×

A catching-up algorithm for multibody dynamics with impacts and dry friction. (English) Zbl 1440.70005

Summary: In the beginning of the 80s, a rigorous mathematical framework was developed for the dynamics of multibody systems with perfect unilateral contacts, particularly due to the contributions of Schatzman and Moreau. Efficient numerical methods have been proposed, for instance J. J. Moreau’s NonSmooth Contact Dynamics (NSCD) [ibid. 177, No. 3–4, 329–349 (1999; Zbl 0968.70006)], which was then extended by M. Jean to cases with friction [ibid. 177, No. 3–4, 235–257 (1999; Zbl 0959.74046)]. But the algorithm, in the latter case, is no longer the time discretization of an evolution problem. In this work, we derive a new algorithm from the time discretization of an evolution problem for multibody dynamics with contacts and friction. Our algorithm has many points in common with the one of Jean and Moreau, but it converges reliably and fixes some energetic inconsistencies. The similarities and differences between the algorithms are illustrated on three planar archetypal examples.

MSC:

70E55 Dynamics of multibody systems
65L05 Numerical methods for initial value problems involving ordinary differential equations
70F35 Collision of rigid or pseudo-rigid bodies
70F40 Problems involving a system of particles with friction

Software:

Siconos; Meschach
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Moreau, J.-J., Numerical aspects of sweeping process, Comput. Methods Appl. Mech. Engrg., 177, 329-349 (1999) · Zbl 0968.70006
[2] Jean, M., The non-smooth contact dynamics method, Comput. Methods Appl. Mech. Engrg., 177, 235-257 (1999) · Zbl 0959.74046
[3] Dubois, F.; Acary, V.; Jean, M., The contact dynamics method: A nonsmooth story, C. R. Mec. (2017)
[4] F. Dubois, M. Jean, M. Renuf, R. Mozul, A. Martin, M. Bagneris, LMGC90, in: CSMA 2011, 2011..; F. Dubois, M. Jean, M. Renuf, R. Mozul, A. Martin, M. Bagneris, LMGC90, in: CSMA 2011, 2011..
[5] Acary, V.; Pérignon, F., An introduction to siconos, Tech. Rep. 0340 (2007), INRIA
[6] Cunedall, P.; Strack, O., A discrete numerical model for granular assemblies, Geotechnique, 29, 1, 47-65 (1979)
[7] Alart, P., How to overcome indetermination and interpenetration in granular systems via nonsmooth contact dynamics. an exploratory investigation, Comput. Methods Appl. Mech. Engrg., 270, 37-56 (2014) · Zbl 1296.74068
[8] Acary, V., Projected event-capturing time-stepping schemes for nonsmooth mechanical systems with unilateral contact and Coulombs friction, Comput. Methods Appl. Mech. Engrg., 256, 224-250 (2013) · Zbl 1352.74477
[9] Schindler, T.; Rezaei, S.; Kursawe, J.; Acary, V., Half-explicit timestepping schemes on velocity level based on time-discontinuous Galerkin methods, Comput. Methods Appl. Mech. Engrg., 290, 250-276 (2015) · Zbl 1423.74918
[10] Alart, P.; Renouf, M., On inconsistency in frictional granular systems, Comput. Part. Mech. (2017)
[11] Moreau, J.-J., Facing the plurality of solutions in non smooth mechanics, (Nonsmooth/nonconvex Mechanics with Applications in Engineering (2006), Ziti: Ziti Thessaloniki), 3-12
[12] Charles, A.; Ballard, P., The formulation of dynamical contact problems with friction in the case of systems of rigid bodies and general discrete mechanical systems—painlevé and Kane paradoxes revisited, Z. Angew. Math. Phys., 67, 4, 99 (2016) · Zbl 1359.70077
