APPLICATIONOFCOMPLEXMODALDECOMPOSITIONMETHODSTODISPERSIVEMEDIAByRickeyAlfredCaldwellJr.ADISSERTATIONSubmittedtoMichiganStateUniversityinpartialtoftherequirementsforthedegreeofMechanicalEngineering|DoctorofPhilosophy2016ABSTRACTAPPLICATIONOFCOMPLEXMODALDECOMPOSITIONMETHODSTODISPERSIVEMEDIAByRickeyAlfredCaldwellJr.Complexorthogonaldecomposition(COD)isappliedtoanexperimentalbeamtoextractthedispersivewavepropertiesusingresponsemeasurements.Thebeamismadeofsteelandisrectangularwithaconstantcrosssection.Oneendofthebeamisfreeandishungbyasoftelasticcord.Animpulseisappliedtothefreeend.Theotherendisburiedinsandtoabsorbthewaveasittravelsfromtheimpactsiteonthefree-end;thiselypreventsofthewavetheburiedendandemulatesabeam.CODisappliedtothecomplexanalyticdisplacementensembletoobtaincomplexmodalvectorsandassociatedcomplexmodalcoordinates.Thespatialwhirlratesofnearlyharmonicmodalvectorsareusedtoextractthemodalwavenumbers,andthetemporalwhirlratesofthemodalcoordinatesareusedtoestimatethemodalfrequencies.Therelationshipbetweenthefrequenciesandwavenumbersareusedtodescribethedispersionrelationships,whicharecomparedfavorablytothoseofthetheoreticalEuler-Bernoullibeam.Furtherareapplied,andanewmethodologyisdeveloped.Anovelmethodcalledsmoothcomplexorthogonaldecomposition(SCOD)isappliedtoasimulatedbeamandtheaboveinstrumentedexperimentalbeam.Thesemeasurementsareconvertedintocomplexanalyticdisplacementsandvelocitiesensembles,whichareusedtocomputetwocorrelationmatrices.Thesecorrelationmatricesformacomplexgeneralizedeigenvalueproblemwhoseeigenvaluesandeigenvectorsleadtotheextractionsofthefrequenciesandwavenumbersoftheconstituentwavesofthetravelingpulse.SCODdirectlyextractsthefrequenciesofthetravelingwavesfromtheeigenvalues.SCODcanextractthegeometricrelationship,phasevelocity,andgroupvelocityandagreeswithanalyticalpredictions.Applicationsofbothmethodsareappliedtoasimulatedmasschain.Thedispersionrelationshipofadiscretechainofmassesisextractedfromnumericallysimulateddatabyapplyingcomplexmodaldecomposition.Whenanimpulseexcitationisappliedtooneendofamass-springchain,awaveisgeneratedandpropagatesdownthechain.Thiswaveconsistsofvariousmodes.Thetimerecordforthegenerateddataislimitedsuchthatthewavedoesnotreturntothesensedmasses.Forexample,a250-masschainissimulated,andweconsider(orsense)thetimerecordofthe100masses.Thedatacollectedfromthenumericalsimulationconsistsofthedisplacementsofeachmassateachtimestep.ThisdataisthenusedtoextractcomplexmodesusingCODandSCOD.Theextractedcomplexmodesaccommodatemodaltravelingwaves.Wethencomputethefrequenciesandwavenumbersfrommodalcoordinatesandmodeshapes,respectively.TheamplitudesandfrequenciesofthemodesarealsoestimatedusingRayleighquotients.TheCODextracteddispersionrelationshipmatchestheanalyticalpredictionofthedispersioncurveforthelinearmasschain.CopyrightbyRICKEYALFREDCALDWELLJR.2016Idedicatethecompletionofthisachievementtoallthosewhocomebeforeme.TothosewhoenduredmuchtocreatetheopportunitiesIhavetoday.Tothosewhosawpotential,whoopendoorsformeorprovidedmeansformetonavigateobstacles,Irecognizeandthankyouall.Iwillpickupandcarrythetorchaswell.vACKNOWLEDGMENTSFirst,Iwouldliketothankmyfamilyfortheirtirelesssupport.ConsiderablegratitudetoDr.BrainFeenyandmyCommittee,theDiversityProgramsDr.PiercePierre,Dr.BarbaraO'KellyandfacultyandoftheDepartmentofMechanicalEngineeringandCollegeofEngineering.viTABLEOFCONTENTSLISTOFTABLES....................................ixLISTOFFIGURES...................................xChapter1LiteratureReviewandIntroduction................11.1LiteratureReview.................................11.2Introduction....................................41.3Background....................................61.3.1WavesinaEuler-BernoulliBeam....................71.3.2TheEigenvalueProblemandRayleigh'sQuotient...........91.3.3ProperOrthogonalDecomposition....................111.4ThesisPreview..................................131.5Contributions...................................13Chapter2BeamExperiment............................142.1Equipment.....................................142.2Beam........................................162.3Procedure.....................................162.4DataProcessing..................................172.5Discussion.....................................19Chapter3ComplexOrthogonalDecompositionofanExperimentalBeam233.1Introduction....................................233.2ComplexOrthogonalDecomposition......................253.3Background....................................273.3.1Euler-BernoulliBeam...........................273.3.2Euler-BeamSolutionintheFourierDomain..............293.4Experiment....................................313.4.1Setup...................................313.4.2DataProcessing..............................323.5ResultsandDiscussions.............................333.5.1UsingCOCstoExtractModalAmplitudes...............433.6Conclusion.....................................44Chapter4SmoothComplexOrthogonalDecompositionintheTimeDomain..................................484.1Introduction....................................484.1.1PrimerofSmoothOrthogonalDecomposition.............494.2ApplicationofSmoothOrthogonalDecomposition...............544.3MathematicalDevelopmentforSmoothComplexOrthogonalDecomposition55vii4.4SimulatedEuler-BernoulliBeam....................624.4.1DataProcessing..............................634.4.2ResultsofSimulatedBeam........................634.5ResultsofBeamExperiment..........................644.6Conclusions....................................66Chapter5ExplorationinSpatialSmoothComplexOrthogonalDecomposition..............................705.1Introduction....................................705.2TravelingPulse..................................715.3Results.......................................755.3.1FutureWorks...............................76Chapter6LinearMassSpringSystem......................786.1Introduction....................................786.2AnalyticalModelforanPeriodicChain................796.3ModalDecompositionAppliedtoaMassChain................816.3.1CODAppliedtoaLinearMassChain.................836.3.2SCODRayleighQuotient........................846.4Conclusions....................................84Chapter7ConclusionsandFutureWork....................867.1SummaryofResults...............................867.2....................................887.3FutureWorks...................................88BIBLIOGRAPHY....................................90viiiLISTOFTABLESTable1.1:Examplesofgeometricdispersionequations,phasevelocities,andgroupvelocities,in(a)wavestravelingthroughtheEarth'score,(b)ripplesonapond,(c)lightwavesthroughaprism,and(d)inauniformbeam.Inexample(a)thephaseandgroupvelocityareequal.In(c),thegroupvelocityisslowerthanthephasevelocity.In(d),thegroupvelocityisfasterthanthephasevelocity.............................6Table2.1:DataAcquisitionSystem...........................15Table2.2:BeamDimensions...............................16Table4.1:SimulatedBeamDimensions.........................62ixLISTOFFIGURESFigure2.1:Experimentalsetup..............................15Figure2.2:Accelerationdataforallofthesensors....................18Figure2.3:Accelerationsfromsensors1,16,and31...................19Figure2.4:Velocitydataforallofthesensors......................19Figure2.5:Velocitiesfromsensors1,16,and31.....................20Figure2.6:Displacementdataforallofthesensors...................20Figure2.7:Displacementsfromsensors1,16,and31..................21Figure3.1:Schematicdiagramofexperimentalsetup.Topview............29Figure3.2:Accelerationdatavspositionalongthebeam...............34Figure3.3:Accelerationsfromsensors1,16,and31...................35Figure3.4:Displacementsfromsensors1,16,and31..................35Figure3.5:Sensor1....................................36Figure3.6:Sensor16...................................37Figure3.7:Sensor31...................................38Figure3.8:FFTofthemodalimpacthammersignal..................39Figure3.9:SelectedCOMs.Topleft:2ndmodalCOM.Topright:6thCOMwhichislesscircularthansecond.Thistrendscontinuesandisillustratedinthelowerleft:7thCOM.Lowerright:10thCOM................40Figure3.10:SelectedCOCs.Topleft:2ndmodalCOC.Topright:6thCOCwhichshowsattenuationaftersomeinitialoscillations.Thistrendscontinuesandisillustratedinthelowerleft:7thCOC.Lowerright:10thCOC..41Figure3.11:Realpartofcomplexorthogonalcoordinatenumber4...........42xFigure3.12:Realpartofcomplexorthogonalcoordinatenumber8...........43Figure3.13:Experimentalresults.TheoreticaldispersionrelationshipfortheEuler-Bernoullibeam(Solidline).CODextracteddatapoints().............44Figure3.14:Phasevelocity.................................45Figure3.15:Groupvelocity................................46Figure3.16:CODextractedmodalamplitudevsfrequency()comparedtotheory..47Figure4.1:Displacementsvs.timefromtheanalyticalsolutionforsensor1,sensor16,andsensor31................................64Figure4.2:Velocitiesvs.timefromnumericalderivativesofthedisplacementsforsensor1,sensor16,andsensor31............65Figure4.3:GeometricdispersionrelationobtainedusingSCODforasimulatedEuler-Bernoullibeam.............................66Figure4.4:Examplesofsmoothcomplexorthogonalmodes(SCOMs)fromthebeamexperiment.Thelowerrightplotillustratesandexampleofspuriousmode.67Figure4.5:DispersionrelationshipderivedfromSCODappliedtoanexperimentalbeam......................................68Figure4.6:PhaseVelocityderivedfromSCODappliedtoanexperimentalbeam..69Figure4.7:GroupVelocityderivedfromSCODappliedtoanexperimentalbeam..69Figure5.1:ContrivedexampleoftheFFTofanarrowbandtravelingpulse.....72Figure5.2:Simulatednarrowbandtravelingpulse...................73Figure5.3:FFTofsimulatednarrowbandtravelingpulse...............74Figure5.4:Dispersionrelationshipforatravelingnarrowbandpulsewithaspeedofc0=1:5....................................75Figure5.5:Dispersionrelationshipforatravelingnarrowbandpulsewithaspeedofc0=12....................................76Figure6.1:masschain.Therelaxedpositionanddisplacementofmassmjaredenotedby~xjandujrespectively.Inthiscasethespringsarelinearwithrelaxedlengthh.............................79xiFigure6.2:Displacementsformasses1,31,61,and91.................81Figure6.3:Complexorthogonalvalues..........................82Figure6.4:Linearsystem'sgeometricdispersionrelationshipusingthemodefromthe30highestCOVsinitialvelocityonthemass=1.0..........82Figure6.5:LineardispersionrelationshipusingRayleighquotientandSCOD....83xiiChapter1LiteratureReviewandIntroduction1.1LiteratureReviewAgreatportionofthisworkfocusesonwavestravelinginmedia,inparticular,Euler-Bernoulli(EB)beams,EBbeams,andamasschain.Eulerderivedamodelforbeamoscillationsin1744[1].HisworkseemedtohavebeenmotivatedbythequestionsofBernoulliandisdocumentedinlettersbetweenthetwoSwiss.DanielBernoullipublishedhisworksonthesubjectin1751[2]andtodaytheEBbeammodelisstillinuse.TheEBmodelassumessmallnegligibleshear,andthatplanarcrosssectionsremainplane.AnotherpopularmodelistheTimoshenkobeammodel[3]whichallowsforthewarpingofcrosssectionsandshear.Itwasdevelopedintheearly20thcenturyandisanextensionofEuler'sandBernoulli'swork.FourierwasthetoderivethesolutionfortheEBbeamwithinitialdisplacementandzeroinitialvelocity[4].Threeandone-halfscoreyearslaterBoussinesq[5]derivedasolutionfortheEBbeamwithaninitialvelocity.ContinuinginchronologicalorderfromFourier,[6]notesthatin1822Cauchydevelopedelastictheoryandtheequationsofmotionsforelasticsolids.totheoriginalpaperswerethwartedbyalanguagebarrierasmostofCauchy'spaperswereinGerman.alsostatesthePoisson1publishedworkdescribingwavesinelasticmediain1828.Poisson(1828)[7]andCauchywereabletogetsimilarresults.todescribewavesinsolidscontinued.ThenextnoteworthydevelopmentrelatingtothissubjectwasPochammer'swhodevelopedfrequencyrelationshipsforwavestravelinginrods.Later,Rayleigh'sworkonsurfacewavesin1885[8]predictedtheexistenceofwhatarecalledRayleigh-Lambwaves.Physically,thesewavestravelnearthesurfaceofthickplates.Acommonexampleareearthquakewaves.Lambpublishedconsiderableworksongroupvelocitiesandwavesinplates[9,10].TheseworkswerefurtheredbyPaoandMindlin[11]inthe1960's.Jumpingbacktotheearly1940's,workinstatisticsanddynamicswouldlenditselftostructuralanalysis.Properorthogonaldecomposition(POD)isastatisticaltooldevelopedindependentlybymanytpeoplefromallovertheglobespanningfrom1943to1955;theprogenitorsincludeKosambi,Karhunen,Pougachev,Oboukhov[12,13]andLumley[14].PODisatoolforstatisticallyseparatingtwoindependentvariablessuchastimeandspace.PODhasequivalentformulationssuchasprincipalcomponentanalysis,Karhunen-Loevedecomposition(KLD),empiricalorthogonalfunction(EOF)[15],andsingularvaluedecomposition(SVD)[13,16].PODwasinitiallyappliedtocharacterizespatialstructuresinws,particularlyturbulentws,turbulentwsaroundobjects,andturbulentwsnearboundaries[14,17{20],andnowisusedinmanysuchascontrols[21{24],signalprocessing,[25,26],[27]andstructures[28,29].Amongthestructuralapplications,modalanalysisandnondestructiveevaluationhavereceivedagreatdealofattention[29{31].Intherealmofmodalanalysis,PODisusedtoextractlinearnormalmodes(LNMs)fromdiscreteandcontinuoussystems[32{34]excitedbyamodalimpacthammerorby2randomexcitation[35].ImprovementsweresoughtandothermethodsbasedonPODweredevelopedsuchasenhancedPOD[33],reduce-ordermass-weightedPOD[33,36],andmanymore.Newformulationswithadditionalensemblesweredevelopedsuchassmoothorthogonaldecoosition(SOD)byChelidzeandZhouin2006[37]andsimilarlythesmoothKarhunen-Loevedecomposition(SKLD)in2009byBellizziandSampaio[38].AbofSODisthataprioriknowledgeofthemassmatrixisnotneededandmodeshapesandfrequenciescanbedirectlyextracted[37,39,40];anddampingratioifusingSKLD[38].ThisfromPODinwhichrequirescomputingthefastFouriertransformofthemodalcoordinatestoextractnaturalfrequencies,andusinglogarithmicdecrementonthemodalcoordinatestoextractdamping.TheideaofPODinthecomplexplanewasdevelopandpublishedsimultaneouslybyGeorgiouandPapadopoulosin2006[41],calledC-POD,andbyFeenyattheASMEIMECEconference,laterpublishedandcalledthecomplexorthogonaldecomposition(COD)in[42].CODwasusedtodecomposewavemotionintoitsprimarycomponentsandparameterssuchaswavenumber,frequency,phasevelocity,andgroupvelocity.Additionally,withCOD,travelingandstandingwavescouldbediscerned,andthedegreeoftravelingwavescouldbequanusingthetravelingorsynchronicityindex[43].FeenyetalappliedCODtobio-locomotionofnematodeworms[44]in2009andofin[45]andwasabletodiscerntmechanismusedbytheorganismtotravelforward,backwards,andtoturn.Esquivel[15]appliedCODtosimulateddataandpowersystemdatatomeasureinpowersystemsin2009.CODiswellsuitedtodescribeinspaceandtime.Whenrelatedtowavevelocitythisrelationshipiscalledthedispersionrelationship.3Thisresearchstartswiththevof[46]withabeamexperimentandcreatesanewformulationrelatedtoCODandSOD,whichallowsforthedirectextractionsofwavenumbersandfrequencies.Thisformulationiscalledsmoothcomplexorthogonaldecomposition(SCOD)andisappliedtoasimulation,beamexperiment,andtoasimulatedmasschain.1.2IntroductionWavesareubiquitous.Pressurewavesassoundandelectromagneticwavesthatweseeaslight,arejustoftwoexamplesofwaves.Mechanicalwaves(likesound)andstresswaves,requiresamediumtotravelfrompointAtopointB.Thesewavescanbeofgreatengineeringimportance,andunderstandinghowtheypropagateiscritical.Forexample,wavesfromexplosionsatconstructionsitesandinminesareusedtoloosenandremovematerialfromalocalizedarea.Thatsameexplosiveforceisusedinimprovisedexplosivedevices(IEDs)orasatoolforterroristwhichcancauseinjury,death,anddestructionofproperty.havebeenmadetounderstandhowtheseblastwavesinteractwiththehumanbody,andhowtomitigatethesewavestopreventdamagetohumantissueandproperty.However,wavesusedforgoodcanassistinsuchthingsaslocatingburiedmines,indiagnosticdevicesinthemedical(magneticresonanceimaging,X-rays,ultrasound),nondestructiveevaluationandtesting,orinarchaeologicallytoburiedruinsusinggroundpenetratingradar.Allofthesemodernequipmentarepossiblebyunderstandingwavesandhowwavesinteractwithobjects.Thetoolsdevelopedinthisdissertationcanbeusedtocharacterizehowwavesinteractwithobjects.4Theobjectiveofthisdissertationistousemeasuredresponsestothegeometricdispersionrelation(GDR)inengineeringstructures.Dispersionrelationshipsareequationsthatdescribetherelationshipbetweenfrequency(!),wavenumber(k),geometry,andmaterialproperties.ExamplesoftsystemsandtheirgeometricrelationshipsareinTable1.1below:(a)wavestravelingthroughtheEarth'score[47],whereBsistheradialcomponent(cylindricalcoordinatesystem)ofthemagnetic0ispermeabilityoffreespace,andˆ0isthereferencedensity.Example(b)describesripplestravelingonthesurfaceofashallowpond[48],where=h20620isanon-dimensionalparameterh0isundisturbedwaterdepthand0isthetypicalwavelength.Example(c)islightthroughaprism,wherecisthespeedoflightinavacuum,andnistherefractiveindex.Example(d)iswavesinabeam,whereEisthemodulusofelasticity,Iisareamomentofinertiaofthecross-section,ˆisthemassperunitlengthofthebeam,andAisthecross-sectionalarea.AlongwitheachsystemisitsGDR,fromwhichthephasevelocityandthegroupvelocitycanbedetermined.Thephasevelocity,cp,isthevelocityofapointonasingle-harmonicwaveinspace,forexample,asinglepeak.Thegroupvelocity,cg;isthevelocityoftheenvelopeofagroupofwaves.Theequationsforthetwovelocitiesarecp=!=kandcg=d!=dk.Dispersionisacharacteristicwheregrouporphasevelocityisdependentonthefrequency.Examples(a)and(c)arenon-dispersiveexample(b)and(d)aredispersive.Sp,wewilluseCODandSCODtoestimatethewavenumber,k,andfrequency,!,oftheconstituentwavestravelinginasimulatedandexperimentalbeam,anda100mass,mass-chainsystem.Fromthewavenumbersandfrequencies,wecandeterminetheGDR,thephasevelocity,andgroupvelocityofeachsystem.5ExampleDispersionRelationshipPhasevelocityGroupvelocity(a)!=krB2s0ˆ0cp=rB2s0ˆ0cg=rB2s0ˆ0(b)!=k4ˇ23cp=14ˇ22cg=112ˇ22(c)!=ckcp=c=ncg=c!=(n!+ck)(d)!=k2qEIˆAcp=kqEIˆAcg=2kqEIˆATable1.1:Examplesofgeometricdispersionequations,phasevelocities,andgroupvelocities,in(a)wavestravelingthroughtheEarth'score,(b)ripplesonapond,(c)lightwavesthroughaprism,and(d)inauniformbeam.Inexample(a)thephaseandgroupvelocityareequal.In(c),thegroupvelocityisslowerthanthephasevelocity.In(d),thegroupvelocityisfasterthanthephasevelocity.ToprovideafoundationalunderstandingabriefreviewofwavesinaEuler-Bernoullibeam,Rayleigh'squotients,andproperorthogonaldecomposition(POD)willbepresented1.3BackgroundAbriefintroductionstothemajorconceptsandtoolsusedinthisresearchwillbereviewed.First,theequationsofmotionsforauniformEuler-Bernoullibeamwillsolvedforsimpleharmonicwaves.Theresultsofthiswillleadtothedispersionrelationship.Examplesofdispersionrelationshipforttypesofsystemswillbepresentedaswellasparametersderivedfromthedispersionrelationshipcallthegroupvelocityandthephasevelocity.Togetherthedispersionrelationship,groupvelocity,andphasevelocityarethepropertiesthisresearchseektomeasureinusingCODandSCOD.61.3.1WavesinaEuler-BernoulliBeamThederivationoftheEuler-Bernoullibeamequationiswellknownandcanbefoundinmostvibration-theorytextbookssuchas[49],andwillnotbereviewed.TheequationofmotionforauniformEuler-Bernoullibeam(EB)isˆAy(x;t)+EI@4@x4[y(x;t)]=0;(1.1)wherexisthepositionalongthebeam,tistime,y(x;t)isthetransversedisplacement,ˆisthemassdensity,Aistheareaofbeam'scross-section,Eisthemodulusofelasticity,andIistheareamomentofinertiaofthebeam'scross-sectionabouttheneutralaxis.TounderstandhowtravelingwavesbehaveinanEBlet'ssubstituteasolutiontothewaveequationintotheEuler-Bernoullibeamequation.Following[6],insertingy(x;t)=ei(kx!t);(1.2)whichisawavetravelinginthepositivex-direction,intoEqn.(1.1),andsolvingfor!,leadstothegeometricdispersionrelationship!=ak2;(1.3)7wherea=sEIˆA:(1.4)InEqn.(1.3)!isthetemporalfrequencywithunitsofrad/s,andkisthespatialfrequencymeasuredinunitsofrad/m.ThisequationdescribestherelationshipbetweenthefrequencyandwavenumberofawavetravelingthroughauniformhomogeneousEuler-Bernoullibeam.Thephasevelocityandgroupvelocityarederivedfromthegeometricdispersionrelationship.Thephasevelocityisascp=!=kandfortheEuler-Bernoullibeamthatisequaltoc=ak.FortheEBbeamthespeedofapropagatingharmonicwaveincreaseslinearlywithitswavenumber.Thisphasevelocityduetothedispersionrelationshipisquadratic.Othersystemsmayhaveconstantphasevelocityoftheformc=a.Thegroupvelocityisascg=d!=dk.FortheEBbeamthegroupvelocityiscg=2ak.ThegroupvelocityistwicethephasevelocityfortheEBbeam.Phaseandgroupvelocitiesfortsystemscanbeequal,positiveandunequal,orhaveoppositesigns.Insolvingthepartialtialequation(PDE)inEqn.(1.1)separationofvariablesisusedasastandardtechnique,whichleadstoaneigenvalueproblem(EVP).Eigenvalueproblems(EVP)orageneralizedeigenvalueproblems(GEVP)areusedextensivelyinthiswork.TheyappearinthesolvingofEqn.(1.1),POD,SOD,COD,andSCOD.Rayleigh'squotientisanimporttoolformakingapproximationsbasedonanEVP.Therefore,abriefoverviewofthesetopicsfollows.81.3.2TheEigenvalueProblemandRayleigh'sQuotientAtersereviewofthepropertiesofeigenvalueproblemsisneeded.Aeigenvalueproblem(EVP)hastheformAv=v(1.5)whereAisasquarematrix,viscalledtheeigenvector,andisascalarcalledtheeigenvalue.TheeigenvectorisacharacteristicvectorofAwiththepropertythatwhenAoperatesonvtheoutputisascalarmultipleofv,asindicatedinEqn.(1.5).Geometrically,whenAperformsatransformationonvtheresultingtransformationcanonlychangevinlengthanddirection(sign,)andnottheorientationofv.(Insomecases,however,andvarecomplex,andtheirrealscalingpropertyisnotobviousintherealspace,asthecomplexactionisinrealspaceasarotation.)SomepropertiesoftheEVParediscussedhere.IfAisreal,thentheeigenvaluesarerealorhavecomplexconjugatepairs.Eachcomplexconjugatepairofeigenvalueshaveacorrespondingcomplexconjugatepairofeigenvectors.IfAissymmetric(AT=A)andpositiveitseigenvaluesarei0.IftherankofAisfull(positivecase)andtheeigenvaluesaredistinct,thentheeigenvectorsformabasisforA.Also,ifeacheigenvectorisnormalizedtounitlength,thentheeigenvectorsformanorthonormalbasisforAsuchthatvivj=ij,whereijistheKroneckerdelta.Theeigenvectorsplusv=0createstheeigenspaceofA.Ifaneigenvalueisequalto0thenthematrixissingularandnoninvertible.9ReturningtotheoriginalEVPinEqn.(1.5)foraparticulareigenvaluevi,pre-multiplyingbyvTileadstovTiAvi=ivTivisuchthati=vTiAvivTivi:(1.6)Thisratiocanbeconstructedusinganarbitraryvector,ui,theRayleigh'squotient(RQ)asR(A;u)=uTAuuTu:ThenitisclearfromEqn.(1.6)thati=R(A;vi).ItcanbeshownthatthevaluesoftheRQarestationaryintheneighborhoodsofeigenvectors[50],andthatvalueoftheRQisboundedbythehighestandlowesteigenvalues,suchthatmin(i)R(A;v)max(i).ThustheRQisagoodwaytoapproximateeigenvaluesgivenanassumedeigenvector.AgeneralizedEVPhastheform,Av=Bv:(1.7)SomepropertiesoftheGEVPwenowreview.IfBisinvertiblethenB1Av=vwhere=1.Inthiscase,theGEVPwillhaveallthepropertiesoftheEVP.Also,ifAisinvertiblethenv=A1B.IfAisrealandsymmetricandBisreal,symmetricandpositive(althoughA1BandB1Aarenotsymmetric)thenallneigenvaluesarereal,andtheeigenvectorsareorthogonalthroughBsothatviBvj=ij.Theeigenvectoris10anyvectorthat,whentransformedbyAandB;theresultingvectorsofbothtransformationsarecollinear.TheRQofaGEVPisasi=R(A;B;v)=vTiBvivTiAvi:(1.8)ForboththeEVPandtheGEVPifAorBiscomplexthenT;thetranspose,isreplacedwithH;theHermitianoperation,orthecomplexconjugatetranspose.1.3.3ProperOrthogonalDecompositionThisthesismakesuseofmethodsthataregeneralizationsofproperorthogonaldecomposition(POD).Hence,PODisintroducedhereforbackground.PODwasusedinmechanicsbyLumleytostudyturbulenceindynamics[14].LaterBerkoozetal.