Autoresonance in weakly dissipative Klein–Gordon chains

Abstract

The problem of response enhancement (autoresonance) in a weakly dissipative Klein–Gordon chain subjected to harmonic forcing with a slowly-varying frequency may be of interest for both a fundamental standpoint as well as practical applications, with special attention to the problems of vibration control in macro-, micro- and nano-structures. Previous investigations have given main attention to the emergence and stability of autoresonance in non-dissipative quasi-linear and strongly-nonlinear chains. In this work, we extend recent results to analytical and numerical studies of the influence of dissipation on autoresonance in a weakly dissipative quasi-linear array. 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 successive failure of resonance of all particles in the chain starting from the ones most distant from the source of energy. The developed asymptotic procedures give rise to simple quasi-steady autoresonant solutions and allow estimating the duration of autoresonance, together with the quantitative assessment of critical dissipation. It is shown that the system dynamics agrees with the theoretical predictions for the entirely autoresonant and entirely non-resonant chains but the intermediate cases of autoresonant energy localization in a part of the chain should be examined separately.