多自由度非线性随机系统的响应与稳定性

响应与稳定性分析一直是随机动力学研究的热点, 发展预测随机响应及判定系统响应性态的方法具有重要的科学意义与广阔的应用前景. 本文综述了有关多自由度非线性随机系统的响应与稳定性的研究. 首先简介用于随机系统响应预测的Fokker-Planck-Kolmogorov方程法、随机平均法、等效线性化法、等效非线性系统法和Monte Carlo模拟法, 评述其优缺点, 进而讨论了多自由度非线性随机系统响应的精确平稳解、近似瞬态解的研究现状. 然后介绍了随机系统稳定性分析的两类方法, 即Lyapunov函数法及Lyapunov指数法,并综述了多自由度非线性随机系统稳定性分析的研究现状. 最后给出几点发展建议.

RESPONSE AND STABILITY OF MULTI-DEGREE-OF-FREEDOM NONLINEAR STOCHASTIC SYSTEMS

The response prediction and stability analysis are always hot topics of research in stochastic dynamics. Developing the method for response prediction of nonlinear stochastic systems and determining the qualitative behavior of the system response are of important significance and extensive application potential. This paper reviews response and stability of multi-degree-of-freedom nonlinear stochastic systems. Firstly, main methods for the response prediction of stochastic systems are outlined, such as Fokker-Planck-Kolmogorov equation, stochastic averaging method, equivalent linearization, equivalent nonlinear system procedure and Monte Carlo simulation. The advantages and disadvantages of these methods are discussed, respectively. The state-of-the-art of the exact stationary solutions and approximately nonstationary solutions is also illustrated for the multi-degree-of-freedom nonlinear stochastic systems. Then, two effective procedures to evaluate the stochastic stability, i.e., the Lyapunov function and Lyapunov exponent, are briefly presented. Based on these two methods, the stochastic stability of multi-degree-of-freedom nonlinear stochastic systems is outlined. Finally, some suggestions are given for further research on the response and stability of nonlinear stochastic systems.