# American Institute of Mathematical Sciences

December  2013, 6(6): 1609-1619. doi: 10.3934/dcdss.2013.6.1609

## A velocity-based time-stepping scheme for multibody dynamics with unilateral constraints

 1 PRES Université de Lyon, UJM F-42023, CNRS UMR 5208, Institut Camille Jordan, 23 rue du Docteur Paul Michelon, 42023 Saint-Etienne Cedex 2, France

Received  June 2012 Revised  September 2012 Published  April 2013

We consider a system of rigid bodies subjected to some non penetration conditions characterized by the inequalities $f_{\alpha} (q) \ge 0$, $\alpha \in \{1, \dots, \nu\}$, $\nu \ge 1$, for the configuration $q \in \mathbb{R}^d$. We assume that there is no adhesion and no friction during contact and we model the behaviour of the system at impact by a Newton's law. Starting from the mechanical description of the problem, we derive two mathematical formulations, using either the configuration or the generalized velocity as unknown. Then a velocity-based time-stepping scheme, inspired by the catching-up algorithms, is presented and its convergence in the multi-constraint case (i.e $\nu \ge1$) is stated.
Citation: Laetitia Paoli. A velocity-based time-stepping scheme for multibody dynamics with unilateral constraints. Discrete and Continuous Dynamical Systems - S, 2013, 6 (6) : 1609-1619. doi: 10.3934/dcdss.2013.6.1609
##### References:
 [1] P. Ballard, The dynamics of discrete mechanical systems with perfect unilateral constraints, Archive for Rational Mechanics and Analysis, 154 (2000), 199-274. doi: 10.1007/s002050000105. [2] A. Bressan, Questioni di regolarità e di unicità del moto in presenza di vincoli olonomi unilaterali, Rend. Sem. Mat. Univ. Padova, 29 (1959), 271-315. [3] B. Brogliato, "Nonsmooth Mechanics: Models, Dynamics and Control," 2nd edition, Springer, London, 1999. [4] B. Brogliato, A. A. ten Dam, L. Paoli, F. Génot and M. Abadie, Numerical simulation of finite dimensional multibody nonsmooth mechanical systems, ASME Applied Mechanics Reviews, 55 (2002), 107-150. doi: 10.1115/1.1454112. [5] R. Dzonou and M. Monteiro Marques, Sweeping process for inelastic impact problem with a general inertia operator, Eur. J. Mech. A Solids, 26 (2007), 474-490. doi: 10.1016/j.euromechsol.2006.07.002. [6] R. Dzonou, M. Monteiro Marques and L. Paoli, A convergence result for a vibro-impact problem with a general inertia operator, Nonlinear Dynamics, 58 (2009), 361-384. doi: 10.1007/s11071-009-9484-1. [7] M. Mabrouk, A unified variational model for the dynamics of perfect unilateral constraints, Eur. J. Mech. A Solids, 17 (1998), 819-842. doi: 10.1016/S0997-7538(98)80007-7. [8] B. Maury, A time-stepping scheme for inelastic collisions. Numerical handling of the nonoverlapping constraint, Numer. Math., 102 (2006), 649-679. doi: 10.1007/s00211-005-0666-6. [9] M. Monteiro Marques, "Differential Inclusions in Nonsmooth Mechanical Problems. Shocks and Dry Friction," Progress in Nonlinear Differential Equations and their Applications, 9, Birkhäuser Verlag, Basel, 1993. [10] J.-J. Moreau, Décomposition orthogonale d'un espace hilbertien selon deux cônes mutuellement polaires, C. R. Acad. Sci. Paris, 255 (1962), 238-240. [11] J.-J. Moreau, Liaisons unilatérales sans frottement et chocs inélastiques, C. R. Acad. Sci. Paris Série II Méc. Phys. Chim. Sci. Univers. Sci. Terre, 296 (1983), 1473-1476. [12] J.-J. Moreau, Standard inelastic shocks and the dynamics of unilateral constraints, in "Unilateral Problems in Structural Analysis" (eds. G. Del Piero and F. Maceri), CISM Courses and Lectures, Vol. 288, Springer Verlag, (1985), 173-221. [13] J.