非线性随机系统稳态响应预测的新方法

The stochastic response prediction is a central issue in the nonlinear stochastic dynamics. This project is going to develop a new method to predict the stationary response of nonlinear stochastic system，mainly for a new method to solve the probability density function (PDF) of stationary response of nonlinear system under Gaussian white noise excitation, Poisson white noise excitation, or combined Gaussian and Poisson white noise excitations, respectively. As for the Gaussian white noise excited nonlinear system, an assumed solution of the reduced FPK equation is constructed first. Then, the undetermined coefficients in the assumed solution are worked out by using the method of weighted residue. Finally, a progressively iterative procedure is introduced to gradually optimize the weighting function, which will improve the accuracy of the solutions obtained with the method of weighted residue. By extending the procedure for solving the reduced FPK equation to the reduced generalized FPK equation, the stationary PDF of nonlinear system under Poisson white noise excitation can be obtained. Based on the procedures for the Gaussian white noise excited system and Poisson white noise excited system, a new method is developed to solve the stationary PDF of nonlinear system under combined Gaussian and Poisson white noise excitations. As an application, the stationary response and stochastic bifurcation due to the changes of parameters of classical nonlinear systems in science and engineering are studied and then compared with the solutions obtained from the Monte Carlo simulation, stochastic averaging method and exponential polynomial closure method. It should be noted that the investigation of the present project is full of theoretical innovation, which can effectively promote the development of the theory of nonlinear stochastic dynamics.

随机响应预测是非线性随机动力学研究的一个中心问题。本项目旨在发展一种预测非线性随机系统稳态响应的新方法，主要是高斯白噪声激励、泊松白噪声激励或高斯与泊松白噪声共同激励下非线性系统的稳态响应概率密度函数的求解新方法。针对高斯白噪声激励情形，先构造FPK方程稳态解的一种假设形式，再借助加权残值法计算出假设解中的待定系数，最后引入渐进迭代的步骤，逐步优化权函数，提高加权残值法的精度。推广稳态FPK方程的求解思路至稳态广义FPK方程，求解泊松白噪声激励情形时的稳态响应概率密度函数。以高斯白噪声激励情形和泊松白噪声激励情形为基础，发展高斯与泊松白噪声共同激励情形时的稳态响应概率密度函数的求解新方法。预测科学与工程中的经典非线性系统的稳态响应，研究参数变化诱导的随机分岔，再用蒙特卡罗模拟法、随机平均法和指数多项式闭合法进行对比验证。研究成果具有很强的理论创新性，能有力地促进非线性随机动力学理论的发展。

提出一种稳态 FPK 方程的求解新思路,发展随机激励下非线性系统的稳态响应概率密度函数的求解新方法。具体工作如下：针对单自由度情形，将对应的稳态FPK方程的概率流设为概率势流、概率环流以及关于系统状态变量多项式三者之间的线性组合，采用加权残值法确定其中的待定系数，最后通过迭代的方式提高解的精度；应用迭代加权残值法分别研究了强非线性范德坡系统、干摩擦系统、水平地震激励下经典Bouc-Wen滞迟系统和广义Bouc-Wen滞迟系统、基于瞬时冲击碰撞模型的单边（双边）碰撞振动系统、基于Hertz模型的双边碰撞振动系统、非高斯噪声激励下拉索的面内非线性随机振动系统的稳态响应概率密度函数近似闭合解，得到的近似闭合解具有较高的精度，可作为一个基准解，检验其他非线性随机振动方法的精度；在单自由度情形的基础上，考察了多自由度非线性系统，对应稳态FPK解的是试函数形式与单自由度情形一致，然后基于数据科学理论的最小二乘法来确定试函数中的未知系数；算例研究表明可高效地获得具有高精度的近似闭合解，甚至精确解。 另外，结合随机振动的其他一些方法，考察了一些无法直接FPK方程法求解的复杂非线性随机动力学系统，包括含时滞PD控制的强非线性系统与自复位结构系统,理论解析解也具有较高的精度。发表(或录用)学术论文22篇。完成既定的研究任务。

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