Momentum and energy propagation in tapered granular chains

TL;DR

This paper investigates how momentum and energy propagate in tapered chains of spherical granules, using a binary collision approximation and a correction algorithm to improve predictions of particle velocities and energy transfer.

Contribution

The authors develop a mixed numerical/analytical correction method to enhance the binary collision approximation for better quantitative predictions.

Findings

Binary collision approximation captures energy trends but lacks quantitative accuracy.

The correction algorithm improves velocity, momentum, and energy predictions.

Effective for various tapering configurations like linear and exponential.

Abstract

We study momentum and energy propagation in 1D tapered chains of spherical granules which interact according to a Hertz potential. In this work we apply the binary collision approximation, which is based on the assumption that transfer of energy along the chain occurs via a succession of two-particle collisions. Although the binary theory correctly captures the trends of increase or decrease of kinetic energy and momentum, the actual values of these quantities are not in good quantitative agreement with those obtained by numerically integrating the full equations of motion. To address this difficulty we have developed a mixed numerical/analytical correction algorithm to provide an improved estimate of the velocity of the particles during pulse propagation. With this corrected velocity we are in turn able to correctly predict the momentum and kinetic energy along the chain for several tapering configurations, specifically for forward linear, forward exponential, backward linear and backward exponential tapering.