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1、KinematicsofRigidBodiesApplications:Analysisofcams,gears,shafts,linkages,connectingrods,etc.Definition:Arigidbodyisaspecialcaseofasystemofparticleswhereinthedistancesbetweenallparticlesremainunchanged.Forasystemofparticleswithgeneralthree-dimensionalmo
2、tionprescribedforeachparticle,thereareupto3NpossibledegreesoffreedomforasystemofNparticles.WiththerestrictionondistancebetweentwopointsAandBsothatj~rA=Bj=constant,theonlypossiblechangewithrespecttimefor~rA=Bisduetorotation.Sincetherotationaldegreesoff
3、reedomcanalsoberepresentedbyavector,thegeneraldescriptionofrigidbodymotioncanbereducedto(i)3translationsand3rotationsforthree-dimensionalproblemsand(ii)2translationsand1rotationfortwo-dimensional(plane)problems.MotionofanArbitraryPointonaRigidBodyIfth
4、etranslationcomponentsofarigidbodyisgivenatareferencepointBandtherotationvectorisAalsoknownfortherigidbody,thenthemotionatanyarbitrarypointAontherigidbodycanbedetermined~rA/Bcompletely.~rAConsiderfirsttherelationshipB~rB~rA=~rB+~rA=B;Oinwhich~rBisthekn
5、ownmotionofthereferencepointBwhile~rAisthemotionofthearbitrarypointAtobedetermined.Therelativedisplacement~rA=Bhastheadditionalrestrictionofj~rA=Bj=constantforarigidbody.ToobtainthevelocityatpointA,takederivativeofthepositionvector~rAwithrespecttotime
6、toyieldd~rAd~rBd~rA=B=+or~vA=~vB+~vA=Bdtdtdtwheretherelativevelocity~vA=Bisdefinedasd~rA=B~vA=B==!~~rA=Bdtforarigidbodybecausej~rA=Bj=constantandthechangeoccursonlyduetorotation.Hence,forarigidbody~vA=~vB+!~~rA=B:Theaboveequationindicatesthatonce~vBa
7、nd!~areknown,thevelocityeverywhereontherigidbodycanbecalculatedeasily.Toobtaintheaccelerationvector,performanothertimederivativetoyieldd~vAd~vBd~!d~rA=B=+~rA=B+!~dtdtdtdtor~aA=~aB+~~rA=B+!~(!~~rA=B):–6/1–Example–PlaneRigidBodyKinematicsArigidbarA
8、PisconnectedtoslidersAandBasshowninthefigure.SliderAPisrestrictedtomoveonlyinthehorizontal300mmdirectionwhilesliderBcanmoveonlyintheverticaldirection.Ifattheinstantshown,sliderAismovingtotheleftwithavelocityof4mm/sec,find~vB,!~and~vP.200mmB45oAS
