【机械类毕业论文中英文对照文献翻译】小弯曲刚度电梯钢丝绳的振动
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机械类毕业论文中英文对照文献翻译
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【机械类毕业论文中英文对照文献翻译】小弯曲刚度电梯钢丝绳的振动,机械类毕业论文中英文对照文献翻译,机械类,毕业论文,中英文,对照,文献,翻译,弯曲,刚度,电梯,钢丝绳,振动
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JOURNAL OFSOUND ANDVIBRATION/locate/jsviJournal of Sound and Vibration 263 (2003) 679699Letter to the EditorVibration of elevator cables with small bending stiffnessW.D. Zhu*, G.Y. XuDepartment of Mechanical Engineering, University of Maryland Baltimore County, 1000 Hilltop Circle, Baltimore,MD 21250, USAReceived 27 September 2002; accepted 3 October 20021. IntroductionWhile cables are employed in diverse engineering applications including suspension bridges 1,elevators 2, power transmission lines 3, and marine towing and mooring systems 4, they aresubject to vibration due to their high flexibility and low intrinsic damping. Irvine and Caughey 5and Triantafyllou 6 studied the dynamics of suspended cables with horizontal and inclinedsupports. Sergev and Iwan 7 and Cheng and Perkins 8 analyzed the vibration of cables withattached masses. Simpson 9, Triantafyllou 10, and Perkins and Mote 11 studied the in-planeand three-dimensional vibration of travelling cables. Wickert and Mote 12 and Zhu and Mote13 analyzed the dynamic response of travelling cables with attached payloads. While the bendingstiffness of cables is neglected in most studies, it was included in the models in Refs. 14,15 toavoid the singular behaviors associated with vanishing cable tension. Bending stiffness was alsoaccounted for when cables are subjected to external moments 3,16 or when their local bendingstresses need to be determined 17.Vibration of elevator cables has been studied by several researchers 2,1821. Chi and Shu 2calculated the natural frequencies associated with the longitudinal vibration of a stationary cableand car system. Roberts 18 used lumped mass approximations to model the longitudinaldynamics of hoist and compensation cables in high-rise elevators. Yamamoto et al. 19 analyzedthe free and forced lateral vibration of a stationary string with slowly, linearly varying length.Terumichi et al. 20 examined the lateral vibration of a travelling string with slowly, linearlyvarying length and a mass-spring termination. Zhu and Ni 21 analyzed the dynamic stability oftravelling media with variable length. The vibratory energy of the media was shown to decreaseand increase in general during extension and retraction, respectively.Due to its small bending stiffness relative to the tension, the moving hoist cable was modelled asa travelling string in Ref. 21. By including the bending stiffness in the models for the stationaryand moving hoist cables with different boundary conditions, the effects of bending stiffness andboundary conditions on their dynamic characteristics are investigated here. Convergence of the*Corresponding author. Tel.: +1-410-455-3394; fax: +1-410-455-1052.E-mail address: wzhu (W.D. Zhu).0022-460X/03/$-see front matter r 2002 Elsevier Science Ltd. All rights reserved.doi:10.1016/S0022-460X(02)01468-2models is examined. The optimal stiffness and damping coefficient of the suspension of the caragainst its guide rails are identified for the moving cable.2. Stationary cable models2.1. Basic equationsWe consider six models of the stationary hoist cable to evaluate the effects of bending stiffnessand boundary conditions on its dynamic characteristics. Since the vertical cable has no sag, it ismodelled as a taut string and a tensioned beam. Shown in Fig. 1 are the beam and string models ofthe cable with the suspension of the car against its guide rails assumed to be rigid. Shown in Fig. 2are the beam and string models of the cable with the suspension of the car against the guide railsmodelled by a resultant stiffness keand damping coefficient ce: In all the cases the mass of the caris denoted by me: While the car can have finite dimensions in Fig. 1, it is modelled as a point massin Fig. 2. When the cable is modelled as a tensioned beam, as shown in Figs. 1(a) and (b), and 2(a)and (b), its free lateral vibration in the xy plane is governed byryttx;t ? Pxyxx;t?x EIyxxxxx;t 0;0oxol;1where the subscript denotes partial differentiation, yx;t is the lateral displacement of the cableparticle at position x at time t; l is the length of the cable, r is the mass per unit length, EI is thebending stiffness, and Px is the tension at position x given byPx me rl ? x?g;2in which g is the acceleration due to gravity. The boundary conditions of the cable with fixed ends,as shown in Fig. 1(a), arey0;t yx0;t 0;yl;t yxl;t 0:3xlyemyememy(a)(c)(b)Fig. 1. Schematic of the stationary hoist cable with the suspension of the car against its guide