## 5 Application in Order Tracking for Fault Diagnosis of Gearbox
As evident from Figure 21c,e, both Method 2 and Method 3 exhibit numerous estimation errors in the rotational speed  
Figure 19: The spectrum of the in situ accelerometer under speed-up working conditions: (**a**) raw signal and (**b**) zoomed-in plot. Figure 20: Acquired signals of the bearing housing accelerometer under speed-up working conditions: (**a**) time domain and (**b**) frequency spectrum. ridge curves. Consequently, these estimation errors give rise to non-smooth regions in Figure 21d,f, ultimately leading to resampling errors in the subsequent steps. Utilizing the Choi-Williams method, the TFR of the bearing housing vibration signals has been derived and is presented in Figure 21g. While a refined TFR image (red color) is observable, it is noted that the estimated TF ridge (blue dashed line) exhibits heightened sensitivity due to the superior TFR resolution compared to that achieved using short-time Fourier transform (STFT). Consequently, significant estimation errors in the rotation angle are discernible (the real rotational speed is plotted by black solid line), as illustrated in Figure 21h. Furthermore, the rotational speed experiences an instantaneous acceleration from one speed regime to another, with the transient period between successive speeds being approximately 0.05 s in the six acquired datasets. In cases where the transition duration is exceedingly brief, it can exacerbate the time resolution error, stemming from the inherent limitations of the STFT method. Consequently, the time resolution of the analysis may be adversely affected, introducing potential inaccuracies in the subsequent interpretation of the results. Figure 21: Cont. Comparisons between the calculated angle domain and time domain waveforms of the four methods are performed, as depicted in Figure 22. Some samples are denoted in both the time domain and calculated angle domain. It has been observed that the conversion errors from the time domain to the angle domain are lower for the proposed Method 1 compared to Methods 2, Method 3 and Method 4. This finding aligns well with the results presented in Figure 21, indicating a higher accuracy in the estimated rotation frequency of Method 1. Additionally, due to the window length of 0.5 s in Methods 2 and 3, signals within the final 0.5 s period cannot be converted into the angle domain. In addition, the order spectrum and order envelope spectrum are obtained from the derived angle domain signal from the three methods, as shown in Figure 23, where one order equals the rotating frequency of the middle shaft. In Method 1, it can be observed that the spectrum density concentrates at 25, 39 and 50 orders, where the 39 th order is the 1 st harmonics of the meshing frequency of the 1 st gear pair, and the 50 th order is the 2 nd harmonics of the meshing frequency of the 2 nd gear pair. The modulation sidebands can be observed in the zoomed-in plot around 25 orders, where 25 1 orders are obvious, which is in accordance with the characteristic frequency of the spalling fault, as shown in Figure 23a. The order envelope spectrum of Method 1 is illustrated in Figure 23b where distinct spectral components of order 1 and its harmonics can be found. In Method 2, it is observed that  
Figure 21: The (**a**) reconstructed time-varying gravity acceleration signal and (**b**) estimated rotational angle based on Method 1, (**c**) the estimated rotational speed and (**d**) angle based on Method 2, and (**e**) the estimated rotational speed and (**f**) angle based on Method 3, (**g**) the estimated rotational speed and (**h**) angle based on Method 4. Figure 22: A comparison between the original time domain signal and the resampled angle domain signal based on (**a**) Method 1, (**b**) Method 2, (**c**) Method 3 and (**d**) Method 4.  
the spectrum density is concentrated at 39 and 50 orders. Although modulation sidebands are identifiable, a frequency smearing phenomenon persists, indicating conversion errors from the time domain to the angle domain, as shown in Figure 23c.