摘要
轴向行进系统主要用于动力传递和物质运输,在制造、交通、航天、国防等行业有着广泛的应用背景,其动力学理论与应用研究具有重要的科学价值,是具有挑战性的国际前沿领域。振动和稳定性分析是动力学研究的基本问题,在工程行进机构的开发中起着至关重要的作用。准确地预计响应的参数稳定域,特别是失稳临界运行速度的大小,不仅能够避免事故的发生,还可以为工程设计提供正确的理论依据。时至今日,轴向行进系统的动力学问题已经获得了长足的发展,由多种形式的激励如简谐荷载、支座运动、移动荷载以及弹性地基、支座的干摩擦等引起的行进弦及索横向振动问题已被国内外学者广为研究,新的分析方法、新的现象层出不穷,有关振动发生机理的研究也日益深刻和准确,为非线性动力学的深入发展提供了强劲的推动力。然而,关于轴向行进系统在复杂环境荷载、约束或耦合条件下的动力学及稳定性问题的理论研究较为匮乏,工程上一般根据非行进系统的理论准则来设计行进系统,很难有效预测行进速度对幅频特性的影响,预报主共振、分岔特性等关键行为。为满足工程需要,亟需开展相关的研究工作。
行进弦及索属于最基本的连续陀螺系统,承受因相对坐标系流动而产生的Colioris力作用,动力学方程为二阶非线性偏微分方程,其中包含对时间和空间变量的混合偏导数项(即对流加速度算子),致使系统产生了依赖于速度的复模态。对非线性振动问题,一般难以找到振动的闭合解。本论文利用Galerkin方法对偏微分方程进行离散,结合半解析和数值计算的方法,在时域内分析了耦合条件下的轴向行进弦及索横向振动的周期运动、稳定性、共振响应和分岔特性等,主要开展了以下几方面的研究工作：
(1)基于空气动力学的准定常理论,建立了轴向行进弦在定常风荷载作用下横向振动的动力学模型,将拟合的气动自激力非线性表达式引入弦线的动力学方程。分析了弦线平衡构形的稳定性,利用Routh-Hurwitz判据确定了平衡点的稳定域,给出了多参数下的Hopf分岔点及产生稳定极限环的显式条件。采用增量谐波平衡法深入研究了轴向行进弦自激和受迫振动的稳态周期响应,并结合Floquet理论确定了周期响应的稳定性。对受迫系统,揭示出周期解还可能通过次Hopf分岔继续失稳,失稳后系统的运动变为准周期运动。采用数值延拓方法计算了所有的周期运动强共振Codim-2分岔点。给出了外激励幅值和激励频率对准周期解平息和同步运动的影响以及准周期运动发生的机理。此外,还发现超临界行进速度的轴向行进弦存在低维的混沌吸引子。对于超临界行进弦的一阶截断系统,通过比较数值积分和Melnikov方法得到的结果,发现Melnikov函数有简单零点的条件不适合预测混沌运动的发生。基于可积哈密顿系统的扰动理论,结合判定相轨迹与同宿轨道的横截、相切性条件和数值积分的方法,从稳态响应的Poincare映射、最大Lyapunov指数和L(t0,kT-函数等角度系统讨论了弦线的局部周期运动、碰擦运动、全局周期运动、准周期运动和混沌运动与行进速度等设计参数的关系,并给出了混沌运动的发生机理。为进一步研究风荷载中脉动成分对行进弦的作用,将风速看作是随时间缓慢变化的参数,采用渐近匹配展开法重点讨论了风速缓慢经过Hopf分岔点时平凡解和周期解之间的变迁过程,计算了匹配的慢变平衡解及边界层的尺度。指出对自激系统,当风速变动较大时,由初始条件激发的动力学响应将不再是周期解,而是慢变准周期解,这是有别于时不变风荷载作用下行进弦的动力学特征。
(2)用改进的L-P方法研究了具有小垂跨比的轴向行进索在外激励作用下的横向面内、面外模态发生强烈耦合的内共振响应。线性特征值分析表明,当垂跨比或轴向张力在一定范围时,行进索的横向面内模态基频接近面外模态基频的二倍或三倍,可能导致2：1或3：1内共振的发生。在只有面内激励作用时,稳态响应中,除了纯面内响应之外,还存在着大幅的非纯平面响应,即内共振响应,说明面内、面外模态间发生了强烈的能量交换。进一步的分析指出,外激励幅值和激励频率决定了内共振的发生范围,而外激励幅值和行进速度对内共振初发段的能量交换速度有着显著的影响。
(3)通过构造轴向行进弦附带多个弹簧-质量振子耦合系统的Green函数,解决了耦合系统的特征值问题。指出耦合系统各阶特征值的实部和虚部都随着振子的移动而变化。采用特征值的最大变化定量地表征了振子同行进弦各阶模态的耦合强度。通过比较Galerkin近似解和精确解,发现随着展开阶数的增加,Galerkin近似解趋于精确解。从特征值的角度解释了高阶Galerkin方法能够用来获得行进弦附带移动振子系统的瞬态动力学响应。在此基础上,基于高阶Fourier级数扩展的Galerkin方法研究了风荷载作用下轴向行进弦附带单弹簧-质量振子的瞬态动力学问题。
(4)引入线性共置的传感器和激励机,对直接时滞速度反馈控制器作用下轴向行进弦的横向振动问题进行了研究。将Belair定理推广到了N维系统,证明了对于含有时滞的指数多项式形式的超越特征值方程,当时滞连续变化时,只有当特征值穿越虚轴时特征值实部大于零的数目才会发生改变,并指出亚临界速度的轴向行进弦的平衡点在时滞速度控制器作用下会通过Hopf分岔发生失稳。确定了平衡点在时滞和反馈速度增益参数域内的稳定性划分,发现存在多个稳定的参数区域并且平衡构形对小时滞反馈具有鲁棒性。应用泛函分析和中心流形约化的方法,重点研究了行进弦在单Hopf分岔点附近的局部动力学行为,特别是时滞对稳态周期运动、稳定性和分岔特性的影响。这些结果表明,以时滞大小做为控制参数可以有效抑制行进弦平衡构形的振动以及弦线稳态响应幅值的大小。
Axially moving systems are extensively applicable to manufacturing, transportation, aerospace and national defense industries for mass transfer and power transmission. The dy-namical theory and application on traveling systems are challenging and of great scientific merit, and have already been international frontiers for dynamical systems. The fundamental issue about axially moving systems is vibration analysis and dynamic stability which plays an important role in designing practical mechanisms of transportation. To avoid accidents and to provide guidance for engineers, it is important to accurately predict parametrically stable re-gions of dynamical responses, especially the critical transport speed. So far rapid progress has been made in the studies of axially moving systems and fruitful achievement has been re-ported in literature regarding linear and nonlinear oscillations of axially moving materials. Transverse vibration problems of strings developed due to various excitations such as support motions, moving forces, constraints from discrete or distributed elastic foundations and dry friction of eyelets have been extensively studied. New analysis approaches and phenomenon are emerging one after another. The understanding of what causes oscillation to the system is increasingly deep and accurate, and provides a strong boost to the development of nonlinear dynamics. However, few investigations have been found regarding the dynamics and stability of axially moving materials with complicated environmental excitations, constraints and coupling conditions. Generally, axially moving systems are still designed empirically based on experience from non-traveling systems to this date. It remains difficult to analyze the in-fluence of moving speed on the amplitude-frequency characteristics and capture significant dynamics of the traveling strings, such as principal resonances and bifurcations with varying design parameters. It is indeed an urgent need to initiate the related investigations.
