桥梁资料

usingtunedmassdampers

J.F.Wanga,C.C.Lin

aba,*,B.L.Chen

bDepartmentofCivilEngineering,NationalChung-HsingUniversity,Taichung,Taiwan,402ROCDepartmentofCivilEngineering,NationalLien-HoInstituteofTechnology,Miaoli,Taiwan,360ROC

Received3July2001;receivedinrevisedform4October2002

Abstract

Thispaperdealswiththeapplicabilityofpassivetunedmassdampers(PTMDs)tosuppresstrain-inducedvibrationonbridges.ArailwaybridgeismodeledasanEuler–Bernoulibeamandatrainissimulatedasseriesofmovingforces,movingmassesormovingsuspensionmassestoinvestigatetheinfluenceofvariousvehiclemodelsonthebridgefea-tureswithorwithoutPTMD.Accordingtothetrainloadfrequencyanalysis,theresonanteffectswilloccurasthemodalfrequenciesofthebridgesareclosetothemultipleoftheimpactfrequencyofthetrainloadtothebridge.AsinglePTMDsystemisthendesignedtoalterthebridgedynamiccharacteristicstoavoidexcessivevibrations.Nu-mericalresultsfromsimplysupportedbridgesofTaiwanHigh-SpeedRailway(THSR)underGermanI.C.E.,JapaneseS.K.S.andFrenchT.G.V.trainsshowthattheproposedPTMDisausefulvibrationcontroldeviceinreducingbridgeverticaldisplacements,absoluteaccelerations,endrotationsandtrainaccelerationsduringresonantspeeds,asthetrainaxlearrangementisregular.ItisalsofoundthattheinnerspaceofbridgeboxgirderofTHSRiswideanddeepenoughfortheinstallationandmovementofPTMD.Ó2003ElsevierScienceLtd.Allrightsreserved.

Keywords:High-speedrailwaybridge;Tunedmassdampers;Trainloads;Resonanttrainspeed;Vibrationsuppression

1.Introduction

Ingeneral,transportationinfrastructureisanimportantfactoraffectingthedevelopmentofanationaleconomy.Becauseofspaceandterrainlimitations,moretransportationstructures,suchashighwaysandrailways,havebeenconstructedasbridgesinurbanareas.Withtherapidadvancesinthefieldofhighperformancematerialsandconstructiontechniques,thesebridgeshaveatrendtowardslongandflexibleasthoseofthehigh-risebuildings.Whenexcessiveexternalloadsoccur,thesebridgesmaysufferlargede-flectionsandevencausedamagesthatwillendangerhumanlifeandproperty.Inordertounderstandthedynamicbehaviorofbridgesundernaturalloadssuchaswindorearthquakeexcitations,considerable

Correspondingauthor.Tel./fax:+886-4-22851992.

E-mailaddresses:jerfu@ms16.hinet.net(J.F.Wang),cclin3@dragon.nchu.edu.tw(C.C.Lin),blchen@mail.nlhu.edu.tw(B.L.Chen).

0020-7683/02/$-seefrontmatterÓ2003ElsevierScienceLtd.Allrightsreserved.PII:S0020-7683(02)00589-9

*466J.F.Wangetal./InternationalJournalofSolidsandStructures40(2003)465–491

NomenclatureAðxÞCbcbjcsCyðxÞdEFbFbjgHgjFðxÞHvsFðxÞIðxÞKbkbjksLMb󰀁ðxÞmmbjmsmvkNNvpkPbðx;tÞRdvjrfjUðÀÞttkvvcvsxsxyðx;tÞzsðtÞzvðtÞdðÀÞgjðtÞgðtÞlsjnjbridgesectionareabridgemodaldampingmatrixjthmodaldampingratioofbridgePTMDdampingcofficientdampingcoefficientofbridgeatsectionxspacingoftrainloadsYoungÕsmodulusmodaltrainloadvectorjthmodaltrainloadgravityaccelerationtransferfunctionofthejthmodaldislpacementofbridgetransferfunctionofPTMDstrokemomentofinertiaofthebridgebridgemodalstiffnessmatrixjthbridgemodalstiffnesscoefficientPTMDÕsstiffnesscoefficientspanlengthofthebridgebridgemodalmassmatrixbridgemassperunitlengthatsectionxjthmodalmassofbridgePTMDmassmassofthekthtrainloadinthemovingsuspensionmassmodelnumberofmodetobeconsiderednumberoftrainloadmagnitudeofthekthtrainloadinthemovingforcemodeldistributedforceappliedonthebeamjthbridgemodalresponseratiofrequencyratioofPTMDtothecontrolledbridgemodeunitstepfunctioninstantaneoustimetimeofthekthtrainloadreachingthebridgetrainspeedresonanttrainspeedPTMDstrokepositionofPTMDonthebridgeinthelongitudinaldirectionlongitudinalpositionmeasuredfromsupportsofbridgebridgeverticaldisplacementPTMDverticaldisplacementverticaldisplacementofthekthtrainloadDiracdeltafunctionjthmodaldisplacementofbridgemodaldisplacementvectorofbridgePTMDmassratiojthmodaldampingratioofbridgeJ.F.Wangetal./InternationalJournalofSolidsandStructures40(2003)465–491467

nsxjxs/jðxÞUðxÞCPTMDdampingratiojthmodalfrequencyofbridgePTMDnaturalfrequencyjthmode-shapefunctionofbridgemode-shapefunctionmatrixofbridgemodalparticipationfactorofthebridge-PTMDsystemnumericalandexperimentaleffortshavebeenmadeoverthepasttwodecades(Abdel-GhaffarandRubin,1982;DumanogluandSevern,1990;Boonyapinyoetal.,1994).

