Coefficient of Restitution Interpreted as Damping in Vibroimpact

TLDR

During impact, relative motion of two bodies is often modeled as a damped sine wave per the Kelvin‑Voigt model, which is shown to be a special case of the proposed law and applies when impacts are absent. The study introduces a damping term λxⁿẋ, with n=3/2 for a sphere impacting a plate, to reinterpret the coefficient of restitution. The mechanism applies a damping term λxⁿẋ with n=3/2 for sphere‑plate impacts, based on Hertzian contact theory. The analysis shows that the Kelvin‑Voigt model predicts unphysical tension and a quadratic velocity dependence, whereas experiments confirm the proposed cubic velocity loss law.

Abstract

During impact the relative motion of two bodies is often taken to be simply represented as half of a damped sine wave, according to the Kelvin-Voigt model. This is shown to be logically untenable, for it indicates that the bodies must exert tension on one another just before separating. Furthermore, it denotes that the damping energy loss is proportional to the square of the impacting velocity, instead of to its cube, as can be deduced from Goldsmith’s work. A damping term λxnx˙ is here introduced; for a sphere impacting a plate Hertz gives n = 3/2. The Kelvin-Voigt model is shown to be approximated as a special case deducible from this law, and applicable when impacts are absent. Physical experiments have confirmed this postulate.