Axially moving strings and cables belong to gyroscopic continuous systems with infinite degrees of freedoms. They are loaded by Colioris force induced by the flow of relative refer-ence. Mathematically, the motion of strings and cables is governed by a second-order partial differential equation. Notice that the partial derivatives with respect to both temporal and spa-tial variables, known as the term of convective acceleration, render complex, speed-dependent modes, which brings great challenge in finding closed-form solutions for nonlinear systems. In this dissertation, the Galerkin's procedure is adopted to discretize the partial differential equation. Combining the analytical and numerical techniques, the nonlinear transverse vibra- tions of axially moving strings and cables with complicated loadings and constraints are in-vestigated in time-domain, such as periodic motion and its stability, resonance response and bifurcation. The main contents are as follows:
(1) Based on the principles of quasi-steady aerodynamics, the dynamical model of trans-verse vibration of axially moving strings with aerodynamic forces is established. The wind loadings are introduced into the partial differential equation by nonlinear curve-fitting. The stability of static configuration of the string is considered. Explicit conditions are derived for stability region of equilibrium of the string based on the Routh-Hurwitz criterion and for gen-eration of stable limit cycles via the Hopf bifurcation in multi-parameter spaces. The periodic motions for self-excited and forced self-excited vibration are determined using the Incremen-tal Harmonic Balance method, with stability analysis carried out by eigenvalue computation of Floquet multipliers. It is demonstrated that for forced systems the stability of the periodic motion may be lost due to more successive bifurcations. The periodic motion becomes qua-si-periodic after secondary Hopf bifurcations. Further, continuation software MatCont is adopted to analyze the bifurcation scenario of the limit cycle, including fold bifurcation, Neimark-Sacker (NS) bifurcation as well as codim-2 bifurcations of NS-NS and the 1:1,1:3 and 1:4 strong resonance. The effects of excitation amplitude and frequency on the quench and synchronized quasi-periodic motions are illustrated. In addition, it is found that low-dimensional chaotic motions exist for the string when the moving speed is supercritical. It is demonstrated that the explicit criterion based on the Melnikov's method can be mislead-ing when the criteria is used to predict the occurrence of chaotic motions. In fact, the chaotic motions are restricted in a chaotic belt region. Based on the perturbation theory of Hamilto-nian systems, the analytical conditions for global transversality and tangency of the periodic motions to the homoclinic orbits are presented and verified by numerical simulation. To un-derstand the geometric structure of steady-state responses for a periodically forced axially moving string, the Poincare map, the largest Lyapunov exponent and the Z(t0,kT)-function (i.e. increment of the first integral) are used. The periodic, quasi-periodic and chaotic motions are illustrated in phase space. The grazing periodic solutions are analytically predicted. It is shown that an increasing mean velocity of the wind decreases the possibility of chaotic res-ponses. Further, The string exhibits quasi-periodic solutions for large wind speed. As the moving speed or excitation amplitude is increased, axially moving string in the supercritical regime has both period-doubling and intermittent routes to chaos. To reveal the effect of time-varying of wind velocity on the dynamical behavior, the slow transition across the point of Hopf bifurcation between trivial solution and steady-state solution is analyzed by means of the matched asymptotic expansions method by taking the wind speed as the slowly varying parameter. The slowly varying equilibrium solution and the size of boundary layer are deter- mined. It is indicated that for self-excited system, there are no periodic motions but slowly varying quasi-periodic motions for almost all initial conditions.
(2) The coupled nonlinear response of the in-plane and out-of-plane modes is investi-gated for a small sagged axially moving cable by using the modified Lindstedt-Poincare me-thod. The frequency analysis shows that internal resonance occurs for a range of small sag-to-span ratio and cable tension. Under an in-plane harmonic excitation, the three-to-one internal resonance between the in-plane and out-of-plane primary modes is analyzed using the modified Lindstedt-Poincare method. Several branches of large frequency-amplitude response curves of the out-of-plane motion are found in the region of internal resonance. The influence of excitation amplitude and transport speed on the energy exchange between the in-plane and the out-of-plane modes is discussed.
(3) The eigenvalue problem of an axially moving string attached by multiple mass-spring oscillators is solved through the Green's function method. The explicit Green's function is obtained by the Construction Theorem of Green's Function and the closed-form transcenden-tal equations of the eigenvalues are presented. The maximum variance rate is adopted to ex-press the dynamical interaction between subsystems of the string and the oscillators. It is found out that the dynamical interaction between the subsystems mainly happens to the first two modes of the string when the eigen-frequency of the oscillator is close to the first fre-quency of the string. The Galerkin's discretization method is analyzed so as to determine the approximate eigenvalues for large numbers of oscillators. It is found that the solution with Galerkin's method tends to be accurate with increasing expansion number. In this way, an ex-planation in terms of eigenvalues is given to show the validity of large series of Flourier ex-pansion in solving transient dynamical responses of axially moving string with attached mass-spring oscillators. Numerical results also illustrate how the eigenvalues are affected by spring stiffness and transport speed.