Thevibrationofabridgestructureduetothepassageofvehiclesisalsoanimportantconsiderationinbridgedesign.Tocomprehendthecomplexinteractionsbetweenthevehicleandthebridgeandtodeveloprationaldesignprocedures,anumberofanalyticalandexperimentalinvestigationshavebeencarriedoutoverthepastfewdecades.Inthosestudies,oneofthemainsubjectswasthesimulationofvehiclesystems.Traditionally,thevehiclewasmodeledasamovingforcethatassumedconstantvehicularloadinthebridgeatanylocation(Vellozzi,1967;HayashikawaandWatanabe,1981,1982;BryjaandSniady,1988).Then,moreaccuratemodels,suchasmovingmass(WilsonandBarbas,1980;InbanathanandWieland,1987;AkinandMofid,1989),movingsuspensionmass(HumarandKashif,1995)andcomplicatedtwo-dimensional(VeletsosandHuang,1970;HuttonandCheung,1979;HuangandWang,1992;Greenetal.,1995)orthree-dimensional(Wangetal.,1992;Chatterjeeetal.,1994;KouandDeWolf,1997;HuangandWang,1998;Huangetal.,1998)vehiclebodies,weredevelopedtorespectivelytaketheinertialforce,thesuspensionsystem,andthecomplexdynamicmechanismsofvehiclesintoaccount.Mostofthepreviousresearchesinvolvedhighwaybridges,wheresinglevehicleorrandommovingvehicleswereusedastheexternalforces.Investigationsinvolvingthedynamicbehaviorofbridgesunderperiodicmovingloadswhichrepresenttheimpactofatrainarerelativelyfew(KlasztornyandLanger,1990;Fr󰀂yba,1996),es-peciallyforthedynamicsofrailwaybridgesunderhigh-speedtrains.ChenandLi(2000)calculatedthedynamicresponsesofTHSRelevatedrailwaybridgessubjectedtotheFrenchT.G.V.,theGermanI.C.E.,andtheJapaneseS.K.S.trainloadswiththemaximumoperationspeedof350km/h.Thebridgeiscon-structedasasingle-spanorthree-equal-spanboxgirdersupportedonpiers.Yangetal.(1997)obtainedtheconditionsofresonanceandcancellationforasimplebeamduetotrainloadsandproposedtheoptimalspanlengthforbridgesforaspecifiedspacingoftrainloads.Theyfoundthatiftheoptimalspanlengthwasnotused,thebridgewouldundergoresonanteffectsbecauseoftheperiodicarrangementofpassengercarsasthetraintraveledatcertainspeeds.Fr󰀂yba(2001)derivedanalyticalformulastoobtaintheresonanttrainspeedswhichappearactuallyonhigh-speedlinesattodayÕstrainspeed.Generally,theseresonancecon-ditionswillresultinlargeresponsesandarenotexpectedforbridges.Theywillseriouslyaffecttrainsafetyoperations,thecomfortofpassengers,theservicelifeofbridgesandtheutilizationofthesurroundingland,andevenendangerthesafetyofsupportingstructures.Therefore,itisessentialtofindanappropriatewaytoreducetheexcessivevibrationofsuchbridgesundertrainloads.

Inthefieldofstructuralengineering,vibrationcontrolsystemshavebeenappliedtoreducethedynamicresponsesofstructuressincethe1980s.Oneofthetechniquesusedistheactive/passivetunedmassdamper(ATMD/PTMD),whichcanbeincorporatedintoanexistingstructurewithlessinterferenceascomparedwithothercontroldevices.Sofar,thisdevicehasbeeninstalledinover300high-risebuildingsagainstwindandearthquakesintheworld.Thecorrespondingtheoreticalandexperimentalinvestigationsforthistechniquearestillundercontinuousdevelopment.MostofthepreviousresearchesaboutPTMDsystemsareconcernedwiththemitigationofbuildingvibrations.OnlyafewresearchershaveinvestigatedthepracticalapplicationsofPTMDinreducingthedynamicvibrationsofbridgesduetomovingvehicles.Kajikawaetal.(1989)utilizedasinglePTMDonhighwaybridgesandconcludedthatthispassivecontrol

468J.F.Wangetal./InternationalJournalofSolidsandStructures40(2003)465–491

devicecouldnotcompletelysuppresstraffic-inducedvibrationssincethedynamicresponsesofabridgearefrequency-variantduetovehiclemotion.Kwonetal.(1998)inspectedthePTMDcontroleffectivenessonahigh-speedrailwaycontinuousbridgewiththreespansinKorea.Intheirpaper,asinglePTMDwiththeparametersproposedbyDenHartog(1956)wasused.ThenumericalresultsforthebridgesubjectedtotheFrenchT.G.V.trainwithaarbitrarytrainspeedof300km/hshowedthatthePTMDwithmassratioof1%wasabletoreducethebridgeverticalfreedisplacementresponseforabout21%butlesseffectiveinsup-pressingtheverticalaccelerationofthepassengercarbecausethevehiclepassagetimeonthebridgeistooshort.

Presently,theplanningandconstructionofahigh-speedrailwayarethemostimportantinfrastructureprojectinTaiwan.Thetotallengthoftherouteisabout345km.Inordertosavelandvalueandcon-structiontime,mostoftheroutewillbeconstructedasaseriesofbridges.ForTHSRbridges,fourdynamicresponselimitationsforthebridgesandthetrainsmustbesatisfied:(1)theverticaldisplacementofthebridge,(2)theverticalaccelerationofthebridge,(3)theendrotationofthebridge,and(4)theverticalaccelerationofthetrain.TheobjectiveofthispaperistoinvestigatethePTMDvibrationcontroleffec-tivenessforthesimplysupportedbridgesproposedintheoriginalTHSRdesignproposalsubjectedtotheGermanI.C.E.,FrenchT.G.V.,andJapaneseS.K.S.trainloads.AnoptimalsinglePTMD,whichisasingle-degree-of-freedom(s.d.o.f.)systemwithmass,damping,andstiffness,wasdevelopedtoreducetheabovefourdynamicresponses.Further,notonlythedynamicbehaviorcharacteristicsofthebridgesunderhigh-speedtrainloadsbutalsothePTMDdetuningeffectresultingfromtheinteractionbetweenthebridgeandtrainareextensivelyexaminedinthefollowingsections.

2.Bridge-PTMDsystemsundertrainloads2.1.Modelingofbridgeandtrain

Incomparisonwithbuildingssubjectedtowindorearthquakeexcitations,thelocationoftrainloadsonbridgesistime–variant.Further,becauseoftheinteractioneffectbetweenthetrainandbridge,themag-nitudeofthetrainloadisdependentupontheresponseofthebridge.Therefore,itwouldbedifficulttoestablishaclearcorrelationbetweenthegoverningparametersandbridgeresponsesifprecisetrainmodelswereusedintheanalyticalstudies.Toclearlyidentifythedominantparametersandtoobtaintheanalyticalsolutions,simplifiedmodelswereusuallyemployedinmanyresearches(Vellozzi,1967;HayashikawaandWatanabe,1981,1982;BryjaandSniady,1988).Oncethebasicparametershavebeenidentified,itispossibletorefinethemodeltoincludeothervariablesforadvancedwork.Inthisstudy,severalassumptionsweremadetomaketheproblemeasier,asfollows:(1)Thebridgeisregardedasastraightbeammadeofhomogeneous,elastic,isotropicmaterial.ThesupportsofthebridgearerigidandtheshapeofthebridgeÕscross-sectionisunchangedduringvibration.Therailirregularityisnegligible.(2)Inordertounderstandthedynamicresponsesofthepassengercars,thetrainismodeledasaperiodicseriesofplanarmovingforcesormovingsuspensionmasses.Thetrainloadsareappliedatthecenterlineofthebeamandmovealongthelongitudinaldirectionwithaconstantspeed.

