LNDYN-L03-DampingModalSuperposition

DampingandModalSuperpositionLecture3L3.2LinearDynamicswithAbaqusOverview•Introduction•DampinginDirectSolutions•DampinginModalSubspaceProjectionSolutions•DampinginModalSuperpositionProcedures•MaterialDamping•ElementDamping•GlobalDamping•ModalDamping•DampingControls•SummaryIntroductionL3.4LinearDynamicswithAbaqusIntroduction•Everydynamicsystemexhibitssomeformofenergyloss•Materialnonlinearity(e.g.,inelasticdissipation)•Internalfriction(e.g.,materialbehavior)•Externalfriction(e.g.,jointbehavior)•Example:DampinginaVehicle•Noisereductionmaterials(heavilydamped)•Dampersbetweenthesuspensionsystemandthevehicle.•Etc.L3.5LinearDynamicswithAbaqusIntroduction•TypesofdampinginAbaqus•TwoprimarytypesofdampingareavailableinAbaqus*:•Velocity-proportionalviscousdamping•Displacement-proportionalstructuraldamping(imaginarystiffness),•Usedinfrequencydomaindynamicsandinmode-basedtransientdynamics.*Athirdtypeofdampingknownascompositedampingservesasameanstocalculateamodelaveragecriticaldampingusedinmode-basedprocedures.L3.6LinearDynamicswithAbaqusIntroduction•FourwaystointroducedampinginanAbaqusmodel•Materialdamping•Throughmaterialdefinitions•Elementdamping•Throughelementdefinitions(e.g.,dashpots,springs,connectors,etc.)•Globaldamping•Allowsyoutospecifyviscousandstructuraldampingfactorstoanentiremodelforalineardynamicanalysisstep•Modaldamping•Inmodallineardynamicsanalysis,dampingmaybeapplieddirectlytothemodesofthesystem.•EachsourcecanincludebothviscousandstructuraldampingeffectsDampinginDirectSolutionsL3.8LinearDynamicswithAbaqusDampinginDirectSolutions•SingleDegreeofFreedom(SDOF)system•FortheSDOFproblem:•Undampedradiannaturalfrequency=•Undampedcyclicnaturalfrequency=•Viscousdampingcoefficientforcriticaldamping=•Thefractionofcriticaldamping(viscousdampingratio)=•Theradianfrequencyfordampedfreevibration=MuCuKuP=/nKM=2nnF=22cnCMKM==cCC=21dn=L3.9LinearDynamicswithAbaqusDampinginDirectSolutions•MultiDegreeofFreedom(MDOF)system•TheequationaboverepresentsthefullsetofnodalDOF.•Directdynamicsolutionsoperateonthefullsetofequations.•Directdynamictransientsolutionscanaccountfornonlinearbehavior.•ThedampingmatrixC:•representsanequivalentviscousdampingmodel.•canbefullypopulated.•canbefrequencyandstrainratedependent.•ThestiffnessmatrixK*=KreiKim:•Imaginarycomponentrepresentsstructuraldamping;selisthestructuraldampingfactorforagivenelement,andi=√1.=*MuCuKuPreelelimelelels==KKKKL3.10LinearDynamicswithAbaqusDampinginDirectSolutions•Dampingindirectsolutionlineardynamics•TheequivalentviscousdampingmatrixCiscreatedwith:•Rayleighmaterialdamping(m)•Discretedashpotandconnectorelements(e)•Viscoelasticmaterials(v)•Thus:C=CmCeCv•ThestructuraldampingmatrixIm(K*)iscreatedwith:•Materialdamping•Complexspringsandconnectorelements•Inaddition,globaldampingmaybedefined(viscousorstructural)•Appliedtoentiremodel,inconjunctionorinplaceoftheabove=*MuCuKuPL3.11LinearDynamicswithAbaqusDampinginDirectSolutions•Summary•Directsteadystatedynamicscanincludeviscousandstructuraldampingviathefollowingsources•Material•Element•Global•Thematerialand/orelementdampingdependsontheforcingfrequency.DampinginModalSubspaceProjectionSolutionsL3.13LinearDynamicswithAbaqusDampinginModalSubspaceProjectionSolutions•ModalSubspaceProjectionusestheeigenvectors(modeshapes)oftheundampedsystemtoreducetheequationsofmotionfromthefullsetofnodalDOFstoamuchsmallersetofunknowns.•ThedynamicsolutiontotheMDOFproblemisassumedtobealinearsummationofuser-selectedmodeshaperesponses.•u(t)istheunknownnodalsolutionvectorasafunctionoftime.•φisthemodeshapefortheselectedmode.•q(t)istheunknowngeneralizeddisplacementforselectedmode,whichissolvedforasafunctionoftime.Sindicatesthesummationisperformedoverallselectedmodes.()()tqt=uL3.14LinearDynamicswithAbaqusDampinginModalSubspaceProjectionSolutions•SimilartohowFEMshapefunctionsareusedtoapproximatethebehaviorofindividualelements,themodeshapesareusedtoapproximatethedynamicbehaviorofanentiremodel.•Creatingamodalmatrixwhereeachcolumnrepresentsoneoftheselectedmodesallowsthelinearsummationequationtoberewrittenas:wherefisthemodalmatrix(eachcolumnisamodeshape)andqisavectoroftheselectedmodegeneralizeddisplacements.•ThenodalDOFtimederivativescanbeexpressedas:=uq=uq=uqL3.15LinearDynamicswithAbaqusDampinginModalSubspaceProjectionSolutions•SubstitutingthelinearsummationequationsintotheMDOFmatrixequationsofmotionandpre-multiplyingbythetransposeofthemodalmatrixwillresultinareducedequationofmotion:•Theunknownsarenowthegeneralizeddisplacementsofthemodeshapesthatwereselectedforthemodalsummation.•Thereducedformfortheequationofmotionisgenerallyordersofmagnitudesmallerinsizethantheoriginalfullsetofequations.•Caution:Themodalsummationapproachisonlyvalidwhentheactualstructuraldisplacementscanbeaccuratelyrepresentedbyasummationoftheselectedmodes.•Toimproveaccuracy,includemoremodesintheselectedset.TTTT=MqCqKqPL3.16LinearDynamicswithAbaqusDampinginModalSubspaceProjectionSolutions•Thereducedequationofmotioncanbeexpressedinmodal