The Painlevé paradox (also called by Jean Jacques Moreau frictional paroxysms) is a well-known example by Paul Painlevé in rigid-body dynamics that showed that rigid-body dynamics with both contact friction and Coulomb friction is inconsistent. This result is due to a number of discontinuities in the behavior of rigid bodies and the discontinuities inherent in the Coulomb friction law, especially when dealing with large coefficients of friction.[1] There exist, however, simple examples which prove that the Painlevé paradoxes can appear even for small, realistic friction.

Modelling rigid bodies and friction greatly simplifies such applications as animation, robotics and bio-mechanics, it is only an approximation to a full elastic model requiring complex systems of partial differential equations. Rigid body assumption also allows one to clarify many features that would otherwise remain hidden; Painlevé paradoxes are one of them. Moreover, the rigid body models can be reliably and efficiently simulated, avoiding stiff problems and issues related to the estimation of compliant contact/impact models, which is often quite a delicate matter.

Solution

The physical paradox was mathematically resolved in the 1990s by David E. Stewart.[2] The Painlevé paradox has not only been solved by D. E. Stewart from the mathematical point of view (i.e. Stewart has shown the existence of solutions for the classical Painlevé example that consists of a rod sliding on a rough plane in 2-dimension), but it has been explained from a more mechanical point of view by Franck Génot and Bernard Brogliato.[3] Génot and Brogliato have studied in great detail the rod dynamics in the neighborhood of a singular point of the phase space, when the rod is sliding. The dynamical equations are then a particular singular ordinary differential equation with vector field f(x)/g(x), where both f and g may vanish at a certain point (angle and angular velocity). One of the results is that at this singular point the contact force may grow unbounded, however its impulse remains always bounded (this may explain why time-stepping numerical methods like Moreau's scheme can well handle such situations since they estimate the impulse, not the force[4]). Hence the infinite contact force is not at all an obstacle to the integration. Another situation (different from the first one) is that the trajectories may attain a zone in the phase space, where the linear complementarity problem (LCP) that gives the contact force, has no solution. Then the solution (i.e. the angular velocity of the rod) has to jump to an area where the LCP has a solution. This creates indeed a sort of "impact" with velocity discontinuity. Interested readers may also have a look at Section 5.5 in Brogliato's book[5] and at figure 5.23 therein where the various important areas of the dynamics are depicted.

It is noteworthy that J. J. Moreau has shown in his seminal paper[6] through numerical simulation with his time-stepping scheme (afterwards called Moreau's scheme) that Painlevé paradoxes can be simulated with suitable time-stepping methods, for the above reasons given later by Génot and Brogliato.

Walter Lewin drawing a dotted line with chalk, showing the bouncing effect

Since mechanics is above all an experimental science, it is of utmost importance that experiments validate the theory. The classical chalk example is often cited (when forced to slide on a black board, a chalk has the tendency to bounce on the board). Since the Painlevé paradoxes are based on a mechanical model of Coulomb friction (multivalued at zero tangential velocity) that is perhaps a simplified model of contact but which nevertheless encapsulates the main dynamical effects of friction (like sticking and slipping zones), it should logically possess some mechanical meaning and should not be just a mathematical fuss. Painlevé paradoxes have been experimentally evidenced several times, see for instance.[7]

References

1. ^ Paul Painlevé (1895). "Sur le lois frottement de glissemment". C. R. Acad. Sci. 121: 112–115.
2. ^ Stewart, David E. (2000). "Rigid-Body Dynamics with Friction and Impact". SIAM Review. 42 (1): 3–39. Bibcode:2000SIAMR..42....3S. doi:10.1137/S0036144599360110.
3. ^ Franck Génot, Bernard Brogliato (1999). "New results on Painlevé paradoxes" (PDF). European Journal of Mechanics A. 18 (4): 653–678. Bibcode:1999EJMS...18..653G. doi:10.1016/S0997-7538(99)00144-8.
4. ^ Vincent Acary, Bernard Brogliato (2008). Numerical Methods for Nonsmooth Dynamical Systems. Lecture Notes in Applied and Computational Mechanics. 65. Heidelberg: Springer Verlag.
5. ^ Brogliato, Bernard (2016). 3rd (ed.). Nonsmooth Mechanics. Communications and Control Engineering. London: Springer Verlag.
6. ^ Moreau, J. J . (1988). "Unilateral Contact and Dry Friction in Finite Freedom Dynamics". In Moreau, J.J.; Panagiotopoulos, P.D. (eds.). Nonsmooth Mechanics and Applications. International Centre for Mechanical Sciences (Courses and Lectures). 302. Vienna: Springer.
7. ^ Zhen, Zhao; Liu, Caishan; Ma, Wei; Chen, Bin; et al. (2008). "Experimental Investigation of the Painlevé Paradox in a Robotic System". Journal of Applied Mechanics. 75 (4): 041006. Bibcode:2008JAM....75d1006Z. CiteSeerX 10.1.1.1027.4938. doi:10.1115/1.2910825.