In terms of elasticity theory, a problem is solved for an annular region with dry friction conditions specified on the outer boundary. On the basis of the friction theory suggested by Ishlinskii-Kragel 'skii, the effect of the friction force and the time of contact is taken into account. It is shown that relaxational vibrations arise only if the constant velocity of the inner ring boundary is low. There is a limit value ω-1$/ of the angular speed of the inner boundary after which the outer boundary moves without stopping.