Abstract
The dynamics of mechanical systems with a finite number of degrees of freedom (discrete mechanical systems) is governed by the Lagrange equation which is a second-order differential equation on a Riemannian manifold (the configuration manifold). The handling of perfect (frictionless) unilateral constraints in this framework (that of Lagrange’s analytical dynamics) was undertaken by Schatzman and Moreau at the beginning of the 1980s. A mathematically sound and consistent evolution problem was obtained, paving the road for many subsequent theoretical investigations. In this general evolution problem, the only reaction force which is involved is a generalized reaction force, consistently with the virtual power philosophy of Lagrange. Surprisingly, such a general formulation was never derived in the case of frictional unilateral multibody dynamics. Instead, the paradigm of the Coulomb law applying to reaction forces in the real world is generally invoked. So far, this paradigm has only enabled to obtain a consistent evolution problem in only some very few specific examples and to suggest numerical algorithms to produce computational examples (numerical modeling). In particular, it is not clear what is the evolution problem underlying the computational examples. Moreover, some of the few specific cases in which this paradigm enables to write down a precise evolution problem are known to show paradoxes: the Painlevé paradox (indeterminacy) and the Kane paradox (increase in kinetic energy due to friction). In this paper, we follow Lagrange’s philosophy and formulate the frictional unilateral multibody dynamics in terms of the generalized reaction force and not in terms of the real-world reaction force. A general evolution problem that governs the dynamics is obtained for the first time. We prove that all the solutions are dissipative; that is, this new formulation is free of Kane paradox. We also prove that some indeterminacy of the Painlevé paradox is fixed in this formulation.
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References
Ballard P.: The dynamics of discrete mechanical systems with perfect unilateral constraints. Arch. Ration. Mech. Anal. 154, 199–274 (2000)
Ballard P.: Formulation and well-posedness of the dynamics of rigid-body systems with perfect unilateral constraints. Philos. Trans. R. Soc. A 359, 2327–2346 (2001)
Ballard P.: Dynamics of rigid bodies systems with unilateral or frictional constraints. In: Gao, D.Y., Ogden, R. (eds) Advances in Mechanics and Mathematics, Chap. 1, vol. 1, pp. 3–87. Kluwer Academic Publishers, Dordrecht/Boston/London (2002)
Charles A., Ballard P.: Existence and uniqueness of solutions to dynamical unilateral contact problems with Coulomb friction: the case of a collection of points. Math. Model. Numer. Anal. 48(1), 1–25 (2014)
Coulomb A.: Théorie des machines simples. Bachelier, Paris (1821)
Glocker C.: Energetic consistency conditions for standard impacts, part i: Newton-type inequality impact laws and Kane’s example. Multibody Syst. Dyn. 29, 77–117 (2013)
Godbillon C.: Géométrie Différentielle et Mécanique Analytique. Hermann, Paris (1969)
Jean M.: The non-smooth contact dynamics method. Comput. Methods Appl. Mech. Eng. 177, 235–257 (1999)
Kane, T.R.: A dynamics puzzle. Stanf. Mech. Alumni Club Newsl. 6 (1983)
Kane T.R., Levinson D.A.: Dynamics: Theory and Applications. McGrawHill, New-York (1985)
Lecornu L.: Sur la loi de Coulomb. Comptes Rendus De l’a adémie Des Sciences) 140, 847–848 (1905)
Michałovsky R., Mróz Z.: Associated and non-associated sliding rules in contact friction problems. Arch. Mech. 30(3), 259–276 (1978)
Monteiro Marques M.D.P.: Differential Inclusions in Nonsmooth Mechanical Problems—Shocks and Dry Friction, Progress in Nonlinear Differential Equations and Their Applications, vol. 9. Birkhäuser, Basel (1993)
Moreau J.J.: Standard inelastic shocks and the dynamics of unilateral constraints. In: Del Piero, G., Maceri, F. (eds) Unilateral Problems in Structural Analysis, pp. 173–221. Springer, Wien (1983)
Moreau J.J.: Une formulation du contact à frottement sec ; application au calcul numérique. Comptes Rendus De l’académie Des Sciences (Paris), série II 302, 799–801 (1986)
Moreau, J.J.: Bounded variation in time. In: Moreau, J.J., Panangiatopoulos, P.D. (eds.) Topics in Non-smooth Mechanics, Chap. 1, pp. 1–74. Birkhäuser, Basel (1988)
Moreau J.J.: Unilateral contact and dry friction in finite freedom dynamics. In: Moreau, J.J., Panagiotopoulos, P.D. (eds) Non-smooth Mechanics and Applications, pp. 1–82. Springer, Wien (1988)
Painlevé P.: Sur les lois du frottement de glissement. Comptes Rendus De l’académie Des Sciences 121, 112–115 (1895)
Paoli, L.: Time-stepping approximation of rigid-body dynamics with perfect unilateral constraints. I and II. Arch. Ration. Mech. Anal. 198, 457–503 and 505–568 (2010)
Paoli L.: A proximal-like algorithm for vibro-impact problems with a non smooth set of constraints. J. Differ. Equ. 250, 476–514 (2011)
Pasquero, S.: Nonideal unilateral constraints in impulsive mechanics: a geometric approach. J. Math. Phys. 49(4), 042902 (2008)
Percivale D.: Uniqueness in the elastic bounce problem, i. J. Differ. Equ. 2(2), 206–215 (1985)
Percivale D.: Uniqueness in the elastic bounce problem, ii. J. Differ. Equ. 90, 304–315 (1991)
Schatzman M.: A class of nonlinear differential equations of second order in time. Nonlinear Anal. Theory Methods Appl. 2(2), 355–373 (1978)
Stewart D.E.: Convergence of a time-stepping scheme for rigid-body dynamics and resolution of painlevé’s problem. Arch. Ration. Mech. Anal. 145, 215–260 (1998)
Stewart D.E.: Rigid-body dynamics with friction and impact. SIAM Rev. 42(1), 3–39 (2000)
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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, 99 (2016). https://doi.org/10.1007/s00033-016-0688-1
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DOI: https://doi.org/10.1007/s00033-016-0688-1