## 5 Application in Order Tracking for Fault Diagnosis of Gearbox
Additionally, when comparing the order envelope spectrum, it appears that the characteristic frequencies associated with spalling faults are less pronounced in Method 2 compared to Method 1, as demonstrated in Figure 23d. In Method 3, the gear meshing frequency and its harmonics do not dominate the signal components. Although modulation sidebands can be observed around the meshing frequency of the second gear pair, their presence is insufficient, as demonstrated in Figure 23e. Consequently, the characteristic frequencies associated with spalling faults are not clearly evident in the corresponding order envelope spectrum, as shown in Figure 23f. In Method 4, the resonance frequencies dominate the signal. Almost no modulation sidebands can be observed around the meshing frequency due to the large conversion error, as demonstrated in Figure 23g. Consequently, the characteristic frequencies associated with spalling faults are not clearly evident in the corresponding order envelope spectrum, as shown in Figure 23h. Figure 23: The (**a**) order spectrum and (**b**) its envelope spectrum based on Method 1, (**c**) the order spectrum and (**d**) its envelope spectrum based on Method 2, (**e**) the order spectrum and (**f**) its envelope spectrum based on Method 3, and (**g**) the order spectrum and (**h**) its envelope spectrum based on Method 4. Based on the analysis presented, it can be concluded that the proposed method exhibits superior performance in rotational speed estimation and order tracking within gearboxes. Furthermore, it is effective in assisting with the diagnosis of localized faults under non-stationary operating conditions. These findings suggest that the proposed method offers good insights for enhancing fault detection and diagnostic capabilities in non-stationary applications. To demonstrate the superior performance of the proposed method compared to the other two methods, the mean peak ratio (MPR) [24] is employed as a metric to assess the fault-related components in the demodulation spectrum. A higher MPR value indicates superior demodulation performance. The MPR is defined as follows:  
\[\text{MPR}=20\log_{10}\frac{\sum\limits_{i=1}^{N_{h}}(P_{i}-A_{s})}{A_{s}} \tag{25}\]  
Figure 21.0  
where \(N_{h}\) denotes the number of harmonics of the fault characteristic frequency required for computing an MPR, and it is fixed at 4 in this study. Additionally, \(P_{i}\) represents the amplitude of the spectral peak that corresponds to the \(i\)th harmonic of the fault characteristic frequency. The symbol \(A_{s}\) denotes the average value of spectral components within the range from the \(a\)-th order to the \(b\)-th order. Lastly, \(S_{k}\) signifies the amplitude of the \(k\)th spectral component. The MPR values obtained from the four methods across the six datasets are presented in Figure 21.1 A noteworthy observation emerges from the analysis, indicating that the proposed Method 1 significantly outperforms Method 2, Method 3 and Method 4 in terms of MPR values across the entire datasets of six working conditions. To be precise, Method 1 achieves an average gain of 5.36 dB over Method 2, 4.06 dB over Method 3 and a substantial 7.27 dB over Method 4. This remarkable performance can be primarily attributed to the enhanced accuracy of speed estimation achieved by Method 1 compared to the three alternative methods.