Resonances in nonlinear structure vibrations under multifrequency excitations

Abstract

The response of a single-degree-of-freedom system with quadratic, cubic and quartic nonlinearities subjected to a sinusoidal excitation that involves multiple frequencies is considered. The method of multiple scales is used to construct a first order uniform expansion yielding two first-order nonlinear ordinary differential equations that are derived for the evolution of the amplitude and phase. These oscillations involve a subharmonic oscillation of order one-fourth and superharmonic oscillation of order two. Steady state responses and their stability are computed for selected values of the system parameters. The effects of these (quadratic, cubic, and quartic) nonlinearities on these oscillations are specifically investigated. With this study, it has been verified that the qualitative effects of these nonlinearities are different. Regions of hardening (softening) behaviour of the system exist for the case of subharmonic resonance. The response curve is not affected by decreasing the damping factor for the case of superharmonic resonance. It is shown that the response curve contracts or expands as the parameters vary. The multivalued region increases or decreases when some parameters vary.

Dynamics of Microbeams under Multi-Frequency Excitations

“…A methodology was introduced in [ 12 ] to provide analytical expressions for the mode shapes and natural frequencies of a coupled microbeam resonator filter. The response of a single-degree-of-freedom model under different types of nonlinearities was considered with the method of multiple scales for subharmonic resonance [ 13 ]. Softening and hardening behaviors were shown.…”

Ghost-vibrational resonance

“…For example, it was found in the sudden dropouts exhibited by a semiconductor laser [12], two laser systems coupled bidirectionally [13], vertical-cavity surface emitting lasers [14], monostable Schmitt trigger electronic circuit [15], an excitable Chua's circuit [16], a chaotic Chua's circuit [17] and a system of n-coupled neurons [18]. Subharmonic resonance behaviour in a nonlinear system with a multi-frequency force containing the fundamental frequency in the absence of a high-frequency input signal is studied in [19].…”

Periodic solutions and stability for a weakly damped nonlinear Mathieu equation

“…El-Bassiouny [19] analyzed the nonlinear vibration of a post-buckled beam subjected to external and parametric excitations. El-Bassiouny and Abd El-Latif [20] investigated the response of a single-degree-of-freedom system with quadratic, cubic and quartic nonlinearities subjected to a sinusoidal excitation that involves multiple frequencies.…”