# Parameter Determination of a Minimal Model for Brake Squeal
## 2 Materials and Methods
### 2-DoF Minimal Model for Brake Squeal  
Referring to [25], a 2-DoF mechanical model is established in Figure 4. In the model, two pads are symmetrically arranged on two sides of the brake disc and have the same motion. \(m_{1}\) denotes the generalized mass of the two pads, and it has a normal mode in \(Y\) direction. \(m_{2}\) denotes the generalized mass of the disc, and it has a torsional mode in \(X\) direction. To couple the normal mode with the torsional mode, an angle \(\theta\) is considered between the friction surface and horizontal direction. \(v\) is the rotational velocity of the brake disc. Friction coefficient \(\mu\) of the contact surface is assumed to be a constant. \(F_{\text{brake}}\) is the braking force applied on the pads. \(c_{1}\) and \(c_{2}\) are the damping of the two modes respectively. \(k_{1}\) and \(k_{2}\) are, respectively, the contact stiffness and torsional stiffness of the disc. Notice that the stiffness is simplified to be linear as an approximation; nevertheless, in reality it should be nonlinear [27]. The dynamic differential equation of the braking system in Figure 5 can be expressed as  
\[\left\{ \begin{array}{l}m_{1}\ddot{Y}+c_{1}(\dot{Y}-\dot{y})+k_{1}(Y-y)=-F_{ \text{brake}}\\ m_{2}\ddot{X}+c_{2}\dot{X}+k_{2}X=-N\sin\theta+T\cos\theta\\ m_{2}\ddot{y}-c_{1}(\dot{Y}-\dot{y})-k_{1}(Y-y)=N\cos\theta+T\sin\theta\end{array} \right. \tag{1}\]  
where  
\[y=X\tan\theta \tag{2}\]  
\[T=\mu N \tag{3}\]  
where \(N\) and \(T\) are the normal force and Coulomb-type sliding friction force subjected by \(m_{2}\), respectively. and \(y\) is the vertical component of \(v\). Substituting Equations (2) and (3) into Equation (1), the force \(N\) can be eliminated, so a reduction form with two-dimensional matrices of Equation (1) can be written as  
\[\mathbf{M}\ddot{\mathbf{x}}+\mathbf{C}\dot{\mathbf{x}}+\mathbf{K}\mathbf{x}= \mathbf{F} \tag{4}\]  
Figure 4: Complex eigenvalue analysis (CEA) results of the unstable mode: (**a**) the real part of eigenvalues; (**b**) the frequency of eigenvalues; (**c**) 3D plot of the coalescence patterns of the finite element mode at 7168.6 Hz. where the displacement vector is \(\mathbf{x}=\left(X,Y\right)^{\mathrm{T}}\), the mass, damping and stiffness matrices of the brake system are denoted by \(\mathbf{M}\), \(\mathbf{C}\), \(\mathbf{K}\) respectively, and \(\mathbf{F}\) is the braking force vector. Their expressions are deduced as follows:  
\[\mathbf{M}=\left[ \begin{array}{cc}m_{2}(\tan\theta^{2}+1)&0\\ 0&m_{1}\end{array} \right] \tag{5}\]  
\[\mathbf{C}=\left[ \begin{array}{cc}c_{1}(\tan\theta^{2}-\mu\tan\theta)+c_{2} (1+\mu\tan\theta)&c_{1}(\mu-\tan\theta)\\ -c_{1}\tan\theta&c_{1}\end{array} \right] \tag{6}\]  
\[\mathbf{K}=\left[ \begin{array}{cc}k_{1}(\tan\theta^{2}-\mu\tan\theta)+k_{2} (1+\mu\tan\theta)&k_{1}(\mu-\tan\theta)\\ -k_{1}\tan\theta&-k_{1}\end{array} \right] \tag{7}\]  
\[\mathbf{F}=\left[ \begin{array}{cc}0\\ -F_{\mathrm{brake}}\end{array} \right] \tag{8}\]  
Assuming a small perturbation \(\mathbf{\bar{x}}=\left(\overline{X},\overline{Y}\right)^{\mathrm{T}}\) at the equilibrium point \(\mathbf{x}_{0}=\left(X_{0},Y_{0}\right)^{\mathrm{T}}\), which satisfies the following condition:  
\[\mathbf{K}\mathbf{x}_{0}=\mathbf{F} \tag{9}\]  
Equation (4) is simplified as  
\[\mathbf{M}\mathbf{\bar{x}}+\mathbf{C}\mathbf{\bar{x}}+\mathbf{K}\mathbf{\bar{ x}}=0 \tag{10}\]  
For convenience, a vector \(\mathbf{\varphi}==\left(\mathbf{\bar{x}},\mathbf{\bar{x}}\right)^{\mathrm{T}}\) is introduced. Then Equation (10) can be transformed into state space equation:  
\[\dot{\mathbf{\varphi}}+\mathbf{A}\mathbf{\varphi}=\mathbf{0} \tag{11}\]  
where  
\[\mathbf{A}=\left[ \begin{array}{cc}\mathbf{0}&-\mathbf{I}\\ \mathbf{M}^{-1}\mathbf{K}&\mathbf{M}^{-1}\mathbf{C}\end{array} \right] \tag{12}\]  
Figure 5: Two degrees of freedom (2-DoF) model for brake squeal.