2.2.Equationsofmotionforbridge-PTMDsystemundertrainloads

Letas.d.o.f.PTMDbeinstalledatabeam-likebridgewithlengthLatpositionx¼xs,asshowninFig.1.Whenatrain,consistingofNvnumberofmovingloads,ispassingoverthebridgewithconstantspeedv,thegoverningequationsforthebridge-PTMDsystemaregivenasfollows:(1)theverticalmotionofthebridgeatmasscenter,

J.F.Wangetal./InternationalJournalofSolidsandStructures40(2003)465–491469

Fig.1.BridgescombinedwithPTMDundertrainloadssimulatedas:(a)movingforcemodel;(b)movingmassmodel;(c)movingsuspensionmassmodel. !

o2yðx;tÞoyðx;tÞo2yðx;tÞo2yðx;tÞ󰀁ðxÞþþCyðxÞEIðxÞm¼Pbðx;tÞ

ot2otox2ox2

ð1aÞ

470J.F.Wangetal./InternationalJournalofSolidsandStructures40(2003)465–491

wherethetrainloadingfunction

Pbðx;tÞ¼À

NvXk¼1

_ðxs;tÞ󰀇gÁdðxÀxsÞpkd½xÀvðtÀtkÞ󰀇Hðt;tkÞþfks½zsÀyðxs;tÞ󰀇þcs½z_sÀyð1bÞ

andHðt;tkÞ¼UðtÀtkÞÀU½tÀðtþv=LÞ󰀇;UðÀÞanddðÀÞrepresenttheunitstepfunctionandtheDirac

deltafunction,respectively,whicharedefinedas

&Z1

1;x¼0

dðxÞdx¼ð2Þ

0;x¼0À1

UðtÞ¼

&1;0;

tP0t<0

ð3Þ

(2)theverticalmotionofPTMD,

_ðxs;tÞ󰀇þks½zsðtÞÀyðxs;tÞ󰀇¼0ms€zsðtÞþcs½z_sðtÞÀy

ð4Þ

󰀁ðxÞ,CyðxÞandEIðxÞrepresentthemassperunitlength,dampingcoefficientoftheInEqs.(1)and(4),m

flexuralmotionandrigidityoftheflexuralmotionofthebridgeatsectionx,respectively.ms,csandksrepresentthemass,dampingcoefficient,stiffnesscoefficientofthePTMD.yðx;tÞandzsðtÞindicatetheverticaldisplacementofthebridgeandthePTMD.InEq.(1b),theexpressionforthetrainload,Pbðx;tÞ,wasproposedbyYangetal.(1997),wherepkisthemagnitudeofkthloadandtkdenotesthetimewhenthekthloadreachesthebridge.Tocalculatethetrainloads,dðxÞandUðtÞareintroducedtolocatethepositionofeachloadonthebridge.Itiseasilyrecognizedthat,thesecondtermoftherighthandsideofthesummationsigninEq.(1b)determinesthelocationofkthloadonthebridge,whereasthethirdtermdetermineswhetherthekthloadisonthebridgeornot.Itshouldbenotedthatthevalueofpkisdependentuponthetrainmodel.Forthemovingforcemodel,asshowninFig.1(a),

pk¼mvkg

ð5aÞ

whichistheweightofthekthload.Ifamovingmassmodel(Fig.1(b))andmovingsuspensionmassmodel(Fig.1(c))areused,pkiswrittenas

pk¼mvkÁfgþ€y½vðtÀtkÞ;t󰀇gand

pk¼mvkÁ½gþ€zvðtÞ󰀇

ð5cÞ

respectively,wheremvkandzvðtÞarethemassandtheverticaldisplacementofthekthtrainload.Comparingthethreetrainmodels,itisobservedthattheinertialforceofthevehicleisneglectedinthefirstmodelandtakenintoaccountintheothermodelsindifferentmanners.Actually,themovingforceandmovingmassmodelsareparticularcasesofmovingsuspensionmassmodel.Whenthesuspensionstiffnessisrigid,thevehicleacceleration,€zvðtÞ,isequaltotheaccelerationwherethevehicleislocated,andthusEq.(5c)willturnouttobeEq.(5b).Ontheotherhand,whenthesuspensionstiffnessisverysoft,€zvðtÞwillapproachtozerotheoretically.Eq.(5c)willbeequaltoEq.(5a),whichrepresentsthemovingforcemodel.Obviously,thevehicularmovementwillalterthedynamiccharacteristicsoftheentiresystemsincethetrainloadsarerelativetothebridgeresponseasprecisemodelsareemployed.2.3.Modaldecouplingtosolvepartialdifferentialequations

ToanalyticallysolveEqs.(1)and(4),modalanalysiswasemployedtoseparatethegoverningpara-meters.Theverticaldisplacementyðx;tÞcanbeexpressedastheproductofthebeamverticalvibration

ð5bÞ

J.F.Wangetal./InternationalJournalofSolidsandStructures40(2003)465–491471

modeshapefunctioninvolvingonlythespatialcoordinate,x,andmodalresponsefunctionsinvolvingthevariabletime,t.Usingtheconceptofmodalsuperposition,yðx;tÞcanberepresentedas

yðx;tÞ¼

NXj¼1

/jðxÞgjðtÞ¼UTðxÞgðtÞ

ð6Þ

whereNisthenumberofmodestobeconsideredforthebridge.UðxÞrepresentsthemode-shapematrix

andgðtÞisthemodal-responsevector.LetvsðtÞ¼zsðtÞÀyðxs;tÞbethePTMDstroke(i.e.,thePTMDdisplacementrelativetothebridgewherethePTMDislocated).SubstitutingEq.(6)intoEq.(1),andpre-multiplyingUðxÞandintegratingfrom0toLateachsideofEq.(1),itbecomes

ZL! ZL! ZL!

€ðtÞþ_ðtÞþ󰀁ðxÞUTðxÞdxgUðxÞmUðxÞCyðxÞUTðxÞdxgU00ðxÞEIðxÞU00TðxÞdxgðtÞ

0

0

NvXk¼1

0

¼À_sþksvsÞpkU½vðtÀtkÞ󰀇Hðt;tkÞþUðxsÞðcsvð7aÞ

orinmatrixformas

€ðtÞþCbg_ðtÞþKbgðtÞ¼FbðtÞþUðxsÞðcsv_sþksvsÞMbg

ð7bÞ

whereMb,CbandKbareNbyNmatricesrepresentingthemodalmass,dampingandstiffnessmatricesof

thebridgeverticalmotions.ItiswellknownthatthesematricesbecomediagonalaftertheorthogonalityofUðxÞisapplied.Therefore,Eq.(7)canbedecoupledintoNnumberofmodesandthejthmodalequationofmotionisexpressedas

