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NonlinearNonlin

    Nonlinear

JournalofZhejiangUniversitySCIENCEA

    ;ISSN1673565X(Print);ISSN1862?1775(Online)

    ;www,zju.educn/~us:www,springedink.com

    ;E-mail:jZUS@ZjU.edu.cn

    ;Xieeta1./JZhejiangUnivSciA20089f9J:11931200

    ;Nonlineardynamicresponseofstaycables

    ;underaxialharmonicexcitation

    ;XuXIE

    ;,HeZHANC~ZhichengZHANG

    ;(DepartmentofCivilEngineering,ZhejiangUniversity,Hangzhou31002ZChina) ;Email:xiexu@zju.edu.cn

    ;ReceivedDec.19,2007;revisionacceptedMar.28,2008

    ;1193

    ;Abstract:ThisPaDerproposesanewnumericalsimulationmethodforanalyzingtheparametricvibrationofstaycablesbasedon

    ;thetheoryofnonlineardynamicresponseofstructuresundertheasynchronoussupportexcitation.Theeffectsofimportantpa

    ;rametersrelatedtoparametricvibrationofcables,i…echaracteristicsofstructure,excitationfrequency,excitationamplitude,

    ;dampingeffectoftheairandtheviscousdampingcoefficientofthecables.wereinvestigatedbyusingtheproposedmethodforthe

    ;cableswithsignificantlengthdifferenceasexamples.Theanalysisresultsshowthatnonlinearfiniteelementmethodisapowerful

    ;techniqueinanalyzingtheparametricvibrationofcables.thebehaviorofparametricvibrationofthetwocableswithdifferentIrvine

    ;parametershassimilarproperties,theamplitudesofparametricvibrationofcablesarerelatedtothefrequencyandamplitudeof

    ;harmonicsupportexcitationsandtheeffectofdistributedviscousdampingonparametricvibrationofthecablesisverysmal1.

    ;Keywords:Staycables,Parametricvibration,Nonlinearvibration,Viscousdamping ;doi:10.1631/jzus.A0720132Documentcode:ACLCnumber:U448.27

    ;INTRODUCTION

    ;Thevibrationofcableinacablestayedbridge

    ;resultsinfatigueofthecableanchorage.shortensthe

    ;service1ifeofcablestayedbridges.significantly

    ;affectsthesmoothnessofthemovingvehiclesandthe

    ;safetyofthestructures.ThestrongvibrationOfstav

    ;cablesandtheirsevereconsequenceswereobserved

    ;inmanybridges(LilienandPinto.1994;Chenand

    ;SunetaL,2003;YangandChen,2005).Therefore, ;vibrationcontro1Ofstavcablesisaveryimportant ;issueinthedesignoflongspancablestayedbridges.

    ;Theparametricvibrationofstaycablesisa ;typicalphenomenonforcablestayedbridges.It

    ;happenswhentheexcitationfrequencyisaninteger ;multiplierofthecablefrequencies,andtheperiodical ;varyingcabletensionwil1inducelargeamplitude ;vibrationsofthecables.

    ;Parametricvibrationofcableswasobservedlong ;Project(No.50578141)supportedbytheNationalNaturalScience

    ;FoundationofChina

    ;timeago.However,researchonthistopicstarted ;mainlyinthe1980s.TalahashiandKonishi(1987a;

    ;1987b)proposedamethodf0rcalculatingthenonlin

    ;earfreevibrationofcablesbasedontheGalerkin ;methodandHarmonicWaveEquilibriummethod. ;andtheyalsousedtheeigenvaluemethodtoestimate ;therangeofparametersthatmayinducepotential ;out.of-planeunstablevibrationofcablesdueto

    planeharmonic1oading.Furthermore.Talahashi ;in

    ;(1991)analyzedtheregionofparametersthatresultin ;unstableparametricvibrationofcablesduetoperi- ;odicallyvaryingaxialforces.wuefa1.(2003)studied ;therelationshipbetweenfrequenciesofacablestayed

