剛體的平面運(yùn)動(dòng)課件

剛體的平面運(yùn)動(dòng)1

§9–1Introductiontoplanemotionofarigidbody

§9–2Planemotioncanbedecomposedinto translationandrotation·Equationsof planemotion

§9–3Velocityofapointinaplanefigure

§9–4Accelerationofapointinaplanefigure·InstantaneouscenterofzeroaccelerationLessonforproblemsolvingChapter9:Planemotionofarigidbody2

§9–1剛體平面運(yùn)動(dòng)的概述§9–2平面運(yùn)動(dòng)分解為平動(dòng)和轉(zhuǎn)動(dòng)·

剛體的平面運(yùn)動(dòng)方程§9–3平面圖形內(nèi)各點(diǎn)的速度§9–4平面圖形內(nèi)各點(diǎn)的加速度·

加速度瞬心的概念習(xí)題課剛體的平面運(yùn)動(dòng)3Inengineering,weoftenencountertheplanemotionofarigidbody,whichismorecomplicated．Investigationintothiskindofmotionisconductedonthebasisoftranslationandrotationofarigidbodyusingcompositionanddecompositionofmotions.Decomposingaplanemotionintotranslationandrotation,andemployingthetheoryofcompositionofmotions，theformulaeforfindingthevelocityandaccelerationofpointintherigidbodycanbederived.

§9-1Introductiontoplanemotionofarigidbody1.Definitionofplanemotionofarigidbody

Thedistancebetweenanypointinarigidbodyandafixedplanealwayskeepsunchangedduringitsmotion.Inotherwords,anypointintherigidbodymovesinaplaneparalleltothefixedplane.Themotiondescribedaboveiscalledplanemotionofarigidbody.Kinematics4

剛體的平面運(yùn)動(dòng)是工程上常見的一種運(yùn)動(dòng)，這是一種較為復(fù)雜的運(yùn)動(dòng)．對(duì)它的研究可以在研究剛體的平動(dòng)和定軸轉(zhuǎn)動(dòng)的基礎(chǔ)上，通過運(yùn)動(dòng)合成和分解的方法，將平面運(yùn)動(dòng)分解為上述兩種基本運(yùn)動(dòng)．然后應(yīng)用合成運(yùn)動(dòng)的理論，推導(dǎo)出平面運(yùn)動(dòng)剛體上一點(diǎn)的速度和加速度的計(jì)算公式．運(yùn)動(dòng)學(xué)§9-1剛體平面運(yùn)動(dòng)的概述一．平面運(yùn)動(dòng)的定義

在運(yùn)動(dòng)過程中，剛體上任一點(diǎn)到某一固定平面的距離始終保持不變．也就是說，剛體上任一點(diǎn)都在與該固定平面平行的某一平面內(nèi)運(yùn)動(dòng)．具有這種特點(diǎn)的運(yùn)動(dòng)稱為剛體的平面運(yùn)動(dòng)．5Forexample:Examinethemotionoftherodinacrank-rodmechanism.SincepointAmovesinacircularpath,andpintBmovealongastraightline，themotionoftherodABisneithertranslationnorrotationaboutafixedaxis，butplanemotion.Kinematics6例如:曲柄連桿機(jī)構(gòu)中連桿AB的運(yùn)動(dòng)，A點(diǎn)作圓周運(yùn)動(dòng)，B點(diǎn)作直線運(yùn)動(dòng)，因此，AB桿的運(yùn)動(dòng)既不是平動(dòng)也不是定軸轉(zhuǎn)動(dòng)，而是平面運(yùn)動(dòng)．運(yùn)動(dòng)學(xué)7KinematicsPleaselookattheanimation8運(yùn)動(dòng)學(xué)請(qǐng)看動(dòng)畫9

2．Simplificationofaplanemotion

Aplanemotionofarigidbodycansimplifiedtoamotionofaplanefigureintheplaneitself.Asindicatedinthefigure,whenstudyingaplanemotionofarigidbody,wedonotneedtoconsideritsgeometricalshapeandsizearenotneeded，andinstead,consideringthemotionofaplanefigureisenoughtodeterminethevelocityandaccelerationofanypointintherigidbody.Kinematics10運(yùn)動(dòng)學(xué)

二．平面運(yùn)動(dòng)的簡(jiǎn)化

剛體的平面運(yùn)動(dòng)可以簡(jiǎn)化為平面圖形S在其自身平面內(nèi)的運(yùn)動(dòng)．即在研究平面運(yùn)動(dòng)時(shí)，不需考慮剛體的形狀和尺寸，只需研究平面圖形的運(yùn)動(dòng)，確定平面圖形上各點(diǎn)的速度和加速度．11§9–2Planemotioncanbedecomposedintotranslationandrotation·Equationsofplanemotion

1.Equationsofplanemotion

Todeterminethepositionofaplanefigure,whichrepresentstheplanemotionofarigidbody,onlythepositionofalinesegmentinthisfigureisneededtobedetermined.ThepositionofalinesegmentABcanbedeterminedbythecoordinatesofpointAandtheanglebetweenABandtheaxisx.Therefore,thepositionofthefigureScanbedeterminedbythesethreeindependentvariables.Hence,wehave

