Autoresonance in a strongly nonlinear chain driven at one end

Abstract

This work examines the emergence of autoresonance (AR) in a one-dimensional chain of strongly nonlinear oscillators subjected to a harmonic force with a slowly varying frequency applied at the end of the chain. The dynamics of the chain is studied assuming 1:1 (fundamental) resonance, when the response of each nonlinear oscillator has a dominant harmonic component with the frequency close to the frequency of the external excitation. Explicit asymptotic equations describing the amplitudes and the phases of the oscillations are derived. These equations demonstrate that, in contrast to the chain with a linear attachment, the strongly nonlinear chain can be entirely captured into resonance provided that its structural and excitation parameters exceed certain critical thresholds but the frequency detuning rate is small enough. It is shown that at large times the amplitudes of all oscillators captured into AR converge to a common monotonically growing quasisteady backbone curve. This implies asymptotic equipartition of energy between the oscillators under the condition of AR. Numerical simulations have been performed for two-, four-, and 12-particle arrays with fixed forcing and coupling parameters and different detuning rates. The obtained results demonstrate the existence of two intervals of the rate's values corresponding to AR and small-amplitude oscillations in the entire chain, respectively, and a narrow gap between these two intervals, wherein each oscillator may escape from resonance individually or in combination with neighboring particles. This implies a negligibly small interval of energy localization in a part of the chain adjacent to the source of energy compared to the interval of the emergence of AR in the entire chain.