【作者】 吕乐丰；

【导师】 刘迎曦； 王跃方；

【作者基本信息】 大连理工大学 ， 工程力学， 2010， 博士

【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。更多还原

【Abstract】 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.更多还原