    ;bridgeandthoseOfstaVcables,thedynamicre

    ;sponsesofcablesundersimpleharmonicaxialexci

    ;tationandthecharacteristicsofparametricvibration. ;UtilizingtheGalerkinmethod,Uhrig(1993)analyzed ;andproposedaparameterplaneofunstablevibration ;ofthecablesduetotowersupportexcitation.Lilien ;andPintofl994)alsousedtheGalerkinmethodto ;studytheparametricvibrationofcables.Georgakis ;andTaylor(2005a;2005b)studiedthenonlineardy

    ;

    ;1194X/eetal,/dZhefiangUnivSciA20089l9):11931200

    ;namicresponseofcablesusingfiniteelementmethod ;fFEM)andsimulatedthestaycablesbylinkelements. ;TheGalerkinmethodwasalsowidelyusedby ;otherresearchers.YangandChen(2005)investigated ;theparametricvibrationproblemofSutongBridgein ;China.KangandZhongfl998)studiedthemecha

    ;nismsofparametricvibrationofcablesusingatwo

    ;degree.of-freedommodel,andanalyzedthedynamic

    ;characteristicsofparametricvibrationOfstaycables ;usinganexampleofasimplecablestayedbridge.

    ;Cheneta/.(2002)andChenandSunf2003)studied ;thenonlineardynamicresponseofparaboliccable ;duetoaxialexcitationconsideringtheeffectsofin

    ;clinationangle,dampingandsagofcables.Sunet

     ;a/.(2003)derivedtheequationsofparametricvibra

    ;tionofstaycablesduetocablesuppo~excitation,and ;studiedtheeffectsofinclination.initialtension.ex

    ;citationfrequenciesandvibrationcontroltechniquein ;detail.Itisobservedthattheregularitiesofparametric ;vibrationsofstaycablesarenotverysimilaramong ;theresultsobtainedbydifferentresearchers,andthe ;correspondingexplanationsaredifferenteither. ;Amongtheabovementionedresearchesusingthe ;Galerkinmethod,trigonometricfunctionwasem

    ;ployedasthetrialfunction.Comparedwiththe ;GalerkinmethOd,FEMismoreadaptive,sincethe ;structuralstatusatdifferenttimestepsaredetermined ;bytheequilibriumequationswithoutintroducingthe ;assumptionsusedintheGalerkinmethod.Andthe ;FEMismorefeasibleinconsiderationsofwindload ;andtheeffectofdampersinstalledonthecable. ;Thispaperproposesanewmethodforanalyzing ;theparametricvibrationofstaycablesbasedon ;nonlinearvibrationtheory.TheapplicationOfthis ;methodwasdemonstratedbytwoexamplesofstay ;cableswitIldifferentspans.andtheeffectsofdis

    ;tributedviscousdampingofthecables,theexcitation ;frequencyandtheamplitudeofsupportexcitationon ;nonlinearvibrationsofcableswerethoroughlyin

    ;vestigated.

    ;MTH0D

    ;Nonlinearvibrationequationofflexiblecableele. ;ment

    ;Thenonlinearequationofmotionforthefiexible ;cablesmaybewrittenasthefollowingincrement ;formbasedonupdatedLagrangeformulation ;+cA.ic+kAx=Af,(1)

    ;wheremisthemassmatrix,cthedampingmatrix, ;thetangentstiffnessmatrix,Axthedisplacement ;incrementofthecable.aftheloadincrement, ;andarerespectivelythefirstandsecondderiva. ;fivesofAxrelativetotime.