(4) The local dynamics of an axially moving string under aerodynamic forces is investi-gated with a collocated time-delayed velocity feedback controller. The Belair Theorem is ad-vanced to a more generalized theorem for any polynomial-exponential equations with con-stant time delay. It is proved that as the time delay varies, the number of solutions of the cha-racteristic equation can only be changed when the eigenvalue passes through the imaginary axis. Thus, it is inferred that the static configuration of the string losses its stability only through Hopf bifurcation for sub-critically moving strings. The Hopf bifurcation curves are presented in the space of controlling parameter. With the aid of the center manifold reduction, a functional analysis is carried out to reduce the modal equation to a single ordinary differen-tial equation in one complex variable on the center manifold. The approximate analytical so-lutions in the vicinity of Hopf bifurcations are derived in the case of primary resonance. The curves of excitation-responses and frequency-response are shown with the effect of time delay. The stability analysis for steady-state periodic solutions of the reduced system indicates the onset of local control parameter for vibration control and response suppression. Moreover, the Poincare-Bendixon theorem and energy-like function are used to investigate the existence and characteristics of quasi-periodic solutions when the periodic solution becomes unstable. The validity of the analytical prediction is demonstrated through numerical results. Two different kinds of quasi-periodic solutions are reported.
引文
[1]Skutch R. Uber die Bewegung eines gespannten Fadens, weicher gezwungun ist, durch zwei feste Punkte,mit einer constanten Geschwindigkeit zu gehen, und zwischen denselben in-transversal-Schwingungen von gerlinger Amplitude versetzt wird. Annalen der Physik und Chenmie,1897,61:190-195.
[2]Hansell G D Ⅲ.Moving threadline oscillation experiments. M. M. E. Thesis, University of Delaware, Newark, Delaware,1966.
[3]Swope R D and Ames W F. Vibrations of a moving threadline. Journal of the Franklin Institute,1963,275(1):36-55.
[4]Zaiser J N. Nonlinear vibrations of a moving threadline. Ph. D. Dissertation, University of Delaware, Newark, Delaware,1964.
[5]Mote C D Jr. Dynamic stability of axially moving materials. The Shock and Vibration Digest, 1972,4(4):2-11.
[6]Wickert J A, Mote C D Jr. Current research on the vibration and stability of axially-moving materials. The Shock and Vibration Digest,1988,20(5):3-13.
[7]Wang K W, Liu S P. On the noise and vibration of chain drive systems.The Shock and Vibration Digest,1991,23(4):8-13.
[8]Abrate A S. Vibration of belts and belt drives. Mechanism and Machine Theory,1992,27(6): 645-659.
[9]陈立群.轴向行进弦线的纵向振动及其控制.力学进展,2001,31(4)：535-546.
[10]Chen L Q. Analysis and control of transverse vibrations of axially moving strings. Applied Mechanics Reviews,2005,58:91-116.
[11]国家自然科学基金委员会数学物理科学部.力学学科发展研究报告,北京：科学出版社,2007.
[12]金福临译.非线性振动理论中的渐近方法.上海：上海科学技术出版社,1963.
[13]Dimarogonas A 0. The origins of vibration theory. Journal of Sound and Vibration,1990, 140:181-189.
[14]Nayf eh A H and Balachandran B. Applied nonlinear dynamics, analytical, computational, and experimental methods. New York:Wiley,1995.
[15]陈立群,刘延柱.振动力学发展历史概述.上海交通大学学报,1997,31(7)：132-136.
[16]刘延柱,陈文良,陈立群编著.振动力学.北京：高等教育出版社,1998.
[17]刘延柱,陈立群编著.非线性振动.北京：高等教育出版社,2001.
[18]陈树辉著.强非线性振动系统的定量分析方法.北京：科学出版社,2006.
[19]Sack R A. Transverse oscillations in travelling strings. British Journal of Applied Physics,1954,5:224-226.
[20]Chubachi T. Lateral vibration of axially moving wire or belt materials. Bulletin of JSME, 1958,1(1):24-29.
[21]Archibald F R, Emslie A G. The vibration of a string having a uniform motion along its length. Journal of Applied Mechanics,1958,25(3):347-348.
[22]Miranker W L. The wave equation in a medium in motion. IBM Journal of Research and Development,1960,4(1):36-42.
[23]Wickert J A, Mote C D Jr. Classical vibration analysis of axially moving continua. Journal of Applied Mechanics,1990,57(3):738-744.
[24]Wickert J A, Mote C D Jr. Response and discretization method for axially moving materials. Applied Mechanics Review,1991,44(11):279-284.
[25]任西春,王跃方.基于Hamilton体系的弹性行进索精确模态分析.振动工程学报,2004,17(s)：744-746.
[26]Wang Y F, Huang L H and Liu X T.Eigenvalue and stability analysis for transverse vibrations of axially moving strings based on Hamiltonian dynamics. Acta Mechanica Sinica, 2005,21(5):485-494.
[27]Renshaw A A, Mote C D Jr. Local stability of gyroscopic systems near vanishing ei-genvalues. Journal of Applied Mechanics,1996,63(1):116-120.
[28]Kao I, Wei S and Chiang F P. Vibration analysis of wiresaw manufacturing processes and wafer surface measurements, the NSF Grantees Conference,1998:427-429.
[29]Wei S, Kao I. Vibration analysis of wire and frequency response in the modern wiresaw manufacturing process. Journal of Sound and Vibration,2000,231(5):1383-1395.
[30]Jha R K, Parker R G. Spatial discretization of axially moving media vibration problems. Journal of Vibration Acoustics,2000,122:290-294.
[31]Mote C D Jr. A study of band saw vibration. Journal of the Franklin Institute,1965,279(6): 430-444.
[32]Perkins N C. Linear dynamics of a translating string on an elastic foundation. Journal of Vibration Acoustics,1990,112(1):2-7.
[33]Bhat R B, Xistris G D and Sankar T S.Dynamic behavior of a moving belt supported on an elastic foundation. Journal of Mechanical Design,1982,104(1):143-147.
[34]Tan C A, Yang B and Mote C D Jr. On the vibrational of a translating string coupled to hydrodynamic bearings. ASME Journal of Vibration and Acoustics,1990,112(3):337-345.
[35]Tan C A, Yang B and Mote C D Jr. Dynamic response of an axially moving beam coupled to hydrodynamic bearings. ASME Journal of Vibration and Acoustics,1993,115(1):9-15.
[36]Wickert J A. Response solutions for the vibration of a traveling string on an elastic foundation. Journal of Vibration and Acoustics,1994,116(1):137-139.
[37]Xiong Y and Hutton S G. Vibration and stability analysis of a multi-guided rotating string. Journal of Sound and Vibration,1994,169(5):669-683.