rails assumed to be rigid:(a) fixedfixed beam model, (b) pinnedpinned beam model, and (c) string model.W.D. Zhu, G.Y. Xu / Journal of Sound and Vibration 263 (2003) 679699680The boundary conditions of the cable with pinned ends, as shown in Fig. 1(b), arey0;t yxx0;t 0;yl;t yxxl;t 0:4For the cable models in Fig. 2(a) and (b), the boundary conditions at x 0 are the same as thosein Eqs. (3) and (4), respectively, and the boundary conditions at x l areyxxl;t 0;EIyxxxl;t Plyxl;t meyttl;t ceytl;t keyl;t:5Note that the bending moment at x l vanishes in the first equation in Eq. (5) because the rotaryinertia of the car is not considered. The governing equation for the models in Figs. 1(c) and 2(c) isgiven by Eq. (1) with EI 0; and the boundary condition at x 0 is y0;t 0: The boundarycondition at x l for the model in Fig. 1(c) is yl;t 0 and the boundary condition at x l forthe model in Fig. 2(c) is given by the second equation in Eq. (5) with EI 0: Due to vanishingslope of the cable at the fixed ends in Figs. 1(a) and 2(a), the models in Figs. 1(c) and 2(c) cannotbe obtained from the models in Figs. 1(a) and 2(a), respectively, by setting EI 0: In addition toproviding a nominal tension meg; the mass of the car results in an inertial force in the secondequation in Eq. (5) for the models in Fig. 2.Galerkins method and the assumed modes method are used to discretize the governing partialdifferential equations for the models in Figs. 1 and 2, respectively. The solution of Eq. (1) isassumed in the formyx;t Xnj1qjtfjx;6where fjx are the trial functions, qjt are the generalized coordinates, and n is the number ofincluded modes. The trial functions for the models in Fig. 1 satisfy all the boundary conditionsand those for the models in Fig. 2 satisfy all the boundary conditions except the force boundaryem/2ek/2ek/2ec/2ecy/2ek/2ek/2ec/2ecemylx/2ek/2ek/2ec/2ecyem(a)(b)(c)Fig. 2. Schematic of the stationary hoist cable where the car is modelled as a point mass meand its suspension againstthe guide rails has a resultant stiffness keand damping coefficient ce: (a) beam model with a fixed end at x 0; (b) beammodel with a pinned end at x 0; and (c) string model.W.D. Zhu, G.Y. Xu / Journal of Sound and Vibration 263 (2003) 679699681condition in Eq. (5). Substituting Eq. (6) into Eq. (1) and the second equation in Eq. (5),multiplying the governing equation by fix (i 1; 2;y;n), integrating it from x 0 to l; andusing the resulting boundary condition yields the discretized equations for the models in Fig. 2(a)and (b):M. qt C qt Kqt 0;7where q q1;q2;y;qn?Tis the vector of generalized coordinates and M, K, and C are thesymmetric mass, stiffness, and damping matrices, respectively, with entriesMijZl0rfixfjxdx mefilfjl;8KijZl0Pxf0ixf0jxdx Zl0EIf00ixf00jxdx kefilfjl;9Cij cefilfjl;10in which the prime denotes differentiation with respect to x: The discretized equations for themodel in Fig. 2(c) are given by Eqs. (7)(10) with EI 0 in Eq. (9). The discretized equations forthe models in Fig. 1(a) and (b) are given by Eqs. (7)(10) with me 0 in Eq. (8) and ke ce 0 inEqs. (9) and (10); the discretized equations for the model in Fig. 1(c) are given by Eqs. (7)(10)with me 0 in Eq. (8) and ke EI ce 0 in Eqs. (9) and (10). While the discretized equationsfor the models in Fig. 1(a) and (b) have the same form, the trial functions used satisfy differentboundary conditions. This also holds for the models in Fig. 2(a) and (b).The eigenfunctions of a fixedfixed beam and those of a fixedfixed beam under uniformtension T meg are used as the trial functions for the model in Fig. 1(a). The eigenfunctions of apinnedpinned beam, which are identical to those of a pinnedpinned beam under uniformtension, are used as the trial functions for the model in Fig. 1(b). The eigenfunctions of a fixedfixed string, which are identical to those of a pinnedpinned beam, are used as the trial functionsfor the model in Fig. 1(c). Due to the same trial functions the discretized equations for the modelin Fig. 1(c) can be obtained from those for the model in Fig. 1(b) by setting EI 0: Theeigenfunctions of a cantilever beam and those of a fixedfree beam under uniform tension T meg are used as the trial functions for the model in Fig. 2(a). The eigenfunctions of a pinnedfreebeam and those of a pinnedfree beam under uniform tension T meg are used as the trialfunctions for the model in Fig. 2(b). The eigenfunctions of a fixedfree string are used as the trialfunctions for the model in Fig. 2(c). Note that a pinnedfree beam has a rigid-body mode and afixedfree string does not. The discretized equations for the model in Fig. 2(c) cannot be obtainedas a special case from those for the model in Fig. 2(b) due to the different trial functions used. Allthe trial functions are