论文题目：多自由度非线性机械系统的全局分叉和混沌动力学研究 作者简介：姚明辉，女， 1971年11月出生，2002年09月师从于北京工业大学张伟教授，于2006年06月获博士学位。
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在机械系统中，有许多问题的数学模型和动力学方程都可用高维非线性系统来描述，对于高维非线性动力系统来说，其研究难度比低维非线性动力系统要大得多，不仅理论方法上有困难，几何描述和数值计算都有困难。目前研究高维非线性系统的全局分叉和混沌动力学的理论方法还不是很多，国际上处于发展阶段，国内尚处于起步阶段，因此发展处理高维非线性动力学系统的理论研究方法是非常重要和迫切的。在高维非线性动力学的全局分叉和混沌动力学问题中，除了单脉冲混沌运动外，还有多脉冲混沌运动，目前研究多脉冲混沌运动的解析方法主要有两种，即广义Melnikov方法和能量相位法。
本论文改进和推广了Kovacic、Haller和Wiggins等人提出的广义Melnikov方法和能量相位法，利用这两种全局摄动解析方法首次研究了非线性非平面运动悬臂梁、粘弹性传动带非平面运动和面内载荷和横向载荷联合作用下四边简支薄板的多脉冲轨道和Shilnikov型混沌运动。理论研究发现这些系统存在多脉冲混沌运动；利用数值方法模拟、验证了理论分析的结果。论文的研究内容及取得的创新性成果有以下几个方面。
(1) 综述了高维非线性系统的分叉和混沌动力学的国内外研究现状；简要介绍了Melnikov方法的发展，高维Melnikov方法的应用，以及广义Melnikov方法的提出和建立；概括了能量相位法的国内外主要研究进展；介绍了研究高维非线性系统的全局分叉和混沌运动的其它方法。总结了能量相位法和广义Melnikov方法的研究进展、成果及存在的不足和有待深入研究的问题。
(2) 介绍了由Haller和Wiggins提出的能量相位法；以及由Kovacic等人提出的广义Melnikov方法。由于能量相位法和广义Melnikov方法提出和发展的时间较短，而且一直是独立的两种解析方法，在本论文中，首次详细地研究了两种全局摄动解析方法的区别和联系。
(3) Haller和Wiggins提出的能量相位法在计算能量差分函数时，所引入的变换改变了原来系统的拓扑结构。为了使原来系统的拓扑结构不发生变化，我们改进了能量相位法。利用改进的能量相位法，首次研究了非线性非平面运动悬臂梁、粘弹性传动带和四边简支薄板的全局分叉和混沌动力学，发现这些系统存在多脉冲混沌运动。
(4) 由于广义Melnikov方法在理解、计算和开折条件的证明上，存在很大的难度，因此，一直没有应用到实际工程中分析一些具体的模型。本文首次把广义Melnikov方法推广到实际工程中，利用广义Melnikov方法研究具有实际工程背景的三个高维非线性机械系统，从理论上给出了这些系统产生Shilnikov型混沌运动的必要条件。
(5) 首次研究了非线性非平面运动悬臂梁的多脉冲异宿轨道和混沌动力学。在主共振-主参数共振-1:2内共振情形的平均方程的基础上，利用规范形理论进行化简；利用能量相位法，首次从理论上得到了非线性非平面运动悬臂梁产生Shilnikov型混沌的必要条件，发现在这个系统中存在着Shilnikov型混沌运动。数值分析表明非线性非平面运动悬臂梁的平均方程确实存在Shilnikov型多脉冲混沌运动，发现系统的阻尼和激励两个参数对系统出现多脉冲混沌运动影响较大，进一步验证了理论分析的结果，在三维相空间里存在Shilnikov型多脉冲混沌运动轨线。
(6) 首次研究了变张力粘弹性传动带非平面运动时多脉冲同宿轨道和混沌动力学。建立了粘弹性传动带非平面运动的偏微分方程，应用Galerkin法和多尺度方法得到主参数共振-1:1内共振情形的平均方程，利用规范形理论化简平均方程；首次利用能量相位法研究粘弹性传动带的多脉冲同宿轨道和混沌动力学，验证Shilnikov多脉冲轨道的存在性。数值模拟了粘弹性传动带的多脉冲同宿轨道的混沌运动，数值计算脉冲个数、区域直径和相位漂移之间的关系，发现随着脉冲个数的增加，Shilnikov型多脉冲轨道的区域直径减小。
(7) 首次研究了面内载荷和横向载荷联合作用下四边简支矩形薄板的多脉冲异宿轨道和混沌动力学。在四边简支矩形薄板的运动偏微分方程基础之上，应用Galerkin法和多尺度方法得到主参数共振-基本参数共振-1:2内共振情形的平均方程，利用规范形理论进行化简，首次利用能量相位法研究薄板的Shilnikov型多脉冲异宿轨道和混沌动力学，理论分析发现系统存在多脉冲跳跃而导致的Smale马蹄意义的混沌。数值分析表明四边简支矩形薄板的平均方程存在Shilnikov型多脉冲混沌运动，发现系统的阻尼和激励两个参数对系统出现多脉冲混沌运动影响较大，进一步验证了理论研究的结果，在三维相空间里存在Shilnikov多脉冲混沌运动。
(8) 首次利用近可积Hamilton系统的广义Melnikov方法研究悬臂梁的多脉冲同宿轨道和混沌动力学，得到了在共振情况下判断非线性非平面运动悬臂梁产生多脉冲混沌运动的广义Melnikov函数，求解满足开折条件的零点。从理论上给出了这个系统产生Shilnikov型混沌的必要条件。数值模拟了非线性非平面运动悬臂梁的多脉冲混沌运动。
(9) 利用近可积Hamilton系统的广义Melnikov方法首次研究了粘弹性传动带空间运动和面内载荷与横向载荷联合作用下四边简支矩形薄板的多脉冲异宿轨道和混沌动力学。得到了在共振情况下判断这些系统产生多脉冲混沌运动的广义Melnikov函数，求解满足开折条件的零点，从理论上给出了这些系统产生Shilnikov型混沌的必要条件。理论分析发现这些系统存在多脉冲跳跃而导致的Smale马蹄意义的混沌。数值结果说明了理论结果的正确性，并且发现一些参数和初始条件对于这些系统产生多脉冲混沌运动有着较大的影响。
(10) 用数值方法研究了一个二自由度机械系统的多脉冲混沌运动，发现了一种新的多脉冲混沌吸引子。
能量相位法和广义Melnikov方法提出和发展的时间较短，理论体系较新而复杂，能量相位法是从多脉冲跳跃轨道的能量耗散方面来研究多脉冲混沌运动，而广义Melnikov方法则是从多脉冲奇异横截面中的稳定流形和不稳定流形来研究多脉冲混沌运动。研究表明，这两种方法分别只研究了多脉冲轨道的一个方面，如果能够把两者结合起来研究多脉冲混沌运动，则其结论将更加完整。
本论文的创新点有以下几个方面。
(1) 首次利用能量相位法和广义Melnikov方法研究了非线性非平面运动悬臂梁、粘弹性传动带非平面运动和面内载荷与横向载荷联合作用下四边简支薄板的多脉冲轨道和Shilnikov型混沌运动，发现在三个机械系统中存在着Shilnikov型混沌运动。
(2) Haller与Wiggins利用能量相位法计算能量差分函数时，他们所引入的变换改变了原系统的拓扑结构。为了使原系统的拓扑结构不发生变化，我们改进了能量相位法。
(3) 由于广义Melnikov方法在理解、计算和开折条件的证明上，存在很大的难度，因此，一直未应用于实际工程系统。本文首次把广义Melnikov方法应用于三个机械系统，从理论上给出了这些系统产生Shilnikov型混沌运动的必要条件。
(4) 用数值方法研究了一个二自由度非线性机械系统，在这个系统中发现了一种新的多脉冲混沌吸引子。
本论文利用能量相位法和广义Melnikov方法研究了非线性非平面运动悬臂梁、粘弹性传动带非平面运动和面内载荷与横向载荷联合作用下四边简支矩形薄板的多脉冲轨道和混沌动力学。通过本文的研究，发现能量相位法和广义Melnikov方法有一些有待于进一步改进和完善的方面。下述几个问题值得进一步的研究。
(1) 如何把能量相位法和广义Melnikov方法推广到高维非自治系统和高于四维的更高维非线性系统。
(2) 利用能量相位法分析非线性系统的多脉冲轨道和混沌动力学的关键在于定义耗散因子，而耗散因子是阻尼与外激励的比值。目前，能量相位法只能用来分析单阻尼、单激励单耗散因子的系统，如何把能量相位法扩展到多阻尼、多激励多耗散因子的系统，有待进一步的研究。
