Engineering Basic Mechanics Ⅱ Dynamics 工程基礎(chǔ)力學(xué) Ⅱ 動(dòng)力學(xué) 課件 Chapter 5 Dynamics of Particle

Chapter5

DynamicsofParticleMainContents§5.1Newton’sLawsofMotion§5.2DifferentialEquationsofmotionofaParticle

§5.3ThetwotypesofbasicproblemsofparticlekineticsKineticsofaParticle

Instatics,westudytheforceactingonabody,andstudytheproblemsofequilibriumofbodiesthatareactedonforces,butdon’tstudythemotionofabodyactedonunbalancedforces;Inkinematics,weonlystudythemotionofabodyfromthegeometricalaspects,anddonotaccountfortherelationsbetweenkinematicsofaparticleandtheforcethatactsontheparticle;Inkinetics,wewilldeterminetherelationsbetweenkinematicsofaparticleandtheforcethatactsontheparticle.Comparedwiththestaticsandkinematics,kineticsisthestudyofthemoregeneralruleofthemechanicalmotionofamaterialbody.Inkinetics,weusuallyusetwomodelsofmechanics:particleandparticles.Whentheshapeandsizeofabodyisnotsignificant;arigidbodywithtranslationalmotionmaybedefinedasaparticle,theparticleconcentrateallmassoftherigidbody,andlieinthecenterofmassoftherigidbody;Sometimes,thebodymaybemodeledasaparticle,andtherotationalmotionofthebodycanbeignored;Aparticle:aparticlehasamassbutnegligiblesizeandshape.Specifically,atwhatpointcanabodybeabstractedandsimplifiedasaparticle?Forexample,instudyingthemotionoftheeartharoundthesun,itispermissibletoconsidertheearthasaparticle.KineticsofaParticleAsystemofparticles:asystemcomposedoffiniteorinfiniteparticlescontactedeachother.Ifabodycannotbestudiedasaparticle,itmustbeaccountedasasystemofparticles.Theconceptofasystemofparticlesisveryuniversal,itincludesarigidbody,adeformablebody,andasystemcomposedofmanyparticlesandbodies.

KineticsofaParticle

Kineticscanbedividedintotwoparts:kineticsofaparticleandkineticsofasystemofparticles(includekineticsofrigidbodies).

Inthischapterwediscusskineticsofaparticle,thatistodeterminetherelationsbetweenkinematicsofaparticleandtheforcethatactsontheparticle.Asfundamentalsofothertheoryofkinetics,kineticsofaparticlebasedonthefundamentalsofNewton’sthreelaws.Thischapteremphasizehowtosolvethetwotypesofproblemsofparticlekineticsapplyingbasicequationsofkineticsandusingmethodsofdifferentialandintegralcalculus.KineticsofaParticle§5.1Newton’sLawsofMotionNewtonfirstlaw(inertialaw):

Intheabsenceofappliedforces,aparticleoriginallyatrestormovingwithconstantspeedinastraightlinewillremainatrestorcontinuetomovewithconstantspeedinastraightline.Inertia:apropertyofmatterbywhichitcontinuesinitsexistingstateofrestoruniformmotioninastraightline,unlessthatstateischangedbyanexternalforce.Therefore,thislawisalsocalledtheinertialaw.Newtonsecondlaw：Ifaparticleissubjectedtoaforce,theparticlewillbeaccelerated.Theaccelerationoftheparticlewillbeinthedirectionoftheforce,andthemagnitudeoftheaccelerationwillbeproportionaltothemagnitudeoftheforceandinverselyproportionaltothemassoftheparticle.Newtonsecondlawmaybeexpressedmathematicallyasfollows.§5.1Newton’sLawsofMotionorTheaboveformulaisthebasicequationforsolvingdynamicproblems,whichiscalledthebasicdynamicequation.Thisformuladiscribestherelationshipbetweenthemotionofaparticleandtheforcesactingontheparticle.

Newtonthirdlaw(thelawofactionandreaction):Foreveryaction,thereisanequalandoppositereaction;thatis,theforcesofinteractionbetweentwoparticlesareequalinmagnitudeandoppositeindirection,andcollinear.Itshouldbenotedthatthefirsttwolawsinthefundamentalequationsofdynamicsonlyapplyininertialcoordinates.Newtonthirdlawhasnothingtodowiththeselectionofcoordinatesystems,anditappliestoallcoordinatesystems.§5.1Newton’sLawsofMotionMechanicalsystemofunits

Inmechanics，weusuallyuseInternationalsystemofunits(SI).IntheSIsystem，allunitsaredividedintothreecategoriesbaseunits,derivedunitsandauxiliaryunits.ThebasedimensionsintheSIsystemaremass,lengthandtime,andthebaseunitsarekilogram(kg),meter(m)andsecond(s).Aunitofforceisderivedunits,beingcalledNewton(N).OneNewtonforcemakeonekilogramofmassgenerateonemeter/second2ofaccelerate,thatisRadianisanauxiliaryunit,canbeusedtoformderivedunits,forexampleangularvelocityunitandangularaccelerationunit,andsoon.§5.1Newton’sLawsofMotionInengineering,weoftenuseengineeringsystemofunits.Thebasedimensionsinengineeringsystemofunitsareforcelengthandtime,andthebaseunitsarekilogramforce(kgf),meter(m)andsecond(s).Massunitisderivedunits,whenonekilogramforcemakeabodygenerateonemeter/second2ofaccelerate,themassofthebodyisoneengineeringunitmass.Thatis§5.1Newton’sLawsofMotionWhenonekilogramforce(9.80665Newton)generateacceleration,themassis9.80665kilogram，hence1massofengineeringunitmass=9.80665kilogram≈9.8kilogramOnekilogramforce（kgf）isthegravityactingonabodythathasonekilogramofmassatalatitudeofofthesea.Hence§5.1Newton’sLawsofMotionThefundamentalequationsofmotionarerepresentedasequationsindifferentialform,knownasthedifferentialequationsofmotionofaparticle.1.VectorformWhenaparticlemovesinanarbitraryspatialcurve,itspositionisrepresentedbythevectordiameter

derivedfromanarbitraryspatialfixedpointO,asshowninthefigure.§5.2DifferentialEquationsofmotionofaParticle

