强非线性动力系统周期解分析

Analysis of Periodic Solutions for Strongly Non-Linear Dynamical Systems

给出一类强非线性动力系统周期解存在性、唯一性和稳定性的简易判别法以及周期解的摄动法。本判别法把问题归结为干扰力在相应的未扰系统振动周期上的功函数及其导数的讨论。其限制条件比现有结果弱。本摄动法可以认为是经典Ｌｉｎｄｓｔｅｄｔ－Ｐｏｉｎｃａｒｅ（Ｌ－Ｐ）法在强非线性振动系统的推广。它与Ｌ－Ｐ法的主要区别在于假设系统的振动频率为相角的非线性函数。

A simple and easy discrimination method of existence,uniqueness and stability of the periodic so-lution and a perturbation method(GLP)for a strongly non-linear dynamical system are presented. This discrlmination method summarizes the questions into a discussions for the work function and itsderivatives over one cycle of the undisturbed system by the disturbing forces,Its greatest advantagesare explicit in the meaning of mechanics,simple in the computations,convenient in the applicationsand weak in the limit conditions.This perturbation method could be thought of a generalization of theLindstedt-Poincare(L-P)method on the strongly non-linear systems.It differs from the classieal L-Pmethod on that the vibration frequency of the system is supposed to be a non-linear function of thephase angle;This method is not only applicable to the strongly non-linear systems but also gives outthe better precision to the weakly non-linear systems.