    ;Themass,dampingandtangentstiffnessmatri ;cesforeachelementcanbederivedusing

    ;isoparametricelementmethodandvirmalworkprin ;ciple,andexpressedas ;ke=EAIB~BLds+lTGds, ;m.:历『?Nds,

    ;ce=『?Nds,

    ;whereSisthelengthoftheelement,Tthecableten ;sion,EAthetensilestiffness,andthemassand

    ;viscousdampingperunitlength,respectively.Forthe

    ;elementshowninFig.1,theshapefunctionNand

    ;matricesBLandGcanbewrittenas(Xieeta1.,2008)

    ;and

    ;专『_OsL

    ;

    ;P

    ;G=

    ;?2N34]

    ;?2?3P]/R,

    ;

    ;OA

    ;

    ;ut

    ;R

    ;OAu

    ;oS

    ;OAuAut

    ;R

    ;PJ7,r.

    ;f?l=(1)/2,{?2:l:

    ;,

    ;J?3=(+0/2,

    ;(5)

    ;,

    ;]?,j

    ;00

    ;00

    ;00

    ;00

    ;00

    ;00

    ;00

    ;00

    ;00

;....,...................,..L=

    ;?

    ;

    ;P=[PtPes]=l12

    ;f12

    ;Le31e32

    9):11931200 ;X/eeta/./JZhejiangUnivSciA20089

    ;(7)

    ;wherefisthedimensionlesslocalcoordinatesystem, ;andthecoordinatesofthetwoendsoftheelementare ;-

    ;1and+1.esisthetangentialdirectionvector,enthe ;directionvectornormaltothecurvatureplane,etthe ;directionvectorincurvatureplane;effisthedirec. ;tiona1cosineofloca1coordinate,inglobalcoordinate ;:Risthecurvatureradiusofthecable.

    ;D

    ;.

    ;Z

    ;Fig.1Three-nodeisoparametriccableelement ;ThedragforceFoactingonthecableintroduced ;bytherelativemotionofthecabletotheairisex

    ;pressedas

    ;1,.

    ;.,(8)

    ;where#istheouterdiameterofthecable,cDthedrag ;coefficient,whichisassumedtobeindependentofthe ;attackangle,thevelocityinYdirection,Pthe ;densityofair.

    ;Nonlineardynamicresponseofcablesundersup- ;portexcitation

    ;Cablevibrationduetoarbitrarysupportexcita. ;tioncanbeanalyzedwiththemethodforthecalcula. ;tionofstructuralresponseduetogroundmotionex. ;citation.AsshowninFig.2,thecablewithspanLand ;verticaldistanceYbetweencableendsisexcitedby ;asynchronoussupportmovement.Thedisplacement ;ofstructurecanbeexpressedas

    ;=

    ;(9)

    ;wheresubscripts”a”and’’b”representdisplacement

    ;1195

    ;ofthenonsupportnodeandthesupportnode,re? ;spectively.Then,Eq.(1)canberewrittenas

;[ma][]+ca,Co1F~]

    ;+

    ;=]

    ;

    ;Fig.2Cableundersupportexcitation

    ;(10)

    ;Theequationrelatedtodynamicdisplacement ;canbederivedfromEq.(10)as

    ;m+CaAJca

    ;+Ax

    ;=

    ;AZ(t)mab?.CabAxb一露abAxb(11)

    ;Ifthecableonlyexperiencestheendsupportex. ;citations,then(iscalculatedaccordingtoFD.

    ;Eq.f11)canbesolvedbyiterationtechniqueus. ;ingcertainconvergencecriteria.ThecabletensionT ;ineverystepisObrainedfromtheequationofmotions ;andupdatedautomatically.

    ;EXAMPIES

    ;Cableparameters

    ;Theparametersofthetwocables.whichareex. ;tractedfromaproposedcable.stayedbridgewith ;mainspanof1400m,areshowninTlab1el(Xieeta1., ;2008).whereListhecablespan,whichisthehori. ;zontalprojectionlengthofthecable,Yistheheight ;differenceofthecablesupports,wtheweightperunit ;length,theratioofsagtocablespan,andthe ;non.dimensiona1Irvineparameter(Irvine,1981).The ;freevibrationfrequenciesofthecablesareshownin ;abe2.

