多自由度强非线性随机系统的响应与稳定性研究
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摘要
响应与稳定性分析一直是随机动力学研究的热点,但对于多自由度强非线性随机系统而言,上述分析具有极大的复杂性,本文即针对这一问题的某些方面进行深入探讨。
第一部分分析响应预测。在精确平稳解方面,本文利用外微分方法和逆解法得到多自由度强非线性随机系统的具非能量依赖的精确平稳解。新得到的精确平稳解包含已有的能量依赖精确平稳解为其特例,是迄今为止最广泛的一类精确平稳解。除平稳解外,很多系统往往还需得到其瞬态响应,本文利用基于广义谐和函数的随机平均法,并将平均FPK方程的解近似表示为变系数的多重Laguerre正交基函数的级数和,由Galerkin法得到幅值响应的近似瞬态概率密度,进而导出位移及速度的近似联合概率密度。详细研究了Gauss白噪声激励的多自由度强非线性随机系统及具有时滞反馈控制的强非线性随机系统响应的近似瞬态概率密度。
第二部分进行稳定性分析,在已有工作基础上应用Lyapunov函数法研究了多自由度拟Hamilton系统的概率为1渐近稳定性。对拟完全可积共振或部分可积系统,Lyapunov函数取为系统的独立对合首次积分的最优线性组合。结合拟Hamilton系统随机平均法导出系统的关于Lyapunov函数的平均It(?)随机微分方程,从而得到系统的概率为1渐近稳定的充分条件。值得指出的是,上述的稳定性分析最终可转化为求相应平均It(?)方程线性化漂移系数矩阵的特征值和特征向量的问题。
第三部分对工程领域中具重要意义的含分数阶导数随机系统进行全面研究。对于线性系统,应用基于Laplace变换和加权广义Mittag-Leffler函数得到的Green函数,导出含两项任意阶次的分数阶导数的随机系统响应的Duhamel积分型封闭表达式,从而解析得到系统响应的统计量。还基于广义谐和函数建立了Gauss白噪声激励、阻尼包含分数阶导数的强非线性系统的随机平均法,并应用该平均法近似得到响应的平稳概率密度和最大Lyapunov指数。
The resporse prediction and stability analysis are always hot topics of research in stochastic dynamics. However, these topics are much more complicated for multi-degree-of-freedom (MDOF) strongly nonlinear system under stochastic excitations. Some of the topics are investigated deeply and comprehensively in the present dissertation.
In the first part, the response prediction of nonlinear stochastic system is studied. By using exterior differentiation and inverse method, exact stationary solutions of strongly nonlinear MDOF systems subject to Gaussian white noises are obtained, which are generally independent of energy. The obtained exact stationary solutions are the most general class of the exact stationary solutions so far, and some classes of the known ones dependent on energy belong to the special cases of them. Besides stationary solutions, the nonstationary responses of some systems are also studied. The stochastic averaging method based on generalized harmonic functions is applied and the nonstationary solution is approximately expressed as a multiple series expansion in terms of Laguerre orthogonal basis functions with time-dependent coefficients. Then the approximate nonstationary probability density for amplitude response can be obtained by using the Galerkin method, from which the approximate nonstationary probability densities for displacement and velocity can be derived. The proposed procedure is applied to study the nonstationary probability densities of MDOF nonlinear systems and strongly nonlinear systems with time-delayed feedback control subject to Gaussian white noise excitations.
In the second part, the stability of quasi Hamiltonian system is analyzed. Base on the existing researches, the asymptotic Lyapunov stability with probability one of MDOF quasi Hamiltonian systems is studied by using Lyapunov function. For quasi-integrable and resonant Hamilton systems and quasi-partially-integrable Hamiltonian systems, the optima linear combination of the independent first integrals in involution is taken as the Lyapunov function. Then, the Ito stochastic differential equation for the Lyapunov function can be obtained by using stochastic averaging method. And the sufficient condition for the asymptotic Lyapunov stability with probability one of MDOF quasi Hamiltonian systems can be determined. It should be pointed out that determining the asymptotic Lyapunov stability with probability one of the system is finally transformed into the problems of evaluating the eigenvalues and eigenvectors associated with the linearized drift coefficients of the averaged Ito equation.
In the last part, the response and stability of stochastic system containing fractional derivative which is of great significance in engineering field are investigated. For linear system, the response of stochastic system including two terms of fractional derivative with real and arbitrary orders can be described as a Duhamel integral-type close-form expression by using Green's functions obtained based on a Laplace transform approach and the weighted generalized Mittag-Leffler function. The statistical behavior of the system is subsequently obtained. For nonlinear system, a stochastic averaging procedure for strongly nonlinear system with light damping modeled by fractional derivative under stochastic excitations is developed by using the so-called generalized harmonic functions. With the application of the developed averaging method, the stationary probability density for system response and the largest Lyapunov exponent are approximately obtained.
