# Dynamics of mechanical systems with two sliding contacts: new facets of Painlevé’s paradox

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## Abstract

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 indeterminacy are identified and new phenomena related to Painlevé’s non-existence and non-uniqueness paradoxes are discovered. Among other results, we show that the non-existence paradox is not fully resolvable by considering impulsive contact forces. These findings 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.

## Keywords

Contact mechanics Dry friction Painlevé paradox Contact regularization Impact without collision## Notes

### Acknowledgments

The author thanks Alan Champneys whose insights helped to significantly improve this paper. This work has been supported by the National, Research, Development and Innovation Office of Hungary under Grant 104501.

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