-4-2剛體一般運(yùn)動(dòng)的動(dòng)力學(xué)例:

已知半徑為R

質(zhì)量為

m

的圓盤(pán)可繞柱鉸鏈C

以角速度w

1

轉(zhuǎn)動(dòng)，T形框架繞z

軸以角速度w

2

轉(zhuǎn)動(dòng)。求鉸鏈C

的約束力。xyzcw

2o1wA2、受力分析：AC=L＝2RFCzFCyCzMMCy3、建立動(dòng)力學(xué)方程：eiCFma

==eC

i

C

dtM

(F

)dLr解：取圓盤(pán)為研究對(duì)象1、運(yùn)動(dòng)分析：w

=w1

+w

2剛體一般運(yùn)動(dòng)mg大小為常量w1,w

215§6-4-2剛體一般運(yùn)動(dòng)的動(dòng)力學(xué)xyzc2wmgow

1AAC=L＝2RFCzFCyCzMMCyaC質(zhì)心運(yùn)動(dòng)定理：maC=

mg

+

FCa

=

Lw

2

=

2Rw

2C 2

2-

maC

=

FCy0

=

FCz

-

mg2F

=

-2mRw

2CyFCz

=

mg相對(duì)質(zhì)心的動(dòng)量矩定理：dte

C

=

MC

(Fi

)dLr21w

=

w

+w16§6-4-2剛體一般運(yùn)動(dòng)的動(dòng)力學(xué)xyzcw

2mgow

1A1M2Mjy'y'

y'

x'x'

x'

x'Mz'=

M=M-

Jz'

)w

x'w

z'

y'z'

-

Jy'

)w

y'w

z'Jz'wz'

+(Jy'

-

Jx'

)w

x'w

y'

=x'

J

w

+(JJ

w

+(Jw

z

'

=

w1x

'

2

x

'

2

1w

z

'

=

0w

y

'

=

w

2

cos

j

,

w

y

'

=

-w

2w1

sin

j

,w

=

w

sin

j

,

w

=

w

w

cos

j

,0.5mR2w

w

cosj

=

M

sinj

-

M

cosj1

2

1

2Mx'

=

M1

sinj

-

M

2

cosjM

y'

=

M1

cosj

+

M

2

sinjMz

'

=

02-

0.5mR

w1w

2

sinj

=

M1

cosj

+

M

2

sinj1

222121=

-

mR

w

wM

=

0,

M17§6-4-2剛體一般運(yùn)動(dòng)的動(dòng)力學(xué)xzcyjmgoy'x'My例:已知半徑為R質(zhì)量為m的圓盤(pán)可繞AC軸自由轉(zhuǎn)動(dòng)，OA軸在力偶M

的作用下繞鉛垂軸轉(zhuǎn)動(dòng)，忽略所有摩擦。建立系統(tǒng)運(yùn)動(dòng)微分方程。設(shè)AC=L＝2R，OA軸對(duì)z軸的轉(zhuǎn)動(dòng)慣量為Jz'方法一：應(yīng)用拉格朗日方程方法二：應(yīng)用動(dòng)量矩定理18方法一：應(yīng)用拉格朗日方程xzcyjmgoz'x'My22

21

1C2C+

T

r+

mvT

=

Jy2222Cz

'

z

'2Cy

'

y

'2Cx

'

x

'CT

r+

1

J

w+

1

J

w=

1

J

w12z

'J

=14mR2mR2

Jy

'x'=

J

=Cwx'

=

-y

sinjw

y

'

=

-y

cosjw

z

'

=

jv

=

Lyyjy'T

=

1

Jy

2

+

17

mR

2y

2

+

1

mR

2j

22

8

4系統(tǒng)的動(dòng)能：19方法一：應(yīng)用拉格朗日方程T

=

1

Jy

2

+

17

mR

2y

2

+

1

mR

2j

22

8

4'jq=

Qj?q

j

-dt

?d

?T

?T(

j

=1,2)=

M

,

Qj

=

0417(

J

+mR

2

)y

=

M21

mR

2j

=

0xzcyjmgo

z'

Qyy'x'My20方法二：應(yīng)用動(dòng)量矩定理xzcyjmgoz'y'x'MyO

OC C

C·

(

m

v

)

+

LrL

=

Jyk

+

rLrCz'

z'Cy'

y'C

Cx'

x'w

j'+J

w

k'=

J

w

i'+J研究整體w

x'

=

-y

sin

jw

y

'

=

-y

cosjw

z

'

=

jOzL

=

Jy

+

mL2y+(-JCx

'w

x'

sinj)w

cosj

)Cy

'

y

'+(-J174mR2

)yOzL

=

(J

+Oz=

MLy21方法二：應(yīng)用動(dòng)量矩定理xzcyjmgox'My研究圓盤(pán)：應(yīng)用相對(duì)質(zhì)心的動(dòng)量矩定理Jz'wz'

=

Mz'JCz

'w

z

'

=

JCz

'j

=

0j

=

0=

0LrC

Cx'w

x'i'+JCy

'w

y

'

j'+JCz

'w

z

'k'=

Jw

=

-y

sin

jx'w

=

-y

cosjy

'z'

w

=

j

z

'

z

'

M

=

0y'

x'x'

x'

y'

z'M

y'M)w

w

=Jz'w

z'

+(Jy'

-

Jx'

)w

x'w

y'

=

Mz'Jy'w

y'

+(Jx'

-

Jz'

)w

x'w

z'

=J

w

+(Jz'

-

J

y'22§6-4-2剛體一般運(yùn)動(dòng)的動(dòng)力學(xué)例：已知不計(jì)質(zhì)量的OA軸繞鉛垂軸轉(zhuǎn)動(dòng)，半徑為R的圓盤(pán)與碾盤(pán)無(wú)滑動(dòng)，若在轉(zhuǎn)軸上作用一個(gè)力偶M，圖示瞬時(shí)轉(zhuǎn)軸的角速度為w1，求圓盤(pán)在該瞬時(shí)的角加速度。（不計(jì)圓盤(pán)的厚度）2RMw

1OyCPrwjz'y'x'解：w

a

=

w

1

+

w

r=

yk

-

jk

'aw=

y

k

-

j

k

'

-

jk

'

k

'

=

w

1

·

k

'w

a=

y

k

-

j

k

'

+

w

1

·

w

r·

k

'

)-

jk

'

=

-j

(w

1=

w

1r1=

w

·

w·

(-jk

'

)23§6-4-2剛體一般運(yùn)動(dòng)的動(dòng)力學(xué)2Rw

1MOyCPvp

=

vc

+