The Painlevé paradox in contact mechanics

Abstract

The 120-year-old so-called Painlevé paradox involves the loss of determinism in models of planar rigid bodies in point contact with a rigid surface, subject to Coulomb-like dry friction. The phenomenon occurs due to coupling between normal and rotational degrees-of-freedom such that the effective normal force becomes attractive rather than repulsive. Despite a rich literature, the forward evolution problem remains unsolved other than in certain restricted cases in 2D with a single contact point. Various practical consequences of the theory are revisited, including models for robotic manipulators and the strange behaviour of chalk when pushed rather than dragged across a blackboard. Reviewing recent theory, a general formulation is proposed, including Poisson or energetic impact laws. The general problem in 2D with a single point of contact is discussed and cases of inconsistency or indeterminacy enumerated. Strategies to resolve the paradox via contact regularization are discussed from a dynamical systems point of view. By passing to the infinite stiffness limit and allowing impact without collision, inconsistent and indeterminate cases are shown to be resolvable for all open sets of conditions. However, two unavoidable ambiguities that can be reached in finite time are discussed in detail, so called dynamic jam and reverse chatter. A partial review is given of 2D cases with two points of contact showing how a greater complexity of inconsistency and indeterminacy can arise. Extension to fully 3D analysis is briefly considered and shown to lead to further possible singularities. In conclusion, the ubiquity of the Painlevé paradox is highlighted and open problems are discussed.