Kinematics12§9-2平面運(yùn)動(dòng)分解為平動(dòng)和轉(zhuǎn)動(dòng)·

剛體的平面運(yùn)動(dòng)方程運(yùn)動(dòng)學(xué)

一．平面運(yùn)動(dòng)方程為了確定代表平面運(yùn)動(dòng)剛體的平面圖形的位置，我們只需確定平面圖形內(nèi)任意一條線段的位置．

任意線段AB的位置可用A點(diǎn)的坐標(biāo)和AB與x軸夾角表示．因此圖形S的位置決定于三個(gè)獨(dú)立的參變量．所以13

2．Planemotioncanbedecomposedintotranslationandrotation

WhenpointAinthefigureSkeepsstatic，therigidbodyrotatesaboutafixedaxisWhentheangle

inthefigureSkeepsunchanged，therigidbodyhastranslatorymotion.Therefore,aplanemotioncanbeviewedasthecompositionofatranslationandrotation．EquationsofplanemotionThecanbefoundfromthisequationatanyinstantt,andthusthelocationoftheplanefigureScanbedetermined.Kinematics14

二．平面運(yùn)動(dòng)分解為平動(dòng)和轉(zhuǎn)動(dòng)當(dāng)圖形Ｓ上Ａ點(diǎn)不動(dòng)時(shí)，則剛體作定軸轉(zhuǎn)動(dòng)當(dāng)圖形Ｓ上

角不變時(shí)，則剛體作平動(dòng)．故剛體平面運(yùn)動(dòng)可以看成是平動(dòng)和轉(zhuǎn)動(dòng)的合成運(yùn)動(dòng)．運(yùn)動(dòng)學(xué)平面運(yùn)動(dòng)方程對(duì)于每一瞬時(shí)

t

，都可以求出對(duì)應(yīng)的，圖形S在該瞬時(shí)的位置也就確定了。15Example

Motionofawheel．

Theplanemotionofthewheelcanbeviewedasthecompositionofthetranslationwiththevehiclebodyandtherotationrelativetothevehicle.Planemotionofthewheelrelativetothestaticreferencesystem(absolutemotion)Translationofthevehiclebody(movingsystemAx

y

)relativetostaticsystem (convectedmotion)Rotationofthewheelrelativetothevehicle(movingsystemAx

y

) (relativemotion)Kinematics16運(yùn)動(dòng)學(xué)例如車輪的運(yùn)動(dòng)．車輪的平面運(yùn)動(dòng)可以看成是車輪隨同車廂的平動(dòng)和相對(duì)車廂的轉(zhuǎn)動(dòng)的合成．

車輪對(duì)于靜系的平面運(yùn)動(dòng)（絕對(duì)運(yùn)動(dòng)）車廂（動(dòng)系A(chǔ)x

y

)相對(duì)靜系的平動(dòng)（牽連運(yùn)動(dòng)）車輪相對(duì)車廂（動(dòng)系A(chǔ)x

y

）的轉(zhuǎn)動(dòng)（相對(duì)運(yùn)動(dòng)）

17ThearbitrarilyselectedpointAiscalledapole.Thus,PlanemotionofawheelTranslationwiththepoleARotationaboutthepoleAAplanemotionofarigidbodycanbedecomposedintoatranslationwithapoleandarotationaboutthepole．Kinematics18運(yùn)動(dòng)學(xué)

我們稱動(dòng)系上的原點(diǎn)Ａ為基點(diǎn)，于是車輪的平面運(yùn)動(dòng)隨基點(diǎn)A的平動(dòng)繞基點(diǎn)A'的轉(zhuǎn)動(dòng)剛體的平面運(yùn)動(dòng)可以分解為隨基點(diǎn)的平動(dòng)和繞基點(diǎn)的轉(zhuǎn)動(dòng)．19

Anotherexample:planefigureＳmovesfrompositionItopositionIIinainterval

t.SelectpointAasapole:TranslatethefiguretogetherwithAtothepositionA'B'',andthenrotateitaroundAthroughanangletothefinalpositionA'B‘.SelectpointAasapole:TranslateittogetherwithBtothepositionA''B',andthenrotateitaroundBthroughanangletothefinalpositionA'B'Clearly,AB

A'B''

A''B'，wehaveKinematics20運(yùn)動(dòng)學(xué)

再例如:平面圖形Ｓ在

ｔ時(shí)間內(nèi)從位置I運(yùn)動(dòng)到位置II

以A為基點(diǎn):

隨基點(diǎn)A平動(dòng)到A'B''后,繞基點(diǎn)轉(zhuǎn)角到A'B'

以B為基點(diǎn):

隨基點(diǎn)B平動(dòng)到A''B'后,繞基點(diǎn)轉(zhuǎn)角到A'B'圖中看出：AB

A'B''