normalized and given in Appendix A. By the orthogonality relations themass matrix for the models in Fig. 1 is a diagonal matrix. If the initial displacement and velocityof the cable in Figs. 1 and 2 are given by yx;0 and ytx;0; respectively, the initial conditions forthe generalized coordinates areqj0 Zl0fjxyx;0dx; qj0 Zl0fjxytx;0dx:11W.D. Zhu, G.Y. Xu / Journal of Sound and Vibration 263 (2003) 679699682The energy of the models in Fig. 1(a) and (b) isEvt 12Zl0ry2t Py2x EIy2xxdx12and that of the models in Fig. 2(a) and (b) isEvt 12Zl0ry2t Py2x EIy2xxdx 12mey2tl;t 12key2l;t:13The energy of the models in Figs. 1(c) and 2(c) is given by Eqs. (12) and (13), respectively, withEI 0: Substituting Eq. (6) into Eqs. (12) and (13) yields the discretized energy expression for themodels in Figs. 1 and 2:Evt 12 qTtM qt qTtKqt?;14where M and K are the corresponding mass and stiffness matrices. Differentiating Eqs. (12) and(13) and using the governing equations and boundary conditions yieldsEvt 0 for the models inFig. 1 andEvt ?cey2tl;t for the models in Fig. 2. The discretized expression ofEvt for themodels in Fig. 2 isEvt ? qTtC qt:2.2. Results and discussionThe parameters used here are similar to those in Refs. 21,22: r 1:005kg=m; EI 1:39 N m2;me 756kg; g 9:81m=s2; l 171m; and ke 2083N=m: The undamped natural frequencies,oi; and modes, xi; of the system in Eq. (7) are obtained from the eigenvalue problem, Kxio2iMxi(i 1;2;y;n). Using the aforementioned trial functions and various numbers of terms inEq. (6), the first three natural frequencies of the models in Figs. 1 and 2 are calculated as shown inTable 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 beam eigenfunctions,respectively. As the natural frequencies converge from above, the use of the tensioned beameigenfunctions significantly accelerates the convergence of the natural frequencies of the model inFig. 1(a). The untensioned beam eigenfunctions yield improved estimates of the naturalfrequencies of the models in Fig. 2(a) and (b) for n 1: Due to the rotational constraints at thefixed ends the natural frequencies of the models in Figs. 1(a) and 2(a) are slightly higher thanthose of the models in Figs. 1(b) and 2(b), respectively. Due to small bending stiffness the naturalfrequencies of the model in Fig. 1(b) are identical to those of the model in Fig. 1(c) within theaccuracy shown for all n: The natural frequencies of the models in Fig. 1(b) and (c) converge atsimilar rates as the natural frequencies of the model in Fig. 1(a) using the tensioned beameigenfunctions. The natural frequencies of the model in Fig. 2(c) converge at similar rates as thenatural frequencies of the models in Fig. 2(a) and (b) using the tensioned beam eigenfunctions.The natural frequencies of the models in Fig. 2 approach those of the corresponding models inFig. 1 when keapproaches infinity.Consider the undamped (i.e., ce 0) responses of the models in Figs. 1 and 2 to the initialdisplacements given in Appendix B and zero initial velocity. The initial displacement for themodels in Figs. 1(a) and 2(a) is the static deflection of a fixedfixed beam under uniform tensionmeg; subjected to a concentrated force at x a resulting in a displacement d at x a: The initialW.D. Zhu, G.Y. Xu / Journal of Sound and Vibration 263 (2003) 679699683displacement for the models in Figs. 1(b) and 2(b) is the static deflection of a pinnedpinned beamunder the same tension, subjected to a concentrated force at x a resulting in a displacement d atx a: The initial displacement for the models in Figs. 1(c) and 2(c) is the static deflection of afixedfixed string subjected to a concentrated force at x a with a displacement d at x a: Theabove initial displacements for a 100m and d 0:07 m are shown in Fig. 3; the resultingdisplacements and velocities of a particle of the models in Figs. 1 and 2 at x 156m are shown inFigs. 4 and 5, respectively, for 0ptptf 38 s; where tfis the final time of the moving cable inSection 3.2. Note that the small bending stiffness leads to the boundary layers in the deflections ofthe beams in the vicinity of the fixed ends and concentrated force to ensure satisfaction of theboundary and internal conditions. Due to small bending stiffness the responses of the models inTable 1The first three natural frequencies (in rad/s) of the models in Figs. 1 and 2 calculated using different numbers of terms inEq. (6), where T is the tension of the beams whose eigenfunctions are used as the trial functions for the models inFigs. 