Mechanisms of friction are known as an important source of vibrations in a large variety of engineering systems, where the emergence of oscillations is noisy and can cause severe damage to the system. The reduction or elimination of these vibrations is then an industrial issue that requires the attention of engineers and researchers together. Friction-induced vibrations have been the matter of several investigations, considering experimental, analytical, and numerical approaches. An aircraft braking system is a complex engineering system prone to friction-induced vibrations, and is the subject herein. By considering experimental observations and by evaluating the mechanisms of friction involved, a complete nonlinear model is built. The nonlinear contact between the rotors and the stators is considered. The stability analysis is performed by determining the eigenvalues of the linearized system at the equilibrium point. Parametric studies are conducted in order to evaluate the effects of various system parameters on stability. Special attention will be given to the understanding the role of damping and the associated destabilization paradox in mode-coupling instabilities.