The properties of solutions of the fundamental problem of dynamics in systems with non-ideal constraints

The problem of determining the accelerations and reactions of constraints in systems with Coulomb friction is discussed. It is shown that a realistic solution of the problem is possible provided only that allowance is made for deformations in the bodies comprising the system. For this purpose, the initial system is expanded by including local deformations among the generalized coordinates. Asymptotic methods are used to divide the expanded system into a “slow” initial subsystem and a “fast” subsystem that serves to determine the reactions. Analysis of the fast subsystem is the key to understanding the dynamics on the whole. The results obtained for the number of steady solutions and their stability are invariant with respect to the viscoelastic characteristics of the contact pairs. For every stable steady solution there is a realizable motion of the initial system, but in order to select the true motion one has to know the initial deformations and their derivatives with respect to time. Along with steady solutions, the “fast” subsystem may have stable oscillatory solutions. It is proved that to the stable limit cycle of the “fast” subsystem there correspond motions of the initial system in which the reactions oscillate at a high frequency about their mean values. The Painlevé-Klein system and the problem of braking a wheel with two brake shoes are considered as examples.