1(a), and 2(a) and (b)Number of modes n12310203050100150Fig. 1(a)T 01st1.8591.8581.7741.7101.6871.6791.6731.6681.6662nd3.5983.5963.4123.3723.3583.3453.3363.3333rd5.3035.1285.0615.0385.0195.0045.000T meg1st1.6661.6641.6641.6641.6641.6641.6641.6641.6642nd3.3333.3283.3283.3283.3283.3283.3283.3283rd5.0024.9914.9914.9914.9914.9914.991Fig. 1(b)1st1.6651.6631.6631.6631.6631.6631.6631.6631.6632nd3.3323.3273.3273.3273.3273.3273.3273.3273rd5.0004.9904.9904.9904.9904.9904.990Fig. 1(c)1st1.6651.6631.6631.6631.6631.6631.6631.6631.6632nd3.3323.3273.3273.3273.3273.3273.3273.3273rd5.0004.9904.9904.9904.9904.9904.990Fig. 2(a)T 01st1.6361.5331.5321.5061.5011.4991.4981.4971.4962nd1.8981.8871.8511.8441.8421.8401.8381.8383rd3.4803.3973.3743.3663.3603.3553.354T meg1st1.5961.5591.5391.5101.5031.5001.4991.4971.4972nd1.9861.9171.8571.8471.8441.8411.8391.8383rd3.6713.4243.3873.3753.3653.3583.356Fig. 2(b)T 01st1.6191.5091.4981.4961.4961.4961.4961.4961.4962nd1.8401.8371.8371.8371.8371.8371.8371.8373rd3.3983.3513.3513.3513.3513.3513.351T meg1st1.5961.5591.5391.5101.5031.5001.4981.4971.4972nd1.9851.9171.8571.8471.8431.8411.8391.8383rd3.6703.4243.3863.3743.3653.3583.355Fig. 2(c)1st1.5961.5591.5391.5101.5031.5001.4981.4971.4972nd1.9851.9171.8571.8471.8431.8411.8391.8383rd3.6703.4243.3863.3743.3653.3583.355W.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. While the maximum displacement in Fig. 5 is larger than that in Fig. 4, the energiesshown by the horizontal lines in Figs. 11(a) and 12(a) are essentially the same. Convergence of theresponses of the various models is similar to that of the natural frequencies discussed earlier.Under the above initial conditions the transverse force at the lower end (i.e., x l) of eachmodel in Fig. 1 is shown in Fig. 6(a). While the transverse force at x l is given by the shear force?EIyxxxl;t for the model in Fig. 1(a) because yxx;l 0; by the transverse component of thetension Plyxl;t for the model in Fig. 1(c), and by both terms ?EIyxxxl;t Plyxl;t for themodel in Fig. 1(b), it has essentially the same value for the three models. The bending moment atthe lower end of the model in Fig. 1(a) is shown in Fig. 6(b); the bending moment at the two endsof the model in Fig. 1(b) vanishes identically. The bending moment at an interior point (e.g.,x 156m) of the models in Fig. 1(a) and (b) has an amplitude that is orders of magnitude smaller0.000.020.040.060.08050100150200x (m)0.069920.069960.0700099.9100100.10.E+004.E-068.E-0600.0050.010.E+005.E-061.E-05170.99170.995171y(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 theboundaries and dashed lines and dots in the expanded view near x 100 m:(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 156 m in the models in Fig. 1 under thecorresponding initial displacements shown in Fig. 3: dashed lines, Fig. 1(a); dashdotted lines, Fig. 1(b); solid lines,Fig. 1(c). The energies of the models are shown in Fig. 11(a). The tensioned beam eigenfunctions are used for theresponse of the model in Fig. 1(a) and n 30 in all the cases.W.D. Zhu, G.Y. Xu / Journal of Sound and Vibration 263 (2003) 679699685than those at the fixed ends of the model in Fig. 1(a). Only the tensioned beam eigenfunctionscan be used to estimate the bending moment and shear force at the fixed ends of the model inFig. 1(a); the untensioned beam eigenfunctions will lead to slowly convergent series for the higherorder derivatives, yxxand yxxx: Both the tensioned and untensioned beam eigenfunctions can beused to determine the transverse force at an interior point of the model in Fig. 1(a) and any pointof the model in Fig. 1(b) because it is dominated by the transverse component of the tension,which involves the first order derivative yx: The transverse force and bending moment arerelated to the transverse and bending stresses, respectively, in the analysis of cumulative fatiguedamage.(a)-0.04-0.0200.020.04(b)-0.1-0.0500.050.1010203040t (s)y(156, t) (m) yt(156, t) (m/s) Fig. 5. The displacement (a) and velocity (b) of the particle at x 156 m in the models in Fig. 2 under thecorresponding initial displacements shown in Fig. 3: dashed lines, Fig. 2(a); dash-dotted lines, Fig. 2(b); solid lines,Fig. 2(c). The energies of the models are shown in Fig. 12(a). The untensioned beam eigenfunctions are used for themodels in Fig. 2(a) and (b) and n 30 in all the cases.