强非线性非对称动力系统的两项谐波法

提出了一类强非线性动力系统的两项谐波法,用Ritz-Galerkin法,将描述动力系统的二阶常微分方程,化为以频率、振幅和偏心距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、振幅和偏心距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.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.应用两项谐波法求解了一个带有参数的强非线性非对称哈密顿系统的例子,实例表明,两项谐波法方法简单,取较少的谐波数目就可以达到比较高的精度.

Two Harmonics Method for Unsymmetrically Dynamic Systems with Strong Nonlinearity

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. Two har-monics method is presented for strongly nonlinear dynamic-system. in a periodic oscillation, the periodic solutions canbe expressed in the form of basic harmonics and bifurcate harmonics. Thus, an oscillation system which is described asa second order ordinary differential equation, can be expressed as a set of non-linear algebraic equations with a frequency,amplitudes and central-offset as the independent variables using Ritz-Galerkin's method. Considering binding equationof initial conditions, they constitutes a complete set of non-linear algebraic equations with a frequency, amplitudes andcentral-offset as the independent variables. For examples, a unsymmetrical Hamilton's systems of strong nonlinearitywith a parameter are solved by two harmonics method. The results are compared with analytic method, and the agreementsare very good too. Two harmonics method combine together the method of harmonic balance and the method of equivalentlinearization. It overcoming two weakness and absorbed two advantages. Two harmonics method has excellence that themethod is briefness. we can attain the higher accuracy using a number of harmonics that takes the less.