Dynamics of mechanical systems with multiple sliding contacts: new faces of Painlev\'e's paradox

We investigate the dynamics of finite degree-of-freedom, planar mechanical systems with multiple sliding, unilateral frictional point contacts. A complete classification of systems with 2 sliding contacts is given. The contact-mode based approach of rigid body mechanics is combined with linear stability analysis using a compliant contact model to determine the feasibility and the stability of every possible contact mode in each class. Special forms of non-stationary contact dynamics including "impact without collision" and "reverse chattering" are also investigated. Many types of solution inconsistency and the indeterminacy are identified and new phenomena related to Painlev\'e"s non-existence and non-uniqueness paradoxes are discovered. Among others, we show that the non-existence paradox is not fully resolvable by considering impulsive contact forces. These results contribute to a growing body of evidence that rigid body mechanics cannot be developed into a complete and self-consistent theory in the presence of contacts and friction.