€jðtÞþcbg_jðtÞþkbgjðtÞ¼FbjðtÞþ/jðxsÞðcsv_sþksvsÞmbgwhere

mbj¼and

FbjðtÞ¼À

NvXk¼1

ð8aÞ

Z

0

Z

0

L

󰀁ðxÞ/2mjðxÞdx;

cbj¼

Z

0

L

CyðxÞ/2jðxÞdx;

L

kbj¼

EIðxÞ½/00jðxÞ󰀇dx

2

ð8bÞ

pk/jðvtÀvtkÞHðt;tkÞ

ð8cÞ

Moreover,thecoordinateinEq.(4)canberearrangedintothePTMDstrokeas

_sðtÞþksvsðtÞ¼Àms€yðxs;tÞms€vsðtÞþcsvSubstitutingEq.(6)intoEq.(9),itbecomes

€ðtÞ_sðtÞþksvsðtÞ¼ÀmsUTðxsÞgms€vsðtÞþcsv

ð10Þð9Þ

CombiningEqs.(7b)and(10),thecoupledequationsofmotioninmodalspacearegiveninmatrixformas23

'' ' '&!&!&Mb0&

_€gðtÞgðtÞÀcUðxÞÀkUðxÞðtÞCKFgðtÞbssbssb4msUTðxsÞms5þTþT¼_sðtÞ€vvsðtÞ000csksvsðtÞ

ð11Þ

where0representsazerovector.Inthisstudy,directnumericalintegrationisappliedtosolveEq.(11),and

theverticaldisplacementresponseofthebridge,yðx;tÞ,isthencalculatedfromEq.(6).

472J.F.Wangetal./InternationalJournalofSolidsandStructures40(2003)465–491

3.Dynamiccharacteristicsofbridgeandtrain

Itisnotedthatatrainisgenerallycomposedofaseriesofmovingcarswiththesameproperties.Suchanexcitationactslikeasteadycontinuousimpactuponabridgewhenthetrainismovingataconstantspeed.Thevibrationfrequencyofthebridgewouldthenbestationaryataspecifiedvalue.Thissituationisdif-ferentfromthatofahighwaybridgesubjectedtosinglecarloadinashorttime.Therefore,itisinterestingtounderstandthepropertiesoftrainloadssothatthedynamiccharacteristicsofthebridgesundertrainloadscanbepredicted.3.1.Resonantspeeds

Assumingthatabridgeisconstructedwithsimplesupports.ItÕsmode-shapefunction,UðxÞ,canbeexpressedas

ÈÉjpxx2px

UðxÞ¼sinpð12ÞsinÁÁÁsinÁÁÁLLLWiththisdefinition,thespectrumofthejthmodaltrainload,FbjðxÞ,couldbefoundthroughtakingtheFouriertransformofFbjðtÞas

)Z1Z1(XNv

11

FbjðxÞ¼FbjðtÞeÀixtdt¼pk/jðvtÀvtkÞHðt;tkÞdtÀ

2pÀ12pÀ1

k¼1

&' !ZNv1

1XjpvðtÀtkÞ¼Àpksinð13ÞHðt;tkÞdt

2pk¼1LÀ1Asstatedearlier,iftheinteractivesystemmodelwasemployed,thevalueofpkwouldberelatedtoboththe

bridgeandthevehicleresponses.Inordertoexplorethetrainloadanalytically,assumingthateachmovingloadhasthesamemagnitude(p1¼p2¼ÁÁÁ¼pk¼p0)andsamespacingd,thentheanalyticalexpressionofFbjðxÞforasimplysupportedbridgecanbederivedas

\"ÀxdÁ#p0pvjxd1sin2vNvÀ2vjx2LÀxdÁFbjðxÞ¼ð14ÞþÁ½ðÀ1ÞÁeÀixL=vÀ1󰀇ÀjpvÁ22sin2v1À

xL

ItwasfoundthatthevalueofFbjðxÞwouldbelargeassinðxd=2vÞ%0orasx¼2pnv=d(n¼1;2;3;...),

wherev=distheimpactfrequencyofthewheelloadstothebridge.Forinstance,foraseriesofmovingforceswithd¼25mandfiftyloadspassinga28.4mbridgeataspeedofv¼82km=h,themagnitudeofthespectrumofthefirstmodalforce,jFb1ðxÞj,isillustratedinFig.2(a).Itisobviouslyseenthatthereexistspeakswhentheexcitationfrequency,x,isequalto2pv=d¼ð0:91HzÞ,4pv=dð¼1:82HzÞ,6pv=dð¼2:73HzÞ;...;etc.,asexpectedforNv¼50.Inthiscondition,ifthemodalfrequencyofthebridge,xj,isclosetothemultipleoftheimpactfrequencyofthemovingloads,theresonanteffectwilloccur.Inotherwords,thebridgewillundergoresonantresponseswhen

v¼vc¼

xjd2np

ð15Þ

wheren¼1;2;3;...Thesespeedsarecalledtheresonantspeeds.InEq.(14),itisalsoshownthattheresonantconditionoccursnotonlyathightrainspeedbutalsoatmediumtrainspeeds.ExaminationofEq.(14)indicatesthatothercriticalconditionsoccurwhen1Àðjpv=xLÞ2¼0orx¼npv=L.Thatis,

v¼

xjLnp

ð16Þ

J.F.Wangetal./InternationalJournalofSolidsandStructures40(2003)465–491473

Fig.2.Fouriertransformofthefirstmodaltrainloadwithvariousnumbersofloads:(a)fiftyloads;(b)oneload.ComparedwithEq.(15),onecanobservethatthemajorcriticalcondition(forn¼1)isonlyencounteredwhenthetrainspeedisseveraltimesthefirstresonantspeed.ThisseemstobeimpossibleforgeneralHSRbridges.ThesefindingsarethesameasthosefoundbyYangetal.(1997).TheconditionsinFig.2(b)aresameasthoseinFig.2(a)exceptthatthenumberofmovingloadsisone.Apparently,theredoesnotexistanyobviousexcitationfrequency.Likeasinglecarpassingthroughabridge,noresonantresponsewillbeproduced.

