# Influence of Material-Dependent Damping on Brake Squeal in a Specific Disc Brake System
## 3 Pad-on-Disc System Description
### Results  
From the set of calculated eigenvalues corresponding to the region of interest 0-10 kHz and the default material properties, two unstable modes characterized by the positive real part of their eigenvalues were identified, as shown in Figure 10.  
Comparing the mode shape in Figure 9a, representing the calculated (4,0) disc-bending mode, with the calculated unstable mode shape in Figure 10a, we concluded that two (4,0) modes coalesce at the unstable eigenfrequency of 5966 Hz. This frequency is close to the measured squeal frequency from Figure 6b. Since there was no other pair of modes in the vicinity of the measured squeal frequency, the unstable mode was predicted correctly. The second calculated unstable mode resulted from the coupling of two (5,0) disc-bending modes; however, it was not experimentally justified; therefore, it was considered to be an over-prediction.  
Stability sensitivity to change in the disc damping and FM was studied further for the first experimentally verified unstable mode. The full factorial numerical experiment involving 49 simulations, evaluated for different values of material-dependent damping of  
Figure 10: Calculated unstable mode shape of the system at (**a**) 5966 Hz (experimentally verified) and (**b**) 8079 Hz (over-predicted).  
Figure 9: (**a**) Numerically obtained and (**b**) experimentally obtained mode shape corresponding to the (4,0) bending mode of the disc.  
the disc and the FM component, revealed the system's stability regions. The intervals of the used damping ratio values in this study are provided in Table 3. Figure 11a shows the maximum of all calculated real parts of eigenvalues \(\lambda\) for different damping ratios of the disc \(\zeta_{d}\) and the FM \(\zeta_{p}\). The green surface separates the stable and unstable solutions, where all solutions above this surface are unstable and all solutions below the surface are stable.  
Figure 11b shows a detailed 2D view of the obtained stability areas. Solid lines represent the boundary between stable and unstable areas. The dashed line represents a proportionally damped system, where the damping of the disc \(\zeta_{d}\) and the FM \(\zeta_{p}\) have the same values. The red dot on the graph represents the default damping for the disc and the FM. The results showed that the behavior of the system with default damping is unstable. If the system was proportionally damped, it would be stabilized at a damping ratio value of about \(\zeta_{d}=\zeta_{p}=0.2\%\); increasing it would make the system more stable, which agrees with the theory of proportional damping playing a purely stabilizing role in nonconservative systems. Based on the actual situation, the given system and the given unstable mode could be stabilized by the suitable adjustment of the damping of the FM and the disc. If the damping of the disc is maintained, the damping of the FM would have to be reduced to bring the system into a stable area. If the damping ratio of the FM is reduced below \(\zeta_{p}=1.5\%\), the system would become stable, which is contra-intuitive. If constant FM damping is considered, the disc damping would have to reach a value of almost \(\zeta_{d}=0.8\%\) for the system to be stable.