几类非线性随机动力学系统的近似瞬态响应

项目名称： 几类非线性随机动力学系统的近似瞬态响应

项目编号： No.11202181

项目类型： 青年科学基金项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 金肖玲

作者单位： 浙江大学

项目金额： 24万元

中文摘要： 非线性随机动力学系统的平稳响应已经得到了深入系统的研究，然而相当部分涉及国家安全的随机动力学系统，例如运载火箭等，更加关注其瞬态响应的分析。瞬态响应是这些系统设计、安全评估及性能控制的重要依据，因此研究非线性随机动力学系统的瞬态响应具有重要的理论意义及工程应用价值。本项目以非共振/共振单/多自由度非线性随机系统为研究对象，研究随机系统的近似瞬态解，并考虑了外共振、时滞、粘弹性、滞迟等重要因素的影响。基本研究方法是：推广基于广义谐和函数的随机平均法，对非共振与共振情形分别得到相应的平均Fokker-Planck-Kolmogorov方程；将上述方程的解表示为指数形式的复合函数，对非共振情形，将内函数近似表示为变系数的多元正交基函数的级数和，对共振情形，则采用变系数的两组不同的基函数的级数和；应用Galerkin法将上述问题转化为一阶线性常微分方程组问题，进而得到系统的近似瞬态响应。

中文关键词： 瞬态响应；非线性随机动力学系统；随机平均法；Galerkin 法；Mellin变换

英文摘要： The stationary response of the nonlinear stochastic dynamical system has been studied systematically and profoundly. However, for a considerable number of stochastic dynamical systems relating to national security, such as launch vehicle and so on, the transient response is more of interest. It is an important basis of design, safety assessment and control of the corresponding system. Thus, the prediction of transient response for nonlinear stochastic dynamical system has important significances and engineering application values. In the present project, approximate transient solutions of non-resonant/resonant single/multi-degree-of- freedom nonlinear stochastic systems are studied, where some significant factors, including external resonance, time delay, viscoelasticity, hystereticity and so on, are considered. The main procedures are as follows. Firstly, by using the stochastic averaging method based on the generalized harmonic functions, the averaged Fokker-Planck-Kolmogorov equation can be derived for the non-resonant and resonant cases. Then, the solution of the equation is represented by an exponential compound function. For non-resonant case, the internal function is approximately expressed as a series expansion in terms of a set of multivariable polynomials with time-dependent coefficients, while, for re

英文关键词： transient response；nonlinear stochastic dynamical system；stochastic averaging；Galerkin method；Mellin transformation