# Nonlinear Dynamics Analysis of Disc Brake Frictional Vibration
## 3 Dynamic Analysis of Disc Brakes
### Effect of Initial Braking Speed on the Nonlinear Characteristics of the System  
The bifurcation of the brake pad and disc movements at different braking speeds were plotted, as shown in Figures 12 and 13. With the increase in braking speed, the amplitude of brake pad vibration increases, and the chaotic region in the figure increases; the vibration characteristics of the brake disc are similar to those of the brake pad, and the system vibration characteristics are enhanced. The Lyapunov diagram of the system at different braking speeds was plotted, as shown in Figure 14. At different braking speeds, the maximum Lyapunov exponent of the system is greater than 0 and the system is in a chaotic state of motion.  
Figure 11: Phase plane diagram \((p=6000\,\mathrm{N})\).  
Figure 12: Bifurcation diagram (v = 10 km/h).  
Figure 10: Phase plane diagram \((p=1000\,\mathrm{N})\).  
The time domain diagrams of the brake pad and disc movements at different braking speeds were plotted for a selected braking pressure of 4000 N, as shown in Figures 15 and 16. When the braking speed is 10 km/h, the brake pad and disc appear to vibrate randomly, and with the increase in time, both eventually reach a steady state of motion; when the braking speed is 60 km/h, the brake pad and disc gradually change from an unstable state at first to a stable state. As the braking speed increases, the time for the system to reach the stable state becomes shorter and shorter. However, the amplitude of both brake pad and brake disc showed a tendency to increase with speed, and the vibration characteristics of the system were enhanced and the vibration was more violent.  
Figure 14: Lyapunov diagram of the braking system at different braking speeds.  
Figure 13: Bifurcation diagram (v = 60 km/h).  
The frequency spectrum diagrams of the brake pad and disc movements at different braking speeds were plotted for a selected braking pressure of 4000 N, as shown in Figures 17 and 18. When the initial braking speed is 10 km/h, the brake pad and brake disc produces different vibrations in the range of 0-500 Hz; when the initial braking speed is 60 km/h, the vibration frequencies of both gradually concentrate to the main frequency, and the main vibration frequency gradually stands out and the amplitude gradually increases, and the random vibration characteristics of the system increase. This is because as the initial braking speed gradually increases, the stick-slip vibration cycle of the system shortens, and the slope of the brake pad displacement growth during the sticking phase becomes greater. When the speed increases to a certain level, the viscous slip motion gradually evolves into a pure slip motion, and the vibrations that are different from the main frequency disappear.  
The braking pressure was chosen to be 4000 N, and the phase plane diagrams were plotted in turn for initial braking speeds of 10 km/h and 60 km/h. As shown in Figures 19 and 20, when the braking speed is 10 km/h, the phase trajectory line is confused, and the brake pad and brake disc display obvious stick-slip phenomena; when the braking speed is 60 km/h, the brake pad and disc are already in the stable movement stage. With the increase in braking speed, the phase trajectory gradually becomes orderly and neat, and the system is in a relatively stable state of motion; at the same time, the number of turns of the phase trajectory line gradually decreases, indicating that the time for the system to reach stable motion decreases. The results show that increasing the braking speed increases the amplitude of the system vibration, but shortens the time for the system to enter the steady state.  
Figure 16: Time domain diagram (v = 60 km/h).  
Figure 17: Frequency spectrum diagram (v = 10 km/h).