1FreeandForcedVibrationResponseofTwoDegreeofFreedomSystems

withDamping2SystemswithViscousDampingExtendingtheprevioussectionstoincludetheeffectsofviscousdamping(dashpots)3ViscousDampinginMDOFSystemsTwobasicchoicesforincludingdampingModalDampingAttributesomeamounttoeachmodebasedonexperience,i.e.,anartfulguessorEstimatedampingduetoviscoelasticityusingsomeapproximationmethodModelthedampingmechanismdirectly(hardandstillanareaofresearch-goodforphysicistsbutengineersneedmodelsthatarecorrectenough).4ModalDampingMethodSolvetheundampedvibrationproblemasbeforeHerethemodeshapesandeigenvectorsarerealvaluedandformorthonormalsets,evenforrepeatednaturalfrequencies(knownbecauseissymmetric)5ModalDamping(cont)DecouplesystembasedonMandK,i.e.,usethe“undamped”modesAttributesomezi(zeta)toeachmodeofthedecoupledsystem(aguess.Notknownbeforehand.Canbetestedwithgrossdatalikex):

Alternately:here6TransformBacktoGetPhysicalSolutionUsemodaltransformtoobtainmodalinitialconditionsandcomputeAiandFi:Withr(t)known,usetheinversetransformtorecoverthephysicalsolution:7ModalDampingbyModeSummationCanalsousemodesummationapproachAgain,modesarefromundampedsystemThehigherthefrequency,thesmallertheeffect(becauseoftheexponentialterm).Sojustfewfirstmodesareenough.8Computeq(t),TransformbackTogettheproperinitialconditionsuse:Usetheabovetocomputeq(t)andthen:theresponseinphysicalcoordinates.9ExampleConsider:Subjecttoinitialconditions:ExperimentsdonotgiveC.Theyprovidezeta(inmodalcoordinates)bythehalfpowermethod.Computethesolutionassumingmodaldampingof:10

Computethemodaldecomposition

L=sqrt(M)Computethemodalinitialconditions:11Computethemodalsolutions:Yields:Thenusex(t)=Sr(t)12So,firstseparatesolutionsinthemodalcoordinateswerefoundandthenthemodeswereassembledbytheuseofS.Theresponseinthephysicalcoordinatesisthereforeacombinationofthemodalresponsesjustasintheundampedcase.13LumpedDampingmodelsInsomecases(FEM,machinemodeling),thedampingmatrixisdetermineddirectlyfromtheequationsofmotion.Thenouranalysismuststartwith:Subjectto14GenericExample:IfthedampingmechanismsareknownthenSumforcestofindtheequationsofmotionFreeBodyDiagram:15MatrixformofEquationsofMotion:TheCandKmatriceshavethesameform.Itfollowsfromthesystemitselfthatconsisteddampingandstiffnesselementsinasimilarmanner.16AQuestionofmatrixdecouplingCanwedecouplethesystemwiththesamecoordinatetransformationsasbefore?Ingeneral,thesecannotbedecoupledsinceKandCcannotbediagonalizedsimultaneously17ALittleMatrixTheory18MoreMatrixStuffandNormalModeSystems19ProportionalDamping20ProportionalDamping(cont)21ForcedResponse:theresponseofan2dofsystemtoaforcingtermk1m1x1m2x2k2F1F2c1c2

22IfthesystemofequationsdecouplethenthemethodsofSDOFcanbeappliedSDOF23Withthemodalequationinhandthegeneralsolutionisgiven24Theappliedforceisdistributedacrosstheallofthemodesexceptinaspecialcase.AnexcitationonasinglephysicalDOFmay“spread”toallmodalDOFs(oneFgeneratesmanyf’s)ItisactuallypossibletodriveaMDOFsystematoneofitsnaturalfrequenciesandnotexperienceresonantresponse(anunusualcircumstance)25Example:

A2-dofsystem26Computethemassnormalizedstiffnessmatrixanditseigensolution27Transformthedampingmatrix,theforcingfunctionandwritedownthemodalequations28ComputethemodalvaluesusingthesingledegreeoffreedomformulasThemodaldampingratiosanddampednaturalfrequenciesarecomputedusingtheusualformulasandthecoefficientsfromthetermsinthemodalequations:29UseSDOFformulafortheparticularsolutionNowtransformbacktophysicalcoordinatesNotethattheforceeffectsbothdegreesoffreedomeventhoughitisappliedtoone.30TheFrequencyResponseofeachmodeisplotted:012345-30-20-1001020Frequency(w)Amplitude(dB)R1(w)/f1(w))R2(w)/f2(w))Thisgraphshowstheamplitudeofeachmodeduetoaninputmodalforcef1andf2.Aforceappliedtomass#2F2willcontributetobothmodalforces!31Thefrequencyresponseofeachdegreeoffreedomisplotted012345-50-40-30-20-10010Frequency(w)Amplitude(dB)X1(w)/F2(w))X2(w)/F2(w))Thisgraphshowstheamplitudeofeachmassduetoaninputforceonmass#2.Eachmassisexcitedbytheforceonmass#2Bothmassesareeffectedbybothmodes32ResonanceformultipledegreeoffreedomsystemscanoccurateachofthesystemsnaturalfrequenciesNotethatthefrequencyresponseofthepreviousexampleshowstwopeaks

IfintheoddcasethatbisorthogonaltooneofthemodeshapesthenresonanceinthatmodemaynotoccurIfthemodesarestronglycoupledtheresonantpeaksmaycombine(seeX1/F2inthepreviousslide)andbehardtonoticeSpecialcases:33Example:Illustratingtheeffectoftheinputforceallocation34Calculatingthenaturalfrequenciesandmodeshapesyields:Themassnormalizedeigenvectorsare:35Transformandcomputethemodalequations:36Homework:37LagrangeEquation38TypicalVibrationAnalysisSteps39TypicalVibrationAnalysisSteps40D’AlembertPrinciple41D’AlembertPrincipleApplyNewton’slawtoeachmassVirtualwork42D’AlembertPrincipleVirtualworkdonebynetforcethroughanadmissibleinfinitesimalvirtualdisplacementiszero.GeneralizedcoordinatesIndependentAdmissiblemotionCompletelyfixeverypartsEqualstonumberofDOF43Hamilton’sPrincipleConsider44Hamilton’sPrincipleConsider2ndterm45Hamilton’sPrincip