# Parameter Determination of a Minimal Model for Brake Squeal
## 3 Computational Design and Optimization Results  
The steepest ascent design was applied to approach the response region of the optimization objectives. Table 2 gives the response results of real part versus \(k_{1}\) and \(k_{2}\). As seen obviously that the response value of computation No. 3 is closest to the optimization objective. Therefore, the following central composite computation is designed with \(k_{1}=2.37\times 10^{9}\) N/m, \(k_{2}=2.1\times 10^{9}\) N/m as level 0, \(k_{1}+\delta\), \(k_{2}+\delta\) as level 1 and \(k_{1}-\delta\), \(k_{2}-\delta\) as level \(-1\), where \(\delta\) equals 0.03\(\times 10^{9}\) N/m. The center composite computation involves 2 variables and 3 levels, so 9 sets of computations are needed. The results are listed out in Table 3. It should be noted that the response values of \(\theta\) in each set are determined according to the method shown in Figure 7 because \(\theta\) has a predominant influence on Hopf bifurcation point, whereas the response value of the real part and the coupling frequency is calculated by CEA, as mentioned in Section 2.2. The variables and response values in Table 3 are then fitted using Equation (16). The response surface functions of the real part, frequency and slope angle \(\theta\) versus stiffness \(k_{1}\) and \(k_{2}\) are obtained as  
\[\begin{array}{llll}\mathrm{Real\ Part}&=4.42\times 10^{4}-3.66\times 10^{-5}\,k_{ 1}-2.38\times 10^{-7}\,k_{2}+4.60\times 10^{-15}\,k_{1}{}^{2}\\ &-1.03\times 10^{-15}k_{2}{}^{2}+4.53\times 10^{-15}k_{1}k_{2}+\e \begin{table}
\begin{tabular}{c c c c} \hline
**Computation Number** & \(k_{1}\) **(10\({}^{9}\) N/m)** & \(k_{2}\) **(10\({}^{9}\) N/m)** & **Real Part** \\ \hline
1 & 2.27 & 2.2 & 1963.3 \\
2 & 2.32 & 2.15 & 1369.0 \\
3 & 2.37 & 2.1 & 777.4 \\
4 & 2.42 & 2.05 & 205.1 \\
5 & 2.47 & 2.0 & 357.8 \\ \hline \end{tabular}
\end{table} 
Table 2: Steepest ascent computational design and measured responses. \begin{table}
\begin{tabular}{c c c c c} \hline
**Computation Number** & **Level of \(k_{1}\)** & **Level of \(k_{2}\)** & **\#/rad** & **Real Part** & **Frequency/Hz** \\ \hline
1 & \(-1\) & \(-1\) & 0.00200 & 757.2 & 7135.4 \\
2 & \(-1\) & 0 & 0.00312 & 941.5 & 7161.9 \\
3 & \(-1\) & 1 & 0.00449 & 1122.7 & 7188.4 \\
4 & 0 & \(-1\) & 0.00122 & 596.6 & 7157.6 \\
5 & 0 & 0 & 0.00210 & 777.4 & 7184.2 \\
6 & 0 & 1 & 0.00322 & 958.3 & 7210.9 \\
7 & 1 & \(-1\) & 0.00064 & 431.4 & 7181.7 \\
8 & 1 & 0 & 0.00132 & 623.6 & 7206.1 \\
9 & 1 & 1 & 0.00225 & 813.2 & 7232.3 \\ \hline \end{tabular}
\end{table} 
Table 3: Central composite computational design and the measured response. \[ \begin{array}{ll}\text{Frequency}&=2.45\times 10^{3}+1.70\times 10^{-6}\ k_{1}+8.75 \times 10^{-7}\ k_{2}+8.54\times 10^{-17}\ k_{1}{}^{2}\\ &+3.64\times 10^{-17}k_{2}{}^{2}-6.46\times 10^{-16}k_{1}k_{2}+\varepsilon\\ \theta=&0.1471-1.60\times 10^{-10}\ k_{1}+4.28\times 10^{-11}\ k_{2}+1.36\times 10^{-19} \ k_{1}{}^{2}\\ &1.36\times 10^{-19}k_{2}{}^{2}-2.44\times 10^{-19}k_{1}k_{2}+\varepsilon\\ \end{array}  \tag{20}\]  
And the response surface plots are shown in Figure 8.  
Ignore the effect of fitting error \(\varepsilon\), and let Equations (19) and (20) equal to the optimization objectives of finite element results, respectively. Thus, the following equations are obtained:  
\[\left\{ \begin{array}{l}4.42\times 10^{4}-3.66\times 10^{-5}\ k_{1}-2.38\times 1 0^{-7}\ k_{2}+4.60\times 10^{-15}\ k_{1}{}^{2}\\ -1.03\times 10^{-15}k_{2}{}^{2}+4.53\times 10^{-15}k_{1}k_{2}=713.4\\ 2.45\times 10^{3}+1.70\times 10^{-6}\ k_{1}+8.75\times 10^{-7}\ k_{2}+8.54 \times 10^{-17}\ k_{1}{}^{2}\\ +3.64\times 10^{-17}k_{2}{}^{2}-6.46\times 10^{-16}k_{1}k_{2}=7168.6\\ \end{array} \right. \tag{22}\]  
Values of \(k_{1}\) and \(k_{2}\) which meet both optimization objectives of the frequency and real part can be solved by Equation (22). The value of \(\theta\) can be further obtained from Equation (21).