第一部分分析响应预测。在精确平稳解方面,本文利用外微分方法和逆解法得到多自由度强非线性随机系统的具非能量依赖的精确平稳解。新得到的精确平稳解包含已有的能量依赖精确平稳解为其特例,是迄今为止最广泛的一类精确平稳解。除平稳解外,很多系统往往还需得到其瞬态响应,本文利用基于广义谐和函数的随机平均法,并将平均FPK方程的解近似表示为变系数的多重Laguerre正交基函数的级数和,由Galerkin法得到幅值响应的近似瞬态概率密度,进而导出位移及速度的近似联合概率密度。详细研究了Gauss白噪声激励的多自由度强非线性随机系统及具有时滞反馈控制的强非线性随机系统响应的近似瞬态概率密度。
第二部分进行稳定性分析,在已有工作基础上应用Lyapunov函数法研究了多自由度拟Hamilton系统的概率为1渐近稳定性。对拟完全可积共振或部分可积系统,Lyapunov函数取为系统的独立对合首次积分的最优线性组合。结合拟Hamilton系统随机平均法导出系统的关于Lyapunov函数的平均It(?)随机微分方程,从而得到系统的概率为1渐近稳定的充分条件。值得指出的是,上述的稳定性分析最终可转化为求相应平均It(?)方程线性化漂移系数矩阵的特征值和特征向量的问题。
第三部分对工程领域中具重要意义的含分数阶导数随机系统进行全面研究。对于线性系统,应用基于Laplace变换和加权广义Mittag-Leffler函数得到的Green函数,导出含两项任意阶次的分数阶导数的随机系统响应的Duhamel积分型封闭表达式,从而解析得到系统响应的统计量。还基于广义谐和函数建立了Gauss白噪声激励、阻尼包含分数阶导数的强非线性系统的随机平均法,并应用该平均法近似得到响应的平稳概率密度和最大Lyapunov指数。
The resporse prediction and stability analysis are always hot topics of research in stochastic dynamics. However, these topics are much more complicated for multi-degree-of-freedom (MDOF) strongly nonlinear system under stochastic excitations. Some of the topics are investigated deeply and comprehensively in the present dissertation.
In the first part, the response prediction of nonlinear stochastic system is studied. By using exterior differentiation and inverse method, exact stationary solutions of strongly nonlinear MDOF systems subject to Gaussian white noises are obtained, which are generally independent of energy. The obtained exact stationary solutions are the most general class of the exact stationary solutions so far, and some classes of the known ones dependent on energy belong to the special cases of them. Besides stationary solutions, the nonstationary responses of some systems are also studied. The stochastic averaging method based on generalized harmonic functions is applied and the nonstationary solution is approximately expressed as a multiple series expansion in terms of Laguerre orthogonal basis functions with time-dependent coefficients. Then the approximate nonstationary probability density for amplitude response can be obtained by using the Galerkin method, from which the approximate nonstationary probability densities for displacement and velocity can be derived. The proposed procedure is applied to study the nonstationary probability densities of MDOF nonlinear systems and strongly nonlinear systems with time-delayed feedback control subject to Gaussian white noise excitations.
In the second part, the stability of quasi Hamiltonian system is analyzed. Base on the existing researches, the asymptotic Lyapunov stability with probability one of MDOF quasi Hamiltonian systems is studied by using Lyapunov function. For quasi-integrable and resonant Hamilton systems and quasi-partially-integrable Hamiltonian systems, the optima linear combination of the independent first integrals in involution is taken as the Lyapunov function. Then, the Ito stochastic differential equation for the Lyapunov function can be obtained by using stochastic averaging method. And the sufficient condition for the asymptotic Lyapunov stability with probability one of MDOF quasi Hamiltonian systems can be determined. It should be pointed out that determining the asymptotic Lyapunov stability with probability one of the system is finally transformed into the problems of evaluating the eigenvalues and eigenvectors associated with the linearized drift coefficients of the averaged Ito equation.
In the last part, the response and stability of stochastic system containing fractional derivative which is of great significance in engineering field are investigated. For linear system, the response of stochastic system including two terms of fractional derivative with real and arbitrary orders can be described as a Duhamel integral-type close-form expression by using Green's functions obtained based on a Laplace transform approach and the weighted generalized Mittag-Leffler function. The statistical behavior of the system is subsequently obtained. For nonlinear system, a stochastic averaging procedure for strongly nonlinear system with light damping modeled by fractional derivative under stochastic excitations is developed by using the so-called generalized harmonic functions. With the application of the developed averaging method, the stationary probability density for system response and the largest Lyapunov exponent are approximately obtained.
引文
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