[13] Lecornu, L., Sur la loi de coulomb, C. R. Acad. Sci. (Paris), 140, 847-848 (1905) · JFM 36.0786.03
[14] Painlevé, P., Sur les lois du frottement de glissement, C. R. Acad. Sci. (Paris), 121, 112-115 (1895) · JFM 26.0916.04
[15] Payr, M.; Glocker, C., Oblique frictional impact of a bar: Analysis and comparison of different impact laws, Nonlinear Dynam., 41, 4, 361-383 (2005) · Zbl 1142.74363
[16] Schatzman, M., A class of nonlinear differential equations of second order in time, Nonlinear Anal., 2, 2, 355-373 (1978) · Zbl 0382.34003
[17] Moreau, J.-J., Standard inelastic shocks and the dynamics of unilateral constraints, (Piero, G. D.; Eds, F. M., Unilateral Problems in Structural Analysis (1983), Springer-Verlag: Springer-Verlag Wien, New-York), 173-221 · Zbl 0619.73115
[18] Ballard, P.; Charles, A., An overview of the formulation, existence and uniqueness issues for the initial value problem raised by the dynamics of discrete systems with unilateral contact and dry friction, C. R. Mec. (2017)
[19] Moreau, J.-J., Unilateral contact and dry friction in finite feedom dynamics, (Moreau, J.-J.; Panangiatopoulos, P.-D., Nonsmooth Mechanics and Applications. Nonsmooth Mechanics and Applications, CISM Courses and Lectures, vol. 302 (1988), Springer-Verlag: Springer-Verlag Wien, New-York), 1-82 · Zbl 0652.00016
[20] Pfeiffer, F.; Glocker, C., Multibody Dynamics with Unilateral Contacts (2004), Wiley · Zbl 0922.70001
[21] Salvat, A. B.N.; Legrand, M., Two-dimensional modeling of unilateral contact-induced shaft precessional motions in bladed-disk/casing systems, Int. J. Non-Linear Mech., 78, 90-104 (2016)
[22] Radjai, F.; Jean, M.; Moreau, J.; Roux, S., Force distributions in dense two-dimensional granular systems, Phys. Rev. Lett., 77, 274 (1996)
[23] Nguyen, N. S.; Brogliato, B., Multiple Impacts in Dissipative Granular Chains (2014), Springer
[24] A. Derouet-Jourdan, F. Bertails-Descoubes, G. Daviet, J. Thollot, Inverse dynamic hair modeling with frictional contact, in: ACM SigGraph Asia 2013, 2013..; A. Derouet-Jourdan, F. Bertails-Descoubes, G. Daviet, J. Thollot, Inverse dynamic hair modeling with frictional contact, in: ACM SigGraph Asia 2013, 2013..
[25] Issanchou, C., Vibrations non linéaires de cordes avec contact unilatéral. A pplication aux instruments de musique (2017), Université Pierre et Marie Curie, (Ph.D. thesis)
[26] Acary, V.; Bonnefon, O.; Brogliato, B., Nonsmooth Modeling and Simulation for Switcher Circuits (2011), Springer · Zbl 1208.94003
[27] Awrejcewicz, J.; Lamarque, C.-H., Bifurcation and Chaos in Nonsmooth Mechanical Systems (2003), World Scientific · Zbl 1067.70001
[28] Nordmark, A.; Dankowicz, H.; Champneys, A., Discontinuity-induced bifurcations in systems with impacts and friction: Discontinuities in the impact law, Int. J. Non-Linear Mech., 44, 10, 1011-1023 (2009) · Zbl 1203.70027
[29] Nordmark, A.; Dankowicz, H.; Champneys, A., Friction-induced reverse chatter in rigid-body mechanisms with impacts, IMA J. Appl. Math., 76, 1, 85-119 (2011), arXiv:/oup/backfile/content_public/journal/imamat/76/1/10.1093/imamat/hxq068/2/hxq068.pdf. http://dx.doi.org/10.1093/imamat/hxq068. · Zbl 1385.70030
