# Parameter Determination of a Minimal Model for Brake Squeal
## 2 Materials and Methods  
### Finite Element Analysis of Brake Squeal  
As shown in Figure 1, a simple finite element brake model with a disc, two pads and two back plates is established in ABAQUS. Table 1 shows the material properties of the three brake components [26], where the brake pads are made of orthotropic material, and its nine elastic constants are given. The back plates are constrained using YASYMM boundary condition, namely, constraining the X, Z direction of the displacement and rotation around the Y axis, and the bolt holes on the disc are constrained with PINNED boundary condition, namely, constraining the X, Y, Z direction of the displacement. A uniform pressure of 1 MPa is applied to the back plates to simulate the actual braking hydraulic load.  
The complex modal analysis of the model is carried out, and the rotation effect is applied to the brake disc in the input file. Finally, eigenvalues of the first 55 complex modes of the model are extracted. Figure 2 shows the frequencies and real parts of the eigenvalues of unstable modes.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline  & **Disc** & **Back Plate** & \multicolumn{6}{c}{**Pad**} \\ \hline  & & & D1111 & 5940 & D1133 & 980 & D1212 & 2590 \\ Young’s Modulus (MPa) & 135,000 & 182,000 & D1122 & 760 & D2233 & 980 & D1313 & 1180 \\  & & & D2222 & 5940 & D3333 & 2270 & D2323 & 1180 \\ \hline Density (t/mm\({}^{3}\)) & 7.2\(\times 10^{-9}\) & 7.8\(\times 10^{-9}\) & \multicolumn{6}{c}{2.51\(\times 10^{-9}\)} \\ \hline Poisson’s ratio & 0.27 & 0.3 & \multicolumn{6}{c}{} \\ \hline \end{tabular}
\end{table}
Table 1: Material properties of brake components.  
Figure 1: Finite element model of brake.  
Figure 2: Unstable modes of the brake.  
It can be seen from Figure 2 that the brake system has several unstable modes which may lead to brake squeal. With the increase of the friction coefficient, the number of unstable modes increases; therefore, the propensity of brake squeal increases. For the friction coefficient \(\mu=0.9\), 8 unstable modes are predicted in the system. Their frequencies are 4060.2 Hz, 5525.5 Hz, 7213.0 Hz, 8429.0 Hz, 9042.1 Hz, 11,056 Hz, 11,198 Hz and 11,496 Hz respectively.  
For instance, the unstable mode with seven node diameters is shown in Figure 3. Figure 4a shows the evolution of real parts of the mode with respect to the friction coefficient from 0.2 to 0.9, namely the Hopf bifurcation diagram. The bifurcation point of this mode is observed for \(\mu_{0}=0.4\). For friction coefficients greater than 0.4, the real parts of this mode are positive, which signifies an unstable mode. Moreover, the maximum real part value is 713.41 for \(\mu=0.9\). Figure 4b illustrates the evolution of frequencies of this mode versus the friction coefficient. With the increase of friction coefficient, two modal frequencies gradually get closer until coalescence occurs. The coalescence frequency at the bifurcation point is 7168.6 Hz. In Figure 4c, a three-dimensional graph including real parts and frequencies of mode coupling is given, where the red line represents the unstable mode, and the blue line represents the stable mode.  
Figure 4: _Cont._  
Figure 3: Modal shape of 7168.6 Hz.  
Then, CEA results of the unstable mode with seven modal diameters will be referred to as a standard in order to establish a minimal model with two DoFs for brake squeal.