含外激励van der Pol-Mathieu方程的非线性动力学特性分析

本文针对含有自激励, 参数激励和外激励等三种激励联合作用下van der Pol-Mathieu方程的周期响应和准周期运动进行分析, 发现其准周期运动的频谱中含有均匀边频带这一新的特性. 首先, 采用传统的增量谐波平衡法(IHB法)分析了van der Pol-Mathieu方程的周期响应, 得到了其非线性频率响应曲线; 再利用Floquet理论对周期解进行稳定性分析, 得到了两种类型的分岔及它们的位置. 然后, 基于van der Pol-Mathieu方程准周期运动的频谱中边频带相邻频率之间是等距的且含有两个不可约的基频的特性(其中一个基频是已知的, 另一个基频事先是未知的), 推导了相应的两时间尺度IHB法, 精确计算出van der Pol-Mathieu方程的准周期运动的另一个未知基频和所有的频率成份及其对应的幅值, 尤其在临界点附近处的准周期运动响应. 得到的准周期运动结果和利用四阶龙格-库塔(RK)数值法得到的结果高度吻合. 最后, 研究发现了含外激励van der Pol-Mathieu方程在不同激励频率时的一些丰富而有趣的非线性动力学现象. 

NONLINEAR DYNAMIC ANALYSIS OF A VAN DER POL-MATHIEU EQUATION WITH EXTERNAL EXCITATION

The periodic responses and quasi-periodic motions of a van der Pol-Mathieu equation subjected to three excitations, i.e., self-excited, parametric excitation, and external excitation, are studied in this paper. A new characteristic is observed that the spectra of the quasi-periodic motions contain uniformly spaced sideband frequencies. Firstly, the traditional incremental harmonic balance (IHB) method is used to obtain periodic responses of the van der Pol-Mathieu equation and to trace their nonlinear frequency response curves automaically. Then the Floquet theory is used to analyze stability of the periodic responses and their bifurcations. Based on the characteristic that the spectra of quasi-periodic motions contain two incommensurate basic frequencies, i.e., the excitation frequency and a priori unknown frequency related to uniformly spaced sideband frequencies. Then the IHB method with two time-scales basing on the two basic frequencies is formulated to accurately calculate all frequency components and their corresponding amplitudes even at critical points. All the results obtained from the IHB method with two time-scales are in excellent agreement with those from numerical integration using the fourth-order Runge-Kutta method. Finally, this investigation reveals rich dynamic characteristics of the van der Pol-Mathieu equation in a range of excitation frequencies.