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thefootbridgeisperceptible,itbecomesmorecomfortableforthepedestrianstowalkinsynchronizationwiththehorizontalmotionofthebridge.Becauselateralmotionaffectsbalance,pedestrianstendtowalkwithfeetfurtherapartandattempttosynchronizetheirfootstepswiththemotionofthesurface.Thepedestrians ndthishelpsthemmaintaintheirlateralbalance.So,oncethebridgestartstovibrate,morepedestrianssynchronizedwiththebridgevibration,whichfurtherincreasesitsresponse.Thisin-stinctivebehaviorofpedestriansensuresthatthedynamicforcesareappliedataresonantfrequencyofthebridge(Fitzpatricketal.2001).Inaddition,theparametricexcitationmechanismof ex-iblefootbridges(suchastheLondonMillenniumFootbridge)wasdiscussedbyPiccardoandTubino(2008).Itwasfoundthatthelateralvibrationof exiblefootbridgesoccursdependingontheratiobetweenthestructuralandexcitationfrequencies,ontheratioofthestructuralandpedestrianmasses,andonthestructuraldamping.

Theproblemofdeterminingtheforceexertedbysynchronizedpedestriansisgenerallytackledusinganempiricalapproach.Mostofthemodelsfoundintheliterature(Zivanovicetal.2005;Newland2004)aredeterministictime-domainmodels,basedontheassump-tionthatbothfeetproduceexactlythesameperiodicforce.Venutietal.(2007)proposedaloadmodeltotakeintoaccountsomeim-portantfeaturesofthesynchronouslateralexcitationphenomenon,suchasthedistinctionbetweensynchronizationamongpedes-trians,causedbycrowddensity,andbetweenthepedestriansandthestructure,causedbydeckoscillations:thetriggeringofthelock-inphenomenonanditsself-limitednature.Themodi edhybridVanderPol/RayleighoscillatormodelwasproposedbyErlicheretal.(2010)topredictthelateralwalkingforceofapedestrianbasedontheexperimentallyobservedphenomena:thewalkingmotionisself-sustained.

Basedontheexperimentaltests,adynamicmodelwasproposedbyDallardetal.(2001b)tosimulatethislateralvibrationproblemontheLondonMillenniumBridge,inwhichtheexternallateralforceinducedbypedestriansisproportionaltothegirderlateralvelocity.Nevertheless,whenthelateralforceislargerenough,thebridgeresponseincreasesin nitelyaccordingtothismethod(Nakamura2004),whichisnotinaccordancewithobservations.Infact,thepedestrianswillreducewalkingspeedorstopwalkingwhentheresponseofthebridgebecomessuf cientlylarge,andthenthebridgeresponsedoesnotincreasein nitely.Therefore,Nakamura(2004)modi edDallard’sloadmodeltoallowschematizationoftheself-limitingnatureofthesynchronizationphenomenon.However,Nakamura’smodelisnotpredictivebecauseofthecoef cientsthatneedtobeestimatedfromexperimentaltestsand eldmeasure-ments.Roberts(2003,2005)proposedadynamicalmodeltocon-sidertheinteractionbetweenthepedestriansandthefootbridge.Byassumingthatsynchronizationbetweenpedestriansandfootbridgeoccurswhenthedisplacementofthefootbridgeisgreaterthanthatofthepedestrians,onecanobtainalimitingnumberofpedestriansonthefootbridge.InRoberts’smodel,theinteractiveforceishar-monicallyvaryingwithfrequencyofabout1Hz;thus,largevibrationsappearonlyforfootbridgeswithanaturalfrequencyclosedtothatvalue.AdifferentmodelwasdevelopedbyMacdonald(2009)todescribethepedestrian-structureinteraction,basedonasimplemodelofhumanbalancefromthebiomechanics eld.Themodelgivesareasonablerepresentationofthelateralpedestrianforcesappliedtothebridgeintheabsenceofbridgemotioncom-paredwithpreviousmeasurements;however,withthechosenparameters,themodeldoesnotaccuratelymatchthenumericalresultswiththefull-scalemeasurements.Tore nethemodel,furtherdataarerequiredonpedestrianbehaviorwhenperturbedbybridgemotion.

