英文翻译Vibration of elevator cables with small bending stiffness.pdf

JOURNAL OFSOUND ANDVIBRATION/locate/jsviJournalofSoundandVibration263(2003)679699LettertotheEditorVibrationofelevatorcableswithsmallbendingstiffnessW.D.Zhu*,G.Y.XuDepartment of Mechanical Engineering, University of Maryland Baltimore County, 1000 Hilltop Circle, Baltimore,MD 21250, USAReceived27September2002;accepted3October20021. IntroductionWhilecablesareemployedindiverseengineeringapplicationsincludingsuspensionbridges1,elevators2,powertransmissionlines3,andmarinetowingandmooringsystems4,theyaresubjecttovibrationduetotheirhighexibilityandlowintrinsicdamping.IrvineandCaughey5and Triantafyllou 6 studied the dynamics of suspended cables with horizontal and inclinedsupports.SergevandIwan7andChengandPerkins8analyzedthevibrationofcableswithattachedmasses.Simpson9,Triantafyllou10,andPerkinsandMote11studiedthein-planeandthree-dimensionalvibrationoftravellingcables.WickertandMote12andZhuandMote13analyzedthedynamicresponseoftravellingcableswithattachedpayloads.Whilethebendingstiffnessofcablesisneglectedinmoststudies,itwasincludedinthemodelsinRefs.14,15toavoidthesingularbehaviorsassociatedwithvanishingcabletension.Bendingstiffnesswasalsoaccountedforwhencablesaresubjectedtoexternalmoments3,16orwhentheirlocalbendingstressesneedtobedetermined17.Vibrationofelevatorcableshasbeenstudiedbyseveralresearchers2,1821.ChiandShu2calculatedthenaturalfrequenciesassociatedwiththelongitudinalvibrationofastationarycableand car system. Roberts 18 used lumped mass approximations to model the longitudinaldynamicsofhoistandcompensationcablesinhigh-riseelevators.Yamamotoetal.19analyzedthefreeandforcedlateralvibrationofastationarystringwithslowly,linearlyvaryinglength.Terumichi et al. 20 examined the lateral vibration of a travelling string with slowly, linearlyvaryinglengthandamass-springtermination.ZhuandNi21analyzedthedynamicstabilityoftravellingmediawithvariablelength.Thevibratoryenergyofthemediawasshowntodecreaseandincreaseingeneralduringextensionandretraction,respectively.Duetoitssmallbendingstiffnessrelativetothetension,themovinghoistcablewasmodelledasatravellingstringinRef.21.Byincludingthebendingstiffnessinthemodelsforthestationaryandmovinghoistcableswithdifferentboundaryconditions,theeffectsofbendingstiffnessandboundaryconditionsontheirdynamiccharacteristicsareinvestigatedhere.Convergenceofthe*Correspondingauthor.Tel.:+1-410-455-3394;fax:+1-410-455-1052.E-mail address: (W.D.Zhu).0022-460X/03/$-seefrontmatterr2002ElsevierScienceLtd.Allrightsreserved.doi:10.1016/S0022-460X(02)01468-2modelsisexamined.Theoptimalstiffnessanddampingcoefcientofthesuspensionofthecaragainstitsguiderailsareidentiedforthemovingcable.2. Stationary cable models2.1. Basic equationsWeconsidersixmodelsofthestationaryhoistcabletoevaluatetheeffectsofbendingstiffnessandboundaryconditionsonitsdynamiccharacteristics.Sincetheverticalcablehasnosag,itismodelledasatautstringandatensionedbeam.ShowninFig.1arethebeamandstringmodelsofthecablewiththesuspensionofthecaragainstitsguiderailsassumedtoberigid.ShowninFig.2arethebeamandstringmodelsofthecablewiththesuspensionofthecaragainsttheguiderailsmodelledbyaresultantstiffness keanddampingcoefcient ce:Inallthecasesthemassofthecarisdenotedbyme:WhilethecarcanhavenitedimensionsinFig.1,itismodelledasapointmassinFig.2.Whenthecableismodelledasatensionedbeam,asshowninFigs.1(a)and(b),and2(a)and(b),itsfreelateralvibrationinthe xy