A Velocity-Based Time-Stepping Method for Frictional Dynamics

The dynamics of a mechanical system (a particle or a rigid body) submitted to one unilateral constraint possibly with friction may be formulated as a so-called measure differential inclusion. This was shown by J. J.Moreau (see e.g. [

2

]), who also introduced and used with great success several numerical timestepping methods for a variety of progressively more complex problems. A convergence proof and existence result may be found in [

1

], which concerns the case of a constant inertia matrix, equal to the identity.

Here, we extend that study, by considering general state-dependent inertia matrices as well as anisotropic Coulomb friction. A convergence result for the corresponding time-stepping scheme, velocity-based “a la Moreau”, is obtained. A thorough analysis of the limit friction law, including the situations leading to the well-known Painleve’s “paradoxes”, is also given. For the single constraint case, the present study also extends and completes the results of D. E. Stewart [

3

].