Engineering Basic Mechanics Ⅱ Dynamics 工程基础力学 Ⅱ 动力学 课件 Chapter 4 Planar Motion of a Rigid Body

Chapter4

PlanarMotionofaRigidBody§

4.1Basicconceptanddecompositionofrigidbodyplanarmotion

Maincontents§4.2

Velocityofanypointinaplanarmotion§4.3

Accelerationofanypointinaplanarmotion1.Whatisplanarmotionofarigidbody?Thedistancebetweenanypointinarigidbodyandafixedplanealwayskeepsunchangedduringitsmotion.Thismotionofrigidbodyiscalled

planarmotionofarigidbody.4.1Basicconceptanddecompositionofrigidbodyplanarmotion2.SimplificationofaplanarmotionTheplanarmotionofarigidbodycanbesimplifiedtoamotionofaplanegraphintheplaneitselfwithoutconsideringitsthickness.

(a)Connectingrodmotion(b)Simplificationofconnectingrodmotion4.1Basicconceptanddecompositionofrigidbodyplanarmotion3.EquationsofplanarmotionSTodeterminethemotionofaplanegraph,choosethefixedreferencesystemOxy,anarbitrarypointO'intheplanegraphS,anarbitrarylinesegmentO'M.Todeterminetheplanarmotionofarigidbody,onlythepositionofthelinesegmentO'Minthisgraphisneededtobedetermined.EquationsofplanarmotionAplanemotioncanberegardedasthecompositionofa

translation

androtation．4.1Basicconceptanddecompositionofrigidbodyplanarmotion4.Planarmotioncanbedecomposedintotranslationandrotation

Aplanemotionofarigidbodycanbedecomposedintoa

translationwithabasicpointanda

rotation

aboutanaxisthatpassesthroughthebasicpoint.Thevelocityandaccelerationofthe

translation

withabasicpoint

intheplanegraphdependson

theselectionof

thebasicpoint,however,theangularvelocityandaccelerationoftherotationabouttheselectedbasicpoint

doesn’tdependon

thechoiceofthebasicpoint．4.1BasicconceptanddecompositionofrigidbodyplanarmotionAThevelocityofpointAintheplanegraphSis,andtherotationalvelocityoftheplanegraphis.SelectAasthebasicpoint;ThemovingreferencesystemattachedtopointA;Thetransportmotionistranslationwiththebasicpoint

A;Therelativemotionisrotationaboutthebasicpoint

A.(1)Basicpointmethod

·BDeterminethevelocityofpointBintheplanegraph.4.2VelocityofanypointinaplanarmotionABTheorem：Forplanarmotionofarigidbody,thevelocityofanypointinthegraphcanbeobtainedasthevectorsumofthevelocityofthebasicpointandtherelativerotationalvelocitywithrespecttothebasicpoint.4.2Velocityofanypointinaplanarmotion

isverticaltothelinkofABallthetime,sotheprojectionofonABisvanish.Thevelocityprojectiontheorem:thevelocityprojectionsofanytwopointinaplanegraphonthelinelinkingthesetwopointsareidentical．(2)VelocityprojectiontheoremAB4.2Velocityofanypointinaplanarmotiona.Background

Ifapointwhosevelocityiszeroisselectedasthebasicpoint,theprocessoffindingthevelocityofanypointwillbegreatlysimplified.Therefore,itisnaturaltoaskifsuchapointexistsinanyinstant.Ifitdoesexist,howtofindsuchapoint?b.InstantaneouscenterofvelocityAtanyinstant,itmustexistasolepointwhosevelocityiszerointheplanegraphoritsexpandingarea，whichiscalledtheinstantaneousvelocitycenterofthisplanegraphatthisinstant.Foraplanegraph,itsinstantaneousvelocitycenteralwaysexistsuniquely.