planeisgovernedbyryttx; tC0Pxyxx; tC138x EIyxxxxx; t0; 0oxol; 1wherethesubscriptdenotespartialdifferentiation, yx; t isthelateraldisplacementofthecableparticleatposition x attime t; l isthelengthofthecable,risthemassperunitlength, EI isthebendingstiffness,and Px isthetensionatposition x givenbyPxmerl C0 xC138g; 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 / Journal of Sound and Vibration 263 (2003) 679699680Theboundaryconditionsofthecablewithpinnedends,asshowninFig.1(b),arey0; tyxx0; t0; yl; tyxxl; t0: 4ForthecablemodelsinFig.2(a)and(b),theboundaryconditionsat x 0arethesameasthoseinEqs.(3)and(4),respectively,andtheboundaryconditionsat x l areyxxl; t0; EIyxxxl; tPlyxl; tmeyttl; tceytl; tkeyl; t: 5Notethatthebendingmomentatx l vanishesintherstequationinEq.(5)becausetherotaryinertiaofthecarisnotconsidered.ThegoverningequationforthemodelsinFigs.1(c)and2(c)isgivenbyEq.(1)with EI 0; andtheboundaryconditionat x 0isy0; t0: Theboundaryconditionat x l forthemodelinFig.1(c)is yl; t0andtheboundaryconditionat x l forthemodelinFig.2(c)isgivenbythesecondequationinEq.(5)with EI 0: DuetovanishingslopeofthecableatthexedendsinFigs.1(a)and2(a),themodelsinFigs.1(c)and2(c)cannotbeobtainedfromthemodelsin Figs.1(a)and2(a),respectively,bysetting EI 0:Inadditiontoprovidinganominaltension meg; themassofthecarresultsinaninertialforceinthesecondequationinEq.(5)forthemodelsinFig.2.Galerkinsmethodandtheassumedmodesmethodareusedtodiscretizethegoverningpartialdifferential equations for the models in Figs.1 and 2, respectively. The solution of Eq.(1) isassumedintheformyx; tXnj1qjtfjx; 6wherefjx arethetrialfunctions, qjt arethegeneralizedcoordinates,and n isthenumberofincludedmodes.ThetrialfunctionsforthemodelsinFig.1satisfyalltheboundaryconditionsandthoseforthemodelsinFig.2satisfyalltheboundaryconditionsexcepttheforceboundaryem/2ek / 2ek/2ec /2ecy/2ek / 2ek/2ec /2ecemylx/2ek / 2ek/2ec /2ecyem(a) (b) (c)Fig.2. Schematicofthestationaryhoistcablewherethecarismodelledasapointmass meanditssuspensionagainsttheguiderailshasaresultantstiffnesskeanddampingcoefcientce:(a)beammodelwithaxedendat x 0;(b)beammodelwithapinnedendat x 0; and(c)stringmodel.W.D. Zhu, G.Y. Xu / Journal of Sound and Vibration 263 (2003) 679699 681condition in Eq.(5). Substituting Eq.(6) into Eq.(1) and the second equation in Eq.(5),multiplyingthegoverningequationby fix (i 1; 2;y; n),integratingitfrom x 0tol; andusingtheresultingboundaryconditionyieldsthediscretizedequationsforthemodelsinFig.2(a)and(b):M.qtCqtKqt0; 7where q q1; q2;y; qnC138Tis the vector of generalized coordinates and M, K, and C are thesymmetricmass,stiffness,anddampingmatrices,respectively,withentriesMijZl0rfixfjxdx mefilfjl; 8KijZl0Pxf0ixf0jxdx Zl0EIf00ixf00jxdx kefilfjl; 9Cij cefilfjl; 10inwhichtheprimedenotesdifferentiationwithrespectto x: ThediscretizedequationsforthemodelinFig.2(c)aregivenbyEqs.(7)(10)with EI 0inEq.(9).ThediscretizedequationsforthemodelsinFig.1(a)and(b)aregivenbyEqs.(7)(10)withme0inEq.(8)andke ce0inEqs.(9)and(10);thediscretizedequationsforthemodelinFig.1(c)aregivenbyEqs.(7)(10)with me0inEq.(8)and ke EI ce0inEqs.