(a)-15-10-50510(b)-0.2-0.100.10.2010203040t (s)Transverse force (N)EIyxx(l,t) (Nm) Fig. 6. (a) The transverse force at the lower end of each model in Fig. 1 under the corresponding initial displacementshown in Fig. 3: dashed lines, Fig. 1(a); dash-dotted lines, Fig. 1(b); solid lines, Fig. 1(c). (b) The bending moment at theupper end of the model in Fig. 1(a). The tensioned beam eigenfunctions are used for the model in Fig. 1(a) and n 100in all the cases.W.D. Zhu, G.Y. Xu / Journal of Sound and Vibration 263 (2003) 679699686Under the above initial conditions the transverse force at the upper end (i.e., x 0) of eachmodel in Fig. 2 is shown in Fig. 7(a). Similar to the case in Fig. 6(a), the transverse force at x 0;though given by different expressions, has essentially the same value for the three models. Thebending moment at the upper end of the model in Fig. 2(a) is shown in Fig. 7(b). Note that theeigenfunctions of a fixedfree beam under uniform tension T meg rgl; which is the cabletension at x 0; are used to calculate the shear force and bending moment at the upper end of themodel in Fig. 2(a). Both the tensioned and untensioned beam eigenfunctions can be used tocalculate the transverse force at an interior point of the model in Fig. 2(a) and any point of themodel in Fig. 2(b). Only the untensioned beam eigenfunctions can be used to determine thetransverse force at the lower end of the models in Fig. 2(a) and (b) because they satisfy a morerealisticboundarycondition,EIyxxxl;t 0;thanthetensionedbeameigenfunctions,EIyxxxl;t Tyxl;t: The transverse force at the lower end of the model in Fig. 2(c) cannot bedetermined here because the trial functions satisfy f0jl 0:3. Moving cable models3.1. Basic equationsShown in Figs. 8 and 9 are the six models of the moving hoist cable corresponding to thestationary cable models in Figs. 1 and 2. During its motion the cable has a variable length lt andan axial velocity vt lt: When the cable is modelled as a travelling tensioned beam, as shownin Figs. 8(a) and (b), and 9(a) and (b), its free lateral vibration relative to the fixed coordinatesystem, xy; is governed by 21rD2yx;tDt2? Px;tyxx;t?x EIyxxxxx;t 0;0oxolt;15(a)-15-5515(b)-0.2-0.100.10.2010203040Transverse force (N)EIyxx(0,t) (Nm) t (s)Fig. 7. (a) The transverse force at the upper end of each model in Fig. 2 under the corresponding initial displacementshown in Fig. 3: dashed lines, Fig. 2(a); dashdotted lines, Fig. 2(b); solid lines, Fig. 2(c). (b) The bending moment atthe upper end of the model in Fig. 2(a). In all the cases n 100:W.D. Zhu, G.Y. Xu / Journal of Sound and Vibration 263 (2003) 679699687whereD2Dt22t2 2vt2xt v2t2x2 vtx;16yx;t is the lateral displacement of the cable particle instantaneously located at position x at timet; Px;t is the tension given byPx;t me rlt ? x?g ? vt?;17and the other variables are defined in Section 2.1. The governing equation for the models inFigs. 8(c) and 9(c) is given by Eq. (15) with EI 0: The boundary conditions for each model inFigs. 8 and 9 are given by those for the corresponding stationary cable model in Section 2.1 with lFig. 8. Schematic of the moving hoist cable with the suspension of the car against its guide rails assumed to be rigid:(a)(c) as in Fig. 1.Fig. 9. Schematic of the moving hoist cable where the car is modelled as a point mass meand its suspension against theguide rails has a resultant stiffness keand damping coefficient ce: (a)(c) as in Fig. 2.W.D. Zhu, G.Y. Xu / Journal of Sound and Vibration 263 (2003) 679699688replaced with lt and Pl; ytl;t and yttl;t in Eq. (5) replaced with Plt;t; Dylt;t=Dt andD2ylt;t=Dt2; respectively, where D=Dt yt vtyxand D2=Dt2is defined in Eq. (16). Themodels in Figs. 8(c) and 9(c) cannot be obtained as a special case from the models in Figs. 8(a)and 9(a), respectively. The mass of the car provides a tension meg ? v and an inertial force in theforce boundary condition for the models in Fig. 9.Galerkins method and the assumed modes method are modified to discretize the governingpartial equations for the models in Figs. 8 and 9, respectively. The solution of Eq. (15) is assumedin the formyx;t Xnj1qjtfjx;t;18where fjx;t are the time-dependent trial functions and the other variables are defined in Section2.1. Following Ref. 21 the instantaneous eigenfunctions of stationary beams and strings withvariable length lt are used as the trial functions for the models in Figs. 8 and 9. They satisfy thesame boundary conditions as the trial functions for the corresponding stationary cable models inSection 2.1 and are normalized so thatRlt0f2jx;tdx 1: Note that the trial functions for themodel in Fig. 9(b) contain a rigid-body mode. Because the trial functions for each model can beexpressed as 21,22fjx;t 1ffiffiffiffiffiffiffiltpcjx;19where x x=lt and cjx; given in Appendix A, are the normalized eigenfunctions of thecorresponding stationary beam or string with unit length, dependence of the system matrices inEq. (20) on time appears in the coefficient of their component matrices, which greatly simplifiesthe analysis. The normalized, instantaneous eigenfunctions of the tensioned beams with variablelength lt cannot be decomposed in the same manner as Eq. (19) and are not used as the trialfunctions for the models in Figs. 8(a), and 9(a) and (b). Substituting Eqs. (18) and (19) intoEq. (15) and the force boundary condition at x lt; multiplying the governing equation bycix=ffiffiffiffiffiffiffiltp; integrating it from x 0 