The objective of this paper is to present a numerical scheme for applying nonlinear modal synthesis to study the dynamic performance of assembled structures with a frictional interface. A generalized Masing model is introduced to describe the frictional mechanism in steady state to enable obtaining the harmonic components of the friction force. By employing the concept of nonlinear modes based on normal form theory, a reduction technique based on branch modes can be applied to obtain a reduced order model. Both complex nonlinear modes and real nonlinear modes depending on the modal amplitude are used in the nonlinear modal synthesis approach, leading to a particular damping term representing energy dissipation due to friction. The implementation of the nonlinear modal synthesis combined with the generalized Masing model provides a simple, fast and efficient numerical method for describing the nonlinear performance of structures with dry friction devices. The proposed approach is applied to three increasingly complex case studies. Numerical simulations are carried out to demonstrate the efficiency and reliability of the method.