分数阶van der Pol振子的超谐共振

以含分数阶微分项的van der Pol振子为对象,研究其超谐共振时的动力学特性.首先,通过平均法得到了系统的一阶近似解,提出了超谐共振时等效线性阻尼和等效线性刚度的概念,研究了分数阶微分项的系数和阶次以等效线性阻尼和等效线性刚度的形式对系统动力学特性的影响.随后,建立了超谐共振时定常解的幅频曲线的解析表达式,得到了超谐共振周期响应的稳定性判断准则并提出等效非线性阻尼和非线性稳定性条件参数的概念.最后,通过数值仿真比较了分数阶与整数阶系统的幅频曲线,分析了分数阶微分项的系数和阶次对响应幅值、幅频曲线以及系统稳定性的影响.

Super-harmonic resonance of fractional-order van der Pol oscillator

The dynamical characteristics of super-harmonic resonance of van der Pol oscillator with fractional-order derivative are studied. First the approximate analytical solution are obtained by the averaging method, and the definitions of equivalent linear damping and equivalent linear stiffness for super-harmonic resonance are established. Effects of the fractional-order parameters on the dynamical characteristics of the system are also studied through the equivalent linear damping and equivalent linear stiffness. Moreover, the amplitude-frequency equation and the stability condition for the steady-state solution are analytically presented, and the definitions of equivalent nonlinear damping coefficient and nonlinear stability parameter are also established. Finally, the comparisons of the fractional-order and the traditional integer-order van der Pol oscillators are carried out by numerical simulation. The effects of the parameters in fractional-order derivative on the steady-state amplitude, the amplitude-frequency curves, and the system stability are also analyzed.