Friction is a complex phenomenon and since its occurrence is tied to numerous other phenomena, its treatment requires a truly multidisciplinary research effort. The subject of friction and its relation to vibration poses one of the most challenging engineering problems. Friction-vibration coupling is the source of many difficulties in brake and clutch noise problems including wear of mechanical components and leakage in mechanical seals, etc. It is realized that the phenomena of noise and vibration caused by frictional contact in mechanical systems must be related to the properties of the contact. This work provides theoretical and computational investigations of the mechanical interaction of two surfaces in frictional contact. The aim of this project is to develop a better understanding of friction and vibration phenomenon in mechanical systems. This study is to demonstrate the methodology through a dynamic analysis of interaction of two disks based on those models. Another goal is to develop an enhanced dynamic finite element (FE) model with friction coupling to be applied for analyzing the design of disc brake system with squeal noise reduction. The FE model is built-up from the individual brake component representations. Its structural connections and boundary conditions are determined by correlating to a set of measured frequency response functions. The proposed friction coupling formulation produces an asymmetric system stiffness matrix that yields a set of complex conjugate eigenvalues. The analysis shows that eigenvalues possessing positive real parts tend to produce unstable modes with the propensity towards the generation of squeal noise. Using the FE model, a selected set of parametric studies is performed to evaluate different design concepts. The optimized best brake system design which produces the least amount of squeal response is attained.