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JOURNALOFSOUNDANDVIBRATIONwww.elsevier.com/locate/jsviJournalofSoundandVibration2632003679–699LettertotheEditorVibrationofelevatorcableswithsmallbendingstiffnessW.D.Zhu,G.Y.XuDepartmentofMechanicalEngineering,UniversityofMarylandBaltimoreCounty,1000HilltopCircle,Baltimore,MD21250,USAReceived27September2002accepted3October20021.IntroductionWhilecablesareemployedindiverseengineeringapplicationsincludingsuspensionbridges1,elevators2,powertransmissionlines3,andmarinetowingandmooringsystems4,theyaresubjecttovibrationduetotheirhighflexibilityandlowintrinsicdamping.IrvineandCaughey5andTriantafyllou6studiedthedynamicsofsuspendedcableswithhorizontalandinclinedsupports.SergevandIwan7andChengandPerkins8analyzedthevibrationofcableswithattachedmasses.Simpson9,Triantafyllou10,andPerkinsandMote11studiedtheinplaneandthreedimensionalvibrationoftravellingcables.WickertandMote12andZhuandMote13analyzedthedynamicresponseoftravellingcableswithattachedpayloads.Whilethebendingstiffnessofcablesisneglectedinmoststudies,itwasincludedinthemodelsinRefs.14,15toavoidthesingularbehaviorsassociatedwithvanishingcabletension.Bendingstiffnesswasalsoaccountedforwhencablesaresubjectedtoexternalmoments3,16orwhentheirlocalbendingstressesneedtobedetermined17.Vibrationofelevatorcableshasbeenstudiedbyseveralresearchers2,18–21.ChiandShu2calculatedthenaturalfrequenciesassociatedwiththelongitudinalvibrationofastationarycableandcarsystem.Roberts18usedlumpedmassapproximationstomodelthelongitudinaldynamicsofhoistandcompensationcablesinhighriseelevators.Yamamotoetal.19analyzedthefreeandforcedlateralvibrationofastationarystringwithslowly,linearlyvaryinglength.Terumichietal.20examinedthelateralvibrationofatravellingstringwithslowly,linearlyvaryinglengthandamassspringtermination.ZhuandNi21analyzedthedynamicstabilityoftravellingmediawithvariablelength.Thevibratoryenergyofthemediawasshowntodecreaseandincreaseingeneralduringextensionandretraction,respectively.Duetoitssmallbendingstiffnessrelativetothetension,themovinghoistcablewasmodelledasatravellingstringinRef.21.Byincludingthebendingstiffnessinthemodelsforthestationaryandmovinghoistcableswithdifferentboundaryconditions,theeffectsofbendingstiffnessandboundaryconditionsontheirdynamiccharacteristicsareinvestigatedhere.ConvergenceoftheCorrespondingauthor.Tel.14104553394fax14104551052.Emailaddresswzhuumbc.eduW.D.Zhu.0022460X/03/seefrontmatterr2002ElsevierScienceLtd.Allrightsreserved.doi10.1016/S0022460X02014682modelsisexamined.Theoptimalstiffnessanddampingcoefficientofthesuspensionofthecaragainstitsguiderailsareidentifiedforthemovingcable.2.Stationarycablemodels2.1.BasicequationsWeconsidersixmodelsofthestationaryhoistcabletoevaluatetheeffectsofbendingstiffnessandboundaryconditionsonitsdynamiccharacteristics.Sincetheverticalcablehasnosag,itismodelledasatautstringandatensionedbeam.ShowninFig.1arethebeamandstringmodelsofthecablewiththesuspensionofthecaragainstitsguiderailsassumedtoberigid.ShowninFig.2arethebeamandstringmodelsofthecablewiththesuspensionofthecaragainsttheguiderailsmodelledbyaresultantstiffnesskeanddampingcoefficientceInallthecasesthemassofthecarisdenotedbymeWhilethecarcanhavefinitedimensionsinFig.1,itismodelledasapointmassinFig.2.Whenthecableismodelledasatensionedbeam,asshowninFigs.1aandb,and2aandb,itsfreelateralvibrationinthexyplaneisgovernedbyryttðxtÞC0½PðxÞyxðxtÞC138xþEIyxxxxðxtÞ¼00oxolð1Þwherethesubscriptdenotespartialdifferentiation,yðxtÞisthelateraldisplacementofthecableparticleatpositionxattimetlisthelengthofthecable,risthemassperunitlength,EIisthebendingstiffness,andPðxÞisthetensionatpositionxgivenbyPðxÞ¼½meþrðlC0xÞC138gð2Þinwhichgistheaccelerationduetogravity.Theboundaryconditionsofthecablewithfixedends,asshowninFig.1a,areyð0tÞ¼yxð0tÞ¼0yðltÞ¼yxðltÞ¼0ð3ÞxlyemyememyacbFig.1.Schematicofthestationaryhoistcablewiththesuspensionofthecaragainstitsguiderailsassumedtoberigidafixed–fixedbeammodel,bpinned–pinnedbeammodel,andcstringmodel.W.D.Zhu,G.Y.Xu/JournalofSoundandVibration2632003679–699680Theboundaryconditionsofthecablewithpinnedends,asshowninFig.1b,areyð0tÞ¼yxxð0tÞ¼0yðltÞ¼yxxðltÞ¼0ð4ÞForthecablemodelsinFig.2aandb,theboundaryconditionsatx¼0arethesameasthoseinEqs.3and4,respectively,andtheboundaryconditionsatx¼lareyxxðltÞ¼0EIyxxxðltÞ¼PðlÞyxðltÞþmeyttðltÞþceytðltÞþkeyðltÞð5ÞNotethatthebendingmomentatx¼lvanishesinthefirstequationinEq.5becausetherotaryinertiaofthecarisnotconsidered.ThegoverningequationforthemodelsinFigs.1cand2cisgivenbyEq.1withEI¼0andtheboundaryconditionatx¼0isyð0tÞ¼0Theboundaryconditionatx¼lforthemodelinFig.1cisyðltÞ¼0andtheboundaryconditionatx¼lforthemodelinFig.2cisgivenbythesecondequationinEq.5withEI¼0DuetovanishingslopeofthecableatthefixedendsinFigs.1aand2a,themodelsinFigs.1cand2ccannotbeobtainedfromthemodelsinFigs.1aand2a,respectively,bysettingEI¼0InadditiontoprovidinganominaltensionmegthemassofthecarresultsinaninertialforceinthesecondequationinEq.5forthemodelsinFig.2.GalerkinsmethodandtheassumedmodesmethodareusedtodiscretizethegoverningpartialdifferentialequationsforthemodelsinFigs.1and2,respectively.ThesolutionofEq.1isassumedintheformyðxtÞ¼Xnj¼1qjðtÞfjðxÞð6ÞwherefjðxÞarethetrialfunctions,qjðtÞarethegeneralizedcoordinates,andnisthenumberofincludedmodes.ThetrialfunctionsforthemodelsinFig.1satisfyalltheboundaryconditionsandthoseforthemodelsinFig.2satisfyalltheboundaryconditionsexcepttheforceboundaryem/2ek/2ek/2ec/2ecy/2ek/2ek/2ec/2ecemylx/2ek/2ek/2ec/2ecyemabcFig.2.Schematicofthestationaryhoistcablewherethecarismodelledasapointmassmeanditssuspensionagainsttheguiderailshasaresultantstiffnesskeanddampingcoefficientceabeammodelwithafixedendatx¼0bbeammodelwithapinnedendatx¼0andcstringmodel.W.D.Zhu,G.Y.Xu/JournalofSoundandVibration2632003679–699681conditioninEq.5.SubstitutingEq.6intoEq.1andthesecondequationinEq.5,multiplyingthegoverningequationbyfiðxÞi¼12yn,integratingitfromx¼0tolandusingtheresultingboundaryconditionyieldsthediscretizedequationsforthemodelsinFig.2aandbM.qðtÞþCqðtÞþKqðtÞ¼0ð7Þwhereq¼½q1q2yqnC138TisthevectorofgeneralizedcoordinatesandM,K,andCarethesymmetricmass,stiffness,anddampingmatrices,respectively,withentriesMij¼Zl0rfiðxÞfjðxÞdxþmefiðlÞfjðlÞð8ÞKij¼Zl0PðxÞf0iðxÞf0jðxÞdxþZl0EIf00iðxÞf00jðxÞdxþkefiðlÞfjðlÞð9ÞCij¼cefiðlÞfjðlÞð10ÞinwhichtheprimedenotesdifferentiationwithrespecttoxThediscretizedequationsforthemodelinFig.2caregivenbyEqs.7–10withEI¼0inEq.9.ThediscretizedequationsforthemodelsinFig.1aandbaregivenbyEqs.7–10withme¼0inEq.8andke¼ce¼0inEqs.9and10thediscretizedequationsforthemodelinFig.1caregivenbyEqs.7–10withme¼0inEq.8andke¼EI¼ce¼0inEqs.9and10.WhilethediscretizedequationsforthemodelsinFig.1aandbhavethesameform,thetrialfunctionsusedsatisfydifferentboundaryconditions.ThisalsoholdsforthemodelsinFig.2aandb.Theeigenfunctionsofafixed–fixedbeamandthoseofafixed–fixedbeamunderuniformtensionT¼megareusedasthetrialfunctionsforthemodelinFig.1a.Theeigenfunctionsofapinned–pinnedbeam,whichareidenticaltothoseofapinned–pinnedbeamunderuniformtension,areusedasthetrialfunctionsforthemodelinFig.1b.Theeigenfunctionsofafixed–fixedstring,whichareidenticaltothoseofapinned–pinnedbeam,areusedasthetrialfunctionsforthemodelinFig.1c.DuetothesametrialfunctionsthediscretizedequationsforthemodelinFig.1ccanbeobtainedfromthoseforthemodelinFig.1bbysettingEI¼0Theeigenfunctionsofacantileverbeamandthoseofafixed–freebeamunderuniformtensionT¼megareusedasthetrialfunctionsforthemodelinFig.2a.Theeigenfunctionsofapinned–freebeamandthoseofapinned–freebeamunderuniformtensionT¼megareusedasthetrialfunctionsforthemodelinFig.2b.Theeigenfunctionsofafixed–freestringareusedasthetrialfunctionsforthemodelinFig.2c.Notethatapinned–freebeamhasarigidbodymodeandafixed–freestringdoesnot.ThediscretizedequationsforthemodelinFig.2ccannotbeobtainedasaspecialcasefromthoseforthemodelinFig.2bduetothedifferenttrialfunctionsused.AllthetrialfunctionsarenormalizedandgiveninAppendixA.BytheorthogonalityrelationsthemassmatrixforthemodelsinFig.1isadiagonalmatrix.IftheinitialdisplacementandvelocityofthecableinFigs.1and2aregivenbyyðx0Þandytðx0Þrespectively,theinitialconditionsforthegeneralizedcoordinatesareqjð0Þ¼Zl0fjðxÞyðx0Þdxqjð0Þ¼Zl0fjðxÞytðx0Þdxð11ÞW.D.Zhu,G.Y.Xu/JournalofSoundandVibration2632003679–699682
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