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Inthispaper,thefocusisonNakamura’smodel.Nakamura(2004)andNakamuraandKawasaki(2006,2009)intendedtoes-tablishapracticalmethodinsteadofapuremathematicalmethod.Thereareothermethodsthatcontainharmonictermsintheloadingfunction(Strogatzetal.2005).ThismaybemoremathematicallyaccuratethanNakamura’smethod;however,suchamethodusuallycontainsahypotheticalequationandisnoteasierduringapplication.Conversely,thecoef cientsofNakamura’smodelweredeterminedbymanylaboratorytestsand http://www.wenkuxiazai.comingNakamura’smethod,predictedresponsescanbemoreeasilycomparedwithmeasuredresponses.Therefore,al-thoughNakamura’smodelmaynotbestateoftheartinthisarea,thusfaritmaybethemostpractical.Nevertheless,somecoef cientsinNakamura’smodelcannotbedecidedpreciselybecauseofthelackofdataavailableonthelateralloadbehaviorsofwalkingpeople.ThelateralgirderresponsespredictednumericallybyNakamura’smodelarealmostalwaysstrongerthanthemeasureddata(NakamuraandKawasaki2009).Thedifferencesofsomecasesaresolargethatitseemsunwisetoentirelyattributetheproblemtotheerrorsof eldmeasurements.Inaddition,themovingpedestriansactasvelocityfeedback,increasingtheresponseofthefootbridgeinNakamura’smodel.Inreality,whenthelateralforceofthepedes-triansisappliedtothedeck,thevelocityofthefootbridgedoesnotvaryimmediately.Thechangeofthevelocityofthefootbridgealwaystakesplaceafteratimedelay.Partofthesourceofthetimedelaybetweenpedestriansandthefootbridgeishumanreactiontime.Theotherpartisthereactiontimeofthefootbridge,whichdependsontheparametersofthefootbridges,suchasthemass,length,andtypeoffootbridge.ItisworthinvestigatingwhetherthetimedelayignoredinNakamura’smodelaffectsthelateralvibrationofthefootbridge.

TheaimofthispaperistomodifyNakamura’smodeltopref-erablyagreewiththemeasureddatabyanalyzingsomecoef cientsofthemodelintheaspectofnonlineardynamics.Therestofthepaperisorganizedasfollows: rst,Nakamura’smodelisintroduced;then,somepossiblereasonswhythenumericalresultsofNakamura’smodelarelargerthanthemeasureddataarediscussedandamodi edmodelisgiven; nally,numericalsimulationsarecarriedouttoverifythistheoreticalanalysis.

Nakamura’sModel

ThebasicequationinNakamura’smodelistheequationofmotion(Nakamura2004)

BþCBx_BþKBxB¼FpðtÞMBx

ð1Þ

whereMB5modalmass,CB5modaldamping,andKB5modal

stiffnessofthebridge.Furthermore,xB5modaldisplacementofthe

B5modalaccel-girder,x_B5modalvelocityofthegirder,andx

erationofthegirder.FpðtÞisthemodallateraldynamicforceappliedbyallthepedestrianstothebridge,givenby

FpðtÞ¼k1k2MpgGðfBÞHðx_BÞ

ð2Þ

whereHðx_BÞ5x_B/ðk31jx_BjÞ.Thecoef cientk1istheratioofthe

lateralforcetothepedestrian’sweight.Thecoef cientk2isthepercentageofpedestrianswhosynchronizetothegirdervibration.

_BÞisthefunctionMpgisthemodalself-weightofpedestrians.Hðx

todescribethepedestrians’synchronizationnature.Nakamuraas-sumedthatthepedestrianssynchronizeproportionallywiththe

JOURNALOFBRIDGEENGINEERING©ASCE/FEBRUARY2013/123

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