车辆工程课件II－Vibration of Vehicle System

§3VibrationofVehicleSystemFreedoms,coordinateandvibrationmodesVibrationofvehiclewithonlysecondarysuspensionMethodofderivingdifferentialequationofvehiclesystemVibrationofvehiclewithwithprimaryandsecondarysuspensionsByzhou

jinsong2004

TongjiUniversity§3VibrationofVehicleSystemFreedoms,coordinateandvibrationmodesDegrees-offreedom:-1bodywith5dofs=52bogieswith5dofs=104wheelsetswith2dofs=8Total=26,i.e.52states§3VibrationofVehicleSystemFreedoms,coordinateandvibrationmodesBodyframe(6DOF)Bogieframes(2*6DOF)Wheelsets(4*6DOF,bounceandrollmodesareconstrainedbyrail)BodyForwardBounceLateralRollYawPitchSuspensionsWheel-railcreepforcesBodyFlexibilitiesOthercomponentsorsub-systems§3VibrationofVehicleSystemFreedoms,coordinateandvibrationmodesBodyxzypitchlateral§3VibrationofVehicleSystemFreedoms,coordinateandvibrationmodesxzybounceyawBody§3VibrationofVehicleSystemFreedoms,coordinateandvibrationmodesxzyforwardBodyroll§3VibrationofVehicleSystemVibrationofvehiclewithonlysecondarysuspensionFreeVibrationTopologyrelationsintheresearchedsystem(includingconnectinginformation,dimension)CoordinatesdefinitionPhysicalpropertiesWhenbuildingsystemmotionequations,especiallypayattentiontothefollowingmainpoints:§3VibrationofVehicleSystemPlot,makecleartheconnectionrelationsbetweeneachpartDefinecoordinatesMarkthephysicalproperties,anddimensionsaccordingly,stepsofderivingmotionequationsare:

TheMostImportantstepistheFirststep,whichsimplifytherealsystemandisthebaseoftheoryanalysis.VibrationofvehiclewithonlysecondarysuspensionFreeVibration§3VibrationofVehicleSystemUseNewton’ssecondlaw,yields:VibrationofvehiclewithonlysecondarysuspensionFreeVibration§3VibrationofVehicleSystemWithviscousdamperinthesuspension,onehas:VibrationofvehiclewithonlysecondarysuspensionFreeVibrationparameters:Mc=36000kgJcy=2300000kg.m^2Ksz=0.36e6N/mCsz=?§3VibrationofVehicleSystemVibrationofvehiclewithonlysecondarysuspensionFreeVibration§3VibrationofVehicleSystemVibrationofvehiclewithonlysecondarysuspensionFreeVibrationNoviscousdamper,stepinputresponse:0.08mStepinput

f=0.8554T=1.1690Csz=0displacement§3VibrationofVehicleSystemVibrationofvehiclewithonlysecondarysuspensionFreeVibrationNoviscousdamper,stepinputresponse:

