Dynamic contact of a beam against rigid obstacles: Convergence of a velocity-based approximation and numerical results

Motivated by the study of vibrations due to looseness of joints, we consider the motion of a beam between rigid obstacles. Due to the non-penetrability condition, the dynamics is described by a hyperbolic fourth order variational inequality. We build a family of fully discretized approximations of this problem by combining some classical space discretizations with velocity based time-stepping algorithms for discrete mechanical systems subjected to unilateral constraints. We prove the stability and the convergence of these numerical methods. Finally we propose some examples of implementation using either Hermite or B-spline finite element approximations.