# 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  
Since experimental analyses of structural damping impact on squeal are complicated and beyond the scope of this study, numerical calculations of brake squeal were adopted. The finite element method (FEM) was used to define a mathematical model of a given  
Figure 5: (**a**) Averaged FRF of the free disc with highlighted mode shapes and (**b**) comparison of FRF of the attached disc with different pad loading variations in the 5800–6200 Hz region. Figure 6: (**a**) Sensor positions for brake squeal measurement and (**b**) single-side amplitude spectrum of the measured squeal signal [27]. physical problem. FEM discretization results in a system of equations of motion with constant coefficient matrices. System stability can be studied using a solution to the general eigenvalue problem in the same way as for the simple 2 DoF system above. The geometry of the system is shown in Figure 7a and its FEM representation in Figure 7b. All calculations were realized in ANSYS software. Mesh convergency analysis was performed for several mesh density configurations. The model counting 184,255 nodes and 72,520 elements to solve 545,466 equations was chosen. In comparison with a significantly more complex model counting 580,754 nodes, 146,351 elements to solve 1,729,380 equations, the model exhibits a difference of 0.1% in the imaginary part of the calculated eigenvalue (damped eigenfrequency) of the (4,0) mode and 4.8% difference in the calculated real part of a given eigenvalue (stability), with 8.4 times less CPU time needed for the solution. The FE model was defined and its modal properties were tuned following the experimental modal analyses (EMAs) on a physical model. The numerical value of the damping ratio \(\zeta_{i}=0.4\%\) of the disc composed of structural steal was determined from FRF measurements for a couple of potentially unstable (4,0) modes. In ANSYS FEM code, the material-dependent damping can be modeled directly through the damping ratio or can be represented by Rayleigh damping factors, the latter of which we applied. Rayleigh damping factors \(\alpha\) and \(\beta\) applied into the FE material model were determined from the equations  
\[\zeta=\frac{1}{2}\Big{(}\frac{\alpha}{\omega}+\beta\omega\Big{)}, \tag{8}\]  
\[\Big{(}\frac{d\zeta}{d\omega}\Big{)}_{\omega=\omega_{2}=\omega_{1+1}}=0. \tag{9}\]  
Under the assumption that the single components are proportionally damped and their damping ratios \(\zeta_{i}\) of the \(i\)th and \((i+1)\)th modes are known, for a couple of two modes and their corresponding eigenfrequencies, two equations of two unknown factors \(\alpha\) and \(\beta\) can be obtained from (8). In the case of a symmetric disc structure, the studied couple of eigenfrequencies, related to the unstable (4,0) mode shape, are assumed to be \(\omega=\omega_{i}=\omega_{i+1}\) and their damping ratios \(\zeta_{i}=\zeta_{i}=\zeta_{i+1}\). In this case, unknown coefficients are calculated satisfying Equations (8) and (9) as \(\alpha=\zeta_{i}\omega_{i}\) and \(\beta=\zeta_{i}/\omega_{i}\). The damping ratio value for nonabsestos organic-type FM containing novolac resin as a binder and various ingredients including metallic fillers can vary in composition from 2%  
Figure 7: (**a**) Geometry of the simplified pad-on-disc brake model, (**b**) FEM representation of the model counting 75,520 elements and 184,225 nodes. to 5% [18]. The damping ratios of the remaining materials in the model were considered to be the same as for the disc component. Table 3 lists all used material parameters. Using FE approximation of a given continuous physical system, the behavior of the system around its equilibrium position can be described by a finite set of differential equations with time-independent coefficient matrices. The static equilibrium position was achieved by prestressing nonlinear static analysis.