f=0.9632T=1.0382displacementdisplacement§3VibrationofVehicleSystemVibrationofvehiclewithonlysecondarysuspensionFreeVibrationNoviscousdamper,stepinputresponse:CenterplateResponseofthispointdisplacement§3VibrationofVehicleSystemVibrationofvehiclewithonlysecondarysuspensionFreeVibrationviscousdamper,stepinputresponse:parameters:Mc=36000kgJcy=2300000kg.m^2Ksz=0.36e6N/mCsz=?0.08mStepinput§3VibrationofVehicleSystemVibrationofvehiclewithonlysecondarysuspensionFreeVibrationviscousdamper,stepinputresponse:Czs=2500Zeta_Zc=0.0258Czs=2500Zeta_Phic=0.0291displacementdisplacement§3VibrationofVehicleSystemVibrationofvehiclewithonlysecondarysuspensionFreeVibrationviscousdamper,stepinputresponse:Czs=2500N.m/sdisplacementdisplacement§3VibrationofVehicleSystemVibrationofvehiclewithonlysecondarysuspensionFreeVibrationviscousdamper,stepinputresponse:Czs=10000N.m/sZeta_zc=0.1034Czs=10000N.m/sZeta_zc=0.1164displacementdisplacement§3VibrationofVehicleSystemVibrationofvehiclewithonlysecondarysuspensionFreeVibrationWithviscousdamper,stepinputresponse:displacementdisplacement§3VibrationofVehicleSystemVibrationofvehiclewithonlysecondarysuspensionFreeVibrationWithviscousdamper,Thesystemstepinputresponseisasfollowing:Czs=20000N.m/sCzs=0N.m/saccelerationacceleration§3VibrationofVehicleSystemVibrationofvehiclewithonlysecondarysuspensionForcedVibrationWhere:a-amplitudew-trackirritatingfrequencyLr-trackwavelengthV-speed§3VibrationofVehicleSystemVibrationofvehiclewithonlysecondarysuspensionForcedVibrationTrackcircularfrequencyTimedelayDisplacementatthecenteroffirsttruckcanbewrittenas:Displacementatthecenterofsecondtruck:§3VibrationofVehicleSystemVibrationofvehiclewithonlysecondarysuspensionForcedVibrationUseNewton’ssecondlaw,thedifferentialequationsofmotion:Simplifyupperequations,onehas:So,yields:§3VibrationofVehicleSystemVibrationofvehiclewithonlysecondarysuspensionForcedVibrationParameters:Csz=0N.m/sZt=sin(t)displacementacceleration§3VibrationofVehicleSystemVibrationofvehiclewithonlysecondarysuspensionForcedVibrationParameters:Csz=10000N.m/sZt=sin(t)displacementacceleration§3VibrationofVehicleSystemMethodsofderivingdifferentialequationsofvehiclesystemMainmethodstoderivethemotionequationareNewton’ssecondlawLagrange’sequationHamiltons’sprincipleInfluencecoefficientmethodBodyPlot,makecleartheconnectionrelationsbetweeneverypartDefinecoordinatesMarkthephysicalproperties,anddimensionsBeforeusethesemethods,onestillhastodofollowingworkstodefinethesystem:§3VibrationofVehicleSystemMethodsofderivingdifferentialequationsofvehiclesystemBodyV§3VibrationofVehicleSystemMethodsofderivingdifferentialequationsofvehiclesystemInfluencecoefficientmethodBodyDifferentialmotionequationsusuallyarewritteninmatrixformasfollowing:where:M–inertiamatrixofsystemC-dampingmatrixK-stiffnessmatrixF-forcematrixAllmatricesaredefinedinthesamecoordinate,globalcoordinate.influencecoefficientmethoddirectlyderivestheMmatrix,C,Kmatrix,canbeobtainedattheaidofcomputer.Farewrittenmanuallylatter.§3VibrationofVehicleSystemMethodsofderivingdifferentialequationsofvehiclesystemInfluencecoefficientmethodFirst,derivetheKmatrix

definecompressingispositive,pullingisnegative.0000000000000000000000Generalizedcooridnatesstiffnesselements§3VibrationofVehicleSystemMethodsofderivingdifferentialequationsofvehiclesystemInfluencecoefficientmethod

Itscoefficientmatrixis:§3VibrationofVehicleSystemMethodsofderivingdifferentialequationsofvehiclesystemInfluencecoefficientmethod

stiffnessmatrixis:§3VibrationofVehicleSystemMethodsofderivingdifferentialequationsofvehiclesystemInfluencecoefficientmethodthen,Cmatrixalsodefinecompressingispositive,pullingisnegative.000000Generalizedcooridnatesdampingelements§3VibrationofVehicleSystemMethodsofderivingdifferentialequationsofvehiclesystemInfluencecoefficientmethod