[38]Parker P G. On the eigenvalues and critical speed stability of gyroscopic continua. Journal of Applied Mechanics,1998,65:134-140.
[39]Riedel C H and Tan C A. Dynamic characteristics and mode localization of elastically constrained axially moving strings and beams. Journal of Sound and Vibration,1998,215(3): 455-473.
[40]Parker P G. Supercritical speed stability of the trivial equilibrium of an axially moving string on an elastic foundation. Journal of Sound and Vibration,1999,221 (2):205-219.
[41]Cheng S P and Perkins N C. The vibration and stability of a friction-guided translating string. Journal of Sound and Vibration,1991,144(2):281-292.
[42]Chen J S. Natural frequencies and stability of an axially travelling string in contact with a stationary load system. Journal of vibration and Acoustic,1997,119(2):152-157.
[43]Tan C A and Ying S. Dynamic analysis of the axially moving string based on wave propagation. Journal of Applied Mechanics,1997,64(2):394-400.
[44]Saeed H M. Disturbance propagation paths in a constrained axially moving string. Journal of Sound and Vibration,1999,228(1):218-225.
[45]Le-Ngoc L and McCallion H. Dynamic stiffness of an axially moving string. Journal of Sound and Vibration,1999,220(4):749-756.
[46]Zhu W D and Mote C D Jr. Free and forced response of an axially moving string transporting a damped linear oscillator. Journal of Sound and Vibration,1994,177(5):591-610.
[47]Hsiang P C. Vibrations of a moving threadline having a geometric nonlinearity. M. M. E. Thesis, University of Delaware, Newark, Delaware,1966.
[48]Mote C D Jr. On the nonlinear oscillation of an axially moving string. Journal of Applied Mechanics,1966,33:463-464.
[49]Ames W F, Lee S Y and Zaiser J N. Non-linear vibration of a traveling threadline. International Journal of Non-Linear Mechanics,1968,3(4):449-469.
[50]Kim Y I and Tabarrok B. On the nonlinear vibration of travelling strings. Journal of the Franklin Institute,1972,293(6):381-399.
[51]Koivurova H and Salonen E M. Comments on non-linear formulations for travelling string and beam problems. Journal of Sound and Vibration,1999,225(5):845-856.
[52]Thurman A L and Mote C D Jr. Free, periodic, nonlinear oscillation of an axially moving strip. ASME Journal of Applied Mechanics,1969,36:83-91.
[53]Perkins N C and Mote C D Jr. Three-dimensional vibration of travelling elastic cables. Journal of Sound and Vibration,1987,114:325-340.
[54]Perkins N C and Mote C D Jr. Theoretical and experimental stability of two translating cable equilibria.Journal of Sound and Vibration,1989,128:397-410.
[55]Wickert J A. Nonlinear vibration of a travelling tensioned beam. International Journal of Nonlinear Mechanics,1992,27:503-517.
[56]Hwang S J and Perkins N C. Supercritical stability of an axially moving beam, Part I: model and equilibrium analysis. Journal of Sound and Vibration,1992,154:381-396.
[57]Luo A C J and Mote C D Jr. Equilibrium solutions and existence for traveling, arbitrarily sagged, elastic cables. ASME Journal of Applied Mechanics,2000,67:148-154.
[58]Luo A C J and Wang Y F. On the rigid-body motion of traveling, sagged, elastic cables. On the rigid-body motion of traveling, sagged, elastic cables. Proceedings of IMECE 2002, ASME International Mechanical Engineering Congress and Exposition, New Orleans,USA 2002, IMECE2002-32420:1-7.
[59]Wang Y F and Luo A C J. Dynamics of traveling, inextensible cables. Communications in Nonlinear Science and Numerical Simulation,2004,9:531-542.
[60]Moustafa M A and Salman F K. Dynamic properties of a moving threadline. Journal of Engineering Industrial,1976,98:868-875.
[61]Chen L Q and Zhao W J. A numerical method-for simulating transverse vibrations of an axially moving string. Applied Mathematics and Computations,2005,160(2):411-422.
[62]Chen L Q, Zhao W J and Jean W Z. Simulations of transverse vibrations of an axially moving string:a modified difference approach. Applied Mathematics and Computation,2005,
166(3):596-607.
[63]Nayfeh A H, Nayfeh J F and Mook D T. On methods for continuous systems with Quadratic and cubic nonlinearities. Nonlinear Dynamics,1992,3:145-162.
[64]Pakdemirli M. A comparison of two perturbation methods for vibrations of systems with quadratic and cubic nonlinearities. Mechanics research communications,1994,21:203-208.
[65]Pakdemirli M and Boyaci H. Comparison of direct perturbation methods with discre-tization perturbation methods for nonlinear vibrations. Journal of Sound and Vibration,1995, 186(5):837-845.
[66]Pakdemirli M, Nayfeh S A and Nayfeh A H. Analysis of one-to-one autoparametric resonances in cable:discretization vs. direct treatment. Nonlinear Dynamics,1995,8(1):65-83.
[67]Nayfeh A H, Nayfeh S A and Pakdemirli M. On the discretization of weakly nonlinear spatially continuous systems. Stochastic Modelling and Nonlinear Dynamics-Applications to Mechanical Systems, Boca Raton:CRC Press,1993.
[68]Nayfeh A H and Lacarbonara W. On the discretization of distributed-parameter systems with quadratic and cubic nonlinearities. Nonlinear Dynamics,1997,13:203-220.
[69]Chin C M and Nayfeh A H. Three-to-One internal resonances in hinged-clamped beams. Nonlinear Dynamics,1997,12:129-154.
[70]Nayfeh A H, Lacarbonara W'and Chin C M. Nonlinear normal modes of buckled beams: three-to-one and one-to-one internal resonances. Nonlinear dynamics,1999,18:253-273.
[71]Pellicano F and Vestroni F. Complex dynamics of high-speed axially moving systems. Journal of Sound and Vibration,2002,258(1):31-44.
[72]Bapat V A and Srinivasan P. Nonlinear transverse oscillations on traveling strings by a direct linearization method. Journal of Applied Mechanics,1967,34:775-777.
[73]Mochensturm E M, Perkins N C and Ulsoy A G. Stability and limit cycles of parametrically excited, axially moving strings. Journal of Vibration and Acoustics,1996,118(3):346-351.
[74]Wang K W. On the stability of chain drive systems under periodic sprocket oscillations. Journal of Vibration and Acoustics,1992,114:119-126.
