On the Use of Continuous Sigmoid Functions to Smooth Out the Coulomb Friction Force Law for Dynamical Systems

Abstract Coulomb friction force law is an empirical formulation which was derived by analyzing the relative motion between two bodies initially in a stationary state. It represents the friction as a constant force acting in the opposite direction of the relative tangential velocity between two sliding objects. Further, the magnitude of the friction force is equal to the product of the normal contact force and the coefficient of friction. It is known that the implementation of the Coulomb force law presents some challenges, such as the absence of an explicit formulation when the contacting bodies are in a sticking regimen, where the sliding velocity is null or near zero. This scenario can lead to numerical instability and convergence problems, which can eventually lead to inappropriate solutions from a physical point of view. This study aims at examining the use of sigmoid functions to smooth or regularize the friction force for sticking phases and transitions between sliding and sticking regimens. The set of sigmoid functions employed provide continuous and explicit representations of the friction force as function of the sliding velocity. The effectiveness of the approach utilized in this work is assessed through computational simulations of a benchmark one degree-of-freedom mass-spring system. The obtained results demonstrate that the regularization strategy using continuous sigmoid functions significantly improves the numerical stability and convergence of the dynamical systems subject to Coulomb friction with sticking and sliding regimens.