Time-Stepping Approximation of Rigid-Body Dynamics with Perfect Unilateral Constraints. I: The Inelastic Impact Case

We consider a discrete mechanical system with a non-trivial mass matrix, subjected to perfect unilateral constraints described by the geometrical inequalities ${f_{\alpha} (q) \geqq 0, \alpha \in \{1, \dots, \nu\} (\nu \geqq 1)}$ . We assume that the transmission of the velocities at impact is governed by Newton’s Law with a coefficient of restitution e = 0 (so that the impact is inelastic). We propose a time-discretization of the second order differential inclusion describing the dynamics, which generalizes the scheme proposed in Paoli (J Differ Equ 211:247–281, 2005) and, for any admissible data, we prove the convergence of approximate motions to a solution of the initial-value problem.