多自由度非线性系统的同伦分析方法

Homotopy Analysis Method for Multi-degree of Freedom Nonlinear Systems

大部分工程实际问题可以用多自由度非线性系统来描述,这些系统的数学模型是许多个耦合的两阶常微分方程。一般地,要精确求解这些方程非常困难,因此可以考虑它们的解析近似解。同伦分析方法是解非线性系统响应的有用工具,本文将它应用于多自由度非线性系统的求解中。利用求两自由度耦合van del Pol振子周期解的实例,展示了同伦分析方法的有效性和巨大潜力。同时,把得到的解析近似解与系统的Runge-Kutta数值解作了比较,结果表明同伦分析方法是求解多自由度非线性系统的有效方法。

Most of practical engineering problems can be described by multi-degree of freedom nonlinear systems. The mathematical model of such systems is given by many second order coupled differential equations. In general, it is extremely difficult to find their exact solutions. Therefore, the efforts have been mainly concentrated on approximate analytical solutions. The homotopy analysis method is the useful analytic tool for solving nonlinear systems. The method was applied to obtain the analytical approximation solution for multi-degree of freedom system. The periodic solutions for the coupled van del Pol oscillators of two degree of freedom were applied to illustrate the validity and great potential of the method. Comparisons were made between results of present method and Runge-Kutta method. The results demonstrate that the homotopy analysis method is an attractive and effective technique for nonlinear multi-degree of freedom systems.