The Painleve paradox in contact mechanics

TL;DR

This paper reviews the Painleve paradox in contact mechanics, proposing a general formulation and exploring solutions to resolve indeterminacies and inconsistencies in models of rigid body contact with friction.

Contribution

It introduces a comprehensive formulation including impact laws, analyzes the paradox in 2D and 3D, and discusses strategies like contact regularisation to resolve indeterminacies.

Findings

Inconsistent and indeterminate cases can be resolved via impact laws and infinite stiffness limit.

Dynamic jam and reverse chatter are key ambiguities in finite time.

Complexity increases with multiple contact points and three-dimensional analysis.

Abstract

The 120-year old so-called Painleve 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 single contact points. 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 a Poisson or energetic impact law. The general problem in 2D with a single point of contact is discussed and cases or inconsistency or indeterminacy enumerated. Strategies to resolve the paradox via contact regularisation 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 three-dimensional analysis is briefly considered and shown to lead to further possible singularities. In conclusion, the ubiquity of the \pain paradox is highlighted and open problems are discussed.