两自由度强非线性振动系统的渐近解及分岔分析  

ASYMPTOTIC SOLUTIONS AND BIFURCATION ANALYSIS OF THE STRONGLY NONLINEAR OSCILLATION SYSTEM WITH TWO DEGREES OF FREEDOM

作　　者：王炜[1] 张琪昌[1] 田瑞兰[1] 

机构地区：[1]天津大学机械学院力学系,天津300072

出　　处：《振动与冲击》2008年第5期130-133,共4页Journal of Vibration and Shock

基　　金：高等学校博士学科点专项科研基金资助课题(编号:20060056005);天津大学优秀博士论文基金资助项目

摘　　要：改进了传统规范形理论,使其适用于研究两自由度强非线性振动系统的渐近响应并进行了相应的分岔分析。通过将待定固有频率法引入规范形求解过程,获得了两自由度立方Duffing-Vander Pol强非线性振动子的规范形及稳态渐近解。参照Hopf分岔定理的形式给出了系统周期解的存在条件,通过算例对比了不同方法所得结果之间的差异,证明了方法的可行性与有效性。最后利用Mathematica编程绘制了一类强非线性振动系统的Lyapunov指数谱,验证了在特定参数值附近具有混沌吸引子。The conventional normal form theory is improved to deal with the asymptotic response of the strongly nonlinear oscillation system with two degrees of freedom, in addition, the bifurcation analysis is also carried out. The normal form and stable asymptotic solutions of the 3:1 internal resonance of Duffing-Van der Pol cubic nonlinear system are obtained by jointly using the undetermined fundamental frequency method and normal form theory. Referring to the Hopf bifurcation theorem the general parameters satisfying the conditions of periodic solutions near the critical point are identified, and the results reveal the difference between the distinct approaches: the modified normal form method may lead to a better result than the former normal form theory, as compared with the result of numerical simulation. The Lyapunov characteristic exponent spectrum of this kind of 2-codimensional continual dynamical systems is plotted by taking advantage of the Mathematica algebra. It shows the existence of chaotic attractor near some special parameter conditions.

关 键 词：规范形 强非线性振动 渐近解 Hopf分岔定理 混沌吸引子 

分 类 号：O322[理学—一般力学与力学基础]