(3)Instantaneouscenterofvelocitymethod4.2Velocityofanypointinaplanarmotionc.InstantaneouscenterofvelocitymethodConsideraplanegraph.TheinstantaneousvelocitycenterisP,andtheangularvelocityoftheplanegraphis.SelectinstantaneousvelocitycenterPisabasicpoint,thevelocityofanarbitrarypointAintheplanegraph:4.2Velocityofanypointinaplanarmotiond.MethodstodeterminetheinstantaneousvelocitycenterPA(1)Whenthevelocityofapointandtheangularvelocity

oftheplanegraphareknown,theinstantaneousvelocitycenter(pointP)canbedetermined,

pointPisinthedirectionofthelineformedbyrotatingthethrough90ºinthedirectionof

aroundpointA．4.2Velocityofanypointinaplanarmotion(2)Whenaplanegraphrollsalongafixedsurfacewithoutslipping,thecontactpointPbetweenthegraphandthefixedsurfacewillbetheinstantaneousvelocitycenter.

(3)WhenthedirectionsofthevelocitiesattwopointsAandBinagraphareknown,andisnotparallelto,drawlinesfromAandBperpendiculartorespectively,andthecrosspointPofthesetwolineswillbetheinstantaneousvelocitycenter.ABP4.2Velocityofanypointinaplanarmotion(4)WhenthevelocitiesoftwopointsAandBaregivenatanyinstant,and.Therearethreecases:ABP

Whenandpointtothesamedirection,but.DrawtheextensionlineofAB,thelinkinglineoftheendingsofand,thecrosspointofthesetwolineswillbetheinstantaneousvelocitycenter.Therotationdirectionofcanbedetermined,anditsmagnitudeis:◆ω

◆Whenandhaveoppositedirections,drawthelinkinglineoftheendingsofand,andthelineconnectingAB.Thecrosspointofthesetwolineswillbetheinstantaneousvelocitycenter.Therotationdirectionofcanbedetermined,anditsmagnitudeis:

ω4.2VelocityofanypointinaplanarmotionBPAB(5)ThevelocitiesoftwopointsAandBpointtothesamedirectionatanyinstant,,,buttheyarenotperpendiculartolineAB.Inthiscase,theinstantaneousvelocitycenterisindefinitelyfaraway,andtheangularvelocity

=0,i.e.allpointinthefigurehavethesamevelocityatthisinstantoftime.Suchamotioniscalledinstantaneoustranslation,buttheiraccelerationsarenotequal.

When,

theinstantaneousvelocitycenterisindefinitelyfaraway.Theplanegraphhasinstantaneoustranslation,=0,allpointsinthegraphhavethesamevelocityatthisinstantoftime,buttheiraccelerationsarenotequal.◆ω4.2VelocityofanypointinaplanarmotionAABAAttheinstant,theangularvelocityofthegraphis,angularaccelerationis,accelerationofapointAis

.DeterminetheaccelerationofanarbitrarypointBinthegraph.·

4.3

AccelerationofanypointinaplanarmotionBA1.

：

4.3

AccelerationofanypointinaplanarmotionBAB(1)thetangentialacceleration

(2)thenormalacceleration2.

：hastwocomponents:

4.3

AccelerationofanypointinaplanarmotionTheabsoluteaccelerationofpointB:Theorem:

Theaccelerationofanarbitrarypointisequaltothevectorsumofaccelerationofthebasicpoint,thetangentialandnormalaccelerationsoftheplanegraphrotatingaboutthebasicpoint.

4.3

Accelerationofanypointinaplanarmotionω1ⅠⅡO1OABCAnexternaltoothingplanetgearmechanismshowninthefigure.ThelinkingbarO1O=l，rotatesaboutaxisO1withauniformangularvelocityω1.ThebiggergearIIisfixed,theplanetgearIofradiusrrollsalongthegearIIwithoutsliding.AandBaretwopointsontheedgegearI,showninthefigure.FindtheaccelerationsofpointsAandB.Example

4-1

4.3

AccelerationofanypointinaplanarmotionBω1ⅠⅡO1ACωOSolution：GearIhasplanarmotion,thevelocityandaccelerationofitscentreO:TheinstantaneousvelocitycenterofgearⅠisC,

itsangularvelocity:1.Solvetheaccelerati