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JOURNALOFSOUNDANDVIBRATIONWWWELSEVIERCOM/LOCATE/JSVIJOURNALOFSOUNDANDVIBRATION2632003679–699LETTERTOTHEEDITORVIBRATIONOFELEVATORCABLESWITHSMALLBENDINGSTIFFNESSWDZHU,GYXUDEPARTMENTOFMECHANICALENGINEERING,UNIVERSITYOFMARYLANDBALTIMORECOUNTY,1000HILLTOPCIRCLE,BALTIMORE,MD21250,USARECEIVED27SEPTEMBER2002;ACCEPTED3OCTOBER20021INTRODUCTIONWHILECABLESAREEMPLOYEDINDIVERSEENGINEERINGAPPLICATIONSINCLUDINGSUSPENSIONBRIDGES1,ELEVATORS2,POWERTRANSMISSIONLINES3,ANDMARINETOWINGANDMOORINGSYSTEMS4,THEYARESUBJECTTOVIBRATIONDUETOTHEIRHIGHflEXIBILITYANDLOWINTRINSICDAMPINGIRVINEANDCAUGHEY5ANDTRIANTAFYLLOU6STUDIEDTHEDYNAMICSOFSUSPENDEDCABLESWITHHORIZONTALANDINCLINEDSUPPORTSSERGEVANDIWAN7ANDCHENGANDPERKINS8ANALYZEDTHEVIBRATIONOFCABLESWITHATTACHEDMASSESSIMPSON9,TRIANTAFYLLOU10,ANDPERKINSANDMOTE11STUDIEDTHEINPLANEANDTHREEDIMENSIONALVIBRATIONOFTRAVELLINGCABLESWICKERTANDMOTE12ANDZHUANDMOTE13ANALYZEDTHEDYNAMICRESPONSEOFTRAVELLINGCABLESWITHATTACHEDPAYLOADSWHILETHEBENDINGSTIFFNESSOFCABLESISNEGLECTEDINMOSTSTUDIES,ITWASINCLUDEDINTHEMODELSINREFS14,15TOAVOIDTHESINGULARBEHAVIORSASSOCIATEDWITHVANISHINGCABLETENSIONBENDINGSTIFFNESSWASALSOACCOUNTEDFORWHENCABLESARESUBJECTEDTOEXTERNALMOMENTS3,16ORWHENTHEIRLOCALBENDINGSTRESSESNEEDTOBEDETERMINED17VIBRATIONOFELEVATORCABLESHASBEENSTUDIEDBYSEVERALRESEARCHERS2,18–21CHIANDSHU2CALCULATEDTHENATURALFREQUENCIESASSOCIATEDWITHTHELONGITUDINALVIBRATIONOFASTATIONARYCABLEANDCARSYSTEMROBERTS18USEDLUMPEDMASSAPPROXIMATIONSTOMODELTHELONGITUDINALDYNAMICSOFHOISTANDCOMPENSATIONCABLESINHIGHRISEELEVATORSYAMAMOTOETAL19ANALYZEDTHEFREEANDFORCEDLATERALVIBRATIONOFASTATIONARYSTRINGWITHSLOWLY,LINEARLYVARYINGLENGTHTERUMICHIETAL20EXAMINEDTHELATERALVIBRATIONOFATRAVELLINGSTRINGWITHSLOWLY,LINEARLYVARYINGLENGTHANDAMASSSPRINGTERMINATIONZHUANDNI21ANALYZEDTHEDYNAMICSTABILITYOFTRAVELLINGMEDIAWITHVARIABLELENGTHTHEVIBRATORYENERGYOFTHEMEDIAWASSHOWNTODECREASEANDINCREASEINGENERALDURINGEXTENSIONANDRETRACTION,RESPECTIVELYDUETOITSSMALLBENDINGSTIFFNESSRELATIVETOTHETENSION,THEMOVINGHOISTCABLEWASMODELLEDASATRAVELLINGSTRINGINREF21BYINCLUDINGTHEBENDINGSTIFFNESSINTHEMODELSFORTHESTATIONARYANDMOVINGHOISTCABLESWITHDIFFERENTBOUNDARYCONDITIONS,THEEFFECTSOFBENDINGSTIFFNESSANDBOUNDARYCONDITIONSONTHEIRDYNAMICCHARACTERISTICSAREINVESTIGATEDHERECONVERGENCEOFTHECORRESPONDINGAUTHORTEL14104553394;FAX14104551052EMAILADDRESSWZHUUMBCEDUWDZHU0022460X/03/SEEFRONTMATTERR2002ELSEVIERSCIENCELTDALLRIGHTSRESERVEDDOI101016/S0022460X02014682MODELSISEXAMINEDTHEOPTIMALSTIFFNESSANDDAMPINGCOEFfiCIENTOFTHESUSPENSIONOFTHECARAGAINSTITSGUIDERAILSAREIDENTIfiEDFORTHEMOVINGCABLE2STATIONARYCABLEMODELS21BASICEQUATIONSWECONSIDERSIXMODELSOFTHESTATIONARYHOISTCABLETOEVALUATETHEEFFECTSOFBENDINGSTIFFNESSANDBOUNDARYCONDITIONSONITSDYNAMICCHARACTERISTICSSINCETHEVERTICALCABLEHASNOSAG,ITISMODELLEDASATAUTSTRINGANDATENSIONEDBEAMSHOWNINFIG1ARETHEBEAMANDSTRINGMODELSOFTHECABLEWITHTHESUSPENSIONOFTHECARAGAINSTITSGUIDERAILSASSUMEDTOBERIGIDSHOWNINFIG2ARETHEBEAMANDSTRINGMODELSOFTHECABLEWITHTHESUSPENSIONOFTHECARAGAINSTTHEGUIDERAILSMODELLEDBYARESULTANTSTIFFNESSKEANDDAMPINGCOEFfiCIENTCEINALLTHECASESTHEMASSOFTHECARISDENOTEDBYMEWHILETHECARCANHAVEfiNITEDIMENSIONSINFIG1,ITISMODELLEDASAPOINTMASSINFIG2WHENTHECABLEISMODELLEDASATENSIONEDBEAM,ASSHOWNINFIGS1AANDB,AND2AANDB,ITSFREELATERALVIBRATIONINTHEXYPLANEISGOVERNEDBYRYTTX;TC0PXYXX;TC138XEIYXXXXX;T0;0OXOL;1WHERETHESUBSCRIPTDENOTESPARTIALDIFFERENTIATION,YX;TISTHELATERALDISPLACEMENTOFTHECABLEPARTICLEATPOSITIONXATTIMET;LISTHELENGTHOFTHECABLE,RISTHEMASSPERUNITLENGTH,EIISTHEBENDINGSTIFFNESS,ANDPXISTHETENSIONATPOSITIONXGIVENBYPXMERLC0XC138G;2INWHICHGISTHEACCELERATIONDUETOGRAVITYTHEBOUNDARYCONDITIONSOFTHECABLEWITHfiXEDENDS,ASSHOWNINFIG1A,AREY0;TYX0;T0;YL;TYXL;T03XLYEMYEMEMYACBFIG1SCHEMATICOFTHESTATIONARYHOISTCABLEWITHTHESUSPENSIONOFTHECARAGAINSTITSGUIDERAILSASSUMEDTOBERIGIDAfiXED–fiXEDBEAMMODEL,BPINNED–PINNEDBEAMMODEL,ANDCSTRINGMODELWDZHU,GYXU/JOURNALOFSOUNDANDVIBRATION2632003679–699680THEBOUNDARYCONDITIONSOFTHECABLEWITHPINNEDENDS,ASSHOWNINFIG1B,AREY0;TYXX0;T0;YL;TYXXL;T04FORTHECABLEMODELSINFIG2AANDB,THEBOUNDARYCONDITIONSATX0ARETHESAMEASTHOSEINEQS3AND4,RESPECTIVELY,ANDTHEBOUNDARYCONDITIONSATXLAREYXXL;T0;EIYXXXL;TPLYXL;TMEYTTL;TCEYTL;TKEYL;T5NOTETHATTHEBENDINGMOMENTATXLVANISHESINTHEfiRSTEQUATIONINEQ5BECAUSETHEROTARYINERTIAOFTHECARISNOTCONSIDEREDTHEGOVERNINGEQUATIONFORTHEMODELSINFIGS1CAND2CISGIVENBYEQ1WITHEI0;ANDTHEBOUNDARYCONDITIONATX0ISY0;T0THEBOUNDARYCONDITIONATXLFORTHEMODELINFIG1CISYL;T0ANDTHEBOUNDARYCONDITIONATXLFORTHEMODELINFIG2CISGIVENBYTHESECONDEQUATIONINEQ5WITHEI0DUETOVANISHINGSLOPEOFTHECABLEATTHEfiXEDENDSINFIGS1AAND2A,THEMODELSINFIGS1CAND2CCANNOTBEOBTAINEDFROMTHEMODELSINFIGS1AAND2A,RESPECTIVELY,BYSETTINGEI0INADDITIONTOPROVIDINGANOMINALTENSIONMEG;THEMASSOFTHECARRESULTSINANINERTIALFORCEINTHESECONDEQUATIONINEQ5FORTHEMODELSINFIG2GALERKIN’SMETHODANDTHEASSUMEDMODESMETHODAREUSEDTODISCRETIZETHEGOVERNINGPARTIALDIFFERENTIALEQUATIONSFORTHEMODELSINFIGS1AND2,RESPECTIVELYTHESOLUTIONOFEQ1ISASSUMEDINTHEFORMYX;TXNJ1QJTFJX;6WHEREFJXARETHETRIALFUNCTIONS,QJTARETHEGENERALIZEDCOORDINATES,ANDNI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