Theoretical and numerical study of vibrational resonance in a damped softening Duffing oscillator

Abstract

We study the possibility of occurrence of vibrational resonance in a softening Duffing oscillator in the underdamped and overdamped cases both theoretically as well as numerically. The oscillator is driven by two periodic forces. Numerically we find that in the underdamped case two oscillatory solutions are obtained in a limited range of the parameters considered (damping coefficient and amplitude of the high frequency force) for a fixed frequency and amplitude of the low frequency periodic force depending on the initial conditions. These solutions have distinct response amplitude to the low frequency force. When damping is gradually increased, only one oscillatory solution is observed. Vibrational resonance is observed in both regions of oscillation. The analytical approximation yields only one oscillatory solution for all damping values. Analytically, the peak in the area bounded by the phase portrait as a function of the amplitude of the high frequency force is connected to vibrational resonance. Also, the values of the frequency of the low frequency forcing and the amplitude of the high frequency forcing at which vibrational resonance is found to occur are obtained. In the overdamped case, vibrational resonance is not observed for the softening Duffing oscillator thus showing a marked contrast to the overdamped bistable oscillator.