约束轮对随机非线性动力学理论研究
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摘要
随着国内外高速铁路和高速列车的大力发展,由此带来了对车辆、轨道系统动力学各方面更为严峻的挑战和要求。确定性的观点在物理、工程技术、生物和经济领域中的应用是众所周知的,然而随着科学技术的发展,要求对实际问题的描述越来越精确。因此,随机因素的影响就不能轻易被忽视,于是对某些实际过程的分析也就有必要从通常的确定性观点转到随机的观点。对于现代轨道车辆而言,由于高速度已成为高速铁路高新技术的核心,随机因素影响更应受到重视,它对系统的运动稳定性、运行平稳性、脱轨安全性、结构服役可靠性以及列车空气动力学问题等均有重要作用。由于对轨道车辆的研究以往主要集中于确定性框架,而对随机非线性动力学的研究也主要集中于响应问题,对定性行为、可靠性等少有涉及,所以本文尝试在这方面作一些工作,内容主要包括：
(1)物理模型和数学模型的建立：考虑轨道随机不平顺激励(根据作用机理主次分为随机外激和随机参激两种类型)与结构自身的频变随机参激作用等,把弹性约束轮对系统的建模从Lagrange体系转换到了Hamilton体系,用Hamilton函数的形式(即从能量的角度,把多因素的随机响应问题转化为单因素能量的分析)来对动力学行为进行研究,并建立了弹性约束轮对系统带Hamilton函数形式的Ito型随机微分方程组,运用随机平均法把该Ito型随机微分方程组表示为一维扩散过程,同时得到了支配该过程的平均Ito随机微分方程。
(2)随机稳定性、分岔以及实验测定方法研究：运用拟不可积Hamilton系统相关理论和Oseledec乘性遍历定理求解了系统的最大Lyapunov指数,得到了系统的随机局部稳定性条件；通过分析一维扩散的奇异边界形态,得到了系统随机全局稳定性条件；依据系统响应的平稳概率密度和联合概率密度,得到了系统的随机Hopf分岔类型,以及D-分岔(动态分岔)和P-分岔(唯象分岔)的分岔条件,并对随机稳定性和确定性稳定性进行了比较分析；另外给出了分岔点以及脱轨安全条件的实验测定方法,并把该方法应用到了具体的实验数据分析中,实验结果和理论分析结果得到了较好吻合,验证了方法的正确性和线路应用的可行性。
(3)首次穿越失效可靠性研究：在对随机稳定性和随机分岔分析的基础上,求得了弹性约束轮对系统发生首次穿越失效可靠性破坏的条件,得到了系统可靠性函数所满足的后向Kolmogorov方程,首次穿越平均时间满足的广义Pontryagin方程以及首次穿越时间条件概率密度方程,并结合初始条件和边界条件,运用相关的数值方法对其进行了分析,研究了首次穿越失效对系统形态的影响以及失效后的动力学行为。
(4)随机非线性最优控制研究：依据随机动态规划方法,以系统可靠度更高和首次穿越时间更长为目标,对弹性约束轮对系统进行了随机非线性最优控制分析,并对控制效果和策略进行了详细讨论,另外还对弹性约束轮对系统随机稳定性化的非线性随机最优控制进行了研究。
With the rapid development of domestic and international high-speed rail and high-speed trains, much more severe challenges and requirements are put forward for vehicle-track system dynamics. As well known, deterministic view is widely used in physical, engineering, biological and economic fields, but with the development of science and technology, descriptions of the practical problems are required to be more accurate. Therefore, the influence of stochastic factors cannot be ignored easily, for some actual process analysis it is necessary to investigate in the stochastic view instead of deterministic view. For the modern rail vehicles, since high speed has become the core of high-speed railway technology, stochastic factors should be taken seriously, which play an important role in system stability, ride comfort, derailment safety, structural reliability, and train air dynamics issues. Because the rail vehicle research in the past mainly focused on the deterministic framework, while the stochastic nonlinear dynamics studies mainly focused on the system response and qualitative behaviors are rarely involved, this thesis attempts to do further research in this area, mainly including:
(1) the building of physical and mathematical model:this paper considers track stochastic irregularity excitation based on primary and secondary action mechanism, that is, stochastic external excitations and randomly parametric excitation, and the structure frequency dependent stochastic parametric excitations. The modeling of elastic restraint wheelset system is transferred from Lagrange system to Hamilton system. With the Hamiltonian (from the energy view, the stochastic response problem analysis of multi-factor is transformed into single-factor energy analysis), dynamic behavior research is conducted. Meanwhile Ito stochastic differential equations with Hamiltonian of the elastic constraints wheelset system are established, which are represented as one-dimensional diffusion process using stochastic averaging method, and average Ito stochastic differential equation which dominated the process has been obtained.