2.Formsinrectangularcoordinates§5.2DifferentialEquationsofmotionofaParticle

3.Formsinpathn-tcoordinates§5.2DifferentialEquationsofmotionofaParticle

Applyingthedifferentialequationsofmotionofaparticlewecansolvetwotypesofproblemsofparticlekinetics.§5.3ThetwotypesofbasicproblemsofparticlekineticsThefirstbasicproblem：knowingthemotionofaparticle,todeterminetheforceactingontheparticle.Thatis,knowingequationsofmotionofaparticle,thesecondderivativewithrespecttotimeofthepositionvectoriscalledtheacceleration,whichissubstitutedintothefundamentalequationsofmotionofaparticle,weobtaintheforceactingontheparticle.Thesecondbasicproblem：knowingtheforceactingonaparticle,todeterminethemotionoftheparticle.(forexample,todeterminethevelocity,trajectoryandequationsofmotionofaparticleandsoon).Inthefundamentalequationsofmotionofaparticle,knowingtheforceactingonaparticle,wecanobtaintheaccelerationofthemotionoftheparticle,todeterminethevelocity,trajectory,equationsofmotionofaparticlebytheaccelerationistheintegralcalculationproblem。Example

5-1

§5.3ThetwotypesofbasicproblemsofparticlekineticsThefollowingexamplesarehowtosolvetwotypesofproblemsofparticlekineticsbyapplyingthedifferentialequationsofmotionofaparticle.，Example

5-1Solution:Firstly,choosetheparticleMastheobjectofstudy,theparticleMdoesplumb-rectilinearmotion,choosethetrajectorylineastherectangularcoordinateaxis,andthedownwarddirectionispositive.ThenputtheparticleMonthegeneralpositionofthemotiontodrawitsforcediagram.TheforcesontheparticleinthispositionaregravityPanddielectricresistanceR.ThenthedifferentialequationofmotionoftheparticleMinrectangularcoordinateiswherePxandRxaretheprojectionsofPandRontheOxaxis,respectively.wehave§5.3ThetwotypesofbasicproblemsofparticlekineticsFromtheknownequationofmotionofthemassExample

5-1ThedifferentialequationofmotioncanthenbewrittenasFromtheknownequationofmotionoftheparticlewegetso,therewas§5.3ThetwotypesofbasicproblemsofparticlekineticsExample

5-2Aparticleofmassmunderahorizontalforce

movesalongthehorizontallinefromrest.Determinetheequationofmotionoftheparticle.when

，aswellasThusgetting

Solution:Fortheproblem,forceisknown,andweneedtosolvemotion.Theforceisadiscontinuoustimefunction.Theobjectofstudyisaparticle,whichisrectilinearlymoving,andtheequationispresentedalongthedirectionofmotion.

§5.3ThetwotypesofbasicproblemsofparticlekineticsExample

5-2Thus,theequationofmotionoftheparticleiswhen

,thevelocityoftheparticle

,positionoftheparticle

thesearetheinitialconditionsat.when

,

,so

Fromtheinitialconditionsat

，ThusTheforceinthisproblemisadiscontinuousfunctionoftime,sotheanalysisshouldbesegmented,whilepayingattentiontotheinitialconditionsofeachsegment.§5.3ThetwotypesofbasicproblemsofparticlekineticsExample5-3

Solution:Taketheinitialpositionoftheobjectastheorigin,andbuildthecoordinatesystemalongthedirectionoftheobject'smotion.Motionanalysis:rectilinearmotion.

§5.3ThetwotypesofbasicproblemsofparticlekineticsEstablishdifferentialequationsofmotion:Example

5-3useSubstitutingintothedifferentialequationofmotionyieldsthusso

§5.3ThetwotypesofbasicproblemsofparticlekineticsthusExample

5-3Integrateoncemoreandget

soConsideringthatx=0att=0,wegetC3=9

§5.3ThetwotypesofbasicproblemsofparticlekineticsxyOABφβωExample

5-4Acrank-guidemechanismisshowninfigure.TheangularvelocityofthecrankOAisconstant,lengthOA=r,lengthoftheconnectingrodisAB=l.Ifλ=r/lissmaller，thepointOistheoriginofthecoordinatesystem,theequationofmotionofsliderBmaybewrittenapproximatelyasfollows:Themassofthesliderisdenotedbynotationm,

NeglectfrictionandthemassoftheconnectingrodAB，whenand,respectively,determinetheforceactingattheconnectingrodAB.§5.3ThetwotypesofbasicproblemsofparticlekineticsxBβxyOABφβωExample

5-4SoSolution：ConsideringthesliderB.Whenφ=ωt,thefree-bodydiagram(FBD)ofthesliderBshowninfigure,wheretherod(AB)istwoforcemember.Writingtheequationofmoti