A''B'，于是有21

Inconclusion，thetranslationinaplanemotiondependsontheselectionofthepole，however,therotationabouttheselectedpoleDOESNOTdependonthechoiceofapole．(Inotherwords,theinstant,ofanyrotationsaboutANYpolesarethesame）Theselectionofthepoleisarbitrary.(Weusuallyselectapointwithknownmotionasthepole.)Kinematics22運(yùn)動(dòng)學(xué)

所以，平面運(yùn)動(dòng)隨基點(diǎn)平動(dòng)的運(yùn)動(dòng)規(guī)律與基點(diǎn)的選擇有關(guān)，而繞基點(diǎn)轉(zhuǎn)動(dòng)的規(guī)律與基點(diǎn)選取無關(guān)．(即在同一瞬間，圖形繞任一基點(diǎn)轉(zhuǎn)動(dòng)的,都是相同的）基點(diǎn)的選取是任意的。(通常選取運(yùn)動(dòng)情況已知的點(diǎn)作為基點(diǎn))23Crank-rodmechanismRodABhasplanemotionDecompositionoftheplanemotion（pleaselookattheanimation）Kinematics24運(yùn)動(dòng)學(xué)曲柄連桿機(jī)構(gòu)AB桿作平面運(yùn)動(dòng)平面運(yùn)動(dòng)的分解（請(qǐng)看動(dòng)畫）25§9-3velocityofanypointinaplanefigure

Employthetheoremofvelocitycomposition

thevelocityatpointBcanbeexpressedasThevelocityofpointAinfigureSandtherotationalvelocity

ofthefigurearegiven,find.SelectAasthepole,andfixthemovingreferencesystemtopointA.Themotionofthemovingsystemistranslation.ConsiderthemovingpointB,itsmotioncanbeviewedasthecompositionofthetranslation,theconvectedmotion,andtherotation,therelativemotion pointtotherotationdirection.Kinematics1.pole-basedmethod(compositionmethod)26§9-3平面圖形內(nèi)各點(diǎn)的速度

運(yùn)動(dòng)學(xué)根據(jù)速度合成定理則Ｂ點(diǎn)速度為：

一．基點(diǎn)法（合成法）取B為動(dòng)點(diǎn),則B點(diǎn)的運(yùn)動(dòng)可視為牽連運(yùn)動(dòng)為平動(dòng)和相對(duì)運(yùn)動(dòng)為圓周運(yùn)動(dòng)的合成已知：圖形S內(nèi)一點(diǎn)A的速度，圖形角速度

求：指向與

轉(zhuǎn)向一致．取A為基點(diǎn),將動(dòng)系固結(jié)于A點(diǎn),動(dòng)系作平動(dòng)。27SincepointAandBarearbitrarilyselected,theequation

givestherelationshipbetweenthevelocitiesofanytwopointinthefigure.Notingthat

,projectingthisequationtoAB，gives——theoremofvelocityprojectionI.e.,thevelocityprojectionsofanytwopointinafigureonthelinelinkingthesetwopointsareidentical．Thismethodforfindthevelocityofapointiscalledvelocityprojectionmethod．Thatis,thevelocityofanypointinthefigureisobtainedasthegeometricsumofthevelocityofthepoleandtherelativerotationalvelocitywithrespecttothepole.Suchamethodoffindingthevelocityiscalledpole-basedmethod,orcompositionmethod,whichisabasicmethodtofindthevelocityofapointinafigure．2．VelocityprojectionmethodKinematics28由于A,B點(diǎn)是任意的，因此表示了圖形上任意兩點(diǎn)速度間的關(guān)系．由于恒有，因此將上式在AB上投影，有—速度投影定理即平面圖形上任意兩點(diǎn)的速度在該兩點(diǎn)連線上的投影彼此相等．這種求解速度的方法稱為速度投影法．運(yùn)動(dòng)學(xué)即平面圖形上任一點(diǎn)的速度等于基點(diǎn)的速度與該點(diǎn)隨圖形繞基點(diǎn)轉(zhuǎn)動(dòng)的速度的矢量和．這種求解速度的方法稱為基點(diǎn)法，也稱為合成法．它是求解平面圖形內(nèi)一點(diǎn)速度的基本方法．二．速度投影法293.Instantaneousvelocitycentermethod

(1)BackgroundIfapointwhosevelocityiszeroisselectedasthepole,theprocessoffindingthevelocityofanypointwillbegreatlysimplified.Hence,itisnaturaltoaskifsuchapointexistsinanyinstant.Ifitdoesexist,howtofindsuchapoint?

