Transient resonant interactions of finite linear chains with essentially nonlinear end attachments leading to passive energy pumping

Abstract

We study resonance capture phenomena leading to energy pumping in systems with multiple degrees of freedom (DOF), composed of N linear oscillators weakly coupled to strongly nonlinear attachments possessing essential (nonlinearizable) cubic stiffness nonlinearities. First we present numerical evidence of energy pumping in the systems under consideration, i.e., of passive, one-way (irreversible) transfer of externally imparted energy to the nonlinear attachments, provided that the energy is above a critical level. To obtain a better understanding of the energy pumping phenomenon we reduce the dynamics governing the chain–attachment interaction to a single, nonlinear integro-differential equation that governs exactly the transient dynamics of the strongly nonlinear attachment. By introducing an approximation based on Jacobian elliptic functions we derive an approximate set of two nonlinear integro-differential modulation equations that govern the time evolution of the amplitude and phase of the motion of the attachment. This set of modulation equations is studied both analytically and numerically. We then perform a perturbation analysis in an O( ε ) neighborhood of a 1:1 resonant manifold of the system in order to study the attracting region in the reduced phase space of the system, that is responsible for resonance capture and nonlinear energy pumping. This analysis provides a justification of the numerical findings.