JOURNALOFSOUNDANDVIBRATION/locate/jsviJournalofSoundandVibration263(2003)679699LettertotheEditorVibrationofelevatorcableswithsmallbendingstiffnessW.D.Zhu*,G.Y.XuDepartmentofMechanicalEngineering,UniversityofMarylandBaltimoreCounty,1000HilltopCircle,Baltimore,MD21250,USAReceived27September2002；accepted3October20021.IntroductionWhilecablesareemployedindiverseengineeringapplicationsincludingsuspensionbridges1,elevators2,powertransmissionlines3,andmarinetowingandmooringsystems4,theyaresubjecttovibrationduetotheirhighexibilityandlowintrinsicdamping.IrvineandCaughey5andTriantafyllou6studiedthedynamicsofsuspendedcableswithhorizontalandinclinedsupports.SergevandIwan7andChengandPerkins8analyzedthevibrationofcableswithattachedmasses.Simpson9,Triantafyllou10,andPerkinsandMote11studiedthein-planeandthree-dimensionalvibrationoftravellingcables.WickertandMote12andZhuandMote13analyzedthedynamicresponseoftravellingcableswithattachedpayloads.Whilethebendingstiffnessofcablesisneglectedinmoststudies,itwasincludedinthemodelsinRefs.14,15toavoidthesingularbehaviorsassociatedwithvanishingcabletension.Bendingstiffnesswasalsoaccountedforwhencablesaresubjectedtoexternalmoments3,16orwhentheirlocalbendingstressesneedtobedetermined17.Vibrationofelevatorcableshasbeenstudiedbyseveralresearchers2,1821.ChiandShu2calculatedthenaturalfrequenciesassociatedwiththelongitudinalvibrationofastationarycableandcarsystem.Roberts18usedlumpedmassapproximationstomodelthelongitudinaldynamicsofhoistandcompensationcablesinhigh-riseelevators.Yamamotoetal.19analyzedthefreeandforcedlateralvibrationofastationarystringwithslowly,linearlyvaryinglength.Terumichietal.20examinedthelateralvibrationofatravellingstringwithslowly,linearlyvaryinglengthandamass-springtermination.ZhuandNi21analyzedthedynamicstabilityoftravellingmediawithvariablelength.Thevibratoryenergyofthemediawasshowntodecreaseandincreaseingeneralduringextensionandretraction,respectively.Duetoitssmallbendingstiffnessrelativetothetension,themovinghoistcablewasmodelledasatravellingstringinRef.21.Byincludingthebendingstiffnessinthemodelsforthestationaryandmovinghoistcableswithdifferentboundaryconditions,theeffectsofbendingstiffnessandboundaryconditionsontheirdynamiccharacteristicsareinvestigatedhere.Convergenceofthe*Correspondingauthor.Tel.:+1-410-455-3394；fax:+1-410-455-1052.E-mailaddress:(W.D.Zhu).0022-460X/03/$-seefrontmatterr2002ElsevierScienceLtd.Allrightsreserved.doi:10.1016/S0022-460X(02)01468-2modelsisexamined.Theoptimalstiffnessanddampingcoefcientofthesuspensionofthecaragainstitsguiderailsareidentiedforthemovingcable.2.Stationarycablemodels2.1.BasicequationsWeconsidersixmodelsofthestationaryhoistcabletoevaluatetheeffectsofbendingstiffnessandboundaryconditionsonitsdynamiccharacteristics.Sincetheverticalcablehasnosag,itismodelledasatautstringandatensionedbeam.ShowninFig.1arethebeamandstringmodelsofthecablewiththesuspensionofthecaragainstitsguiderailsassumedtoberigid.ShowninFig.2arethebeamandstringmodelsofthecablewiththesuspensionofthecaragainsttheguiderailsmodelledbyaresultantstiffnesskeanddampingcoefcientce:Inallthecasesthemassofthecarisdenotedbyme:WhilethecarcanhavenitedimensionsinFig.1,itismodelledasapointmassinFig.2.Whenthecableismodelledasatensionedbeam,asshowninFigs.1(a)and(b),and2(a)and(b),itsfreelateralvibrationinthexyplaneisgovernedbyryttx；tC0Pxyxx；tC138xEIyxxxxx；t0；0oxol；1wherethesubscriptdenotespartialdifferentiation,yx；tisthelateraldisplacementofthecableparticleatpositionxattimet；listhelengthofthecable,risthemassperunitlength,EIisthebendingstiffness,andPxisthetensionatpositionxgivenbyPxmerlC0xC138g；2inwhichgistheaccelerationduetogravity.Theboundaryconditionsofthecablewithxedends,asshowninFig.1(a),arey0；tyx0；t0；yl；tyxl；t0:3xlyemyememy(a)(c)(b)Fig.1.Schematicofthestationaryhoistcablewiththesuspensionofthecaragainstitsguiderailsassumedtoberigid:(a)xedxedbeammodel,(b)pinnedpinnedbeammodel,and(c)stringmodel.W.D.Zhu,G.Y.Xu/JournalofSoundandVibration263(2003)679699680Theboundaryconditionsofthecablewithpinnedends,asshowninFig.1(b),arey0；tyxx0；t0；yl；tyxxl；t0:4ForthecablemodelsinFig.2(a)and(b),theboundaryconditionsatx0arethesameasthoseinEqs.