Coefficient of restitution of colliding viscoelastic spheres

TL;DR

This paper derives a theoretical model for the coefficient of restitution in viscoelastic sphere collisions, providing explicit formulas and a Padé-approximation that accurately matches experimental data for ice particles.

Contribution

It introduces a simple theory to explicitly calculate the velocity dependence of the restitution coefficient and develops a practical approximation valid over a wide impact velocity range.

Findings

Derived explicit formulas for  and  coefficients.

Constructed a Pade9-approximation for (g) valid for various impact velocities.

Accurately reproduces experimental data for ice particle collisions.

Abstract

We perform a dimension analysis for colliding viscoelastic spheres to show that the coefficient of normal restitution $\epsilon$ depends on the impact velocity g as \epsilon=1-\gamma_1g^{1/5}+\gamma_2g^{2/5}\mp..., in accordance with recent findings. We develop a simple theory to find explicit expressions for coefficients \gamma_1 and \gamma_2. Using these and few next expansion coefficients for \epsilon(g) we construct a Pad\'e-approximation for this function which may be used for a wide range of impact velocities where the concept of the viscoelastic collision is valid. The obtained expression reproduces quite accurately the existing experimental dependence \epsilon(g) for ice particles.