一种新的求解非线性系统周期解方法

A New Periodic Solution of Nonlinear Dynamics

本文利用切比雪夫多项式的若干良好性质,对非自治强非线性动力系统进行分析。将状态矢量在主周期上先展开谐波级数的形式,再将各谐波展开为切比雪夫多项式的形式,从而将求周期解的问题转变为非线性代数方程组的求解问题,得出一种可以方便、迅速地获得近似周期解的解析方法。此方法不依赖于小参数假设,可以用于分析强非线性问题和高维问题,而且对参数激励系统同样有效。以Duffing系统周期解的计算为例,通过与标准谐波平衡方法和四阶Runge-Kutta数值积分结果比较,说明此方法的有效性。

A heteronomy strong nonlinear dynamics system was analyzed by using the good properries Chebyshev polynomials. The state vectors were expanded in terms of the harmonic progressions over principal period. Each harmonic progression was expanded in terms of Chebyshev polynomials. Such an expansion reduces the original problern of getting periodic solution to a set of nonlinear algebraic equations from which the solution in one period can be obtained. This new method does not need to be based on the assumption of small parameters and can be used to analyze strong nonlinear problems. It is also convenience for the analysis of systems with periodical varying coefficients or high dimensional problems. As illustration example, the analytical results of Dulling equation was compared with those obtained via a Runge-Kutta integration algorithm and the standard Harmonic Balance Method. The results indicate that the suggested approach is extremely accurate and effective.