## 3 Experimental Verification
### Bearing Outer Race Fault Diagnosis  
Figure 6 shows the test rig for vibration signal collection from the bearing with a local outer race fault, which was a pre-seeded fault (it was created by the manufacturer and the dimensions of the fault are unknown). The shaft is driven by a motor controlled by an AC inverter. Two ER16K bearings are mounted to support the 1-inch steel shaft and the load. The faulty bearing is mounted on the left side. An 623C01 ICP accelerometer (PCB Piezotronics, NY, USA) is mounted on the platform of the test rig (far away from the faulty bearing) in order to challenge the effectiveness of the proposed method. The frequency range of the accelerometer is 0.5-10 kHz, with sensitivity of (\(\pm\) 10%) 100 mV/g. The data are transferred to an NI USB-6212 BNC data Acquisition Module (National Instruments, TX, USA) and recorded by a computer. The diameter ratio of the sheaves is 1:2.6 and the number of pinion teeth is 18. Then, the gear meshing frequency can be computed as the shaft IF divided by the diameter ratio of the sheaves and subsequently multiplied by the number of the pinion teeth. In addition, a tachometer was used to collect shaft IF for comparison purposes. The shaft IF increased from 30.25 Hz to 60.5 Hz. The sampling frequency in this experiment was 20 kHz. The measured vibration signal and its TFR are presented in Figure 7a,b, respectively. From Figure 7b, it can be seen that the TFR of the signal is dominated by an interfering signal, i.e., the instantaneous gear meshing frequency (=6.92\(f_{s}\)). No clear IF ridges related to IFCF are observable. The lower frequency band signal, which is obtained via a low-pass filter with cut off frequency being \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Bearing Type** & **Pitch Diameter** & **Ball Diameter** & **Number of** & **IFCF of Outer** & **IFCF of Inner** \\  & **(mm)** & **(mm)** & **Balls** & **Race Fault** & **Race Fault** \\ \hline ER16K & 38.52 & 7.94 & 9 & 3.57\(f_{r}\) & 5.43\(f_{r}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of the bearings used in the test. Figure 5: The flowchart of multiple IF ridge extraction. Figure 6: Experimental setup for outer race fault test. set at 500 Hz, is shown in Figure 7c. The envelope of the resonant frequency band signal obtained by the SK method is shown in Figure 7d. Then, STFT is used to get TFRs of the lower and resonant frequency band signals, as shown in Figure 8a,b. From Figure 8a, no clear shaft IF trajectories can be recognized. The IFCF of the outer race fault can be seen in Figure 8b; however, its second harmonic suffers from the smearing problem in the first second. Thus, IF ridges cannot be accurately extracted from such TFRs using the FPO algorithm. As can be seen in Figure 8c,d, the extracted IF ridges in the first second still suffer from slight jumps due to the smearing problem of TFRs. The extracted IF ridges from STFT-resulting TFRs using FPO are named pre-IFs, which are utilized to guide the base construction for FMLT. Figure 8: Outer race fault case: (**a**) TFR of lower frequency band signal, (**b**) TFR of envelope of resonant frequency band signal, (**c**) extracted IF ridges from (**a**) using FPO, and (**d**) extracted IF ridges from (**b**) using FPO. Figure 7: Outer race fault case: (**a**) Raw vibration signal, (**b**) TFR of the raw vibration signal, (**c**) lower frequency band signal, and (**d**) envelope of the resonant frequency band signal. The constructed base frequencies guided by the pre-\(f_{2}\) are illustrated in Figure 9. It can be seen that the base frequencies vary with the shaft IF and can match the frequencies of signal components from the bearing with a local outer race fault. With such bases, FMLT can be performed and the resulting TFRs are shown in Figure 10a,b, from which the shaft IF and IFCF related ridges can be easily recognized. Accordingly, the more accurate IF ridges can be extracted from the enhanced TFRs, as shown in Figure 10c,d.