[17]reviewedthepropertiesofPODforturbulence.PODwasappliedtostructuraldynamics[51{53],andlatertoextractmodalparameters[32,44,54,55].PODisanimportantandwidelyusedtool.Forexample,anISIcitationsearchonNovember2015,yieldsmorethan2900papers.PODisequivalenttosingularvaluedecomposition(SVD)[16,55{57].Ithasseveraltvariationsfortheextractionoftparametersofinterest.Thesevariationsincludemass-weightedreducedorderproperorthogonaldecomposition[36,40],Ibrahimtime-domaindecomposition[58],smoothorthogonaldecomposition(SOD)[37,39,40],state-variablemodaldecomposition(SVMD)[59]and,complexorthogonaldecomposition(COD)[46,60].ToapplyPODtoastructuretheexperimentersensesastructureandcapturesresponsesignals(forexampledisplacements)attpointssimultaneouslyonthestructureduring11itsresponsetosometypeofexcitation,forexampleanimpulse[54].Nextanensemblematrix,X2RMN,iscreatedfromthesensormeasurements,whereMisthenumberofsensorsandNisthenumberoftimesamplestaken.Sp,X=hxT1xT2xTMiT,wherexi=[xi(0)xiT)xiT)xi((NT)]T,suchthateachrowoftheensemblematrixisthedatafromonesensorandtheindexofeachcolumnisasinglemeasurementsample:X=26666666664x1(0)x1t)x1((Nt)x2(0)x2t)x2((Nt)............xM(0)xMt)xM((Nt)37777777775Oftenthemeanissubtractedfromeachsensor'stimehistory.Nextacorrelationmatrix,R=XXTN,iscomputed,suchthatR2RMM.Finally,aneigenvalueproblemisformulatedasRv=v.Theeigenvectorsarecalledproperorthogonalmodes(POMs)andforlightlydampedstructuresvibratinginthelinearelasticrangewithamassmatrixproportionaltotheidentity,thePOMsconvergetolinearnormalmodes(LNMs)[32,55,61].Theeigenvaluesarecalledproperorthogonalvalues(POVs)andrelatetomodalenergy[17,44].Thenaturalfrequenciesofthebeamcanbeestimatedfromtheproperorthogonalmodalcoordinates(POC)asqi=vTX.TypicallythefastFouriertransform(FFT)ofthePOCareusedtoestimatethefrequencyofeachmode.Complexorthogonaldecompositionandacomplexversionofsmoothorthogonaldecompositionaretobeappliedinthisresearch,andwillbediscussedlater.121.4ThesisPreviewTherearesixremainingchaptersinthisbodyofwork.Chapter2describesthebeamexperiment.TheexperimentalsetupisbasedonthepriorworkofOnsayandHaddow[62].Thegoaloftheexperimentwastocreateatestingrigwhichallowedwaves,generatedfromanimpact,totraveldownabeamandnotofoneofthefreeends.Thiswasdonetoemulateabeam.Chapter3introducesCODandtheapplicationofCODtoabeamexperiment.Alternativeformulationsweresoughttoachievedirectcomputationofwavenumbersandfrequencies.InChapter4anoveldecompositionmethodwascreatedcalledthesmoothcomplexorthogonaldecomposition(SCOD).SCODisformulatedinthetimedomaintodirectlycomputefrequencies.TheninChapter5,SCODiscastinthespatialdomaintodirectlycomputewavenumbers.Finally,chapter6appliesbothCODandSCODtoamass-chainsystemtoexploretheircapabilityonlinearmasschain.1.5ContributionsContributionstothecoveredinthisthesisincludeapplyingCODtoabeamexperiment.Thisreinforcesthesimulationdoneinpriorwork,andshowsthatCODcanextractthedispersionrelationshipfromabeaminrealworldconditionsonrealsystems.BydevelopingSCODseveralcontributionsweremade.First,atheoreticalframeworkforSCODhasbeenestablished.Seconddirectfrequencyestimationfromaneigenvalueproblemcanbemade.CODandSCODtogetherarenewtoolsforwavepropagationanalysisandcontributetothestate-of-the-art.Thestudyonamasschainsystemcontributesthescienceofdispersioninparticlesandlumped-masssystems,insightsandatoolforextractingadispersionrelationshipinlatticesystemwithapplicationinmetamaterials.13Chapter2BeamExperimentThisexperimentissimilarinsetuptoanexperimentconductedbyOnsayandHaddow[62]inwhichtheyusedawavelettransformtoextractthepropertiesofawavetravelinginanelasticmedium.Inthisexperimentoneendofabeamwaspartiallyburiedincoarseunpackedsandandtheotherendofthebeamwassuspendedfreelywithanelasticcord.Thebeamwassensedwith31accelerometersplacedequidistantlyontheunburiedpartofthebeam.Thebeamwasstruckwithaminiimpacthammernearthefreeend.Astheimpulsewavestraveledalongthebeam,startingatimpactlocationnearthefreeendandmovedtowardsthesandbox,thewavesweresensedbyeachaccelerometerastheypropagated.Oncethewavesreachedtheburiedend,thesandabsorbedthewaves,whichresultedinanechoictermination.Whatfollowsisadetaileddescriptionoftheexperimentalongwiththepurposeofthedesign.2.1EquipmentThedataacquisition(DAQ)equipmentislistedinTable2.1.TheNationalInstrumentsDAQsystemhad32availablechannels,anexpressPCIcardwhichwasconnectedtoaDELLInspironlaptopcomputerrunninga32bitWindowsXPoperatingsystem,andwaslinkedtotheNIChassisviaacontrollercard.TwoDAQcardswerepluggedintothechassis.Eachcardhadan16channelcapacity.Thesamplingratewassetto25kHzbasedonthe14maximummeasurablewavenumber(asexplainedinthesectionondataprocessing).ThemaximumsamplingrateoftheDAQsystemwas32kHz.Thirty-onechannelswereusedforaccelerometers,andonechannelwasusedfortheimpacthammer.Althoughtheimpacthammerwasnotusedinanyoftheoutputonlydecompositionmethodcomputations,seeingtheimpactdatahelpedtoensurequalityresultsbyhavingthecapabilityofinspectingforbadimpacts(forexample,doubletaps,sliding,sensorsaturation,andnon-impulsivewaves).Thenumberofchannelslimitedthespatialresolutionofthebeamexperiment.Figure2.1:Experimentalsetup.DataAcquisitionEquipmentChassisNIPXI-1042QCardsPCIexpresscardNIPXI-8360DAQcardNIPXI-4496ControllercardNIPXI-8361AccelerometerPCB352B10Accelerometermass0.0007kgImpacthammerPCBmodel086C80HammertipsFlatmetalPCBmodel084B03VinylcapPCBmodel085A07PointedmetalconeunknownTable2.1:DataAcquisitionSystem152.2BeamTheexperimentalbeamwasmadeofcarbonsteel,anditsdimensionsarelistedinTable2.2.Thebeamwascleanedwithsandpaperandwipeddownwithacetonetoremoverustandchemicalresidues.OneendofthebeamwassupportedbyanelasticcordandtheotherendwasburiedinsandasshowninFigure2.1.ThesandwasplaygroundsandpurchasedatHomeDepot.Theboxthatheldthesandwasmadefrom1/4inchorientedstrainboard(OSB)plywoodboards.Thedimensionsoftheboxwere3ft1ft1ft.A1/2inchslotwascutintothe1ft1ftfaceoftheboxwhichwasusedtoinsertthebeamintothesand.Carewastakentoensurethatthebeamwassuspendedinthesandanddidnottouchthebox.Thetotalbeamlengthwas2.038mofwhich0.61mwasburiedinthesand.Theremaining1.43munburiedportionofthebeamwassensedwithaccelerometers.Theaccelerometerswereattachedwithbee'swaxalongtheneutralaxisofthebeam,0.0458mapart,foratotalsensornetworklengthof1:42m.Themassofeachaccelerometerwas0.0007kg.ExperimentalBeamDimensionsLength2:038mBase0:069mHeight0:0045mDensity7870kg/m3ModulusofElasticity200GPaTable2.2:BeamDimensions2.3ProcedureThebeamwastappedlightlywiththeminiimpacthammer,suchthatthebendingmodewasnotvisiblydetectedwiththenakedeye.Theimpulsetraveleddownthebeam,andthe16inducedaccelerationsweredetectedandmeasuredbytheaccelerometersandrecordedbythedataacquisitionsystem.2.4DataProcessingThedataincluded100samplesbeforethehammerimpactand300samplesaftertheimpactforatotalof400samples.First,thedatafromeachsensorwasforwardandbackwardwithahigh-passwithafrequencyof100Hz.Second,themeanofeachsensor'stimehistorywassubtractedfromitssamplesusingMatlab's"detrend"witha"constant"moThird,anylineartrendsintimewereremovedusingthesamecommandasabovewitha"linear"moFourth,thesignalsweretranslatedonthetimeaxissuchthatthesamplebeforethehammerimpacthadatimeandforcevalueofzero.Finally,theandlast100samplesweretruncatedleavingthestartoftheimpactplus200samples.AccelerationssignalsareshowninFigure2.2andFigure2.3.Togetvelocities,thesignalswerenumericallyintegratedusingthe"cumtrapz"command.Themeansweresubtractedfromthevelocities,whichwerethenhighpasstoremovelowfrequencyintegrationdrift,andintegratedoncemoretogetdisplacements.ThevelocitiesareshowninFigure2.4andFigure2.5.Themeansweresubtractedfromthedisplacements,andthedisplacementswereforatime.ThedisplacementsarepicturedinFigure2.6andFigure2.7.Basedonthesamplingrateofratef=25,000HzandtimerecordofN=f=0:008s,themaximumdetectablefrequencyisfmaxt=12,500Hzandtheminimumdetectablefrequency17isfmint=125Hz.Theminimumdetectablewavenumberisbythespanofsensors,L,suchthatkmins=2ˇL=4:4rad/m.ThemaximumdetectablewavenumberisbyaspatialNyquistcriteriaaskmaxs=ˇx=68:6rad/m.Makinguseofthedispersionrelation!=ak2,wherek=p2ˇf=a,wethatthetemporalsamplingparameterscorrespondtowavenumberlimitsofkmaxt=109rad/mandkmint=10:9rad/m.Thetotalwavenumberlimitsarethuskmax=min(kmaxs;kmaxt)=68:0rad/m,andkmin=max(kmins;kmint)=10:9rad/m.Thus,theupperlimitonextractablewavenumber(andhencefrequency)isdeterminedbythespatialsamplingintervalNyquistcriterion,andthelowerlimitonextractablewavenumberisdeterminedbythelengthofthetimerecord.Figure2.2:Accelerationdataforallofthesensors.18Figure2.3:Accelerationsfromsensors1,16,and31.Figure2.4:Velocitydataforallofthesensors.2.5DiscussionTheaccelerationhistoriesshownin.2.2showtheensemblesignalenergydecayastheresponsepropagatesintothesandpit,presumablywithasmalldampingintheexposedbeamaswell.Figure2.3showstheaccelerationhistoryofsensor1,16,and31.Theplot19Figure2.5:Velocitiesfromsensors1,16,and31.Figure2.6:Displacementdataforallofthesensors.showsthatthewavehitssensor1withathatresemblesapulse.Thedispersivewavehitssensor16next,withhighfrequenciesarrivingaheadofthelowfrequencies,ashigherfrequenciespropagatefasterindispersivebeams.Thefrequenciesspreadoutmore20Figure2.7:Displacementsfromsensors1,16,and31.bythetimetheresponsereachessensor31.Theresidualhighfrequencywigglewasduetothepulsebackandforththroughthewidthofthebeamandasmalleramplitudecanbeseeninthevelocitiesplotsaswell.ThesetrendswerealsoobservedbyOnsayandHaddow[62].Varioustipswereusedintheexperiment:ametalpointedtip,metaltip,andanylonroundedtip.Thebothmetaltipsproducedsignalswithmuchlarger-amplituderesidualringing.Thepointedmetaltipresultedinthemostextremering,followedbythemetaltip.Thenylonroundedtipprovidedthedatashowninthispaper.Thefrequencyofringingmatchedthemodeofvibrationsacrosstheheightofthebeamwhenviewedasaplate.ThiswasalsoobservedbyOnsayandHaddow.Lowpasscanalsohavetheofsofterhammertips.Inreality,thevelocityanddisplacementshouldstartatzerountilthewavereachesthesensorlocation.However,theremovalofmeans,andlineartrendsusingthe"detrend"command,leavesthevelocityanddisplacementsignalswiththislow-frequencydistortion.21Distortionoflowfrequenciesoutsidetherangeusedforthecalculationofthedispersionfrequenciesandwavenumbersdoesnotthecharacterizationofthedispersionrelationinthefrequencyrangeofthecalculation.22Chapter3ComplexOrthogonalDecompositionofanExperimentalBeam3.1IntroductionComplexorthogonaldecomposition(COD)[63]isinthefamilyofoutput-onlymodaldecompositionmethodsrelatedtoproperorthogonaldecomposition(POD).PODwasusedinmechanicsbyLumleytostudyturbulenceindynamics[14].Berkoozetal.[17]reviewedthepropertiesofPODforturbulence.PODhasbeenappliedtostructuraldynamics[51{53],andlatersptoextractmodalparameters[32,44,54,55].PODisequivalenttosingularvaluedecomposition[16,55{57],andhasseveraltvariationsfortheextractionoftparametersofinterest.Thesevariationsincludemass-weightedreduced-orderproperorthogonaldecomposition(MWPOD)[36,40],Ibrahimtime-domaindecomposition[58],smoothorthogonaldecomposition(SOD)[37,39,40],state-variablemodaldecomposition(SVMD)[40,59]and,thetopicofthispaper,COD.ToseetheconnectionbetweenPODandCOD,next,wewillexplainPODfollowedbyCOD.First,toperformPOD,theanalystcapturesmeasurementsignals(forexampledisplacements)attpointssimultaneouslyonthestructureduringitsresponsetosometypeof23excitation,forexample,animpulse[54].Next,anensemblematrix,X2RMN,iscreatedfromthesensormeasurements,whereMisthenumberofsensorsandNisthenumberoftimesamplestaken.Sp,X=hxT1xT2xTMiT;wherexi=[xi(0)xiT)xiT)xi((NT)]T,suchthateachrowoftheensemblematrixisthesampleddatafromonesensorandeachcolumnisasingletimesample.Oftenthemeanissubtractedfromeachsensorsignal.Nextacorrelationmatrix,R=XXTN,iscomputed,suchthatR2RMM.Finally,aneigenvalueproblemisformulatedasRv=v.Theeigenvectorsarecalledproperorthogonalmodes(POMs).Forlightlydampedlinearvibrationsystemswithamassmatrixproportionaltotheidentity,theseconvergetolinearnormalmodes[32,55].Theeigenvaluesarecalledproperorthogonalvalues(POVs)andrelatetomodalenergy[17,44].MWPOD,SOD,SVMD,andCODallinvolvevariationsonthisalgorithm.Eachmethodextractstmodalparameters.ComplexorthogonaldecompositionusessimultaneousmeasurementsmuchlikePOD.Howeverwemustextendthesesignalsintocomplexanalyticform.Suchsignalsdonothavenegativefrequencycontent,andcanbecreatedbymanipulationinthefrequencydomainorbyusingtheHilberttransform[63,64].Thereareseveralparameterestimationalgorithmsthattakeplaceinthecomplexdomain,whichhavebeenusedintheofelectromagnetics,anduseresponseensemblesandeigenvalueproblemtoestimateparameters,includingMUSIC[65]andESPRIT[66].ThedetailsoftheCODalgorithmarepresentedinthenextsection.ThusfarCODhasbeenappliedtonematodeposturing[67],whitinglocomotion[45],anddiscerningtravelingandstandingmodalwaves[63].CODhasalsobeenappliedtoextractthedispersionrelationfromasimulatedbeam[46].24Inafollowuptothesimulatedbeamstudy,thispaperreportsontheapplicationofCODtoanexperimentalbeamtoextractthegeometricdispersionrelationship.Inthefollowing,Section3.2givesadetailedoverviewofCOD.Section3.3providesabriefbackgroundofwavesinanEuler-Bernoullibeam.Section3.4describestheexperimentanddataprocessing.Section3.5showstheresultsofCODappliedtoapropagatingwaveinabeam,withinterpretations.3.2ComplexOrthogonalDecompositionComplexorthogonaldecompositionisanextensionofPODwithaverysimilarcomputation,withthenotableexceptionthatCODrequiresananalyticsignal.First,astructureisinstrumented,forexamplewithaccelerometersdistributedevenlyonthestructure.Next,thestructureisexcited,inthisexperimentwithanimpacthammer,andthesensorsaresampledsimultaneously.OncethemeasurementdataisacquiredthedataisarrangedinameasurementensembleXasdescribedinSection3.1,andthentheanalyticformofXiscomputedtogetZ.TocomputeZ,theanalyticformofX,wetakethefastFouriertransform(FFT)ofeachrowofXfromthetimedomaintofrequencydomaintogeteX.TheFFTensembleinthediscretefrequencydomaincanbetoroughlycoverthespectrumfromapproximately!nyto!ny,morepreciselyasindexedfrom1toN=2(forexampleinthecasethatNiseven).EachrowofeXistheFFTofasensorsignalandeachcolumnisafrequencysample,andtheelementsofeXareeXijwithi=1;;Mandj=(N21);;N=2.Thenthenegativespectrumisnandthepositivespectrumisdoubled,suchthattheelements25ofeZijofeZareeZij=8>>><>>>:0ifj<02eXijifj0:(3.1)Thecomplexanalyticensembleisobtained,usingtheinverseFFT(IFFT),asZ=IFFT(eZ).NowthatthemeasurementensembledatahasbeenconvertedintoanalyticsignalsthecorrelationmatrixRiscomputedasR=ZZHN,wheresuperscriptHdenotestheHermitianoperation(conjugatetranspose).OncethecorrelationmatrixiscomputedtheeigenvalueproblemisformulatedsuchthatRv=v.visacomplexorthogonalmode(COM)andisthecorrespondingcomplexorthogonalvalue.Indeed,thecorrelationmatrixcanalsobeformedinthefrequencydomainaseR=eZeZH=N,toproducethesameCOMs[46].Thecomplexmodalcoordinate(COC)ensemble,Q,iscomputedsuchthatQ=V1Z,whereeachcolumnofVisaneigenvectorvandV1=VH,ifnormalized,duetoorthogonalityoftheHermitianeigenvalueproblem.EachrowqTiofQisasampledCOC,suchthatQ=[q1q2qM]T,Q2CMN.FornearlyharmonicCOCs,therateofchangeofthephasewithrespecttothetimesampleisequaltothefrequency,!,suchthattheinstantaneousfrequencyoftheithmodalcoordinateis!i=ddtf\qigwhere,inthesampledcase,thetimederivativeisappliednumerically.Similarly,theCOMsarecomplexandtherateofchangeofthephaseofnearlyharmonicCOMswithrespecttothespatialposition,x,alocalwavenumberoftheithmodeaski=ddxf\vig.UsingtheCOMsandQ;themeanwavenumber,ki,andfrequency,!i,canbeextractedforeachmodei.26BeforeapplyingCODtothebeamexperimentwereviewtheEuler-Bernoullibeammodel.3.3Background3.3.1Euler-BernoulliBeamApplyingatravelingwavesolutiontotheEuler-Bernoullibeamequationprovidesatheoreticalframeworkfortheexperimentalresults.TheequationofauniformEuler-BernoullibeamisˆAy(x;t)+EI@4@x4[y(x;t)]=0;(3.2)wherexisthepositionalongthebeam,tistime,y(x;t)isthetransversebendingdisplacement,ˆisthemassdensity,Aistheareaofbeam'scrosssection,Eisthemodulusofelasticity,andIistheareamomentofinertiaofthebeam'scrosssectionabouttheneutralaxis.Anbeamhasboundaryconditionsofboundeddisplacementandslope.Following[6],insertingatravelingwavesolutionincomplexharmonicform,y(x;t)=ei(kx!t);(3.3)intoEq.