## 4 Validation of the Rotational Speed Estimation Principle Using GAD
### Speed Estimation Result Comparisons
During the initial startup phase, the presence of oscillations and noticeable differences among the CEEMDAN, ICEEMDAN and EWT methods are observed. Notably, the EWT method exhibits superior noise suppression performance. However, it is also observed that there are some reconstruction errors  
Figure 5.0 The calculated correlation coefficients of the IMFs and local embedded acceleration signals under Test 01: (**a**) CEEMDAN; (**b**) ICEEMDAN; (**c**) EWT. Figure 5.1 The decomposed IMFs of the filtered local embedded acceleration signals: (**a**) IMFs 1–9 and (**b**) IMFs 10–18 based on CEEMDAN; (**c**) IMFs 1–8 and (**d**) IMFs 9–15 based on ICEEMDAN; (**e**) IMFs 1–8 and (**f**) IMFs 9–15 based on EWT. before the actual startup, which may be attributed to noise or interference. In conclusion, the utilization of the CEEMDAN, ICEEMDAN and EWT methods for decomposition and reconstruction reveals minimal differences among the three approaches. As a result, these minor variations are likely to have a negligible impact on the estimation errors of the rotational speed. This suggests that the choice of decomposition technique, within the range of these three methods, may not significantly affect the overall accuracy of rotational speed estimation. Therefore, the reconstruction results based on the CEEMDAN method are still used as the example. The acquired encoder signals under the Test 01 working condition are shown in Figure 5.2 There are two kinds of square waves of the encoder: encoder-phase Z and encoder-phase A, corresponding to 1 trigger per revolution and 360 triggers per revolution. Thus, the average speed per revolution and instantaneous speed of the disc can be calculated from encoder-phase Z and encoder-phase A, respectively. It is found that the frequency of the square wave trigger grows over time for both encoder-phase Z and encoder-phase A, corresponding to the startup process of Test 01. Based on Hilbert transformation and the phase angle calculation method in Section 3.2, the average rotational speed per revolution and instantaneous rotational speed are estimated from the reconstructed gravity acceleration component. It is noted that the average rotational speed per revolution is determined by the average of the instantaneous RPMs per revolution. The estimated average and instantaneous rotational speeds for the Test 01 working condition and their relative errors are illustrated in Figure 5.3 The analysis is conducted on the time segment spanning from 2 s to 10 s. Within this period, it is observed that the estimated average rotational frequency underwent a significant increase, specifically from 2.44 Hz to 11.59 Hz. This translates to a speed range varying from 146.56 r/min to 695.37 r/min. Notably, this estimated increase in rotational frequency and corresponding speed  
Figure 5.4 The acquired encoder signal under the Test 01 working condition: (**a**) encoder signal; (**b**) zoomed-in plot. Figure 5.5 The (**a**) reconstructed time-varying gravity component under Test 01 based on the three methods and (**b**) the zoomed-in plot. From Figure 12, it can be found that the gravity acceleration components are well reconstructed and most of the oscillations are suppressed. Similarly, some oscillations can still be observed at a low-speed range. The magnitude of the reconstructed sinusoidal periodic component is approximately equal to the gravity. In Figure 13, the analysis is conducted on the time segment spanning from 0 s to 7 s. Within this period, it is observed that the estimated average rotational frequency underwent a significant decrease, specifically from 10.37 Hz to 2.41 Hz. This translates to a speed range varying from 622.19 r/min to 144.80 r/min. Notably, this estimated decrease in rotational frequency and corresponding speed range also exhibits minimal deviation from the results obtained from the encoder, as shown in Figure 13a. Similarly, the estimated instantaneous rotation speed also achieves good agreement with the encoder instantaneous speed, as illustrated in Figure 13b.