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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
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