Traveling waves for the Mass in Mass model of granular chains

TL;DR

This paper studies traveling wave solutions in a granular chain with an internal resonator, establishing existence under specific conditions and exploring non-monotonic waves through numerical analysis.

Contribution

It rigorously proves the existence of bell-shaped traveling waves in the mass-in-mass granular chain under anti-resonance conditions, extending prior results.

Findings

Bell-shaped traveling waves exist under anti-resonance conditions.

Non-monotonic waves with tails can occur when conditions are not met.

Numerical simulations support theoretical and experimental observations.

Abstract

In the present work, we consider the mass in mass (or mass with mass) system of granular chains, namely a granular chain involving additionally an internal resonator. For these chains, we rigorously establish that under suitable "anti-resonance" conditions connecting the mass of the resonator and the speed of the wave, bell-shaped traveling wave solutions continue to exist in the system, in a way reminiscent of the results proven for the standard granular chain of elastic Hertzian contacts. We also numerically touch upon settings where the conditions do not hold, illustrating, in line also with recent experimental work, that non-monotonic waves bearing non-vanishing tails may exist in the latter case.