A Semi-Implicit Time-Stepping Model For Frictional Compliant Contact Problems

In this paper, we formulate a semi-implicit time-stepping model for multibody mechanical systems with frictional, distributed compliant contacts. Employing a polyhedral pyramid model for the friction law and a distributed, linear, viscoelastic model for the contact, we obtain mixed linear complementarity formulations for the discrete-time, compliant contact problem. We establish the existence and finite multiplicity of solutions, demonstrating that such solutions can be computed by Lemke’s algorithm. In addition, we obtain limiting results of the model as the contact stiffness tends to infinity. The limit analysis elucidates the convergence of the dynamic models with compliance to the corresponding dynamic models with rigid contacts within the computational time-stepping framework. Finally, we report numerical simulation results with an example of a planar mechanical system with a frictional contact that is modeled using a distributed, linear viscoelastic model and Coulomb’s frictional law, verifying empirically that the solution trajectories converge to those obtained by the more traditional rigid-body dynamic model.