    ;]?,??????J

    ;

    ;1196Xieeta1./JZhejiangUnivSciA20089(9):1193-1200 ;Table1Parametersofthesteelcables

    ;Table2Frequenciesofthecables(Hz)

    ;Assumethatthelowersupportendisexperi

    ;encingharmonicexcitationinthedirectionofcable ;chord,theverticalandhorizontalcomponentsofthe ;excitationcanbewrittenas

    ;f=sin(2nft)cos0,

    ;1”=sin(2nil)sin0,

    ;whereAniStheamplitude,whichiSassumedtobe ;1/5000,1/10000and1/50000ofthecablespan,re

    ;spectively.fistheexcitationfrequency,andtwo

    ;valuesareusedinthisstudy:f=Aand2A,wherejqis ;thefirstinplanevibrationfrequency.0istheangle ;betweenthehorizontalaxisandcablechord.YaxisiS ;positiveifdownward.

    ;ThedampingofthecableiSassumedtobe

    ;zeroifthereiSnoexplicitindication.Thedragcoef- ;ficientCD=0.7.

    ;Nonlineardynamiccharacteristicsofthelongca

    ;bleS1

    ;Fig.3showstheverticaldisplacementresponse ;andcorrespondingfrequencyspectraatthemiddle ;pointofthelongercableS1duetoexcitationof

    ;frequency.Thevibrationlastsfor2500S.Because ;oftheeffectofdragforce,theamplitudeofdynamic ;displacementexperiencesaperiodicvariationand ;becomessmoothandstationary.

    ;Patternofcablevibrationduetosupportexcita

    ;tionswichdifferentamplitudesiSidentica1.whichiS ;initiallynonlinearforcedvibrationwithperiodic ;varyingamplitudeandultimatelyparametricvibra

    ;tionwithsteadyamplitude.Theamplitudeofpara

    ;metricvibrationiSproportionaltothesupportexcita. ;tionamplitude,sincetheparametricvibrationofthe ;20

    ;O

    ;20

    ;2O

    ;1O

    ;O

    ;O

    ;20

    ;20

    ;O

    ;20

    ;O

    ;

    ;??J??…

    ;.I-

    ;O5001000150020002500

    ;f(s)

    ;(c)

    ;I

    ;

    ;--.

;.--Ao=L/5000

    ;J

    ;-

    ;-Ao=L/10000

    ;

    ;.__An=L/50000_

    ;_

    ;1

    ;J-

    ;f(Hz)

    ;(d1

    ;Fig.3Verticaldisplacementresponseandspectrumat

    ;themiddlepointofthecableS-Iduetoexcitationof

    ;frequencywithoutdamping.(a)Ao=L/5000;(b) ;Ao=L/IO000;(C)Ao=L/50000;(d)Spectrumofvertical

    ;displacement

    ;cableisaffectedbycabletension,andthesupport ;excitationwithabigamplitudewillinducesignifcant

    ;changesofcabletension,thusresultinaremarkable

    ;variationofthecableparameters. ;Onthecontrary,itis

    ;hardforsmallamplitudeexcitationtoinducepara

    ;?

    ;?

    ;

    ;O

    ;??

    ;Or

    ;(

    ;?

    ;O

    ;O

    ;

    ;O

    ;

    ;O

    ;

    ;

    ;一目)0lIq暑《

    ;

    ;Xieeta1./dZhejiangUnivSciA200899):1193-1200

    ;metricvibrationofcable.Thevibrationwithperiodic

    ;varyingamplitudesisthe’’beat”phenomenon.

    ;Fig.3dshowsthecorrespondingfrequency

    ;spectraofthedisplacementvstimehistoriesin ;Figs.3a-3c.Theresultsdemonstratethatthecharac

    ;teristicsofthedynamicresponsesareidenticaldueto ;thethreedifferentsupportexcitations.Twomodes ;withfrequenciesand2tiareincludedincablevi

    ;brationandthedominantoneisthemodewith. ;Fig.4depictstheverticaldisplacementresponse ;andcorrespondingfrequencyspectraatthemiddle ;pointofcableS1duetoexcitationoffrequency2.

