Abstract
We consider a discrete mechanical system subjected to perfect time-dependent unilateral constraints, which dynamics is described by a second order measure differential inclusion. The transmission of the velocity at impacts is given by a minimization property of the kinetic energy with respect to the set of kinematically admissible post-impact velocities. We construct a sequence of feasible approximate positions by using a time-stepping algorithm inspired by a kind of Euler discretization of the differential inclusion. We prove the convergence of the approximate trajectories to a solution of the Cauchy problem and we obtain as a by-product a global existence result.
Highlights
We consider a mechanical system with a finite number of degrees of freedom subjected to frictionless unilateral constraints
We prove the convergence of the approximate trajectories to a solution of the Cauchy problem and we obtain as a by-product a global existence result
If we denote by u(t) its representative point in generalized coordinates and by K (t) the set of admissible positions at any instant t, the dynamics of the system is described by a second order differential equation combined with the condition u(t) ∈ K (t), leading to a measure differential inclusion
Summary
We consider a mechanical system with a finite number of degrees of freedom subjected to frictionless unilateral constraints. In [25] an existence result is established, by considering a generalization of the Yosida-type approximation already proposed in [20] This technique transforms the differential inclusion into a sequence of very stiff ordinary differential equations which are not well suited for implementation (see [21] for a more detailed discussion). More recently another existence result, based on a time-stepping approximation of the problem, has been obtained [2] when the sets K (t) are defined as a finite intersection of complements of convex sets. We introduce a time-stepping scheme inspired by some implicit Euler’s type discretization and the main steps of the convergence proof are outlined
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