A simple nonlinear system generating the self-synchronization is developed in order to clarify the occurrence mechanism of the phenomena. This system consists of two nonlinear self-excited oscillators with Coulomb friction and the stick-slip motion is generated. These two oscillators are directly coupled in series by a coil-spring and a dashpot. The self-synchronized phenomena generated in this system are investigated analytically and experimentally. The validity of the analytical model and the computational method based on the shooting method are verified by comparing the computational results and the experimental results. It is clarified that two types of the self-synchronized solutions exist in this system and the vibration patterns of the self-synchronized solutions are closely related to the undamped free vibration characteristics of the model without Coulomb friction. In addition, the differences between the occurrence mechanisms of the self-synchronized solutions are analytically confirmed by examining the energy transmission between two oscillators through the coupling element. It is also proved that the unstable regions caused by the internal resonance exist in the solution branch in which two oscillators vibrate nearly in phase.