(2)ConceptofInstantaneousvelocitycenter

ConsideraplanefigureS.ThevelocityofpointAis

,andtheangularvelocityofthefigureis.Take

alineALalongthedirectionof,andthenturnit90ointhedirectionof

to

AL‘a(chǎn)ndfindthepointPinAL‘byletting,wehave Kinematics30

三．瞬時(shí)速度中心法（速度瞬心法）

1.問題的提出若選取速度為零的點(diǎn)作為基點(diǎn)，求解速度問題的計(jì)算會(huì)大大簡(jiǎn)化．于是，自然會(huì)提出，在某一瞬時(shí)圖形是否有一點(diǎn)速度等于零？如果存在的話，該點(diǎn)如何確定？運(yùn)動(dòng)學(xué)

２．速度瞬心的概念平面圖形S，某瞬時(shí)其上一點(diǎn)A速度,圖形角速度

，沿方向取半直線AL,然后順

的轉(zhuǎn)向轉(zhuǎn)90o至AL'的位置,在AL'上取長(zhǎng)度則： 31Atanyinstant,theremustexistasolepointwhosevelocityiszero，whichiscalledtheinstantaneousvelocitycenterofthisfigureatthisinstant.3.MethodstodeterminetheInstantaneousvelocitycenter①Whenthevelocityofapointandtheangularvelocity

ofthefigureareknown,theinstantaneousvelocitycenter(pointP)canbedetermined.

andpointPisinthedirectionofthelineformedbyrotatingthethrough90ointhedirectionofaroundpointA．②Whenaplanefigurerollsalongafixedsurfacewithoutslipping,thecontactpointPbetweenthefigureandthefixedsurfacewillbetheinstantaneousvelocitycenter.

Kinematics32

即在某一瞬時(shí)必唯一存在一點(diǎn)速度等于零，該點(diǎn)稱為平面圖形在該瞬時(shí)的瞬時(shí)速度中心，簡(jiǎn)稱速度瞬心．運(yùn)動(dòng)學(xué)３．幾種確定速度瞬心位置的方法

①已知圖形上一點(diǎn)的速度和圖形角速度

，可以確定速度瞬心的位置．（P點(diǎn)）且Ｐ在順

轉(zhuǎn)向繞A點(diǎn)轉(zhuǎn)90o的方向一側(cè)．

②已知一平面圖形在固定面上作無滑動(dòng)的滾動(dòng),則圖形與固定面的接觸點(diǎn)P為速度瞬心．33

④ThemagnitudesofthevelocitiesoftwopointsAandBatanyinstantaregiven,and .(b)(a)

③WhenthedirectionsofthevelocitiesattwopointsAandBinafigureareknown,and,drawlinesfromAandBperpendiculartorespectively,andthecrosspointPofthesetwolineswillbetheinstantaneousvelocitycenter.Kinematics34

運(yùn)動(dòng)學(xué)

④

已知某瞬時(shí)圖形上A,B兩點(diǎn)速度大小,且(b)(a)

③已知某瞬間平面圖形上A,B兩點(diǎn)速度的方向，且 。過A,B兩點(diǎn)分別作速度的垂線,交點(diǎn)

P即為該瞬間的速度瞬心。35

Inaddition,ifvA＝vB

incase

④

，themotionisinstantaneoustranslationtoo.

⑤ThevelocitiesoftwopointsAandBpointtothesamedirectionatanyinstant,buttheyarenotperpendiculartolineAB.

Inthiscase,theinstantaneousvelocitycenterisindefinitelyfaraway,andtheangularvelocity

=0,i.e.allpointinthefigurehavethesamevelocityatthisinstantoftime.Suchamotioniscalledinstantaneoustranslation,(buttheiraccelerationsarenotidentical).Kinematics36

運(yùn)動(dòng)學(xué)另：對(duì)

種(a)的情況，若vA＝vB，則是瞬時(shí)平動(dòng)．⑤已知某瞬時(shí)圖形上A,B兩點(diǎn)的速度方向相同，且不與AB連線垂直．此時(shí),圖形的瞬心在無窮遠(yuǎn)處,圖形的角速度

=0,圖形上各點(diǎn)速度相等,這種情況稱為瞬時(shí)平動(dòng).(此時(shí)各點(diǎn)的加速度不相等)37

Example:Attheinstantoftimeshowninthefigure,therodBCinthecrank-rodmechanismhasinstantaneoustranslation.TheangularvelocityoftherodBC,ThevelocitiesofallpointsinBCareidentical,buttheiraccelerationsarenot.If

isuniform

，thenButthedirectionof

isalongAC,

Instantaneoustranslationisdifferentfromtranslation.Kinematics38

例如:曲柄連桿機(jī)構(gòu)在圖示位置時(shí)，連桿BC作瞬時(shí)平動(dòng)．此時(shí)連桿BC的圖形角速度，BC桿上各點(diǎn)的速度都相等.但各點(diǎn)的加速度并不相等．設(shè)勻

，則而的方向沿AC的，瞬時(shí)平動(dòng)與平動(dòng)不同運(yùn)動(dòng)學(xué)39４.InstantaneousvelocitycentermethodThemethodforfindingthevelocityofapointinafigureusinginstantaneousvelocitycenteriscalledinstantaneousvelocitycentermethod.

Atanyinstantoftime,themotionofafigurecanbeviewedastherotationaroundtheinstantaneousvelocitycenter.

IfPistheinstantaneousvelocitycenter,thevelocityofanypointA anditsdirection

AP,pointingthesamedirectionwith.

５.Cautions

Thepositionofthetheinstantaneousvelocitycenterisnotfixedatalltime，itchangesinstantlywithtime,andexistsuniquelyatanyinstantoftime.

Onlythevelocityattheinstantaneousvelocitycenteriszero,butitsaccelerationisnotcertainlyzero.Thisdiffersfromtherotationaboutafixedaxis.

