Chapter9d’AlembertPrincipleandVirtualDisplacementPrinciple§9.4Constraint,virtualdisplacement,virtualwork§9.5Principleofvirtualdisplacement§9.1InertialForceandd’AlembertPrincipleofaParticle§9.2d’AlembertPrincipleofaSystemofParticles§9.3ReductionofaSystemofInertialForcesofaRigidBodyMainContentsInthischapter,wewilldiscussd’Alembertprinciple,itprovidesageneralmethodtosolvethekineticproblemofaparticleandasystemofparticles,themethodisthatthemethodsofstaticsareappliedtosolvekineticsproblems,thuskineticproblemscanbetransformedformallytoanequivalentstaticproblems,theycanbesolvedbytheoremofequilibrium.Thusthismethodiscalledthekinetic-staticmethod.Applyingthekinetic-staticmethodwecandeterminethemotion,forexampleaccelerationangularacceleration;canalsodeterminetheforce.D’Alembert’sPrincipleApplyingNewtonsecondlaw,wehave§9.1Inertialforceandd’AlembertprincipleofaparticleAssumingmassofaparticleis,accelerateis,activeforceactingontheparticleis,constraintforceis,showninfigure.AboveequationistransposedandwrittenasMaking

Wehave

hasthedimensionofforce,iscalledtheinertialforceofparticle：itsmagnitudeisequaltotheproductofmassandaccelerationofparticle，itsdirectioniscontrarytothedirectionofparticleacceleration.Theactiveforce,constraintforceandvirtualinertialforceactingontheparticlecomposedformallyequilibratedsystemofforces，thisisd’Alembertprincipleofaparticle.OlθExamole

9-1§9.1Inertialforceandd’AlembertprincipleofaparticleShowninfigure,aconicalpendulum.Aballofmassm=0.1kgtiesaropeoflengthl=0.3m,oneendoftheropetiestoafixedpointO,andtheanglewiththeleadstraightlineisθ=60o.Ifthesmallballmakeuniformcircularmotioninthehorizontalplane,determinethevelocityoftheballvandthemagnitudeoftensionFoftherope.OlθenetebmgF*Example9-1FSolution:choosethesmallballastheparticletostudy.Theparticlemakesuniformcircularmotion,onlyhavenormalacceleration,theforcesactingontheparticleincludesgravitymg,pullingforceFofropeandnormalinertialforceF*,showninfigure.Accordingtod’Alembertprinciple,thethreeforcescomposedformallyequilibratedsystem,thatisTakingtheprojectionformulaofaboveequationinnaturalaxis,wehave：§9.1InertialforceandD’Alembert’sprincipleofaparticleExample

9-1OlθenetebmgF*FSolutionis:§9.1InertialforceandD’Alembert’sprincipleofaparticleAssumingssystemofparticlescomposedofnparticles，massofanyparticleiis,

accelerationis,allforcesactingontheparticleisdividedintoresultantforceofactiveforce,resultantforceofconstraintforce,theparticleisimaginarilyplusitsinertialforce,accordingtod’Alembertprincipleofaparticle,wehaveAboveequationshows,theactiveforce,constraintforceanditsinertialforceactingoneveryparticleofthesystemcomposedformallyequilibratedsystemofforces,thisisd’Alembertprincipleofasystemofparticles.Thisshows,externalforce,internalforceanditsinertialforceactingoneveryparticleofthesystemcomposedformallyequilibratedsystemofforces.§9.1InertialforceandD’Alembert’sprincipleofaparticleAllforcesactingtheithparticlearedividedintoresultantforceofexternalforce,

resultantforceofinternalforce,andaboveequationcanbewrittenasBystaticsweknowthatnecessaryandsufficientconditionofequilibriumofspacialgeneralforcesystemisthattheprincipalvectoroftheforcesystemandtheprincipalmomentaboutanypointisequaltozero,thatisAboveequationshows,externalforceactingonsystemofparticlesandinertialforcevirtualaddingoneveryparticlecomposeformallyequilibratedsystemofforces,thisisanotherrepresentationofd’Alembertprincipleofasystemofparticles.§9.2D’Alembert’sprincipleofasystemofparticlesSinceInternalforcesofthesystemofparticlesalwaysexistinpairs

