Geometric characterization of frictional impacts by means of breakable kinetic constraints

TL;DR

This work addresses frictional impacts in impulsive mechanics by avoiding direct contact-force analysis and instead leveraging a geometric framework called RGIMS, which combines a unilateral constraint S with a breakable instantaneous kinetic constraint B. A constitutive map I(p_L) assigns reactive impulses to admissible left velocities, producing a right velocity p_R = p_L + I(p_L) and enabling ideal, non-ideal, and Coulomb-like friction characterizations through orthogonal velocity components. Three progressively complex examples (point, disk, rod) illustrate stick-slip behavior and show how friction redistributes velocity between normal and tangential directions while maintaining energy balance. The approach offers a deterministic, geometry-driven route to impulsive multi-body impacts with friction and provides a foundation for extensions to two-body contacts and analyses of phenomena like the Painlevé paradox within a unified framework.

Abstract

In the context of geometric Impulsive Mechanics of systems with a finite number of degrees of freedom, we model the roughness of a unilateral constraint ${\mathcal S\/}$ by introducing a suitable instantaneous kinetic constraint ${\mathcal B\/}\subset {\mathcal S\/}$. A constitutive characterization of ${\mathcal B\/}$ based only on the geometric properties of the setup and on the dry friction laws can then be introduced to model the frictional behavior of ${\mathcal S\/}$ in an impact of the system. Such a model restores determinism and avoids the analysis of frictional forces in the contact point, with all its associated theoretical problems of causality. Three examples of increasing complexity, showing a natural stick--slip behavior of the impact, are presented.