Abstract : The thesis consists of two parts presented in four chapters. The first one deals with the modelling, the simulations and the experimental validations of a beam model with a unilateral linear spring under a periodic excitation. It is a simplified mechanical model of a satellite solar array and an elastic bumper during the launch stage. The system is under a harmonic excitation given as an imposed acceleration or a punctual force. The model is validated with experimental sequences on an aluminum beam in contact with a Solithane bumper. The results show a good agreement with the numerical simulations. The second part is focused on the nonlinear normal modes of mechanical systems. A new formulation is then presented to find these modes as zeros of a nonlinear mapping. An algorithm based on the continuation of periodic solutions is performed using existent algorithms. The perturbation technique using multiple scales method for the calculation of approximate analytical solutions of a differential equation with a unilateral term is introduced. We use then this technique for the calculation of the nonlinear normal modes of n degrees of freedom autonomous system with a unilateral contact. We also deal with the case of forced systems, thus we obtain a simple procedure for the calculation of the nonlinear normal modes. All these techniques provide different validated mathematical tools for a modal analysis of the mechanical system treated in the first part of the thesis.