3.2.Influenceoftrainmodels

FromEq.(15),itisseenthattheresonantspeedsofatrainaredependentupontwofactors,themodalfrequenciesofthebridge,xj,andtheloadspacingofthetrain,d.Asstatedearlier,theinteractionbetweenthebridgeandtrainwillexistasprecisetrainmodelsareapplied.Forexample,applyingamovingmass

474J.F.Wangetal./InternationalJournalofSolidsandStructures40(2003)465–491

modelforthetrain(usingEqs.(5b)and(6))andwithouttheinstallationofPTMD,theequationofmotionofthebridge,Eq.(7),becomes

€ðtÞþCbg_ðtÞþKbgðtÞ¼ÀMbgor

½Mbþ

€ðtÞM0bðtÞ󰀇g

_ðtÞþKbgðtÞ¼ÀþCbg

NvXk¼1

NvXk¼1

no

T

€ðtÞU½vðtÀtkÞ󰀇Hðt;tkÞmvgþU½vðtÀtkÞ󰀇g

ð17aÞ

mvgU½vðtÀtkÞ󰀇Hðt;tkÞð17bÞ

PNv

T

whereM0bðtÞ¼k¼1mvU½vðtÀtkÞ󰀇U½vðtÀtkÞ󰀇Hðt;tkÞ.Itisapparentthatthemassmatrixofthebridgesystemwillbealteredbecauseoftheadditionofthetrain.Moreover,thevariationinthemassvarieswithtime.Thismeansthatthemodalfrequencyofthebridgewillchangewithtimeduringthepassageofthetrain.Fig.3showsthetimevariationoffirstmodalfrequencyofthebridge,B2(whosepropertiesareshowninTable1),subjectedtoaFrenchT.G.V.train(whosepropertiesareshowninTable2)withcertainspeedsimulatedasamovingmassmodel.Sincetheoverallmassofthebridgesystemincreasesasthetrainmovesontothebridge,thesystemnaturalfrequenciesaresmallerthantheiroriginalvalues.Moreover,thevariationofthefirstmodalfrequencyatdifferenttimeissmallandalmoststableatanaveragevalue,ðx1Þavg,asthetrainsteadilyactsonthebridge.Thisphenomenonwillleadtoachangeintheresonanttrainspeeds.Forvarioustrainspeeds,thecorrespondingvaluesofðx1Þavg(solidline)areillustratedinFig.4.Itisshownthatthegreatertrainspeed,thesmallerðx1Þavg,buttheirdifferencesareverysmall.AccordingtoEq.(15),thelinerelatingthefirstmodalfrequencyofthebridge-trainsystemandthemainresonanttrainspeed(j¼1andn¼1)isalsoillustratedinFig.4.Thislineintersectsðx1Þ0andðx1Þavgattwopoints,whichabscissasdenotetheresonanttrainspeeds(vcandv0c)formovingforceandmovingmasstrainmodels,respectively.Itisseenthatv0cissmallerthanvc,becauseofðx1Þavg<ðx1Þ0.But,theirdifferencesareabout2.5%.Itindicatesthattheapplicationofdifferenttrainmodelsdoesnotleadtosignificantchangeofresonanttrainspeeds.

Fig.3.TimevariationofthefirstmodalfrequencyofthebridgeB2underaFrenchT.G.V.trainsimulatedasmovingmassmodel;ðx1Þ0:theoriginalvalue;ðx1Þavg:theaveragevalueasthetrainsteadilytravelsoverbridge.J.F.Wangetal./InternationalJournalofSolidsandStructures40(2003)465–491

Table1

PropertiesoftwoTHSRbridgesPropertiesE(t=m2)I(m4)m󰀁(t=m)nj(%)x1(Hz)

BridgeB1(L¼30m)2:83Â1067.8441.742.54.64

BridgeB2(L¼40m)2:87Â10617.9038.242.53.56

475

Table2

PropertiesofGermanI.C.E.,JapaneseS.K.S.andFrenchT.G.V.HSRtrainsTrainpropertiesNo.ofbogie

Carspacing,d(m)ðvcÞn¼1(km/h)BridgeðvcÞn¼2(km/h)BridgeðvcÞn¼1(km/h)BridgeðvcÞn¼2(km/h)Bridgemv(kg)cv(Ns/m)kv(N/m)mb(kg)Ib(kgm2)cb(Ns/m)kb(N/m)mw(kg)lw(m)

I.C.E.5226.444022033816927,00022,700660,0003000400078,4002,360,00018001.5

S.K.S.6425.041820932016020,87590,200530,0003040393078,4002,360,00017801.25

T.G.V.5218.731115624012027,00096,700664,0003000400078,4002,360,00018001.5

B1B1B2B2

Fig.4.AveragefirstmodalfrequencyofthebridgesystemversustrainspeedandthechangeofresonanttrainspeedforaT.G.V.traintravelingoverbridgeB2.476J.F.Wangetal./InternationalJournalofSolidsandStructures40(2003)465–491

4.Optimalpassivetunedmassdampers

ThePTMDconceptdatesbackto1909(Frahm,1911).Sincethen,muchresearchhasbeencarriedouttoexamineitseffectivenessforvariousdynamicloadapplications.Thisdeviceisinstalledtoabsorbtheenergytransmittedfromtheprimarystructurebytuningitsfrequencytothestructuralfrequencywithoutaddi-tionalpower.Thesuccessofsuchasysteminreducingwind-excitedstructuralresponsesisnowwelles-tablished.However,becauseofthecomplexcharacteristicsofearthquakeexcitations,therestillhasnotbeenageneralagreementontheeffectivenessofPTMDsystemsinsuppressingseismicresponses.ThePTMDeffectivenessforsuppressingtrafficresponseshasalsonotbeenclearlyinvestigatedandconfirmed.4.1.DesignphilosophyandcontroleffectivenessofPTMD

ThedesignphilosophyforPTMDgenerallyistoalterthecharacteristicsoftheprimarystructuresothatthesystemisapplicabletothechangesinexternalexcitations.Inordertounderstandthisprinciple,onecanusethetransferfunctiontoexaminethePTMD.Fig.5showsthetypicaltransferfunctionofthespecifiedmodeofastructurewithandwithoutPTMD,designedforseismicresponsecontrolbasedontheopti-mizationprocedurebyLinetal.(1994).ItisseenthatthesetwocurvesintersectatpointsPandQ.ItisknownthatthepositionsofPandQvarywiththePTMDÕsnaturalfrequency,xs,andtheirelevationsareindependentofthePTMDÕsdampingratio,ns.ItisalsoseenthatthetransferfunctionofastructurewithPTMDdecreasesinthefrequenciesbetweenPandQ(operatingrange),butdoesnotreduceormayevenamplifyintheotherfrequencies.Thus,itisexpectedthatthePTMDwillnotproducevibrationreductionunlessthefrequencycontentoftheexcitationsiswithintheoperatingrange.Otherwise,thePTMDwillbeineffective.Thisphenomenoncanbeverifiedinthetimedomainusingtwos.d.o.f.structuresA(xp¼1:13Hz,np¼2%)andB(xp¼3:0Hz,np¼2%)subjectedtothehorizontalearthquakeaccelerationrecordedatthecampusofNationalChung-HsingUniversity(calledNCHUrecord)duringthe1999TaiwanChi-Chiearthquake.Fig.6showstheFourieramplitudespectrumoftheNCHUearthquakerecordandtheope-ratingrangesofstructuresAandB.Itisclearlyseenthatthemajorfrequencycontentoftheexciting

Fig.5.TypicaltransferfunctionofastructurewithandwithoutoptimalPTMD.J.F.Wangetal./InternationalJournalofSolidsandStructures40(2003)465–491477

