# Influence of Material-Dependent Damping on Brake Squeal in a Specific Disc Brake System
## 3 Pad-on-Disc System Description
### Mathematical Description of Pad-on-Disc System
Loading of the system was achieved by the boundary conditions defined in Figure 8. The bottom area of the hub was fixed, while on the square yellow areas of the thin plate, displacement of \(8\times 10^{-2}\) mm, corresponding to the reaction force of 200 N, was applied. After the equilibrium position was achieved, the pseudo rotation of the disc using the ANSYS CMROTATE command was applied in a further step of the analysis. This command sets the constant rotational velocity on the nodes of the disc element component, despite any deformation at the nodes. This feature is primarily used for generating sliding contact at frictional contact interfaces in brake-squeal analysis. Due to the sliding contact, the asymmetric coefficient matrices, in the linear differential equations set (10), are obtained. \[\mathbf{M\ddot{q}}+[\mathbf{C}+\mathbf{C_{ns}}(\mu)]\mathbf{\dot{q}}+[\mathbf{ K}+\mathbf{K_{ns}}(\mu)]\mathbf{q}=\mathbf{0} \tag{10}\]  
The terms \(\mathbf{K_{ns}}(\mu)\mathbf{q}\) and \(\mathbf{C_{ns}}(\mu)\mathbf{\dot{q}}\) represent the nonconservative friction forces. The matrices \(\mathbf{K_{ns}}\) and \(\mathbf{C_{ns}}\) are nonsymmetric and smoothly depend on friction parameter \(\mu\). Symmetric positively definite matrices \(\mathbf{C}\) with respect to \(\mathbf{K}\) represent a system structural damping with respect to stiffness. The matrix \(\mathbf{C}\) is modeled nonproportionally to the \(\mathbf{K}\) and \(\mathbf{M}\) matrices due to the definition of different material damping for FM and the rest of the assembly. Gyroscopic effects were neglected in this study. For further analyses of the system stability, complex eigenvalues \(\lambda_{i}\) and corresponding complex displacement eigenvectors \(\psi_{\mathbf{D}i}\) of the original system (10) were obtained by solving the eigenvalue problem (11) and transformation of the complex eigenvector \(\ddot{\psi}_{i}\) from modal into generalized coordinates (13)  
\[(\mathbf{A}-\lambda_{i}\mathbf{I})\ddot{\psi}_{i}=\mathbf{0}, \tag{11}\]  
\[\mathbf{A}= \begin{bmatrix}\mathbf{0}&\mathbf{I}\\ -\mathbf{\Omega}^{\top}\mathbf{K_{ns}}\mathbf{\Phi}&-\mathbf{\Phi}^{\top}( \mathbf{C}+\mathbf{C_{ns}})\mathbf{\Phi}\end{bmatrix} , \tag{12}\]  
\[\psi_{\mathbf{D}i}=\mathbf{\Phi}\mathbf{T}\ddot{\psi}_{i^{\prime}}\mathbf{T}= [\mathbf{I},\mathbf{0}]. \tag{13}\] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{6}{l}{**Friction Material**} \\ \hline \(T\) (°C) & \(\rho\) (kg/m\({}^{3}\)) & \(E_{x}\) (Pa) & \(E_{y}\) (Pa) & \(E_{z}\) (Pa) & \(v_{xy}\) (\(-\)) & \(v_{yz}\) (\(-\)) \\ \hline
23 & 2930 & \(8.7\times 10^{9}\) & \(8.7\times 10^{9}\) & \(2.5\times 10^{9}\) & 0.24 & 0.1 \\
100 & 2930 & \(5.9\times 10^{9}\) & \(5.9\times 10^{9}\) & \(1.7\times 10^{9}\) & 0.24 & 0.1 \\ \hline  & \(v_{xz}\) (\(-\)) & \(G_{xy}\) (Pa) & \(G_{yz}\) (Pa) & \(G_{zz}\) (Pa) & \(\bar{\zeta}_{p}\) (\%) & \\ \hline
23 & 0.35 & \(3.5\times 10^{9}\) & \(1.6\times 10^{9}\) & \(1.6\times 10^{9}\) & 0–10 (5) * & \\
100 & 0.35 & \(2.4\times 10^{9}\) & \(1.1\times 10^{9}\) & \(1.1\times 10^{9}\) & 0–10 (5) * & \\ \hline \multicolumn{6}{l}{**Disc**} \\ \hline \(T\) (°C) & \(\rho\) (kg/m\({}^{3}\)) & \(E\) (Pa) & \(v\) (\(-\)) & \(\bar{\zeta}_{d}\) (\%) & \\ \hline
23 & 7683 & \(1.9\times 10^{11}\) & 0.25 & 0–0.8 (0.4) * & \\ \hline \multicolumn{6}{l}{**Thin Plate, Backing Plate, Hub**} \\ \hline \(T\) (°C) & \(\rho\) (kg/m\({}^{3}\)) & \(E\) (Pa) & \(v\) (\(-\)) & \(\bar{\zeta}\) (\%) & \\ \hline
23 & 7800 & \(2.1\times 10^{11}\) & 0.3 & 0.4 & \\ \hline \hline \end{tabular}  
* default value.  
\end{table} 
Table 3: Material properties used in the complex eigenvalues calculation. The modal \(\mathbf{\Phi}=\{\phi_{i}\}\) and the spectral \(\mathbf{\Omega}^{2}=diag\{\omega_{i}^{2}\}\) matrixes were obtained by solving the generalized eigenvalue problem  
\[(\mathbf{K}-\omega_{i}^{2}\mathbf{M})\phi_{i}=\mathbf{0}. \tag{14}\]