Breather arrest in a chain of damped oscillators with Hertzian contact

Matteo Strozzi,Oleg V Gendelman

doi:10.1016/j.wavemoti.2021.102779

Abstract

Breather propagation in a damped oscillatory chain with Hertzian nearest-neighbour coupling is investigated. The breather propagation exhibits an unusual two-stage pattern. The first stage is characterized by power-law decay of the breather amplitude. This stage extends over finite number of the chain sites. Drastic drop of the breather amplitude towards the end of this finite fragment is referred to as breather arrest. At the second stage, the breather exhibits very small amplitudes with hyper-exponential decay. Numeric results are rationalized by considering a simplified model of two damped linear oscillators coupled by Hertzian contact forces. Initial excitation of one of these oscillators results in a finite number of beating cycles in the system. This simplified model reliably predicts main features of the breather arrest. More general coupling potentials and effect of pre-compression on the breather propagation are also discussed.

Highlights

Based on the findings presented below, the breather arrest (BA) can be defined as abrupt crossover from power-law to hyper-exponential decay of the maximum breather amplitude, leading to a negligibly small amplitude after penetration of the breather to finite depth in the lattice [31]
We demonstrated that the combination of strongly nonlinear coupling and damping leads to qualitatively new patterns of the breather propagation in the lattice — namely, the crossover between two asymptotic regimes
The simplified model of two oscillators allows an insight into this stage and yields reasonable predictions of the power-law decay and the scaling of the penetration depth with the initial excitation amplitude and damping

Summary

Introduction

Localized excitations propagating in nonlinear lattices, such as solitary waves and breathers, have attracted a lot of interest in recent years, from purely theoretical viewpoint, as well as in view of versatile applications including electro-mechanical devices, e.g. shock and energy absorbers [1,2,3,4], actuators and sensors [5,6], acoustic lenses and diodes [7,8]. It was demonstrated that the breathers in nonlinear lattices with Hertzian contact between the oscillators in the absence of pre-compression reveal themselves due to existence of parabolic on-site potentials [35] These theoretical models are directly related to recent experiments on the pulse propagation in Hertzian chains mounted on elastic flexures [36] or embedded into a viscoelastic medium [37,38]. This last system is simple enough to allow direct analytic exploration and offers quantitative explanations to the observed regularities of the breather propagation.

Description of the model and numeric results

Simplified model of the breather propagation

Conclusions