planar kinematics of a rigid body review刚体平面运动的回顾

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1、ME316Lecturenote2LectureNote(2)AGeneralMethodforKinematicsofPlanarMechanismsTopic:1.DerivativeofaVectorwhenitrotates2.Ageneralequation3.ExamplesRemarks:Alsorefertothetext(9thEdition)(pp354-369).1.DerivativeofaVectorwhenitrotatesLetbeavector.Itsmagnitudeisconstant.Vectorrotatesaboutanaxisperpendi

2、culartothesheetplanewithanangularvelocity;seeFigure2.1a.Notethatintheplanarmotionthedirectionoftheangularvelocitywillbeperpendiculartothesheetplane;seeFigure2.1b.(b)(a)xyzFigure2.1Page16of16ME316Lecturenote2Wecanprovethefollowingequation:(2.1)Proof:ThederivativeofthevectorAwithrespecttotimecanbe

3、writtenasfollows:wherethemagnitudeofcanbeexpressedas()A;seeFigure2.2.Thedirectionofthelimitcanbeexpressedbywherethefirstunitvectoristherotationaxisperpendiculartothesheetplaneandoutward,whilethesecondunitvectoristhedirectionof.ThereforewecanobtainFigure2.2Endoftheproof.Notethattheaboveproofcanbe

4、foundfromthetext(P355,9thEdition).Page16of16ME316Lecturenote22.AGeneralEquationWebeginwiththepositionanalysisfortwopointsAandBonthebody(Figure2.3).Figure2.3ashowsarigidbodywithtwopointsorparticles(AandB).Thebodyrotateswiththeanglevelocity.Inthiscasetherelativevelocitybetweenthetwopoints(AandB),w

5、hichisrelatedtothepositionvector,is(2.2)InderivingEquation(2.2),wehaveappliedEquation(2.1)(Notice:themagnitudeofis,inthiscase,unchangedwithrespecttotime).Now,weconsiderthesituationthatthedistancebetweenAandBchangeswithrespecttotimewhenthebodyrotates(Figure2.3b).ABFixedChangewithrespecttotime(a)A

6、B(b)xyxyFigure2.3Positionanalysisleadsto:Page16of16ME316Lecturenote2(2.3)ThefirstorderderivativeofEquation(2.3)onbothsideswithrespecttimeleadsto:(2.4)TheseconditemontherightsideofEquation(2.4)isfurtherwritteninto:(2.5)whereisanunitvectorrepresentingthedirectionofvector.Thephysicalmeaningoftheite

7、msontherightsideofEquation(2.5)canbeexplained.Thefirstitem:therateofchangeinthedistancebetweenAandB,i.e.,therelativetranslationvelocitybetweentwopoints(A,B).Theseconditem:therateofchangeofaunitvectorwhoseorientationchangesbu