Time discretization of vibro‐impact

Abstract

We consider vibro–impact problems, i.e. mechanical systems with a finite number of degrees of freedom submitted to perfect unilateral constraints. The dynamics is basically described by a second–order measure differential inclusion for the unknown position completed with a constitutive impact law. Another formulation of the problem as a frictionless sweeping process is possible: the unknown velocity belongs to an appropriate functional space and satisfies a first order measure differential inclusion. The equivalence of these two formulations is studied. They lead to time–discretizations written in terms of positions or in terms of velocities, respectively. We present these different schemes and we compare them on the simple test–problem of a bouncing ball. We recall the convergence results in the single constraint case. Moreover, an example of implementation of the scheme derived from the basic description of the dynamics is presented. Finally, in the multi–constraint case, we point out some theoretical and computational difficulties.