(3) 能量相位法和广义Melnikov方法理论体系比较复杂，不利于工程科学家用来解决工程实际问题。如何进一步改进和简化这两种方法，提出新的多脉冲轨道和混沌动力学的判定准则，使这两种全局摄动方法更好地应用于工程实际问题。
关键词：
广义Melnikov方法，能量相位法， Shilnikov型多脉冲轨道，全局分叉，混沌动力学，规范形，悬臂梁，粘弹性传动带，薄板
Studies on Global Bifurcations and Chaotic Dynamics in Multi-Degree of Freedom Nonlinear Mechanical Systems
Yao Minghui ABSTRACT
The governing equations of motion for a number of engineering problems can be described by high-dimensional nonlinear systems.Comparing with low-dimensional nonlinear systems, the theory method, geometrical description and numerical simulation on the complicated dynamic behavior of high dimensional nonlinear systems were more difficult.The global bifurcations and chaotic dynamics of high-dimensional nonlinear systems have been at the forefront of nonlinear dynamics for the last two decades.Due to lack of analytical tools and methods to study the global bifurcations and chaotic dynamics for high-dimensional nonlinear systems, it is extremely challenging to develop the theories of the global bifurcations and chaotic dynamics for high-dimensional nonlinear systems and to give systematic applications to engineering problems.Therefore, the global bifurcations and chaotic dynamics of high-dimensional nonlinear systems are important theoretical problems in science and engineering applications as they can reveal the instabilities of motion and complicated dynamical behaviors in high-dimensional nonlinear systems.Besides the Shilnikov type single-pulse global bifurcations and chaotic dynamics of high-dimensional nonlinear systems, the Shilnikov type multi-pulse homoclinic and heteroclinic bifurcations and chaotic dynamics were investigated.Two main methods for studying the Shilnikov type multi-pulse homoclinic and heteroclinic orbits in high-dimensional nonlinear systems are the energy-phase method and the generalized Melnikov method.In this diertation, we improve and expand the energy-phase method and the generalized Melnikov method presented by Haller, Kovacic and Wiggins.These two methods are utilized to investigate the Shilnikov type multi-pulse heteroclinic and homoclinic bifurcations and chaotic dynamics for three high-dimensional nonlinear mechanical systems which the nonlinear non-planar oscillations of a cantilever beam, a parametrically excited viscoelastic moving belt and a parametrically and externally excited thin plate.The analysis of global dynamics indicates that there exist the multi-pulse jumping orbits in the perturbed phase space of the averaged equation for three high-dimensional nonlinear mechanical systems.These results show that the multi-pulse Shilnikov orbits chaotic motions can occur for three high-dimensional nonlinear mechanical systems.Numerical simulations are also given to verify the analytical predictions. The research contents and the innovative contributions of this diertation are as follows: (1) We give a review of the researches on the global bifurcations and chaotic dynamics of high-dimensional nonlinear systems and summarize the developments and results achieved on studies the Shilnikov type multi-pulse chaotic dynamics with the energy-phase method and the generalized Melnikov method in the past two decades.We indicate the unsolved problems at present and the developing directions in the energy-phase method and the generalized Melnikov method in the future.