The numerical simulation and control design of many systems is hampered by the presence of dry (Coulombic) friction. In particular, the discontinuous nature of the friction law leads to computational difficulties whenever sticking occurs across the frictional interfaces. This paper uses results from Variable Structure System (VSS) theory to develop a model useful for simulating stick-slip motion in a generic system with multiple frictional interfaces. The Variable Structure framework is also used to analyze the errors incurred in the new simulation model. A second approach, motivated by the alleviation of chatter in VSS systems, uses a saturation-type nonlinearity to represent the frictional interface. The accuracy of this approach is analyzed using VSS and singular perturbation theory. The accuracy and numerical efficiency of each method is demonstrated in two numerical examples.