## 5 Results and Discussion  
Comparing the calculated unbalance in Table 16 and the measured results in Table 17, the simulation results were in good agreement with the experimental results. As shown in Table 17, when the \(\theta_{22}\), \(\theta_{23}\), and \(\theta_{24}\) were 120\({}^{\circ}\), 105\({}^{\circ}\), and 120\({}^{\circ}\), respectively (i.e., the calibrated screw holes of the LPC, HPC, and back shaft should rotate four, seven, and four distributed angles), the unbalance of the four-stage rotor reached the minimum value (184.6 g\(\cdot\)mm). When the \(\theta_{22}\), \(\theta_{23}\), and \(\theta_{24}\) were 150\({}^{\circ}\), 60\({}^{\circ}\), and 60\({}^{\circ}\), respectively (i.e., the calibrated screw holes of the LPC, HPC, and back shaft should rotate five, four, and two distributed angles), the unbalance of the four-stage rotor reached the maximum value (404.1 g\(\cdot\)mm). When the \(\theta_{22}\), \(\theta_{23}\), and \(\theta_{24}\) were all 0\({}^{\circ}\) (i.e., the initial state), the unbalance of the four-stage rotor was 404.1 g\(\cdot\)mm. Compared with the maximum unbalance and the initial unbalance, the minimum unbalance was notably reduced by 54.3% and 43.3%, respectively. These phenomena revealed that the proposed method of unbalance optimization for multi-stage rotor in this paper was effective. The contents of the unbalance optimization method were mainly reflected in three aspects. First, an accurate assembly error propagation model is important to calculate the unbalance of a multi-stage rotor after assembly. The relevant simulation results in Section 3.1 showed that the assembly error propagation model developed in Section 2.1 was able to accurately predict the coordinates of any point in the assembled rotor. Moreover, the distribution of the screw holes of each single-stage rotor was not considered in the existing assembly error propagation models. In the actual assembly process of the multi-stage rotor, the assembly orientations of each single-stage rotor are finite and discrete. This is not simply to turn the continuous variables (0\(-\)180\({}^{\circ}\)) into the discrete variables (i.e., 180\({}^{\circ}\) divided by the number of the screw holes). In order to calculate the assembly cumulative error under any corresponding relationship of the screw holes of rotors at different stage, the sampling angles (\(\varphi_{n}\)) between the calibrated screw hole and the highest point of each single-stage rotor must be known. Therefore, the sampling angles were introduced into the error propagation model in our study to calculate the optimal corresponding relationship of each rotor. With comparison to the existing studies, the distribution of the screw holes of rotors at different stages was directly introduced into the model for the first time, which increases the suitability for guidance on actual assemblies. The second aspect is concerned with the selection and calculation of the assembly datum for unbalance optimization. Considering the actual operating conditions of dynamic balancing for a rotor, two-plane unbalance measured by the dynamic balancing machine must be perpendicular to the axis of rotation of the measured rotor. Such an axis of rotation is simply a line connecting the midpoints of the journals of the front and back shaft. In current studies, the assembly datum for calculating unbalance of a multi-stage rotor was selected as the same as the measured datum for geometric parameter. It is clearly unreasonable that the simulated assembly axis is inconsistent with the axis of rotation of the dynamic balancing test. In addition, the results of the simulation are not comparable with the results of the experiment. In this study, we solved the problem by selecting the \"orientation-varying axis\" in Figure 3 as the assembly datum for unbalance optimization. The proposed assembly datum can ensure the constraint conditions of the unbalance optimization have good consistency with those of the dynamic balance test.