Mathematical formulation of a dynamical system with dry friction subjected to external forces

We consider the response of a one-dimensional system with friction. Shaw (1986) introduced the set up of different coefficients for the static and dynamic phases (also called stick and slip phases). He constructs a step by step solution, corresponding to an harmonic forcing. In this paper, we show that the theory of variational inequalities (V.I.) provides an elegant and synthetic approach to obtain the existence and uniqueness of the solution, avoiding the step by step construction. We then apply the theory to a real structure with real data and show that the model qualitatively agrees with the real data. In our case, the forcing motion comes from dilatation, due to temperature.