New results on Painleve paradoxes

In this note, we deal with the dynamics of a mechanical system subject to u nilateral constraints. In particular, we shall focus on the integration of such a system that is known to be closely related to so-called linear compl ementary problem (LCP). Except a in very simple case like codimension 1, fr ictionless constraints, the problem of well-posedness (existence and unique ness of solution) to such hybrid dynamic systems (smooth dynamics + LCP + s hock dynamics) is a big challenge. We concentrate on the well-known Painlev e example the dynamics of which in a sliding regime may be singular, depend ing on the friction coefficient. A new critical friction coefficient is pre sented below, the contact forces of which remain bounded. Moreover, a detai led analysis of the vector field near the singularity shows that the eventu al divergence of the contact forces does not call into question the well-po sedness of the model. (C) Elsevier, Paris.