Analysis of an Oscillatory Painlevé–Klein Apparatus with a Nonholonomic Constraint

Abstract

In dynamics, both the concepts of rigid body and Coulomb's law of friction are well established, although it is known at least since Painlevé's time that they may lead to irregularities and contradictions, such as loss of uniqueness or existence of the solution of the equations of motion. The problem is still of very actual interest, since it can be of practical significance also for the industrially used rigid body codes. One of the simplest mechanical systems in which these difficulties can be well described is the Painlevé-Klein apparatus. As most other systems discussed in this context in the literature, this is a holonomic system. In the present note, we briefly examine a nonholonomic oscillatory system which is an extension of the classical Painlevé-Klein apparatus and we study its dynamics with respect to the Painlevé paradox. Both the borders of paradoxical regions and their reachability are addressed.

The Painlevé paradox in contact mechanics

“…There is less in the literature on the Painlevé paradox occurring in problems with multiple contacts. The Russian literature tends to use as the canonical model, not the CPP but the so-called Painlevé-Klein problem, see [32,52,88,24,3]. A good summary discussion of work on this problem is given in Figure 7: (Adapted from [85]).…”

On the Normal Force and Static Friction Acting on a Rolling Ball Actuated by Internal Point Masses

“…For the implementation of the nonholonomic constraint, that coefficient is taken to be very large (or even infinite). For discussion and resolution of these highly complex issues of contact dynamics, see [14,15,16,17]. Our mechanical system is substantially more complex than the one considered by Painlevé, and thus careful treatment of the dynamics' continuation through detachment is beyond the scope of this paper.…”