    ;,entheamplitudeoftheexcitationis1/50000of ;thecablespan,thevibrationamplitudeissmallerthan ;0.2mandparametricvibrationwithbigamplitude ;doesnotoccuLthuscorrespondingresultsarenot ;presented.

    ;Accordingtotheresults,thepaaemofcablevi

    ;brationsduetosupportexcitationsoffrequencyis ;similartothatduetosupportexcitationsoffrequency ;.Thebeatpattemoccursintheearlystageandis ;followedbyparametricvibrationswithsteadyam

    ;plitudes.Theexcitationfrequencyisnotthenatural

     ;frequencyofthecable,sotheinitialvibrationampli

    ;tudeandvibrationofcabletensionarerelativelysmal1. ;Astheexcitationcontinues,thevariationofcable ;tensioninducesself-motivatedvibration,whichis ;calledparametricvibration.

    ;Fig.4cshowsthecorrespondingspectraofthe ;dynamicresponses.Itisobservedthatthemodewith ;frequencyisstillthedominantmodeofthevibra

    ;tionalthoughtheexcitationfrequencyis.

    ;Thecharacteristicsofcablevibrationdueto ;supportexcitationobtainedinthisstudyaresimilarto ;thatinWu(2002)’sresearch.Thevibrationofthe

    ;cablevariesfromnonlinearvibrationduetosupport ;excitationtoparametricvibrationwithsteadyampli

    ;tudeinducedbyperiodicvariationofthecabletension. ;Theamplitudeofparametricvibrationisrelatedtothe ;amplitude0ofsupportexcitation.WhenA0isrela

    ;tivelybig.thebeatvibrationwill1astforashorttime, ;andthesteadyamplitudeofparametricvibrationis ;muchbigger.

    ;Nonlineardynamiccharacteristicsoftheshort ;cableS-2

    ;Fig.5showstheverticaldisplacementresponse ;atthemiddlepointofcableS-2.Whentheamplitude

;

    ;

    ;20

    ;0

    ;20

    ;lo

    ;o

    ;lo

    ;0500l000150020002500

    ;tfs)

    ;(a)

    ;.??L?_…——

    ;,r1r1…

    ;0500l000l50020002500

    ;t(s)

    ;(b)

    ;f(Hz)

    ;(c)

    ;11g7

    ;Fig.4Verticaldisplacementresponseandspectrumat ;themiddlepointofthecableS-1duetoexcitationof ;frequencyf=ZAwithoutdamping.(a)Ao=LISO00;(b) ;Ao=L/10000;(c)Spectrumofverticaldisplacement ;oftheexcitationis1/50000ofthecablespanandthe ;excitationfrequencyis2,;,theparametricvibration ;withabigamplitudehasnothappenedwiththeex

    ;citationdurationof2500s,thereforethecorre

    ;spondingresultsarenotpresentedinthispaper. ;ItisobservedfromFig.5thatthevibrationof ;cableS2issimilartothatofcableS.1.However, ;becausethestructuralpropertiesofthetwocablesare ;different,theamplitudesandtheperiodforbeatvi

    ;brationbetweenthetwocablesaredifferent. ;Accordingtotheresults,althoughtheIrvine ;parametersbetweenthecablesS1andS2aredif-

    ;ferent,ofwhichthecharacteristicsofparametri.(a)Ao=L/5000,;(b)Ao=

    ;L/10000,;(c)Ao=L/50000,;(d)Ao=L/5000,;

    ;(e)Ao=L/IO000,

    ;Effectsofviscousdamping

    ;Inordertodiscussthedampingeffectofdis

    ;tributeddampingonparametricvibrationofcables, ;nonlineardynamicresponsesareobtainedfordier.

    ;entdistributeddampingcoecients.

    ;Assumethattheuniformlydistributeddamping

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