# Contribution Analysis of Assembled Brake System to Reduce Squealing
## 2 Stability Analysis of Assembled Brake System Model  
The squealing issue can be addressed with a theoretical brake model using several methods such as transient analysis or quasi-stationary methods. In this study, the complex eigenvalue method was adopted to analyze squealing using MSC Nastran software. To demonstrate the stability of the assembled brake system, the complex eigenvalues were plotted in the s-plane. The imaginary part (frequency) was plotted against the real part (damping coefficient). A positive damping coefficient causes the amplitude of the oscillations to increase with time so that the system becomes unstable when the damping coefficient is negative [9; 10; 11]. Figure 1 shows the FE model of the assembled brake system for complex eigenvalue analysis. The constraints for brake assembly based on each FE model of the disk brake parts were explained in a previous article [12]. The contact stiffness between the disk and pad was derived in an additional experiment [12]: 1.0 \(\times\) 10\({}^{4}\) N/mm\(-\)5.0 \(\times\) 10\({}^{4}\) N/mm for brake pressures between 10-40 bar. The friction coefficient between the disk and pad was assumed to be 0.387, based on the measurement data from a previous study [11; 12]. Adjacent parts of the assembled brake system were used as the fixed contact between them, i.e., the knuckle-caliper hub, hub-disk, brake pad-backplate-caliper, housing-piston, and piston-backplate. The connecting arm locations of the knuckle part to the suspension linkage were all fixed for a 6 degree-of-freedom setup. The instability of the assembled brake system was caused by the negative values of the off-diagonal terms in the stiffness matrix, as shown in Figure 2. The stiffness matrix between the disk and the brake pad was generated with Direct Matrix Input at Grid Points (DMIG) supported in NASTRAN/MSC software, and matrices #1 and #2 and matrices #3 and #4 were formulated in Equations (1) and (2), respectively. The nature of the friction behavior at the brake and disk parts' contact surface was assumed to be linear with respect to the brake action between them. Previous studies have focused on using stochastic techniques to overcome the non-linear behavior observed in friction modelling [22; 23], however, such non-linear effects were not considered in this study. Here,  
If the stiffness matrix is defined as \(K(\mu)\), then accounting for the formulation in Equations (1) and (2), the governing equation of the assembled brake system is:  
\[M\ddot{X}+D\dot{X}+(K+K(\mu))X=0, \tag{3}\]  
where \(K(\mu)\) is the mass matrix, \(D\) is the damping matrix, \(K\) is the stiffness matrix without considering the contact condition between the pad and disk, and \(X\) is the displacement vector of the assembled brake system. The formula in Equation (3) was built under the linear assumption of the assembled brake system. This governing equation can be solved using complex eigenvalue analysis. No damping coefficients (\(D=0\)) were applied for all subparts of the assembled brake system, and the brake pressure was applied for four cases from 10 to 40 bar. The material properties of the main components are listed in Table 1. A complex eigenvalue analysis was conducted for four brake pressures, and the results are illustrated in Figure 3. Instability of the assembled brake system was found at two frequency points related to the squealing: 3.4 kHz under a brake pressure from 10 to 30 bar and 6.4 kHz under a 40-bar brake pressure. The contribution of the subparts according to different brake pressures can be understood according to the mode shapes of the assembled brake system, as illustrated in Figure 4. For 10-30 bar, the mode shape of the disk is remarkable, whereas the mode shape of the brake pad or caliper appears most remarkable at 40 bar. The dramatic change in the instability frequency from 3.4 kHz to 6.4 kHz was caused by the different stiffness coefficients in Equations (1) and (2).