to lt; and using the resulting boundary condition andthe orthonormality relations for cjx yields the discretized equations for the models in Fig. 9(a)and (b):Mt. qt Ct Gt? qt Kt Ht?qt 0;20where entries of the symmetric mass, stiffness, and damping matrices areMij rdij mel?1tci1cj1;21Kij14rl?2tl2tdij? rl?2tl2tZ101 ? x2c0ixc0jxdx rl?1tg ?.lt?Z101 ? xc0ixc0jxdx mel?2tg ?.lt?Z10c0ixc0jxdx EIl?4tZ10c00ixc00jxdx me34l2tl?3t ?12.ltl?2t?12cel?2tlt kel?1t?ci1cj1;22W.D. Zhu, G.Y. Xu / Journal of Sound and Vibration 263 (2003) 679699689Cij cel?1tci1cj1 ? mel?2tltci1cj1;23in which dijis the Kronecker delta, and entries of the skew-symmetric gyroscopic and circulatorymatrices areGij rl?1tlt 2Z101 ? xcixc0jxdx ? dij?;24Hij rl?2tl2t ? l?1t.lt?12dij?Z101 ? xcixc0jxdx?:25The discretized equations for the model in Fig. 9(c) are given by Eqs. (20)(25) with EI 0 inEq. (22). The discretized equations for the models in Fig. 8(a) and (b) are given by Eqs. (20)(25)with me 0 in Eq. (21), the last term involving ci1cj1 in Eq. (22) set to zero, and ce me 0in Eq. (23). The discretized equations for the model in Fig. 8(c) are given by those for the modelsin Fig. 8(a) and (b) with EI 0 in the entries of K: The discretized equations for the model inFig. 8(c) can be obtained from those for the model in Fig. 8(b) by setting EI 0; the discretizedequations for the model in Fig. 9(c) cannot be obtained as a special case from those for the modelin Fig. 9(b). If the initial displacement and velocity of the cable in Figs. 8 and 9 are given by yx;0and ytx;0; respectively, where 0oxol0; the initial conditions for the generalized coordinatesareqj0 ffiffiffiffiffiffiffiffil0pZ10yxl0;0cjxdx;26 qj0 ffiffiffiffiffiffiffiffil0pZ10ytxl0;0cjxdx v0l0Xni1qi0Z10xc0ixcjxdx v02l0qj0:27The vibratory energy of the models in Fig. 8(a) and (b) isEvt 12Zlt0ryt vyx2 Py2x EIy2xx? dx28and that of the models in Fig. 9(a) and (b) isEvt 12Zlt0ryt vyx2 Py2x EIy2xx? dx12meytlt;t vyxlt;t?212key2lt;t:29The vibratory energy of the models in Figs. 8(c) and 9(c) is given by Eqs. (28) and (29),respectively, with EI 0: Substituting Eqs. (18) and (19) into Eq. (28) yields the discretizedexpression of the vibratory energy of the models in Fig. 9(a) and (b):Evt 12 qTtMt qt qTtRt qt qTtStqt?;30W.D. Zhu, G.Y. Xu / Journal of Sound and Vibration 263 (2003) 679699690where entries of M are given by Eq. (21)Rij ?rl?1tltdij 2rl?1tltZ101 ? xcixcjxdx ? mel?2tltci1cj1;31Sijr ?14l?2tl2tdijl2tl?2tZ101 ? x2c0ixc0jxdx? l?1tg ?.lt?Z101 ? xc0ixc0jxdx? kel?1tci1cj1 EIl?4tZ10c00ixc00jxdx mel?2tg ?.lt?Z10c0ixc0jxdx 14mel?1tl2tci1cj1:32The discretized expression of Evt for the model in Fig. 9(c) is given by Eqs. (30)(32) with EI 0in Eq. (32). The discretized expression of Evt for the models in Fig. 8(a) and (b) is given byEqs. (30)(32) with me 0 in the entries of M and Eq. (31) and with the last term in Eq. (32) set tozero. The discretized expression of Evt for the model in Fig. 8(c) is given by the discretizedexpression of Evt for the model in Fig. 8(b) with EI 0 in the entries of S: The rate of change ofthe vibratory energy of each model in Figs. 8 and 9 can be calculated from the control volume andsystem viewpoints following Refs. 21,23. The rate of change of Evt from the control volumeviewpoint can characterize the dynamic stability of the system under consideration and isobtained by differentiating Evt using Leibnitzs rule. It is given byEvt ?vt2EIy2xx0;t ?12. vtZlt0me rlt ? x?y2xdx? ceytlt;t vtyxlt;t?233for the model in Fig. 9(a) and byEvt ?vt2P0;t ? rv2t?y2x0;t EIvtyx0;tyxxx0;t?12. vtZlt0me rlt ? x?y2xdx ? ceytlt;t vtyxlt;t?234for the model in Fig. 9(b). Similarly,Evt for the model in Fig. 9(c) is given by Eq. (34) withEI 0 andEvt for each model in Fig. 8 is given by that for the corresponding model in Fig. 9with ce 0: The discretized expression ofEvt isEvt 12 qTtFt qt qTtUtqt qTtWtqt?;35W.D. Zhu, G.Y. Xu / Journal of Sound and Vibration 263 (2003) 679699691whereFij ?cel?1tci1cj1;Uij cel?2tltci1cj136for the models in Fig. 9, Fij Uij 0 for the models in Fig. 8,Wij12rl?1t. vtZ10xc0ixc0jxdx ?12me rlt?l?2t. vtZ10c0ixc0jxdx?14cel?3tl2tci1cj1 EIl?5tltc0i0c000j0?12l?3tltfme rlt?g ?.lt? ? rl2tgc0i0c0j037for the model in Fig. 9(b), Wijfor the model in Fig. 9(c) is given by Eq. (37) with EI 0; WijforthemodelinFig. 