ON THE THEORY OF FRICTION SYSTEMS WITH HEREDITARY TYPE FRICTION

Abstract

The work is devoted to the study of the nonlinear dynamics of frictional vibration systems, which take into account, when developing mathematical models of vibration systems, an essentially nonlinear function that describes the work of friction forces with memory. The problem considered in the work combines two well-known Painleve? problems and the problem of studying the friction forces of hereditary friction. In both cases, we are talking about the idealization of real elastic bodies and flexible bonds with ideal absolutely hard and rigid ones. Using the well-known experimentally proven hypothesis of A.Yu. Ishlinsky and I.V. Kragelsky about the hereditary dependence of the coefficient of friction of relative rest at low speeds of movement of bodies, the dynamic system under study is transformed from autonomous to non-autonomous, in which new types of movements that have not previously been observed in the study of vibration systems appear. In particular, taking into account the Painleve? paradox entails the previously unexpected possibility of the occurrence of instability and self-oscillations, and taking into account the heredity of the friction coefficient leads to the appearance of chaotic movements of bodies. A numerical and analytical method has been developed for studying the behavior of the dynamic characteristics of friction vibration systems depending on the chosen model of the relative static friction coefficient. Based on the main parameters, the new bifurcation diagrams presented in this work demonstrate the presence of arbitrarily complex periodic regimes of body motion, including chaotic ones. It has been established that the transition to chaotic modes of motion occurs according to the well-known period doubling scenario. Analytical relations are given for point mappings of Poincare? surfaces, which made it possible to obtain succession functions illustrating arbitrarily complex modes of body motion, including chaotic ones. It is noted that such body movements were not found during the study of such vibration systems.