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 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 Painlevé paradox is highlighted and open problems are discussed.

Dynamics of mechanical systems with two sliding contacts: new facets of Painlevé’s paradox

“…To determine the consistency of each contact mode, we combine the equations of motion (1) with the relevant equality constraint (2) or (5) The consistency conditions summarized in column 3 of Table 1 This observation enables us to classify systems by a three digit code ijk where i  {1,2…,4}; j,k  {1,2…,6}. The first two digits will be used to define a coarse classification (from now: classes), previously described by Ivanov [11].…”

“…they do not induce contact sticking or reversal of sliding direction. Under these restrictions, almost all simple models of frictional, planar, single-point impact become identical, see for example the model of Chatterjee and Ruina [2]; Kane and Levinson [29]; Pfeiffer and Glocker [23]; Routh (or Poisson) [24]; and Stronge [26]. We will use two important features of these impact models:…”

“…A fundamental shortcoming of the contact mode-based approach was already discovered in the late 19 th century [12], [20]: the number of consistent solutions may be zero, as well as greater than 1. These observations became known as Painlevé's nonuniqueness and non-existence paradox [1] [2]. For planar systems with a single contact, all possible types of indeterminacy and inconsistency have been identified and listed.…”

Control of Painlevé Paradox in a Robotic System

“…Most people have experienced at least once the annoying high-pitched sound that chalk may produce when pressed against a blackboard. As it is now well known [1], the sound is the result of fast vibrations of the piece of chalk that quickly and repeatedly detaches from the blackboard and comes back into contact with it. This phenomenon is paradoxical as the more one presses the chalk against the surface, the more likely bouncing motion becomes.…”

“…Namely, the solution to the differential equations describing the motion of the stick may become indeterminate or inconsistent, in the sense that the model would predict the stick to penetrate the rigid surface, which clearly is not realistic. In the following years, many mathematicians and scientists have been interested in the study of this paradoxical phenomenon, but, as pointed out by Champneys in [1], to this date, all the ways in which the stick can enter the inconsistent or indeterminate solution modes have not been determined analytically. For this reason, most of the research follows a numerical or experimental approach in the investigation of the problem.…”