[30] Leine, R.; Nijmeijer, H., Dynamics and Bifurcations of Non-smmto Mechanical Systems (2004), Springer-Verlag · Zbl 1068.70003
[31] Leine, R.; van de Wouw, N., Stability and Convergence of Mechanical Systems with Unilateral Constraints (2007), Springer Verlag
[32] Brogliato, B.; Lozano, R.; ans, B. M.; Egeland, O., Dissipative Systems Analysis and Controle (2007), Springer · Zbl 1121.93002
[33] Marques, M. D.M., (Brezis, H., Differential Inclusions in Nonsmooth Mechanical Problems - Shocks and Dry Friction. Differential Inclusions in Nonsmooth Mechanical Problems - Shocks and Dry Friction, Progress in Nonlinear Differential Equations and their Applications, vol. 9 (1993), Birkhäuser: Birkhäuser Basel, Boston, Berlin) · Zbl 0802.73003
[34] Paoli, L.; Schatzman, M., A numerical scheme for impact problems II. The multidimensional case, SIAM Numer. Anal., 40, 2 (2002) · Zbl 1027.65093
[35] Paoli, L., A proximal-like algorithm for vibro-impact problems with a non smooth set of constraints, J. Differential Equations, 250, 476-514 (2011) · Zbl 1323.70056
[36] Paoli, L., Existence and approximation for vibro-impact problems with a time-dependent set of constraints, Math. Comput. Simulation, 118, C, 302-309 (2015) · Zbl 07313424
[37] Anitescu, M.; Potra, F. A.; Stewart, D. E., Time-stepping for three-dimensional rigid body dynamics, Comput. Methods Appl. Mech. Engrg., 177, 3, 183-197 (1999) · Zbl 0967.70003
[38] Stewart, D. E., Convergence of a time-stepping scheme for rigid-body dynamics and resolution of Painlevé’s problem, Arch. Ration. Mech. Anal., 145, 3, 215-260 (1998) · Zbl 0922.70004
[39] Glocker, C., Energetic consistency conditions for standard impacts, Multibody Syst. Dyn., 29, 1, 77-117 (2013) · Zbl 1271.74365
[40] Glocker, C., Dynamik von starrkörpersystemen mit reibung und stößen (1995), TU München, (Ph.D. thesis)
[41] Ballard, P., The dynamics of discrete mechanical systems with perfect unilateral constraints, Arch. Ration. Mech. Anal., 154, 199-274 (2000) · Zbl 0965.70024
[42] de Lagrange, J. L., Méchanique analytique (1788), La Veuve Desaint
[43] Germain, P., (Mécanique I. Mécanique I, Ecole Polytechnique (1986), Ellipse)
[44] Ballard, P., Dynamics of rigid bodies systems with unilateral or frictional constraints, (Gao, D. Y.; Ogden, R. W., Advances in Mechanics and Mathematics (2002), Springer US: Springer US Boston, MA), 3-87 · Zbl 1058.70017
[45] Abraham, R.; Marsden, J., Fundations of mechanics (1978), Addison-Wesley publishing Compagny
[46] Godbillon, C., Géométrie différentielle et mécanique analytique (1969), Hermann · Zbl 0174.24602
[47] Pasquero, S., Nonideal unilateral constraints in impulsive mechanics: a geometric approach, J. Math. Phys., 76, 4 (2008) · Zbl 1152.81577
[48] Moreau, J.-J., Bounded variation in time, (Moreau, J.-J.; Panangiatopoulos, P.-D.; Strang, G., Topics in Non-smooth Mechanics (1988), Birkhaüser Verlag: Birkhaüser Verlag Basel-Boston-Berlin), 1-74 · Zbl 0657.28008
[49] Lötstedt, P., Mechanical systems of rigid bodies subject to unilateral constraints, SIAM J. Appl. Math., 42, 2 (1982) · Zbl 0489.70016
[50] Rockafellar, R. T., Convex Analysis (1972), Princeton University Press · Zbl 0224.49003
[51] Moreau, J.-J., On unilateral constraints, friction and plasticity, (Capriz, G.; Stampacchia, G., New variational Techniques in Mathematical Physics (1974), Edizioni Cremonese), 173-322