Fig.6.FourieramplitudespectrumoftheNCHUrecordduring1999TaiwanChi-ChiearthquakeandthePTMDoperatingrangeforstructuresAandB.earthquakefallsintheoperatingrangeofstructureA,butnotinthatofstructureB.Thestructuraldis-placementtimehistoriesareillustratedinFig.7(a)and(b)whichshowthatthePTMDperformanceisexcellentforstructureA,butisnotgoodforstructureB,asexpectedinthefrequencydomain.ThisresultimpliesthatthePTMDeffectivenessishighlyfrequency-dependent.Itoperatesefficientlyonlyunderres-onantconditions.Inpreviousresearches,PTMDwasintuitivelythoughttobeanineffectiveseismiccontroldevicebecauseofthisphenomenon.However,althoughPTMDislessusefuloutsideresonantconditions,thestructuralresponseissmallinthissituation.Therefore,thePTMDisausefulvibrationcontroldeviceinreducingthe‘‘excessive’’responses,whichoccurduringresonantconditions.AsstatedinSection3.1,itisknownthatthespectrumoftrainloadsisanarrow-bandedandwellseparatedlikeasimpleharmonicforce,andtheresonantpeaksoccurwhenthetrainistravelingatresonantspeeds.Therefore,thePTMDcanbeexpectedtobeabletoreducethebridgeresponsesundertheseconditions.4.2.OptimalPTMDparametersforsimplysupportedbridges

Aftertheequationsofmotionforthebridge-PTMDsystemareestablished,thePTMDparametersaredeterminedsothatthebridgeresponsesaresmallerthanthosewithouttheinstallationofPTMD.Sincethetrainfactors,suchasspeed,weight,axialdistance,etc.,areallunknown,itisusuallynotdesiredtoincludetheseuncertaintiesintothePTMDparameterdetermination.ThemaingoalindesigningasinglePTMDistoalteronlythedynamiccharacteristicsofthebridge.

SinceasinglePTMDsystemhasasinglenaturalfrequency,onlyonevibrationmodeofthebridgecanbecontrolled.AccordingtoEq.(11),selectingthejthmodeofthebridgeascontrolledmodeandcombiningitwiththePTMD,theequationofmotionforthesystemcouldbegiveninthematrixformas

'' ' '& !&!&!&

mbj0€jðtÞ_jðtÞcbjÀcs/jðxsÞkbjÀks/jðxsÞFbjðtÞgggjðtÞ

þþ¼

ms/jðxsÞms€_sðtÞvsðtÞ0csv0ksvsðtÞ0

ð18Þ

478J.F.Wangetal./InternationalJournalofSolidsandStructures40(2003)465–491

Fig.7.RelativedisplacementtimehistoryofstructuresAandBwithandwithoutPTMDundertheNCHUrecordof1999TaiwanChi-Chiearthquake.Dividingthefirstandsecondrowsbymbjandmsrespectively,Eq.(18)becomes

!&!&!&' ' '&'

10€jðtÞ_jðtÞgðtÞgg2njxjÀ2nsxsusj1uxjÀx2jssj

þþ¼FðtÞ=mbj

2/jðxsÞ1€_sðtÞvsðtÞvvsðtÞ02nsxs0bj0xs

ð19aÞ

or

€ðtÞþCZ_ðtÞþKZðtÞ¼CFðtÞ;MZ

ð19bÞ

wherensandxsrepresentthedampingratioandnaturalfrequencyofthePTMD,respectively,whereas

lsj¼/jðxsÞms=mbjmeansthemodalmassratioofthePTMD.TakingtheFouriertransformofEq.(19)yields

&'HðxÞÀ1gjF

ZðxÞ¼ðÀx2MþixCþKÞCFðxÞ¼AÀ1CFðxÞ¼FðxÞð20Þ

HvsFðxÞ

J.F.Wangetal./InternationalJournalofSolidsandStructures40(2003)465–491479

where

Àx2þixð2njxjÞþx2j

A¼2

Àx/jðxsÞ

Àixð2nsxslsjÞÀx2slsjÀx2þixð2nsxsÞþx2s

!

ThefunctionsHgjFðxÞandHvsFðxÞrepresentthetransferfunctionsofthejthmodaldisplacementofthe

bridgeandthePTMDstrokecorrespondingtothemodaltrainloadFðxÞ,respectively.Whenatraintravelsataconstantspeed,themainexcitationfrequencyisusuallynarrowbanded,asshowninFig.2(a).Therefore,thesuppressionofpeakvalueofthetransferfunctionisimportantratherthantheentireareaofthetransferfunction.ConsideringthecasesofabridgewithandwithouttheinstallationofPTMD,aresponseratioisdefinedas

Rdvj¼

peakvalueofjHgjFðxÞjPTMDpeakvalueofjHgjFðxÞjNOPTMD

ð21Þ

toevaluatethevibrationcontroleffectivenessofPTMD.TheindexRdvjisafunctionofnj,/jðxsÞ(bridgeparameters),lsj,nsandrfj(PTMDparameters),whererfj¼xs=xjisthemodalfrequencyratio.Forgivenvaluesofnjand/jðxsÞ,theoptimalPTMDparameterscanbeobtainedbydifferentiatingRdvjwithrespecttorfj,nsandlsj,andequatingtozero,respectively,tominimizeRdvj.Thesevaluesmaybefoundbysolvingthefollowingequationssimultaneously

oRdvj

¼0;olsj

oRdvj

¼0;ons

oRdvj

¼0orfj

ð22Þ

Ingeneral,thebridgeparametersaredeterminedthroughotherapproaches,forexample,bysystemidentificationtechniquesforexistingbridges.Theoptimallocation,xs,fortheplacementofPTMDisattheplacewherethecontrolledmode-shapevalueismaximal(Linetal.,1994),thatis,atthemiddleofaspan(i.e.,xs¼0:5L)forsimplysupportedbridgeswhenthefirstmodeistobecontrolled.Meanwhile,inpractice,themodalmassratio,lsj,isgenerallyassignedbyconsideringbotheconomyandthebridgecapacity,andfinally,theoptimalsetofðrfj;nsÞarefoundthroughsolvingtheremainingtwoequationsin(22).

