# Parameter Determination of a Minimal Model for Brake Squeal
## 4 Discussion on Stability  
The existence of modal coupling phenomenon in squeal instability have been confirmed by extensive studies. Actually, when mode coupling occurs in the system, simultaneously the friction coefficient appears to have notable nonlinear variation. Experiments have also found the friction coefficient of different materials to be a function of the relative speed between the friction components [31; 32]. In addition, brake squeal usually occurs at relatively low speed in a real traffic jam. This all means that the relative speed between the disc and pad plays an important role in brake squeal. So the effect of variation of friction coefficient with braking velocity is discussed in the stability analysis of this section. TA is used to solve the dynamic response of the system. Compared with CEA, TA can obtain the amplitude of vibration velocity and displacement, which is conducive to observe the variation law of friction coefficient. However, the feasibility of TA using finite element method is reduced because the method is time-consuming. So the 2-DoF model is adopted for the following analysis. It is more efficient to investigate the effect of different braking velocities on the stability. On the basis of the linear model mentioned in Section 3, a negative slope friction characteristic between the brake disc and pad are introduced. The equation provided by [PERSON] [19] is  
\[\mu=\mu_{\mathrm{s}}-\mu_{\mathrm{k}}\Big{|}v-\dot{X}/\cos\theta\Big{|} \tag{23}\]  
Figure 9: Evolution of real parts and frequencies versus \(\mu\): (**a**) 3D plot of the coalescence patterns of the minimal model at 7168.6 Hz; (**b**) Comparison of the real part; (**c**) Comparison of the frequency. where \(\mu_{\rm s}\) is the static friction coefficient, \(\mu_{\rm k}\) is the negative slope coefficient which represents the descent rate of friction coefficient, \(v\) is the rotation velocity of the disc which is a constant, \(\dot{X}/\cos\theta\) is the velocity projection of \(m_{2}\) in horizontal direction. Subsisting Equation (23) into Equation (3) yields a more complex friction model as the static friction coefficient is introduced. Here a simplification is made for the friction law that the stick phase (e.g., the friction force is \(T<\mu_{\rm s}N\)) is not considered, so that nonlinearity of discontinuity does not involve in the friction model, and the system can be easily integrated by a standard ODE-solver available in MATLAB [33]. Then the time-history response of the equation is solved via the classic fourth-order Runge-Kutta algorithm. The initial conditions are set as \(X_{0}=Y_{0}=\dot{X}_{0}=\dot{Y}_{0}=0\), and \(\mu_{\rm s}=0.4625\), \(\mu_{\rm k}=0.025\), so that when the velocity of brake disc equals to 2.5 m/s, the initial dynamic friction coefficient is \(\mu=0.4\), just at the bifurcation area. The effect of brake pressure on the stability is not the interest of this paper, and \(F_{\rm brake}=100\) N in this case. Figure 10a gives the velocity response over time in the \(Y\) direction at \(v=2.5\) m/s, and Figure 10b is the response with timescale zoomed in. Since the dynamic response in the \(X\) direction is similar, it is not shown. In Figure 10b, the steady state response in the \(Y\) direction is a periodic signal with changing amplitude. The reason for this phenomenon is known from Equation (23), that during oscillation process of the system, the relative velocity \(\left|v-\dot{x}\right|\) is constantly changing. This causes the friction coefficient to change repeatedly around the bifurcation point; namely, the system is in constant conversion between stable and unstable. When the friction coefficient exceeds the bifurcation point, the system becomes unstable, so the velocity response increases. At the same time, the friction coefficient reduces below the bifurcation point as the relative velocity increases, then the vibration response decreases.