Irreversible energy transfer, localization and non-reciprocity in weakly coupled, nonlinear lattices with asymmetry

Abstract

A semi-infinite network of two coupled semi-infinite nonlinear lattices is studied, having the capacity for passive irreversible energy transfer (redirection) from an “excited lattice” (forced by an impulse) to an “absorbing lattice” (that is not directly forced). This is achieved by breaking the symmetry through a spatial variation of the grounding stiffnesses of the excited lattice from the second unit cell onwards, and assuming weak coupling. We show analytically that irreversible energy transfer is caused by the macroscopic analogue (in space) of the Landau–Zener tunneling (LZT) effect, which originated in quantum mechanics in the context of linear parametric oscillators (in time). By constructing a reduced-order model (ROM) of two coupled oscillators it is possible to theoretically model the LZT-induced irreversible energy redirection from the excited to the absorbing lattice. A computational study reveals that the LZT effect is realized only in a critical band of energy, while outside this band there occurs energy localization. The lower and upper energy bounds of this critical band are theoretically approximated by constructing appropriate ROMs of the coupled lattices and studying the bifurcations in their dynamics with energy. This analysis sheds physical insight on the different regimes of the acoustics, and theoretical predictions agree well with direct numerical simulations of the full lattice network. Finally, we show that the LZT effect induces strong non-reciprocity in the dynamics of the finite asymmetric lattice network, in the sense that the impulsive response of this network depends crucially on the forcing-measurement locations and energy. The implications and the possible applications of these results are discussed.