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 & 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.  Google Scholar

[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.  Google Scholar

[3]

B. Brogliato, "Nonsmooth Mechanics: Models, Dynamics and Control," 2nd edition, Springer, London, 1999. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

J.-J. Moreau, Décomposition orthogonale d'un espace hilbertien selon deux cônes mutuellement polaires, C. R. Acad. Sci. Paris, 255 (1962), 238-240.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[13]

J.-J.Moreau, Dynamique de systèmes à liaisons unilatérales avec frottement sec éventuel, essais numériques, preprint 85-1, LMGC Montpellier, 1986. Google Scholar

[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.  Google Scholar

[15]

J.-J. Moreau, Some numerical methods in multibody dynamics: application to granular materials, European J. Mechanics A Solids, 13 (1994), 93-114.  Google Scholar

[16]

L. Paoli, "Analyse Numérique de Vibrations avec Contraintes Unilatérales," Ph.D thesis, University Lyon 1, 1993. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

R. T. Rockafellar, "Convex Analysis," Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, NJ, 1970.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[3]

B. Brogliato, "Nonsmooth Mechanics: Models, Dynamics and Control," 2nd edition, Springer, London, 1999. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

J.-J. Moreau, Décomposition orthogonale d'un espace hilbertien selon deux cônes mutuellement polaires, C. R. Acad. Sci. Paris, 255 (1962), 238-240.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[13]

J.-J.Moreau, Dynamique de systèmes à liaisons unilatérales avec frottement sec éventuel, essais numériques, preprint 85-1, LMGC Montpellier, 1986. Google Scholar

[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.  Google Scholar

[15]

J.-J. Moreau, Some numerical methods in multibody dynamics: application to granular materials, European J. Mechanics A Solids, 13 (1994), 93-114.  Google Scholar

[16]

L. Paoli, "Analyse Numérique de Vibrations avec Contraintes Unilatérales," Ph.D thesis, University Lyon 1, 1993. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

R. T. Rockafellar, "Convex Analysis," Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, NJ, 1970.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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