[75]Moon J and Wickert J A. Nonlinear vibration of power transmission belts. Journal of Sound and Vibration,1997,200(4):419-431.
[76]Ariaratnam S T and Asokanthan S F. Dynamic stability of chain drives. ASME Journal of Mechanisms, Transmissions, and Automation in Design,1987,109(3):412-418.
[77]Huang J S, Fung R F and Lin C H. Dynamic stability of a moving string undergoing three-dimensional vibration. International Journal of Mechanical Sciences,1995,37(2):145-160.
[78]Fung R F, Wu S L. Dynamic stability of a three-dimensional string subjected to both magnetic and tensioned excitations. Journal of Sound and Vibration,1997,204(1):171-179.
[79]Pellican F, Vestroni F and Fregolent A. Experimental and theoretical analysis of a power transmission belt. Proceeding of the ASME Conference on Vibration and Control of Continuous Systems, Florida,2000,107:71-78.
[80]Fung R F, Huang J S and Chen Y C. The transient amplitude of the viscoelastic traveling string:an integral constitutive law. Journal of Sound and Vibration,1997,201(2):153-167.
[81]Fung R F, Huang J S, Chen Y C et al. Nonlinear dynamic analysis of the viscoelastic string with a harmonically varying transport speed. Computers and Structures,1998,66(6): 777-784.
[82]Pakdemirli M and Batan H. Dynamic stability of a constantly accelerating string. Journal of Sound and Vibration,1993,168(2):371-378.
[83]Pakdemirli M, Ulsoy A G and Ceranoglu A. Transverse vibration of an axially accelerating string. Journal of Sound and Vibration,1994,169(2):179-196.
[84]Chen L Q, Zhang N H and Zu J W. Bifurcation and chaos in nonlinear vibrations of a moving viscoelastic string. Mechanics Research Communications,2002,29:81-90.
[85]Chen L Q, Zhang N H and Zu J W. The regular and chaotic vibrations of an axially moving viscoelastic string based on 4-order Galerkin truncation. Journal of Sound and Vibration, 2003,261:764-773.
[86]Chen L Q, J. Wu and Zu J W. The chaotic response of the viscoelastic traveling string: an integral constitutive law. Chaos Solitons and Fractals,2004,21:349-357.
[87]Chen L Q, J. Wu and Zu J W. Asymptotic nonlinear behaviors in transverse vibration of an axially accelerating viscoelastic string. Nonlinear Dynamics,2004,35:347-360.
[88]Chen L Q and Yang X D. Transverse nonlinear dynamics of axially accelerating vis-coelastic beams based on 4-term Galerkin truncation, Chaos. Solitons and Fractals,2006,27: 748-757.
[89]Zhang N H and Chen L Q. Nonlinear dynamics of axially moving viscoelastic strings based on translating eigenfunctions. Chaos, Solitons and Fractals,2005,24:1065-1074.
[90]Zhang N H. Dynamic analysis of an axially moving viscoelastic string by the Galerkin method using translating string eigenfunctions. Chaos, Solitons, and Fractals,2008,35: 291-302.
[91]Hammond D S and McEachern J F. A dynamic model for deploying cable systems. Oceans (New York), IEEE Oceanic Engineering Society,1991,1:620-626.
[92]Rega G. Non-linearity, bifurcation and chaos in the finite dynamics of different cable models. Chaos, Solitons and Fractals,1996,7(10):1507-1536.
[93]Garziera R and Amabili M. Damping effect of winding on the lateral vibrations of axially moving tapes. ASME Journal of Vibration and Acoustics,2000,122:49-53.
[94]Hu H Y and Jin D P.Non-linear dynamics of a suspended travelling cable subject to transverse fluid excitation. Journal of Sound and Vibration,2001,239(3):515-529.
[95]李映辉,高庆.轴向运动小垂度索动力响应.西南交通大学学报,2002,37(2)：117-120.
[96]Chen L Q, Zu J W and Wu J. Steady-state response of the parametrically excited axially moving string constituted by the Boltzmann superposition principle. Acta Mechanica,2003,162: 143-155.
[97]Beikmann R S, Perkins N C and Ulsoy A G. Nonlinear coupled response of serpentine belt drive systems. Journal of Vibration Acoustics,1996,118(4):567-574.
[98]Zhang L and Zu J W. One-to-one auto-parametric resonance in serpentine belt drive systems. Journal of Sound and Vibration,2000,232(4):783-806.
[99]Zu J W and Zhang L. Two-to-one internal resonance in serpentine belt drive systems. International Journal of Nonlinear Sciences and Numerical Simulation,2000,1(3):187-198.
[100]Riedel C H and Tan C A. Coupled, forced response of an axially moving strip with internal resonance.International Journal of Non-Linear Mechanics,2002,37:101-116.
[101]吴俊,陈立群.轴向变速运动弦线的非线性振动的稳态响应及其稳定性.应用数学和力学,2004,25(9)：917-926.
[102]杨晓东,陈立群.变速度轴向运动粘弹性梁的动态稳定性.应用数学和力学,2005,26(8)：905-910.
[103]陈立群,吴俊.轴向变速运动粘弹性弦线的横向振动分岔.固体力学学报,2005,26(1)：83-86.
[104]陈立群,吴俊.轴向运动粘弹性弦线的横向非线性动力学分析.工程力学,2005,22(4)：48-51.
[105]李晓军,陈立群.关于两端固定轴向运动梁的横向振动.振动与冲击,2005,24(1)：22-23.
[106]陈树辉,黄建亮,佘锦炎.轴向运动梁横向非线性振动研究.动力学与控制学报,2004,2(1)：40-45.
[107]黄建亮,陈树辉.轴向运动体系横向非线性振动的联合共振.振动工程学报,2005,18(1)：19-23.
[108]Chen S H, Huang J L and Sze K Y. Multidimensional Lindstedt-Poincare method for non-linear vibration of axially moving beams. Journal of Sound and Vibration,2007,306:1-11.
[109]Zhang W and Yao M H. Multi-pulse orbits and chaotic dynamics in motion of parametrically excited viscoelastic moving belt. Chaos, Solitons and Fractals,2006,28:42-66.
[110]Wickert J A and Mote C D Jr. Linear transverse vibration of an axially moving string-particle system. The Journal of the Acoustical Society of America,1988,84(3):963-969.
[111]Wickert J A and Mote C D Jr. Traveling load response of an axially moving string. Journal of Sound and Vibration,1991,149(2):267-284.