,

andisequalinmagnitudeandoppositeindirection,andcollinear,

thenwehaveand

,henceInstatics,

iscalledtheprincipalvector,

istheprincipalmomentaboutpointO,nowiscalledtheprincipalvectorofinertialforcesystem,

istheprincipalmomentofinertialforcesystemaboutpointO.AccordingtoD’Alembert’sprincipleofasystemofparticles,thisisformallyaequilibratedsystemofforces,

hencewecanapplymethodofstaticsforsolvingvariousequilibratedforcesystemtosolvekineticproblem.§9.2D’Alembert’sprincipleofasystemofparticlesOABrExample

9-2Showninfigure，theradiusofpulleyisr,massmuniformlydistributedintherim,canrotatearoundthehorizontalaxis.Bothendsofthesoftropeacrosstherimhangheavybodyofmass

m1andm2,andm1>m2.Neglectweightofrope,thereisnorelativeslidingbetweenropeandpulley,neglectbearingfriction.Determinetheaccelerationofheavybody.§9.2D’Alembert’sprincipleofasystemofparticlesOABryExample

9-2aam1gmgm2gFNSolution：choosepulleyandthetwoheavybodiesasthesystemofparticlestobestudied.Theexternalforcesactingonthesystemincludegravitym1g,m2g,mgandbearingconstraintforces

FN.Eachparticleofthesystemisvirtuallyaddedinertialforce,wecanapplyd’Alembertprinciple.Weknowm1>m2,thenthedirectionofaccelerationaofheavybodyshowninfigure.Thedirectionofinertialforceofheavybodyisoppositetothedirectionofaccelerationa,magnitudearerespectively：§9.2D’Alembert’sprincipleofasystemofparticlesorExample

9-2OABraam1gmgm2gFNymiApplyingequationofmomentofforceaboutrotatingaxis,weobtain

§9.2D’Alembert’sprincipleofasystemofparticlesMassofeachpointonpulleyedgeismi,magnitudeoftangentialinertialforceis,directionisalongtherimtangentline,pointasshowninfigure.Whenthereisnorelativeslidingbetweenropeandpulley,;magnitudeofnormalinertialforceis,directionisalongradiusanddeparturefromthecenter.

sinceSolutionisExample

9-2OABraam1gmgm2gFNymi§9.2D’Alembert’sprincipleofasystemofparticles§9.3ReductionofasystemofinertialforcesofarigidbodyThisexpressionisestablishedaboutanymotionofanysystemofparticles,alsoappliestotherigidbodythatmakestranslation,fixedaxisrotationandplanemotion.Inthefollowingweintroducereductionofasystemofinertialforcesinthreecommoncases.Applyingd’Alembertprincipleofasystemofparticlestosolvekineticproblemofthesystem,

eachparticleofthesystemisaddeditsinertialforce,

theseinertialforcesformasystemofforces,

whichiscalledinertialforcesystem.Ifusingsimplifiedtheoryofforcesysteminstatics,

todeterminetheprincipalvectorandtheprincipalmomentintheinertialforcesystem,

substituteinertialforceaddedtoeachparticlewhenwespecificallysolve,

itwillbringconveniencetosolveproblem.Inthefollowingweonlydiscussreductionofinertialforcesystemintranslationofrigidbody,

fixedaxisrotationandplanemotion.representstheprinciplevectorofinertialforcesystem,

accordingtoandtheoremofmotionofmasscenter,

wehave1.RigidbodyintranslationRigidbodyisintranslation,

ateveryinstantaccelerationofanyparticleiinrigidbodyisthesameasaccelerationofmasscenter,

here,

inertialforcesystemofrigidbodydistributesinfigure,

arbitrarilychooseapointOassimplifiedcenter,

representstheprincipalmoment,

wehaveWhenrigidbodyisintranslation,theprinciplemomentofinertialforceaboutarbitrarypointisgenerallynotequaltozero.Ifchoosemasscenterassimplifiedcenter,itsprincipalmomentiszero,simplifiedasaresultantforce.Henceweconclude:inertialforcesystemoftranslationalrigidbodycanbesimplifiedtoresultantforcethroughmasscenter,itsmagnitudeisequaltotheproductofmassofrigidbodyandacceleration,thedirectionofresultantforceisoppositetothedirectionofacceleration.§9.3ReductionofasystemofinertialforcesofarigidbodyWhere,

isradiusvectorfrommasscenterCtosimplifiedcenterO,theprinciplemomentisgenerallynotequaltozero.IfchoosemasscenterCassimplifiedcenter,representtheprincipalmoment,then,

wehave2.Fixedaxisrotationofarigidbody§9.3ReductionofasystemofinertialforcesofarigidbodyInertialforceofparticlecanbedividedintotangentialinertialforceandnormalinertialforce