(3)and(4),respectively,andtheboundaryconditionsatxlareyxxl；t0；EIyxxxl；tPlyxl；tmeyttl；tceytl；tkeyl；t:5NotethatthebendingmomentatxlvanishesintherstequationinEq.(5)becausetherotaryinertiaofthecarisnotconsidered.ThegoverningequationforthemodelsinFigs.1(c)and2(c)isgivenbyEq.(1)withEI0；andtheboundaryconditionatx0isy0；t0:TheboundaryconditionatxlforthemodelinFig.1(c)isyl；t0andtheboundaryconditionatxlforthemodelinFig.2(c)isgivenbythesecondequationinEq.(5)withEI0:DuetovanishingslopeofthecableatthexedendsinFigs.1(a)and2(a),themodelsinFigs.1(c)and2(c)cannotbeobtainedfromthemodelsinFigs.1(a)and2(a),respectively,bysettingEI0:Inadditiontoprovidinganominaltensionmeg；themassofthecarresultsinaninertialforceinthesecondequationinEq.(5)forthemodelsinFig.2.GalerkinsmethodandtheassumedmodesmethodareusedtodiscretizethegoverningpartialdifferentialequationsforthemodelsinFigs.1and2,respectively.ThesolutionofEq.(1)isassumedintheformyx；tXnj1qjtfjx；6wherefjxarethetrialfunctions,qjtarethegeneralizedcoordinates,andnisthenumberofincludedmodes.ThetrialfunctionsforthemodelsinFig.1satisfyalltheboundaryconditionsandthoseforthemodelsinFig.2satisfyalltheboundaryconditionsexcepttheforceboundaryem/2ek/2ek/2ec/2ecy/2ek/2ek/2ec/2ecemylx/2ek/2ek/2ec/2ecyem(a)(b)(c)Fig.2.Schematicofthestationaryhoistcablewherethecarismodelledasapointmassmeanditssuspensionagainsttheguiderailshasaresultantstiffnesskeanddampingcoefcientce:(a)beammodelwithaxedendatx0；(b)beammodelwithapinnedendatx0；and(c)stringmodel.W.D.Zhu,G.Y.Xu/JournalofSoundandVibration263(2003)679699681conditioninEq.(5).SubstitutingEq.(6)intoEq.(1)andthesecondequationinEq.(5),multiplyingthegoverningequationbyfix(i1；2；y；n),integratingitfromx0tol；andusingtheresultingboundaryconditionyieldsthediscretizedequationsforthemodelsinFig.2(a)and(b):M.qtCqtKqt0；7whereqq1；q2；y；qnC138TisthevectorofgeneralizedcoordinatesandM,K,andCarethesymmetricmass,stiffness,anddampingmatrices,respectively,withentriesMijZl0rfixfjxdxmefilfjl；8KijZl0Pxf0ixf0jxdxZl0EIf00ixf00jxdxkefilfjl；9Cijcefilfjl；10inwhichtheprimedenotesdifferentiationwithrespecttox:ThediscretizedequationsforthemodelinFig.2(c)aregivenbyEqs.(7)(10)withEI0inEq.(9).ThediscretizedequationsforthemodelsinFig.1(a)and(b)aregivenbyEqs.(7)(10)withme0inEq.(8)andkece0inEqs.(9)and(10)；thediscretizedequationsforthemodelinFig.1(c)aregivenbyEqs.(7)(10)withme0inEq.(8)andkeEIce0inEqs.(9)and(10).WhilethediscretizedequationsforthemodelsinFig.1(a)and(b)havethesameform,thetrialfunctionsusedsatisfydifferentboundaryconditions.ThisalsoholdsforthemodelsinFig.2(a)and(b).TheeigenfunctionsofaxedxedbeamandthoseofaxedxedbeamunderuniformtensionTmegareusedasthetrialfunctionsforthemodelinFig.1(a).Theeigenfunctionsofapinnedpinnedbeam,whichareidenticaltothoseofapinnedpinnedbeamunderuniformtension,areusedasthetrialfunctionsforthemodelinFig.1(b).Theeigenfunctionsofaxedxedstring,whichareidenticaltothoseofapinnedpinnedbeam,areusedasthetrialfunctionsforthemodelinFig.1(c).DuetothesametrialfunctionsthediscretizedequationsforthemodelinFig.1(c)canbeobtainedfromthoseforthemodelinFig.1(b)bysettingEI0:TheeigenfunctionsofacantileverbeamandthoseofaxedfreebeamunderuniformtensionTmegareusedasthetrialfunc