## 3 Results and Discussion
### Simulation and Experiment
#### 3.1.1 Simulation Results  
The structure of the high-speed winding rotor is shown in Figure 1, and its structural parameters are listed in Table 2. The winding rotor speeds up to the maximum 7730 rpm from the stationary state with equal acceleration, as shown in Figure 9.  
Figure 10: Experimental test of the winding rotor. (**a**) Experimental scheme; (**b**) measuring device. In this paper, the unbalanced mass-radius product is calculated according to the dynamic balance standard and part mass, which acts at the position where imbalance is likely to occur. The positions of node 30 and node 46 are likely to generate unbalance during the assembling process. Therefore, the residual unbalance was added in node 30 and node 46 in this article, respectively. When the speed-up time is 60 s, the rotational speed of the winding rotor increases from 0 to 7730 rpm, and the displacement amplitudes of investigated point A (node 51), point B (node 45) and point C (node 39) in the \(X\)-axis direction were obtained through the simulation, respectively, as shown in Figure 11b. It can be found that the winding rotor passed through the first-order and second-order critical speed points during speeding up, and two critical speed points are 988 rpm and 1373 rpm, respectively. In order to study the dynamic characteristics of the winding rotor during speeding up, the velocity amplitude and acceleration amplitude were further obtained, as shown in Figure 11c,d. \begin{table}
\begin{tabular}{c c c} \hline \hline
**ID** & **Label** & **Value** & **Unit** \\ \hline
1 & Supporting arm length & 1230 & (mm) \\
2 & Shaft length & 1340 & (mm) \\
3 & Sleeve length & 1780 & (mm) \\
4 & Outer diameter of sleeve & 107 & (mm) \\
5 & Interference coupling length & 75 & (mm) \\
6 & Bearing support span between I and II & 380 & (mm) \\
7 & Bearing support span III & 400 & (mm) \\
8 & Support flexible I & 15 & (mm) \\
9 & Support flexible II & 30 & (mm) \\
10 & Support flexible III & 505 & (mm) \\
11 & Maximum rotational speed of winding rotor & 7730 & (rpm) \\
12 & Number of rear bearings & 2 & (set) \\
13 & Number of front bearings & 2 & (set) \\
14 & Mass of shaft & 8.66 & (kg) \\
15 & Mass of sleeve & 47.19 & (kg) \\
16 & Mass of supporting arm & 19.38 & (kg) \\
17 & Mass of support flexible I & 1.6 & (kg) \\
18 & Mass of support flexible II & 3.0 & (kg) \\
19 & Mass of support flexible III & 4.5 & (kg) \\
20 & Unbalance of node 30 & \(1.42\times 10^{-5}\) & (kg\(\cdot\)m) \\
21 & Unbalance of node 46 & \(7.41\times 10^{-5}\) & (kg\(\cdot\)m) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters of the high-speed winding rotor. The displacement amplitude, velocity amplitude and the acceleration amplitude of the investigated points were analyzed in Figure 11, and the results show:  
* During speeding up, two critical speed points of the winding rotor have appeared. The first-order and second-order critical speed points are mainly caused by the flexible support of the rubber O-ring. Meanwhile, taking into account the influence of automatic centering of the winding rotor, the investigated points have a larger vibration displacement amplitude. However, due to the low rotational speed and the short excitation time of the residual unbalanced force acting on the winding rotor, the acceleration amplitude of the investigated points is relatively small;
* As the rotational speed increases, the exciting force of the residual unbalance formed by the winding rotor becomes stronger, which causes the investigated points on the winding rotor to present a larger vibration displacement amplitude. As seen in Figure 11c,d, the velocity amplitude and acceleration amplitude of the three investigated points increase correspondingly. In addition, when the rotational speed of the rotor exceeds 2000 rpm and is less than 7730 rpm, the winding rotor has a small vibration amplitude and a high degree of stability.