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Time discretization of vibro-impact. (English) Zbl 1067.70012
Summary: 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.

MSC:
70F35 Collision of rigid or pseudo-rigid bodies
70-08 Computational methods for problems pertaining to mechanics of particles and systems
74M20 Impact in solid mechanics
34A37 Ordinary differential equations with impulses
74M15 Contact in solid mechanics
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