LNDYN-L08-RandomResponse

IntroductiontoRandomResponseLecture8L8.2LinearDynamicswithAbaqusIntroductiontoRandomResponse•Overview•ExcitationandOutput•RandomResponseUsage•BaseMotionExample•Steady-StateDynamicsAlternativeApproach•WorkshopOverviewL8.4LinearDynamicswithAbaqusOverview•Objectives•ThislectureprovidesageneraldescriptionoftherandomresponseanalysiscapabilitiesavailableinAbaqus.•AcompletediscussionoftheunderlyingtheoryandAbaqusimplementationisbeyondthescopeofthislecture.•ThistypeofinformationisavailableintheAbaqusAnalysisUser’sManualandtheAbaqusTheoryManual.•Therandomresponseanalysisprocedurewillbedescribedandillustratedinthecontextofanexampleofprimarybasemotionrandomexcitation.•Asteady-statedynamicspostprocessingapproachforobtainingpoint-by-pointrandomresponseanalysisresultswillalsobeillustrated.L8.5LinearDynamicswithAbaqusOverview•Randomexcitation•Loading(forces,motions,etc.)isunknownasafunctionoftime.•Theexactvalueoftheexcitationatanyparticularpointintimecannotbepredictedbaseduponpasthistory.•However,thebasicnatureoftheloadingisknownandcanbedescribedinastatisticalsense.•Thesamestatisticalmeasuresusedtodescribetherandomloadingarealsousedtodescribethestructuralresponse.L8.6LinearDynamicswithAbaqusOverview•Simplifyingassumptions•Theexcitationisstatisticallyconstantintime.•Thetimeassociatedwiththeexcitationis“long.”•Short-termtransients(i.e.,start-up/shut-down)areignored.•“Long”shouldbeconsideredrelativetothelowestresponsefrequencyofthestructure(longestresponseperiod).•Theexcitationisstationaryintime.•Itsrelevantstatisticalpropertiesdonotvarywithtime.•Forseveraltimehistorysamples,thestatisticalpropertiesareidentical.•Theexcitationisergodic.•Forseveraltimehistorysamplesoftheexcitation,thetimeaverageofeachsampleisthesame.L8.7LinearDynamicswithAbaqusOverview•RandomresponseinAbaqus•Randomresponseanalysisisafrequency-domainlinearperturbationprocedure.•Linearbehaviorofthestructuralsystemisassumed.•Therandomresponseisobtainedintermsoftheeigenmodes.•Subspaceprojectionisnotavailable.•Dampingcanbemodal,structural,Rayleighorcomposite.•Steady-statedynamictypecalculationsareembeddedwithintherandomresponseprocedure•Modalfrequencyresponsefunctionsaregeneratedaspartofthesolutionprocess.L8.8LinearDynamicswithAbaqusOverview•Statisticaldefinitions•TheMean(Expected)Valueofarandomvariablez(t)isthetimeaveragedvalueofthevariable.•Sincethedynamicresponseiscomputedaboutastaticbasestate,themeanvalueofanyrandomloadingorresponsevariablemustalwaysbezero./2/21()lim()TTTEzztdtT()0EzL8.9LinearDynamicswithAbaqusOverview•TheVariancemeasuresthetimeaveragedsquareddifferencebetweentherandomvariableanditsMeanValue.•SincetheMeanValueiszeroforthisapplicationtheVarianceisthesameastheMeanSquareValue.•TheStandardDeviation(RMS-RootMeanSquare)valueis:/222/21lim|()()|TzTTztEzdtT/222/21lim|()|TzTTztdtT2zzL8.10LinearDynamicswithAbaqusOverview•PowerSpectralDensity(PSD)functionofarandomvariablez:•z(t)containsalargenumberoffrequencycomponents.•ThePSDcurvecharacterizestheamountofpower(inthesenseofameansquaredvalue)containedinz(t),perunitfrequencyasafunctionoffrequency(frequencydomain).•TheMeanSquareValue(Variance)oftherandomvariablez(t)canbeexpressedintermsofthePSDas:where,Sz(f)isthedesignationforthePSDcurve.20()zzSfdfL8.11LinearDynamicswithAbaqusOverview•Simpleexample•ThePSDcurveSzforarandomexcitationvariable,suchasabasemotionacceleration,isprovideasadesignspecification.•Forabasemotionacceleration,thePSDcurvehasunitsofaccelerationsquaredperfrequency.Forexample:or,whereg=gravitationalacceleration.•ThefrequencyresponsefunctionH(f)duetoaunitaccelerationbasemotioniscomputedforanoutputquantitythatisofinterest.•H(f)canberepresentativeofadisplacement,acceleration,stress,force,etc.withunitsofoutput/acceleration(Forexample,inches/g)24m/secHz2gHzL8.12LinearDynamicswithAbaqusOverview•ThePSDcurvefortherandomresponsevariablex,withunitsofoutput2/Hzis:•TheMeanSquareValue(Variance)oftherandomresponsevariablexcanthenbeestimatedbynumericallyintegratingtheresponsePSD.•TheRMSresponse(onestandarddeviation)fortherandomresponseoutputvariableisthen:2xxz()()()SfHfSfmaxF2xx0()Sfdf2xxExcitationandOutputL8.14LinearDynamicswithAbaqusExcitationandOutput•Excitationloads•Concentratedpointloads•Distributedloads•Connectorelementloads•Primarybasemotions(*BASEMOTIONviaKeywordsEditor)•TYPE=ACCELERATIONmustbeused(default).•SCALEandAMPLITUDEparametersshouldbeomitted.•Secondarybasemotionisnotsupported.•Usercanindependentlydefinea“bigmass”andforceexcitation.•Amplitudecurvesassociatedwithloadsareignored.•“Unit”forcingconditionsarealwaysassumed.•FrequencydependenceisviatheuserprovidedexcitationPSD.L8.15LinearDynamicswithAbaqusExcitationandOutput•Multipleexcitationscanbeactive.•Eachrandomexcitationisassignedtoaloadcase.•Loadcasesareactivatedintheanalysisviathe*CORRELATIONoption(discussionwillfollow).•Loadcase1isreservedforalldistributedloads.•Concentratedpointloads,connectorelementloads,andbasemotionscanbeassignedtoanumberedloadcasewiththeLOADCASEparameterontheappropriatekeywordoption.•TheL