【作者】 周强；

【导师】 黄志龙；

【作者基本信息】 浙江大学 ， 工程力学， 2010， 硕士

【摘要】 高斯白噪声下作用下的非线性系统响应的瞬态概率密度一直是随机动力学研究的热点与难点,本文针对这一问题的分为两个不同的非线性系统进行了有益的探讨。第一部分研究高斯白噪声激励的有强非线性刚度带分数阶导数的单自由度振子的近似瞬态响应。应用基于广义谐和函数的随机平均法,并将幅值的瞬态概率密度的近似解表示为拉盖尔正交基函数的级数和,其中系数是随时间变化的,用Galerkin法可得到幅值的近似瞬态概率密度,从而得到状态变量响应的近似瞬态概率密度,通过与使用Monte Carlo数值模拟得到的结果比较表明该方法的有效性。第二部分研究受高斯白噪声作用下的非线性阻尼耦合的两个杜芬-范德波振子响应的近似瞬态概率密度。使用本文介绍的方法将幅值的瞬态概率密度的近似解表示为系数随时间变化的拉盖尔正交基函数的级数和的形式,通过取权函数积分可得到幅值的近似瞬态概率密度和状态变量响应的近似瞬态概率密度,数值模拟结果表明该方法有很好的适用性及精度。最后总结前面的分析,说明本文使用的方法具有较好的精度与可行性并可推广应用于更一般的非线性随机系统。更多还原

【Abstract】 The nonstationary probability densities for responses of Nonlinear Stochastic Systems subject to Gaussian white noise excitations are interesting and difficult topics in the field of stochastic dynamical systems. The approximate nonstationary responses of a single-degree-of-freedom(SDOF) nonlinear stochasic system with fractional derivative and the coupled two-degree-of-freedom(TDOF) nonlinear stochastic system are investigated in the present thesis.The first part is mainly about the nonstationary probability densities of SDOF stochastic system with fractional derivative which is of great significance in engineering field. Firstly, the stochastic averaging method based on generalized harmonic functions is applied. Then, the nonstationary probability density for amplitude response is approximately expressed as a series expansion in terms of a set of Laguerre orthogonal basis functions with time-dependent coefficients. Finally, the approximate solution of the nonstationary probability density for amplitude response is derived with the application of the Galerkin method, and the results from analytical solution are compared to with numerical simulation of the original system.In the second part of the thesis, the approximate nonstationary probability densities for responses of two Duffing-Van der Pol oscillators with coupled nonlinear damping subject to Gaussian white noise excitations are investigated. Firstly, the stochastic averaging method based on generalized harmonic functions is applied. Then, the nonstationary probability density for amplitude response is approximately expressed as a series expansion in terms of a set of Laguerre orthogonal basis functions with time-dependent coefficients. Finally, the approximate solution of the nonstationary probability density for amplitude response is derived with the application of the Galerkin method. The results obtained from proposed procedures are compared with those obtained by Monte Carlo simulation of the original systems, and it is shown that the proposed method is of high precision and applicability. It is concluded that the method in the thesis can be applied to obtain nonstationary probability densities of nonlinear Stochastic Systems subject to Gaussian white noise excitations.更多还原