(3.2),andsolvingfor!,leadstothegeometricdispersionrelationship!=ak2,wherea=sEIˆA:(3.4)27Thephasevelocityisthenasc=!=k=ak.Thus,fortheEuler-Bernoullibeam,thespeedofanindividuallypropagatingharmonicwaveincreaseslinearlywithitswavenumber.Thegroupvelocityiscg=d!=dk=2akforthebeam.Inthepreviousstudy[46],theCODwasappliedtoasimulatedresponsetoaninitialGaussiandisplacementdistributionstartingatrest[6].TheextractedtravelingwavemodescouldberelatedtoaformthatresembledEq.(3.3),fromwhich!andkwereextractedasdiscussedabove,andwereshowntobeconsistentwiththedispersionrelation.Thisanalysiswasrobusttoaddedsensornoise.Inthiswork,weapplythisanalysistoanexperimentthatemulatesabeamwithanimpulseexcitationattheend.TransversevibrationsofaEuler-Bernoullibeamwithanendimpacthavebeenaddressedby[26]bymeansofLaplaceandFouriertransforms,resultinginananalyticalexpressionofthedisplacementasafunctionoftime.Bussow[68]alsotreatedtheproblemwithslightlytboundaryconditionstoobtainthedisplacementinthefrequencydomain.IntheBackgroundSection3.3.2,weincludeananalysistoexpressthedisplacementandaccelerationresponsesasfunctionsofspaceandfrequency,whichcanbeusedwhenevaluatingtheresponsedata.Wekeepinmindthatourbeaminthesandpitwassetupwithafocusonextractingthebeam'sdispersionbehavior.Althoughtheconstraintofthesandonthebeamatlargexcouldcausesomedeviationbetweentheresponsesoftheidealtheoryof[6],[68],andtheBackgroundSection3.3.2,fromtheexperimentalbeam,evenintheunconstrainedregionsofthebeam,wewouldexpectsomeconsistencyinsomeaspectsoftheresponsebehavior.28Figure3.1:Schematicdiagramofexperimentalsetup.Topview.3.3.2Euler-BeamSolutionintheFourierDomainTheEuler-Bernoullibeamexcitedwithaimpulseatx=0describedbyF(0;t)=^F(t)issolvedusingtheFouriertransform.ThePDEandboundaryconditions(BCs)arelistedbelow:ˆAy(x;t)+EI@4@x4[y(x;t)]=0y00(0;t)=0y0000(0;t)=^F(t)EI(3.5)forx0,wherethedotsindicatepartialderivativeswithrespecttotandtheprimesindicatepartialderivativeswithrespecttox,andwithtwoadditionalconditionsofboundeddisplacementsandwavesthatareonlyallowedtotravelinthedirectionofthepositivex-axis.29TakingtheFouriertransformofthePDEandBCsyields!2eY(x;!)+4eY0000(x;!)=0eY00(0;!)=0eY(0;!)=^FEIwhere4=EIˆA,whilerememberingtheboundeddisplacementsandone-directionalwavecondition.NowwehaveanODEoftheformeY0000eY=0where=!24.SubstitutinginthestandardtrialsolutioneY=erxandsolvingforryieldsr=!1=2;i!1=2;whichleadstoeY(x;!)=A1ep!x+A2ep!x+A3eip!x+A4eip!x:Sincethedisplacementsarebounded,A1=0,andsincewavescanonlytravelinthepositivexdirection,A3=0,whichcanbeseenbyconsideringthattheinverseFouriertransformcombinesthesetermswithei!x.Applyingtheothertwoboundaryconditions,A2=A4are30determined,resultingineY(x;!)=^F4pEI(ˆA)3=4!3=2(i1)(ep!x+eip!x):(3.6)Thisprovidesthefrequencydomaindescriptionofthedisplacementresponse.Alooselydescribedderivationcanbefoundin[68].TheaccelerationresponseinthefrequencydomainisthuseA(x;!)=!2^F4pEI(ˆA)3=4!3=2(i1)(ep!x+eip!x):(3.7)3.4Experiment3.4.1SetupThetestspecimenwasarectangularsteelbeamwithaconstantcrosssection.Intoemulateafreebeam,thebeamwassuspendedwithelasticcords,suchthatoneendwasfree,andtheotherwasembeddedinasandpit,asdonebyOnsayandHaddow[62].Thesandabsorbsthewaveandpreventstheburiedendofthebeam.AschematicoftheexperimentisshowninFigure3.1.Inthiscase,thesandboxwaswithunpackedcoarsesand.Thebeamhada0:0045m0:0698mcrosssection.Thelengthofthebeamwas2:04m.Theunburiedpartofthebeammeasured1:43m,suchthatapproximately0:609mwasburiedinthesand.Thedensityandmodulusofelasticityofthebeamwere7870kg=m3and200GPa,respectively,basedonpublishedvaluesforsteel.FromthegeometryandmaterialpropertiesthetheoreticalvaluefromEq.(3.4)isa=6:548m2=s.31Thebeamwassensedwith31accelerometersplacedatadistancex=0:0458mapartoveradistanceofL=1:4198mandwassampledatf=25;000HzusingaNationalInstrument'sPXIdataacquisitionsystem.ThebeamwasstruckwithaPCBmodel086C80miniimpacthammer,lightlysuchthatbendingwerenotvisibletothenakedeye.3.4.2DataProcessingThedataincluded100samplesbeforethehammerimpactand300samplesaftertheimpactforatotalof400samples.Weaimtointegratetheaccelerationdatatoobtainvelocityanddisplacement.Numericalintegrationcanbeproblematicbecauseanintegrationconstantisintroduced,andlow-frequencynoiseisandcancausetheintegratedsignaltodrift.Toreducethesethefollowingstepsweretaken.Firstthedatawasforwardandbackwardwithahigh-passwithafrequencyof100Hz.Second,themeanofeachsensor'stimehistorywassubtractedfromitssamplesusingMatlab's"detrend"witha"constant"moThird,anylineartrendswereremovedusingthesamecommandasabovewitha"linear"moFourth,thesignalsweretranslatedonthetimeaxissuchthatthesamplebeforethehammerimpacthadatimeandforcevalueofzero.Theandlast100samplesweretruncatedleavingthestartoftheimpactplus200samples.Next,togetvelocities,thesignalswerenumericallyintegratedusingthe"cumtrapz"command.Themeansweresubtractedfromthevelocities,whichwerethenhighpassandintegratedoncemoretogetdisplacements.Themeansweresubtractedfromdisplacementsandthedisplacementswereforatime.TheseN=200samplesofdisplacementswerethenusedforCOD.32Theminimumdetectablewavenumberbythespanofsensors,L,iskmins=2ˇL=4:4rad/m.ThemaximumdetectablewavenumberisbyaspatialNyquistcriterionaskmaxs=ˇx=68:6rad/m.Basedonthesamplingratef=25;000HzandtimerecordofN=f=0:008s,themaximumdetectablefrequencyisfmaxt=12;500Hzandtheminimumdetectablefrequencyisfmint=125Hz.Makinguseofthetheoreticaldispersionrelation!=ak2,ork=p2ˇf=a,wethatthetemporalsamplingparameterscorrespondtotheoreticalwavenumberlimitsofkmaxt=109rad/mandkmint=10:9rad/m.Theapproximatetotalwavenumberlimitsarethuskmax=min(kmaxs;kmaxt)=68:6rad/m,andkmin=max(kmins;kmint)=10:9rad/m.Thus,theupperlimitonextractablewavenumbers(andhencefrequencies)isdeterminedbythespatialsamplingintervalNyquistcriterion,andthelowerlimitonextractablewavenumbersisdeterminedbythelengthofthetimerecord.3.5ResultsandDiscussionsTheaccelerationhistoriesinFigure3.2showtheensemblesignalenergydecayastheresponsepropagatesintothesandpit,presumablywithasmalldampingintheexposedbeamaswell.Figure3.3showstheaccelerationhistoryofsensors1,16,and31.Theplotshowsthatthewavehitssensor1withathatresemblesapulse.Thedispersivewavehitssensor16next,withhighfrequenciesarrivingaheadofthelowfrequencies,ashigherfrequenciespropagatefasterindispersivebeams.Thefrequenciesspreadoutmorebythetimetheresponsereachessensor31.Theresidualhighfrequencywiggleisduetothepulsebackandforththroughthewidthofthebeam.Thesetrendswerealsoobserved33Figure3.2:AccelerationdatavspositionalongthebeambyOnsayandHaddow[62].Varioustipswereusedintheexperiment:ametalpointedtip,metaltip,andanylonroundedtip.Themetaltipsexcitedhigherfrequenciesandthusproducedsignalswithmuchlarger-amplituderesidualringingthroughthebeamwidth.Thepointedmetaltipresultedinthemostextremering,followedbythemetaltip.Ourinterestisinextractingbehaviorpredictedbytheone-dimensionalbeammodel,andhenceweusedasoftertip.Thenylonroundedtipgeneratedthedatashowninthisthesis,andsomeringingwasstillexcited.Thefrequencyofringingmatchedthemodeofvibrationsacrosstheheightofthebeam,whenviewedasaplate.Lowpasscanalsohavetheofsofterhammertips.Theaccelerationswereandintegratedtwice,accordingtotheprocessofSection3.4.2,34Figure3.3:Accelerationsfromsensors1,16,and31Figure3.4:Displacementsfromsensors1,16,and31toproducethedisplacementsinFigure3.4,wheresensors1,16,and31,only,areshownforclarity.Inreality,thedisplacementshouldstartatzerountilthewavereachesthesensorlocation.However,despitethehigh-passandremovalofmeansandlineartrendstogether,thesignalswereleftwithsomelow-frequencydistortion.Thisdistortion,however,isoutsideofthefrequencyrangeusedforthecalculationofthedispersionfrequenciesandwavenumbers,andthereforedoesnotthemodaldynamicsnorthecharacterizationofdispersionwithinthisrangeofcalculation.Nonetheless,wecanseesomekeyfeaturesinFigure3.3carryingovertoFigure3.4,showingthehighfrequencycomponentsofwave35Figure3.5:Sensor1displacementarrivinginsensors16and31aheadoflower-frequencycomponents,consistentwithbeamdispersion.Black(dark)solidlinesinFigs.3.5-3.7showtheFFTsofaccelerometerssignalsforsensors1,16,and31.Thedottedlinesshowthescaledtheoreticalaccelerations.Figure3.8showstheFFToftheimpacthammersignal.Wecancomparetheresponseswiththetheoreticalpredictions.IfweconsiderthattheresponseofEqn.(3.2)underanendimpactloadingis~Y(x;!)asgiveninEqn.(3.6)inthebackgroundSection3.3.2,then~H(x;!)=~Y(x;!)=^Fcanbeconsideredasaunitimpulseresponsefunctioninthefrequencydomain,thatis,afrequencyresponsefunctionbetweenthedisplacementy(x;t)andtheendinput,suchthatforthecaseofanidealimpulseattheendpoint,where~F(!)=^F,then~Y(x;!)resultsintheEqn.(3.6).However,Figure3.8showsthatthe36Figure3.6:Sensor16impactisnotideal,andtheassociatedinputF(!)istoabandwidth.TheresponsetothisnonidealimpactisY(x;!)=~H(x;!)F(!).TheaccelerationresponseislikewiseA(x;!)=~A(x;!)F(!)=^F,where~A(x;!)isgivenasEqn.(3.7)inthebackgroundSection3.3.2.Wehaveevaluatedj~A(x;!)jatthesensorlocationsxi,andatthefrequenciesoftheFFT,andthenobtainedj^A(xi;!)j=j~A(xi;!)jjF(!)j=^Fwithanormalizedinputsignal(\normalized"tooptimizetheaccuracyforsensor16).TheresultsarethedottedlinesintheplotsofFigures3.5-3.7.Thus,relativetothenormalizedinput,theresultsareingoodqualitativeagreement.Themagnitudeoftheplotforsensor1isverysensitive,asthetheorypredictsaverysteepchangeinresponseamplitudeasxgetssmall.Theseplotsshowthat,althoughthebeamisembeddedinasandpitdownstreamofthewave,theexposedregionof37Figure3.7:Sensor31thebeambehaves,inthebandwidthofourexcitation,similarlyasaifitwereabeam.ItalsoshowsthattheanalysisoftheEuler-Bernoullibeamunderanendimpactprovidesusefulpredictionsofbehaviorinthefrequencyandspacedomains,particularlyiftheinputisquanCODwasapplied,andsomeoftheCOMs(Figure.3.9)andCOCs(Figure3.10)areshownforillustrationpurposes.TheCOMsdepictedinFigure3.9showtherealandimaginarypartsofselectedextractedcomplexmodes,normalizedsuchthateachmodalvectorhasunitamplitude.Theplotsareparameterizedinthesensorlocationindex.Thus,thereare31realandimaginaryorderedpairsplottedandconnectedwithstraightlines.Aperfectwhirlwouldindicateapurelyspatiallyharmonicwavemodewhoserealandimaginarypartsare90degreesoutofphase.Thus,thespatiallywhirlingextractedmodesresembleharmonic38Figure3.8:FFTofthemodalimpacthammersignal.wavemodes,andthespatialwhirlingratecanthenbeusedtoestimatethecomplexmodalwavenumber,aslongasthespatialsamplingintervalissmallenoughtoaccuratelyestimateawavenumber.TheCOCsplottedinFigure3.10showtherealandimaginarypartsofcorrespondingcomplexmodalcoordinates.Thenormalizationofthecomplexmodes,andtheCOCmodalenergy(theCOVs),theamplitudeofthecomplexmodalcoordinates.Theseplotsareparameterizedinthetimeindex.Thus,thereare200pointsplottedineachgraph,connectedwithlines,suchthatthegraphsappearrathersmoothforthelowerfrequencymodes.Inthistimeparameterization,thetypicaltrendisthatthemodalcoordinatebeginsasanearlyharmonicoscillation,andthendecaysasthemodalcomponenttravelsofthemeasurementzone.ExamplesoftherealpartsasfunctionsoftimeareshowninFigs.3.11and3.12.Sincethehigherfrequenciestravelfasterthroughthebeam,thehigherfrequencymodalcoordinateshaveshorterdurationsofnearlyharmonicoscillation.Thistrendwasalsoobservedindecompositionsofnumericallysimulatedwaves[46].Intheexperiment,the39Figure3.9:SelectedCOMs.Topleft:2ndmodalCOM.Topright:6thCOMwhichislesscircularthansecond.Thistrendscontinuesandisillustratedinthelowerleft:7thCOM.Lowerright:10thCOM.modesarenotpureharmonics,andthereisnoiseandmodelingerrorthatcontributetosomemodalpollutionapparentasaloweramplitude,lowerfrequencyoscillationafterthestrongharmonichasleftthemeasurementzone.Theintervalofthestrongnearlyharmonicoscillationisthenusedtoextractthemodalcoordinatetemporalwhirlrate,whichrepresentsthefrequencyofthemode.Themechanismforhowexperimentalerrorleadstomodalpollutionwasanalyzedpreviously[67].TheextractedgeometricdispersionrelationshipisshowninFigure3.13.Inthisvaluesof!andkareplottedintherangesforwhichthemodeshapesandmodalcoordinatesexhibitedintervalsofwellwhirls.InFigure3.13itcanbeseenthattheexperimentaldatashowgoodagreementwiththeory.Fromtheorywehave!