(2) We give a briefly description on the energy-phase method and the generalized Melnikov method based on the research work given by Haller, Kovacic and Wiggins et al.in the theoretical frame.Due to the short time of the development and independence of the two methods, we analyze the difference and relation between the two global singular perturbation methods in detail for the first time.(3) Based on research obtained in this diertation, we think that the symplectic transformations used by Haller et al.do not have topological equivalence because they will change the topology of the phase space and the types of multi-pulse connections.The energy-phase method is further improved to ensure the equivalence of topological structure for the phase portraits.The multi-pulse Shilnikov orbits and chaotic dynamics with the energy-phase method in three high-dimensional nonlinear mechanical systems are studied in this diertation for the first time.These results show that the multi-pulse Shilnikov orbits chaotic motions can occur for three high-dimensional nonlinear mechanical systems.(4) Due to difficulties of comprehension and computation of the generalized Melnikov method, it is not always applied to engineering problems.We expand and apply the generalized Melnikov method to study the Shilnikov type multi-pulse orbits to resonance bands in three high-dimensional nonlinear mechanical systems for the first time.These results show that the multi-pulse Shilnikov orbits chaotic motions can occur for three high-dimensional nonlinear mechanical systems.(5) The many pulses orbits with the energy-phase method chaotic dynamics for the nonlinear non-planar oscillations of a cantilever beam are studied in this diertation for the first time.The resonant case considered here is principal parametric resonance-1/2 sub-harmonic resonance for the first mode and fundamental parametric resonance-primary resonance for the second mode.Based on normal form obtained, the improved energy-phase method is utilized to analyze the multi-pulse global heteroclinic bifurcations and chaotic dynamics for the nonlinear non-planar oscillations of the cantilever beam for the first time.The chaotic motions of the nonlinear non-planar oscillations of a cantilever beam are also found by using numerical simulation.
(6) The multi-pulse orbits and chaotic dynamics of parametrically excited viscoelastic moving belt are studied in detail for the first time.Using Kelvin-type viscoelastic constitutive law, the equations of motion for viscoelastic moving belt with the external damping and parametric excitation are determined.The four-dimensional averaged equation under the case of 1:1 internal resonance and primary parametric resonance is obtained by directly using the method of multiple scales and Galerkin’s approach to the partial differential governing equation of viscoelastic moving belt.From the averaged equations obtained here, the theory of normal form is used to give the explicit expreions of normal form with a double zero and a pair of pure imaginary eigenvalues.Based on the normal form, the improved energy-phrase method is employed to analyze the global homoclinic bifurcations and chaotic dynamics in parametrically excited viscoelastic moving belt.The global analysis indicates that there exist the Shilnikov type multi-pulse orbits in the averaged equation.The results obtained above mean the existence of the chaos for the Smale horseshoe sense in motion of parametrically excited viscoelastic moving belt.The chaotic motions of viscoelastic moving belts are also found by using numerical simulation.It is also found from the results of numerical simulation of the relationship of the width of the layers and the lowest number of pulses that the width of the layers decreases with the augment of the lowest number of pulses.
