Squeal and chatter phenomena are often generated in bicycle disc brakes. The squeal is in-plane unstable vibration along the disc surface caused by dry friction with negative slope with respect to the relative velocity. The squeal is generated in the vibrating system including the brake unit, spokes and hub. According to experiments, the chatter is generated in a certain limited range at high disc temperatures, and is nonlinear frictional vibration in which the out-of-plane unstable vibration of the disc due to Coulomb friction and the squeal are combined through the internal resonance relationship between the out-of-plane and in-plane vibrations caused by the increase in disc temperature during braking. First, we propose a simple nonlinear system with three degrees of freedom which possesses the essential vibration factors included in a bicycle disc brake system to investigate whether or not the coupled nonlinear vibration due to the internal resonance occurs. Then, we numerically confirm the mechanism of the chatter generated in a bicycle disc brake by nonlinear analysis.