## 5 Application in Order Tracking for Fault Diagnosis of Gearbox
The time intervals T\({}_{1}\) and T\({}_{2}\) are consistent with the rotation frequencies of the middle shaft under the working conditions of 500 rpm and 700 rpm. The corresponding spectrum of the time domain waveform is shown in Figure 19; the dominant frequencies mainly concentrate at 172.3 Hz, 442.8 Hz, 664.2 Hz, etc., none of which are meshing frequency components at either 500 rpm or at 700 rpm. In the zoomed-in spectrum, characteristic frequencies of 500 rpm, 700 rpm and 900 rpm working conditions are found together, which increase the difficulty in accurate fault diagnosis. In addition, the \"frequency smearing\" phenomenon is found due to the non-stationary condition of the rotation speed, as depicted in Figure 19b. Furthermore, Figure 20 depicts the time domain waveform and spectrum of the vibration signals acquired from the ICP accelerometer. It is noteworthy that, analogous to the previous observations, there is a discernible upward trend in the amplitude with increasing rotation speed in the time domain. However, it is crucial to point out that the maximum amplitude recorded is approximately one-fifth of that observed in the MEMS sensor. In the corresponding spectrum, it is observed that the dominant frequencies primarily cluster around 2300 Hz. Additionally, the phenomenon of \"frequency smearing\" is evident, which is attributed to the non-stationary nature of the rotation speed. Figure 17: (**a**) A schematic diagram of the seeded spalling fault (**b**) photograph of the seeded spalling fault. Figure 18: Time domain waveforms of the in situ accelerometer under sped-up working conditions: (**a**) raw signal and (**b**) zoomed-in plot. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Load** & **Small Fault** & **Medium Fault** & **Large Fault** \\ \hline
20 Nm & \#1 & \#2 & \#3 \\
40 Nm & \#4 & \#5 & \#6 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The acquired six datasets. In order to show the superiority of the proposed GAD method, the traditional vibration-based tacholess method in Ref. [19] is compared, in which the ridge of the instantaneous rotating frequency can be detected from the time-frequency representation (TFR) of the vibration signal based on the STFT method. The proposed GAD method is designated as Method 1, the tacholess method utilizing in situ MEMS signals is designated as Method 2 [19] and the tacholess method reliant on bearing housing vibration signals is designated as Method 3. Furthermore, it is known that the spectrogram has some disadvantages that cannot give all information existing images, and it is a little difficult to determine the time step and overlap of the STFT method. So, one more tacholess method reliant on bearing housing vibration signals based on the Choi-Williams method is also compared, which is designated as Method 4 [31]. The estimated rotational speed and angle based on the three methods are depicted in Figure 21. The reconstructed time-varying gravity acceleration signal and the estimated rotation angle are shown in Figure 21a,b. The maximum magnitude of the time-varying gravity acceleration signal nearly reaches 10, and its frequency exhibits a direct correlation with the rotational speed, increasing as the latter increases. This correlation is precise and consistent with the rotational speed of the middle shaft. Figure 21b illustrates the estimated rotational angle plot, clearly demonstrating the slope variations resulting from changes in rotational speed. Notably, the absence of pronounced non-smooth regions is detrimental to the process of angle resampling. In Method 2 and Method 3, the TFR is derived based on the STFT method with a time overlap of 0.05 s and a window length of 0.5 s. This results in a time resolution of 0.05 s and a frequency resolution of 2 Hz in the estimation of rotational speed. Within the TFR, the gear meshing frequency of the first gear pair is chosen as the tracking frequency.