物理非线性粘弹性杆动力分析研究及混沌动力系统仿真软件的开发
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摘要
非线性科学已成为各门学科的研究前沿,固体力学将呈现出以非线性力学为核心的发展趋势,其所代表的无穷维动力系统的研究也已经获得了很大的进展。目前,混沌科学虽然在基础理论方面取得了很大的进展,但还没有取得根本性的突破,还有许多问题没有解决,数值计算仍然是研究混沌现象的一项基本手段。鉴于上述情况,本文主要作了以下几方面的研究工作:
1.对固体力学领域中混沌运动的研究现状和方法作了整体上的概述,并针对基本结构元件非线性直杆的动力学相关研究作了简要的回顾,指出了其中一些值得注意的问题。
2.针对数值计算在非线性动力学研究中的重要性,利用VisualC++6.0面向对象编程语言,开发了微分动力系统的仿真分析软件,集成了对已知微分动力系统和实验数据的各种混沌表征分析功能,其中主要包括时程曲线,相平面轨迹(或重构),功率谱分析,Poincare映射以及混沌的定量特征参数李雅谱诺夫指数的计算等。实现了人机交互操作和动态直观显示,可望为动力系统的研究提供一个极为便利的分析工具。软件主要特点有:(1)软件内植入了求解微分方程初值问题的自适应伪弧长算法,有利于对刚性和奇异性方程的求解;(2)独特的符号解析功能实现了微分动力系统输入的任意性,使其不再是一个局限于内嵌固定方程的演示软件而成为一种工具,从而极大地拓宽了软件的应用范围;(3)软件部分模块(Lyapunov指数谱)的计算中,利用Matlab软件提供的接口协议调用了Matlab符号Jacobi矩阵运算功能,实现了软件与Matlab的后台通信。另外通过一些算例的比较,验证了软件计算的正确性和准确性。
3.杆件作为力学与工程应用中最常用的基本构件,在一般的静
Nonlinear dynamics has become frontier topic almost in all branches of science. Unexceptionally, the development of solid mechanics also exhibits the same tendency. Although a great progress has been made in the Research of the infinite dimensional dynamic system which is frequently represented by continues mechanics, many theoretical problems remain unsettled. However, Numerical calculation, as a basic approach, is always a powerful tool to study the chaos. Major work in this dissertation include following aspects:1. An overview of the researching of chaos in solid mechanics field is made, and specially, a brief review of studies about the rod is also present, which is usually a fundamental structural member in mechanics and engineering. Some remarkable problems are addressed.2. Considering the important role played by numerical computation in the research of nonlinear dynamics, simulation software for chaotic dynamic system is developed by Visual C++ 6.0—a sort of excellent programming language. This program gathers almost all of the functions to character a chaotic response for a dynamic system or experiment data, such as time-history curve, phase portrait, power spectrum analysis, Poincare mapping or section, and Lyapunov exponents—a unique approach to descript the chaos quantificationally. This simulation software is a man-machine interactive system, where dynamic evolutionary process of the system can be demonstrated. It
might provide an effective tool for the studying of the dynamic system. Some distinguishing features are deserved to be noted: (1) a selectable arch-length integration method which is a proved better method to integrate a stiff or singular differential equation with initial values is immigrated into the software; (2) symbol resolution function embedded in the system allows an arbitrary input of differential equation. That makes the software become a powerful tool more than a demonstrating one, and significantly enlarges the scope of the applicability of the software. (3) During the calculating of Lyapunov exponents' spectrum, the powerful calculating capacity of symbol matrix provided in Matlab is invoked through background communication between VC and Matlab according to the Matlab interface protocol. Finally, the effectivity and validity of the software are confirmed through comparing the calculating results of some classical model.As a widely used structural member, rods are studied extensively whose strength, rigidity and stability are always our focus. In this paper, the basic longitudinal wave equation is derived by virtual work principle for a quadratic and cubic physical nonlinear Keilven-Voigt visco-elastic rod under the consideration of transverse inertia. Then based on the ideas that the solitary wave and the shock wave existed in a nonlinear evolutionary equation are matched with the homo-clinic and hetero-clinic orbits existed in a ordinary differential equation derived from the former, the solitary wave, the shock wave and the elliptical periodic wave solutions for a nonlinear rod are found in different material cases, and the influence of the nonlinearity and the dispersion on the properties of the nonlinear wave above are presented, such as the velocity, the amplitude and the width. Furthermore, the exact solutions of MKdV-Burgers equation for nonlinear visco-elastic rod are also obtained by hyperbolic tanh-function method. Physical signification of the results is emphasized. Since the emergence of solitary wave is
always accompanied with the gathering of energy, so the results presented in this dissertation may provide a reference for dynamic design or dynamic failure.4. A physical nonlinear viscoelastic rod with one end fixed and another end subjected to an axial periodical excitation was studied under the consideration of transverse inertia. Galerkin method is applied to transform an infinite dimensional dynamic system into a single freedom equation; critical conditions of chaos occurrence were established by use of Melnikov method. Finally numerical results are carried out by using the simulation software mentioned above, a clear image of strange abstractor is presented in Poincare mapping, whose fractal dimension and Lyapunov exponents are also obtained. The periodic doubling routine to chaos is demonstrated vividly in a product space spaned by response value and the parameter of exciting amplitude and damping coefficient respectively, from which all of the bifurcation value and threshold to chaos can be get easily, while the curves of Lyapunov exponents spectrum varying with the parameters above are also given, which completely match with the bifurcation graph.5. Although the rationality of this simplification hasn't been proved, Galerkin method becomes an extensively adopted method for studies of structure dynamic problems. In this paper, Galerkin method is applied to a nonlinear viscoelastic rod system, controlling equations are derived in different truncation orders, each of which are analyzed and computed for its response behavior, the effect of the truncation orders on computation of bifurcation and chaos is discussed. It is thought that Galerkin method should been applied carefully. Especially in a nonlinear system with quadric terms, at least 2-orders truncation rather than only 1-order truncation should be taken into account so that a safety value can be given relatively in engineering application at the consideration of its first instable point.