Whenarigidbodyisininstantaneoustranslation,althoughthevelocitiesatallpointsinitareidentical,buttheiraccelerationsarenotnecessarilyidentical.Thisisdifferentfromthetranslatorymotion.Kinematics40４.速度瞬心法利用速度瞬心求解平面圖形上點(diǎn)的速度的方法,稱為速度瞬心法.平面圖形在任一瞬時(shí)的運(yùn)動(dòng)可以視為繞速度瞬心的瞬時(shí)轉(zhuǎn)動(dòng)，速度瞬心又稱為平面圖形的瞬時(shí)轉(zhuǎn)動(dòng)中心。若P點(diǎn)為速度瞬心，則任意一點(diǎn)A的速度方向

AP，指向與

一致。

運(yùn)動(dòng)學(xué)５.注意的問題

速度瞬心在平面圖形上的位置不是固定的，而是隨時(shí)間不斷變化的。在任一瞬時(shí)是唯一存在的。

速度瞬心處的速度為零,加速度不一定為零。不同于定軸轉(zhuǎn)動(dòng)

剛體作瞬時(shí)平動(dòng)時(shí)，雖然各點(diǎn)的速度相同，但各點(diǎn)的加速度是不一定相同的。不同于剛體作平動(dòng)。41Solution:Inthismechanism,OArotatesaboutafixedaxis,ABhasplanemotion,andpistonBhastranslation.

Pole-basedmethod（compositionmethod）

ConsiderABandtakeAasthepole.Itsdirectionisshowninthefigures.（）[Example1]Inacrank-rodmechanism,OA=AB=l,andthecrankOArotateswithauniform.Find:when

=45o,thevelocityofpistonBandtheangularvelocityofrodAB．KinematicsSince DrawtheparallelogramofvelocitiesatpointB,asshowninthefigure.42解：機(jī)構(gòu)中,OA作定軸轉(zhuǎn)動(dòng),AB作平面運(yùn)動(dòng),滑塊B作平動(dòng)。

基點(diǎn)法（合成法）研究AB，以A為基點(diǎn)，且方向如圖示。（）運(yùn)動(dòng)學(xué)[例1]

已知：曲柄連桿機(jī)構(gòu)OA=AB=l，取柄OA以勻

轉(zhuǎn)動(dòng)。求：當(dāng)

=45o時(shí),滑塊B的速度及AB桿的角速度．根據(jù)在Ｂ點(diǎn)做速度平行四邊形，如圖示。43（）Trytocomparethesethreemethods.Employingtheoremofvelocityprojection ,WehaveWecannotfind.

Velocityprojectionmethod.ConsiderAB.,anditsdirection

OA,

isalongBO.

InstantaneousvelocitycentermethodConsiderAB.Sincethedirectionsofareknown,wecandeterminetheinstantaneouscenterofvelocityatpointP.Kinematics44（）試比較上述三種方法的特點(diǎn)。運(yùn)動(dòng)學(xué)根據(jù)速度投影定理不能求出

速度投影法研究AB,

,方向

OA,方向沿BO直線

速度瞬心法研究AB，已知的方向，因此可確定出P點(diǎn)為速度瞬心45§9-4Accelerationofapointinaplanefigure·InstantaneouscenterofaccelerationTakeAasthepole,andfixthemovingreferencesystemontoA.TakeBasthemovingpoint,themotionofpointBcanbedecomposedintoarelativemotion(circularmotion)andaconvectedmotion(translation)withthepole.Hence,employingthetheoremofaccelerationcomposition ,thefollowingformulacanbeobtained:1.Pole-basedmethod(compositionmethod)Ataninstantoftime,theaccelerationofapointAinafigureS,

and

aregiven.FindtheaccelerationofanypointBinthefigure.Kinematics46§9-4平面圖形內(nèi)各點(diǎn)的加速度加速度瞬心的概念取A為基點(diǎn)，將平動(dòng)坐標(biāo)系固結(jié)于A點(diǎn)取B動(dòng)點(diǎn)，則B點(diǎn)的運(yùn)動(dòng)分解為相對(duì)運(yùn)動(dòng)為圓周運(yùn)動(dòng)和牽連運(yùn)動(dòng)為平動(dòng)．于是,由牽連平動(dòng)時(shí)加速度合成定理可得如下公式．運(yùn)動(dòng)學(xué)一.基點(diǎn)法(合成法)已知：圖形S內(nèi)一點(diǎn)A的加速度和圖形的

,

（某一瞬時(shí)）。求：該瞬時(shí)圖形上任一點(diǎn)B的加速度。47where

,direction

AB,pointingtothesamedirectionwith

;whichisalongABandpointstoA.

