This paper deals with the computation of non-linear dynamic steady-state solutions of autonomous non-smooth contact systems prone to mono-instability. The addressed issue is the use of the shooting method in order to determine periodic solutions of self-excited mechanical systems subject to friction-induced vibrations. The method is tested in the case of a non-smooth contact dynamical system (non-regularized Signorini unilateral contact and Coulomb friction laws) with damping and planar friction. In order to initiate the shooting algorithm, an initial solution is calculated using an original approach combining the results of the linear stability analysis for the shapes and the period with a non-linear power balance for the amplitude. It significantly enhances the computational efficiency of the method since convergence is reached in a few iterations. Steady-state limit cycles exhibiting adhesion or separation behaviors (i.e. stick-slip or contact-separation phenomena) are in good agreement with those provided by a full time integration method. It demonstrates the potential of the proposed method to estimate the self-sustained vibrations of non-smooth contact dynamical systems for which loss of contacts and inelastic shocks occur.