# Influence of Material-Dependent Damping on Brake Squeal in a Specific Disc Brake System
## 2 Mathematical Description of the Disc Brake Minimal Model  
### Two Degrees of Freedom (DoF) Disc Brake Model  
The purpose of the minimal model, as well as the complex model (discussed below) was to demonstrate the basic principle of dissipation-induced instability and emphasize the need for damping influence analysis on squeal occurrence, regardless of the complexity of the model. The 2 DoF disc brake model with a wobbling disc is considered in this section. The model depicted in Figure 1 was introduced and analyzed by [PERSON] [26]. The system consists of a rigid disc rotating with angular velocity \(\Omega\), which can conduct 2 independent rotations corresponding to the general coordinates \(q_{1}\) and \(q_{2}\). Two pins, attached to the ground by a spring element with stiffness coefficient of the pad \(k\), represent the brake pads. The pins are perpendicular to the \(x_{1}\), \(x_{2}\) plane, influencing only the \(q_{1}\) coordinate because the pin contacts lay on the \(x_{2}\) axis, and they are in point frictional contact with the rigid disc. The difference from the original [PERSON]'s model is the damping matrix \(\mathbf{C}\) definition used to control the proportionality of system damping, allowing us to perform dissipation-induced instability analysis. In addition, the gyroscopic effect was neglected. [PERSON]'s law was employed as friction model for pin-disc contact. The equation of motion of the model shown in Figure 1 is:  
\[\mathbf{M\ddot{q}}+\mathbf{C\dot{q}}+\mathbf{Kq}=\mathbf{0}, \tag{1}\]  
where  
\[\mathbf{M}= \begin{bmatrix}I&0\\ 0&I\end{bmatrix} \mathbf{C}=\beta\mathbf{K} \begin{bmatrix}b_{q_{1}}&0\\ 0&b_{q_{2}}\end{bmatrix} \mathbf{K}= \begin{bmatrix}k_{t}+2 kr^{2}+N_{0}h&\frac{ 1}{2}\mu N_{0}\frac{b^{2}}{r}\\ -\mu(khr+2N_{0}r)&k_{t}+(1+\mu^{2})N_{0}h\end{bmatrix} \mathbf{q}= \begin{bmatrix} q_{1}\\ q_{2}\end{bmatrix} . \tag{2}\]  
The parameter \(I\) represents the inertia momentum of the rigid disc to the axis \(x_{1}\), \(x_{2}\); \(\mu\) is the friction coefficient; \(h\) is the thickness of the disc; \(r\) is the radius of the disc; the distance of the pin-disc contact point to the center, \(N_{0}\), is a prestress applied by pads on the disc; \(k_{t}\) is the stiffness coefficient of a disc bearing; and \(k\) is the stiffness coefficient of a pad. The stiffness matrix \(\mathbf{K}\) in Equation (1) can by split into the symmetric matrix \(\mathbf{K_{s}}\) and skew-symmetric matrix \(\mathbf{K_{ss}}\), as introduced in [19]  
\[\mathbf{K_{s}}=\mathbf{K_{disc}}+\mathbf{K_{pin}}\ +\mathbf{K_{ friction}} \tag{3}\]  
\[\mathbf{K_{s}}= \begin{bmatrix}k_{t}&0\\ 0&k_{t}\end{bmatrix} + \begin{bmatrix}2 kr^{2}+N_{0}h&0\\ 0&N_{0}h\end{bmatrix} + \begin{bmatrix}0&\frac{1}{4}\mu N_{0}\frac{b^{2}}{r}- \frac{1}{2}\mu(khr+2N_{0}h)\\ \frac{1}{4}\mu N_{0}\frac{b^{2}}{r}-\frac{1}{2}\mu(khr+2N_{0}h)&\mu^{2}hN_{0} \end{bmatrix} . \tag{4}\]  
Figure 1: Minimal model of rigid disc brake including friction force between the pads and a wobbling disc. \[\mathbf{K_{ss}}= \begin{bmatrix}0&\frac{1}{4}\mu N_{0}\frac{\mathrm{i}^{2}}{r}+ \frac{1}{2}\mu(klhr+2N_{0}h)\\ -\frac{1}{4}\mu N_{0}\frac{\mathrm{i}^{2}}{r}-\frac{1}{2}\mu(klhr+2N_{0}h)&0 \end{bmatrix} , \tag{5}\]  
where \(\mathbf{K_{disc}}\) signifies the disc bearing's stiffness contribution, \(\mathbf{K_{pin}}\) is the pin stiffness contribution, and \(\mathbf{K_{friction}}\) is the contribution of friction to the symmetric stiffness matrix \(\mathbf{K_{s}}\). Equation (4) represents a simple expansion of the matrices in Equation (3). The stiffness proportional damping model (for \(b_{q_{1}}=b_{q_{2}}\)) is used to define the damping matrix \(\mathbf{C}\), where \(\beta\) represents the stiffness damping coefficient corresponding to the stiffness matrix \(\mathbf{K}\); and \(b_{q_{1}},b_{q_{2}}\) are general dimensionless proportionality damping factors, which are used to bring the nonproportionality of the damping matrix into the system.