# Gear Wear Process Monitoring Using a Sideband Estimator Based on Modulation Signal Bispectrum
## 4 Results and Discussion
### MSB-SE Characteristics of Gearbox Vibration Signals and Monitoring Indicator  
Figure 3 shows typical MSB-SE results at early operating phases--specifically at the fourth hour under higher load cases. The MSB-SE magnitude and its corresponding MSB-SE results are presented in the region \(f_{r}<75\) Hz to include the sidebands up to 3\(f_{r1}\) (72 Hz) and \(f_{m}<4400\) Hz to include the carrier frequency up to 3\(f_{r1}\) (4192 Hz). In the graphs, the significant peaks at the mesh frequency of 2\(f_{r1}\) (2795 Hz) and 3\(f_{r1}\) can be seen for all load cases. In particular, the MSB-SE peak magnitudes at \(f_{r1}\) (24 Hz) and its harmonics are \begin{table}
\begin{tabular}{c c c} \hline \hline
**Gear Parameters** & **1 st Stage** & **2 nd Stage** \\ \hline Number of teeth & 58/47 & 13/59 \\ Meshing frequency (Hz) & \(f_{m1}=1421\) & \(f_{m2}=393.04\) \\ Shaft frequency (Hz) & \(f_{r1}=24.5,f_{r2}=30.23,f_{r3}=6.66\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Specification of two-stage helical gearbox. Figure 2: Scuffing on gear tooth surfaces of the tested gear. (**a**) Scuffing marks on one tooth surface; (**b**) Scuffing markers on the other tooth surface. very dominant. In the meantime, all the bifrequency peaks are supported by MSB-SEc peaks, shown in the second column of Figure 3. In addition, the MSB-SE peaks at \(f_{r2}\) (30 Hz) and \(f_{r3}\) (7 Hz) and their responding harmonics around mesh frequency of \(2f_{r1}\) and \(3f_{r1}\) are also visible. However, their MSB-SEc magnitudes do not have significant corresponding peaks, showing lower amplitudes than that of fundamental mesh frequency of \(f_{r1}\) (1397 Hz), to confirm the nonlinear interactions. Therefore, these peaks are from a weaker nonlinear interaction than that between \(f_{m1}\) and \(f_{r1}\) and their harmonics. In contrast, the MSB-SE coherence slice at mesh frequency of \(f_{m2}\) (386.5 Hz) shows significant peaks at the position of all shaft frequencies and their harmonics, even though their corresponding MSB-SE magnitudes are relatively small. A similar result can be found at the MSB-SEc slice of \(f_{m1}\) (1397 Hz). Moreover, the MSB-SEc of all shaft frequencies and their harmonics at the mesh frequency of \(2f_{m2}\) (773 Hz) and \(3f_{m2}\) (1160 Hz) for all load cases are relatively obvious. In general, these bispectral peaks show that MSB-SEc can reveal the nonlinear interaction between the mesh frequencies and shaft frequencies as the modulating components. This is in agreement with the proposed analysis made in Section 2. In addition to the MSB-SEc peaks at gear shaft rotation frequencies and their harmonics, many other MSB-SEc peaks can be also observed undoubtedly at the MSB-SE spectrum; e.g., MSB-SEc peaks at \(f_{r2}\) + \(f_{r3}\) (36 Hz), \(2f_{r1}\) + \(f_{r3}\) (55 Hz) and \(f_{r1}\) + \(f_{r2}\) + \(f_{r3}\) (60 Hz). These peaks indicate that the harmonics of the shaft rotation frequencies also cause nonlinear interactions. The reason for these couplings is that the gear wear-induced vibration has two main paths to go from its source to the sensor through the solid mechanical components and their contacts [17]. The first path is from vibration signal origin to the shaft, then to the gearbox casing, and then reaches the sensor. Whereas through the second  
Figure 3: Modulation signal bispectrum-based sideband estimator (MSB-SE) magnitude and its coherence spectrum of different meshing frequency at the fourth hour under different loads. (**a**) MSB-SE spectrum at Load = 80%; (**b**) MSB-SE spectrum at Load = 90%. path, the vibration signal follows a longer path, from its origin to the middle shaft, from the middle shaft going through the second gear pair, the third shaft, after that to the gearbox casing, and finally to the sensor. These peaks also indicate that MSB-SE slices can provide reliable and sufficient information about the modulation effect of gear transmission. So, the peaks of MSB-SE slices will be used for diagnosis feature development.