=ak2wherea=6:548m2=s.Aleastsquares(LSQ)40Figure3.10:SelectedCOCs.Topleft:2ndmodalCOC.Topright:6thCOCwhichshowsattenuationaftersomeinitialoscillations.Thistrendscontinuesandisillustratedinthelowerleft:7thCOC.Lowerright:10thCOCofthethegeometricdispersionrelationusingCODextractedkand!leadstoavalueofafit=6:4431m2=s,whichgivesanunderestimationof1.61%.Basedonregressionerroranalysis[69],theextractedvaluedof6:44m2=sleadstoameansquarederrorof6:1667104rad/s,andits95%intervalisa=[6:2638;6:26224]m2=s.WhenthedataprocessinglaidoutinSection3.4.2wasdonewithoutthehigh-passi.e.wherethedatajusthadthemeanssubtractedfollowedbyintegrationrepeatedasneededtogetdisplacements,theleastsquaresestimateofawasafit=6:8m2=s,whichwasanoverestimationof3.8%.ThephasevelocityisshowninFigure3.14,comparingtheequationc=akwithc=!=kcomputedfromCOD-extracted!andk.ThesolidlineistheoryandthecirclesareCODextracteddatapoints.Similarly,thegroupvelocityisshowninFigure3.15,fromvaluesthatwerecomputedusingforwardontheCODextracted!andkfromFigure3.13.41Figure3.11:Realpartofcomplexorthogonalcoordinatenumber4.Severalthingsareworthnotingatthispoint.WethattheCOVsincreasewithdecreasingmodalwavenumber(decreasingfrequency).Thismaybesensibleifweconsiderthattheimpulseexcitationisslightlystrongerforlowerfrequencies(seeFigure3.8).Furthermore,thehigherfrequenciespropagatefasteraccordingtothebeamtheory,suchthatlowerfrequencymodesareactiveinthemeasurementzoneforlongertimeintervals.Theexcitationbandwidthandmodalactivitydurationcontributetohighermeansquaredamplitudes(COVs)forlowermodes.WealsoseethattheextracteddatainFigure3.13fallswithinthetheoreticallimitsofkmin,kmax,and!maxdeterminedbysamplingparameters.Spatialresolutionhasagreatapproximatelythelowest1/3oftheCOMs(thosewithhighestCOVsandlowerwavenumbers)havegoodwhirlpropertieswithhighenoughspatialsamplingresolutionto42Figure3.12:Realpartofcomplexorthogonalcoordinatenumber8.allowforgoodkextraction.ThesecondCOMshowninFigure3.9hadthebestcircularwhirling.AstheCOVsdecreasetheCOMsbecomelesscircularinthecomplexplane.Thisispartlybecausethespatialresolutionbecomescoarseasthemodalwavenumberincreases.HowevertheCOCsaretoalesserdegreebyresolutionlimitationsthantheCOMs,whichmakessensebecausethespatialsamplingdistanceandtimerecordwerefoundtobetheparametersthatlimitedtheaccessiblefrequencyrange,andthetemporalsamplingwasthusabundantlyfast,suchthattheaccessiblemodalcoordinatesweresmoothlysampled.3.5.1UsingCOCstoExtractModalAmplitudesItmaybeofinteresttodeterminetheamplitudeofthewavetravelingthroughthebeamasafunctionoffrequency.ToachievethisthefastFouriertransform(FFT)ofthetheCOCswerecomputedandthemaximummagnitudeanditsfrequencywasrecordedtoderiveCOD43Figure3.13:Experimentalresults.TheoreticaldispersionrelationshipfortheEuler-Bernoullibeam(Solidline).CODextracteddatapoints().extractedjeAcod(x;!)jshownastheredcirclesinFigure3.16.Inordertocomparethiswiththeory,eA(x;!)wascomputedforeachsensorlocationiandtheneacheA(xi;!)wasmultipliedbytheFFTofthemodalimpacthammersignalshowninFigure3.8togetA(xi;!)wherei=1;;31.AcompositewascreatedsuchthatU(!)=PA(xi;!)MwascomputedandisshownasthesolidlineinFigure3.16.U(!)issimplytheaverageofthescaledtheoreticalaccelerationforeachsensor.TheplotoftheeAcod(x;!)(circles)andU(!)(line)isshowninFigure3.16andshowsgreatagreement.3.6ConclusionExperimentswereperformedonathinbeamsuspendedinthesandtoemulateabeam.Thebeamwasinstrumentedwithaccelerometers,excitedwithanimpulse,andthemeasuredresponseswereintegratedintodisplacementsignalsandthenanalyzedusing44Figure3.14:PhasevelocityCODtoextracttheunderlyingcomplexmodes.Truetobeamtheory,higherfrequencywavecomponentstraveledfaster,andthusremainedactiveforshortersegmentsofthetimerecord,thanlowerfrequencycomponents.Themeasuredaccelerationresponsesalsoagreedqualitativelywiththeoreticalresponsestothemeasuredinputpulse.Thelower-frequencyextractedcomplexmodesresembledharmoniccomplexwaveforms.Theassociatedmodalcoordinatesweredominantlyharmonicduringatimeintervaldictatedbythewavespeedofthecorrespondingcomponentofthewaveform,andthelengthofthemeasurementzoneonthebeam.Thenearlyharmonicnatureofthedecomposedmodeshapesandmodalcoordinatescontainedinformationonmodalwavenumberandfrequencies,whichcouldbeestimated.Withthisapproach,weextractedthedispersioncharacteristicsoverafrequencyand45Figure3.15:Groupvelocitywavenumberinterval,aswellastheamplitudeofthewavestravelingthroughthebeam.TheresultswereconsistentwithEuler-Bernoullibeamtheory,andwithapreviousanalysisofsimulatedresponsedata.TheworkshowsthatCODisafastandsimpletoolthatcanbeusedtoextractthegeometricdispersionrelationshipbetweenthefrequency,phasevelocity,orgroupvelocity,thewavenumber,andwaveamplitudesforwavestravelinginauniformstructure.46Figure3.16:CODextractedmodalamplitudevsfrequency()comparedtotheory47Chapter4SmoothComplexOrthogonalDecompositionintheTimeDomain4.1IntroductionDuringthemiddleofthe20thcenturyvariousresearchersindependentlydevelopedproperorthogonaldecomposition(POD)[14,17,70,71]whichwasappliedtostatisticsandturbulentws.PODlatercaughttheinterestofstructuralengineersandwasusedtoextractmodeshapesfromvibratingstructures[58,72{74].OthergeneralizationsofPODweredeveloped,suchasthesmoothorthogonaldecomposition(SOD)forthenaturalfrequenciesandlinearnormalmodes[37,39],andthestatevariablemodaldecomposition(SVMD)[59]fornaturalfrequencies,normalmodes,and(intheory)modaldamping.SoonafterPODwasexpandedforextractingtravelingwavemodesusingcomplexorthogonaldecomposition[63].Complexorthogonaldecomposition(COD)[46]canextractnonsynchronousandstandingwavesofvibratingstructures,andwhenwavesaretravelingthroughanelasticmediumCODcanbeusedtoextractthewavenumbers,frequencies,andthedispersion48relationshipofthewaves.IntheapplicationofCODthewavenumberisextractedfromtheeigenvector,andthefrequencyisextractedfromthemodalcoordinate.AnewgeneralizationofCODandSODisoutlinedherecalledthesmoothcomplexorthogonaldecomposition(SCOD).ItwillbeshownthatwithSCODthewavenumberandfrequencycanbeextractedfromtheeigenvectorandeigenvalueoftheSCODeigenvalueproblem(EVP).AmathematicaldevelopmentforSCODwillbecoveredinSection4.2.SCODwillbeappliedtoasimulatedbeaminSection4.4,andexperimentallytoarectangularbeaminSection4.4.2.AprimerforSODisreviewedinthefollowingsubsection.4.1.1PrimerofSmoothOrthogonalDecompositionWewillstartbyrelatingSOD[37]tothesymmetricemass-springsystem,following[39].Giventhegeneralundampedlinearfreevibrationmass-springequationMx+Kx=0(4.1)andsubstitutingintheassumedsolutionofx=˚ei!tleadstothegeneralizeeigenvalueproblemintheformofM˚=K˚;(4.2)wheretheeigenvalues=!2aremodalfrequenciessquared,andtheeigenvectors˚arethemodeshapes.Inmatrixform,wecanwriteM=;(4.3)49whereisamatrixconsistingofcolumns˚iandisadiagonalmatrixwhoseelementsare!2i.Wewillshowtherelationshipbetweenthemass-springsystemandthesmoothorthogonaldecomposition.SODisbyR =S ;(4.4)orinmatrixform,=;(4.5)whereisasmoothorthogonalvalue(SOV),whichwillbeseentoapproximatethemodalfrequencysquared,isadiagonalmatrixwhoseelementsarethei, isthesmoothorthogonalmode,andisamatrixconsistingofcolumns i.SODinvolvestwocorrelationmatrices,RandS,whereR=XXTN(4.6a)S=VVTN;(4.6b)andwhereXisameasurementensembleofdisplacements,andVisameasurementensembleofvelocities.Typicallythemeanofthetimehistoryofeachsensorissubtractedfromeachofthedatasamplesofthatsensor.Thegeneralformofallensemblesusedinthisbodyof50workis,fortheexampleofthedisplacements,X=26666666664x1(0)x1t)x1((Nt)x2(0)x2t)x2((Nt)............xM(0)xMt)xM((Nt)37777777775(4.7)whereMisthenumberofsensorsandNisthenumberofsamples.Thetimehistoryforeachsensorisorganizedinrows,suchthateachcolumnisasetofsamplesateachsamplinginterval.StartingwiththeSODeigenvalueproblem,as=andsubstitutingintheexpressionsforRandS,leadstoXXT=VVTChedizeandZhou[37]usedtheapproximationV˘=XDT,andshowedthatXDDTXT˘=XAT,whereDisamatrixthatperformsasimplenumericalderivative.Hence,S=VVTˇXAT:Hence,NS=VVTˇXAT:PluggingthisintoS,theEVPbecomesXXT=XAT:51NownotingthatAisanensembleofaccelerations,thenfromMA+KX=0,wehaveA=M1KX.ThenXXT=X[M1KTX]TorXXT=XXTKTMT:AssumingXXThasfullrank,andisinvertible,wehave=KTMT:TheassumptionthatXXTisfullrankcorrespondstoafullymultimodalmotionwithN=M(oratnoiselevel).Takingtheinversetransposeoftheaboveequation,h=KTMTiTwehaveT=T:(4.8)ComparingEqn.(4.8)withEqn.(4.3)weseethattheSCODeigenvaluesrepresentthestructuraleigenvalues,i.e.themodalfrequenciessquared,andthatthematrixofSODeigenvectors,,andthelinearnormalmodes(LNMs),,ofthemassspringsystemarerelatedas=T.Theseconclusionswerereachedin[37,75].AnotherrelationshipworthnotingcanbeshownifwestepbacktoEqn.(4.8)andlet52=.ThenK()T=M()TKMTT=MMTT:Usingsymmetry,wehaveKM1T=TM1T=K1T:Takingtheinversetransposeofbothsidesoftheequationandusingsymmetry,=1=Therefore,andarerelatedthroughMas=M1:(4.9)Anotherwaytoseethisistonotethat,byorthogonalityinsymmetricsystems,T=I,whichcanbemanipulatedintotheequationT=:Bythepreviousconclusion,that=T,wehave=:ThisrelationshipwillbeimportantinthemathematicaldevelopmentofSCOD.Insummary,forlinearundamped,symmetric,multimodalfreevibrationsystems,the53tworelationshipsfortheeigenvectorsofthemassspringsystemandsmoothorthogonaldecompositionwerejustestablishedas=Tand=.Furthermore,theSODeigenvaluesapproximatethesquaresofthemodalfrequencies.Theserelationshipswillbeexploitedinapproximationinexperiments,sinceagreementwiththemodelisanidealization.Smoothcomplexorthogonaldecompositionwillbeformulatedsimilarly.However,sincewewillbeworkingwithcomplexanalyticsignals,thecorrelationmatriceswillbegeneralizedasR=XXTN;(4.10)S=VVTN;(4.11)wheretheoverbardenotesacomplexconjugate.Thesubtlegeneralization,incomparisontoequations(4.6a)and(4.6b)istheuseofthecomplexconjugate.Sincewewillbeworkingwithacontinuousbeam,inalatersectionwewillconnecttheSCODformulationtodistributedparametervibrationmodels.4.2ApplicationofSmoothOrthogonalDecompositionTheapplicationofSCODinvolvesgatheringstatemeasurements.Accelerometerswereusedinthisbodyofwork.Aftertheaccelerationsaremeasuredthedataispost-processedtoderivevelocitiesanddisplacements.Thevelocitiesarecollectedinavelocityensemble,V,anddisplacementsareorganizedinaensemblematrix,X.Theensemblesareorganizedsoeachsensorisallocatedtoarowsuchthatallthedatafromtheithsensorareintheithrow.Measurementsamplesarethenineachcolumn.Forexample,Xijcontainstheithsensor's54jthsample,fori=1;;Mandj=1;;N,whereMisthenumberofsensorsandnisthenumberoftimesamples.Typically,themeanofeachrowissubtractedfromeachelementofthatrow.Thetwoensemblesareconvertedtoanalyticmatrices,ZandZV.Next,twocorrelationmatricesarecomputed,onefromtheanalyticdisplacementsR=ZZT=NandtheotherfromanalyticvelocitiesS=ZVZTV=N.FinallytheSCODeigenvalueprobleminthetimedomainisposedasR=S.Then=Hprovidesanestimateofthemodalmatrix.Thefrequenciesarethesquarerootsofthediagonalsin.Thewavenumbersarecomputedfromthe:Eachcolumnofisusedtoextractawavenumberwherethewavenumberisthegradientofthephaseangle.4.3MathematicalDevelopmentforSmoothComplexOrthogonalDecompositionInthissectionwewilldeveloptheframeworkforSCODinthetimedomainfordistributedparametersystems.Startingwithalinearself-adjointpartialtialequation(PDE)foraone-dimensionalmedium,itseigenvalueproblemwillbeexpressed.TheSCODeigenvalueproblemwillbeandthenrelatedtothePDEeigenvalueproblem.Assuch,relationshipsbetweentheeigenvaluesandeigenfunctionsofSCODandthePDEeigenvalueproblemwillbeuncovered.Butwewilllookatthecorrelationfunctionsinthetimedomainandmakeobservationsabouttheuseoftheoperatorintheassociateddiscretecorrelationmatrices.55ChelidzeandZhou[37]hadapproximatedV˘=XDT;whereDisasimpleoperatormatrix,andhadshownthat,forSOD,VVT˘=XDDTXT˘=XAT:WiththiswenowturntoSCODfollowingEqn.