(9)and(10).WhilethediscretizedequationsforthemodelsinFig.1(a)and(b)havethesameform,thetrialfunctionsusedsatisfydifferentboundaryconditions.ThisalsoholdsforthemodelsinFig.2(a)and(b).The eigenfunctions of a xedxed beamand those of a xedxed beamunder uniformtension T meg areusedasthetrialfunctionsforthemodelinFig.1(a).Theeigenfunctionsofapinnedpinned beam, which are identical to those of a pinnedpinned beam under uniformtension,areusedasthetrialfunctionsforthemodelinFig.1(b).Theeigenfunctionsofaxedxedstring,whichareidenticaltothoseofapinnedpinnedbeam,areusedasthetrialfunctionsforthemodelinFig.1(c).Duetothesametrialfunctionsthediscretizedequationsforthemodelin Fig.1(c) can be obtained fromthose for the model in Fig.1(b) by setting EI 0: Theeigenfunctionsofacantileverbeamandthoseofaxedfreebeamunderuniformtension T meg areusedasthetrialfunctionsforthemodelinFig.2(a).Theeigenfunctionsofapinnedfreebeamand those of a pinnedfree beamunder uniformtension T meg are used as the trialfunctionsforthemodelinFig.2(b).TheeigenfunctionsofaxedfreestringareusedasthetrialfunctionsforthemodelinFig.2(c).Notethatapinnedfreebeamhasarigid-bodymodeandaxedfreestringdoesnot.ThediscretizedequationsforthemodelinFig.2(c)cannotbeobtainedasaspecialcasefromthoseforthemodelin Fig.2(b)duetothedifferenttrialfunctionsused.AllthetrialfunctionsarenormalizedandgiveninAppendixA.BytheorthogonalityrelationsthemassmatrixforthemodelsinFig.1isadiagonalmatrix.IftheinitialdisplacementandvelocityofthecableinFigs.1and2aregivenbyyx;0andytx;0;respectively,theinitialconditionsforthegeneralizedcoordinatesareqj0Zl0fjxyx;0dx; qj0Zl0fjxytx;0dx: 11W.D. Zhu, G.Y. Xu / Journal of Sound and Vibration 263 (2003) 679699682TheenergyofthemodelsinFig.1(a)and(b)isEvt12Zl0ry2t Py2x EIy2xxdx 12andthatofthemodelsinFig.2(a)and(b)isEvt12Zl0ry2t Py2x EIy2xxdx 12mey2tl; t12key2l; t: 13TheenergyofthemodelsinFigs.1(c)and2(c)isgivenbyEqs.(12)and(13),respectively,withEI 0:SubstitutingEq.(6)intoEqs.(12)and(13)yieldsthediscretizedenergyexpressionforthemodelsinFigs.1and2:Evt12qTtMqtqTtKqtC138; 14whereMandKarethecorrespondingmassandstiffnessmatrices.DifferentiatingEqs.(12)and(13)andusingthegoverningequationsandboundaryconditionsyieldsEvt0forthemodelsinFig.1andEvtC0cey2tl; t forthemodelsinFig.2.ThediscretizedexpressionofEvt forthemodelsinFig.2isEvtC0qTtCqt:2.2. Results and discussionTheparametersusedherearesimilartothoseinRefs.21,22:r 1:005kg=m; EI 1:39Nm2;me756kg; g 9:81m=s2; l 171m; and ke2083N=m: Theundampednaturalfrequencies,oi; and modes, xi; of the systemin Eq.(7) are obtained fromthe eigenvalue problem, Kxio2iMxi(i 1;2;y; n).UsingtheaforementionedtrialfunctionsandvariousnumbersoftermsinEq.(6),therstthreenaturalfrequenciesofthemodelsinFigs.1and2arecalculatedasshowninTable 1. The trial functions for the models in Figs.1(a), and 2(a) and (b), corresponding toT meg and T 0; are referred to as the tensioned and untensioned beameigenfunctions,respectively. As the natural frequencies converge fromabove, the use of the tensioned beameigenfunctionssignicantlyacceleratestheconvergenceofthenaturalfrequenciesofthemodelinFig.1(a). The untensioned beam eigenfunctions yield improved estimates of the naturalfrequenciesofthemodelsinFig.2(a)and(b)for n 1: DuetotherotationalconstraintsatthexedendsthenaturalfrequenciesofthemodelsinFigs.1(a)and2(a)areslightlyhigherthanthoseofthemodelsinFigs.1(b)and2(b),respectively.DuetosmallbendingstiffnessthenaturalfrequenciesofthemodelinFig.1(b)areidenticaltothoseofthemodelinFig.1(c)withintheaccuracyshownforall n: ThenaturalfrequenciesofthemodelsinFig.1(b)and(c)convergeatsimilar rates as the natural frequencies of the model in Fig.1(a) using the tensioned beameigenfunctions.ThenaturalfrequenciesofthemodelinFig.2(c)convergeatsimilarratesasthenaturalfrequenciesofthemodelsinFig.2(a)and(b)usingthetensionedbeameigenfunctions.ThenaturalfrequenciesofthemodelsinFig.2approachthoseofthecorrespondingmodelsinFig.1when