On detachment conditions in the problem on the motion of a rigid body on a rough plane

A. P. Ivanov

Russian journal of nonlinear dynamics, Tome 4 (2008) no. 3, pp. 287-302.

Résumé

The classical mechanical problem about the motion of a heavy rigid body on a horizontal plane is considered within the framework of theory of systems with unilateral constraints. Under general assumptions about the character of friction, we examine the question on the possibility of detachment of the body from the plane under the action of reaction of the plane and forces of inertia. For systems with rolling, we find new scenarios of the appearing of motions with jumps and impacts. The results obtained are applied to the study of stationary motions of a disk. We have showed the following. 1) In the absence of friction, the detachment conditions on stationary motions do not hold. However, if the angle $\theta$ between the symmetry axis and the vertical decreases to zero, motions close to stationary motions are necessarily accompanied by detachments. 2) The same conclusion holds for a thin disk that rolls on the support without sliding. 3) For a disk of nonzero thickness in the absence of sliding, the detachment conditions hold on stationary motions in some domain in the space of parameters; in this case, the angle $\theta$ is not less than 49 degrees. For small values of $\theta$, the contact between the body and the support does not break in a neighborhood of stationary motions.

Détail

Keywords: unilateral constraint, Painlevé paradoxes.
Mots-clés : friction