分数阶Rayleigh-Duffing系统的主共振

Primary resonance of a fractional-order Rayleigh-Duffing system

利用多尺度法研究了谐波激励下分数阶Rayleigh-Duffing系统的主共振。首先,利用多尺度法得到系统的一次近似解析解,通过数值仿真发现解析解与数值解吻合较好,验证了近似解析解的准确性;然后,建立了稳态解的幅频方程及相频方程,基于Lyapunov稳定性理论得到了稳态解的稳定性条件;最后,通过数值仿真并结合幅频曲线,分析发现非线性刚度系数、线性阻尼系数以及分数阶阶次等参数对系统动力学特性有重要影响,这对此类系统的优化和控制有重要意义。

In this paper,the primary resonance of a fractional-order Rayleigh-Duffing system under harmonic excitation is studied by multi-scale method.Firstly,the approximate analytical solution is obtained based on the multi-scale method.The numerical simulation shows that the analytical solution agrees well with the numerical solution,and the accuracy of the approximate analytical solution is verified.Then,the amplitude-frequency and phase-frequency equations for the steady-state solution are established,and its stability conditions are obtained based on the Lyapunov stability theory.Finally,through numerical simulation combined with amplitude-frequency curves,it is found that the parameters such as nonlinear stiffness coefficient,linear damping coefficient,and fractional order have important effects on the system dynamics characteristics,which is of great significance for the optimization and control of such systems.