(2) The study of stochastic stability, bifurcation and experimental determination method:based on quasi non-integrable Hamilton system theories and Oseledec multiplicative ergodic theorem, maximal Lyapunov exponents of the system are obtained thus and local stability conditions of the random system are got. Through the analysis of one-dimensional singular boundary diffusion patterns, global stability conditions of the random system are obtained Using averaged probability density and joint probability density of system response, stochastic Hopf bifurcation types, D-bifurcation (dynamic bifurcation) and P-bifurcation (phenomenological bifurcation) bifurcation conditions are obtained, and the stochastic stability and deterministic stability are compared and analyzed. Experimental determination methods of the bifurcation points and derailment safety conditions are given, which have been applied to realistic experimental data analysis. The experimental results and theoretical analysis results have a good match which verifies the correctness of the methods and the feasibility of application in specific railway lines.
(3) First reliability investigation of transversing derailment failure:based on stochastic stability and stochastic bifurcation analysis, this paper obtains the first passage derailment failure reliability destruction conditions of elastic constraints wheelset system, gets the backward Kolmogorov equation met by the system reliability function, the generalized Pontryagin equation met by averaged first passage time and the conditional probability density equation of first passage, studies the influence of first passage failure on system configurations as well as the dynamical behavior after failure combined with the initial conditions and boundary conditions using numerical methods.
(4)Stochastic nonlinear optimal control study:with stochastic dynamic programming by targeting higher system reliability and longer first-passage time, stochastic nonlinear optimal control analysis is carried out on the elastic constraints wheelset system, and control effectiveness and strategy are discussed in detail. In addition, nonlinear stochastic optimal control of the elastic constraints wheelset on stochastic system stabilization is discussed.
(1)物理模型和数学模型的建立：考虑轨道随机不平顺激励(根据作用机理主次分为随机外激和随机参激两种类型)与结构自身的频变随机参激作用等,把弹性约束轮对系统的建模从Lagrange体系转换到了Hamilton体系,用Hamilton函数的形式(即从能量的角度,把多因素的随机响应问题转化为单因素能量的分析)来对动力学行为进行研究,并建立了弹性约束轮对系统带Hamilton函数形式的Ito型随机微分方程组,运用随机平均法把该Ito型随机微分方程组表示为一维扩散过程,同时得到了支配该过程的平均Ito随机微分方程。
(2)随机稳定性、分岔以及实验测定方法研究：运用拟不可积Hamilton系统相关理论和Oseledec乘性遍历定理求解了系统的最大Lyapunov指数,得到了系统的随机局部稳定性条件；通过分析一维扩散的奇异边界形态,得到了系统随机全局稳定性条件；依据系统响应的平稳概率密度和联合概率密度,得到了系统的随机Hopf分岔类型,以及D-分岔(动态分岔)和P-分岔(唯象分岔)的分岔条件,并对随机稳定性和确定性稳定性进行了比较分析；另外给出了分岔点以及脱轨安全条件的实验测定方法,并把该方法应用到了具体的实验数据分析中,实验结果和理论分析结果得到了较好吻合,验证了方法的正确性和线路应用的可行性。
(3)首次穿越失效可靠性研究：在对随机稳定性和随机分岔分析的基础上,求得了弹性约束轮对系统发生首次穿越失效可靠性破坏的条件,得到了系统可靠性函数所满足的后向Kolmogorov方程,首次穿越平均时间满足的广义Pontryagin方程以及首次穿越时间条件概率密度方程,并结合初始条件和边界条件,运用相关的数值方法对其进行了分析,研究了首次穿越失效对系统形态的影响以及失效后的动力学行为。
(4)随机非线性最优控制研究：依据随机动态规划方法,以系统可靠度更高和首次穿越时间更长为目标,对弹性约束轮对系统进行了随机非线性最优控制分析,并对控制效果和策略进行了详细讨论,另外还对弹性约束轮对系统随机稳定性化的非线性随机最优控制进行了研究。
With the rapid development of domestic and international high-speed rail and high-speed trains, much more severe challenges and requirements are put forward for vehicle-track system dynamics. As well known, deterministic view is widely used in physical, engineering, biological and economic fields, but with the development of science and technology, descriptions of the practical problems are required to be more accurate. Therefore, the influence of stochastic factors cannot be ignored easily, for some actual process analysis it is necessary to investigate in the stochastic view instead of deterministic view. For the modern rail vehicles, since high speed has become the core of high-speed railway technology, stochastic factors should be taken seriously, which play an important role in system stability, ride comfort, derailment safety, structural reliability, and train air dynamics issues. Because the rail vehicle research in the past mainly focused on the deterministic framework, while the stochastic nonlinear dynamics studies mainly focused on the system response and qualitative behaviors are rarely involved, this thesis attempts to do further research in this area, mainly including:
(1) the building of physical and mathematical model:this paper considers track stochastic irregularity excitation based on primary and secondary action mechanism, that is, stochastic external excitations and randomly parametric excitation, and the structure frequency dependent stochastic parametric excitations. The modeling of elastic restraint wheelset system is transferred from Lagrange system to Hamilton system. With the Hamiltonian (from the energy view, the stochastic response problem analysis of multi-factor is transformed into single-factor energy analysis), dynamic behavior research is conducted. Meanwhile Ito stochastic differential equations with Hamiltonian of the elastic constraints wheelset system are established, which are represented as one-dimensional diffusion process using stochastic averaging method, and average Ito stochastic differential equation which dominated the process has been obtained.
(2) The study of stochastic stability, bifurcation and experimental determination method:based on quasi non-integrable Hamilton system theories and Oseledec multiplicative ergodic theorem, maximal Lyapunov exponents of the system are obtained thus and local stability conditions of the random system are got. Through the analysis of one-dimensional singular boundary diffusion patterns, global stability conditions of the random system are obtained Using averaged probability density and joint probability density of system response, stochastic Hopf bifurcation types, D-bifurcation (dynamic bifurcation) and P-bifurcation (phenomenological bifurcation) bifurcation conditions are obtained, and the stochastic stability and deterministic stability are compared and analyzed. Experimental determination methods of the bifurcation points and derailment safety conditions are given, which have been applied to realistic experimental data analysis. The experimental results and theoretical analysis results have a good match which verifies the correctness of the methods and the feasibility of application in specific railway lines.
(3) First reliability investigation of transversing derailment failure:based on stochastic stability and stochastic bifurcation analysis, this paper obtains the first passage derailment failure reliability destruction conditions of elastic constraints wheelset system, gets the backward Kolmogorov equation met by the system reliability function, the generalized Pontryagin equation met by averaged first passage time and the conditional probability density equation of first passage, studies the influence of first passage failure on system configurations as well as the dynamical behavior after failure combined with the initial conditions and boundary conditions using numerical methods.
(4)Stochastic nonlinear optimal control study:with stochastic dynamic programming by targeting higher system reliability and longer first-passage time, stochastic nonlinear optimal control analysis is carried out on the elastic constraints wheelset system, and control effectiveness and strategy are discussed in detail. In addition, nonlinear stochastic optimal control of the elastic constraints wheelset on stochastic system stabilization is discussed.
引文
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