[112]Zhu W D and Mote C D Jr. Free and forced response of an axially moving string tran-sporting a damped linear oscillator. Journal of Sound and Vibration,1994,177(5):591-610.
[113]Pesterev A V and Bergman L A. Response of elastic continuum carrying moving linear oscillator. Journal of Engineering Mechanics,1997,123(8):878-884.
[114]Pesterev A V, Yang B, Bergman L A et al. Response of elastic continuum carrying multiple moving oscillators. Journal of Engineering Mechanics,2001,127(3):260-265.
[115]Pesterev A V and Bergman L A. Response of a nonconservative continuous system to a moving concentrated load. Journal of Applied Mechanics,1998,65:436-444.
[116]Pesterev A V and Bergman L A. An improved series expansion of the solution to the moving oscillator problem. Journal of Vibration and Acoustics,2000,122:54-61.
[117]Yang B, Tan C A and Bergman L A. Direct numerical procedure for solution of moving oscillator problems. Journal of Engineering Mechanics,2000,126(5):462-469.
[118]Shadnam M R, Mof id M and Akin J E. On the dynamic response of rectangular plate with moving mass.Thin-walled Structures,2001,39:797-806.
[119]Pesterev A V, Bergman L A, Tan C A et al. On asymptotics of the solution of the moving oscillator problem. Journal of Sound and Vibration,2003,260:519-536.
[120]Biondi B and Muscolino G. New improved series expansion for solving the moving oscillator problem. Journal of Sound and Vibration,2005,281:99-117.
[121]Siddiqui S A Q, Golnaraghi M F and Heppler G R. Large free vibrations of a beam carrying a moving mass. International Journal of Non-Linear Mechanics,2003,38:1481-1493.
[122]Hikami Y and Shiraishi N. Rain-wind induced vibrations of cables stayed bridges. Journal of Wind Engineering and Industrial Aerodynamics,1988,29(1-3):409-418.
[123]Matsumoto M, Saitoh T, Kitazawa M, Shirato H and Nishizaki T. Response characteristics of rain-wind induced vibration of stay-cables of cable-stayed bridges. Journal of Wind Engineering and Industrial Aerodynamics,1995,57:323-333.
[124]顾明,罗国强.斜拉桥拉索的风(雨)激振及控制.力学季刊,1998,4：281-288.
[125]彭天波.斜拉桥拉索风雨激振的机理研究.上海：同济大学,2000.
[126]彭天波,顾明.斜拉桥拉索风雨激振的机理研究.同济大学学报,2001,29(1)：35-39.
[127]Gu M and Lu Q. Theoretical analysis of wind-rain induced vibration of cables of cable-stayed bridges. Journal of Wind Engineering,2001,89:125-128.
[128]Pei U and Nahrath N. Modeling of rain-wind induced vibrations. Wind and Structures, 2003,6(1):41-52.
[129]符旭晨,周岱,吴筑海.斜拉索的风振与减振.振动与冲击,2004,23(3)：29-37.
[130]Gu M and Du X. Experimental investigation of rain-wind-induced vibration of cables in cable-stayed bridges and its mitigation. Journal of Wind Engineering and Industrial Aerodynamics,2005,93(1):79-95.
[131]Burton D,Cao D Q, et al. On the stability of stay cables under light wind and rain conditions. Journal of Sound and Vibration,2005,279(1-2):89-117.
[132]中华人民共和国国家标准.双线往复式客运架空索道设计规范,GB／T13676-1992.
[133]中华人民共和国国家标准.客运架空索道安全规范,GB／T12352-1990.
[134]Rao G V and Iyengar R N. Internal resonance and non-linear response of a cable under periodic excitation. Journal of Sound and Vibration,1991,149(1):25-41.
[135]Benedettini F, Rega G and Alaggio R. Non-Linear oscillations of a four-degree-of-freedom model of a suspended cable under multiple internal resonance conditions. Journal of Sound and Vibration,1995,182:775-798.
[136]Chang W K, Ibrahim R A and Afaneh A A. Planar and non-planar nonlinear dynamics of suspended cables under random in-plane loading-I:single internal resonance. International Journal of Non-Linear Mechanics,1996,31:837-859.
[137]Lee C L and Perkins N C. Nonlinear oscillations, of suspended cables containing a two-to-one internal resonance. Nonlinear Dynamics,1992,3:465-490.
[138]Zhao Y Y, Wang L H, Chen D L and Jiang L Z. Non-Linear dynamic analysis of the two dimensional simplified model of an elastic cable. Journal of Sound and Vibration,2002, 255(1):43-59.
[139]赵跃宇,王连华,陈得良,蒋丽忠.斜拉索面内振动和面外摆振的耦合分析.土木工程学报,2003,36(4)：65-69.
[140]王连华,赵跃宇.悬索在考虑1：3内共振情况下的动力学行为.固体力学学报,2006,27(3)：230-236.
[141]潘祖梁,陈仲慈.工程技术中的偏微分方程.浙江：浙江大学出版社,1996.
[142]Guckenheimer J and Holmes P. Nonlinear oscillations, dynamical systems, and bifur-cations of vector fields. New York:Springer-Verlag,1983.
[143]Hirsch M W and Smale S. Differential equations, dynamical systems and linear algebra. Boston:Academic Press,1974.
[144]Wiggins S. Introduction to applied nonlinear dynamical systems and chaos. New York: Springer-Verlag,1990.
[145]刘延柱,陈立群.非线性振动.北京：高等教育出版社,2001.
[146]胡海岩.应用非线性动力学.北京：航空工业出版社,2000.
[147]Vialar T. Complex and chaotic nonlinear dynamics:advances in economics and finance, mathematics and statistics. Berlin Heidelberg:Springer-Verlag,2009.
[148]Kuznetsov Y A. Elements of applied bifurcation theory. New York:Springer-Verlag,2004.
[149]Nayfeh A H and Mook D T. Nonlinear Oscillations. New York:John Wiley and Sons,1979.
[150]陈政清.桥梁风工程.北京：人民交通出版社,2005.
[151]Slater J E. Aeroelastic instability of a structural angle section. PhD Thesis, University of British Columbia,1969.
[152]Blevins R D and Iwan W D. The galloping response of a two-degree-of-freedom system. ASME Journal of Applied Mechanics,1974,41:1113-1118.
[153]Chen L Q and Zhao W J. The energetic and the stability of axially moving Kirchhoff strings. Journal of the Acoustical Society of America,2005,117:55-58.
