R. A. Adams, Sobolev spaces, 1975.

J. Ahn and D. E. Stewart, An Euler--Bernoulli Beam with Dynamic Contact: Discretization, Convergence, and Numerical Results, SIAM Journal on Numerical Analysis, vol.43, issue.4, pp.1455-1480, 2005.
DOI : 10.1137/S0036142903432619

P. Alart and A. Curnier, A mixed formulation for frictional contact problems prone to Newton like solution methods, Computer Methods in Applied Mechanics and Engineering, vol.92, issue.3, pp.353-375, 1991.
DOI : 10.1016/0045-7825(91)90022-X

N. J. Carpenter, Lagrange constraints for transient finite element surface contact, International Journal for Numerical Methods in Engineering, vol.1, issue.1, pp.103-128, 1991.
DOI : 10.1002/nme.1620320107

URL : https://hal.archives-ouvertes.fr/hal-01389918

P. G. Ciarlet, The finite element method for elliptic problems, 1978.

P. G. Ciarlet, Basic error estimates for elliptic problems II of Handbook of Numerical Analysis, pp.17-351, 1991.

F. Dabaghi, A. Petrov, J. Pousin, and Y. Renard, Convergence of mass redistribution method for the wave equation with a unilateral constraint at the boundary ESAIM, pp.2-48, 2014.

P. Deuflhard, R. Krause, and S. Ertel, A contact-stabilized Newmark method for dynamical contact problems, International Journal for Numerical Methods in Engineering, vol.1, issue.9, pp.1274-1290, 2007.
DOI : 10.1007/978-1-4612-5152-1

Y. Dumont and L. Paoli, Vibrations of a beam between obstacles. Convergence of a fully discretized approximation, ESAIM: Mathematical Modelling and Numerical Analysis, vol.63, issue.4, pp.2-40, 2006.
DOI : 10.1115/1.2788865

Y. Dumont and L. Paoli, Numerical simulation of a model of vibrations with joint clearance, International Journal of Computer Applications in Technology, vol.33, issue.1, pp.41-53, 2008.
DOI : 10.1504/IJCAT.2008.021884

URL : https://hal.archives-ouvertes.fr/hal-01569160

D. Doyen, Méthodes numériques pour desprobì emes dynamiques de contact et de fissuration, Thèse de l, 2010.

Y. Renard and J. Pommier, An open source generic C++ library for finite element methods, 2016.

C. Hager, S. Hüeber, and B. Wohlmuth, A stable energy-conserving approach for frictional contact problems based on quadrature formulas, International Journal for Numerical Methods in Engineering, vol.40, issue.2, pp.205-225, 2008.
DOI : 10.1007/978-3-642-55814-6

P. Hauret and P. L. Tallec, Energy-controlling time integration methods for nonlinear elastodynamics and low-velocity impact, Computer Methods in Applied Mechanics and Engineering, vol.195, issue.37-40, pp.4890-4916, 2006.
DOI : 10.1016/j.cma.2005.11.005

URL : https://hal.archives-ouvertes.fr/hal-00111458

P. Hauret, Mixed interpretation and extensions of the equivalent mass matrix approach for elastodynamics with contact, Computer Methods in Applied Mechanics and Engineering, vol.199, issue.45-48, pp.199-2941, 2010.
DOI : 10.1016/j.cma.2010.06.004

R. A. Ibrahim, V. I. Babitsky, and M. Okuma, Vibro-Impact Dynamics of Ocean Systems and Related Problems, Lect. Notes Appl. Comput. Mech, vol.44, 2009.
DOI : 10.1007/978-3-642-00629-6

H. B. Khenous, P. Laborde, and Y. Renard, Mass redistribution method for finite element contact problems in elastodynamics, European Journal of Mechanics - A/Solids, vol.27, issue.5, pp.918-932, 2008.
DOI : 10.1016/j.euromechsol.2008.01.001

URL : https://hal.archives-ouvertes.fr/hal-00381443

K. Kuttler and M. Shillor, Vibrations of a beam between two stops, Dynamics of Continuous, Discrete and Impulsive Systems, Ser. B, Appl. Algorithms, vol.8, pp.93-110, 2001.

T. A. Laursen and V. Chawla, DESIGN OF ENERGY CONSERVING ALGORITHMS FOR FRICTIONLESS DYNAMIC CONTACT PROBLEMS, International Journal for Numerical Methods in Engineering, vol.37, issue.5, pp.863-886, 1997.
DOI : 10.1016/0045-7949(90)90324-U

URL : https://hal.archives-ouvertes.fr/hal-01435617

T. A. Laursen and G. R. Love, Improved implicit integrators for transient impact problems?geometric admissibility within the conserving framework, International Journal for Numerical Methods in Engineering, vol.32, issue.2, pp.245-274, 2002.
DOI : 10.1002/nme.1620320107

L. Paoli, Time discretization of vibro-impact, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol.359, issue.1789, pp.2405-2428, 2001.
DOI : 10.1098/rsta.2001.0858

URL : https://hal.archives-ouvertes.fr/hal-01566940

L. Paoli and M. Schatzman, A Numerical Scheme for Impact Problems I: The One-Dimensional Case, SIAM Journal on Numerical Analysis, vol.40, issue.2, pp.702-733, 2002.
DOI : 10.1137/S0036142900378728

L. Paoli and M. Schatzman, Numerical simulation of the dynamics of an impacting bar, Computer Methods in Applied Mechanics and Engineering, vol.196, issue.29-30, pp.2839-2851, 2007.
DOI : 10.1016/j.cma.2006.11.024

URL : https://hal.archives-ouvertes.fr/hal-01635192

A. Petrov and M. Schatzman, Visco??lastodynamique monodimensionnelle avec conditions de Signorini, Comptes Rendus Mathematique, vol.334, issue.11, pp.983-988, 2002.
DOI : 10.1016/S1631-073X(02)02399-3

A. Petrov and M. Schatzman, A pseudodifferential linear complementarity problem related to a one dimensional viscoelastic model with Signorini condition. Archive for Rational Mechanics and Analysis, 2009.

C. Pozzolini and M. Salaün, Some energy conservative schemes for vibro-impacts of a beam on rigid obstacles, ESAIM: Mathematical Modelling and Numerical Analysis, vol.36, issue.6, pp.2-45, 2011.
DOI : 10.1002/nme.1620361211

URL : https://hal.archives-ouvertes.fr/hal-00813464

C. Pozzolini, Y. Renard, and M. Salaün, Vibro-impact of a plate on rigid obstacles: existence theorem, convergence of a scheme and numerical simulations, IMA Journal of Numerical Analysis, vol.33, issue.1, pp.33-261, 2013.
DOI : 10.1093/imanum/drr057

URL : https://hal.archives-ouvertes.fr/hal-00812715

Y. Renard, The singular dynamic method for constrained second order hyperbolic equations: Application to dynamic contact problems, Journal of Computational and Applied Mathematics, vol.234, issue.3, pp.906-923, 2010.
DOI : 10.1016/j.cam.2010.01.058

URL : https://hal.archives-ouvertes.fr/hal-01461799

R. L. Taylor and P. Papadopoulos, On a finite element method for dynamic contact/impact problems, International Journal for Numerical Methods in Engineering, vol.2, issue.12, pp.2123-2140, 1993.
DOI : 10.1115/1.3408787

Y. Mochida, Bounded Eigenvalues of Fully Clamped and Completely Free Rectangular Plates, 2007.