剛體的一般運(yùn)動(dòng)課件

剛體的一般運(yùn)動(dòng)1

§10–1剛體繞平行軸轉(zhuǎn)動(dòng)的合成

2Chapter10:Generalmotionofarigidbody§10-1

Compositionofrotationsabouttwoparallelaxes

Compositionproblemofrotationsaboutparallelaxesisoftenencounteredinmechanicalengineering.Forinstance,theplanetgearwheelinaplanetgearmechanismhasplanemotion.Inpreviousstudy,wedealtwithaplanemotionofaplanetgearbydecomposingitintoatranslationandarotation.However,itismoreconvenienttoviewtheplanemotionofaplanetgearasacompositionoftworotations.

Kinematics3剛體的一般運(yùn)動(dòng)§10-1剛體繞平行軸轉(zhuǎn)動(dòng)的合成剛體繞平行軸轉(zhuǎn)動(dòng)的合成問(wèn)題在機(jī)械中經(jīng)常遇到。例如，行星圓柱齒輪機(jī)構(gòu)，行星輪作平面運(yùn)動(dòng)。前面所研究的平面運(yùn)動(dòng)是把它看成為平動(dòng)和轉(zhuǎn)動(dòng)的合成運(yùn)動(dòng)，但是在分析行星輪系的傳動(dòng)問(wèn)題時(shí)，將行星輪的平面運(yùn)動(dòng)看成為轉(zhuǎn)動(dòng)與轉(zhuǎn)動(dòng)的合成運(yùn)動(dòng)則比較方便。運(yùn)動(dòng)學(xué)4（Lookfortheanimationonthenextpage）Takeaplanetgearmechanismasanexample.StaticsystemisO1xyMovingsystemisO1x'y'Relativemotion:rotationaboutaxisO2,

ristherelativeangularvelocity.

Embroilingmotion:rotationaboutaxisO1,

eistheembroilingangularvelocity.

Kinematics5運(yùn)動(dòng)學(xué)靜系：O1xy動(dòng)系：O1x'y'相對(duì)運(yùn)動(dòng):繞O2軸轉(zhuǎn)動(dòng),

r為相對(duì)角速度。

牽連運(yùn)動(dòng):繞O1軸轉(zhuǎn)動(dòng),

e為牽連角速度。（翻頁(yè)請(qǐng)看動(dòng)畫）例如在行星輪系中6Kinematics7運(yùn)動(dòng)學(xué)8Kinematics9運(yùn)動(dòng)學(xué)10Inconclusion:Theabsoluteangularvelocity

aoftheplanefigure(theplanetgear)equalstothealgebraicsumoftheembroilingangularvelocity

eandtherelativeone

r.When

e

and

r

havesamedirection,thedirectionof

a

isthesameasthem.When

eand

rhaveoppositedirections,thedirectionof

a

isthesameasthebiggerone.

Atmomentt,itisinpositionO2A;Atmomentt+

t,itisinpositionO2'A‘.ThefigureshowsthatDifferentiatingitwithrespecttotgives:Kinematics11

由圖看出對(duì)t求導(dǎo)：即：平面圖形（這里指行星輪）的絕對(duì)角速度

a等于牽連角速度

e

與相對(duì)角速度

r的代表和．當(dāng)

e

與

r

轉(zhuǎn)向相同時(shí)轉(zhuǎn)向與兩者相同．當(dāng)

e與

r轉(zhuǎn)向相異時(shí)轉(zhuǎn)向與大者的相同．運(yùn)動(dòng)學(xué)ｔ時(shí)刻，O2A

位置；

t+

t

時(shí)刻,O2'A'位置12Next,wedeterminethepositionoftheinstantaneousvelocitycenterfortheplanefigureS.

Pis

theinstantaneouscenterofvelocityofthefigure.TheaxisacrosspointPandperpendiculartoaxesO1andO2iscalledinstantaneousaxis,everypointofwhichhaszerovelocity.

Sinceve=vr,andtheyhaveoppositedirections,vp=0,Pistheinstantaneouscenterofvelocity.Hence,

e

and

rhavethesamedirection

e

and

rhavetheoppositedirectionsKinematics13下面來(lái)確定圖形S的瞬心的位置

P點(diǎn)為圖形的速度瞬心,通過(guò)點(diǎn)P且與軸O1、O2平行的軸稱為瞬時(shí)軸,該軸上各點(diǎn)的速度都等于零。運(yùn)動(dòng)學(xué)

由于ve=vr,且方向相反,因此vp=0,P為速度瞬心。此時(shí)

e

與

r同轉(zhuǎn)向

e

與

r同反向14Conclusion：Tworotationsofarigidbodyabouttwoparallelaxescanbecomposedintoarotationaboutainstantaneousaxis.Theinstantaneousaxisisparalleltotheoriginalaxes，andtheratioofitsdistancetooneaxisandthattotheotherisinverselyproportionaltotheratiooftheirangularvelocities.Whentheserotationsturntothesamedirection,theinstantaneousaxisliesbetweenthetwoaxes, .Otherwise,itisintheextensionofthelinesegmentbetweenthetwoaxesandliesinthesideoftheaxiswithlargerangularvelocity .Kinematics15即：剛體繞兩平行軸的轉(zhuǎn)動(dòng)可合成為繞瞬軸的轉(zhuǎn)動(dòng)，瞬軸與原兩軸共面且平行，到兩軸的距離與兩角速度大小成反比。同向轉(zhuǎn)動(dòng)時(shí)，瞬軸在兩軸之間，，轉(zhuǎn)向與兩者相同；反向轉(zhuǎn)動(dòng)時(shí),瞬軸在兩軸之外,在角速度值大的一側(cè),，轉(zhuǎn)向與大者的相同。運(yùn)動(dòng)學(xué)16[Example1]GearandhavethesameradiusR,andtheradiusofgearisr.Theyaremeshedonebyone,andgearisfixed.GearandispinedtothecrankO1O3respectively，andtheycanrotatearoundaxesO2andO3.CrankO1O3rotatewith

0

．Finetheangularvelocity3rofgearIIIrelativetothecrank,itsabsoluteangularvelocity3,andthevelocitiesofpointAandBrespectivelyattheinstantshowninthefigure.Kinematics17運(yùn)動(dòng)學(xué)[例1]

齒輪

、

半徑均為R,齒輪

半徑為r,依次互嚙合,輪

固定不動(dòng)，輪

和輪

裝在曲柄O1O3上，可分別繞O2、O3軸轉(zhuǎn)動(dòng)。設(shè)曲柄O1O3以

0順時(shí)針轉(zhuǎn)動(dòng)．試求齒輪III相對(duì)于曲柄轉(zhuǎn)動(dòng)的角速度

3r

和齒輪

的絕對(duì)角速度

3以及圖示瞬時(shí)A、B

兩點(diǎn)的速度。18Solution：TakeO1O3asthemovingcoordinatesystem,anddenote

1r,

2rand

３rastheangularvelocitiesofgear,andrelativetotheconnectingrod.ThetransmissionratiocanbeexpressedasEmployingthetheorydevelopedabove,wehaveKinematics19解：取系桿O1O3為動(dòng)系,

1r、

2r、

３r

分別是

、

、

輪相對(duì)于系桿的角速度,根據(jù)傳動(dòng)比公式，可得由平行軸轉(zhuǎn)動(dòng)的合成理論，得運(yùn)動(dòng)學(xué)20Itisclearthatgearis