# Nonlinear Dynamics Analysis of Disc Brake Frictional Vibration
## 3 Dynamic Analysis of Disc Brakes
### Effect of Brake Pressure on the Nonlinear Characteristics of the System  
In this paper, by numerically solving the system of Equation (1) with the fourth-order Runge-Kutta method, we first obtain the time-domain vibration response of the system, then plot the bifurcation, Lyapunov, and phase diagrams of the system response using the multiset solution, and finally obtain the frequency-domain diagram of the system response using the fast Fourier transform method. By reviewing the relevant literature, the stiffness and damping coefficients of the disc brake system were obtained, as shown in Table 1. Because the upper and lower brake pads have the same kinematic characteristics and are the axial linear systems, there will be no stable limit ring. The following content only analyzes the tangential motion of the upper brake pads and brake discs and calls the upper brake pads brake pads. The motion bifurcation diagram of brake pads and brake discs was drawn. As shown in Figure 3, when the braking pressure is 1000 N and the braking speed is between 0 km/h and 50 km/h, the brake pads and discs produce random vibration. As shown in Figure 4, when the braking pressure is 6000 N and the braking speed is between 0 km/h and 95 km/h, the brake pads and discs exhibit strong chaotic characteristics. As the brake pressure increases, the amplitude of the brake pad and disc vibration increases significantly, and the corresponding braking speed increases significantly when the system is in chaotic motion. The results show that excessive brake pressure is one of the important factors that induce nonlinear vibration in the system. The Lyapunov exponent is a numerical value used to describe the divergence of neighbouring trajectories in space. When the Lyapunov exponent is less than 0, it indicates that the system is converging in that direction; when the Lyapunov exponent is greater than 0, it indicates that the system is diverging in that direction [18]. With the Lyapunov index, the state of motion of the system can be determined directly. The Lyapunov diagram of the whole system under different braking pressures is drawn, as shown in Figure 5. Under different braking pressures, the Liapunov exponent of the system is always greater than 0, indicating that the system is always in a chaotic motion state. When the braking pressure is 1000 N and the braking speed is between 0 km/h and 50 km/h, the Lyapunov exponent varies randomly and then tends to be stable; when the braking pressure is 6000 N and the braking speed is between 0 km/h and 80 km/h, the Lyapunov exponent varies randomly and then tends to be stable, and the random vibration characteristics of the system are enhanced. Therefore, the higher the braking pressure, the stronger the vibration characteristics of the system. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline \(k_{1}\), \(k_{2}\), \(k_{2x}\), \(k_{2y}\)/(N\(\cdot\)m\({}^{-1}\)) & \(2.5\times 10^{7}\) \\ \(k_{23}\), \(k_{12}\)/(N\(\cdot\)m\({}^{-1}\)) & \(3.9\times 10^{7}\) \\ \(c_{1}\), \(c_{3}\), \(c_{2x}\), \(c_{2y}\)/(N\(\cdot\)s\(\cdot\)m\({}^{-1}\)) & 295 \\ \(c_{23}\), \(c_{12}\)/(N\(\cdot\)m\({}^{-1}\)) & 495 \\ \(m_{1}\), \(m_{22}\)/kg & 0.135 \\ \(m_{3}\)/kg & 8.33 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Disc brake dynamics parameters. Figure 4: Bifurcation diagram \((p=6000\) N). Figure 3: Bifurcation diagram \((p=1000\) N). As shown in Figures 8 and 9, in the range of 0-500 Hz, when the brake pressure is 1000 N, the main frequency of brake pad vibration is 210 Hz, with the increase in brake pressure, the main frequency of vibration is gradually prominent, and the amplitude of vibration increases; the main frequency of brake disc vibration is 300 Hz, and as the braking pressure increases, other vibration frequencies different from the main frequency gradually stand out, the amplitude of vibration increases, and the chaotic characteristics of the system are enhanced.