keapproachesinnity.Consider the undamped (i.e., ce0) responses of the models in Figs.1 and 2 to theinitialdisplacements given in Appendix B and zero initial velocity. The initial displacement for themodelsinFigs.1(a)and2(a)isthestaticdeectionofaxedxedbeamunderuniformtensionmeg; subjectedtoaconcentratedforceat x a resultinginadisplacement d at x a: TheinitialW.D. Zhu, G.Y. Xu / Journal of Sound and Vibration 263 (2003) 679699 683displacementforthemodelsinFigs.1(b)and2(b)isthestaticdeectionofapinnedpinnedbeamunderthesametension,subjectedtoaconcentratedforceatx aresultinginadisplacementd atx a: TheinitialdisplacementforthemodelsinFigs.1(c)and2(c)isthestaticdeectionofaxedxedstringsubjectedtoaconcentratedforceat x a withadisplacement d at x a: Theabove initial displacements for a 100mand d 0:07mare shown in Fig.3; the resultingdisplacementsandvelocitiesofaparticleofthemodelsinFigs.1and2atx 156mareshowninFigs.4and5,respectively,for0ptptf38s; where tfisthenaltimeofthemovingcableinSection3.2.Notethatthesmallbendingstiffnessleadstotheboundarylayersinthedeectionsofthebeamsinthevicinityofthexedendsandconcentratedforcetoensuresatisfactionoftheboundaryandinternalconditions.DuetosmallbendingstiffnesstheresponsesofthemodelsinTable1Therstthreenaturalfrequencies(inrad/s)ofthemodelsinFigs.1and2calculatedusingdifferentnumbersoftermsinEq.(6),where T isthetensionofthebeamswhoseeigenfunctionsareusedasthetrialfunctionsforthemodelsinFigs.1(a),and2(a)and(b)Numberofmodes n 1 2 3 10 20 30 50 100 150Fig.1(a) T 0 1st 1.859 1.858 1.774 1.710 1.687 1.679 1.673 1.668 1.6662nd 3.598 3.596 3.412 3.372 3.358 3.345 3.336 3.3333rd 5.303 5.128 5.061 5.038 5.019 5.004 5.000T meg 1st 1.666 1.664 1.664 1.664 1.664 1.664 1.664 1.664 1.6642nd 3.333 3.328 3.328 3.328 3.328 3.328 3.328 3.3283rd 5.002 4.991 4.991 4.991 4.991 4.991 4.991Fig.1(b) 1st 1.665 1.663 1.663 1.663 1.663 1.663 1.663 1.663 1.6632nd 3.332 3.327 3.327 3.327 3.327 3.327 3.327 3.3273rd 5.000 4.990 4.990 4.990 4.990 4.990 4.990Fig.1(c) 1st 1.665 1.663 1.663 1.663 1.663 1.663 1.663 1.663 1.6632nd 3.332 3.327 3.327 3.327 3.327 3.327 3.327 3.3273rd 5.000 4.990 4.990 4.990 4.990 4.990 4.990Fig.2(a) T 0 1st 1.636 1.533 1.532 1.506 1.501 1.499 1.498 1.497 1.4962nd 1.898 1.887 1.851 1.844 1.842 1.840 1.838 1.8383rd 3.480 3.397 3.374 3.366 3.360 3.355 3.354T meg 1st 1.596 1.559 1.539 1.510 1.503 1.500 1.499 1.497 1.4972nd 1.986 1.917 1.857 1.847 1.844 1.841 1.839 1.8383rd 3.671 3.424 3.387 3.375 3.365 3.358 3.356Fig.2(b) T 0 1st 1.619 1.509 1.498 1.496 1.496 1.496 1.496 1.496 1.4962nd 1.840 1.837 1.837 1.837 1.837 1.837 1.837 1.8373rd 3.398 3.351 3.351 3.351 3.351 3.351 3.351T meg 1st 1.596 1.559 1.539 1.510 1.503 1.500 1.498 1.497 1.4972nd 1.985 1.917 1.857 1.847 1.843 1.841 1.839 1.8383rd 3.670 3.424 3.386 3.374 3.365 3.358 3.355Fig.2(c) 1st 1.596 1.559 1.539 1.510 1.503 1.500 1.498 1.497 1.4972nd 1.985 1.917 1.857 1.847 1.843 1.841 1.839 1.8383rd 3.670 3.424 3.386 3.374 3.365 3.358 3.355W.D. Zhu, G.Y. Xu / Journal of Sound and Vibration 263 (2003) 679699684Figs.1 and 2 are virtually indistinguishable within the scales of the plots in Figs.4 and 5,respectively.WhilethemaximumdisplacementinFig.5islargerthanthatinFig.4,theenergiesshownbythehorizontallinesinFigs.11(a)and12(a)areessentiallythesame.Convergenceoftheresponsesofthevariousmodelsissimilartothatofthenaturalfrequenciesdiscussedearlier.Undertheaboveinitialconditionsthetransverseforceatthelowerend(i.e., x