-J.Moreau, Dynamique de systèmes à liaisons unilatérales avec frottement sec éventuel, essais numériques, preprint 85-1, LMGC Montpellier, 1986. [14] J.-J. Moreau, Unilateral contact and dry friction in finite freedom dynamics, in "Nonsmooth Mechanics and Applications" (eds. J.-J. Moreau and P. Panagiotopoulos), CISM Courses and Lectures, Vol. 302, Springer-Verlag, (1988), 1-82. [15] J.-J. Moreau, Some numerical methods in multibody dynamics: application to granular materials, European J. Mechanics A Solids, 13 (1994), 93-114. [16] L. Paoli, "Analyse Numérique de Vibrations avec Contraintes Unilatérales," Ph.D thesis, University Lyon 1, 1993. [17] L. Paoli, An existence result for vibrations with unilateral constraints: Case of a nonsmooth set of constraints, Math. Models Methods Appl. Sci. (M3AS), 10 (2000), 815-831. doi: 10.1142/S0218202500000422. [18] L. Paoli, An existence result for non-smooth vibro-impact problems, J. of Diff. Equ., 211 (2005), 247-281. doi: 10.1016/j.jde.2004.11.008. [19] L. Paoli, Continuous dependence on data for vibro-impact problems, Math. Models Methods Appl. Sci. (M3AS), 15 (2005), 53-93. doi: 10.1142/S0218202505003903. [20] L. Paoli, Time stepping approximation of rigid body dynamics with perfect unilateral constraints I: the inelastic impact case, Archive for Rational Mechanics and Analysis, 198 (2010), 457-503. doi: 10.1007/s00205-010-0311-0. [21] L. Paoli, Time-stepping approximation of rigid-body dynamics with perfect unilateral constraints II: the partially elastic impact case, Archive for Rational Mechanics and Analysis, 198 (2010), 505-568. doi: 10.1007/s00205-010-0312-z. [22] L. Paoli, A proximal-like method for a class of second order measure-differential inclusions describing vibro-impact problems, J. of Diff. Equ., 250 (2011), 476-514. doi: 10.1016/j.jde.2010.10.010. [23] L. Paoli and M. Schatzman, Mouvement à un nombre fini de degrés de liberté avec contraintes unilatérales: cas avec perte d'énergie, Modèl. Math. Anal. Numér. (M2AN), 27 (1993), 673-717. [24] L. Paoli and M. Schatzman, Ill-posedness in vibro-impact and its numerical consequences, in "Proceedings of European Congress on Computational Methods in Applied Sciences and Engineering" (ECCOMAS), CDRom, 2000. [25] L. Paoli and M. Schatzman, A numerical scheme for impact problems. I: The one-dimensional case, SIAM Journal Numer. Anal., 40 (2002), 702-733. doi: 10.1137/S0036142900378728. [26] L. Paoli and M. Schatzman, A numerical scheme for impact problems. II: The multidimensional case, SIAM Journal Numer. Anal., 40 (2002), 734-768. doi: 10.1137/S003614290037873X. [27] L. Paoli and M. Schatzman, Numerical simulation of the dynamics of an impacting bar, Computer Meth. Appl. Mech. Eng., 196 (2007), 2839-2851. doi: 10.1016/j.cma.2006.11.024. [28] R. T. Rockafellar, "Convex Analysis," Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, NJ, 1970. [29] M. Schatzman, A class of nonlinear differential equations of second order in time, Nonlinear Analysis, 2 (1978), 355-373. doi: 10.1016/0362-546X(78)90022-6. [30] M. Schatzman, Penalty method for impact in generalized coordinates. Non-smooth mechanics, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci., 359 (2001), 2429-2446. doi: 10.1098/rsta.2001.0859. [31] D. Stoianovici and Y. Hurmuzlu, A critical study of the applicability of rigid-body collision theory, J. Appl. Mech., 63 (1996), 307-316. doi: 10.1115/1.2788865.