(7) The multi-pulse Shilnikov orbits and chaotic dynamics in a parametrically and externally excited thin plate are studied in this diertation for the first time.The thin plate is subjected to transversal and in-plane excitations, simultaneously.The formulas of the thin plate are derived from the von Kármán equation and Galerkin’s method.The method of multiple scales is used to find the averaged equation.The theory of normal form, based on the averaged equation, is used to obtain the explicit expreions of normal form aociated with a double zero and a pair of purely imaginary eigenvalues from the Maple program.Based on the normal form obtained above, the diipative version of the improved energy-phase method is utilized to analyze the multi-pulse global heteroclinic bifurcations and chaotic dynamics in a parametrically and externally excited thin plate.The global dynamics analysis indicates that there exist the multi-pulse jumping orbits in the perturbed phase space of the averaged equations for a parametrically and externally excited thin plate.These results show that the chaotic motions of the multi-pulse Shilnikov type can occur in a parametrically and externally excited thin plate.Numerical simulations are given to verify the analytical predictions.It is also found from the results of numerical simulation that the multi-pulse Shilnikov type orbits exist in a parametrically and externally excited thin plate.(8) The generalized Melnikov method of near-integral Hamiltonian system is applied to study the multi-pulse global homoclinic bifurcations and chaotic dynamics for the nonlinear non-planar oscillations of the cantilever beam for the first time.The analysis of global dynamics indicates that there exist the multi-pulse jumping orbits in the perturbed phase space of the averaged equation for the nonlinear non-planar oscillations of the cantilever beam.Numerical simulations are given to verify the analytical predictions.It is also found from the results of numerical simulation in three-dimensional phase space that the multi-pulse orbits exist for the nonlinear non-planar oscillations of the cantilever beam.(9) The generalized Melnikov method of near-integral Hamiltonian system is applied to study the multi-pulse global heteroclinic bifurcations and chaotic dynamics for parametrically excited viscoelastic moving belt and a parametrically and externally excited thin plate for the first time.The analysis of global dynamics indicates that there exist the multi-pulse jumping orbits in the perturbed phase space of the averaged equation for these systems.Numerical simulations are given to verify the analytical predictions.It is also found from the results of numerical simulation in three-dimensional phase space that the multi-pulse orbits exist for these systems.(10) The results of numerical simulation show that the chaotic motion of the new Shilnikov type multi-pulse orbits can occur for a two-degree-of-freedom nonlinear mechanical system.
