# Multi-Objective Optimization of Three-Stage Turbomachine
## 3 Optimization Problem
### Multi-Objective Rotordynamic Optimization
#### 3.1.1 Design Variables  
There are twenty-one design variables for the initial rotor geometry, which are presented in Table 2. The design variables, consisting of different units and dimensions, were normalized to have an upper limit of 1 and a lower limit of 0 to ensure uniform search sensitivity across all variables. The limits corresponding to these values are shown in Table 3. For instance, when the normalized design variable \(x_{1}=1\), it corresponds to \(L_{1}\) being 60 mm in the model. If \(x_{1}=0\), then \(L_{1}\) corresponds to 10 mm in the model. This indicates that \(L_{1}\) has a lower limit of 10 mm and an upper limit of 60 mm and the normalized value takes a value between these limits. In determining the lower and upper limits of the beam length, a reference was made to the benchmark [24]. Here, broader beam length limits were defined for the combustion chamber, while narrower beam length limits were established for the fan section. For the beam thickness, a lower limit of 25% and an upper limit of 50% were set based on the initial value. For the mass, a lower limit of 40% and an upper limit of 30% were defined according to the initial value.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \multicolumn{6}{c}{**Beam Parameters**} \\ \hline \(L_{1}=110\) mm & \(R_{1}=20\) mm & \(L_{4}=110\) mm & \(R_{4}=20\) mm & \(L_{7}=110\) mm & \(R_{7}=20\) mm \\ \(L_{2}=110\) mm & \(R_{2}=20\) mm & \(L_{5}=110\) mm & \(R_{5}=20\) mm & \(L_{8}=110\) mm & \(R_{8}=20\) mm \\ \(L_{3}=110\) mm & \(R_{3}=20\) mm & \(L_{6}=110\) mm & \(R_{6}=20\) mm & \(L_{9}=110\) mm & \(R_{9}=20\) mm \\ \(k_{s}=1.7\) Shape Factor & \multicolumn{2}{c}{\(E=200\) GPa} & \multicolumn{2}{c}{\(\rho=7850\) kg/m\({}^{3}\)} \\ \multicolumn{6}{c}{\(v=0.3\) Poisson Ratio} \\ \hline \multicolumn{6}{c}{**Disk Parameters**} \\ \hline \(M_{1}=6.95\) kg & \(I_{d2}=25,943.7\) kg\(\cdot\)mm\({}^{2}\) & \(I_{p1}=51,A24.1\) kg\(\cdot\)mm\({}^{2}\) & \(g=9.80665\) kg\(\cdot\)m/s\({}^{2}\) \\ \(M_{2}=6.95\) kg & \(I_{d2}=25,943.7\) kg\(\cdot\)mm\({}^{2}\) & \(I_{p2}=51,A24.1\) kg\(\cdot\)mm\({}^{2}\) & \(\Omega=160\) Hz \\ \(M_{3}=6.95\) kg & \(I_{d3}=25,943.7\) kg\(\cdot\)mm\({}^{2}\) & \(I_{p3}=51,A24.1\) kg\(\cdot\)mm\({}^{2}\) & \\ \hline \multicolumn{6}{c}{**Bearing Parameters**} \\ \hline \multicolumn{6}{c}{\(K_{1,2,3}=100,000\) N/mm} & \multicolumn{6}{c}{\(C_{1,2,3}=0.1\) Ns/mm} \\ \hline \end{tabular}
\end{table}
Table 1: Initial rotor parameters.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \multicolumn{6}{c}{**Normalized Design Variables**} \\ \hline \(x_{1}\)\(=\overline{L_{1}}\) & \(x_{5}\)\(=\overline{L_{5}}\) & \(x_{9}\)\(=\overline{L_{9}}\) & \(x_{13}\)\(=\overline{R_{4}}\) & \(x_{17}\)\(=\overline{R_{8}}\) & \(x_{21}\)\(=\overline{m_{3}}\) \\ \(x_{2}\)\(=\overline{L_{2}}\) & \(x_{6}\)\(=\overline{L_{6}}\) & \(x_{10}\)\(=\overline{R_{1}}\) & \(x_{14}\)\(=\overline{R_{5}}\) & \(x_{16}\)\(=\overline{R_{9}}\) & \\ \(x_{3}\)\(=\overline{L_{3}}\) & \(x_{7}\)\(=\overline{L_{7}}\) & \(x_{11}\)\(=\overline{R_{2}}\) & \(x_{15}\)\(=\overline{R_{6}}\) & \(x_{19}\)\(=\overline{m_{1}}\) & \\ \(x_{4}\)\(=\overline{L_{4}}\) & \(x_{8}\)\(=\overline{L_{8}}\) & \(x_{12}\)\(=\overline{R_{3}}\) & \(x_{16}\)\(=\overline{R_{7}}\) & \(x_{20}\)\(=\overline{m_{1}}\) & \\ \hline \end{tabular}
\end{table}
Table 2: Design variables.  
\begin{table}
\begin{tabular}{c c c c c} \hline \multicolumn{6}{c}{**True Design Variables**} \\ \hline \(L_{1}\)\(=10+50x_{1}\) & \(L_{5}\)\(=10+500x_{5}\) & \(L_{9}\)\(=10+100x_{9}\) & \(R_{4}\)\(=15+15x_{13}\) & \(R_{8}\)\(=15+15x_{17}\) & \(M_{3}\)\(=4+5x_{21}\) \\ \(L_{2}\)\(=10+50x_{2}\) & \(L_{6}\)\(=10+500x_{6}\) & \(R_{1}\)\(=15+15x_{10}\) & \(R_{5}\)\(=15+15x_{14}\) & \(R_{9}\)\(=15+15x_{18}\) & \\ \(L_{3}\)\(