含分数阶微分的线性单自由度振子的动力学分析

Dynamical analysis of linear single degree-of-freedom oscillator with fractional-order derivative

研究了一个含分数阶微分的线性单自由度振子,通过平均法得到了系统的近似解析解.在近似解中,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影响着系统的动力学特性,这一点与现有文献中直接将分数阶微分项归类为阻尼进行处理的方法完全不同.比较了近似解析解和数值解,二者的符合精度很高,证明了近似解析解的准确性.分析了分数阶系数和分数阶阶次对系统响应特性的影响,发现分数阶系数和分数阶阶次都既可以通过等效线性阻尼影响系统的共振振幅,又可以通过等效线性刚度影响系统的共振频率.

A linear single degree-of-freedom oscillator with fractional-order derivative is researched by the averaging method, and the approximately analytical solution is obtained. The effects of the parameters on the dynamical property, including the fractional coefficient and the fractional order, are characterized by the equivalent linear damping coefficient and the equivalent linear stiffness, and this conclusion is entirely different from the published results. The comparison of the analytical solution with the numerical results verifies the correctness of the approximately analytical results. The following analysis on the effects of the fractional parameters on the amplitudefrequency is fulfilled, and it is found that the fractional coefficient and the fractional order could affect not only the resonance amplitude through the equivalent linear damping coefficient, but also the resonance frequency by the equivalent linear stiffness.