On the regularization of impact without collision: the Painlev\'e paradox and compliance

TL;DR

This paper investigates how compliance at contact points can regularize impact without collision phenomena in rigid body dynamics with Coulomb friction, providing mathematical conditions and asymptotic expressions for different damping regimes.

Contribution

It introduces a rigorous mathematical framework to recover impact without collision in the Painlevé paradox using compliance, including explicit formulas and asymptotic analysis.

Findings

Impact without collision is recovered for a broad class of rigid bodies.

Explicit conditions separating impact without collision and lift-off are derived.

Asymptotic expressions for small and large damping regimes are provided.

Abstract

We consider the problem of a rigid body, subject to a unilateral constraint, in the presence of Coulomb friction. We regularize the problem by assuming compliance (with both stiffness and damping) at the point of contact, for a general class of normal reaction forces. Using a rigorous mathematical approach, we recover impact without collision (IWC) in both the inconsistent and indeterminate Painlev\'e paradoxes, in the latter case giving an exact formula for conditions that separate IWC and lift-off. We solve the problem for arbitrary values of the compliance damping and give explicit asymptotic expressions in the limiting cases of small and large damping, all for a large class of rigid bodies.