Existence of solutions for a Lipschitzian vibroimpact problem with time-dependent constraints

Samir Adly,Nguyen Nang Thieu

doi:10.1186/s13663-022-00713-y

Abstract

We study a mechanical system with a finite number of degrees of freedom, subjected to perfect time-dependent frictionless unilateral (possibly nonconvex) constraints with inelastic collisions on active constraints. The dynamic is described in the form of a second-order measure differential inclusion. Under some regularity assumptions on the data, we establish several properties of the set of admissible positions, which is not necessarily convex but assumed to be uniformly prox-regular. Our approach does not require any second-order information or boundedness of the Hessians of the constraints involved in the problem and are specific to moving sets represented by inequalities constraints. On that basis, we are able to discretize our problem by the time-stepping algorithm and construct a sequence of approximate solutions. It is shown that this sequence possesses a subsequence converging to a solution of the initial problem. This methodology is not only used to prove an existence result but could be also used to solve numerically the vibroimpact problem with time-dependent nonconvex constraints.

Highlights

Vibroimpact systems are dynamical multibody systems subjected to perfect nonpenetration conditions that generate vibrations and impacts
Vibroimpact systems cannot be modeled by ordinary differential equations, and one uses measure differential inclusions
Paoli [27, 29] proposed a time-stepping approximation scheme for the problem and proved its convergence, which gives as a byproduct a global existence result when the set of admissible positions at any instant is defined by a finite family of C2 functions

Summary

Introduction

Vibroimpact systems are dynamical multibody systems subjected to perfect nonpenetration conditions that generate vibrations and impacts. Under some appropriate regularity assumptions on the data, we will prove the existence of at least one solution to problem (P). For vibroimpact problems with time-dependent constraints (i.e., when the set of admissible positions C(t) depends on time), there are few solution existence theorems. Paoli [27, 29] proposed a time-stepping approximation scheme for the problem and proved its convergence, which gives as a byproduct a global existence result when the set of admissible positions at any instant is defined by a finite family of C2 functions.

Results

Conclusion