分数阶Duffing振子的亚谐共振

研究了含分数阶微分项的Duffing振子的亚谐共振,利用平均法得到了系统的一阶近似解。提出了亚谐共振时等效线性阻尼和等效线性刚度的概念,分析了分数阶微分项的系数和阶次对系统动力学特性的影响。建立了亚谐共振定常解的幅频曲线的解析表达式,并得到了亚谐共振周期响应的存在条件和稳定性判断准则。最后进行了数值解和解析解的比较,证明了解析结果的准确性,并通过数值仿真研究了分数阶微分项的参数对亚谐共振解的存在条件、稳定性条件和系统幅频曲线的影响。

Sub-harmonic resonance of Duffing oscillator with fractional-order derivative

The sub-harmonic resonance of Duffing oscillator with fractional-order derivative is studied, and the first-order approximate solution is obtained by the averaging method. The definitions of equivalent linear damping and equivalent linear stiffness for sub-harmonic resonance are presented, and the effects of the fractional-order parameters on the system dynamical characteristics are also analyzed. The amplitude-frequency equation for steady-state solution is established. The existence condition and stability condition for periodic response of sub-harmonic resonance are derived. At last the comparison between the analytical and numerical solution is fulfilled, and the result verifies the satisfactory precision of the approximate analytical solution. The numerical simulation is also used to analyze the effects of the parameters in fractional-order derivative on the existence condition, the stability condition and the amplitude-frequency curves of sub-harmonic resonance.