COMPARATIVE STUDIES ON MOVING FORCE IDENTIFICATION

JournalofSoundandibration(2000)235(1),87}104doi:10.1006/jsvi.2000.2909,availableonlineat¹heHongKongPolytechnic;niversity,HungHom,Kowloon,HongKong.E-mail:[e-mail](Received30August1999,andin,nalform12February2000)Forceidenticationfromdynamicresponsesofbridgesisanimportantinverseproblem.Theparametersofboththevehicleandthebridgeplayanimportantroleintheforceidentication.Basedonthebendingmomentsmeasuredinthelaboratoryandthefouridenticationmethodsdeveloped,thispaperaimstoinvestigatethee!ectofvariousparametersonthefourmethods.Forthispurpose,abridge}vehiclesystemmodelhasbeendesignedinthelaboratory.Thebendingmomentsandaccelerationresponsesofthemodelbridgearesimultaneouslymeasuredwhenthemodelvehiclemovesacrossthebridgeatdi!erentspeeds.Themovingforcesareidentiedfromthebridgestrainsusingthefourmethods,andtherebuiltresponsesarecalculatedfromtheidentiedforcesforcomparativestudiesonthefourmethods.Assessmentresultsshowthatallfourmethodsaree!ectiveandacceptablewithhigheraccuracytosomeextent.TheTDMisthebestandisstronglyrecommendedforincorporationintoamovingforceidenticationsystem(MFIS).(2000AcademicPress1.INTRODUCTIONManymethodshavebeenpresentedforforceidenticationinrecentyears[1}8].Stevens[9]hasgivenanexcellentsurveyoftheliteratureontheforceidenticationproblemaswellasanoverview.However,mostofthemethodsmentionedmeasureonlystaticaxleloads.O'ConnorandChan[10]suggestedanadvancedforceidenticationmethod*InterpretiveMethodI(IMI)tointerprettheforcehistory,whichisanadvancementoftheweight-in-motionmethodsmentionedaboveandisabletomeasurethedynamicaxleforcesofmulti-axlesystems.Basedonsystemidenticationtheory,theauthorshavedevelopedanothertwomovingforceidenticationmethods,namelythetime-domainmethod(TDM)[11]andthefrequency}time-domainmethod(FTDM)[12].Recently,anewmethodsimilartoIMI,theInterpretiveMethodII(IMII),hasalsobeenpublished[13].Preliminarystudies[14]showedthatallthesefourmethodscouldidentifymovingforceswithacceptableaccuracy.However,eachmethodhasitsmerits,limitationsanddisadvantages.Theyneedtobeimprovedandenhancedforpracticalapplicationineldtests.Itisalsoagoodideatomergethemintoamovingforceidenticationsystem(MFIS).Therefore,comparativestudiesonthefourmethodsbasedonthemeasuredbendingmomentsaredescribedinthispaper.Thee!ectsofvariousparametersontheforceidenticationhavebeencriticallyinvestigatedusingexperimentaldata.Theparametersincludethebridgemodenumbersused,samplingfrequencies,vehiclespeeds,computationaltime,sensornumbersandlocations.Acceptableresultsonidentiedforcesareobtainedandsomesuggestionsaremadefortheidenticationmethods.0022-460X/00/310087#18$35.00/0(2000AcademicPressFigure1.Beam-elementmodel.2.DESCRIPTIONOFIDENTIFICATIONMETHODS2.1.ANALYTICALMODELSThemovingforceidenticationdescribedhereisaninverseofaforwardproblem,wherebystructuralresponsescausedbyasetoftime-varyingforcesrunningacrossabridgearefound.Intheinverseproblem,forcesarededucedfrommeasuredresponsesinstead.Twomodelscanbeusedforthiskindofanalysis.2.1.1.Beam-element-modelAsimplysupportedbridgecanbemodelledasanassemblyoflumpedmassesinterconnectedbymasslesselasticbeamelementsasshowninFigure1,andthenodalresponsesfordisplacementsor/andbendingmomentatanyinstantaregivenbyequations(1)and(2)respectively,MN[A]MPN![I][Dm]MGN![I][C]MQN,(1)MMN[MA]MPN![MI][Dm]MGN![MI][C]MQN,(2)whereMPNisthevectorofwheelloads,[Dm]isthediagonalmatrixcontainingvaluesoflumpedmass,[C]isthedampingmatrix,MMN,MN,MQN,MGNarethenodalbendingmoment,displacement,velocity,accelerationvectorsrespectively,[A],[I]arematricesfornodalforcestoobtainnodaldisplacements,and[MA],[MI]arematricesfornodalforcestoobtainnodalbendingmoments.2.1.2.ContinuousbeammodelThebridgemodelisconsideredasasimplysupportedbeamwithaspanlength¸,constant#exuralsti!nessEI,constantmassperunitlengthoandviscousproportionaldampingC.Thee!ectsofsheardeformationandrotaryinertiaarenottakenintoaccount(Bernoulli}Eulerbeam).IftheforcePmovesfromlefttorightataspeedc,asshowninFigure2,thenanequationofmotioncanbeexpressedasoL2l(x,t)Lt2#CLl(x,t)Lt#EIL4l(x,t)Lx4d(x!ct)P(t),(3)wherel(x,t)isthebeamde#ectionatpointxandtimetandd(x!ct)istheDiracdeltafunction.Basedonmodalsuperposition,ifthenthmodeshapefunctionofthebeamUn(x)sin(nnx/¸),thesolutionofequation(3)canbeexpressedasfollows:l(x,t)=+n/1sinnnx¸qn(t),(4)88T.H.T.CHANE¹A¸.Figure2.Movingforceonasimplesupportedbeam.wherenisthemodenumber,andqn(t)(n1,2,2,R)arethenthmodaldisplacements.Aftersubstitutingequation(4)intoequation(3),integratingtheresultantequationwithrespecttoxbetween0and¸,andthenusingtheboundaryconditionsandthepropertiesoftheDiracdeltafunction,theequationofmotionintermsofthemodaldisplacementqn(t)canbeexpressedasq(n(t)#2mnunqR(t)#u2nqn(t)2o¸pn(t)(n1,2,2,R),(5)whereunn2n2¸2SEIo,mnC2oun,pn(t)P(t)sinAnnxN¸B(6)arethenthmodalfrequency,themodaldampingandthemodalforcerespectively.xNisthedistanceoftheaxlefromtheleft-handsup