强非线性自激振子同宿轨道的摄动分析方法  

Perturbation Method for Analyzing the Homoclinic Orbit of a High-power Strongly Non-linear Self-excited Oscillator

作　　者：陈洋洋[1] 赵卫[2] 陈树辉[3] 

机构地区：[1]广州大学减震控制与结构安全国家重点实验室(培育),广州510405 [2]暨南大学重大工程灾害与控制教育部重点实验室,广州510632 [3]中山大学应用力学与工程系,广州510275

出　　处：《噪声与振动控制》2013年第4期195-199,206,共6页Noise and Vibration Control

基　　金：973国家重点基础研究发展计划(2011CB013606);国家自然科学基金(11102045);广东省自然科学基金(S2011040004039);广东省高校优秀青年创新人才培育项目(LYM10108);广州市高校科研项目(10A024)

摘　　要：在双曲函数摄动法的基础上,推广双曲函数Lindstedt-Poincaré(L-P)法的适用范围,使之适用于定量分析一类含五次强非线性项的自激振子的同宿分岔和同宿解问题。以双曲函数系为基础推导出适用于高次非线性系统的摄动步骤,对极限环的同宿分岔参数进行摄动展开,给出同宿摄动解奇异项的定义,以消除同宿摄动解奇异项作为确定极限环同宿分岔点的条件,给出能够严格满足同宿条件的同宿轨道摄动解。算例表明,在相平面内该方法的结果与Runge-Kutta法数值周期轨道的逼近结果比较吻合。Based on the previous studies on hyperbolic perturbation methods,the hyperbolic Lindstedt-Poincaré（L-P） method was extended for homoclinic solution and homoclinic bifurcation analysis of strongly nonlinear self-excited oscillators.By adopting the hyperbolic functions instead of traditional periodic functions in the L-P method,the perturbation procedure for high-power strongly nonlinear system was derived.The homoclinic bifurcation values for limit cycle were expanded in power of perturbation parameter,the secular terms of the perturbation homoclinic solutions were defined.The homoclinic bifurcation values were then determined by eliminating the secular terms.The homoclinic solutions which satisfy the homoclinic conditions were given.The solutions of the phase planes and bifurcation values of some typical examples were obtained.It showed that the results by the presented method were in agreement with those of the Runge-Kutta numerical method.Thus,the accuracy and efficiency of the present method was verified.

关 键 词：振动与波 双曲函数L-P法 自激振子 同宿分岔 同宿解 

分 类 号：TB53[理学—物理] O322[理学—声学]