用两项谐波法求解强非线性Duffing方程

提出了一类强非线性动力系统的两项谐波法。采用Ritz-Galerkin法,将描述动力系统的二阶常微分方程化为以频率、振幅为变量的非线性代数方程组;考虑初始条件补充约束方程,构成频率、振幅为变量的封闭非线性代数方程组。利用Maple程序可以方便地求解。分析了三种标准类型的Duffing方程,实例表明,两项谐波法方法简单,具有较高的精度。两项谐波法将谐波平衡法与等效线性化方法相结合,克服了二者的缺点吸取了二者的优点,取较少的谐波数目就可以达到比较高的精度。

Two Harmonics Method for Strongly Nonlinear Duffing Equation

Two harmonics method is presented for strongly nonlinear dynamic-system.In a periodic oscillation,the periodic solutions can be expressed in the form of basic harmonics and bifurcate harmonics.Thus,an oscillation system which is described as a second order ordinary differential equation,can be expressed as a set of non-linear algebraic equations with a frequency and amplitudes as the independent variables using Ritz-Galerkin's method.Considering binding equation of initial conditions,they constitutes a complete set of non-linear algebraic equations with a frequency and amplitudes as the independent variables.Two examples are given by two harmonics method.In example one,the results are compared with analytic method.In example two,the results are compared with the numerical integration method.The examples are given at end of this paper.In example one,the phase trajectories of Duffing equation are computed for a hardening spring,a softening spring,and Ueda genre.The result agrees very well with the numerical integration method.