LNDYN-L07-HarmonicResponse

Steady-StateHarmonicResponseLecture7L7.2LinearDynamicswithAbaqusOverview•Introduction•Steady-StateDynamicsSolutionProcedures•ExcitationandOutput•Steady-StateDynamicsUsage•ComparativeExample•WorkshopIntroductionL7.4LinearDynamicswithAbaqusIntroduction•STEADY-STATEDYNAMICS(SSD)analysisproceduresprovidesolutionstothelinearequationsofmotionwhentheloadingisharmonic.•Harmonicloadingrepeatsintimewithaperiodcorrespondingtoacompletecycleofload.ThecyclicfrequencyFoftheappliedloadisF=1/,andtheradianfrequencyisw=2pF=2p/.•Harmonicloadinghastheformofatrigonometricfunction:P(t)=Pmagsin(wt+q)•Thephaseangleqallowstheloadingtobedescribedrelativetoanystartingpointforthecycle(timeaxisposition)t1axist2axisTimePmagL7.5LinearDynamicswithAbaqusIntroduction•Steady-statedynamics•Whenadampedstructurethatisinitiallyatrestisexcitedwithaharmonicload,ithasatransientresponsethatdisappearsratherquicklyandisrarelyofmuchinterest.•Eventuallythestructurereachesasteady-stateconditionthatischaracterizedbyharmonicresponsewiththesamefrequencyastheappliedharmonicload.•Thetrigonometricformoftheresponse(vector,tensorcomponents)isdescribedbyamagnitudeandaphaseanglewhichpositionstheresponserelativetothestartingpointforthesolutioncycle(timet=0).timeaxisTimeUmagU(t)=Umagsin(wt+q)L7.6LinearDynamicswithAbaqusIntroduction•Steady-statedynamicsprocedures...•arefrequencydomainsolutionsthatrepresentasinglecompletecycleofharmonicresponse.•providesolutionsforeachexcitationfrequencythatareindependentoftheotherexcitationfrequencies(noinitialconditions).•excitationinputmayconsistoftwopartscorrespondingtoamagnitudeandaphaseangle.•produceresponseoutputconsistingoftwopartscorrespondingtoamagnitudeandaphaseangle.•havetwicethenumberofactivesolutionvariablescomparedtostaticortransientdynamicsolutions.•Twounknownsarebeingsolvedforeachdegreeoffreedom(magnitudeandphaseangle).L7.7LinearDynamicswithAbaqusIntroduction•Complexrepresentation•Magnitudeandphaseangleexcitationand/oroutputcanalsobeexpressedintermsofcomplexquantitiesconsistingofreal(Re)andimaginary(Im)parts.ForagenericquantityAtherelationshipis:A*=ARe+iAImcomplexvalueARe=Amagcos(q)AIm=Amagsin(q)•Excitationisinputascomplex(realandimaginarycomponents)•Responsecanbeviewedinmultipleformats(magnitude/phaseorrealandimaginarycomponents).AAL7.8LinearDynamicswithAbaqusIntroduction•ComplexplaneRealAxisImaginaryAxisθAmagAReAIm22magReImAAA=+1ImReAtanAq=L7.9LinearDynamicswithAbaqusIntroduction•Complexplane(cont'd)•Thetimevariationofanexcitationoroutputquantityduringacycleofresponseisequaltoitsprojectiononarotating“SolutionAxis.”RealAxis=SolutionAxisatt=0ImaginaryAxisqAmagSolutionAxisradianposition(wt)attimetSolutionAxisrotatesatwradians/secwAattimetofthecycleAL7.10LinearDynamicswithAbaqusIntroduction•Complexplane(cont'd)•Example:Unitforceduetoanimbalanceforaz-axisrotation.•Fx=1+0i,Fy=0+1i(rotatesabout+z-axis)•Fx=1+0i,Fy=01i(rotatesaboutz-axis)xFRealAxis=SolutionAxisatt=0ImaginaryAxisSolutionAxisradianposition(wt)attimetFxandFyattimetofthecycle(eachgoesthrougha+/cycle)Theyare90degreesout-of-phase.FxFywxFyFyFL7.11LinearDynamicswithAbaqusIntroduction•Equationofmotion:•Assumeharmonicresponseandappliedloads:•Accelerationis180degreesout-of-phasewithcomplexdisplacement:•Velocityis90degreesout-of-phasewithcomplexdisplacement:•Loadvectoriscomplex:++=MuCuKuP2=*uui=*uu=*PPLet=excitationfrequencyLet*designateacomplexquantityitUe==*uuL7.12LinearDynamicswithAbaqusIntroduction•Harmonicequationofmotion:•Dampingforcesare90degreesout-of-phasewiththeelasticandinertialforces.•Withoutdampingthesolutionbecomesunboundedwhenequalsasystemnaturalfrequencywn.2i++=****MuCuKuPL7.13LinearDynamicswithAbaqusIntroduction•Forstructuraldampingmodels,thedampingforcesareproportionaltodisplacements,notvelocity,forwhichtheconceptofacomplexstiffnessinsteady-statedynamicsisuseful.Thus,whereKLisalossmodulusandKSisthestoragemodulus.•ThemotionequationintermsofastructuraldampingconstantSiswhereKL=SKS.2i++=****LSMuKuKuP21iS++=***SMuKuPL7.14LinearDynamicswithAbaqusIntroduction•Thebehavioroflightly-dampedsystemsnearresonantfrequencies...•canbedescribedequallywellintermsofeffectiveviscousandstructuraldampingmodels.•Theeffectivestructuraldampingconstantistwicetheeffectiveviscousdampingratio(fractionofcriticaldamping):•Thehalf-powerbandwidthofaresonantresponsecanbeusedtoestimatetheeffectivedamping.•Half-powerpoints(F1,F2)arethosewheretheresponseis0.707oftheresonantpeak.AmeasureofthesharpnessofaresonantconditionisreferredtoastheQualityFactor(Q).2S=21112nFQFFS===Steady-StateDynamicsSolutionProceduresL7.16LinearDynamicswithAbaqusSteady-StateDynamicsSolutionProcedures•Threetypesofsteady-statedynamicsanalysisproceduresareavailableinAbaqus:•Direct•SolvesthecompletesetofmodelDOFsateachexcitationfrequency.•Subspaceprojection•Projectstheequationofmotionontoasubspaceofmodeshapesandsolvesforthegeneralizeddisplacements(GU)•TraditionalandSIMarchitectures.•Modalsuperposition•Approximatesthesystemresponsebyasuperpositionofmodalresponses.•Thereducedsetofeq