NDLI: On the variational formulation of the dynamics of systems with friction

Content Provider	Springer Nature Link
Author	Ivav, Alexander P.
Copyright Year	2014
Abstract	We discuss the basic problem of the dynamics of mechanical systems with constraints, namely, the problem of finding accelerations 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 are considered.For systems with ideal constraints the problem under discussion was 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 one to obtain the solution as the minimum of a quadratic function of acceleration, called the constraint. In 1872 Jellett gave examples of non-uniqueness of solutions in systems with static friction, and in 1895 Painlevé showed that in the presence of friction, the absence of solutions is possible along with the nonuniqueness. Such situations were a serious obstacle to the development of theories, mathematical models and the practical use of systems with dry friction. An elegant, and unexpected, advance can be found in the work [1] by Pozharitskii, where the author extended the Gauss principle to 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 [2–4].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 [5] includes results on the existence and uniqueness, as well as the developed methods of solution.
Starting Page	100
Ending Page	115
Page Count	16
File Format	PDF
ISSN	15603547
Journal	Regular and Chaotic Dynamics
Volume Number	19
Issue Number	1
e-ISSN	14684845
Language	English
Publisher	Pleiades Publishing
Publisher Date	2014-02-09
Publisher Place	Moscow
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	principle of least constraint dry friction Painlevé paradoxes Dynamical Systems and Ergodic Theory
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Statistical and Nonlinear Physics Mathematical Physics Mechanical Engineering Modeling and Simulation

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