A proximal-like algorithm for vibro-impact problems with a non-smooth set of constraints

Pages: 1186 - 1195, Issue Special, September 2011

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Laetitia Paoli - Université de Lyon, Université de Saint-Etienne, Jean Monnet, LaMUSE (Laboratoire de Mathématiques de l'Universitté de Saint-Etienne), 23 rue Michelon, 42023 Saint-Etienne Cedex 2, France (email)

Abstract: We consider a discrete mechanical system subjected to perfect unilateral contraints characterized by some geometrical inequalities $f_\alpha(q) >=0$, $\alpha \in {1,...,v}$, with $v >=1$. We assume that the transmission of the velocities at impacts is governed by a Newton’s impact law with a restitution coefficient $e \in [0, 1]$, allowing for conservation of kinetic energy if $e=1$, or loss of kinetic energy if $e \in [0, 1)$, when the constraints are saturated. Starting from a formulation of the dynamics as a first order measure-differential inclusion for the unknown velocities, time-stepping schemes inspired by the proximal methods can be proposed. Convergence results in the single-constraint case $(v = 1)$ are recalled and extended to the multi-constraint case $(v > 1)$, leading to new existence results for this kind of problems.

Keywords:  Frictionless unilateral constraints, Newton’s impact law, multi-constraint case, time-stepping scheme.
Mathematics Subject Classification:  Primary: 34A60, 34A12, 70E55; Secondary: 65N20

Received: July 2010;      Revised: April 2011;      Published: October 2011.