(4.11).ExpandingS=VVT=NtocreatetheintegralformofthecorrelationmatrixstartswithS=1N2666664v1(t1)v1(tN)......vM(t1)vM(tN)37777752666664v1(t1)vM(t1)......v1(tN)vM(tN)3777775suchthateachelementisSij=1NNXk=1vi(tk)vj(tk)=1NtNXk=1vi(tk)vj(tkt:AssumingtissmallandNislarge,thelimitingintegralformisSij=1TZT0vi(t)vj(t)dt;(4.12)whereT=NT.ThenintegratingbypartsyieldsSij=1T_xixjjT01TZT0xi(t)xj(t)dt:(4.13)56SincethetermdiminishesasTgetslarge,inthelimitwehaveSij=1TZT0xi(t)xj(t)dt:(4.14)TheassociateddiscreteformisSij=1NtNXk=1ai(tk)xj(tk)t=1NAXT=1NXAT;(4.15)thelatterofwhichcomesfromsymmetryinEqn.(4.12)andthestepsthatfollow.ThissuggeststhatifTislargeenough,theformoftheoperatorDdoesnotmatter,andgenerallyVVT=XAT.Thisalsoappliesfordistributedparametersystems.StartingwiththePDEforthebeamequationinthelinearoperatorform,m(x)u+Lu=0(4.16)wherem(x)isthemassperunitlengthandu=u(x;t)isafunctionofspaceandtime.Weassumeu(x;t)iscomplexandanalyticwiththeformu(x;t)=ei!t(c(x)+id(x))=ei!t˚(x):Takingthepartialderivativeofu(x;t)withrespecttottwice,@2@t2fu(x;t)g=!2ei!t˚(x);57andsubstitutingitintoEqn.(4.16),leadstothecontinuouseigenvalueproblemL˚(x)=!2m(x)˚(x);(4.17)wheretheeigenfunction˚(x)isacomplexmodalfunction.Letu(x;t),u(y;t),v(x;t),andv(y;t)becomplexanalyticdisplacementandvelocitytimesignalsevaluatedatpointsxandyonthebeamandsampledattimestk=ktfork=1;;N.WeR(x;y)=1NNXk=1u(x;tk)u(y;tk)S(x;y)=1NNXk=1v(x;tk)v(y;tk)whereu(y;tk)andv(y;tk)arethecomplexconjugates.MultiplyinginsidethesummationanddividingoutsidethesummationbytgivesR(x;y)=1NtXku(x;tk)u(y;tk)tS(x;y)=1NtXkv(x;tk)v(y;tk)tLettingthetbecomewhileholdingT=Ntresultsinr(x;y)=1TZT0u(x;t)u(y;t)dt(4.18a)s(x;y)=1TZT0v(x;t)v(y;t)dt:(4.18b)58Discretizingspatially,Rij=r(xi;yj)andSij=s(xi;yj),andthentemporally,leadsbacktoRandS,hereforcomplexanalyticsignalsasR=1TZT0u(xi;t)u(yi;t)dt˘=1NUUT(4.19a)S=1TZT0v(xi;t)v(yi;t)dt˘=1NVVT(4.19b)wherehereUisthecomplexdisplacementensemblematrix.TheSCODEVPisR =S where isadiscreteeigenvector.Wewillshowthat approximatesadiscretizationofthesystemmodalfunctions.TheEVPcanbewrittenasMXk=1Rik ky=MXk=1Sik ky;(4.20)whereMisthenumberofmeasurementpoints.Eqn.(4.20)canbewrittenasMXk=1r(xi;yk) (yk)y=MXk=1s(xi;yk) (yk)y;andthenforaveryhighspatialresolutionandtimestep,weapproachthecontinuumlimit,Zr(x;y) (y)dy=Zs(x;y) (y)dy;(4.21)whichisthecontinuousSCODinthetimedomain.SubstitutingEqns.(4.18a)and(4.18b)into(4.21)leadstoTZy ZT0u(x;t)u(y;t)dt! (y)dy=1TZy ZT0v(x;t)v(y;t)dt! (y)dy:(4.22)59ItisdesiredtoexpressthecontentsinsidetherighthandparenthesisastheproductofaccelerationanddisplacementaswasdoneinthederivationofdiscreteSOD.EvaluatingxandyatxiandyiinEqn.(4.22)andreferringtoEqn.(4.12),wenotethatthecontentinsidetheparenthesisontherighthandsideequatestoSij.UsingEqn.(4.14),wecanwriteSijintermsofthedesiredproduct.HenceitispossibletorecastEqn.(4.21)and(4.22)asTZy ZT0u(x;t)u(y;t)dt! (y)dy=1TZy ZT0u(x;t)u(y;t)dt! (y)dy:(4.23)AsubtlebutimportantpointshouldbenotedthatinEqn(4.23)themultiplicandhasaover-bar,forexampletheproducthereisv(x;t)v(y;t)andnotvivj.UsingthesymmetrymentionedafterEqn.(4.15),thecontentoftheparenthesisontherighthandsideofequationisHermitianinthecontinuoussense,suchthat1TZT0u(x;t)u(y;t)dt=1TZT0u(x;t)u(y;t)dt:Solving(4.16)foruandsubstitutinginto(4.23)bringsustoZy ZT0u(x;t)u(y;t)dt! (y)dy=Zy ZT0u(x;t)Lu(y;t)m(y)dt! (y)dy:Inthenextstepwewillmoveeverythingontoonesideoftheequation,makeRytheinner60integral,andmoveallthetermsintotheinnerintegral,yieldingZT0Zyu(x;t)u(y;t) (y)u(x;t)1m(y)Lu(y;t) (y)dydt=0ZT0u(x;t)Zyu(y;t) (y)1m(y)Lu(y;t) (y)dydt=0:Applyingtheadjoint<Lq;p>=<q;Lp>tomovethelinearoperator,factoringoutu(y;t),andmovingallthetermsbackintotheinnerintegralproducesZT0Zyu(x;t)u(y;t) (y)L (y)m(y)dydt=0:(4.24)Forthisequationtohaveanon-trivialsolution,theexpressioninsideoftheparenthesismustbeequaltozero,producingthefollowingrelationship:(y)=L (y)m(y):(4.25)Ifthesystemisselfadjoint,L=L,andEqn.(4.25)isrewrittenas(y)=L (y)m(y):(4.26)Nowlet (y)=m(y)˚(y)(4.27)61toget(y)˚(y)=L˚(y):(4.28)Thisshowsthat˚and arerelatedbythemassdistribution.4.4SimulatedEuler-BernoulliBeamTheresponseofanEuler-Bernoullibeamwassimulatedintheworksof[6,63].Theinitialconditionsforthebeamarey(x;0)=f0e x24b20!_y(x;0)=0whichisaGaussiandistributionontheinitialdisplacement,wheref0=1mmandb0=0:01m.TheanalyticalsolutiontotheEuler-Bernoullibeamwiththeinitialconditiongivenaboveis[6]y(x;t)=f0(1+a2t2=b40)1=4e x2b204(b40+a2t2)!cos atx24(b40+a2t2)12arctan atb20!!:(4.29)SimulatedBeamDimensionsLength1Crosssection0:069m0:0045mDensity7870kg/m3Modulusofelasticity200GPaTable4.1:SimulatedBeamDimensionsThesensornetworkonthesimulatedbeamwasdesignedtobethesameasthebeam62experiment,withidenticalsensorspacingofx=0:0461moveralengthofL=1:42875m,wheresensor1islocatedattheorigin.However,thesamplerateinthesimulationwas100,000Hz,whereasinthebeamexperimentthesamplingratewas25,000Hz.4.4.1DataProcessingFirstdisplacementswerecomputedfromtheanalyticalsolutionfromEqn.(4.29),andthemeansweresubtractedfromeach\sensor's"timehistory.Next,usingcentralthevelocityensemble,V,wascomputedandthemeanswereremoved.Centralreducestheensemblematrixdimensionsby2h,wherehistheindexintervalusedinthecentralSo,inordertomakeeachensembledimensionallycompatible,theandlasthrowsofthetimesamplesofdisplacementwereremovedfromtheensemble.Weusedh=1.Theresultingdisplacementsforsensors1,16,and31areshowninFigure4.1.ThesameplotsarealsoshownforvelocitiesinFigure4.2.Next,thesignalswereconvertedintocomplexanalyticsignalsbytakingtheFFTofeachrow,zeroingoutthenegativefrequenciesandmultiplyingtheremainingspectrabytwo.Finally,theIFFTwasappliedtoobtaintheanalyticformofXandVasZandZvrespectively.4.4.2ResultsofSimulatedBeamTheanalyticmeasurementensembleswereusedtocreatecorrelationmatricesfortheSCODEVP.ThenaturalfrequencieswereextracteddirectlyfromSCOD'seigenvalues,,insteadofmodalcoordinatehistoriesasdoneintheapplicationofCOD[46,76].ThecomplexmodalmatrixwascomputedfromtheinverseHermitiantransposeoftheeigenvectormatrixas=H.Thewavenumberswerethancomputedwithanumericalderivativeas63Figure4.1:Displacementsvs.timefromtheanalyticalsolutionforsensor1,sensor16,andsensor31.ki=d\˚i=dx,where˚iisacolumnofthematrix.Aforwardwasusedalongthespatialindexofthediscretizedmodalvectorinplaceofthethederivatived=dx.Thevalueofkiwaschosenasthemeanvaluefromaspatialintervalofuniformvaluesinki.ThecomputedmodalfrequenciesversusmodalwavenumbersareplottedinFigure4.3.TheleastsquaresofthedatainFigure4.3totheequation!=ak2producedasim=6:52m2=scomparedtoatheory=6:56m2=s,whichisa0.41%error.4.5ResultsofBeamExperimentSCODwasappliedtothebeamexperiment,andfrequenciesweredirectlycomputedfromtheeigenvalues,whichsavessomecomputationalwhencomparedtoCODforobtainingfrequencyinformation.SamplesofSCOMsareshowninFigure4.4.SCOMswerenotwhirlingascleanlyasCOMsforhighermodes(see[76]andChapter5).Themodalwave64Figure4.2:Velocitiesvs.timefromnumericalderivativesofthedisplacementsforsensor1,sensor16,andsensor31numberswereestimatedfromthewhirlratesoftheSCOMs,andpairedwiththeassociatedsmoothcomplexorthogonalvalues,,toobtainthedispersionrelationship.ItshouldbenotedthatspuriousmodesareproducedwhenapplyingSCOD.ThisisaresultofcomputingtheeigenvectorsoftheSCODGEVP.SinceRandShavedimensionsMM,MSCOMswillcomputed.WecangenerallyexpectatmostM=2modes.ThiscanbeseenbyconsideringaspatialNyquistcriterionifmodesarenearlyharmonic.ThereforeatleastM=2SCOMsareexpectedtobespurious.Furthermore,manymodestypicallyhaveamplitudeswithinthelevelofexperimentalnoise.Onecandiscerntheactualmodesfromthespuriousmodesbyvisualinspection.Actualmodeswilltypicallybetightlycoiledcirclesorellipses,whilespuriousmodestendtolooklikenoise.Figure4.4showsSCOMsextractedfromtheexperimentalbeam.Thetopleft,top65Figure4.3:GeometricdispersionrelationobtainedusingSCODforasimulatedEuler-Bernoullibeam.right,andbottomleftimaginesdepictactualSCOMs,thebottomrightdepictsaspuriousmode.SCODextractsdataoverasmallerwavenumberrangeforthesamedatawithsimilaraccuracy,whencomparedtoCOD.Figures4.5,4.6and4.7showtheextracteddispersionrelationshipswhenSCODwasappliedtotheexperimentalbeamdata.ThegroupvelocitiesplottedinFigure4.7wereobtainedfromaappliedtothedispersiondataplottedinFigure4.5.TheleastsquaresapproximationofausingtheSCODestimated!andkgivesaSCOD=6:72m2=s,whichisanerrorof2:6%whencomparedtoatheory=6:56m2=s.4.6ConclusionsThispaperoutlinedthemathematicaldevelopmentandapplicationofsmoothcomplexorthogonaldecompositiontoestimatecomplexmodalcharacteristics.InusingSCODin66Figure4.4:Examplesofsmoothcomplexorthogonalmodes(SCOMs)fromthebeamexperiment.Thelowerrightplotillustratesandexampleofspuriousmode.thetimedomainonecandirectlycomputefrequenciesfromtheeigenvalues.ThecomplexstructuralmodesarecomputedfromtheinverseconjugatetransposeoftheSCODeigenvectormatrix.Foranearlyharmonictravelingcomplexmode,thewavenumbercanbeobtainedfromthegradientofthephaseofthemodeshape.SCODwasshowntoaccuratelyextractmodalcomponentsofwavestravelingthroughasimulatedbeam,andanexperimentalbeam.Fromthesemodalcomponents,thedispersionpropertieswereaccuratelyextracted.Currently,arebeingappliedinwhichSCODisappliedspatially,wheredisplacementsandslopeorcurvatureisused,dependingonthemodel'sPDE,toextracttheparametersofwavespropagatinginanelasticmedium.Incontrasttoapplicationinthetimedomain,inwhichsignalsaresummed(integrated)throughtimetoproducecrosscorrelationsbetweenpointsinspace,representedbyRandS,spatial-domainSCODwillinvolveintegrations(sums)ofsignalsthroughspacetoproducecrosscorrelationsbetweenpointsintime.Applied67Figure4.5:DispersionrelationshipderivedfromSCODappliedtoanexperimentalbeam.spatially,insomesystemswavenumbersareextracteddirectlyfromtheeigenvaluesandthefrequenciesareextractedfromthewhirlingratesofthetemporaleigenvectors.68Figure4.6:PhaseVelocityderivedfromSCODappliedtoanexperimentalbeamFigure4.7:GroupVelocityderivedfromSCODappliedtoanexperimentalbeam69Chapter5ExplorationinSpatialSmoothComplexOrthogonalDecomposition5.1IntroductionCODandSCODandhavebeenappliedtosampleddisplacementsfromstructureswithcomplex-separabletemporalandspatialcharacteristics.TheMNsampledensemblematriceshaverowswhicharetimesamplesofdisplacementsatacertainpointonastructure.Thus,thehorizontaldimensionoftheoftheensemblematrixcontainstheevolutionintime,andtheverticaldimensioncontainsvariationsinspace.Theapplicationsintheprevioussectionshadtemporalandspatialbehaviorswhich,whenisolated,wereapproximatelyharmonic.ThetemporalapplicationoftheCODandcomplexSCODinvolvedensemblematrixproductssuchasZZT=N.Thesematrixproductsessentiallyinvolvesummationsthroughtimeandproducespatialcrosscorrelations.TheresultoftheCODorSCODareeigenvectors(modes)whichrepresentspatialoptimizationsofsignalstrength(COD)orsignalfrequency(SCOD),andeigenvalueswhichrepresentmodalstrength(COD)ormodalfrequency(SCOD).Modalcoordinatesthencontainthetemporalmodulationsofthemodes.70Inthissectionweconsiderinvertingtherolesofthetemporalandspatialvariations.Thus,aspatialapplicationofCODand/orSCOD(referredtohereasCOD-xandSCOD-x)isenvisionedsuchthatmatrix-productsumsaremadethroughspace,producingspatialcrosscorrelationsbetweeninstantsintime.Thedecompositionwouldthusproduceeigenvectorsthatrepresentthetemporalmodulations,andeigenvaluesthatrepresentmodalstrength(CODx)andmodalwavenumber(SCODx).Thespatialdecompositionmodalcoordinateswouldcontaininformationaboutspatialmodulation,akintotheusualnotionofamode.Tothisend,inthischapterwewillintroducesmoothcomplexorthogonaldecompositioninthespatialvariabledomainthroughanexample.InthepriorchapterwediscussedSCODinthetimedomainwhichprovidedtheabilitytodirectlycomputethefrequenciesfromtheeigenvaluesoftheSCODgeneralizedeigenvalueproblem(GEVP).ThischapterwillillustratehowtoextractthewavenumbersdirectlyfromtheGEVPbyposingSCODspatially.InthenextsectionSCODwillbeappliedspatiallytoatravelingpulse,withsomemathematicalprefaceinordertoillustrateproofofconcept.5.2TravelingPulseSCODwillbeappliedtoatravelingwavewhoseconstituentwaveshavesimilarfrequenciessuchthattheofthemaximumandminimumfrequenciesaresmall,i.e.f=fmaxfmin,issmall,wheref=!