[52] Brezis, H., Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert (1973), Elsevier · Zbl 0252.47055
[53] Anitescu, M.; Potra, F., Formulating dynamic multi-rigid-body contact problems with friction as solvable linear complementarity problems, Nonlinear Dyn., 14, 3, 231-247 (1997) · Zbl 0899.70005
[54] Stewart, D. E.; Trinkle, J. C., An implicit time-stepping scheme for rigid body dynamics with inelastic collisions and coulomb friction, Int. J. Numer. Methods Eng., 39, 15, 2673-2691 (1996) · Zbl 0882.70003
[55] Stewart, D., Rigid-body dynamics with friction and impact, SIAM Rev., 42, 1, 3-39 (2000) · Zbl 0962.70010
[56] Moreau, J.-J., Sur les lois de frottement, de plasticité et de viscosité, C.R. Acad. Sci. Ser. A, 271, 608-611 (1970)
[57] Génot, F.; Brogliato, B., New results on painlevé paradoxes, Eur. J. Mech. A Solids, 18, 4, 653-677 (1999) · Zbl 0962.70019
[58] Cadoux, F., Méthodes d’optimisation pour la dynamique non-régulière (2009), Université joseph fourier, (Ph.D. thesis)
[59] Glocker, C., Concepts for modeling impacts without friction, Acta Mech., 168, 1, 1-19 (2004) · Zbl 1063.74075
[60] Ballard, P., Formulation and well-posedness of the dynamics of rigid-body systems with perfect unilateral constraints, Phil. Trans. R. Soc., 359, 1789, 2327-2346 (2001), arXiv:http://rsta.royalsocietypublishing.org/content/359/1789/2327.full.pdf. http://dx.doi.org/10.1098/rsta.2001.0854 · Zbl 1014.70005
[61] Glocker, C., On frictionless impact models in rigid-body systems, Phil. Trans. R. Soc. Lond. A (2001) · Zbl 1005.70012
[62] Frémond, M., Rigid bodies collisions, Phys. Lett. A, 204, 1, 33-41 (1995) · Zbl 1020.70501
[63] Stronge, W. J., Rigid body collisions with friction, Proc. Math. Phys. Sci., 431, 1881, 169-181 (1990) · Zbl 0703.70016
[64] Ballard, P.; Basseville, S., Existence and uniqueness for dynamical unilateral contact with Coulomb friction: a model problem, ESAIM Math. Model. Numer. Anal., 39, 59-77 (2005) · Zbl 1089.34010
[65] Charles, A.; Ballard, P., Existence and uniqueness of solutions to dynamical unilateral contact problems with coulomb friction: the case of a collection of points, ESAIM: M2AN, 48, 1, 1-25 (2014) · Zbl 1369.70029
[66] Möller, M., Consistent Integrators for Non-Smooth Dynamical Systems (2011), ETH Zürich, (Ph.D. thesis)
[67] Glocker, C.; Studer, C., Formulation and preparation for numerical evaluation of linear complementarity systems in dynamics, Multibody Syst. Dyn., 13, 4, 447-463 (2005) · Zbl 1114.70008
[68] Casamayou, A.; Cohen, N.; Connan, G.; Fousse, T. D.L.; Maltey, F.; Meulien, M.; Mezzarobba, M.; Pernet, C.; Thiéry, N. M.; Zimmermann, P., Calcul mathématique avec Sage (2013), Amazon.co.uk, Marston Gate
[69] Dzonou, R.; Marques, M. D.M., A sweeping process approach to inelastic contact problems with general inertia operators, Eur. J. Mech. A Solids, 26, 3, 474-490 (2007) · Zbl 1150.74085
[70] Bruls, O.; Acary, V.; Cardona, A., Simultaneous enforcement of constraints at position and velocity levels in the nonsmooth generalized-\( \alpha\) scheme, Comput. Methods Appl. Mech. Engrg., 281, 131-161 (2014) · Zbl 1423.74659
[71] Diestel, J.; Uhl, J., Vector measures, Math. Surveys, 15 (1977) · Zbl 0369.46039
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.