分数阶van der Pol-Mathieu方程的动力学分析

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振动与冲击 ›› 2023, Vol. 42 ›› Issue (8) : 62-68.

论文

郭建斌1，申永军1,2

作者信息 +

Dynamic analysis of the van der Pol-Mathieu equation with fraction-order derivative

GUO Jianbin1，SHEN Yongjun1,2

Author information +

文章历史 +

摘要

以含分数阶微分项的van der Pol-Mathieu方程为对象，研究了谐波激励作用下主共振的动力学行为和稳定性。采用平均法得到了方程近似解析解，通过数值方法验证了解析结果的准确性。建立了系统稳态响应的幅频方程，利用Lyapunov第一方法得到定常解的稳定条件，确定了解的稳定性。在此基础上，分析了参激项、自激项以及分数阶微分项参数对系统幅频特性的影响。结果表明：改变参激项系数主要影响系统的响应幅值和共振频率范围；改变自激项系数主要影响系统响应幅值和多值性；改变分数阶微分项系数和阶次对系统的动力学行为具有双重调节的作用。

Abstract

The dynamics and stability for the primary resonance of the van der Pol-Mathieu equation with fractional-order differential term under harmonic excitation were studied. At first, the approximate analytical solution of the equation was obtained by the averaging method, and the numerical method verified the accuracy of the analytical results. Moreover, the amplitude-frequency equation of the system steady-state response of the was established, and the stability conditions for the steady-state response were obtained through using Lyapunov theory. On this basis, the effects of the parameters of the parametric excitation, self-excitation and fractional-order differential term on the amplitude-frequency characteristics of the system were analyzed. The results show that: The change of the parameter-excited coefficient mainly affects the response amplitude and resonant frequency range of the system. The change of the self-excitation coefficient mainly affects the response amplitude and multi-value property of the system. The change of the coefficient and order of the fractional-order differential term has a double-regulation effect on the dynamic behavior of the system.

导出引用

郭建斌1，申永军1,2. 分数阶van der Pol-Mathieu方程的动力学分析[J]. 振动与冲击, 2023, 42(8): 62-68

GUO Jianbin1，SHEN Yongjun1,2. Dynamic analysis of the van der Pol-Mathieu equation with fraction-order derivative[J]. Journal of Vibration and Shock, 2023, 42(8): 62-68

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