6. A two-freedom Hamilton system is obtained in terms of the free oscillation in double modes of a nonlinear rod system, Melnikov method used for near integrable Hamilton system is adopted to give a critical condition of chaos occurrence. Gradually breaking of KAM torus in different energy level is investigated through the numerical simulation. The evolutionary routine from periodic or quasi-periodic orbits to chaos is illustrated distinctly. It shows that stochastic motion might exist in free oscillation of a nonlinear rod. During the process of dynamic simulation, new phenomena are found that a big KAM torus break into many little torus in a kind of "hanging" mode when energy level is promoted, and these little torus are connective sequentially. The universality for such phenomena needs further research.7. When multiple truncation orders are adopted in Galerkin method, the obtained controlling equation will have a giant "volume" due to too many terms, which will be accumulated rapidly with the truncation orders in a geometrical series mode, so FEM is used to analyze the nonlinear viscoelastic rod system in this dissertation. Principal of virtual work is applied and nonlinear dynamic system represented by a group of coupling ordinary equation is obtained finally. Response of bifurcation and chaos is investigated for a rod subjected to periodic excitation, and the results in different discrete number and different time step is compared in order to verify the practicability of FEM for calculating bifurcation and chaos.8. Melnikov method is widely used as a criterion to check whether subharmonic and ultra-subharmonic or even chaos will occur. In this paper, It is proved that if there exists a periodic solution for a class of non-autonomous differential dynamic system, it can only be subharmonic, ultra-subharmonic periodic solution is impossible. Moreover, the existence of R-type ultra-subharmonic periodic solution defined for a specified planar system is also denied. As an application
of above conclusions, through investigating some typical examples, it is pointed out that the existence of ultra-subharmonic periodic orbits in a planar perturbation system can't be determined by second-order Melnikov method. An explanation is also provided geometrically.
1.对固体力学领域中混沌运动的研究现状和方法作了整体上的概述,并针对基本结构元件非线性直杆的动力学相关研究作了简要的回顾,指出了其中一些值得注意的问题。
2.针对数值计算在非线性动力学研究中的重要性,利用VisualC++6.0面向对象编程语言,开发了微分动力系统的仿真分析软件,集成了对已知微分动力系统和实验数据的各种混沌表征分析功能,其中主要包括时程曲线,相平面轨迹(或重构),功率谱分析,Poincare映射以及混沌的定量特征参数李雅谱诺夫指数的计算等。实现了人机交互操作和动态直观显示,可望为动力系统的研究提供一个极为便利的分析工具。软件主要特点有:(1)软件内植入了求解微分方程初值问题的自适应伪弧长算法,有利于对刚性和奇异性方程的求解;(2)独特的符号解析功能实现了微分动力系统输入的任意性,使其不再是一个局限于内嵌固定方程的演示软件而成为一种工具,从而极大地拓宽了软件的应用范围;(3)软件部分模块(Lyapunov指数谱)的计算中,利用Matlab软件提供的接口协议调用了Matlab符号Jacobi矩阵运算功能,实现了软件与Matlab的后台通信。另外通过一些算例的比较,验证了软件计算的正确性和准确性。
3.杆件作为力学与工程应用中最常用的基本构件,在一般的静
Nonlinear dynamics has become frontier topic almost in all branches of science. Unexceptionally, the development of solid mechanics also exhibits the same tendency. Although a great progress has been made in the Research of the infinite dimensional dynamic system which is frequently represented by continues mechanics, many theoretical problems remain unsettled. However, Numerical calculation, as a basic approach, is always a powerful tool to study the chaos. Major work in this dissertation include following aspects:1. An overview of the researching of chaos in solid mechanics field is made, and specially, a brief review of studies about the rod is also present, which is usually a fundamental structural member in mechanics and engineering. Some remarkable problems are addressed.2. Considering the important role played by numerical computation in the research of nonlinear dynamics, simulation software for chaotic dynamic system is developed by Visual C++ 6.0—a sort of excellent programming language. This program gathers almost all of the functions to character a chaotic response for a dynamic system or experiment data, such as time-history curve, phase portrait, power spectrum analysis, Poincare mapping or section, and Lyapunov exponents—a unique approach to descript the chaos quantificationally. This simulation software is a man-machine interactive system, where dynamic evolutionary process of the system can be demonstrated. It
might provide an effective tool for the studying of the dynamic system. Some distinguishing features are deserved to be noted: (1) a selectable arch-length integration method which is a proved better method to integrate a stiff or singular differential equation with initial values is immigrated into the software; (2) symbol resolution function embedded in the system allows an arbitrary input of differential equation. That makes the software become a powerful tool more than a demonstrating one, and significantly enlarges the scope of the applicability of the software. (3) During the calculating of Lyapunov exponents' spectrum, the powerful calculating capacity of symbol matrix provided in Matlab is invoked through background communication between VC and Matlab according to the Matlab interface protocol. Finally, the effectivity and validity of the software are confirmed through comparing the calculating results of some classical model.As a widely used structural member, rods are studied extensively whose strength, rigidity and stability are