5.Numericalverifications

Basedonthetheoreticalderivationsdescribedpreviously,somenumericalinvestigationsareperformedinthissection.ThebridgesB1andB2whosepropertieslistedinTable1,whichareproposedforTHSR(Lin,1993),areusedforverifyingthePTMDcontroleffectiveness.BridgeB1representsageneralbridgewithalengthof30m,whereasBridgeB2isamediumlongbridgewithalengthof40m,asshowninFig.8.BothbridgesareconstructedassimplysupportedbridgeswithaconstantcrosssectionareaofboxgirderinsidewhichasinglePTMDistobeinstalled.Threevarioustrains,FrenchT.G.V.,GermanI.C.E.,andJapaneseS.K.S.,showninFig.9,areusedastheexternalmovingloadsactingonthebridges.AccordingtotheTHSRdesignspecifications,theverticaldisplacement,accelerationandendrotationofthebridge,andtheverticalaccelerationofthetrainshouldbelimitedtoassurethesafetyandthecomfortofpassengers.ThereductionsofthefourdynamicresponsesofthebridgeduetotheinstallationofproposedoptimalPTMDwillbeclearlyillustratedinthissection.Inaddition,thePTMDdetuningeffect,resultingfromtheinteractionbetweenthevehicleandthebridge,onthePTMDcontroleffectivenesswillalsobeextensivelyexamined.

J.F.Wangetal./InternationalJournalofSolidsandStructures40(2003)465–491481

5.1.PTMDcontroleffectivenessforTHSRbridgesundertrainloads

Sincethespacebetweentwoaxles(3m)isnottoosmallcomparedwiththelengthofB1(30m),twoloadsfortwoaxlesaremodeledtocalculatethedynamicresponsesofbridge-PTMDsystemandFrenchT.G.V.,GermanI.C.E.,andJapaneseS.K.S.trains.Fig.10illustratesthetrainloadmodelwhichconsistsofamass-spring-dashpotsystem(mv;kv;cv)torepresentonehalfofatraincar,onebogiesystemwithtwod.o.f.s.(zb;hb),andtwowheelsets(mw).TheirphysicalpropertiesusedinthispaperarelistedinTable2.AccordingtothecarspacingofeachtrainandthemodalfrequenciesofbridgesB1andB2,thecorre-spondingresonanttrainspeeds(j¼1andn¼1,2)foreachbridgearecalculatedandalsoshowninTable2.Itisfoundthatthemainresonanttrainspeeds(j¼1andn¼1)forbridgeB1arelargerthanthedesigntrainspeedof350km/hexceptfortheFrenchT.G.V.sincebridgeB1ismorerigidthanbridgeB2.ThePTMDisdesignedtocontrolthefirst-moderesponsethatdominatesthetotalresponsesofbridgeandtrainsystemsandisinstalledatthemiddleofthebridgewherethemode-shapevalueismaximal.Themodalmassratiols1isselectedtobe0.5%andthecorrespondingoptimalPTMDparametersbasedontheoptimaldesignproceduresandcriteriainSection4arecalculatedandlistedinTable3.ToinvestigatethePTMDcontroleffectiveness,themaximalvaluesofthefourdynamicresponses,whichwerementionedpreviously,forbridgeswithandwithoutPTMDwithinthedesigntrainspeed(350km/h)arepresentedinTables4and5.BecausethemainresonanttrainspeedsforbridgeB1arelargerthanthedesignspeed,themaximalbridgeresponsesarenotdominantbytheresonantresponsesthatthePTMDisexpectedtosuppress.Therefore,theresponsereductionsforbridgeB1andthetrainsduetothePTMDinstallationarequitesmall.Moreover,partsofthedynamicresponsesareevenamplifiedforthecasesoftheI.C.E.andS.K.S.trainsasweexpected.However,itisseenthattheresponsevaluesshowninTable4areverysmallandfarfromthedesignlimitations.Inthissituation,thevibrationcontroldevicesareactuallynotnecessary.

Fig.10.Illustrationoftrainloadmodel.Table3

OptimalPTMDparametersfortwoTHSRbridgesPTMDparametersForbridgeB1ForbridgeB2

Mass(kg)ms(ls1¼0:5%)58027648

Damping(Ns/m)cs(ns¼5%)1690416978

Stiffness(N/m)ks(rf¼0:991)4:84Â1063:77Â106

482J.F.Wangetal./InternationalJournalofSolidsandStructures40(2003)465–491

Table4

MaximalresponsesoftrainsandbridgeB1withandwithoutPTMDTraintype

PTMDcondition

Maximalverticaldisplacementatmidspan(cm)0.2000.200(0.0%)0.1660.167()0.6%)0.1680.163(3.0%)61.875(L/1600)

Maximalverticalaccelerationatmidspan(g)0.0800.076(5.0%)0.0740.072(2.7%)0.0520.048(7.7%)60.35

Maximalrotationatrightend(10À4radian)2.262.27()0.4%)1.871.90()1.6%)1.981.93(2.5%)65.5

Maximalverticalaccelerationatbogie(g)0.0050.005(0.0%)0.0100.010(0.0%)0.0070.007(0.0%)60.05

I.C.E.

WithoutPTMDWithPTMDWithoutPTMDWithPTMDWithoutPTMDWithPTMD

S.K.S.

T.G.V.

Designrequirements

NumberinbracketÔ()Õ:reductionpercentage.

Table5

MaximalresponsesoftrainsandbridgeB2withandwithoutPTMDTraintype

PTMDcondition

Maximalverticaldisplacementatmidspan(cm)0.2550.247(3.1%)0.2270.189(16.7%)0.3620.281(22.4%)61.905(L/2100)

Maximalverticalaccelerationatmidspan(g)0.0450.041(8.9%)0.0520.034(34.6%)0.0950.055(42.1%)60.35

Maximalrotationatrightend(10À4radian)2.022.05()1.5%)1.781.50(15.7%)2.972.32(21.9%)63.8

Maximalverticalaccelerationatbogie(g)0.0060.006(0.0%)0.0120.008(33.3%)0.0170.011(32.3%)60.05

I.C.E.

WithoutPTMDWithPTMDWithoutPTMDWithPTMDWithoutPTMDWithPTMD

S.K.S.

T.G.V.

Designrequirements

NumberinbracketÔ()Õ:reductionpercentage.

ComparedwithbridgeB1,thePTMDisshowntobemoreeffectiveinreducingthedynamicresponsesofbridgeB2sincetheresonanttrainspeedsareallbelowthedesigntrainspeed.Thefourpeakresponsecurvesundertrainswithspeedsrangingfrom0to350km/hforbridgeB2areillustratedinFigs.11–14.ThesefiguresprovethattheresonantresponsesoccurneartheresonanttrainspeedslistedinTable2.Meanwhile,theresonantresponsesexcitedbytheT.G.V.trainaremoreapparentthanthosebyothertrains.ThisisduetothefactthatthebridgearrangementfortheT.G.V.trainismoreregular.ThePTMDismoreeffectiveinreducingthemaximaldynamicresponsesexcitedbytheT.G.V.train.AsshowninTable5,thereductionsareabout25%and40%formaximaldisplacementandaccelerationrespectively.ThetimehistoriesofverticaldisplacementandverticalaccelerationofbridgeB2,andthemaximalverticalaccelerationofT.G.V.trainwithaspeedof240km/h,areillustratedinFigs.15–17.Itisobviousthatthepeakandoverallresponsesareallreduced.ThetimehistoryofthePTMDstrokeisalsoplottedinFig.18.Itisseenthatthe