[154]Yu P. Closed-form conditions of bifurcation points for general differential equations. International Journal of Bifurcation and Chaos,2005,15(4):1467-1483.
[155]Lau S L and Cheung Y K. Amplitude incremental variational principle for nonlinear vibration of elastic system. ASME Journal of Applied Mechanics,1981,48(4):959-964.
[156]Cheung Y K and Lau S L. Incremental time-space finite strip method for nonlinear structural vibrations. Earthquake Engineering and Structural Dynamics,1982,10(2):239-253.
[157]Leung A Y T and Chui S K. Nonlinear vibration of coupled doffing oscillators by an improved incremental harmonic balance method. Journal of Sound and Vibration,1995,181(4): 619-633.
[158]Raghothama A and Narayanan S. Periodic response and chaos in nonlinear systems with parametric excitation and time delay. Nonlinear Dynamics,2002,27(4):341-365.
[159]Sze K Y, Chen S H and Huang J L. The incremental harmonic balance method for nonlinear vibration of axially moving beams. Journal of Sound and Vibration,2005,281(3-5):611-626.
[160]Doedel E J, Govaerts W and Kuznetsov Y A. Computation of periodic solution bifurcations in ODEs using bordered systems. SIAM Journal on Numerical Analysis,2003,41(2):401-435.
[161]Govaerts W, Kuznetsov Y A and Dhooge A. Numerical continuation of bifurcations of limit cycles in MATLAB. SIAM Journal on Scientific Computing,2005,27(1):231-252.
[162]迪尔比耶,汉森著,薛素铎,李雄彦译.结构风荷载作用.北京：中国建筑工业出版社,2006.
[163]O'Malley R E Jr. Boundary layer methods for nonlinear initial value problems. SIAM Review,1971,13(4):425-434.
[164]Haberman R. Slowly varying jump and transition phenomena associated with algebraic bifurcation problems. SIAM Journal on Applied Mathematics,1979,37(1):69-106.
[165]Erneux T and Mandel P. Imperfect bifurcation with a slowly-varying control parameter. SIAM Journal on Applied Mathematics,1986,46(1):1-15.
[166]Baer S M, Erneux T and Rinzel J. The slow passage through a Hopf bifurcation:delay, memory effects, and resonance. SIAM Journal on Applied Mathematics,1989,49(1):55-71.
[167]Raman A, Bajaj A K and Davies P. On the slow transition across instabilities in non-linear dissipative systems.Journal of Sound and Vibration,1996,192(4):835-865.
[168]Lagerstrom P A. Matched Asymptotic Expansions:Ideas and Techniques. New York:Springer-Verlag,1988.
[169]Poincare H. New methods of Celestial mechanics, Vol.3, English translation. Paris: Gauthier-Zvillars,1967.
[170]Melnikov V K. On the stability of the center for time periodic perturbations. Transactions Moscow Mathematical Society,1963,12:1-56.
[171]Luo A C J. Global tangency and transversality of periodic flows and chaos in a periodically forced, damped duffing oscillator. International Journal of Bifurcation and Chaos,2008,18(1):1-49.
[172]Luo A C J. Discontinuous dynamical systems on time-varying domains. Beijing:Higher Education Press,2009.
[173]Luo A C J. A theory for n-dimensional nonlinear dynamics on continuous vector fields. Communication in Nonlinear Science and Numerical Simulation,2007,12:117-194.
[174]Lau S L, Cheung Y K and Wu S Y. Incremental harmonic balance method with multiple time scales for aperiodic vibration of nonlinear systems. Journal of applied mechanics,1983, 50:871-876.
[175]陈树辉,黄建亮.轴向运动梁非线性振动内共振研究.力学学报,2005,37：57-63.
[176]张伟,温洪波,姚明辉.黏弹性传动带1：3内共振时的周期和混沌运动.力学学报,2004,36：443-454.
[177]Liu Y Q, Zhang W and Chen L H. Non-planar periodic and chaotic oscillation of an axially moving viscoelastic belt with one-to-one internal resonance. International Journal of Nonlinear Sciences and Numerical Simulation,2007,8:589-600.
[178]Wang Y F, Liu X T and Huang L H.Stability analyses for axially moving strings in nonlinear free and aerodynamically excited vibrations. Chaos, Solitons and Fractals,2008, 38:421-429.
[179]Jones S E. Remarks on the perturbation process for certain conservative systems. International Journal of Non-Linear Mechanics,1978,13:125-128.
[180]Chen S H and Cheung Y K. A modified Lindstedt-Poincare method for a strongly nonlinear system with quadratic and cubic nonlinearities. Shock and Vibration,1996,3:279-285.
[181]Cheung Y K, Chen S H and Lau S L. A modified Lindstedt-Poincare method for certain strongly nonlinear oscillators. International Journal of Nonlinear Mechanics,1991,26: 367-378.
[182]He J H. Modified Lindstedt-Poincaremethods for some strongly non-linear oscillations part I:expansion of a constant.International Journal of Non-Linear Mechanics,2002,37: 309-314.
[183]He J H. Modified Lindstedt-Poincare methods for some strongly non-linear oscillations part II:a new transformation. International Journal of Non-Linear Mechanics,2002,37: 315-320.
[184]He J H. A review on some new recently developed nonlinear analytical techniques. International Journal of Nonlinear Sciences and Numerical Simulation,2000,1:51-70.
[185]He J H. An elementary introduction to recently developed asymptotic methods and nanomechanics in textile engineering. International Journal of Modern Physics B,2008,22: 3487-3578.
[186]Nayfeh A H. Introduction of Perturbation Techniques. New York-Wiley,1981.
[187]Meirovitch L. A new method of solution of the eigenvalue problem for gyroscopic systems. AIAA Journal,1974,12(10):1337-1342.
[188]Meirovitch L. A modal analysis for the response of linear gyroscopic systems. Journal of Applied Mechanics,1975,42:446-450.
[189]钟万勰.发展型哈密顿核积分方程.大连理工大学学报,2003,43(1)：1-11.
[190]Stanisic M M. On a new theory of the dynamic behavior of the structures carrying moving masses. Archive of Applied Mechanics,1985,55(3):176-185.
[191]Kukla S. The Green function method in frequency analysis of a beam with intermediate elastic supports. Journal of Sound and Vibration,1991,149(1):154-159.