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##### References:
 [1] P. Ballard, The dynamics of discrete mechanical systems with perfect unilateral constraints, Archive for Rational Mechanics and Analysis, 154 (2000), 199-274. doi: 10.1007/s002050000105. [2] A. Bressan, Questioni di regolarità e di unicità del moto in presenza di vincoli olonomi unilaterali, Rend. Sem. Mat. Univ. Padova, 29 (1959), 271-315. [3] B. Brogliato, "Nonsmooth Mechanics: Models, Dynamics and Control," 2nd edition, Springer, London, 1999. [4] B. Brogliato, A. A. ten Dam, L. Paoli, F. Génot and M. Abadie, Numerical simulation of finite dimensional multibody nonsmooth mechanical systems, ASME Applied Mechanics Reviews, 55 (2002), 107-150. doi: 10.1115/1.1454112. [5] R. Dzonou and M. Monteiro Marques, Sweeping process for inelastic impact problem with a general inertia operator, Eur. J. Mech. A Solids, 26 (2007), 474-490. doi: 10.1016/j.euromechsol.2006.07.002. [6] R. Dzonou, M. Monteiro Marques and L. Paoli, A convergence result for a vibro-impact problem with a general inertia operator, Nonlinear Dynamics, 58 (2009), 361-384. doi: 10.1007/s11071-009-9484-1. [7] M. Mabrouk, A unified variational model for the dynamics of perfect unilateral constraints, Eur. J. Mech. A Solids, 17 (1998), 819-842. doi: 10.1016/S0997-7538(98)80007-7. [8] B. Maury, A time-stepping scheme for inelastic collisions. Numerical handling of the nonoverlapping constraint, Numer. Math., 102 (2006), 649-679. doi: 10.1007/s00211-005-0666-6. [9] M. Monteiro Marques, "Differential Inclusions in Nonsmooth Mechanical Problems. Shocks and Dry Friction," Progress in Nonlinear Differential Equations and their Applications, 9, Birkhäuser Verlag, Basel, 1993. [10] J.-J. Moreau, Décomposition orthogonale d'un espace hilbertien selon deux cônes mutuellement polaires, C. R. Acad. Sci. Paris, 255 (1962), 238-240. [11] J.-J. Moreau, Liaisons unilatérales sans frottement et chocs inélastiques, C. R. Acad. Sci. Paris Série II Méc. Phys. Chim. Sci. Univers. Sci. Terre, 296 (1983), 1473-1476. [12] J.-J. Moreau, Standard inelastic shocks and the dynamics of unilateral constraints, in "Unilateral Problems in Structural Analysis" (eds. G. Del Piero and F. Maceri), CISM Courses and Lectures, Vol. 288, Springer Verlag, (1985), 173-221. [13] J.-J.Moreau, Dynamique de systèmes à liaisons unilatérales avec frottement sec éventuel, essais numériques, preprint 85-1, LMGC Montpellier, 1986. [14] J.-J. Moreau, Unilateral contact and dry friction in finite freedom dynamics, in "Nonsmooth Mechanics and Applications" (eds. J.-J. Moreau and P. Panagiotopoulos), CISM Courses and Lectures, Vol. 302, Springer-Verlag, (1988), 1-82. [15] J.-J. Moreau, Some numerical methods in multibody dynamics: application to granular materials, European J. Mechanics A Solids, 13 (1994), 93-114. [16] L. Paoli, "Analyse Numérique de Vibrations avec Contraintes Unilatérales," Ph.D thesis, University Lyon 1, 1993. [17] L. Paoli, An existence result for vibrations with unilateral constraints: Case of a nonsmooth set of constraints, Math. Models Methods Appl. Sci. (M3AS), 10 (2000), 815-831. doi: 10.1142/S0218202500000422. [18] L. Paoli, An existence result for non-smooth vibro-impact problems, J. of Diff. Equ., 211 (2005), 247-281. doi: 10.1016/j.jde.2004.11.008. [19] L. Paoli, Continuous dependence on data for vibro-impact problems, Math. Models Methods Appl. Sci. (M3AS), 15 (2005), 53-93. doi: 10.1142/S0218202505003903. [20] L. Paoli, Time stepping approximation of rigid body dynamics with perfect unilateral constraints I: the inelastic impact case, Archive for Rational Mechanics and Analysis, 198 (2010), 457-503. doi: 10.1007/s00205-010-0311-0. [21] L. Paoli, Time-stepping approximation of rigid-body dynamics with perfect unilateral constraints II: the partially elastic impact case, Archive for Rational Mechanics and Analysis, 198 (2010), 505-568. doi: 10.1007/s00205-010-0312-z. [22] L. Paoli, A proximal-like method for a class of second order measure-differential inclusions describing vibro-impact problems, J. of Diff. Equ., 250 (2011), 476-514. doi: 10.1016/j.jde.2010.10.010. [23] L. Paoli and M. Schatzman, Mouvement à un nombre fini de degrés de liberté avec contraintes unilatérales: cas avec perte d'énergie, Modèl. Math. Anal. Numér. (M2AN), 27 (1993), 673-717. [24] L. Paoli and M. Schatzman, Ill-posedness in vibro-impact and its numerical consequences, in "Proceedings of European Congress on Computational Methods in Applied Sciences and Engineering" (ECCOMAS), CDRom, 2000. [25] L. Paoli and M. Schatzman, A numerical scheme for impact problems. I: The one-dimensional case, SIAM Journal Numer. Anal., 40 (2002), 702-733. doi: 10.1137/S0036142900378728. [26] L. Paoli and M. Schatzman, A numerical scheme for impact problems. II: The multidimensional case, SIAM Journal Numer. Anal., 40 (2002), 734-768. doi: 10.1137/S003614290037873X. [27] L. Paoli and M. Schatzman, Numerical simulation of the dynamics of an impacting bar, Computer Meth. Appl. Mech. Eng., 196 (2007), 2839-2851. doi: 10.1016/j.cma.2006.11.024. [28] R. T. Rockafellar, "Convex Analysis," Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, NJ, 1970. [29] M. Schatzman, A class of nonlinear differential equations of second order in time, Nonlinear Analysis, 2 (1978), 355-373. doi: 10.1016/0362-546X(78)90022-6. [30] M. Schatzman, Penalty method for impact in generalized coordinates. Non-smooth mechanics, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci., 359 (2001), 2429-2446. doi: 10.1098/rsta.2001.0859. [31] D. Stoianovici and Y. Hurmuzlu, A critical study of the applicability of rigid-body collision theory, J. Appl. Mech., 63 (1996), 307-316. doi: 10.1115/1.2788865.
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