,andtheirdirectionsshowninfigure,magnitudearerespectivelyWhenrigidbodyisinfixedaxisrotation,assumingangularvelocityofrigidbodyis,angularaccelerationis,massofanyparticleinrigidbodyis,thedistancetorotatingaxisis，theninertialforceofanyparticleinrigidbodyis.Forsimplicity,arbitrarilychooseapointO

onrotatingaxisassimplifiedcenter,establishrectangularcoordinatesystemshowninfigure,coordinatesoftheparticleisIftherigidbodyhasaplaneofmasssymmetryandtheplaneisverticaltotherotatingaxisz,andthesimplifiedcenter

ischosentobetheintersectionpointofthisplanewiththerotatingaxisz,thenMomentofinertialforcesystemaboutaxisz

is

Sincenormalinertialforceofeachparticlepassthroughaxisz,

wehave§9.3Reductionofasystemofinertialforcesofarigidbody3.Rigidbodyinplanemotion（paralleltothemasssymmetryplane）§9.3ReductionofasystemofinertialforcesofarigidbodyInengineering,rigidbodyinplanemotionoftenhasmasssymmetryplane,andparalleltotheplanemotion,nowonlyinthiscasewediscussreductionofasystemofinertialforces.Similartorotationofrigidbodyaroundfixedaxis,rigidbodyisinplanemotion,spaceforcesystemcomposedofinertialforcesofeachparticle,canbesimplifiedtoplaneforcesysteminthemasssymmetryplane.Chooseplanefigureinthemasssymmetrypaneasshowninfigure.Bykinematicsweknow,motionofplanefigurecanbedividedintotranslationwiththebasepointandrotationaroundthebasepoint.NowchoosemasscenterCasthebasepoint,assumingtheaccelerationofmasscenteris,angularvelocityofrotationaroundmasscenteris,angularaccelerationis,similartorotationofrigidbodyaroundfixedaxis,nowtheprincipalmomentofreductionofasystemofinertialforcestomasscenterCisWhere

isthemassmomentofinertiaoftherigidbodyabouttheaxiswhichpassesthroughmasscenterandisverticaltothemasssymmetryplane.§9.3ReductionofasystemofinertialforcesofarigidbodySoweconclude:

rigidbodyhavethemasssymmetryplane,

whenmovingparalleltotheplane,

asystemofinertialforcesofrigidbodyisreducedtoaforceandacoupleintheplane.Theforcepassesthroughmasscenter,

itsmagnitudeisequaltotheproductofmassofrigidbodyandaccelerationofmasscenter,

itsdirectionisoppositetothedirectionofaccelerationofmasscenter;

momentofthecoupleisequaltotheproductofthemassmomentofinertiaoftherigidbodyabouttheaxiswhichpassesthroughmasscenterandisverticaltothemasssymmetryplaneandangularacceleration,

rotatingdirectionisoppositetoangularacceleration.xyωm1gm2gCOhφExample

9-3§9.3ReductionofasystemofinertialforcesofarigidbodyShowninfigure,massofstatorofelectricmotorism1,mountedonahorizontalbase.ThedistancebetweenrotatingaxisOandhorizontalplaneish,and

massofrotorism2,itsmasscenterisC,eccentricdistanceOC=e,whenmotionbegins,masscenterCisatthelowestposition.Rotorrotateswithconstantangularvelocityω,determinetheconstraintforceofthebaseactingontheelectricmotor.Example

9-3xyωm1gm2gCOhφFyFxMAF*Solution:choosethewholemotorasobjecttobestudied.Theforcesincludegravitym1gandm2g,constraintforceofbaseandgroundscrewactingontheelectricmotorsimplifiedtopointAasacoupleMandaforceF(showninfigureFxandFy).Thesystemofparticlesisaddedtoinertialforce.RotoruniformlyrotatesaboutfixedaxisOwithangularvelocityω,thesystemofinertialforceisreducedaforcethroughpointO,magnitudeisItsdirectionisoppositetoaccelerationaCofmasscenterC.SinceaCisalongOCandpointstocenterO,

F*isalongOCanddepartsfrompointO.§9.3ReductionofasystemofinertialforcesofarigidbodyExample

9-3xyωm1gm2gCOhφFyFxMAF*Accordingtod’Alembertprinciple,activeforce,constraintforceandinertialforceactingonthesystemofparticlesformallycomposeequilibriumforcesystem,wecanwriteequilibriumequation：Sincerotoruniformlyrotates,φ=ωt

