ON A PROBLEM IN THE THEORY OF RELAXATION OSCILLATIONS

Abstract

We consider a model of a braking device described by a differential equation relating the brake shoe rotation angle and its relative angular velocity. The dry friction torque depends on the rotation angle and angular velocity as a piecewise function, while the moment of inertia of the brake shoe device under consideration is a small quantity. From a mathematical standpoint, this equation reduces to a system of two differential equations, one of which contains a small parameter at the highest derivative, a so-called Tikhonov system. The system under consideration has a single equilibrium state, but it is unstable. It is self-excited, and relaxation self-oscillations will set in. Our goal was to provide an example of such a right-hand side of the equation of motion for which experimental phenomena are sufficiently accurately explained, and to obtain an asymptotic expansion of the solution as a function of time. To find the asymptotic expansion of an arbitrary-order solution to our problem, we used the method of constructing boundary functions. The justification of the asymptotic expansion can be carried out as in classical theory.