Unilateral Contact and Dry Friction in Finite Freedom Dynamics

TLDR

The paper develops an approach to the dynamics of mechanical systems with finite degrees of freedom under unilateral constraints. The method models dynamics using measure differential inclusions in n‑dimensional force and velocity spaces, allowing non‑differentiable velocities, soft dissipative collisions, and a new Coulomb friction law without convexity assumptions. The resulting formulations enable time‑discretized numerical algorithms that effectively handle nonsmooth effects from unilateral constraints and dry friction.

Abstract

An approach to the dynamics of mechanical systems with a finite number of degrees of freedom, involving unilateral constraints, is developed. In the n-dimensional linear spaces of forces and velocities, some classical concepts of Convex Analysis are used, but no convexity assumption is made concerning the constraint inequalities. The velocity is not supposed to be a differentiable function of time, but only to have locally bounded variation, so the role of the acceleration is held by a n-dimensional measure on the considered time interval. Dynamics is then governed by measure differential inclusions, which treat possible velocity jumps on the same footing as smooth motions. Possible collisions are described as soft, thus dissipative. Friction is taken into account under a recently proposed expression of Coulomb's law. These formulations have the advantage of generating numerical algorithms of time-discretization, able to handle, in particular, the nonsmooth effects arising from unilaterality and from dry friction.