## 4 Experiments  
An aero-engine was scaled in a certain proportion and simplified to a four-stage simulation rotor. A section view of the four-stage rotor is shown in Figure 7. As observed, it was composed of a front shaft, a low-pressure compressor (LPC), a high-pressure compressor (HPC), and a back shaft. Each single-stage rotor was assembled step by step through its own axial and radial front edge, and connected by a certain number of screws. The distances between the two pre-set balanced Surfaces A and B from the support point were \(l_{A}\) and \(l_{B}\), respectively. To further prove the validity of the proposed assembly optimization method in this paper, experiments were performed, and the corresponding experimental steps are detailed below:  
Step-1: The geometric parameters of the rotors at different stages were measured by a 3-D CMM. Figure 8 shows the measured actual scene and schematic diagram of the HPC. First, the center of a pair of screw holes on the mounting surface of HPC was calibrated. Taking this center as the starting point of sampling, the concentricity (\(C_{n}\)), the parallelism (\(P_{n}\)), the measured radius (\(D_{n}\)), an angle between this center and the highest point (\(\varphi_{n}\)), and the high point (\(H\)) of the mounting surface of the HPC were measured successively. As shown in Figure 8b, the circle of diameter \(t_{1}\) is the tolerance zone defined by the concen  
Figure 6: Selection of the optimal orientations between two optional orientations. Figure 7: The section view of the four-stage rotor. Step-2: The initial unbalance of the four-stage rotor was artificially added to each rotor. To better reflect the experimental effect of the unbalance optimization, the unbalanced mass was artificially added to the rotors at different stages, which is far greater than the unbalance caused by uneven material distribution. Hexagon-socket-head-screws were used for the connection of the four-stage rotor, as shown in Figure 9. The method was to replace an original M4 \(\times\) 12-screw with a M4 \(\times\) 20-screw at the calibrated screw hole of rotors at different stages to produce an unbalanced mass with a mass difference of about 2 g, whose effect radius and phase are known. The initial unbalance parameters of the four-stage rotor are shown in Table 11. \begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Component**} & **Distributed Numbers** & **Distributed Numbers** \\  & **(Top Mounting Surface)** & **(Bottom Mounting Surface)** \\ \hline Front shaft & 12 & 12 \\ LPC & 24 & 12 \\ HPC & 12 & 24 \\ Back shaft & 12 & 12 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: The distributed numbers of the screw holes of each component. Figure 8: Measurement of the geometric parameters of a high-pressure compressor (HPC). (**a**) Measured scene. (**b**) Schematic diagram. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Component** & **\(u(C_{p})\) [\(\mu\)m]** & **\(u(P_{p})\) [\(\mu\)m]** & **\(u(D_{p})\) [\(\mu\)m]** & **\(u(\varphi_{n})\) [\({}^{\circ}\)]** & **\(u(H_{n})\) [\(\mu\)m]** \\ \hline Front shaft & 0.35 & 0.49 & 0.74 & 0.40 & 1.02 \\ LPC & 1.10 & 0.68 & 1.67 & 0.33 & 1.13 \\ HPC & 1.04 & 0.33 & 0.52 & 0.22 & 1.41 \\ Back shaft & 0.38 & 0.45 & 0.32 & 0.31 & 1.22 \\ \hline \end{tabular}
\end{table} 
Table 9: The standard uncertainties of different measured parameters. Step-3: The unbalance of the four-stage rotor was measured using a dynamic balancing machine. According to the geometric and unbalance parameters obtained in Step-1 and Step-2, the genetic algorithm derived in Section 3.2 was used to calculate the optimal assembly orientations of the rotors at different stages with three assembly strategies as follows:  
Strategy-1: the minimum unbalance was set as the optimization objective;  
Strategy-2: the maximum unbalance was set as the optimization objective; and  
Strategy-3: direct assembly (i.e., the rotors at each stage remained in their initial state).