Theaccelerationofanypointinafigureequalstothegeometricsumoftheaccelerationofthepole,tangentialandnormalaccelerationsofthispointrotatingaboutthepoletogetherwiththefigure.Thismethodtofindtheaccelerationofapointiscallpole-basedmethod,orcompositionmethod,whichisthebasicmethodtofindingtheaccelerationofapointinaplanefigure.Theformulagivenaboveisanequationintermsofplanevectors.Therefore,wecansolvetwounknownsfromitprovidedthattheothervariablesaregiven.Sincethedirectionsof

arealwaysknown,tosolvetheunknowns,onlyfourothervariablesareneeded.Kinematics48其中：，方向

AB，指向與

一致；，方向沿AB，指向A點(diǎn)。運(yùn)動(dòng)學(xué)即平面圖形內(nèi)任一點(diǎn)的加速度等于基點(diǎn)的加速度與該點(diǎn)隨圖形繞基點(diǎn)轉(zhuǎn)動(dòng)的切向加速度和法向加速度的矢量和。這種求解加速度的方法稱為基點(diǎn)法，也稱為合成法。是求解平面圖形內(nèi)一點(diǎn)加速度的基本方法。上述公式是一平面矢量方程。需知其中六個(gè)要素，方能求出其余兩個(gè)。由于方位總是已知，所以在使用該公式中，只要再知道四個(gè)要素，即可解出問題的待求量。49

2.Instantaneouscenterofacceleration

DuetothedependentofonthepointB,wecanalwaysfindapointQinthefigure，atwhichtherelativeaccelerationhasthesamemagnitudebutoppositedirectionwiththeaccelerationinthepolesothattheabsoluteacceleration.SuchapointQiscalledtheinstantaneouscenterofacceleration.Kinematics50

二．加速度瞬心．由于的大小和方向隨B點(diǎn)的不同而不同,所以總可以在圖形內(nèi)找到一點(diǎn)Q，在此瞬時(shí)，相對(duì)加速度大小恰與基點(diǎn)A的加速度等值反向，其絕對(duì)加速度,Q點(diǎn)就稱為圖形在該瞬時(shí)的加速度瞬心．運(yùn)動(dòng)學(xué)51[Note]

Ingeneral,theinstantaneouscenterofaccelerationdoesnotcoincidewiththeinstantaneouscenterofvelocity.

Ingeneral，thereisnosimilarrelationshipbetweentheaccelerationsattwopointsgiveninthetheoremofvelocityprojection.Thatistosay,thefollowingrelationshipdoesnotholdingeneral:Onlyinthespecialcasewhere

=0,thefigureisininstantaneoustranslation,holds.Inotherwords,iftheangularvelocityofaplanefigureiszeroataninstantoftime,theaccelerationprojectionsofanytwopointinafigureonthelinelinkingthesetwopointsareidentical

．Kinematics52運(yùn)動(dòng)學(xué)[注]

一般情況下,加速度瞬心與速度瞬心不是同一個(gè)點(diǎn)．

一般情況下，對(duì)于加速度沒有類似于速度投影定理的關(guān)系式.即一般情況下,圖形上任意兩點(diǎn)A,B的加速度

若某瞬時(shí)圖形

=0,即瞬時(shí)平動(dòng),則有即若平面圖形在運(yùn)動(dòng)過程中某瞬時(shí)的角速度等于零，則該瞬時(shí)圖形上任意兩點(diǎn)的加速度在這兩點(diǎn)連線上的投影相等．53

Sinceitisnoteasytofindtheinstantaneouscenterofaccelerationandtheredoesnotexistsarelationshipsimilartothetheoremofvelocityprojection,instantaneouscenterofaccelerationisnotwidelyusedtofindaccelerationsofagivenpoint.Instead,thepolemethodisoftenemployed.Analysis：

magnitude？√Ｒ

Ｒw

2

direction？√√√Therefore,

and

shouldbesolvedfirst.（）

[Example1]

AwheelwithradiusRrollsonaplanesurfacewithoutslipping.ThevelocityandaccelerationattheitscenterOaregiven.FindtheaccelerationatthecontactpointP.Solution:ThewheelhasplanemotionandPistheinstantaneousvelocitycenter.Kinematics54

由于加速度瞬心的位置不象速度瞬心那樣容易確定，且一般情況下又不存在類似于速度投影定理的關(guān)系式，故常采用基點(diǎn)法求圖形上各點(diǎn)的加速度或圖形角加速度．分析：大小？√Ｒ

Ｒw

2

方向？√√√故應(yīng)先求出

．（）運(yùn)動(dòng)學(xué)

[例1]

半徑為R的車輪沿直線作純滾動(dòng),已知輪心O點(diǎn)的速度及加速度,求車輪與軌道接觸點(diǎn)P的加速度．解：輪O作平面運(yùn)動(dòng)，P為速度瞬心，55Sincetheequationgivenaboveisvalidatanyinstant,andpointOmovesalongastraightline,weobtain（）Itisevidentthattheaccelerationattheinstantaneouscenterofvelocityisnotzero,whichmeansthatpointPisnottheinstantaneouscenterofacceleration.Inthiscase,theaccelerationofattheinstantaneouscenterofvelocityPpointstothecenterofthewheel.

TakingOasthepole，yields

Asshowninthevectordiagramoftheaccelerations,（andareequalinmagnitudebutoppositeindirection）

i.e.

