Interaction of Traveling Waves with Mass-With-Mass Defects within a Hertzian Chain

TL;DR

This paper investigates how a resonant inclusion in a granular chain affects wave transmission, reflection, and energy trapping, revealing tunable energy control and unique trapping capabilities not seen in other defect types.

Contribution

It introduces the concept of a mass-with-mass defect in a Hertzian chain and analyzes its effects on wave dynamics, including energy trapping and tunability.

Findings

Harmonic oscillator can tune transmitted and reflected energy.

System can asymptotically trap energy, unlike other defects.

Wave behavior varies with low and high resonator mass.

Abstract

We study the dynamic response of a granular chain of particles with a resonant inclusion (i.e., a particle attached to a harmonic oscillator, or a mass-with-mass defect). We focus on the response of granular chains excited by an impulse, with no static precompression. We find that the presence of the harmonic oscillator can be used to tune the transmitted and reflected energy of a mechanical pulse by adjusting the ratio between the harmonic resonator mass and the bead mass. Furthermore, we find that this system has the capability of asymptotically trapping energy, a feature that is not present in granular chains containing other types of defects. Finally, we study the limits of low and high resonator mass, and the structure of the reflected and transmitted pulses.