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Ternary Logic of Motion to Resolve Kinematic Frictional Paradoxes

Mechanical Engineering, University of Wisconsin—Milwaukee, 3200 North Cramer St., Milwaukee, WI 53211, USA
Department of Machine Design, St. Petersburg Polytechnic University, 29 Polytechnicheskaya St., 195251 St. Petersburg, Russia
Author to whom correspondence should be addressed.
Entropy 2019, 21(6), 620;
Received: 16 April 2019 / Revised: 19 June 2019 / Accepted: 21 June 2019 / Published: 24 June 2019
(This article belongs to the Special Issue Entropic Methods in Surface Science)
PDF [1977 KB, uploaded 24 June 2019]


Paradoxes of dry friction were discovered by Painlevé in 1895 and caused a controversy on whether the Coulomb–Amontons laws of dry friction are compatible with the Newtonian mechanics of the rigid bodies. Various resolutions of the paradoxes have been suggested including the abandonment of the model of rigid bodies and modifications of the law of friction. For compliant (elastic) bodies, the Painlevé paradoxes may correspond to the friction-induced instabilities. Here we investigate another possibility to resolve the paradoxes: the introduction of the three-value logic. We interpret the three states of a frictional system as either rest-motion-paradox or as rest-stable motion-unstable motion depending on whether a rigid or compliant system is investigated. We further relate the ternary logic approach with the entropic stability criteria for a frictional system and with the study of ultraslow sliding friction (intermediate between the rest and motion or between stick and slip). View Full-Text
Keywords: Painlevé paradoxes; friction; ternary logic; Łukasiewicz logic; entropic stability criteria; ultraslow friction Painlevé paradoxes; friction; ternary logic; Łukasiewicz logic; entropic stability criteria; ultraslow friction

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Nosonovsky, M.; Breki, A.D. Ternary Logic of Motion to Resolve Kinematic Frictional Paradoxes. Entropy 2019, 21, 620.

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