LNDYN-L06-ResponseSpectrum

ResponseSpectrumAnalysisLecture6L6.2LinearDynamicswithAbaqusOverview•Introduction•TheResponseSpectrum•DeterminingPeakModalResponse•ModalSummationMethods•CombiningResultsfromMultipleSpectrums•SpectrumDefinition•ResponseSpectrumUsage•ResponseSpectrumExample•WorkshopIntroductionL6.4LinearDynamicswithAbaqusIntroduction•Responsespectrumanalysisisanapproachtoestimatingthepeakresponse(displacement,stress,etc.)ofastructureduetotransientbasemotions.•Shockanalysisofshipboardequipment(DDAM).•Designingstructuresforseismicevents(earthquakeresistance).•Thebasemotionsareassumedtoconsistonlyoftranslations.•Aresponsespectrumcharacterizesthedynamicenvironmentassociatedwithbasemotions.•ItisthepeaklineardynamicresponseofaSingleDegreeofFreedom(SDOF)oscillatoroverarangeofnaturalfrequencies.•Itisaspectrumofeitheracceleration,velocityordisplacement.•Itisnotabasemotion.•Itcanbederivedfromtransienttestdata.•Itisoftenprovidedaspartofadesignspecification.L6.5LinearDynamicswithAbaqusIntroduction•Theresponsespectrumprocedureisinherentlyanapproximatesolutionmethod:•Typicallythemethodisusedwhenaconservativeestimateofastructure’speakresponseisrequiredfordesignpurposes.•Itsadvantageisthatittakesminimalstafforcomputertimetoobtainasolution,soitisoftenusefulforpreliminarystudies.•Responsespectrumanalysisisamode-basedprocedure.•Itisstrictlyalinearanalysisprocedure.•Itmustbeprecededbyaneigenvalueextraction.•Itcanberunasarestartanalysis.L6.6LinearDynamicswithAbaqusIntroduction•Responsespectrumanalysisisamode-basedprocedure.•Itassumesastructure’snaturalfrequenciesandmodeshapesarerepresentativeofsimpleSDOFoscillators.•Thecomputationofthepeakresponseofeachmodeisderivedfromtheuser-suppliedresponsespectrum.•Thepeakresponseofthestructureisestimatedbyaprocessofsummingthepeakresponsesofasetofuser-selectedmodes.•Severalmodalsummationmethodsareavailable.•Modalsummationproceduresvaryfromindustrytoindustry.•Modalsummationproceduresaredesignedtobeconservative.•Thepeakstructuralresponsesfrom2to3independentresponsespectrumcasescanbecombinedwithinasingleanalysisexecution.TheResponseSpectrumL6.8LinearDynamicswithAbaqusTheResponseSpectrum•TheresponsespectrumrepresentsthepeaklineardynamicresponseduetoatransientbasemotionofaSingleDegreeofFreedom(SDOF)oscillatoroverarangeofnaturalfrequencies.•Itcanbeconstructedforaknowntransientbasemotionby:1.Performinglineartransientdynamicanalysesonanumberofspring-mass-dampersystems,eachrepresentingasingleresponsefrequency.2.Identifyingthepeaktransientresponse(displacement,velocity,acceleration)foreachspring-mass-damper.3.Plottingthepeakresponsesvs.frequency.4.Alternatively,solvetheSDOFConvolutionIntegralequationforthepeakresponseattherequirednaturalfrequenciesviaeitheranalyticalornumericalmeans.(Thomson,W.T.,―TheoryofVibrationwithApplications‖,Prentice-Hall,Inc.)L6.9LinearDynamicswithAbaqusTheResponseSpectrum•Characteristics•Thepeakresponseofaspring-mass-dampersystemonlydependsonthestiffness-to-massratio(frequency)andthedamping(%critical).•Thespring-mass-dampersusedintheidealizationofaresponsespectrumareoftenreferredtoasmasslessoscillators(anymass,evenonenearzerocanbeusedtogeneratethespectrum).basemotionequationRecall:bmucukumu2cncmccc/nkm22nnbcukuuuuuummL6.10LinearDynamicswithAbaqusTheResponseSpectrum•Thetermmasslessoscillatoralsoreflectstheinherentassumptionthatthestructurebeinganalyzedhasinsufficientmasstoinfluencethemotionofthesupportingbase(i.e.,abuildingdoesnotinfluenceearthquakegroundmotion).•Adifferentresponsespectrumisgeneratedfordifferentfractionsofcriticaldamping(viscousdampingratios).•Theeffectofdampingwithinastructureisaccountedforinthegenerationoftheresponsespectrum.•Dampingassociatedwithastructure’seigenmodesisusedtointerpolatethepeakresponsebetweenspectrumsthatcorrespondtodifferentviscousdampingratios.L6.11LinearDynamicswithAbaqusTheResponseSpectrum•Foragivenfractionofcriticaldamping:•Onlyasingleresponsespectrumcanbegeneratedfromagivenbasemotiontransient.•However,manybasemotiontransientsmaybeabletobecharacterizedbyagivenresponsespectrum.•Asadesignanalysistool,aresponsespectrumrepresentsanupperboundresponseforalargenumberofpossibletransients.•Responsespectrumsareoftenbasedonacombinationofanalytical,empirical,andexperimentaldata.L6.12LinearDynamicswithAbaqusTheResponseSpectrum•ThreespectrumtypescanbeusedbyAbaqus:•Adisplacementspectrum(SD)isthepeakdisplacementresponserelativetothesupportingbase.•Avelocityspectrum(SV)isthepeakvelocityrelativetothebase.•Anaccelerationspectrum(SA)isthepeakacceleration.•ResponsespectrumanalysisisbasedontheharmonicbehaviorofsimpleSDOFoscillators.Therefore,thepeakdisplacement,velocityandaccelerationspectrumsarerelatedasfollows:2AVDSSSDeterminingPeakModalResponseL6.14LinearDynamicswithAbaqusDeterminingPeakModalResponse•ConsideraSingleDegreeOfFreedom(SDOF)system.•Theequationgoverningfreevibration(harmonicbehavior)is:•Accelerationanddisplacementare180degreesout-of-phase.•Thepeakaccelerationanddisplacementmagnitudesoccuratthesametimeduringaresponsecycle.•The