On the Stability of the van der Pol-Mathieu-Duffing Oscillator under the Effect of Fast Harmonic Excitation

Abstract

This paper aims to examine the nonlinear dynamics of a van der Pol-Mathieu-Duffing oscillator under the effect of fast harmonic excitation. The governing equations of motion describing the harmonically forced oscillations of the van der-Pol-Mathieu-Duffing oscillator are expressed in terms of the second-order nonhomogeneous nonlinear ordinary differential equation with suitable initial conditions. This paper uses Krylov-Bogoliubov averaging technique for the stability analysis of the system. The frequency response curves under the effect of external excitation, damping, and nonlinearity are obtained at various resonances. Additionally, the stable and unstable regions were identified. It turns out that the damping reduces the amplitude of oscillations and squeezes the instability regions, whereas the stability region grew with the increase in the amplitude of external excitation.