Simulation of spherical rigid bodies subject to friction with multiple impacts

Abstract

This work introduces a novel methodology for simulating multiple impacts between spherical rigid bodies under frictional contact, within the framework of nonsmooth contact dynamics and the finite element method for large rotations. The approach extends a previously developed frictionless impact algorithm based on Newton’s impact law (Cosimo et al., 2020) to incorporate sliding, rolling, and drilling friction effects. The core contribution lies in the sequential resolution of impact problems over vanishing time intervals, redefining the active contact set in both normal and tangential directions. Closed contacts with zero pre-impact velocity are considered inactive, enhancing numerical robustness.The method employs advanced sphere-plane and sphere-sphere contact elements (Cavalieri et al., 2021), and solves the contact problem using an augmented Lagrangian formulation. The equations of motion are integrated with the nonsmooth generalized-α time integration scheme, ensuring stable and accurate results. A billiard break scenario is used as a numerical benchmark to validate the method’s ability to handle simultaneous impacts with and without friction. Two cases are considered: one without rolling resistance and another including a rolling resistance radius of ρ = 0.005 m. Results demonstrate that the proposed method avoids interpenetration, unlike penalty-based formulations, and significantly reduces computational time from 25,000 s to 40 s. This strategy is efficient, fully automatic, and avoids the need for manual sequencing or topological analysis of impact events, making it a promising tool for complex multibody dynamics simulations.