Generalized Melnikov method and the energy-phase method developed in the short time.The energy-phase method studies diipative energy of multi-pulse orbits, while generalized Melnikov method analyses the distance of the stable manifold and unstable manifold of multi-pulse orbits.They have merit and defect respectively.If we can combine these both methods to study multi-orbits, we will draw a conclusion completely.The innovative achievements of this diertation mainly are as follows: (1) The Shilnikov type multi-pulse orbits with the energy-phase method and generalized Melnikov method chaotic dynamics for the nonlinear non-planar oscillations of a cantilever beam, parametrically excited viscoelastic moving belt and a parametrically and externally excited thin plate are studied in this diertation for the first time.The Shilnikov type chaotic dynamics are found in the three high-dimensional nonlinear mechanical systems.(2) Based on research obtained in this diertation, we think that the symplectic transformations used by Haller et al.do not have topological equivalence because they will change the topology of the phase space and the types of multi-pulse connections.The energy-phase method is further improved to ensure the equivalence of topological structure for the phase portraits.(3) Due to difficulties of comprehension and computation of the generalized Melnikov method, it is not always applied to engineering problems.We expand and apply the generalized Melnikov method to study the Shilnikov type multi-pulse orbits to resonance bands in three high-dimensional nonlinear mechanical systems for the first time.These results show that the multi-pulse Shilnikov orbits chaotic motions can occur for three high-dimensional nonlinear mechanical systems.(4) The results of numerical simulation show that the chaotic motion of the new Shilnikov type multi-pulse orbits can occur for a two-degree-of-freedom nonlinear mechanical system.
The Shilnikov type multi-pulse orbits with the energy-phase method and generalized Melnikov method chaotic dynamics for the nonlinear non-planar oscillations of a cantilever beam, parametrically excited viscoelastic moving belt and a parametrically and externally excited thin plate are studied in this diertation.We find the energy-phase method and generalized Melnikov method needing to improve further through our research.The three aspects as follows need further study: (1) How to expand the energy-phase method and generalized Melnikov method to high-dimensional non-autonomous nonlinear systems and high-dimensional which are higher than four dimensional nonlinear systems? (2)Using the energy-phase method to analyze multi-pulse orbits for high-dimensional nonlinear systems is important to define a diipative factor which is the ratio of the damping coefficient to the excited force.Until now the energy-phase method can only study single diipative factor, while can not analyze many diipative factors which are the ratio of the damping coefficients to the excited forces.How to deal with many diipative factors? (3) The theory of the energy-phase method and generalized Melnikov method is too complicated to apply to engineering problems conveniently.How to improve and simplify these two methods to apply engineering field well? How to present new criterion of the multi-pulse orbits?
Key words: Generalized Melnikov method, the energy-phase method, Shilnikov type multi-pulse, global bifurcations, chaotic dynamics, theory of normal form, cantilever beam, viscoelastic moving belt, thin plate