9(a)isgivenbyEq. (37)withthelasttwotermsreplacedwith?12EIl?5tltc00i0c00j0; and Wijfor each model in Fig. 8 is given by that for the correspondingmodel in Fig. 9 with ce 0:3.2. Results and discussionThe parameters used here are the same as those in Section 2.2. Under the upward movementprofile shown in Fig. 10 23, where l0 equals l in Section 2.2, and the same initial conditions asthose for the corresponding stationary cable models, the undamped (i.e., ce 0) responses of themodels in Figs. 8 and 9 are calculated with n 30 and shown in Figs. 11 and 12, respectively. Theresponses of the models in Figs. 8 and 9 are virtually indistinguishable within the scales of theplots in Figs. 11 and 12, respectively. The vibratory energies of the models in Fig. 9 are smallerthan those of the models in Fig. 8 and approach them when keapproaches infinity (e.g.,ke 200; 000N=m).The transverse force at any point other than the fixed ends of the models in Figs. 8(a) and 9(a)and the lower end of the model in Fig. 9(c) can be calculated. The discretized expression ofEvtfor the model in Fig. 8(c) yield essentially the same result as that obtained from Evt throughfinite difference (not shown). The discretized expressions ofEvt for the models in Figs. 8(a) and(a)0100200(b)- 6- 4- 20(c)- 1.0- 0.50.00.51.0010203040t (s)(d)- 1.0- 0.50.00.51.0010203040t (s) l(t) (m) v(t) (m/s) l(t) (m/s2) v(t) (m/s3) . . Fig. 10. Upward movement profile of the hoist cable: (a) lt; (b) vt; (c).lt; and (d) . vt:W.D. Zhu, G.Y. Xu / Journal of Sound and Vibration 263 (2003) 679699692(a)01234(b)-0.10-0.050.000.050.10(c)-1.0-0.50.00.51.0010203040t (s) y(l(t)-15,t) (m) Ev(t) (J) y(l(t)-15,t) (m/s) D Dt Fig. 11. The undamped responses of the models in Fig. 8: (a) vibratory energy, (b) displacement of a fixed particle atx lt ? 15 m; and (c) velocity of a fixed particle at x lt ? 15 m; Dylt ? 15;t=Dt: The horizontal lines in (a)show the energies of the stationary cable models in Fig. 1. Dashed lines, Figs. 8(a) and 1(a); dashdotted lines,Figs. 8(b) and 1(b); solid lines, Figs. 8(c) and 1(c).0123-0.1-0.0500.050.1-1-0.500.51010203040y(l(t)-15,t) (m) Ev(t) (J) D Dt y(l(t)-15,t) (m/s)(a)(b)(c)Fig. 12. The undamped responses of the models in Fig. 9 along with the response of the model in Fig. 9(a) under theoptimal suspension stiffness and damping coefficient: (a)(c) as in Fig. 11. The horizontal lines in (a) show the energiesof the stationary cable models in Fig. 2. Dashed lines, Figs. 9(a) and 2(a); dashdotted lines, Figs. 9(b) and 2(b);thick solid lines, Figs. 9(c) and 2(c). The response of the model in Fig. 9(a) under k?and c?is shown in thin solid linesin (a)(c).W.D. Zhu, G.Y. Xu / Journal of Sound and Vibration 263 (2003) 6796996939(a) cannot be used here because y2xx0;t cannot be determined with the untensioned beameigenfunctions. As the second term in Eq. (34) has a much smaller amplitude than the first term,the discretized expressions ofEvt for the models in Figs. 8(b) and 9(b) yield essentially the sameresults as those obtained from the corresponding Evt through finite difference. While thediscretized expression ofEvt for the model in Fig. 9(c) can be used when ce 0; it cannotbe used when cea0 because yxlt;t cannot be determined with the trial functions satisfyingfjxlt;t 0:Under the above initial conditions and movement profile the dependence of the averagevibratory energy of the model in Fig. 9(a), defined by%Ev 1=tf?Rtf0Evtdt; where tf 38s; onkeand cewith the other parameters unchanged is shown in Fig. 13. With the optimal suspensionstiffness and damping coefficient around k?e 2800N=m and c?e 280N s=m; respectively,%Evisreduced from 1:22 J for the models in Fig. 8 to 0:33 J: The response of the model in Fig. 9(a)under k?eand c?eis shown in thin solid lines in Fig. 12. The optimal suspension stiffness anddamping coefficient are generally independent of the initial conditions. It is interesting to find thatthe vibratory energy of the models in Fig. 9 with ke ce 0 is essentially the same as that of themodels in Fig. 8.4. ConclusionsWhile the string and beam models predict essentially the same gross behaviors for thestationary and moving hoist cables, the maximum bending moment occurs at the fixed ends of thebeam models. The untensioned beam eigenfunctions cannot be used here to determine yxxandyxxxat the fixed ends of the beam models. With the optimal suspension stiffness and dampingcoefficient the average vibratory energy of the models in Fig. 9 during upward movement isreduced from that of the models in Fig. 8 by over 70%.0.3320.340.350.390.440.490.540.60.680.680.680.750.850.951.051.1551015202530102030405060ke (kN/m)ce (x 20 Ns/m)Fig. 13. Contour plot of the average vibratory energy of the model in Fig. 9(a) where its isoline values in J are labelled.In all the calculations n 30:W.D. Zhu, G.Y. Xu / Journal of Sound and Vibration 263 (2003) 679699694AcknowledgementsThis material is based on work supported by the National Science Foundation through awardnumber CMS-0116425.Appendix A. Trial functionsThe eigenfunctions of a fixedfixed beam of length l under uniform tension T arefjx Bjfcosb1jx ?sjgjsin b1jx ?1gjb2jsin b1jl ? b1jcosb1jleb2jx?l b2jsin b1jl b1jcosb1jle?b2jxl? b1je?b2jx? e?b2j2l?x?g;A:1where Bjare determined from