Kinematics56

由于此式在任何瞬時(shí)都成立，且O點(diǎn)作直線運(yùn)動(dòng)，故而（）

由此看出，速度瞬心P的加速度并不等于零，即它不是加速度瞬心．當(dāng)車輪沿固定的直線軌道作純滾動(dòng)時(shí)，其速度瞬心P的加速度指向輪心．運(yùn)動(dòng)學(xué)以O(shè)為基點(diǎn)，有其中：做出加速度矢量圖，由圖中看出：（與等值反向）

即57Solution:(a)ABhastranslation,[Example2]Attheinstantshowninthefigure,O1A=O2BandO1A/O2B.Are

1and

2or

1and

2identicalincase(a)and(b).

(a)(b)Kinematics58解：(a)AB作平動(dòng)，運(yùn)動(dòng)學(xué)[例2]

已知O1A=O2B,圖示瞬時(shí)O1A/O2B

試問(a),(b)兩種情況下

1和

2，

1和

2是否相等？(a)(b)59(b)ABhasplanemotion,anditisininstantaneoustranslationatthisinstantoftime.Therefore .Kinematics60(b)AB作平面運(yùn)動(dòng),圖示瞬時(shí)作瞬時(shí)平動(dòng),此時(shí)運(yùn)動(dòng)學(xué)61[Example3]Inacrank-rodmechanism,theradiusofthewheelisR=15cm,n=60rpmFind

Ｂand

Ｂ

ofthewheelwhen

=60oandOA

AB.Lookfortheanimationonthenextpage

Kinematics62運(yùn)動(dòng)學(xué)[例3]

曲柄滾輪機(jī)構(gòu)滾子半徑R=15cm,n=60rpm求：當(dāng)

=60o時(shí)(OA

AB),滾輪的

Ｂ，

Ｂ．翻頁(yè)請(qǐng)看動(dòng)畫

63LookattheanimationKinematics64請(qǐng)看動(dòng)畫65

Solution：OArotatesaboutafixedaxis,ABandwheelBhaveplanemotions.Consider

rodAB.（）P１istheinstantaneousvelocitycenterofABAnalysis:Tofind

Ｂ,

Ｂ

ofthewheel,vB

and

aB

shouldbeworkedoutfirst.P2P1vBP2istheinstantaneousvelocitycenterofthewheelKinematics66

解：OA定軸轉(zhuǎn)動(dòng),AB桿和輪B作平面運(yùn)動(dòng)研究AB：（）P１為其速度瞬心運(yùn)動(dòng)學(xué)分析:要想求出滾輪的

Ｂ,

Ｂ先要求出vB,

aBP2P1vBP2為輪速度瞬心67TakeAasthepole,PointingtoOMagnitude？√？√Direction√√√√Drawthevectordiagramforaccelerations.ProjectingtheaboveequationontoBA,gives）(）(ConsiderwheelB：its

instantaneousvelocitycenterisP2Kinematics68運(yùn)動(dòng)學(xué)取A為基點(diǎn)，指向O點(diǎn)大??？√？√方向√√√√作加速度矢量圖，將上式向BA線上投影）(）(研究輪B：P2為其速度瞬心69LessonofproblemsolvingforChapter91.Conceptsandcontent

(1)Definitionofplanemotion

Thedistancebetweenanypointinarigidbodyandafixedplanealwayskeepsunchangedduringitsmotion.

(2)Simplificationofplanemotion

ThemotionofaplanefigureintherigidbodyparallelwiththefixedplaneSinitsownplanecanbeusedtorepresenttheplanemotionoftheentirerigidbody.

(3)Decompositionofplanemotion.Itcanbedecomposedinto:

(4)Pole

Theoreticallyspeaking,anyapointinthefigurecanbeselectedasthepole.Weusuallychoseapointwithknownmotionasthepole.Translationtogetherwiththepole（itdependsonthechoiceofpole）

Rotationaroundthepole（itdoesnotdependontheselectionofpole）Kinematics70剛體平面運(yùn)動(dòng)習(xí)題課一．概念與內(nèi)容

1.剛體平面運(yùn)動(dòng)的定義剛體運(yùn)動(dòng)時(shí)，其上任一點(diǎn)到某固定平面的距離保持不變．

2.剛體平面運(yùn)動(dòng)的簡(jiǎn)化可以用剛體上一個(gè)與固定平面平行的平面圖形S在自身平面內(nèi)的運(yùn)動(dòng)代替剛體的整體運(yùn)動(dòng)．

3.剛體平面運(yùn)動(dòng)的分解分解為

4.基點(diǎn)

可以選擇平面圖形內(nèi)任意一點(diǎn),通常是運(yùn)動(dòng)狀態(tài)已知的點(diǎn)．隨基點(diǎn)的平動(dòng)（平動(dòng)規(guī)律與基點(diǎn)的選擇有關(guān)）繞基點(diǎn)的轉(zhuǎn)動(dòng)（轉(zhuǎn)動(dòng)規(guī)律與基點(diǎn)的選擇無關(guān)）運(yùn)動(dòng)學(xué)71(5)Instantaneouscenterofvelocity

Atanyinstantoftime,thereexistsasolepointinthefigureoritsextensionwhosevelocityiszero.

