强非线性振动系统周期解的能量迭代法

Energy Iteration Method for Analytic Periodic Solutions of Full Strongly Nonlinear Vibration Systems

对于完全强非线性系统:x+g(x)+f(x,x)x=0,提出求周期近似解析解以及这些解的稳定性的新方法。式中,g(x)、f(x,x)x分别是x,x、x的非线性函数。方法是基于能量原理,求出其一次近似解析解,然后引进牛顿迭代思想,得到周期系数微分方程,最后根据谐波平衡原理及最小二乘法求其高次近似解,高次近似解的表达式由计算机辅助推导。计算参考文献和中的例题,令其中ε=1,研究该完全强非线性系统的周期解及其稳定性,本文方法与龙格-库塔数值法算得的结果对照如图1—3所示,它们表明本文方法不仅有效而且精度较高。

A new method to find the approximate periodic analytic solutions and study their stability was presented for full strongly nonlinear vibration systems:x + g(x)+ f(x,x)x = 0,in which g(x) and f(x, x) x are nonlinear functions of x, x , x , respectively. The method is that using energy principle to find its first analytic solution expression. The Newton iteration idea has been applied and the nonlinear equation which of periodic coefficient was appeared. According to the harmonic balance principle and least square technique, the high order expression of analytic solution has been gained by computer aid drive. The example of full strongly nonlinear system as reference [2] and [3], in which let ε = 1, has been solved and compared with Runge-Kutta numerical method shown in Fig. 1 to Fig.3. It is conclude that the energy-iteration method is both effective and high accuracy.