The primary and subharmonic simultaneous resonance of Duffing oscillator with fractional-order derivative is studied. Firstly, the approximately analytical solution of the resonance is obtained by the method of multiple scales, and the correctness and satisfactory precision of the analytical solution are verified by numerical simulation. Then, the amplitude–frequency curve equation and phase–frequency curve equation are derived from the analytical solution. The stability condition of the steady-state response is obtained by Lyapunov’s first method, and the state switching between two stable periodic orbits is demonstrated. Finally, the effects of nonlinear factor on the system response are analyzed, and the difference between stiffness softening and stiffness hardening system is demonstrated. The influence of fractional-order term on the system is analyzed in depth, and the effect mechanism of fractional-order term is revealed, i.e., the focus and intensity of effect are determined by the order and coefficient of the fractional-order derivative, respectively.

研究了具有分数阶导数的Duffing振荡器的一次和次谐波同时共振。首先，通过多尺度方法获得了共振的近似解析解，并通过数值模拟验证了解析解的正确性和令人满意的精度。然后，从解析解导出振幅-频率曲线方程和相位-频率曲线方程。利用李雅普诺夫的第一种方法获得稳态响应的稳定条件，并说明了两个稳定周期轨道之间的状态切换。最后，分析了非线性因素对系统响应的影响，并论证了刚度软化与刚度硬化系统之间的区别。