## 3 Results and Discussion
### Simulation and Experiment
#### 3.1.2 Test Measurement  
Before the test, the angular speed of the winding rotor increases at a constant angular acceleration by adjusting the control parameters of the motor, as shown in Figure 9.  
The non-contact measurement device shown in Figure 10 was used to perform the vibration displacement test on the winding rotor. Certainly, the positions of measuring points on the winding rotor were consistent with investigated points in the simulation model shown in Figure 12a.  
During data processing, due to a variety of interference effects, such as white noise, displacement signals obtained have a certain error. Therefore, in this paper, the FIR digital filter is designed by using the ripple Chebyshev method, and the signals were processed by the low-pass filter to enhance the signal-to-noise ratio.  
When speed-up time is 50 s, 60 s and 70 s, respectively, the testing results are shown in Figure 12b-d. By studying the vibration amplitude curve obtained from the test, the following conclusions can be obtained.  
* The displacement amplitude of point B is smaller than that of point A and point C. It is mainly caused because the central position of the sleeve is supported by the supporting arm, as shown in Figure 12a. Due to the cantilever beam structure of the winding rotor and the supporting position of point B, the distance between point B  
Figure 12: Testing results of the winding rotor under different speed-up times **(a)**Measuring points on the winding rotor; (b) speed-up time is 50 s; (c) speed-up time is 60 s; (d) speed-up time is 70 s.  
and point C is larger than the distance between point B and point A, so the vibration displacement amplitude of point C is higher than that of point A near the first-order critical speed point;
* When the speed-up time is 60 s, comparing the simulation results shown in Figure 10b and experimental results shown in Figure 12c, showed that they are in good agreement, which verifies the correctness of the developed model of the winding rotor in this paper. However, the vibration amplitudes of the measured points are larger than that of the investigated points. The main reason is that the large vibration amplitude of the measured points was mainly due to the cylindricity error of the testing position on the sleeve and the greater residual unbalance of the winding rotor. In the same speed-up time, the maximum displacement amplitude is still in point C near the first-order critical speed point. Comparing different speed-up times, as the speed-up time increases, the vibration amplitude increases correspondingly: 0.0915 mm (speed-up time 50 s), 0.0962 mm (speed-up time 60 s) and 0.0997 mm (speed-up time 70 s). Similarly, points B and point C have the same characteristics.
* During speeding up, it can be found that the rotational speed passes through the first-order critical speed point of the winding rotor based on the response amplitude. However, the vibration amplitude caused by passing through the second-order critical speed point cannot be clearly identified from Figure 12. To further analyze the experimental data, the frequency analysis of vibration data for the measured point C is performed under the speed-up time of 50 s, and the spectrum diagram is shown in Figure 13.  
As shown in Figure 13, there is a second critical speed point, and the corresponding speed is 1234.8 rpm, which is close to the simulation second-order speed 1373 rpm. Since the two critical speed points of the winding rotor are close and the vibration amplitude of the first critical speed point is larger than that of the second, the response amplitude caused by the second critical speed point is not obvious.