On the variational formulation of dynamics of systems with friction

Alexander P. Ivanov

Russian journal of nonlinear dynamics, Tome 9 (2013) no. 3, pp. 478-498.

Résumé

We discuss the basic problem of dynamics of mechanical systems with constraints-finding acceleration as a function of the phase variables. It is shown that in the case of Coulomb friction, this problem is equivalent to solving a variational inequality. The general conditions for the existence and uniqueness of solutions are obtained. A number of examples is considered. For systems with ideal constraints discussed problem has been solved by Lagrange in his “Analytical Dynamics” (1788), which became a turning point in the mathematization of mechanics. In 1829, Gauss gave his principle, which allows to obtain the solution as the minimum of a quadratic function of acceleration, called “constraint”. In 1872 Jellett gaves examples of non-uniqueness of solutions in systems with static friction, and in 1895 Painlevé showed that in the presence of friction, together with the non-uniqueness of solutions is possible. Such situations were a serious obstacle to the development of theories, mathematical models and practical use of systems with dry friction. An unexpected and beautiful promotion was work by Pozharitskii, where the author extended the principle of Gauss on the special case where the normal reaction can be determined from the dynamic equations regardless of the values of the coefficients of friction. However, for systems with Coulomb friction, where the normal reaction is a priori unknown, there are still only partial results on the existence and uniqueness of solutions. The approach proposed here is based on a combination of the Gauss principle in the form of reactions with the representation of the nonlinear algebraic system of equations for the normal reactions in the form of a variational inequality. The theory of such inequalities includes the results of existence and uniqueness, as well as the developed methods of solution.

Détail

Keywords: principle of least constraint, dry friction, Painlevé paradoxes.

Bibliographie

[1] Pozharitskii G. K., “Rasprostranenie printsipa Gaussa na sistemy s sukhim treniem”, PMM, 25:4 (1961), 391–406

[2] Matrosov V. M., Finogenko I. A., “O razreshimosti uravnenii dvizheniya mekhanicheskikh sistem s treniem skolzheniya”, PMM, 58:6 (1994), 3–13 | MR | Zbl

[3] Pang J.-S., Trinkle J. C., “Complementarity formulations and existence of solutions of dynamic multi-rigid-body contact problems with Coulomb friction”, Math. Program. Ser. A, 73:2 (1996), 199–226 | DOI | MR | Zbl

[4] Ivanov A. P., “Usloviya odnoznachnoi razreshimosti uravnenii dinamiki sistem s treniem”, PMM, 72:4 (2008), 531–546 | MR | Zbl

[5] Kinderlehrer D., Stampacchia G., An introduction to variational inequalities and their applications, Pure Appl. Math., 88, Academic Press, New York, 1980, 313 pp. | MR | MR | Zbl

[6] Rumyantsev V. V., “O printsipakh Gaussa i Chetaeva dlya sistem s neidealnymi svyazyami”, Teoret. i prikl. mekhan., 5:1 (1974), 9–14

[7] Painlevé P., Leçons sur le frottement, Hermann, Paris, 1895, 111 pp.

[8] Rockafellar R. T., Convex analysis, Princeton Univ. Press, Princeton, 1970, 451 pp. | MR

[9] Pfeiffer F., Glocker Ch., Multibody dynamics with unilateral contacts, Wiley, New York, 1996, 317 pp. | MR | Zbl

[10] Zhukovskii N. E., “Usloviya ravnovesiya tverdogo tela, opirayuschegosya na nepodvizhnuyu ploskost nekotoroi ploschadkoi i moguschego peremeschatsya vdol etoi ploskosti s treniem”: N. E. Zhukovskii, Sobr. soch., v. 1, Gostekhizdat, M., 1948, 339–354

[11] Chernousko F. L., Smyshlyaev A. S., “Usloviya ravnovesiya sterzhnya na sherokhovatoi ploskosti”, PMM, 66:2 (2002), 177–182 | MR

[12] Kozlov V. V., “Printsip Gaussa i realizatsiya svyazei”, Nelineinaya dinamika, 4:3 (2008), 281–285 | MR

[13] Ivanov A. P., “Bifurkatsii v sistemakh s treniem: osnovnye modeli i metody”, Nelineinaya dinamika, 5:4 (2009), 479–498

[14] Ivanov A. P., “Issledovanie razryvnykh bifurkatsii v negladkikh dinamicheskikh sistemakh”, Nelineinaya dinamika, 8:2 (2012), 231–247