The Bounds on the Coefficients of Restitution for the Frictional Impact of Rigid Pendulum Against a Fixed Surface

Abstract

Upper bounds on Newton’s, Poisson’s, and energetic coefficients of normal restitution for the frictional impact of rigid pendulum against a fixed surface are derived, demonstrating that the upper bound on Newton’s coefficient is smaller than 1, while the upper bound on Poisson’s coefficient is greater than 1. The upper bound on the energetic coefficient of restitution, which is a geometric mean of Newton’s and Poisson’s coefficients of normal restitution, is equal to 1. Lower bound on all three coefficients is equal to zero. The bounds on the tangential impact coefficient, defined by the ratio of the frictional and normal impulses, are also derived. Its lower bound is negative, while its upper bound is equal to the kinetic coefficient of friction. Simplified bounds in the case of a nearly vertical impact are also derived.