## 3 Simulation
### GA-Based Unbalance Optimization  
We assumed that the three-stage simulation rotor described in Section 3.1 is made of steel. The initial coordinates of the center of mass of the rotors at different stages were measured using software (see Table 3). The initial unbalance of each rotor is the product of its mass and the vertical distance from the center of mass to the central inertia principal axis, which can be obtained using Equation (6). As shown in Figure 4, the balanced Surfaces A and B of the three-stage rotor are perpendicular to the central principal axis of inertia (i.e., the line connecting the center of the top mounting surface and the bottom surface) and, respectively, pass through the center of the top mounting surface and the bottom surface. \(l_{A}\) is set as 0, and \(l_{B}\) is equal to the distance between the center of the top and bottom mounting surfaces of the three-stage rotor. The unbalance on the balanced Surfaces A and B can be obtained by Equation (11), and the integral unbalance of the three-stage rotor can be obtained by Equation (12). The minimum and maximum unbalance are \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Stage**} & \multirow{2}{*}{\begin{tabular}{c} **Height (\(H_{u}\))** \\ **[mm]** \\ \end{tabular} } & \begin{tabular}{c} **Measured** \\ **Radius (\(D_{n}\))** \\ **[mm]** \\ \end{tabular} & \begin{tabular}{c} **Concentricity** \\ **(\(C_{n}\)) [mm]** \\ \end{tabular} & \
\begin{tabular}{c} **Parallelism** \\ **(\(P_{n}\)) [mm]** \\ \end{tabular} &
\begin{tabular}{c} **Sampling** \\ **Angle (\(\varphi_{n}\))** \\ **[\({}^{\circ}\)]** \\ \end{tabular} \\ \hline
1 & 70 & 100 & 0.005 & 0.005 & 0 \\
2 & 70 & 100 & 0.005 & 0.005 & 0 \\
3 & 70 & 100 & 0.005 & 0.005 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The geometric parameters of the three-stage rotor [17]. Figure 4: The center coordinates of the top mounting surface of the assembled three-stage rotor were measured using software. taken as the optimization objectives, and the genetic algorithm proposed in Section 2.3 is applied to calculate the optimal assembly orientations of the second and third stage rotors. In terms of the first stage rotor, its assembly orientation was designed at 0\({}^{\circ}\) by default; each chromosome was designed to hold two genes, that is (\(\theta_{22}\), \(\theta_{23}\)); and, the number of initial populations was set at 1000. In addition, the selection principle of the internal parameters of the GA referred to the literature [22], and the mutation rate, crossover rate, and iterative times were set as 0.01, 0.9, and 500, respectively. The convergence progress of the GA-based unbalance optimization of the three-stage rotor is shown in Figure 5. When \(\theta_{22}\) and \(\theta_{23}\) were 0\({}^{\circ}\) and 180\({}^{\circ}\), respectively, the corresponding minimum unbalance reached 32.2568 g-mm (minimum unbalance), as shown in Table 4. When \(\theta_{22}\) and \(\theta_{23}\) were 180\({}^{\circ}\) and 180\({}^{\circ}\), respectively, the corresponding maximum unbalance reached 129.6123 g-mm (maximum unbalance). Further considering the distribution of the assembly screw holes, the optional assembly orientations of the rotors at different stage are discrete, and the optimal corresponding relationship of the screw holes of the rotors should be obtained by a secondary calculation according to the above optimal solution. As shown in Figure 6, assuming the optimal orientation obtained by the optimization is \(\theta\), \(\theta_{1}\) and \(\theta_{2}\) are the two optional orientations adjacent the optimal orientation. The fitness in both cases must be calculated again, and the optimal solution can be selected. The angles of the screw holes adjacent to the optimal solution were determined by calculating all possible alignment relations (number: 2\({}^{n-1}\)) for \(n\)-stage rotors.