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. 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. Google Scholar

[3]

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

[12]

J.-J. Moreau, Standard inelastic shocks and the dynamics of unilateral constraints,, in, (1985), 173. Google Scholar

[13]

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

[14]

J.-J. Moreau, Unilateral contact and dry friction in finite freedom dynamics,, in, (1988), 1. Google Scholar

[15]

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

[16]

L. Paoli, "Analyse Numérique de Vibrations avec Contraintes Unilatérales,", Ph.D thesis, (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. 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. 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. 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. 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. 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. 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. Google Scholar

[24]

L. Paoli and M. Schatzman, Ill-posedness in vibro-impact and its numerical consequences,, in, (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. 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. 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. doi: 10.1016/j.cma.2006.11.024. Google Scholar

[28]

R. T. Rockafellar, "Convex Analysis,", Princeton Mathematical Series, (1970). Google Scholar

[29]

M. Schatzman, A class of nonlinear differential equations of second order in time,, Nonlinear Analysis, 2 (1978), 355. 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. 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. 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. 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. Google Scholar

[3]

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

[12]

J.-J. Moreau, Standard inelastic shocks and the dynamics of unilateral constraints,, in, (1985), 173. Google Scholar

[13]

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

[14]

J.-J. Moreau, Unilateral contact and dry friction in finite freedom dynamics,, in, (1988), 1. Google Scholar

[15]

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

[16]

L. Paoli, "Analyse Numérique de Vibrations avec Contraintes Unilatérales,", Ph.D thesis, (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. 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. 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. 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. 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. 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. 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. Google Scholar

[24]

L. Paoli and M. Schatzman, Ill-posedness in vibro-impact and its numerical consequences,, in, (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. 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. 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. doi: 10.1016/j.cma.2006.11.024. Google Scholar

[28]

R. T. Rockafellar, "Convex Analysis,", Princeton Mathematical Series, (1970). Google Scholar

[29]

M. Schatzman, A class of nonlinear differential equations of second order in time,, Nonlinear Analysis, 2 (1978), 355. 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. 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. doi: 10.1115/1.2788865. Google Scholar

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