Thelocationoftheinstantaneouscenterofvelocitychangeswithtime.

Atanyinstant,themotionofaplanefigurecanbeviewedastheinstantrotationaroundtheinstantaneouscenterofvelocity.However,thisinstantrotationisdifferentfromtherotationaboutafixedaxis.

=0,instantaneouscenterofvelocityisindefinitelyfaraway.Inthiscase,allpointshavethesamevelocityandtherigidbodyisininstantaneoustranslation.

Notethatinstantaneoustranslationisdifferentfromtranslation.Kinematics72運(yùn)動(dòng)學(xué)5.瞬心（速度瞬心）

任一瞬時(shí),平面圖形或擴(kuò)大部分都唯一存在一個(gè)速度為零的點(diǎn)

瞬心位置隨時(shí)間改變．

每一瞬時(shí)平面圖形的運(yùn)動(dòng)可視為繞該瞬時(shí)瞬心的轉(zhuǎn)動(dòng)．這種瞬時(shí)繞瞬心的轉(zhuǎn)動(dòng)與定軸轉(zhuǎn)動(dòng)不同．

=0,瞬心位于無窮遠(yuǎn)處,各點(diǎn)速度相同,剛體作瞬時(shí)平動(dòng),瞬時(shí)平動(dòng)與平動(dòng)不同．73(6)Rotationandtranslationaretwospecialcasesofplanemotion.(7)Methodstofindvelocityofanypointinafigure

polemethod：

velocityprojectionmethod：

instantaneousvelocitycentermethod：Inthesemethods,thepolemethodsisthebasicmethod,andtheinstantaneousvelocitycentermethodisaspecialcaseofthepolemethod.Kinematics74運(yùn)動(dòng)學(xué)6.剛體定軸轉(zhuǎn)動(dòng)和平面平動(dòng)是剛體平面運(yùn)動(dòng)的特例．7.求平面圖形上任一點(diǎn)速度的方法

基點(diǎn)法：

速度投影法：

速度瞬心法：其中，基點(diǎn)法是最基本的公式，瞬心法是基點(diǎn)法的特例．75

(8)Methodstofindaccelerationofapointinafigurepolemethod：,Aisthepole.Thisisthemostusefulmethod.Inaddition,when

=0,i.e.instantaneoustranslation,wecanalsoemploy

.Itisthespecialcaseofthepolemethodwhen

=0.(9)Conditionsfortheapplicationsofthemethodsusedinplanemotionandcompositionofmotions

methodsusedinplanemotionareappliedtofindtherelationshipbetweenthevelocities/accelerationsofanytwopointsordeterminetherelationshipamongthevelocity,accelerationofapointandtheangularvelocityandangularaccelerationofONErigidbodyhavingplanemotion.

ThemethodsusedinthecompositionofmotionsareusuallyappliedtodeterminethetransferofmotionatthecontactpointwhereTWOrigidbodiesareincontactandrelativeslippingexists.

Kinematics76

8.求平面圖形上一點(diǎn)加速度的方法 基點(diǎn)法：，A為基點(diǎn),是最常用的方法 此外，當(dāng)

=0,瞬時(shí)平動(dòng)時(shí)也可采用方法 它是基點(diǎn)法在

=0時(shí)的特例。運(yùn)動(dòng)學(xué)9.平面運(yùn)動(dòng)方法與合成運(yùn)動(dòng)方法的應(yīng)用條件

平面運(yùn)動(dòng)方法用于研究一個(gè)平面運(yùn)動(dòng)剛體上任意兩點(diǎn)的速度、加速度之間的關(guān)系及任意一點(diǎn)的速度、加速度與圖形角速度、角加速度之間的關(guān)系．

合成運(yùn)動(dòng)方法常用來確定兩個(gè)相接觸的物體在接觸點(diǎn)處有相對(duì)滑動(dòng)時(shí)的運(yùn)動(dòng)關(guān)系的傳遞．772.Stepsinproblemsolvingandsomekeypoints

(1)Accordingtotheinformationgivenintheproblemandthedefinitionsofthemotionsofarigidbody,determinethetypeofmotionsoftherigidbodiesappearedintheproblem.Notethatonlyonerigidbodyshouldbeconsideredeachtime.

Kinematics(2)Forarigidbodywithplanemotion,，properlychooseasuitablemethodtofindthevelocity(orangularvelocity)ofapointbasedontheknownandunknownconditions.Tofindaccelerations,polemethodisrecommended.

(3)Dothethoroughanalysisforvelocitiesandaccelerations,andfinallysolvetheunknowns.(Polemethod:properlyselectapoleandthendrawtheparallelogramforvelocitiesandthevectordiagramforaccelerations；Velocityprojectionmethod:itslimitationisthat

cannotbefound;Instantaneousvelocitycentermethod:findtheInstantaneousvelocitycenteristhekeystep.)78二．解題步驟和要點(diǎn)

1.根據(jù)題意和剛體各種運(yùn)動(dòng)的定義，判斷機(jī)構(gòu)中各剛體的運(yùn)動(dòng)形式．注意每一次的研