广义分数阶van der Pol-Duffing振子的动力学响应与隔振效果研究

用平均法研究了含分数阶导数项的van der Pol-Duffing振子的动力学行为和力传递率。得到了主共振时振子的一阶解析解、定常解幅频曲线和相频曲线的解析表达式,进一步通过与数值解作对比,验证了解析解的正确性,分析了不同参数对幅频曲线和力传递率的影响。结果表明:解析解与数值解吻合良好;在无量纲情况下,共振区分数阶项系数、非线性参数、分数阶阶次、阻尼比对幅频曲线和力传递率的共振峰值均有抑制作用;不同频率区段参数对隔振效果的影响不同,在低频隔振区非线性参数和幅值越小隔振效果越好,此外阻尼比对力传递率影响很小;在高频隔振区增大非线性参数、幅值和阻尼比有助于提高隔振效果。

Dynamic response and vibration isolation effect of generalized fractional-order van der Pol-Duffing oscillator

Here,dynamic behavior and force transmissibility of van der Pol-Duffing oscillator with fractional-order derivative term were studied with the average method.Firstly,analytical expressions during main resonance of the oscillator’s the first-order analytical solution,steady solution’s amplitude-frequency curve and phase-frequency curve were obtained.Then,the correctness of the analytical solution was verified by comparing it with the numerical solution.Finally,effects of different parameters on amplitude-frequency curve and force transmissibility were analyzed.The results showed that the analytical solution agrees well with the numerical solution;in dimensionless case,fractional-order term coefficient,nonlinear parameters,number of fractional-order and damping ratio within resonance area can have suppressing action on resonance peaks of amplitude-frequency curve and force transmissibility;parameters within different frequency ranges can have different influences on vibration isolation effect,within lower frequency vibration isolation region,the smaller the nonlinear parameters and amplitude,the better the vibration isolation effect;damping ratio has little effect on force transmissibility;within higher frequency vibration isolation region,increasing nonlinear parameters,amplitude and damping ratio can be helpful to improve vibration isolation effect.