Based on the smoothed Stribeck friction model, a nonlinear dynamic model of a two-degree-of-freedom disc braking system was established. The stability of the equilibrium points was analyzed using the Routh-Hurwitz criterion, and the influence of different parameters on the stability of the braking system was discussed. The Hopf bifurcation point was obtained using the Hurwitz criterion, and the first Lyapunov coefficient at the bifurcation point was calculated by introducing the projection method to determine the type of Hopf bifurcation. The theoretical analysis results were verified through numerical simulations. The study shows that when the angular velocity of the brake disc is low, the system remains stable; whereas when the angular velocity is high, increasing the attenuation factor or reducing the dynamic friction coefficient can significantly improve the stability. As the braking force increases and the angular velocity decreases, the unstable region of the system also expands; moreover a reasonable design of the stiffness ratio between the brake disc and the brake pad can optimize the stability of the system. In addition, the system undergoes subcritical Hopf bifurcation under critical parameters, whereby the stability of the equilibrium point changes, an unstable limit cycle is generated, and self-excited vibration is triggered.