,substitutingitintoaboveequations,weobtain:§9.3ReductionofasystemofinertialforcesofarigidbodymAgmgFABCExample

9-4MassofhomogeneousdiscismA,radiusisr.Lengthofslenderrodisl=2r,massism.PointAofrodendhingedsmoothlytowheelcenter,showninfigure.IfpointAsufferedahorizontalpullingforceF,makewheelrollalonghorizontalplane.DeterminethemagnitudeofforceF,whenendBofrodjustlefttheground.Inordertoensurepurerolling,determinecoefficientofstaticslidingfrictionbetweenthewheelandtheground.§9.3ReductionofasystemofinertialforcesofarigidbodyBCmgAF*CFAxFAyamAgmgFABCExample

9-4F*AF*CM*Accordingtokinetic-staticmethod,wewriteequationSolutionis

§9.3ReductionofasystemofinertialforcesofarigidbodySolution:whenslenderrodleftthegrounditisstillintranslation,andconstraintforceofgroundisequaltozero,assumingitsaccelerationisa.Chooserodasobjecttobestudied,theforcesactingonrodandaddinginertialforceasshowninfigure,where

Theforcesactingonthewholesystemandaddinginertialforcesasshowninfigure,whereAccordingtoequationweobtainmAgmgFABCF*AF*CM*FNFsExample

9-4Frictionofground

Inordertodeterminefriction,choosethewheelasobjecttobestudied.Solutionis

§9.3ReductionofasystemofinertialforcesofarigidbodyApplyingequationweobtainAmAgFFNF*AM*FsExample

9-4Thus,coefficientoffrictionofground§9.3ReductionofasystemofinertialforcesofarigidbodyAFNF*AF*CmAgmgFBCM*FsThenchoosethewholesystemasobjecttobestudied,

byequation,weobtainmAgFAFNF*AM*FsPrincipleofvirtualdisplacement:§9.4Constraint,virtualdisplacement,virtualworkToestablishtheequilibriumconditionsforthesystemofmasspointsindependentoftheNewtonianmechanicssystem.Newtoniansystemofmechanics:Vectormechanics,whichdescribesmechanicalquantitiesthatarerepresentedbyvectors,suchasvectordiameter,velocity,acceleration,angularvelocityandangularacceleration.Analyticalmechanicssystem:Scalarmechanics,whichdescribesphysicalquantitiesasscalars,suchasgeneralizedcoordinates,energyandwork.Theprincipleofvirtualdisplacementisbasedonanalyticalmechanicstoestablishthesufficientconditionsfortheequilibriumofthesystem,whichhasawidersignificancethantheequilibriumconditionsestablishedbyNewtonianmechanics.1.Constraintsandtheirclassification(1)Therestrictionsonthemotionofanobjectarecalledconstraints.Expressedasamathematicalequation,whichiscalledconstraintequation.Forexample:xφOyM(x,y)ιPlanependulumconstraintequation§9.4Constraint,virtualdisplacement,virtualwork2.Classificationofconstraints§9.4Constraint,virtualdisplacement,virtualworkGeometricconstraint:restrictonlythegeometricpositionofaparticle.Motionconstraint:theconstraintequationcontainsthederivativeoftheparticlecoordinates(withrespecttotime).Steadyconstraint:theconstraintisindependentoftime,i.e.,theconstraintequationdoesnotcontaintimet.Unsteadyconstraint:

theconstraintisdependentoftime,i.e.,thetimetisincludedintheconstraintequation.Holonomicconstraint:

includinggeometricconstraintsandmotionconstraintsthatcanbereducedtogeometricconstraints.Nonholonomicconstraint:amotionconstraintcannotbereducedtoageometricconstraint.3.

VirtualdisplacementAtacertaininstant,anyinfinitesimaldisplacementthattheparticlesystemmayachieveundertheconditionsallowedbyconstraintsiscalledthevirtualdisplacementoftheparticlesystem(atthatinstant).Thevirtualdisplacementcanbeeitheralineardisplacementoranangulardisplacement.Usually,thevariationalsymbolδisusedtorepresentvirtualdisplacement.Inthefollowingtwoexamples,δφ,δrAand

δrBareallvirtualdisplacement.xφOyMδφδs(+)xBAOyMFδrAδrBδφ§9.4Constraint,virtualdisplacement,virtualwork4.Differencebetweenvirtualdisplacementandrealdisplacement

Realdisplacementisthetruedisplacementachievedbyaparticlesystemwithi