The effect of nonlinear damping on vibrational resonance and chaotic behavior of a beam fixed at its two ends and prestressed

Abstract

This research work is based on the study of the dynamic of one-degree-of freedom nonlinear oscillator representing a built-in clamped–clamped prestressed beam model with a nonlinear damping. First of all, we model this moving structure where we regard the perturbations as a combination of both low-frequency force and high-frequency force. Then, we analyze the occurrence of vibrational resonance, where the response consists of a slow motion and a fast motion respectively with low and high frequencies. Through this, we obtain an approximate analytical expression of the response amplitude and we determine the values of the low frequency and the amplitude of the high-frequency force at which vibrational resonance occurs. The theoretical predictions are found to be in good agreement with numerical results. Moreover, for fixed parameters values of the system, as the nonlinear damping vary, we found appearance and the disappearance of resonance with or without cross-well motion. Secondly, we study the chaotic dynamic of the beam. In this case, critical values of perturbation parameters for the onset of the chaotic motion are specified using Melnikov’s method. Hence, the global dynamical changes of the system have been examined by plotting phase portrait, bifurcation diagram and their corresponding Lyapunov exponent.