Yan Zhi, Liu Xianbin
State Key Laboratory of Mechanics and Control of Mechanical Structures, College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China.
Chaos. 2020 Feb;30(2):023109. doi: 10.1063/1.5133810.
Investigation on linear/nonlinear resonance phenomena and supercritical/subcritical pitchfork bifurcation mechanism is reported in a complex bifractional-order damped system which endures a high-frequency parametric excitation and contains fractional-power nonlinearity. The approximate theoretical expression of the linear response amplitude at the primary frequency and the superharmonic response amplitude at the second and third harmonic frequencies are obtained by utilizing an analytical method and an iterative formula. A numerical approximation scheme based on the Caputo derivative for the simulation of the system is introduced, showing sufficient precision. Due to the parametric excitation, analytical approximation expressions of the stable equilibrium points are given explicitly when the exponent is not an integer so that the pitchfork bifurcation, nonlinear resonance can be studied in an analytical way, exhibiting much more operability than the external excitation case. It is found that the fractional-order derivative may bring new multibifurcation and new multiresonance phenomena, which have not yet been reported before. With the variation of different control parameters of the system, the equivalent slow-varying system can be converted from bistability to monostability and finally to bistability. Unlike the cases of the system excited by bifrequency external excitation, the optimum response amplitude of the parametric excited system is not monotonous with respect to the values of the exponent. For a range of parameters of the system, it is also found that the superharmonic resonance at the second and third harmonic frequencies is affected deeply by the parametric excitation.
报道了在一个承受高频参数激励且包含分数幂非线性的复分数阶阻尼系统中，对线性/非线性共振现象以及超临界/亚临界叉形分岔机制的研究。利用解析方法和迭代公式，得到了一次频率处线性响应幅值以及二次和三次谐波频率处超谐波响应幅值的近似理论表达式。引入了一种基于Caputo导数的数值近似方案来模拟该系统，显示出足够的精度。由于参数激励，当指数不是整数时，明确给出了稳定平衡点的解析近似表达式，从而可以用解析方法研究叉形分岔和非线性共振，比外部激励情况具有更强的可操作性。发现分数阶导数可能带来新的多分岔和新的多共振现象，这些现象此前尚未见报道。随着系统不同控制参数的变化，等效慢变系统可从双稳性转变为单稳性，最终又回到双稳性。与双频外部激励系统的情况不同，参数激励系统的最佳响应幅值相对于指数值并非单调变化。对于系统的一系列参数，还发现二次和三次谐波频率处的超谐波共振受到参数激励的深刻影响。