inertiamatrix000000000000000000000000000000inertiaelementsGeneralizedcooridnates§3VibrationofVehicleSystemMethodsofderivingdifferentialequationsofvehiclesystemInfluencecoefficientmethodBodyso,inertiamatrixis:so,thefreevibrationmotioneq.is:§3VibrationofVehicleSystemMethodsofderivingdifferentialequationsofvehiclesystemInfluencecoefficientmethodBodywhenthereexiststrackirregularity:?§3VibrationofVehicleSystemMethodsofderivingdifferentialequationsofvehiclesystemInfluencecoefficientmethodwhenthereexiststrackirregularity:0000000000000000000000zw100-kpz000zw2000-kpz00zw30000-kpz0zw400000-kpzTrackinput§3VibrationofVehicleSystemMethodsofderivingdifferentialequationsofvehiclesystemInfluencecoefficientmethodBody

so,wecanseethismethodhasverygoodflexibility,whenmoreandelementsaretobetakenintoconsideration,justaugmenttheAmatrix,thecomplexityincreasesonlylinearly!so,thevibrationmotioneq.Withtrackirregularityinputisthesameformas:note,wherethevariablevectoris:§3VibrationofVehicleSystemMethodsofderivingdifferentialequationsofvehiclesystemInfluencecoefficientmethodBodyso,motionequationsshouldbewrittenas:

but,thelastfourwheelsetmotionequationsshouldbedeleted,becausetheyareconstrained,andequaltotrackinputaccordingtoassumption.where:§3VibrationofVehicleSystemMethodsofderivingdifferentialequationsofvehiclesystemInfluencecoefficientmethodBody