J.F.Wangetal./InternationalJournalofSolidsandStructures40(2003)465–491483

Fig.11.MaximaldisplacementsofbridgeB2withandwithoutPTMDatvarioustrainspeeds.484J.F.Wangetal./InternationalJournalofSolidsandStructures40(2003)465–491

Fig.12.MaximalaccelerationsofbridgeB2withandwithoutPTMDatvarioustrainspeeds.J.F.Wangetal./InternationalJournalofSolidsandStructures40(2003)465–491485

Fig.13.MaximalrotationsofbridgeB2withandwithoutPTMDatvarioustrainspeeds.486J.F.Wangetal./InternationalJournalofSolidsandStructures40(2003)465–491

Fig.14.MaximalaccelerationsoftrainspassingoverbridgeB2withandwithoutPTMDatvarioustrainspeeds.J.F.Wangetal./InternationalJournalofSolidsandStructures40(2003)465–491487

Fig.15.TimehistoryofthedisplacementofbridgeB2atthemidspanunderaT.G.V.trainwithconstantspeedof240km/h.Fig.16.TimehistoryofaccelerationofbridgeB2atthemidspanunderaT.G.V.trainwithconstantspeedof240km/h.peakresponseisonly0.81cm,muchsmallerthantheinnerdepthofboxgirder.ThereisenoughspacefortheinstallationandmovementofPTMD.

488J.F.Wangetal./InternationalJournalofSolidsandStructures40(2003)465–491

Fig.17.TimehistoryofmaximalaccelerationofaT.G.V.trainwithconstantspeedof240km/h.Fig.18.TimehistoryofPTMDÕsstrokeforbridgeB2underaT.G.V.trainwithconstantspeedof240km/h.5.2.PTMDdetuningeffect

ThevibrationsuppressionisachievedthroughtuningPTMDfrequencytothevicinityofthecontrolledstructurefrequency.Asstatedpreviously,knowledgeofthecontrolledprimarystructureisrequiredin

J.F.Wangetal./InternationalJournalofSolidsandStructures40(2003)465–491489

Fig.19.MaximalverticaldisplacementofbridgeB2withoutandwithtwotypesofPTMDs(PTMDÃ:consideringthevehiclemodeleffect).ordertoaccuratelycalculatetheoptimalPTMDparameters.Frompreviousstudies,itisgenerallyrec-ognizedthatthestructuralnaturalfrequencyestimationerrorwillsignificantlyaffectthePTMDdesignaswellasitscontroleffectiveness.ThisiscalledthedetuningeffectbecausethePTMDdoesnottunetotherightfrequency.InadditiontothePTMDstructuralpropertyestimationandfabricationerrors,thetime-variantcharacteristicsofthecombinedsystemmayalsoleadtothePTMDdetuningeffect.Thus,variousdynamicinteractionsbetweenthebridgeandtrain,resultingfromvariousvehiclemodels,willaffectthePTMDcontroleffectiveness.SincethePTMDisalinearsystem,examiningtheinfluenceofsuchtime-variantphenomenaonthePTMDcontroleffectivenesswillensureitsreliabilityandapplicability.

Toinvestigatetheproblemdescribedabove,astudyforbridgeB2subjectedtotheT.G.V.trainsim-ulatedasmovingmassmodelisperformed.Fig.19showsthemaximalverticaldisplacementofthebridgewithoutandwithtwotypesofPTMDsunderthetrainwithspeedsrangingfrom100to350km/h.ThesymbolizedlineisthebridgewiththePTMDdesignedbasedontheoriginalnaturalfrequencyofthebridge.Itcanbeseenthattwolocalpeaksofthecurvesarenotbalanced,whichisduetothePTMDdetuningeffect.Thesolidlineisthesamecase,butconsideringthetrainmodeleffect.ThePTMDÃshowninFig.19isdesignedbasedontheaveragenaturalfrequencyofthebridge–trainsystem,asillustratedinFigs.3and4.Obviously,thePTMDÃismoreeffectiveinreducingthetransientresponse.Thebridgemaximaldisplacementreductionincreasesfrom21%to25%.However,thisvariationisrelativelysmall.Theinflu-enceoftheinteractioneffectbetweenthebridgeandtrainonthePTMDcontroleffectivenessisnotsig-nificant.

6.Conclusions

APTMDisausefulvibrationcontroldevicethatdissipatesthestructuralvibrationenergywhentheexternalexcitationisresonantwiththestructure.Forwindandearthquakeloads,thefrequencycontentis

490J.F.Wangetal./InternationalJournalofSolidsandStructures40(2003)465–491

wide-bandandusuallycoversthestructuraldominantfrequencies.Therefore,thePTMDhasgoodcontrolperformance.Forahigh-speedrailwaytrainpassingoverabridgewithconstantspeed,itsinducedexci-tationfrequencycontenthasnarrowbandwidthandcouldbefarawayfromthebridgeÕsnaturalfre-quencies.Thus,aPTMDhasgoodcontrolefficiencyonlywhenthetraintravelsatresonantspeeds.

Thestudyofthetheoreticalandnumericalsimulationresultsinprevioussectionsindicatesthatthefollowingconclusionsmaybedrawn:

(1)Whennaturalfrequenciesofthebridgearemultiplesoftheimpactfrequencyofatrain,theresonant

effectwilloccur,eventhoughthetrainistravelingatmoderatespeeds(lessthan200km/h).

(2)Ifthemaximumdynamicresponsesofthebridgeandtrainaredominatedbytheresonantresponse

withinthedesigntrainspeed,thePTMDhasgoodvibrationcontrolperformance.Thenumericalver-ificationsresultsforamoderatelylongbridge(bridgeB2)showthatthePTMDcansuppressthever-ticaldisplacements,verticalaccelerations,andendrotationsofthebridge,andtheverticaltrainaccelerations.Amongthem,themaximumverticalaccelerationofthebridgecanbereduceduptoabout40%foronly0.5%PTMDmassratio.Inaddition,thePTMDstrokeissmallsothatitcanbeinstalledinsidetheinnerHSRboxgirder.

(3)TheinteractionbetweenthebridgeandtrainwhichleadstothePTMDdetuningeffectwillaffectthe

PTMDcontroleffectiveness.AprecisetrainmodelmustbeconsideredincalculatingtheaccuratePTMDparameters.However,theinfluenceoftheinteractioneffectonthePTMDperformanceisnotsignificant.

Acknowledgements

ThisworkwassupportedbyNationalScienceCounciloftheRepublicofChinaundergrantNSC86-2221-E-005-004.Thissupportisgreatlyappreciated.Theauthorswouldliketothankthereviewersfortheirconstructivecommentsthatimprovedthequalityofthepaper.

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