Selected problems of the mechanics of systems with dry friction. (Избранные задачи механики систем с сухим трением.) (Russian) Zbl 1285.74002

Sumbatov, A. S.; Yunin, E. K.

Moskva: Fizmatlit (ISBN 978-5-9221-1444-8/hbk). 196 p. (2013).

The book has thirteen chapters and starts with the history of dry friction models from the beginning of the twentieth century. It is shown that the rigid body model and Coulomb’s law of dry friction lead to ambiguities and contradictions in some cases. Then the authors discuss the physical nature of dry friction forces and present different approaches to the explanation of paradoxes in the well-known Painlevé-Klein example, based on Russian scientific papers published during a fifteen-year period. The ideas of St. Petersburg mechanician P. A. Zhilin concerning this problem are also discussed here. The book discusses two publications of H. Beghin, the famous French mechanician and founder of the servoconstraint theory. The authors demonstrate that the unusual features of the dynamic behavior of mechanical systems do not always depend on the inadequacy of the law of dry friction. It is shown that all types of Poshekhonov’s pendulum oscillations can be explained within the framework of the classical Coulomb’s law, and no additional hypotheses are required. A graphical method named “the Lerneray constructions” is applied to study self-oscillations of systems with dry friction, and the unusual dynamics of braking is explained by means of friction. An approximate approach to the solution of flexible rod vibrations with friction is proposed. The problem of the brachistochrone plane curve with dry friction is considered in the last chapter together with the classical problem of frictionless brachistochrone, so that the reader can find several new facts and conclusions. The book is self-consistent and can be recommended for independent study of classical mechanics.

Reviewer: Irina Alexandrovna Bolgrabskaya (Donetsk)

MSC:

74-02	Research exposition (monographs, survey articles) pertaining to mechanics of deformable solids
70-02	Research exposition (monographs, survey articles) pertaining to mechanics of particles and systems
74M10	Friction in solid mechanics
74H45	Vibrations in dynamical problems in solid mechanics
70F40	Problems involving a system of particles with friction
70J35	Forced motions in linear vibration theory

Keywords:

rigid body model; Coulomb’s law; self-oscillation; brachistochrone problem