基于平均法的余弦激励分数阶振子系统动力学分析

Dynamic Analysis of Fractional Oscillator System with Cosine Excitation Based on Average Method

提出了一种求解余弦激励下分数阶振子系统位移响应的解析和数值算法。其中解析方法是用平均法来求出系统的稳态响应解和瞬态响应解,总位移响应解即为稳态解和瞬态解的总和。提出一种数值方法,利用Grunwald-Letnikov分数阶导数的定义将系统中的分数阶微分项进行离散化处理,降低了原系统阶数。在一般的周期激励下,系统的近似响应解可以采用傅里叶级数展开法和线性系统叠加原理得到。最后,利用数值模拟对所提方法的有效性和可行性进行了验证,并对分数阶阶次、线性阻尼系数和分数阶导数系数对系统稳态响应振幅和总位移响应的影响进行了分析。

An analytical and numerical algorithm for solving the displacement response of fractional oscillator system under cosine excitation is presented.The analytical method means that steady-state response and transient response solutions of the system can be obtained by the average method.The total displacement response solution is the sum of the steady-state solution and transient solutions.In the numerical method,the Grunwald-Letnikov definition of fractional derivative is used to discretize the fractional differential term in the system,so as to reduce the order of the original system.Considering the general periodic excitation,the approximate response solution of the system can be obtained by using the Fourier series expansion method and the linear system superposition principle.Finally,the effectiveness and feasibility of the proposed method are verified by numerical simulation.The effects of fractional order,linear damping coefficient and fractional derivative coefficient on steady-state response amplitude and total displacement response of the system are analyzed.