Energy localization in damped nonlinear disordered metastructures under superharmonic resonance

Abstract

This paper proposes a framework for energy localization in nonlinear oscillator chains operating in non-fundamental resonance, with emphasis on superharmonic regimes. The system is modeled as a metastructure composed of Duffing oscillators with both linear and nonlinear coupling, incorporating disorder-induced periodicity breaking. Under the assumption of strong excitation, the method of multiple scales is employed to derive the governing equations for soliton dynamics. At first-order perturbation, the classical form of the Nonlinear Schrödinger Equation emerges, whereas second-order analysis yields a previously unreported equation arising from the restitution of time scales. Analytical and numerical results demonstrate the nucleation of solitons in both hardening and softening regimes, based on two approaches: direct time-domain simulations from initially motionless states and numerical continuation in the frequency domain. A key finding is the distinct role of phase in superharmonic resonances compared to the primary resonance; specifically, the coexistence of multiple frequency components in the steady-state response precludes interpreting the soliton directly as a displacement envelope. Instead, the resulting secular terms captures the soliton associated with the resonant contribution, while transient components remain present under superharmonic excitation. Furthermore, robustness against disorder uncertainty is assessed by determining the tolerance levels that preserve the phenomenon. These results support the development of vibration control strategies aimed at mitigating the increase in resonant frequencies associated with the geometric downscaling of mechanical systems.