the normalization relationsRl0f2jxdx 1;gj 2b2je?b2jlsin b1jl ? b1j1 ? e?2b2jl;sj 2b2je?b2jlcosb1jl ? b2j1 e?2b2jl;A:2b1jffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiT2 4EIrO2jq? T2EIvuut;b2jffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiT2 4EIrO2jq T2EIvuut;A:3in which Ojare obtained from the frequency equation4b1jb2je?b2jl? 2b1jb2j1 e?2b2jlcosb1jl b22j? b21j1 ? e?2b2jlsin b1jl 0:A:4When T 0; b1j b2j bjwith bjl being the jth positive root zjof2e?zj? 1 e?2zjcos zj 0:A:5The trial functions for the models in Fig. 1(b) and (c) are fjx ffiffiffiffiffiffiffi2=lpsinjpx=l:The eigenfunctions of a fixedfree beam of length l under uniform tension T arefjx Bjfcosb1jx ?sjgjsin b1jx 1gjb21jcosb1jl ? sin b1jleb2jx?l? b21jcosb1jl sin b1jle?b2jxl? b22je?b2jx? e?b2j2l?x?g;A:6where Bjare determined through normalization,gj 2b21je?b2ilsin b1il b22i1 ? e?2b2il;sj 2b21je?b2jlcosb1jl b22j1 e?2b2il;A:7and b1jand b2jare defined in Eq. (A.3) with Ojobtained from the frequency equation2e?b2jlb31jT EIb21j ? b32jT ? EIb22j? Tb2j? b1j EIb1jb2jb2j b1j?1 e?2b2jl? b1jb2jcosb1jl Tb1j b2j EIb1jb2jb1j? b2j?1 ? e?2b2jlb1jb2jsin b1jl 0:A:8When T 0; b1j b2j bjwith bjl being the jth positive root zjof2e?zj 1 e?2zjcoszj 0:A:9W.D. Zhu, G.Y. Xu / Journal of Sound and Vibration 263 (2003) 679699695The eigenfunctions of a pinnedfree beam of length l under uniform tension T arefjx Bjsin b1jx b21jsin b1jlb22j1 ? e?2b2jleb2jx?l? e?b2jxl?();A:10where Bjare determined through normalization and b1jand b2jare defined in Eq. (A.3) with Ojobtained from the frequency equationb1jT ? EIb22j1 e?2b2jlsin b1jl b2jT EIb21j1 ? e?2b2jlcosb1jl 0:A:11When T 0; fjx for jX2 are given by Eq. (A.1), where b1j b2j bjwith bjl being the(j ? 1)th positive root zj?1of1 ? e?2zjcoszj? 1 e?2zjsin zj 0:A:12The normalized eigenfunction for the rigid-body mode is f1x ffiffiffiffiffiffiffiffiffi3=l3px: The trial functions forthe model in Fig. 2(c) are fjx ffiffiffiffiffiffiffi2=lpsin2j ? 1px=2l?:The functions cjx for the model in Fig. 8(a) arecjx Bjfcoskjpx ?yjljsin kjpx ?1lje?kjp1?xsin kjp ? coskjp e?kjp1xsin kjp coskjp ? e?kjpx e?kjp2?x?g;A:13where kjp is the jth positive root zjof Eq. (A.5) andlj 2e?kjpsin kjp ? 1 e?2kjp;yj 2e?kjpcoskjp ? 1 ? e?2kjp:A:14The functions cjx for the model in Fig. 9(a) arecjx Bjfcoskjpx ?yjljsin kjpx ?1lje?kjp1?xsin kjp ? coskjp e?kjp1xsin kjp coskjp e?kjpx? e?kjp2?x?g;A:15where kjp is the jth positive root zjof Eq. (A.9) andlj 2e?kjpsin kjp 1 ? e?2kjp;yj 2e?kjpcoskjp 1 e?2kjp:A:16The constants Bjin Eqs. (A.13) and (A.15) are both expressed asBjffiffiffiffiffiffiffikjpp1 yjlj? ?2#14sinh 2kjp ? cosh kjpsin kjp? 1 ?yjlj? ?2#14sin 2kjp?(?sinh kjp coskjp?kjp 2yjlj14cos2kjp ? cosh2kjp sin kjpsinhkjp?1=2:A:17Calculation shows that Bj 1: The functions cjxjX2 for the model in Fig. 9(b) arecjx Bjfsinkjpx sin kjp1 ? e?2kjpekjpx?1? e?kjpx1?g;A:18where kjp is the (j ? 1)th positive root zj?1of Eq. (A.12) andBjffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi21 ? e?2kjp21 ? e?2kjp2? 4e?2kjpsin2kjps:A:19W.D. Zhu, G.Y. Xu / Journal of Sound and Vibration 263 (2003) 679699696The associated function for the rigid-body mode is c1x ffiffiffi3px: For the models in Fig. 8(b) and(c), cjx ffiffiffi2psin pjx; and for the model in Fig. 9(c), cjx ffiffiffi2psin 2j ? 1px=2?: Since a largenumber of terms are used in Eqs. (6) and (18), to avoid loss of precision in evaluating thehyperbolic functions with large arguments, the trial functions are expressed above as exponentialfunctions with negative exponents.Appendix B. Initial displacementsThe static deflection of a fixedfixed beam of length l under uniform tension meg; subjected to aconcentrated force P at x a; isYx A1 N1x Q1x;0oxoa;A2 N2x Q2x;aoxol;(B:1whereA1a?2sae?sl sle?sa e?2slsa ? e?sa? e?2slsa 1 ? esa?sl? e?2sl e?sa?sl? sl ? a1 e?2sl?;N1 as2e?sl? e?sa? e?2slsa sl ? a1 ? e?2sl ? 1 ? e?2sl esa?sl e?sa?sl?;Q1x a?1 ? sae?slsx e?sasx? 0:5sle?sasx e?2slsasxe?2slsx? e?sa?slsx sl ? ae?2slsx sae?sl?sx? 0:5sle?sa?sx 1:5sl ? 1e?2slsa?sx? e?sx sl ? ae?sx?;B:2A2asle?sa e?2slsa ? e?sa e?2slsa? sl a1 e?2sl s2la 11 ? e?2sl? esa?sl? e?sa?sl 2sae?sl?;N2 as?e?sa? e?2slsa 1 e?2sl esa?sl e?sa?sl? sa1 ? e?2sl ? 2e?sl?;Q2x ae?sa?slsx e?2slsx? e?slsx sae?slsx? 0:5sle?2slsasx? 1 0:5sle?sa?2slsx sl ? ae?2slsx? esa?sl?sx? e?sx e?sl?sx sae?sl?sx? 0:5sle?sa?sx? 0:5sl ? 1esa?sx sl ? ae?sx?;B:3in whichs ffiffiffiffiffiffiffiffimegEIr;a PEIs34e?sl? 21 e?2sl sl1 ? e?2sl?:B:4The static deflection of a pinnedpinned beam of length l under uniform tension meg; subjected toa concentrated force P at x a; is given by Eq. (B.1) withA1 0;N1Pl ? aEIs2l;W.D. Zhu, G.Y. Xu / Journal of Sound and Vibration 263 (2003) 679699697Q1x P2EIs3?1 e?2sle?sasx e?2slsasx? e?sa?sx? e?2sl
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