Dry Friction and Mechanical System Motion Implicit Equations

It is shown that forces acting on the mechanical system points could depend on accelerations of the system points. Differential equation system of the mechanical system motion appears to be implicit. It is not resolved with respect to senior derivatives. Fundamental mathematical problems appear associated with possibility and uniqueness of these equations' solution with respect to the senior derivatives. Such problems are common in mechanical systems with dry sliding friction and rolling friction. Such problems are missing in the point dynamics. However, such problems are rather typical in more complex mechanical systems appearing in the study of a rigid body motion, which entire mass is concentrated in a single point, as well as in systems with one degree of freedom. Four fairly simple examples of mechanical systems are considered, and their motion is described by implicit differential motion equations. Situations could appear in these systems, when motion equations are not solvable with respect to the senior derivatives (motion equations are missing), as well as situations, when there are several solutions with respect to senior derivatives (there are several different systems of the mechanical system motion equations). At the same time, one of the fundamental principles of mechanics is not fulfilled, i.e., the principle of determinism

References

[1] Golubev Yu.F. Osnovy teoreticheskoy mekhaniki [Fundamentals of theoretical mechanics]. Moscow, MSU Publ., 2019.

[2] Kolesnikov K.S. Kurs teoreticheskoy mekhaniki [Course of theoretical mechanics]. Moscow, BMSTU Publ., 2017.

[3] Nikitin N.N. Kurs teoreticheskoy mekhaniki [Course of theoretical mechanics]. St. Petersburg, Lan’ Publ., 2021.

[4] Geronimus Ya.L. Teoreticheskaya mekhanika [Theoretical mechanics]. Moscow, Nauka Publ., 1973.

[5] Markeev A.P. Teoreticheskaya mekhanika [Theoretical mechanics]. Moscow, Izhevsk, NITs Regulyarnaya i khaoticheskaya dinamika Publ., 2007.

[6] Lapshin V.V. On principle of virtual displacements. Herald of the Bauman Moscow State Technical University, Series Natural Sciences, 2012, no. 2 (45), pp. 59--64 (in Russ.).

[7] Ivanov A.P. Dinamika sistem s mekhanicheskimi soudareniyami [System dynamics with mechanical hittings]. Moscow, Mezhdunarodnaya programma obrazovaniya Publ., 1997.

[8] Painleve P. Lecons sur le frottement. Paris, Hermann, 1895.

[9] Appell P. Traite de mecanique rationnelle. Paris, Guathier-Villars, 1902.

[10] Neymark Yu.I., Fufaev N.A. Painleve paradoxes and dynamics of a brake pad. PMM, 1995, vol. 59, iss. 3, pp. 366--375.

[11] Samsonov V.A. The dynamics of a brake shoe and "impact generated by friction". J. Appl. Math. Mech., 2005, vol. 69, iss. 6, pp. 816--824. DOI: https://doi.org/10.1016/j.jappmathmech.2005.11.002

[12] Rozenblatt G.M. On the settings of problems in dynamics of a rigid body with constraints and Painleve paradoxes. Vestnik Udmurtskogo universiteta. Matematika. Mekhanika. Komp’yuternye nauki [The Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science], 2009, no. 2, pp. 75--88 (in Russ.).

[13] Andronov V.V., Zhuravlev V.F. Sukhoe trenie v zadachakh mekhaniki [Dry friction in mechanical problems]. Moscow, Izhevsk, R&C Dynamics Publ., 2010.

[14] Ivanov A.P. Osnovy teorii sistem s treniem [Theory fundamentals of systems with friction]. Moscow, Izhevsk, NITs Regulyarnaya i khaoticheskaya dinamika Publ., IKI Publ., 2011.

[15] Zhuravlev V.Ph. The "paradox" of a brake pad. Dokl. Phys., 2017, vol. 62, pp. 271--272. DOI: https://doi.org/10.1134/S1028335817050123

[16] Zhuravlev V.F. Uncorrect problems in mechanics. Herald of the Bauman Moscow State Technical University, Series Instrument Engineering, 2017, no. 2 (113), pp. 77--85 (in Russ.). DOI: https://doi.org/10.18698/0236-3933-2017-2-77-85

[17] Matrosov V.M., Figonenko I.A. On solvability of equation for mechanical system motion with sliding friction. J. Appl. Math. Mech., 1994, vol. 58, no. 6, pp. 3--13 (in Russ.).

[18] Ivanov A.P. The conditions for the unique solvability of the equations of the dynamics of systems with friction. J. Appl. Math. Mech., 2008, vol. 72, iss. 4, pp. 372--382. DOI: https://doi.org/10.1016/j.jappmathmech.2008.08.016