强非线性动力系统的频率增量法  

FREQUENCY-INCREMENTAL METHOD FOR STRONGLY NONLINEAR DYNAMICAL SYSTEMS

作　　者：黄赪彪[1] 宗国威[2] 陈兆莹[2] 胡敏[1] 

机构地区：[1]中山大学力学系,广州510275 [2]香港城市大学

出　　处：《力学学报》2001年第2期242-249,共8页Chinese Journal of Theoretical and Applied Mechanics

基　　金：国家自然科学基金!(19772025);广东省自然科学基金!(960029)资助项目

摘　　要：提出一类强非线性动力系统的瞬时频率增量法．将描述动力系统的二阶常微分方程，化为以相位为自变量、瞬时频率为未知函数的积分方程；用谐波平衡原理，将求解瞬时频率的积分问题，归结为求解以频率增量的Ｆｏｕｒｉｅｒ系数为独立变量的线性代数方程组；给出了若干例子．Frequency is one of the essential factors to describe the dynamical property of the periodic oscillation systems. The strongly nonlinear problems are difficult to solve by the classical procedures such as perturbation methods. Their main limitation may be generally due to the unreasonable assumption of the constant frequency. The breakthrough point of a series of the results obtained over the years may be generally due to the instantaneity of the frequency. In a periodic oscillation, the periodic solutions can be expressed in the form of simple harmonics. Thus, an oscillation system which is described as a second order ordinary differential equation, can be expressed as an integral equation with phase angle as the independent variable and its first order derivative as a differential equation. Moreover, the integral equation problem is turned into the problem of solving a set of linear algebraic equations with the Fourier coefficients of the frequency increment as the independent variables using the principle of harmonic balance. The initial values of the incremental method are taken as the solutions of a conservative system. The amplitude and eccentricity are determined by the necessary condition for the existence of a periodic solution. When these algebraic equations airs solved in an iterative way, a semi-analytical solution that satisfies any prescribed precision may be obtained. Two examples are given at end of this paper. In example one, the phase trajectories of van der Pol equation are computed for arbitrary values of the parameter ε = 1, 10, 200, 1000. The result agrees very well with the numerical integration method even for ε = 1000. It is a good explanation for why the performance of the electron tube oscillators described by van der Pol 70 years ago is so stable. In example two, the periodic solutions and their bifurcations of a six-order nonlinear system are studied. The results are compared with the numerical integration method, and the agreements are very good too.

关 键 词：强非线性动力系统 瞬时频率 频率增量法 谐波平衡 增量方程 

分 类 号：O19[理学—数学]