Analyzing the Finite Element Dynamics of Nonlinear

COMPUTATIONALMECHANICSNewTrendsandApplicationsS.Idelsohn,E.O~nateandE.Dvorkin(Eds.)cCIMNE,Barcelona,Spain1998AnalyzingtheFiniteElementDynamicsofNonlinearIn-PlaneRodsbytheMethodofProperOrthogonalDecompositionIoannisT.Georgiou1;??;?andJamalSansour21SpecialProjectforNonlinearScience,Code6700.3NavalResearchLaboratory,WashingtonDC203752DarmstadtUniversityofTechnology,FachbereichMechanik,Hochschulstr.1,64289Darmstadt,GermanyKeywords:Nonlinearrods,niteelementdynamics,chaos,properorthogonalde-composition,activedegrees-of-freedom,invariantmanifolds.Abstract:Thisstudyconcernstheanalysisofspatio/temporalsolutionsoftheniteel-ementprojectionofinnite-dimensionalnonlineardynamicalsystems,Cosseratcontinua,modelingin-planerods.TheniteelementsolutionisanalyzedbyapplyingPOD(properorthogonaldecomposition)techniques.Wendthattheforcedregularandchaoticre-sponseofstraightin-planerodsandshallowin-planearchesisdominatedbyafewactivedegrees-of-freedom.11??ResearchScientist,SAIC-ScienceApplicationsInternationalCorporation,McLean,VA221021IoannisT.GeorgiouandJamalSansour1IntroductionNonlinearpartialdierentialequationsinstructuraldynamicscanbereducedtoanitesetofcoupledoscillatorsbysuccessiveprojectionsontothebasisoffunctionscomposedofthespatialshapesofthenormalmodesofthelinearizedsystem.Theresultingcoupledsetofoscillatorscanbeanalyzedbyapplyingthetheoryofnormalmodesofoscillation1;2;3andthetheoryofgeometricsingularperturbationsandinvariantmanifolds4;5todeterminetheactivedegrees-of-freedom.Giventhefactthatnormalmodesandtheirrealizationastwo-dimensionalinvariantmanifoldsinphasespaceplayafundamentalroleintheanalysisofcoupledoscillators,wewouldliketoexplorethepossibilitytointroducetheseconceptsintheanalysisoftheniteelementdynamicsofhighlynonlinearcoupledpartialdierentialequationsdescribingthemotionsofcontinuainsolidmechanics.Finiteelementmethodologiescantakeintoaccountalmostanytypeofnonlinearity.Thisisincontrastwiththemodaldecompositionapproachwhereonetakesintoaccountsimplenonlinearities,forinstance,quadraticancubic.Thebasicproblemnowistoextractfromtheniteelementsolutionofadynamicalsystem,whosenatureofexactnonlinearitieshasnotbeensacriced,theessentialcharacteristicsofthedynamicssuchasactivedegrees-of-freedom.Andsomehowbringusethenotionofinvariantmanifoldsofmotiontogivedenitemeaningtothegeometricstructureofactivedegrees-of-freedominphasespace.Thisworkattemptstoaddresstheissueofactivedegrees-of-freedomofthedynamicsofCosseratcontinua.Thesecontinuamodelrodsandshellsbytakingintoaccounttheexactnatureofgeometricnonlinearities.Recentlyniteelementschemeshavebeendevelopedtosolvethisinterestingclassofinnitedimensionaldynamicalsystems6;7;8.Wedevelopamethodbasedonproperorthogonaldecompositions9;10;11toanalyzetheniteelementdynamicsofin-planerodsandarchesmodeledasCosseratcontinua.Themethodidentiestheactivedegrees-of-freedom.2NonlinearRodsWeareinterestedinanalyzingthedynamicsofelasticrods.Arodcanbemodeledasone-dimensionalCosseratcontinuum.Whenrestrictedtomoveinaplane,suchacontinuumischaracterizedbytheaxialdisplacementeldu1,thetransversedisplacementeldu2,andtherotationeld!.TheseeldsaremeasuredwithrespecttoareferencecongurationB,parametrizedbyarchlengths,[e-mail]denotetheangleformedbythetangentvectorofthecongurationBandanhorizontalaxis.Ithasbeenshownthatthefollowingkinematicrelations7:U1=cos(!)+cos(+!)@u1@s+sin(+!)@u2@s;2IoannisT.GeorgiouandJamalSansourU2=−sin(!)−sin(+!)@u1@s+cos(+!)@u2@s;K=@!@s(1)provideanaturalmeasureofstraininthecontinuumdeformedbythedistributedaxialforceP1,transverseforceP2andin-planemomentM.Letn1,n2,andmbetheforcesandmomentconjugatetothestrainmeasures(1)inthesensethatΨint=n1U1+n2U2+mKwhereΨintisthestrainenergy.LetTandΨextdenoterespectivelythethekineticenergyandtheworkoftheexternalforces.Themotionofthecontinuumisgovernedbytheprincipleofvirtualwork:J(T−Ψint+Ψext)=ZB(A¨u1u1+A¨u2u2+I¨!!)ds−ZB(n1U1+n2U2+mK)ds+ZB(P1u1+P2u2+M!)ds+(P1u1+P2u2+M!)j@B=0;(2)whereA,Idenoterespectivelythecross-sectionanditssecondmoment.Notethattheexternalforcesmaydependonthevelocityeldtoaccountfordissipation.Wefocusourattentionontheclassofcontinuawithlinearelasticmaterialbehavior,thatis,n1@Ψ@U1=EA(U1−1);n2@Ψ@U2=GAU2;m@Ψ@K=EIK(3)whereE,andGdenoterespectivelythemodulusofelasticity,andtheshearingmodulusofelasticity.Theabovelinearconstitutiverelationsintroducethroughthekinematicstrainrelations(1)theexactnatureofgeometricnonlinearityinthefunctionalJ.ThefunctionalJprovidesthemostprimitivedescriptionofthedynamicsofthemotionofthecontinuum.First,itcanbeusedtoderiveasetofcoupledpartialdierential3IoannisT.GeorgiouandJamalSansourequationstobeintegratedtoobtainthespatio-temporaldynamicsoftheeldsu1,u2and!.Theseequationsarenonlinearandcanbetackledanalyticallytosomeextendbyperturbationmethodsprovidedthatthenonlinearitiesareweakandareapproximatedbytheleadingtermsofapolynomialexpansion.Theclassicalapproachistoturnthesimpliednonlinearequationsintoasetofcoupledoscillatorsbyamodaldecomposition(Galerkinprojection).Thisapproachlea