# Nonlinear Dynamics Analysis of Disc Brake Frictional Vibration
## 1 Introduction  
The automotive braking process is a very complex tribological phenomenon. Disc brakes are widely used in many vehicles because of their superior performance. The braking force of the disc brake system comes from the friction between the brake disc and the brake pad. It is due to the nonlinearity of the contact stress at the contact interface and the nonlinearity of the relative motion speed of the friction pair. The frictional force excites the nonlinear vibration of the brake system, which not only affects the ride comfort of the vehicle, but also reduces the efficiency of the brake system and causes other failures. Therefore, the dynamics analysis based on reducing the frictional vibration characteristics of disc brakes is of great importance. Many scholars have studied the nonlinear dynamics of the system caused by automotive braking friction. [PERSON] [1, 2] studied transient and steady-state nonlinear self-excited vibrations and pointed out that linear stability analysis conditions are not valid in determining transient and steady-state oscillations, and that complex eigenvalue analysis does not predict such instabilities. [PERSON] [3] conducted braking experiments on a typical friction block of a high-speed train. The friction and wear, interface temperature, vibration, and noise generated at the braking interface were studied, and the interrelationship between the vibration response and the interface contact behavior was analyzed. [PERSON] [4] found that the frictional force generated by the brake depends on the frictional characteristics between the brake disc and the brake pad and is influenced by the relative velocity, temperature, and normal pressure of the contact surfaces. [PERSON] [5] found that the impulsive excitation caused by the physical characteristics of the frictional contact is an important cause of the strong nonlinear vibrations in braking systems, axles, and chassis components. [PERSON] [6] used the image-only model to describe the three main motion mechanisms of adhesion-slip, sliding, and modal coupling of the friction pair during the braking of a car, pointing out that the high-frequency white originates from the modal coupling of the friction pair, thetwittering sound is caused by adhesion-slip, and both modal coupling and adhesion-slip cause the squeak. By applying a normal-loaded mass-damped-spring oscillator to a thin circular disc rotating at an equal speed, [PERSON] [7] found that when the rotating disc speed is low to a certain critical speed, the friction pad (mass block) and the brake disc (rotating disc) experience frequent separation-engagement alternations, when the system is destabilized and rich nonlinear dynamic characteristics appear. Others have developed dynamics models with different degrees of freedom for the problem of disc brake braking friction. [PERSON] [8] considered a 6-degree-of-freedom brake system dynamics model using surface spring contact between the brake disc and brake pad, derived the corresponding kinetic differential equations, and constructed a Simulink model. The results show that the initial brake fluid pressure of the braking system has little effect on the magnitude of fluctuations in the braking torque, where as the proposed dynamics model has high accuracy in predicting braking vibrations. [PERSON] [9] established a 6-degree-of-freedom nonlinear dynamics model for the disc brake vibration excitation problem and explored the effects of braking speed, braking pressure, and support stiffness on the vibration characteristics of the system. [PERSON] [10] analyzed a 4-degree-of-freedom torsional dynamics model of a disc brake system and found that the adhesion-slip discontinuity between the brake disc and the friction pad is the source of brake creep and gibberish. [PERSON] [11] established a nonlinear torsional vibration model with four degrees of freedom and investigated its Lyapunov exponent characteristics and Hopf bifurcation properties.