=(2ˇ).InthefrequencydomaintheFFTofthetravelingpulsewouldhaveanarrowbandqualitativelysimilartoFigure5.1[6].Thewaveisconstructedusingy(x;t)=PMi=1Aisin(ki(xc0t))wherec0isthewavespeed,ki=k0+(ik,withk0=4andk=0:0472.TheamplitudesAiarechosensuchthatthepeakofthetravelingwavehasavalueof1.0.Thenarrowbandtravelingpulse71Figure5.1:ContrivedexampleoftheFFTofanarrowbandtravelingpulseandtheFFTofthetravelingpulseatsomeithtimeisshowninFigures5.2and5.3.AnMNensemblematrixwascreatedwhereNisthenumberoftimesamplesandMisthenumberofspatialsensors.ForthisexampleN=M=128.Thedisplacementy(x;t)andslopey0(x;t)aresampledtobuildensemblesYandYslope,whichareconvertedintoanalyticensemblesZandZV.RandSmatricesarecomputedtoproduceNNmatricesR=ZTZM(5.1)S=ZTVZVM:(5.2)72Figure5.2:SimulatednarrowbandtravelingpulseThegeneralizedeigenvalueproblemisformulatedasRV=SV(5.3)wherethewavenumbers,ki,arethesquarerootsofthemagnitudesoftheeigenvalues,whicharethediagonalelementsof,suchthatki=pjjiijj.Equation(5.3)involvesNNmatrices,andyieldsN1eigenvectorsv,packagedinNNmatrixV,andNeigenvalues.The\temporalmodes",thatisthecharacteristictemporalarerepresentedbycolumnsoftheNNmatrixt=VT=VH;(5.4)muchlikewiththespatialmodesfromtheconventionalSCODinChapter4.Assuchamatrixof\modalcoordinates"arecalculatedfromtheinverseconjugatetransposeofthe73Figure5.3:FFTofsimulatednarrowbandtravelingpulseeigenvectorV,fromwhichthefrequencies!icanbeextractedfromthewhirlrateofthephase.InChapter4,thecomplexmodalcoordinateensemblewasrelatedtothemodalmatrixandcomplexanalyticdisplacementensembleviaZ=,whereisMMandQisMN.Similarly,intermsofthequantitiesfromtheSCODxapproach,weexpecttowriteZ=Qxt;(5.5)whereQxisanMNensembleof\spatialmodalcoordinates",andareseenasmodalcoordinatesintermsofthecomputation,butplaytheroleofspatialcharacteristicshapes.ToobtainQx,weperformQx=1t=ZVT=ZVH(5.6)74fromequation(5.4).The\spatialmodalcoordinates"shouldhave\spatialwhirlrates",representingwavenumbers,consistentwiththeeigenvaluesofEquation(5.3).The\spatialmodalcoordinates"Qxcanbeusedinmodalreductionormodalanalogoustowhatmightbedonewithresultsfromtime-domainSCOD.5.3ResultsSCODxwasappliedtowaveswithc0=1:5;3;6;and12;andwavenumberswereobtainedfromtheeigenvalueswhilefrequencieswereobtainedfromthewhirlratesoftheeigenvectors.AleastsquareswasappliedtotheSCODxextractedkiand!itoexactwavespeedsof1.499,2.99,5.99,and11.97.Figuresfortheextracteddispersionrelationshipforc0=1:5andc0=12areshowninFigures5.4and5.5.Figure5.4:Dispersionrelationshipforatravelingnarrowbandpulsewithaspeedofc0=1:5Itisbelievedthatasmootherheadandtailofthetravelingpulsewouldallowforbetterdispersionextraction.Howevertheresultssuggestthatforengineeringpracticewitha75Figure5.5:Dispersionrelationshipforatravelingnarrowbandpulsewithaspeedofc0=12non-smoothtravelingwave,SCODxappliedspatiallytoanarrowbandtravelingpulsecanaccuratelyextractthedispersionrelationshipandwavespeed.Whilethisexampleistrivialitisaproof-of-conceptforSCODx.5.3.1FutureWorksFutureworkinSCODxstartswithaformulationforastring;SCODxisthentobeappliedtoasimulatedstringandpossiblyotherstructures.Thepartialtialequationforauniformstringhastheformm@2u@t2+k@2u@x2=0(5.7)Forthestring,axialrod,andtorsionalrod,thetialoperatorsandindependentvariablesareinterchangeable.Forexample,youcouldrelabeltasx,andxast,andgettheequationm@2@x2+k@2@t2u=0:(5.8)76ThusthetheorythatSCODinthetimedomain,whichincludedEqn(5.7),alsoincludesEqn(5.8),whichamountstoSCODappliedinthespatialdomain.Thepairofequations(5.7)and(5.8)maybeatrivialmotivatingexamplesince,withmandkbeingconstant,thesystemwillnothaveinterestingdispersioncharacteristics.ButSCODxstillshouldwork,atleastforsecond-orderoperators.Soasubsequenttheorywillinquiretowhatextentthisswapindomains"willenabletheSCODinthespatialdomain,withausefulinterpretationforwavenumbersandmodalcoordinatehistories.77Chapter6LinearMassSpringSystem6.1IntroductionInthischapterwewillapplycomplexorthogonaldecomposition(COD)[46,63,77]andsmoothcomplexorthogonaldecomposition(SCOD)[77]forthepurposeofextractingtheparametersofharmoniccomplexstructuralwavestravelingthroughamass-chainsystem.Whenapplied,bothmethodsusemeasurementensemblestoformcorrelationmatrices.CODusesanensembleofanalyticdisplacementmeasurementstocomputeacorrelationmatrixR=ZZH=N,whereZistheanalyticdisplacementensemble,HistheHermitianorconjugatetranspose,andNisthenumberoftimesamples.NexttheCODeigenvalueproblem(EVP)isposedasR˚=˚,orinthematrixform=.Wavenumbersarecomputedfromthe˚asthemeanofkiofki=d\˚i=dx,andthefrequenciesarecomputedfromthemodalcoordinates,Q=1Z,as!i=d\qi=dt,whereqiisarowofQ.WhenusingSCOD,asecondcorrelationmatrix,S,isbuiltusingananalyticensemblematrixfromvelocitiesZv,suchthatS=ZvZvH=N.ASCODgeneralizedEVP(GEVP)isposedas=,whereisadiagonalmatrixofeigenvalues,whoseelementsprovideanestimateofthemodalfrequenciessquared,suchthatii=!2i.ThewavenumbersarecomputedsimilarlyaswithCOD.However,theyarecomputedfrom,wherecolumnsof78=Happroximatethemodeshapes.Thegradientoftheangleofeachcolumnofapproximatestheassociatedmodalwavenumber.ThispaperwillfocusonapplyingCODandSCODtoadisturbancepropagatingthroughasimulatedmass-chainsystem.Theextracteddispersionbeaviorwillbecompaedtothatpredictedbytheory.InSection5.2thedynamicsofthemass-chainsystemwillbesummarized.Section5.3willapplyCOD,Rayleighquotientapproximations,andSCODtothelinearmasschain.Finally,wewillconcludeinSection5.4.6.2AnalyticalModelforanPeriodicChainThewavebehaviorinanuniformlinearmass-springchainwithhasbeenstudiedindetailalongwithanonlinearchainin[78,79].Here,weshowthederivationforalinearchainshowninFigure6.1.Figure6.1:masschain.Therelaxedpositionanddisplacementofmassmjaredenotedby~xjandujrespectively.Inthiscasethespringsarelinearwithrelaxedlengthh.Themass-springchainisarrangedinafashionsuchthateachmassisseparatedbyadistancehfromitsnearestneighbor.hisalsotherelaxedlengthofeachspringbeforeanydeformationoccurs.Therelaxedpositionanddisplacementofmassmjaredenotedbyxj79andujrespectively.Weusetheassumptionthatallthemassesareequal(mj=m)andonlythenearestneighborshavedirectoneachother.Assuch,weconsideralinearspringforcerelativetotheequilibriumstate.Theequationsofmotion(EOM)inphysicalcoordinatescanthenbewrittenasmuj=~(uj+1uj)(ujuj1)(6.1)forj=;2;1;0;1;2;.Letting=~mwegetuj=(uj+1uj)(ujjuj1)(6.2)Thenon-dimensionalizationisdonebyassumingx=~xh,whichresultsinxj=jandxj1=j1.Thenon-dimensionalwavenumberisdenotedby.Weassumeatravelingdispersivewavesolutionatfrequency!andwavenumberandplugintotheequationsofmotion.Letuj(t)=Aei(!t)+Aei(!t):(6.3)Substitutinguj1=eAei(!t)intotheequationsandbalancingleadsto!2=2(1cos)(6.4)fortherequiredrelationshipbetween!and.Thisisthedispersionrelationshipforthelinearchain.Thusforalinearchain,thedispersionisgivenas!=p2(1cos)(6.5)80Figure6.2:Displacementsformasses1,31,61,and91.6.3ModalDecompositionAppliedtoaMassChainA250massmass-springchainwasexcitedwithaunitimpulseonthelefthandside.Thedisturbancepropagatedtowardtheright.Theresponseofthe100masseswererecordedasthedisturbancepassedthe100thmassbutbeforetraveledbackwardtothe100thmass.Essentially,thetimerecordwastruncatedatatimesuchthatnowererecorded.Themeansweresubtractedfromthedisplacementsofeachmass.ThedisplacementsforfourofthemassesareshowninFigure6.2.Itcanbeseenherethatthelowerfrequencywavestravelfasterthanthehigherfrequencywaves,whichisconsistentwiththepredicteddispersionbehaviorofalinearuniformchain.VelocitieswerealsocapturedforuseinSCOD.81Figure6.3:ComplexorthogonalvaluesFigure6.4:Linearsystem'sgeometricdispersionrelationshipusingthemodefromthe30highestCOVsinitialvelocityonthemass=1.082Figure6.5:LineardispersionrelationshipusingRayleighquotientandSCOD6.3.1CODAppliedtoaLinearMassChainThedisplacementsareconvertedtocomplexanalyticsignalsandthenusedtocomputetheco-variancematrix,andtheCODeigenvalueproblem.TheCOVswereusedasaguidetodiscernwhichCOMscontainedextractabledata[44].TheCOVsinFigure6.3showthatabout33oftheextractedmodeshavetmodalenergy.Usingthe30highestCOVsthedispersionrelationshipextractedisshowninFigure6.4.TheCODextracteddispersionrelationshipcorrelateswelltothetheoryqualitatively.COD,likePOD,hasahardtimeuniquelyextractingmodes,andhencewavenumbersandfrequencies,whentheconstituentwaveshavenearlythesameeigenvalue(COV).InordertoimproveonthiswewilluseassumedmodesandthegeneralizedRayleighquotientbasedonthesmoothcomplexorthogonaldecomposition,tobediscussednext.836.3.2SCODRayleighQuotientTheSCODRayleighquotientisbasedontheSCODEVP,andisasRQ()= HaR a HaS a:(6.6)RandSarethecorrelationmatricesfromSCOD, aarecolumnsofasquarematrix=H,wherecolumnsofare˚a,andareassumedmodes,forthisworktobechosenintheform˚a=eikx.Thevalueofkcanbebytheuser.TheRayleighquotientwillapproximateaneigenvalueintheeigenspaceofRandSwhen˚ausedasanapproximateeigenvector.SCODcorrelationmatriceswereusedbecausethesquarerootoftheSCODeigenvaluesarethefrequenciesofthetravelingwaves.TheresultsofusingtheSCODRayleighquotient,incomparisontostraight-forwardSCOD,canbeseeninFigure6.5.6.4ConclusionsThedynamicsofanmass-chainweresummarizedincludingthedispersionrelationship.wassolvedusingthemethodofmultiplescalesforanonlinearmass-chainsystem.Forthiswork,theelementsof˚aarechosenas˚aj=eikxj,consistentwithharmonictravelingwaves.Thechainwassimulatedwithaninitialimpulseappliedtotheendmassofalongchain.First,CODwasappliedtotheanalyticdisplacementsofthemasses.CODwasabletoextractaqualitativelygoodapproximationtothedispersionrelationship.SCODwasalsoappliedtoobtainanapproximationtothelowerfrequencyandlowerwavenumberpartofthedispersionrelation.ImprovementsweresoughtbyusingassumedmodesandthegeneralizedRayleighquotient.ThepredicteddispersionrelationshipusingRayleighquotientswasnearlyexact.84Slightdeviationwasseennearthetheoriginandneark=ˇ.Currentareunderwaytoapplythismethodtothelinearcubicmass-chainsystem.85Chapter7ConclusionsandFutureWork7.1SummaryofResultsCODwassuccessfullyappliedtoabeamexperimentwiththegoalofextractingthedispersionrelationshipforthebeam.Theparametersofthebeam,dataacquisitionequipment,andsignalprocessingwereoutlinedsothatindividualscanrepeattheexperiment.ThewindowofminimumandmaximumdetectablewavenumbersandfrequenciesweredeterminedbythespatialandtemporalNyquistfrequencyandrecordtimelengthcriteria.COD,onceappliedtothemeasureddata,producednearlyharmoniccomplexwavesandmodalcoordinateswithinthesamplingwindow.Thespatialandtemporalwhirlingrateswereobtainedfromthecomplexmodesandmodalcoordinatesandwereusedtoextractthemodalfrequenciesandmodalwavenumbers.Fromthesethedispersionrelationshipwasextractedandmatchedthetheoretical!=ak2.Additionally,wewereabletoanticipatethedispersioncurve'sshapebasedonthebeamparameters.Thecot,a,wasestimatedusingaleastsquareofthedataandwasfoundtobeafit=6:4431m2=s,whichgivesanunderestimationof1.61%.SpuriousmodeswereproducedbyCOD.ActualmodescanbediscernedfromspuriousmodesbyvisuallyinspectingofthewhirlingcomplexmodesorbytheCOVs.SpuriousmodeshaveverysmallCOVscomparedtoactualmodes.86ThemathematicalframeworkforSCODwasoutlinedandSCODwasappliedtemporallytobothasimulatedbeamandanexperimentalbeam(thesamebeamusedaboveandhenceddatafromabove).SCODproducednearlyharmoniccomplexmodeandcomplexeigenvalues.Thewavenumberswerecomputedfromtheinverseofmatrixwhosecolumnswerethecomplexconjugatetransposeofthecomplexmodes.Themodalfrequencieswerecomputeddirectlyfromtheeigenvalues.ThemodalfrequenciesandwaveamplitudescanbecomputeddirectlyfromtheSCOC.UsingtheSCODextractedmodalwavenumbersandmodalfrequenciesthedispersionrelationshipwasextracted.Leastsquareswasusedtotoapproximatethecot,a.WhenappliedtoasimulatedbeamSCODextractedleastsquaresestimatedasim=6:52m2=swhichis0.41%error.SCODappliedtotheexperimentalbeamyieldedaSCOD=6:72m2=s,whichisaerrorof2:6%whencomparedtotheory.AnexploratorystudysuggeststhatitmaybefeasibletoapplyCODandSCODinthespatialdomain.InaprocessreferredtoasSCODx,thespatialcorrelationmatricesforcomplexdisplacementsandvelocitieswerecomputedfortheexampleofanarrow-bandpulse.TheSCODxeigenvalueproblemthenproducedtemporalmodesfromtheeigenvectorsandwavenumbersfromtheeigenvalues.Spatialmodalcoordinateswerealsocomputedtoshowthestrengthofmodalparticipation.Finally,COD,SCOD,andtheSCODRayleighquotientwereappliedtoanmassspringsystem.Thechainconsistedof250massesandonlythedatafromthe100masseswererecorded.CODwasabletoextractthedispersionrelationshipfromthemasschainfor87alinearsystem.However,duethetherelativeclosenessofthefrequenciesofthewavesinthemasschainCODhasahardtimedistinguishthemodesofthewaves.SCODwasemployedandwasslightlybetteratextractingthemodes.ThebestresultscamefromusingSCODRayleighquotientswithassumedmodes.7.2ThisworkpresentsforthetimeanexperimentalvofCODappliedtoabeam.Successfulexperimentalapplicationshowstheofthemethod,andimpliestrobustnessforrealapplications.AmathematicalfoundationforSCODwaspresentedshowingthatSCODcanbeconsideredasatoolfordirectlyobtainingcomplexnormalmodesandmodalfrequencies.SCODwassuccessfullyappliedtoasimulatedtebeamandanexperimentalbeam.Finally,COD,SCOD,andtheSCODRayleighquotientwereappliedtoamass-chain,whichshowedCODandSCODcanbeappliedtoaone-dimensionallattice.Thisworkhasapplicationsinsensing,materialidenmaterialpropertyidenrangedetection,non-destructivetestingandevaluation,andmeta-materials.7.3FutureWorksFutureworksonthesetopicsincludestheapplicationtolongitudinalrods,plates,andmorecomplicatedstructures.Thesetechniqueswillalsobeappliedwhenwavearepresenttoseeifcotsandotheracousticalboundarypropertiescanbe88quanWealsocanconsiderexpandingthemethodtomeasurewaveattenuation.CODxandSCODxwillbeformulatedforarigorousspatialapplicationofmodaldecompositionmethodsinsystemswithoperatorsoforders.ConditionsfortheapplicationandinterpretationofspatialCODxandSCODxwillbeunderstoodfromthisrigoroustheoreticalformulation.TheaimofSCODxwillbetogeneratewavenumbersdirectlyfromeigenvalues,inplaceofanalyzingmodalvectorsfortheirspatialwhirlrates.SCODandSCODxcouldbeusedintandemtoobtain,andcrosscheck,completetemporalandspatialwavecharacteristics.89BIBLIOGRAPHY90BIBLIOGRAPHY[1]Euler,L.,1744.Methodusinveniendilineascurvasmaximiminimiveproprietategaudentes,siveSolutioproblematisisoperimetricilatissimosensuaccepti.apudMarcum-MichaelemBousquet&socios.