astotheFmatrix,discomposetheforcesactedonvehiclepartsintogeneralizedforces,definetheirsignsaccordingtothegeneralizedcoordinates,thenFmatrixformed.asaexample,seethefigureatrightside,FeccentricfromcenterlineofcarbodysothematrixFis:flso,theforcedvibrationdifferentialeq.is:§3VibrationofVehicleSystemMethodsofderivingdifferentialequationsofvehiclesystemLagrange’sequationBodyV§3VibrationofVehicleSystemMethodsofderivingdifferentialequationsofvehiclesystemLagrange’sequationAsasimpleexample,derivethemotioneq.Offollowingsystem:§3VibrationofVehicleSystemMethodsofderivingdifferentialequationsofvehiclesystemLagrange’sequationKineticenergy:Potentialenergy:Dissipationfunction:§3VibrationofVehicleSystemMethodsofderivingdifferentialequationsofvehiclesystemLagrange’sequationKineticenergy:potentialenergy:§3VibrationofVehicleSystemMethodsofderivingdifferentialequationsofvehiclesystemLagrange’sequationDissipationfunction:so:Usethesamemethod,wehave:§3VibrationofVehicleSystemMethodsofderivingdifferentialequationsofvehiclesystemHamilton’sprincipleAnotheralternateapproachforderivingthedifferentialequationsofmotionfromscalarenergyquantitiesisHamilton’sprinciple.OneofthemostapplicablevariationaltechniqueisHamilton’sprinciple,whichstatesthat:WhereTisthesystemkineticenergy,Visthesystempotentialenergy,andisthevirtualworkofnonconservativeforces.§3VibrationofVehicleSystemVibrationofvehiclewithwithprimaryandsecondarysuspensionsAnalysisoffreevibration,withoutviscousdampingBodyV§3VibrationofVehicleSystemVibrationofvehiclewithwithprimaryandsecondarysuspensionsAnalysisoffreevibration,withoutviscousdampingAsanalternativemethod,thetraditionalNewton’ssecondlaw’sareusedtoderivethemotionequationsasthetextbookdemonstrates,themotionequationsare:Simplifyas:§3VibrationofVehicleSystemVibrationofvehiclewithwithprimaryandsecondarysuspensionsAnalysisoffreevibration,withoutviscousdampingWhere:§3VibrationofVehicleSystemVibrationofvehiclewithwithprimaryandsecondarysuspensionsAnalysisoffreevibration,withoutviscousdampingConsiderthecarbodyandtruckmotionequations,theyarecoupled,soareanalyzedtogether:BodyZtMcMt4Ksz8Kpzwhere:Zc§3VibrationofVehicleSystemVibrationofvehiclewithwithprimaryandsecondarysuspensionsAnalysisoffreevibration,withoutviscousdampingTheirsolutionsareassumedas:BodyZtMcMt4Ksz8KpzA,B,amplitudeofcarbodyandtruck,psystemnaturalfrequency,alphi,phaseangle.Substitutethesolutionintomotionequations,yields:ZcCharacteristicfunctionsSamewiththeSDOFsystem,onehas:§3VibrationofVehicleSystemVibrationofvehiclewithwithprimaryandsecondarysuspensionsAnalysisoffreevibration,withoutviscousdampingTwo,therearetwonaturalfrequencies.Solvethecharacteristicfunctions,theloweroneis:BodyZtMcMt4Ksz8KpzZcTheotheroneis:§3VibrationofVehicleSystemVibrationofvehiclewithwithprimaryandsecondarysuspensionsAnalysisoffreevibration,withoutviscousdampingSolutionare:BodyZtMcMt4Ksz8KpzZcModeshape:Lowfrequency:Highfrequency:Atlowfrequency§3VibrationofVehicleSystemVibrationofvehiclewithwithprimaryandsecondarysuspensionsAnalysisoffreevibration,withoutviscousdampingBodyZtMcMt4Ksz8KpzZc§3VibrationofVehicleSystemVibrationofvehiclewithwithprimaryandsecondarysuspensionsAnalysisoffreevibration,withoutviscousdampingBodyZtMcMt4Ksz8KpzZcAsanexample,aVehicleparameters:Mc=36000kgMt=2100kgKsz=260000N/mKpz=300000N/mWhenzt1(0)=0.02,zt2(0)=0.02,zc=0.02*3.0670,thesimulationresultinmatlab/simulinkisasfollowing:fst1=0.1641m,fst2=0.3392fst/fst1=3.0670§3VibrationofVehicleSystemVibrationofvehiclewithwithprimaryandsecondarysuspensionsAnalysisoffreevibration,withoutviscousdampingCarbodyverticaldisplacementTruckframeverticaldisplacement§3VibrationofVehicleSystemVibrationofvehiclewithwithprimaryandsecondarysuspensionsAnalysisoffreevibration,withoutviscousdampingTruckframeverticalacceleration§3VibrationofVehicleSystemVibrationofvehiclewithwithprimaryandsecondarysuspensionsAnalysisoffreevibration,withoutviscousdampingBodyV0.08mStepinputTheresponseisasfollowing:§3VibrationofVehicleSystemVibrationofvehiclewithwithprimaryandsecondarysuspensionsAnalysisoffreevibration,withoutviscousdampingCarbodyverticaldisplacementCarbodypitchangleresponse§3VibrationofVehicleSystemVibrationofvehiclewithwithprimaryandsecondarysuspensionsAnalysisoffreevibration,withoutviscousdampingTruckframeverticaldisplacementTruckframepitchangleresponseIncludingthreefrequenciesinthewaveWhy?§3VibrationofVehicleSystemMulti-DegreeofFreedomSystemInthecaseofundampedfreevibrationofmulti-degreeoffreedomsystems:Assumeasolutionintheform:Substitutetheupperformintothefirstequation,leadto:Whichleadsto:Ascomparing,standardeigenvalueproblemistheformasfollowing:§3VibrationofVehicleSystemMulti-DegreeofFreedomSystem

Thisequationhasanontrivialsolutionifandonlyifthecoefficientmatrixissingular,thatis:Characteristicfunctions

ModeShapes:Associa