非高斯噪声激励下分数阶导数系统的非平稳响应研究

项目名称： 非高斯噪声激励下分数阶导数系统的非平稳响应研究

项目编号： No.11302157

项目类型： 青年科学基金项目

立项/批准年度： 2014

项目学科： 数理科学和化学

项目作者： 李伟

作者单位： 西安电子科技大学

项目金额： 28万元

中文摘要： 噪声激励下分数阶导数系统的理论和应用是随机动力学领域关注的前沿课题之一。由于研究工具的匮乏，分数阶导数系统的研究成果主要局限于高斯白噪声激励及平稳响应讨论。本项目针对分数阶导数随机动力系统的研究现状，旨在从两方面展开工作。一方面，对工程结构领域中的典型振动系统，激励考虑更符合实际的非高斯或非平稳噪声，如具有平稳性的有界噪声、实噪声、宽带噪声以及不具有平稳性的均匀调制噪声等。发展分数阶导数的渐近分析方法，改进随机平均法、小波基变换法等，深入讨论各类噪声的时间关联性和演变功率谱特征，分析各激励方式下分数阶导数系统的响应问题。另一方面，发展不同阶次分数阶导数的有效渐近式，改进Galerkin法、FPK方程的特征函数法、Markov逼近法等，探讨分数阶导数对系统振动的作用机理，建立研究分数阶导数系统非平稳响应的理论框架。本项目研究对了解工程结构系统振动规律、提高系统稳定性具有重要的指导意义。

中文关键词： 分数阶导数；非平稳响应；非高斯噪声；均匀调制噪声；随机平均法

英文摘要： The theories and applications of systems with fractional derivatives under noise excitations are the hot issue in the field of stochastic dynamics. However, the current references are mainly limited in Gaussian noise excitaitons and studies on stationary response due to the lack of efficiently analytical tools. Therefore, the purpose of this project will carry on research from two respects based on the present situation of fractional derivatives in sthochastically dynamical systems. On the one hand, for typical vibration systems in the field of engineering structures, consider non-Gaussian or non-stationary noise as excitations, such as bounded noise, real noise and wide-band noise with stationary properties and uniformly modulated noise without stationary properties, develop asymtotically analytical methods for fractional derivative, improve stochastic averaging method, wavelet-base transformation and so on, so as to discuss deeply the time-depedence and evolutionary power spectrum of all kinds of noises. Furthermore, to analyze the response of fractional derivative systems under different excitations. On the other hand, explore the approximated expressions for fractional derivative with different orders, improve Galerkin method, eigenfunctions of FPK equation, Markov approximation approach and so on, so a

英文关键词： fractional derivative；non-stationary response；non-Gaussian noise；uniform modulated noise；stochastic averaging method