In a previous article  the time-discretization of those relations modeling a class of dynamical systems with friction was discussed. 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 particularly 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. In this article, we discuss the methods in the case of higher number of degrees of freedom elasto-dynamical systems, and the special case of one degree of freedom. The content can be summarized as follows: We discuss first models of the constrained motions under consideration, including a rigorous formulation involving a kind of dynamical multiplier. An iterative method allowing the computation of this multiplier will be discussed. 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.