Discrete Breathers in a Mass-in-Mass Chain with Hertzian Local Resonators

TL;DR

This paper demonstrates the existence and properties of discrete breathers in a one-dimensional mass-in-mass chain with nonlinear Hertzian resonators, combining theoretical predictions, numerical continuation, and direct simulations.

Contribution

It introduces the first detailed analysis of discrete breathers in a mass-in-mass chain with Hertzian local resonators, bridging theory and numerical methods.

Findings

Discrete breathers exist in the studied chain model.

Breather profiles are localized around one or two sites.

Frequency-amplitude relations are characterized through numerical continuation.

Abstract

We report on the existence of discrete breathers in a one-dimensional, mass-in-mass chain with linear intersite coupling and nonlinear Hertzian local resonators, which is motivated by recent studies of the dynamics of microspheres adhered to elastic substrates. After predicting theoretically the existence of the discrete breathers in the continuum and anticontinuum limits of intersite coupling, we use numerical continuation to compute a family of breathers interpolating between the two regimes in a finite chain, where the displacement profiles of the breathers are localized around one or two lattice sites. We then analyze the frequency-amplitude dependence of the breathers by performing numerical continuation on a linear eigenmode (vanishing amplitude) solution of the system near the upper band gap edge. Finally, we use direct numerical integration of the equations of motion to demonstrate the formation and evolution of the identified localized modes, including within settings that may be relevant to future experimental studies.