| The vibrations induced by dry-friction which accompanied with stick-slip motions exist broadly in engineering fields. So it is significant to study the dynamical systems with the characters of stick-slip in theory or practice.The brake system of automobile is typical dry friction system. The main work of this paper is investigation of a typical kind of double-mass brake system with dry-friction. It reveals the main periodical solution, bifurcation and chaos of slip and stick-slip motions by using the cocycle space and numerical method. The main content of this paper includes five aspects as below:1. The typical vibration system with dry-friction and the model of dry-friction are introduced.2. The background of Lyapunov exponent calculation methods is introduced.3. Discussion of the stability of the typical brake system.4. Numerically analyses of the brake system with the Lu-Gre friction model.5. Accounting to analyzing the system equation, get the generalized equation and transitional matrix. Then, summarize the Lyapunov exponent and discuss the result that we've gotten.6. At the end of this paper, the main investigation results are summarized and the next step of work is pointed out.
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