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Regular and Chaotic Dynamics

, Volume 13, Issue 4, pp 355–368 | Cite as

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

Nonholonomic Mechanics

Abstract

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. 1)

    In the absence of friction, the detachment conditions on stationary motions do not hold. However, if the angle θ between the symmetry axis and the vertical decreases to zero, motions close to stationary motions are necessarily accompanied by detachments.

     
  2. 2)

    The same conclusion holds for a thin disk that rolls on the support without sliding.

     
  3. 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 θ is not less than 49 degrees. For small values of θ, the contact between the body and the support does not break in a neighborhood of stationary motions.

     

Key words

unilateral constraint friction Painlevé paradoxes 

MSC2000 numbers

70E18 70E50 70G70 

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Copyright information

© MAIK Nauka 2008

Authors and Affiliations

  1. 1.A. N. Kosygin Moscow State Textile UniversityMoscowRussia

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