Response enhancement and energy localization in autoresonant nonlinear chains

Abstract

The response enhancement (autoresonance) in a weakly dissipative nonlinear chain subjected to harmonic forcing with a slowly-varying frequency may be of interest for both a fundamental standpoint as well as practical applications. However, the cumulative effect of dissipation and forcing parameters on the emergence and stability of autoresonance in an anharmonic array has not been discussed thus far in the literature. Recent theoretical and numerical developments were focused on the study of unbounded autoresonance in a non-dissipative strongly nonlinear chain. In this work, the asymptotic methods are employed to construct the evolutionary equations, which describe the long-term behavior of a weakly dissipative strongly nonlinear chain under the condition of 1:1 resonance. Analytical approximations and numerical examples demonstrate that weak dissipation permits the emergence of autoresonance on a bounded time interval but an increase of dissipation leads to escape from resonance of either the entire chain or a part of the chain distant from the source of energy. Numerical simulations for a series of examples confirm close proximity of the exact (numerical) results to their analytical approximations, together with the dependence of the duration of autoresonance, its localization length, and the amplitudes attainable on dissipation and detuning parameters.