A simple nonlinear system that generates synchronization is developed to clarify the mechanism behind the phenomenon. The present system consists of two nonlinear self-excited oscillators subjected to Coulomb friction, resulting in the generation of stick-slip motion. These oscillators are directly coupled in series by a coil spring and a dashpot. The synchronization generated in this system is studied both analytically and experimentally. The validity of the model and the numerical method based on the shooting method are verified by comparing the numerical and experimental results. As a result, it is found that two types of synchronized solutions exist in this system, and the vibratory patterns of the synchronized solutions are closely related to the undamped free vibration characteristics of the model without Coulomb friction. In addition, differences between the occurrence mechanisms of the synchronized solutions are analytically confirmed by examining the energy transmission between the two oscillators through the coupling element. It is also proved that unstable regions caused by internal resonance exist in the solution branch, in which the two oscillators vibrate nearly in phase.