Binary collision approximation for multi-decorated granular chains

TL;DR

This paper extends the binary collision approximation to analyze pulse propagation in decorated tapered granular chains with multiple small grains, providing analytical predictions that closely match numerical simulations for pulse velocity.

Contribution

It introduces an effective description for decorated chains with arbitrary small grains, enabling analytical calculation of pulse residence times and velocities.

Findings

Analytical predictions agree well with numerical simulations for pulse velocity.

Effective masses and potentials accurately model pulse dynamics in decorated chains.

The method generalizes previous results to chains with multiple small grains.

Abstract

We study pulse propagation along decorated tapered granular chains without precompression. Our goal is to generalize the results obtained in our previous work, by analyzing a decorated chain with an arbitrary number of small grains between the large ones. Making use of an effective description, where the original decorated tapered chain is replaced by a non-decorated tapered chain with effective masses interacting via an effective potential, and applying the binary collisions approximation, we calculate the residence time of the pulse on each effective large grain. We also present the comparison between the numerical integration of the equations of motion and our analytical predictions which show the agreement to be very good for the pulse velocity, albeit only qualitatively for the velocity of the grains.