# Contribution Analysis of Assembled Brake System to Reduce Squealing
## 3 Contribution Analysis of Assembled Brake System
### Contribution Analysis of Assembled Brake System for Squealing  
A contribution analysis of the assembled brake system was conducted using the obtained spectral response vectors for each subpart. The theoretical background was the component contribution using the response spectrum data under the selected excitation situation [11; 12; 19; 20]. Previous studies have performed a contribution analysis on complex brake systems over squealing noise problems. [PERSON] et al. [24] located sensitive parts using substructure modes related to the unstable modes, and [PERSON] et al. [14] and [PERSON] et al. [25] investigated the most sensitive parts using component contribution analysis. All contribution analyses were based on the eigenvectors of the components (or parts) of interest in unstable modes. However, the proposed contribution analysis is based on the response vectors of the parts of interest over the excitation at a disk pad. Because the squealing noise is triggered by the sticking and slipping between the disk and brake pad [1; 2], the proposed response vectors may sufficiently discribe the physical behavior of the squealing's vibration or noise. The proposed contribution analysis formulation can be explained using the spectral response vector provided in Equation (4). If the spectral response vector at a certain subpart (_i_) is assumed to be \(\Phi_{\omega_{e},i}\) at the frequency of interest \(\omega_{e}\), the global vector of the assembled brake system (\(\Phi_{\omega_{e},i}\)) can be formulated by assuming that the total number of subparts is \(N\). \[\Phi_{\omega_{e}}= \begin{bmatrix}\Phi_{\omega_{e}1}&\Phi_{\omega_{e}2}&\cdots& \Phi_{\omega_{e}N-1}&\Phi_{\omega_{e}N}\end{bmatrix}  \tag{4}\]  
The contribution of certain subparts to the squealing of the assembled brake system can be calculated using the inner product of the spectral response vectors between the subpart of interest and the total response. The total response of the assembled brake system is given by Equation (1), and the spectral response of subpart (_i_) can be formulated by Equation (4). The inserted zeros in the vectors in Equation (2) are used to compensate for the difference between the total response vectors (\(\Phi_{\omega_{e}}\)), so that the length of Equation (5) is  
Figure 7: White-noise assignment for the assembled brake system. also \(N\). In this condition, the normalized inner product of the two vectors (\(I(\omega_{e})\)) can be written as Equation (6). \[\[PERSON]^{m}_{\omega_{e},i}= \begin{bmatrix}0&\cdots&\[PERSON]_{\omega_{e},i}&\cdots&0\end{bmatrix}  \tag{5}\]  
\[I_{t}(\omega_{e})=\frac{\[PERSON]_{\omega_{e},i}\cdot\left(\[PERSON]^{m}_{\omega_{e},i} \right)^{T}}{\[PERSON]_{\omega_{e},i}\cdot\left(\[PERSON]_{\omega_{e}}\right)^{T}} \tag{6}\]  
where, '\(\cdot\)' is the inner product between two vectors, and \((A)^{T}\) is the transpose of matrix \(A\). Equation (4) indicates the similarity of the spectral response of certain subparts over the total response of the assembled brake system, so the calculated value may be equivalent to the contribution of the \(i\)-th subpart over the total response of the assembled brake system caused by the squealing. Finally, the contribution index of the \(i\)-th subpart can be defined by Equation (4). Because the assembled brake system consists of several subparts, as illustrated in Figure 8, the contribution of each subpart can be evaluated using the calculated index value in Equation (4). The contribution indices for all subparts were calculated, and the contribution results are illustrated in Figure 9 for frequencies of 3.4 and 6.4 kHz. The contribution results shown in Figure 9 indicate that the disk was most sensitive at 3.4 kHz and the caliper was most influential at the critical frequency of 6.4 kHz. In addition, the brake pad and backplate had the second-highest indices for squealing at 6.4 kHz.