Abstract In a previous article , the authors discussed the time-discretization of those relations modeling a class of dynamical systems with friction. The main goal of this article is to address similar problems using a more sophisticated friction model giving a better description of the system behavior when the velocities are close to zero. These investigations are motivated by the need for more accurate friction models in the software simulating the motion of mechanical systems, such as the remote manipulators of the Space Shuttle or of the International Space Station. As a first step, we shall consider one degree of freedom systems. However, the methods discussed in this article can be easily generalized to higher number of degrees of freedom elasto-dynamical systems; these generalizations will be the object of another publication. The content can be summarized as follows. We first discuss several models of the constrained motions under consideration, including a rigorous formulation involving a kind of dynamical multiplier. Next, in order to treat friction, we introduce an implicit/explicit numerical scheme which is unconditionally stable, and easy to implement and generalize to more complicated systems. Indeed, the above scheme can be coupled, via operator-splitting, to schemes classically used to solve differential equations from frictionless elasto-dynamics. The above schemes are validated through numerical experiments.