Computational methods for complex eigenproblems in

ComputationalmethodsforcomplexeigenproblemsinfiniteelementanalysisofstructuralsystemswithviscoelasticdampingtreatmentsFernandoCorte´s*,Marı´aJesu´sElejabarrietaDepartmentofMechanicalEngineering,MondragonUnibertsitatea,Loramendi4,20500Mondragon,SpainReceived3October2005;receivedinrevisedform18January2006;accepted19January2006AbstractInthispaperefficientnumericalmethodstoapproximatethecomplexeigenvaluesandeigenvectorsinnon-proportionalandnon-vis-coussystemsarepresented.Thesemethodsarespeciallyconceivedforpracticalengineeringapplicationsmakinguseofthefiniteelementanalysistodeterminatetheeffectthatpotentialdampingtreatmentshaveonthenaturalfrequenciesandmodeshapesofstructuralsys-tems.Consideringthesolutionoftheundampedproblem,thecomplexeigenpairisestimatedbyfiniteincrementsusingtheeigenvectorderivatives.Fornon-proportionalviscoussystemswithlowandmediumdamping,asimplesingle-steptechniqueispresentedwhoserapidityandaccuracyisverifiedbymeansofnumericalapplications.Forhigherdampedsystemsanincrementalapproachisproposedthatkeepstheaccuracywithoutsignificantlyincreasingthecomputationaltime.Fornon-viscouslydampedsystemsafastiterativemodalityissuggested,whichallowstoapproximate,inanefficientandsimpleway,thecomplexeigenpair.Asnumericalapplications,thestudyofametallicbeamwithfreelayerdampingtreatmentiscompletedusingfiniteelementprocedures,wherethedampingmaterialismodelizedbyanexponentialmodelwhoseparametersareobtainedfromcurvefittingtoexperimentaldata.2006ElsevierB.V.Allrightsreserved.Keywords:Complexmodes;Dampingtreatments;Finiteelementanalysis;Non-classicaldamping;Non-viscousdamping1.IntroductionViscoelastictreatmentsareextendedproceduresforstructuralvibrationreductionusingdampingmaterials(mono-graphsonthissubjectcanbefoundinliterature,e.g.Refs.[1–3]).Inengineeringapplications,theefficacyofthedampingtreatmentsmaybemeasuredbymeansofthedampedsystemeigenvalues.Indeed,thesymmetriceigenproblemwithvis-cousdampingisgivenbythesecondordersystem[4]s2rMþsrCþKur¼0;ð1ÞwhereM,CandKarethemass,viscousdampingandstiffnessn·nsymmetricmatrices.MispositivedefiniteandKispositivedefiniteorsemi-positivedefinite.Forthegeneralcase,alsocallednon-proportionalcase,thertheigenpair,withr62·n,isgivenbythecomplexeigenvaluesrandthecomplexeigenvectorur,whichappearinconjugatepairs,srþn¼srð2Þand0045-7825/$-seefrontmatter2006ElsevierB.V.Allrightsreserved.doi:10.1016/j.cma.2006.01.006*Correspondingauthor.Tel.:[phone];fax:[phone].E-mailaddresses:[e-mail](F.Corte´s),[e-mail](M.J.Elejabarrieta).(2006)6448–6462urþn¼ur;ð3Þrespectively,whereðÞdenotesthecomplexconjugateoperator.AnexceptionoccurswhentheM,CandKmatricesverifytheconditionpresentedbyCaugheyandO’Kelly[5],KM1C¼CM1K;ð4Þthen,thesystempossessnormalmodes,i.e.theeigenvectorsarerealandtheyarethesameasthatoftheundampedsystem.Acommontechniqueforsolvingthegeneralnon-proportionalcomplexeigenproblemconsistsinthetransformationofEq.(1)intoalinearsystemgivenbythestate-spaceequationsK00Mur_ur¼srCMM0ur_ur;ð5Þwhere_ur¼srur.Withthistechniquethepositivedefinitionofthematricesisnolongerverifiedandthesizeoftheproblemisdoubled,thus,thecomputationaleffortsareseriouslyenlarged.Theeigenvaluessrarethesameasthoseoftheoriginalsystemandtheeigenvectorszrarerelatedwiththeformerbyzr¼ursrur.ð6ÞThelarge-scalesystemsmaybesolvedbytheclassiccomplexLanczos’s[6]orArnoldi’s[7]methods.Analternativeispre-sentedbyLeung[8],whoanalysesacomplexsubspaceiterationmethod.Morerecently,somemethodswhichworkintheoriginalspaceandsolvethequadraticeigenvalueproblemhavebeendeveloped,suchasthecomplexvectoriterationprocesspresentedbyRuge[9]andthecomplexsubspaceiterationprocedurebyFisher[10].Allthesemethodsarenotappropriateinsystemswithnon-viscousdamping,whosemotionequationinfreevibrationisrepresentedintime-domain[11]byM€uðtÞþZt1GðtsÞ_uðsÞdsþKuðtÞ¼0;ð7Þwheretisthetime,sistheretardationtime,G(t)isthedampingfunctionmatrixandu(t),_uðtÞand€uðtÞarethedisplacementvectoranditsfirstandsecondtimederivatives,respectively.Then,theeigenproblemisgivenbys2rMþsreGðsrÞþK ur¼0;ð8Þwhereð~ÞdenotestheLaplacetransformation.Thiseigenproblembecomesanon-linearmatrixsystemduetothetermeGðsrÞ,wherefromtheeigenvaluesmaybecomputed[12]bysolvingthemrootsofthecharacteristicequation,dets2rMþsreGðsrÞþK ¼0;ð9Þwhoseordermisgenerallyhigherthan2·n,m¼2nþp;pP0;ð10Þthatis,thenon-viscouscharacterofthesystemintroducespextrapoles.Consequently,theeigenmodescanbeclassifiedinelasticmodes,relatedtothe2·ncomplexconjugateeigenvalues,andnon-viscousmodes,whichareintroducedbythenon-viscousnatureofthedampingmechanism.Thisfactisamajordifferencebetweentheviscousandnon-viscouslydampedsystems.Forstablepassivesystems,thesepextraeigenvaluesarenegativerealeigenvalues,thus,theassociatenon-viscousmodesareovercriticallydampedmodes,i.e.theyhavenotoscillatorybehaviour.Aninterestingandexhaustiveanalysisofaonedegreeoffreedomnon-viscoussystemispresentedbyMullerinRef.[13].Inengineeringapplicationsaimedatstudyingtheeffectofd