分数阶Duffing振子的组合共振

Combination resonance of Duffing oscillator with fractional-order derivative

研究了2个谐波激励作用下含分数阶微分项的Duffing振子的一类组合共振,利用多尺度法得到了2ω1+ω2型组合共振的一次近似解析解,分析了定常解的稳定性。应用奇异性理论研究了幅频响应分岔方程,得到了开折参数平面的转迁集和所有区间上分岔曲线的拓扑结构。最后通过数值仿真分析了系统参数对组合共振幅频响应的影响。研究表明:分数阶微分项即具有阻尼特性又具有刚度特性,选择合理的分数阶微分项参数可以有效改善系统的响应特性。

The combined resonance of a Duffing oscillator with fractional-order derivative under a bi-harmonic external excitation is studied. The method of multiple scales is utilized to seek the approximately analytical solution, the amplitude-frequency curve and to study its stability conditions. Then, the system＇s singularity is analyzed completely, the transition sets and the bifurcation diagrams in various regions of the parameter plane are obtained. The numerical simulation is also conducted to analyze the effects of the parameters in the fractional system on the steady state responses. It is found that the fractional-order derivatives had both damping and stiffness effects, some of which are applicable to some vibration control problems.