[192]Kukla S and Zamojska I. Frequency analysis of axially loaded stepped beams by Green's function method. Journal of Sound and Vibration,2007,300:1034-1041.
[193]Pesterev A V, Yang B and Bergman L A. Response of elastic continuum carrying multiple moving oscillators. Journal of Engineering Mechanics,2001,127(3):260-265.
[194]Gerlach U H. Linear Mathematics in Infinite Dimensions:Signals, Boundary Value Problems, and Special Functions. Lecture Notes of Ohio State University, http://www. math. ohio-state. edu/~gerlach/math/BVtypset/,2007.
[195]Lee K Y and Renshaw A A. Solution of the moving mass problem using complex eigenfunction expansions. Journal of Applied Mechanics,2000,67:823-827.
[196]Yang B. Eigenvalue inclusion principles for distributed gyroscopic systems. Journal of Applied Mechanics,1992,59:650-656.
[197]张伟.轴向行进弦线横向振动的控制.上海大学博士学位论文,2006.4.
[198]Meirovitch L. Dynamics and Control of Structures. New York:Wiley,1990.
[199]Hughes P C and Skelton R E. Controllability and observability of linear ma-trix-second-order systems. Journal of Applied Mechanics,1980,47(2):415-420.
[200]Yang B and Mote C D Jr. Controllability and observability of distributed gyroscopic systems. Journal of Dynamic Systems, Measurement, and Control,1991,113(1):11-17.
[201]Yang B and Mote C D Jr. Frequency-Domain vibration control of distributed gyroscopic systems. Journal of Dynamic Systems,Measurement, and Control,1991,113(1):18-25.
[202]Yang B and Mote C D Jr. Active vibration control of the axially moving string in the S domain. Journal of Applied Mechanics,1991,58(1):189-196.
[203]Yang B. Noncollocated control of a damped string using time delay. Journal of Dynamic Systems, Measurement,and Control,1991,114(4):736-740.
[204]Chung C H and Tan C A. Active vibration control of the axially moving string by wave cancellation. Journal of Vibration and Acoustic,1995,117(1):49-55.
[205]Tan C A and Ying S.Active wave control of the axially moving string:theory and experiment.Applied Mathematics and Computation,2000,236(5):861-880.
[206]Zhang W and Chen L Q. Vibration control of an axially moving string system:Wave cancellation method. Applied Mathematics and Computation,2006,175:851-863.
[207]Fung R F and Tseng C. Boundary control of an axially moving string via Lyapunov method. Journal of Dynamic System, Measurement, and Control,1999,121:105-110.
[208]Li T C, Hou Z C and Li J F. Stabilization analysis of a generalized nonlinear axially moving string by boundary velocity feedback. Automatica,2008,44:498-503.
[209]Fung R F and Liao C C. Application of variable structure control in the nonlinear string system. International Journal of Mechanical Sciences,1995,37(9):985-993.
[210]Fung R F, Huang J S, Wang Y C and Yang R T. Vibration reduction of the nonlinearly traveling string by a modified variable structure control with proportional and integral compensations.International Journal of Mechanical Sciences,1998,40(6):493-506.
[211]Ramana Reddy D V, Sen A and Johnston G L. Dynamics of a limit cycle oscillator under time delayed linear and nonlinear feedbacks. Physica D,2000,144:335-357.
[212]Campbell S A, Belair J, Ohira T and Milton J. Complex dynamics and multistability in a damped harmonic oscillator with delayed negative feedback. Chaos,1995,5(4):640-645.
[213]Xu J and Chung K W. Effects of time delayed position feedback on a van der Pol-Duffing oscillator. Physica D,2003,180:17-39.
[214]Xu J, Chung K W, Chan C L. An efficient method for studying weak resonant double Hopf bifurcation in nonlinear systems with delayed feedbacks. SIAM Journal on Applied Dynamical Systems,2007,6(1):29-60.
[215]Xu J and Yu P. Delay-induced bifurcations in a nonautonomous system with delayed velocity feedbacks. International Journal of Bifurcation and Chaos,2004,14(8):2777-2798.
[216]Wang Z H and Hu H Y. An energy analysis of nonlinear oscillators with time-delayed coupling. International Journal of Bifurcation and Chaos,2006,16(8):2275-2292.
[217]Ji J C and Zhang N. Additive resonances of a controlled van der Pol-Duffing oscillator. Journal of Sound and Vibration,2008,315(1-2):22-33.
[218]Ji J C and Hansen C H. Stability and dynamics of a controlled van der Pol-Duffing oscillator. Chaos, Solitons and Fractals,2006,28:555-570.
[219]Maccari A. Modulated motion and infinite-periodic homoclinic bifurcation for pa-rametrically excited Lienard systems. International Journal of Non-Linear Mechanics,2000, 35:239-262.
[220]Maccari A. Vibration control for parametrically excited Lienard systems. International Journal of Non-Linear Mechanics,2006,41:146-155.
[221]Maccari A. Vibration amplitude control for a van der Pol-Duffing oscillator with time delay. Journal of Sound and Vibration,2008,317:20-29.
[222]Jack H. Theory of Functional Differential Equations. New York:Spring-Verlag,1977.
[223]秦元勋.带有时滞的动力系统的运动稳定性.北京：科学出版社,1989.
[224]Yang B and Mote C D Jr. On time delay in noncolocated control of flexible mechanical systems. Journal of Dynamic Systems, Measurement, and Control,1990,114(3):409-415.
[225]Belair J and Campbell S A. Stability and bifurcation of equilibria in a multiple-delayed differential equation. SIAM Journal on Applied Mathematics,1994,54(5):1402-1424.
[226]Estermann T. Complex Numbers and Functions. The Athlone Press, University of London, London,1962.
[227]Shampine L F and Thompson S. Solving DDEs in MATLAB. Applied Numerical Mathematics, 2001,37:441-458.
[228]Hassard B D, Kazarinoff N D and Wan Y H. Theory and Applications of Hopf Bifurcation. Cambridge University Press, Cambridge,1981.
[229]Chafee N. A bifurcation problem for a functional differential equation of finitely retarded type. Journal of Mathematical Analysis and Applications,1971,35:312-348.
[230]Chen L Q, Zhang W and Zu J W. Nonlinear dynamics for transverse motion of axially moving strings. Chaos, Solitons and Fractals,2009,40:78-90.
[231]Ravindra B and Zhu W D. Low-dimensional chaotic response of axially accelerating· continuum in the supercritical regime, Archive of Applied Mechanics,1998,68:195-205.