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

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Journal of Vibration and Shock ›› 2023, Vol. 42 ›› Issue (8) : 62-68.

GUO Jianbin1，SHEN Yongjun1,2

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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.

Key words

fractional-order derivative / van der Pol-Mathieu equation / averaging method / stability

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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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