Dynamics of a mechanical network consisting of discontinuous coupled system oscillators with strong irrational nonlinearities: Resonant states and bursting waves

Abstract

The purpose of this work is the investigation of the dynamical behavior of a mechanical network consisting of discontinuous elastically coupled system oscillators with strong irrational nonlinearities. This mechanical network is consisted of a set of N nonlinear oscillators, elastically coupled where nonlinearity in each unit cell is just due to the geometric configuration, which is the inclination of linear strings. By using the Lagrangian formulation, the set of model equations governing the dynamics of the system are established. These set of equations have strong irrational nonlinearities, with smooth or discontinuous characteristics depending just to the inclination angles of strings, and they are used to study the steady states as well as their stabilities analysis. Then the behaviors of the system when its frequency approaches it eigenfrequency are studied, the well-known resonance phenomenon, which shows the appearing of hysteresis as the frequency shift increases while the displacements of masses tend to infinite. Next, the partial differential equation governing the dynamics of continuous signals in the system is derived. This partial differential equation admits for strong amplitude the train of envelope bursting like signal as solution which is very interesting in this work, and for weak amplitude the well-known envelope like pulse or kink soliton as well as elliptic solutions, with their amplitudes just functions of the inclination angle. All these solutions are found as well as their stabilities criteria which are just functions of the inclination angle. Next by using the perturbation method, the nonlinear Schrödinger (NLS) equation governing the small amplitude modulated signal in the network is found and used to seek modulated pulse and dark solitons as approximated solutions of the network equation. Finally, the conditions (threshold amplitude) for which the input signal will propagate in the network for the input frequency belonging to the forbidden band zone is established, the well-known supratransmission criterion, which is checked by numerical investigations.