l)ofeachmodelinFig.1isshowninFig.6(a).Whilethetransverseforceatx l isgivenbytheshearforceC0EIyxxxl; t forthemodelinFig.1(a)because yxx; l0; bythetransversecomponentofthetension Plyxl; tforthemodelinFig.1(c),andbybothtermsC0EIyxxxl; tPlyxl; tforthemodelinFig.1(b),ithasessentiallythesamevalueforthethreemodels.ThebendingmomentatthelowerendofthemodelinFig.1(a)isshowninFig.6(b);thebendingmomentatthetwoendsof the model in Fig.1(b) vanishes identically. The bending moment at an interior point (e.g.,x 156m)ofthemodelsinFig.1(a)and(b)hasanamplitudethatisordersofmagnitudesmaller0.000.020.040.060.080 50 100 150 200x (m)0.069920.069960.0700099.9 100 100.10.E+004.E-068.E-0600.050.010.E+005.E-061.E-05170.99 170.995 171y(x, 0)(m) Fig.3. The initial displacements for the models in: Figs.1(a) and 2(a) (dashed lines); Figs.1(b) and 2(b) (dots);Figs.1(c) and 2(c) (solid lines). The boundary layers are shown in dashed lines in the expanded views near theboundariesanddashedlinesanddotsintheexpandedviewnear x 100m:(a)-0.02-0.010.000.010.02(b)-0.10-0.050.000.050.10010203040t (s)yt(156, t)(m/s) y(156, t)(m) Fig.4. The displacement (a) and velocity (b) of the particle at x 156min the models in Fig.1 under thecorrespondinginitialdisplacementsshowninFig.3:dashedlines,Fig.1(a);dashdottedlines,Fig.1(b);solidlines,Fig.1(c). The energies of the models are shown in Fig.11(a). The tensioned beameigenfunctions are used for theresponseofthemodelinFig.1(a)and n 30inallthecases.W.D. Zhu, G.Y. Xu / Journal of Sound and Vibration 263 (2003) 679699 685thanthoseatthexedendsofthemodelinFig.1(a).OnlythetensionedbeameigenfunctionscanbeusedtoestimatethebendingmomentandshearforceatthexedendsofthemodelinFig.1(a);theuntensionedbeameigenfunctionswillleadtoslowlyconvergentseriesforthehigherorderderivatives, yxxand yxxx: BoththetensionedanduntensionedbeameigenfunctionscanbeusedtodeterminethetransverseforceataninteriorpointofthemodelinFig.1(a)andanypointofthemodelinFig.1(b)becauseitisdominatedbythetransversecomponentofthetension,which involves the rst order derivative yx: The transverse force and bending moment arerelatedtothetransverseandbendingstresses,respectively,intheanalysisofcumulativefatiguedamage.(a)-0.04-0.0200.020.04(b)-0.1-0.0500.050.10 10203040t (s)y(156, t)(m) yt(156, t)(m/s) Fig.5. The displacement (a) and velocity (b) of the particle at x 156min the models in Fig.2 under thecorresponding initial displacements shownin Fig.3: dashedlines, Fig.2(a); dash-dotted lines, Fig.2(b); solid lines,Fig.2(c).TheenergiesofthemodelsareshowninFig.12(a).TheuntensionedbeameigenfunctionsareusedforthemodelsinFig.2(a)and(b)and n 30inallthecases.(a)-15-10-50510(b)-0.2-010203040t (s)Transverse force (N)EIyxx(l,t)(Nm) Fig.6. (a)ThetransverseforceatthelowerendofeachmodelinFig.1underthecorrespondinginitialdisplacementshowninFig.3:dashedlines,Fig.1(a);dash-dottedlines,Fig.1(b);solidlines,Fig.1(c).(b)ThebendingmomentattheupperendofthemodelinFig.1(a).Thetensionedbeameigenfunctionsareusedforthemodelin Fig.1(a)andn 100inallthecases.W.D. Zhu, G.Y. Xu / Journal of Sound and Vibration 263 (2003) 679699686Undertheaboveinitialconditionsthetransverseforceattheupperend(i.e., x 0)ofeachmodelinFig.2isshowninFig.7(a).SimilartothecaseinFig.6(a),thetransverseforceatx 0;thoughgivenbydifferentexpressions,hasessentiallythesamevalueforthethreemodels.Thebendin