always our focus. In this paper, the basic longitudinal wave equation is derived by virtual work principle for a quadratic and cubic physical nonlinear Keilven-Voigt visco-elastic rod under the consideration of transverse inertia. Then based on the ideas that the solitary wave and the shock wave existed in a nonlinear evolutionary equation are matched with the homo-clinic and hetero-clinic orbits existed in a ordinary differential equation derived from the former, the solitary wave, the shock wave and the elliptical periodic wave solutions for a nonlinear rod are found in different material cases, and the influence of the nonlinearity and the dispersion on the properties of the nonlinear wave above are presented, such as the velocity, the amplitude and the width. Furthermore, the exact solutions of MKdV-Burgers equation for nonlinear visco-elastic rod are also obtained by hyperbolic tanh-function method. Physical signification of the results is emphasized. Since the emergence of solitary wave is
always accompanied with the gathering of energy, so the results presented in this dissertation may provide a reference for dynamic design or dynamic failure.4. A physical nonlinear viscoelastic rod with one end fixed and another end subjected to an axial periodical excitation was studied under the consideration of transverse inertia. Galerkin method is applied to transform an infinite dimensional dynamic system into a single freedom equation; critical conditions of chaos occurrence were established by use of Melnikov method. Finally numerical results are carried out by using the simulation software mentioned above, a clear image of strange abstractor is presented in Poincare mapping, whose fractal dimension and Lyapunov exponents are also obtained. The periodic doubling routine to chaos is demonstrated vividly in a product space spaned by response value and the parameter of exciting amplitude and damping coefficient respectively, from which all of the bifurcation value and threshold to chaos can be get easily, while the curves of Lyapunov exponents spectrum varying with the parameters above are also given, which completely match with the bifurcation graph.5. Although the rationality of this simplification hasn't been proved, Galerkin method becomes an extensively adopted method for studies of structure dynamic problems. In this paper, Galerkin method is applied to a nonlinear viscoelastic rod system, controlling equations are derived in different truncation orders, each of which are analyzed and computed for its response behavior, the effect of the truncation orders on computation of bifurcation and chaos is discussed. It is thought that Galerkin method should been applied carefully. Especially in a nonlinear system with quadric terms, at least 2-orders truncation rather than only 1-order truncation should be taken into account so that a safety value can be given relatively in engineering application at the consideration of its first instable point.
6. A two-freedom Hamilton system is obtained in terms of the free oscillation in double modes of a nonlinear rod system, Melnikov method used for near integrable Hamilton system is adopted to give a critical condition of chaos occurrence. Gradually breaking of KAM torus in different energy level is investigated through the numerical simulation. The evolutionary routine from periodic or quasi-periodic orbits to chaos is illustrated distinctly. It shows that stochastic motion might exist in free oscillation of a nonlinear rod. During the process of dynamic simulation, new phenomena are found that a big KAM torus break into many little torus in a kind of "hanging" mode when energy level is promoted, and these little torus are connective sequentially. The universality for such phenomena needs further research.7. When multiple truncation orders are adopted in Galerkin method, the obtained controlling equation will have a giant "volume" due to too many terms, which will be accumulated rapidly with the truncation orders in a geometrical series mode, so FEM is used to analyze the nonlinear viscoelastic rod system in this dissertation. Principal of virtual work is applied and nonlinear dynamic system represented by a group of coupling ordinary equation is obtained finally. Response of bifurcation and chaos is investigated for a rod subjected to periodic excitation, and the results in different discrete number and different time step is compared in order to verify the practicability of FEM for calculating bifurcation and chaos.8. Melnikov method is widely used as a criterion to check whether subharmonic and ultra-subharmonic or even chaos will occur. In this paper, It is proved that if there exists a periodic solution for a class of non-autonomous differential dynamic system, it can only be subharmonic, ultra-subharmonic periodic solution is impossible. Moreover, the existence of R-type ultra-subharmonic periodic solution defined for a specified planar system is also denied. As an application
of above conclusions, through investigating some typical examples, it is pointed out that the existence of ultra-subharmonic periodic orbits in a planar perturbation system can't be determined by second-order Melnikov method. An explanation is also provided geometrically.
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