LNDYN-L01-Introduction

IntroductiontoAbaqusLinearDynamicsLecture1L1.2LinearDynamicswithAbaqusOverview•DynamicResponse•WhentoConsiderDynamicEffects•LinearDynamicsProcedures•LinearDynamicsSoftwareArchitectureDynamicResponseL1.4LinearDynamicswithAbaqusDynamicResponse•Aproblemisdynamicwhentheinertialforcesthatresultfromstructuralaccelerationsareboth:•significantand•varyrapidlyintime.•Inertialforcesareproportionaltoastructure’smassandacceleration.•Iftheloadsactingonastructurevary“slowlyenough”theinertialforceswillbe“small,”andtheresponsecanbeconsideredasquasi-static.•Bloodpressurepulsationloadsactingonmedicalstents.•Sometimeswehavelargeinertialoadsbutcandostaticanalysesbecausetheloadsvaryslowlywithtime.Forexample,•Centrifugalloading(inertialloadsareconstantintime).•Gravity(deadweight)loading.L1.5LinearDynamicswithAbaqusDynamicResponse•Sometimesdynamicvibrationproblemscanbestudiedeffectivelyinthefrequencydomainwhentheexcitationandresponseareassumedtovaryharmonicallywithtime.•Anoutofbalancepumpimpeller.•Motormountdesignsforvibrationisolation.•Dynamicresponseassociatedwithnonlinearstructuralsystemsmustbesolvedwithatimeintegrationalgorithm(timedomain).•Materialnonlinearity(plasticity,largestrains,etc.).•Geometricnonlinearity(largedisplacement,contact,etc.).•Abaqus/StandardandAbaqus/Explicitareavailablefornonlineardynamics.L1.6LinearDynamicswithAbaqusDynamicResponse•Theprimaryemphasisofthiscourseisonthedynamicanalysisoflinearsystems.•Lineardynamicanalysisassumesthatasystem’selasticanddampingforcesdependlinearlyonthenodaldisplacementsandvelocities.•Lineardynamicsalsoassumesthattheexternallyappliedforcesareindependentofthenodaldisplacementsandvelocities.•Linearsystemscanbeanalyzedwithseveraldynamicprocedures.•Eigenvalueextraction(naturalfrequenciesandmodeshapes).•Timedomaintransient(viamodalsuperposition).•Steady-stateresponse(harmonicfrequencydomainanalysis).•Responsespectrum(peakresponsecalculationsforshock).•Randomresponse(vibrationsduetorandomexcitation).L1.7LinearDynamicswithAbaqusDynamicResponse•InAbaqus,theconceptoflinearanalysisisgeneralizedtotheideaoflinearperturbationanalyses.Theideaisthatthelinearresponseis,ingeneral,basedontheanalysisofsmallperturbationsaboutapreloadedstate:uLineardynamicanalysisproceduresinAbaqusallutilizeLinearPerturbationstepsPerturbationanalysisaboutthebasestateu0PLinearperturbationresponseP(u),general,nonlinear,basestateresponse0uL1.8LinearDynamicswithAbaqusDynamicResponse•Thelinearperturbationconceptisparticularlyusefulfor:•rotatingmachineanalysis,wheretheconstantrotationofthesystemcreatesstressstiffeningeffectsthatcanalterthesystem’seffectivestiffnesssignificantly.•investigatingpotentialstructuralcollapsemechanisms,wheretheeffectivestiffnessofastructureapproacheszeronearacriticalload.•Thus,lineardynamicsanalysisinAbaqusallowsthelocallylinearizedresponseofageneral,nonlinearsystemtobestudied.•Suchperturbationanalysiscanberepeatedatdifferentsolutionpointsinthegeneral,nonlinearresponseanalysis.L1.9LinearDynamicswithAbaqusDynamicResponse•Modalsuperposition:•Whenthelinearresponseassumptionsareapplicable,thenthenaturalfrequencycontentofasystemisconstant.•Thedynamicresponseisapproximatedbyalinearcombinationoftheresponseofasubsetofthesystem’snormalmodes(modeshapesfromtheeigenvalueextraction).•Modalsubspaceprojectionissimilartomodalsuperposition,but...•canaccountforsomechangesinasystem’sstiffnessasafunctionofthedeformation(mildnonlinearityfortransients).•canaccountforchangesinasystem’sstiffnessanddampingbehaviorasafunctionoftheharmonicexcitationfrequency.•assumesthepotentialchangesinstiffnessanddampingdonotsignificantlyalterthemodeshapesthatareneededtocharacterizethedynamicresponse.L1.10LinearDynamicswithAbaqusDynamicResponse•Theapproachtodynamicmodelingofasystemdependsstronglyonwhetherornotnonlinearitiesmustbeconsideredindetail.•Ifthesystemhassignificantlynonlinearbehavior,thendirectintegrationofthefullsetofnodalequationsintimeisnecessary.•Alldetailsoftheresponsecanbemodeled.•Theanalysismaybecostly.•Steadyharmonicloadingwillstillrequireatransientsolution.•ForsomeformsofmildnonlinearitysignificantcostsavingscanbeattainedatminimallossofaccuracywithSubspaceProjection.•Ifthesystemcanbelinearizedreasonablyforthedynamicevent,modalsuperpositionmethodscanbeapplied.•Computationallyveryefficient(lowcost).•Greaterinsightintothedynamicbehaviorofthesystemsinceproblemscanbesolvedinboththefrequencyandtimedomains.WhentoConsiderDynamicEffectsL1.12LinearDynamicswithAbaqusWhentoConsiderDynamicEffects•Indynamicanalysis,theimbalancebetweentheexternalandinternalforcesresultsinanacceleration:where=inertialforces(massxacceleration)I=internalforcesP=externalforces•Theresultingaccelerationsgiverisetotheinertialforces.•TheinertialforcesarealsoreferredtoasD’Alembertforces.PIMuMuL1.13LinearDynamicswithAbaqusWhentoConsiderDynamicEffects•Theprecedingequationiscompletelygeneral.•Itisreferredtoastheequationofmotionsinceitrelatestheinternalandexternalforcesthatactuponastructuretothemotionofthestructure(acceleration).•Theequationofmotionisalso