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Asymptotic analysis of Painlevé' s paradox

Zhen Zhao 1 Caishan Liu 2 Chen Bin 2 Bernard Brogliato 3
3 BIPOP - Modelling, Simulation, Control and Optimization of Non-Smooth Dynamical Systems
Inria Grenoble - Rhône-Alpes, LJK [2007-2015] - Laboratoire Jean Kuntzmann [2007-2015], Grenoble INP [2007-2019] - Institut polytechnique de Grenoble - Grenoble Institute of Technology [2007-2019]
Abstract : Painlevé’s paradox is a well-known problem in rigid-body dynamics, of which the forward dynamics equations could have no solution. To handle this situation, an assumption of tangential impact is often introduced. Although the assumption seems to provide a good fix, it still needs to be mathematically examined via analyzing the asymptotic property of a compliance-based model at the limit of rigidity. In this paper, we revisit the paradox using the typical Painlevé’s example of a rod sliding on a rough surface. For convenience, the interaction at the contact point of the rod is represented by a linear spring to scale the local normal compliance, coupled with Coulomb’s law to reflect friction. We perform an asymptotic analysis using the spring stiffness as a perturbation parameter. The rod dynamics in the Painlevé’s paradox, accompanying the variation of friction status, consists of three distinct phases as follows: An initial period of sliding which allows contact force to diverge with the increase of the spring stiffness, a period of sticking which ends at the occurrence of a reverse slip motion, and a reverse slip phase which causes the rod to be detached from the contact surface. As the stiffness goes to infinity, all the time intervals of the three phases converge to zero. This analysis theoretically confirms the assumption of the tangential impact in the paradox of sliding rod dynamics.
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Zhen Zhao, Caishan Liu, Chen Bin, Bernard Brogliato. Asymptotic analysis of Painlevé' s paradox. Multibody System Dynamics, Springer Verlag, 2015, 35 (3), pp.299-319. ⟨10.1007/s11044-014-9448-1⟩. ⟨hal-01232780⟩

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