[2]Bernoulli,D.,1751.\Devibrationibusetsonolaminarumelasticarum".CommentariiAcademiaeScientiarumImperialisPetropolitanae,13,pp.105{120.[3]Timoshenko,S.P.,1921.\Onthecorrectionforshearofthetialequationfortransversevibrationsofprismaticbars".TheLondon,Edinburgh,andDublinPhilosophicalMagazineandJournalofScience,41(245),pp.744{746.[4]Fourier,J.,1818.\Noterelativeauxvibrationsdessurfaceselastiquesetaumouvementdesondes".Bull.Sci.Soc.PhilomathiqueParis,pp.129{136.[5]Boussinesq,J.,1885.Applicationdespotentielsal'etudedel'equilibreetdumouvementdessolideselastiques:principalementaucalculdesdeformationsetdespressionsqueproduisent,danscessolides,desquelconquesexercessurunepetitepartiedeleursurfaceoudeleurinterieur:memoiresuividenotesetenduessurdiverspointsdephysique,mathematiqueetd'analyse,Vol.4.Gauthier-Villars.[6]K.F.,1975.WaveMotioninElasticSolids.CourierDoverPublications.[7]Poisson,S.-D.,1828.Memoiresurl'equilibreetmouvementdescorpselastiques.L'Academiedessciences.[8]Rayleigh,L.,1885.\Onwavespropagatedalongtheplanesurfaceofanelasticsolid".ProceedingsoftheLondonMathematicalSociety,s1-17(1),pp.4{11.[9]Lamb,H.,1904.\Ongroup-velocity".ProceedingsoftheLondonMathematicalSociety,2(1),pp.473{479.[10]Lamb,H.,1904.\Onthepropagationoftremorsoverthesurfaceofanelasticsolid".PhilosophicalTransactionsoftheRoyalSocietyofLondon.SeriesA,ContainingPapersofaMathematicalorPhysicalCharacter,pp.1{42.[11]Pao,Y.-H.,andMindlin,R.,1960.\Dispersionofwavesinanelastic,circularcylinder".JournalofAppliedMechanics,27(3),pp.513{520.91[12]Narasimha,R.,2011.\Kosambiandproperorthogonaldecomposition".Resonance,16(6),June,pp.574{581.[13]Chatterjee,A.,2000.\Anintroductiontotheproperorthogonaldecomposition".CurrentScience,78(7),April,pp.808{817.[14]Lumley,J.,1970.StochasticToolsinTurbulence.NewYork,AcademicPress.[15]Esquivel,P.,2009.\Wide-areawavemotionanalysisusingcomplexempiricalorthogonalfunctions".InElectricalEngineering,ComputingScienceandAutomaticControl,CCE,20096thInternationalConferenceon,IEEE,pp.1{6.[16]Liang,Y.C.,Lee,H.P.,Lim,S.P.,Lin,W.Z.,Lee,K.H.,andWu,C.G.,2002.\Properorthogonaldecompositionanditsapplications|part1:Theory".JournalofSoundandVibration,252(3),pp.527{544.[17]Berkooz,G.,Holmes,P.,andLumley,J.,1967.\Theproperorthogonaldecompositioninanalysisofturbulentws".AnnualreviewofFluidMechanics,25(539-575),pp.137{146.[18]Payne,F.R.,andLumley,J.L.,1967.\Largeeddystructureoftheturbulentwakebehindacircularcylinder".PhysicsofFluids(1958-1988),10(9),pp.S194{S196.[19]Iungo,G.,andLombardi,E.,2011.\Time-frequencyanalysisofthedynamicsoftvorticitystructuresgeneratedfromatriangularprism".JournalofWindEngineeringandIndustrialAerodynamics,99(67),pp.711{717.TheEleventhItalianNationalConferenceonWindEngineering,IN-VENTO-2010,Spoleto,Italy,June30th-July3rd2010.[20]Bakewell,H.P.,andLumley,J.L.,1967.\Viscoussublayerandadjacentwallregioninturbulentpipew".PhysicsofFluids(1958-1988),10(9),pp.1880{1889.[21]Atwell,J.A.,andKing,B.B.,2004.\Reducedordercontrollersforspatiallydistributedsystemsviaproperorthogonaldecomposition".SIAMJournalonComputing,26(1),pp.128{151.[22]Kunisch,K.,Volkwein,S.,andXie,L.,2004.\HJB-POD-basedfeedbackdesignfortheoptimalcontrolofevolutionproblems".SIAMJournalonAppliedDynamicalSystems,3(4),pp.701{22.92[23]Leibfritz,F.,andVolkwein,S.,2006.\ReducedorderoutputfeedbackcontroldesignforPDEsystemsusingproperorthogonaldecompositionandnonlinearprogramming".LinearAlgebraanditsApplications,415(2),pp.542{575.[24]Ly,H.V.,andTran,H.T.,2001.\Modelingandcontrolofphysicalprocessesusingproperorthogonaldecomposition".MathematicalandComputerModeling,33(1),pp.223{236.[25]Gaonkar,A.,andKulkarni,S.,2015.\ApplicationofmultilevelschemeandtwoleveldiscretizationforPODbasedmodelorderreductionofnonlineartransientheattransferproblems".ComputationalMechanics,55(1),pp.179{191.[26]Willcox,K.,andPeraire,J.,2002.\Balancedmodelreductionviatheproperorthogonaldecomposition".AIAAJournal,40(11),pp.2323{2330.[27]Pironneau,O.,2012.\Properorthogonaldecompositionforpricingoptions".TheJournalofComputationalFinance,16(1),Fall,pp.33{VI.[28]Acharjee,S.,andZabaras,N.,2003.\AproperorthogonaldecompositionapproachtomicrostructuremodelreductioninRodriguesspacewithapplicationstooptimalcontrolofmicrostructure-sensitiveproperties".ActaMaterialia,51(18),pp.5627{5646.[29]Galvanetto,U.,andViolaris,G.,2007.\Numericalinvestigationofanewdamagedetectionmethodbasedonproperorthogonaldecomposition".MechanicalSystemsandSignalProcessing,21(3),pp.1346{1361.[30]Galvanetto,U.,Surace,C.,andTassotti,A.,2008.\Structuraldamagedetectionbasedonproperorthogonaldecomposition:experimentalvAIAAJournal,46(7),pp.1624{1630.[31]Banks,H.,Joyner,M.L.,Wincheski,B.,andWinfree,W.P.,2000.\Nondestructiveevaluationusingareduced-ordercomputationalmethodology".InverseProblems,16(4),p.929.[32]Feeny,B.F.,andKappagantu,R.,1998.\Onthephysicalinterpretationofproperorthogonalmodesinvibrations".JournalofSoundandVibration,211(4),pp.607{616.[33]Han,S.,andFeeny,B.F.,2002.\Enhancedproperorthogonaldecompositionforthemodalanalysisofhomogeneousstructures".JournalofVibrationandControl,8(1),pp.19{40.93[34]Feeny,B.F.,2002.\Ontheproperorthogonalmodesandnormalmodesofcontinuousvibrationsystems".JournalofVibrationandAcoustics,124(1),pp.157{160.[35]Feeny,B.F.,andLiang,Y.,2003.\Interpretingproperorthogonalmodesinrandomlyexcitedvibrationsystems".JournalofSoundandVibration,265(5),pp.953{966.[36]Yadalam,V.K.,andFeeny,B.F.,2011.\Reducedmassweightedproperdecompositionformodalanalysis".JournalofVibrationandAcoustics,133(2),p.024504.[37]Chelidze,D.,andZhou,W.,2006.\Smoothorthogonaldecomposition-basedvibrationmodeidenJournalofSoundandVibration,292(3-5),pp.461{473.[38]Bellizzi,S.,andSampaio,R.,2009.\SmoothKarhunen-Loevedecompositiontoanalyzerandomlyvibratingsystems".JournalofSoundandVibration,325(3),pp.491{498.[39]Farooq,U.,andFeeny,B.F.,2008.\Smoothorthogonaldecompositionforrandomlyexcitedsystems".JournalofSoundandVibration,316(3-5),Sep,pp.137{146.[40]CaldwellJr.,R.A.,andFeeny,B.F.,2014.\Output-onlymodalidenofanonuniformbeambyusingdecompositionmethods".JournalofVibrationandAcoustics,136(4),May,p.041010(10pages).[41]Georgiou,I.T.,andPapadopoulos,C.I.,2006.\DevelopingPODoverthecomplexplanetoformadataprocessingtoolforelementsimulationsofsteadystatestructuraldynamics".InInternationalMechanicalEngineeringCongressandExposition,Vol.onDVD-ROM.[42]Feeny,B.F.,2007.\Acomplexorthogonaldecompositionforwaveanalysis.".JournalofSoundandVibration,310(1),pp.77{90.[43]Feldman,M.,2011.\Hilberttransforminvibrationanalysis".MechanicalSystemsandSignalProcessing,25(3),pp.735{802.[44]Feeny,B.F.,2002.\Onproperorthogonalcoordinatesasindicatorsofmodalactivity".JournalofSoundandVibration,255(5),pp.805{817.[45]Feeny,B.F.,andFeeny,A.K.,2013.\Complexmodalanalysisoftheswimmingmotionofawhiting".JournalofVibrationandAcoustics,135(2),p.021004.[46]Feeny,B.F.,2013.\Complexmodaldecompositionforestimatingwavepropertiesinone-dimensionalmedia".JournalofVibrationandAcoustics,135(3),p.031010.94[47]Cox,G.A.,Livermore,P.W.,andMound,J.E.,2014.\Forwardmodelsoftorsionalwaves:dispersionandgeometricGeophysicalJournalInternational,196(3),pp.1311{1329.[48]Hoogstraten,H.W.,1968.\Dispersionofnon-linearshallowwaterwaves".JournalofEngineeringMathematics,2(3),pp.249{273.[49]Thomson,W.,andDahleh,M.,1998.TheoryofVibrationwithApplication5thEdition.Prentice-Hall.[50]Meirovitch,L.,1967.AnalyticalMethodsinVibrations.Macmillan,NewYork.[51]Cusumano,J.C.,andBai,B.,1993.\Perioyperiodicmotion,chaos,andspatialcoherenceina10-degree-of-freedomimpactoscillator".Chaos,Solitions,andFractals,3(5),pp.515{535.[52]FitzSimons,P.,andRui,C.,1993.\Determininglowdimensionalmodelsofdistributedsystems".AdvancesinRobustandNonlinearControlSystems,ASMEDSC-VOl.53,pp.9{15.[53]Azeez,M.F.A.,andVakakis,A.F.,2001.\Properorthogonaldecomposition(POD)ofaclassofvibroimpactoscillations".JournalofSoundandVibration,240(5),Mar,pp.859{889.[54]Riaz,M.S.,andFeeny,B.F.,2003.\Properorthogonaldecompositionofabeamsensedwithstraingages".JournalofVibrationandAcoustics,125(1),pp.129{131.[55]Kerschen,G.,andGolinval,J.C.,2002.\Physicalinterpretationoftheproperorthogonalmodesusingthesingularvaluedecomposition".JournalofSoundandVibration,249(5),pp.849{865.[56]Biglieri,E.,andYao,K.,1989.\Somepropertiesofsingularvaluedecompositionandtheirapplicationstodigitalsignalprocessing".SignalProcessing,18,pp.277{289.[57]Liu,W.,Gao,W.C.,andSun,Y.,2009.\Applicationofmodalidenmethodstospatialstructureusingmeasurementdata".JournalofVibrationandAcoustics,131(3),June.[58]Ibrahim,S.,andMikulcik,E.,1977.\Amethodforthedirectidenofvibrationparametersfromthefreeresponse".ShockandVibrationBulletin,47(4),pp.183{198.95[59]Farooq,U.,andFeeny,B.F.,2012.\Anexperimentalinvestigationofastate-variablemodaldecompositionmethodformodalanalysis".JournalofVibrationsandAcoustics,132(2),p.021017(8pages).[60]Feeny,B.F.,1996.\Acomplexorthogonaldecompositionforanalyzingwavemotion".InASMEInternationalMechanicalEngineeringCongressandExposition,Vol.onDVD-ROM.[61]Kappagantu,R.,andFeeny,B.,1999.\An\optimal"modalreductionofasystemwithfrictionalexcitation".JournalofSoundandVibration,224(5),pp.863{877.[62]Onsay,T.,andHaddow,A.G.,1994.\Wavelettransformanalysisoftransientwave-propagationinadispersivemedium".JournaloftheAcousticalSocietyofAmerica,95(3),pp.1441{1449.[63]Feeny,B.F.,2008.\Acomplexorthogonaldecompositionforwavemotionanalysis".JournalofSoundandVibration,310(1{2),Feb,pp.77{90.[64]Oppenheim,A.V.,andSchafer,R.W.,1989.Discrete-TimeSignalProcessing.PrenticeHall,EnglewoodNJ.[65]Schmidt,R.,1986.\Multipleemitterlocationandsignalparameterestimation".AntennasandPropagation,IEEETransactionson,34(3),Mar,pp.276{280.[66]Roy,R.,andKailath,T.,1989.\ESPRIT-estimationofsignalparametersviarotationalinvariancetechniques".IEEETransactionsonAcoustics,Speech,andSignalProcessing,37(7),Jul,pp.984{995.[67]Feeny,B.F.,Sternberg,P.W.,Cronin,C.J.,andCoppola,C.A.,2013.\Complexorthogonaldecompositionappliedtonematodeposturing".JournalofComputationalandNonlinearDynamics,8(4),p.041010.[68]Bussow,R.,2008.\Applicationsoftheimpulseresponsefunctionsinthetimedomain".ActaAcusticaunitedwithAcustica,94(2),pp.207{214.[69]Beck,J.,andArnold,K.,1977.ParameterationinEngineeringandScience.JohnWileyandSons,NewYork.[70]Kosambi,D.,1943.\Statisticsinfunctionspace".JournalofIndianMathematicalSociety,7,pp.76{88.96[71]Karhunen,K.,1946.\ZurSpektralTheorieStochastischerProzesse".Ann.Acad.Sci.Fennicae,A,pp.1{34.[72]Ibrahim,S.R.,andMikulcik,E.C.,1973.\Atimedomainmodalvibrationtesttechnique".ShockandVibrationBulletin,34(4),pp.21{37.[73]Han,S.,andFeeny,B.F.,2003.\Applicationofproperorthogonaldecompositiontostructuralvibrationanalysis".MechanicalSystemsandSignalProcessing,17(5),pp.989{1001.[74]Brincker,R.,Zhang,L.,andAndersen,P.,2001.\Modalidenofoutput-onlysystemsusingfrequencydomaindecomposition".SmartMaterialsandStructures,10(3),pp.441{445.[75]Feeny,B.F.,andFarooq,U.,2008.\Anonsymmetricstate-variabledecompositionformodalanalysis".JournalofSoundandVibration,310(4-5),March,pp.792{800.[76]Caldwell,R.A.,andFeeny,B.F.,2016.\Characterizingwavebehaviorinabeamexperimentbyusingcomplexorthogonaldecomposition".JournalofVibrationandAcoustics,138(4),p.041007.[77]CaldwellJr,R.A.,andFeeny,B.F.,2016.\Smoothcomplexorthogonaldecompositionappliedtotravelingwavesinelasticmedia".InRotatingMachinery,HybridTestMethods,Vibro-Acoustics&LaserVibrometry,Volume8.Springer,pp.281{293.[78]Narisetti,R.K.,Leamy,M.J.,andRuzzene,M.,2010.\Aperturbationapproachforpredictingwavepropagationinone-dimensionalnonlinearperiodicstructures".ASMEJournalofVibrationandAcoustics,132(3),p.031001.[79]Narisetti,R.K.,Ruzzene,M.,andLeamy,M.J.,2011.\Aperturbationapproachforanalyzingdispersionandgroupvelocitiesintwo-dimensionalnonlinearperiodiclattices".JournalofVibrationandAcoustics,133(6),p.061020.97