## 3 Experimental Verification
### Bearing Inner Race Fault Diagnosis  
The experimental setup for inner race fault signal collection is shown in Figure 11. The gearbox on the platform of the test rig is dis-connected in the inner race test. The accelerometer is mounted the top of the faulty bearing. The accelerometer, tachometer and data acquisition module are the same as those used for the outer race fault test. The sampling rate is 12 kHz. The shaft IF speeds up from 17.3 Hz to 33 Hz and then drops to 24.4 Hz. The tachometer is also used for shaft IF collection for the comparison between real shaft IF and extracted IF.  
The collected raw vibration signal and its corresponding TFR are plotted in Figure 12a,b, respectively, where no information related to inner race fault can be observed. To obtain IF ridges associated with both shaft IF and IFCF, the signal is also divided into the lower frequency band signal (by low-pass filtering) in Figure 12c and the resonance frequency band signal (by fast spectral kurtosis) in Figure 12d. With the traditional STFT, the TFRs of the lower and resonant frequency band signals can be obtained, as displayed in Figure 13a,b. Two pre-IF ridges extracted from the lower frequency band signal by iteratively using the FPO algorithm are presented in Figure 13c. The other two from the resonant frequency band signal are shown in Figure 13d. Due to the smearing effects of TFR, the extracted pre-IF ridges present frequency jumps in the first five seconds, which may influence bearing condition monitoring results. Thus, the proposed FMLT is applied to enhance the readability of the TFR with the assistance of the pre-IF from the lower frequency band signal.  
Figure 11: Experimental setup for inner race fault test.  
The pre-IF is applied to build the bases of FMLT. The constructed bases are presented in Figure 14, from which it can be seen that frequencies of bases match the frequencies of signal components. The FMLT is then performed and the resulting enhanced TFRs are shown in Figure 15a,b. Then, the IF ridges extracted by an iterative FPO operation from the enhanced TFRs can be observed in Figure 15c,d. Similar to the outer race fault case, MREs of IF paths extracted by the FPO algorithm from traditional STFT-resulting TFRs and the FMLT-resulting TFRs are listed in Table 3. It can be observed that the MREs of the IF ridges extracted from the enhanced TFR are less than those of the IF ridges extracted from the STFT-resulting TFR, indicating that the IF ridges extracted from the improved TFR are more accurate than the ones from STFT-resulting TFR.  
Figure 12: (**a**) Raw vibration signal from bearing with a local inner race fault, (**b**) TFR of the raw vibration signal, (**c**) lower frequency band signal, and (**d**) envelope of the resonant frequency band signal.  
Figure 13: Inner 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.  
Then, according to the ratios of any two points at the same time instant from the lower frequency band and resonance frequency band in Figure 15c,d, respectively, the bearing fault type can be diagnosed. As shown in Figure 15c,d, the ratio of a point on IF ridge\(f_{3}\) to the one on pre-\(f_{1}\) equals 2.59 and the ratio of a data point on \(f_{4}\) to the one on pre-\(f_{2}\) is 2.61, which are almost identical to the half of the fault characteristic coefficient of the inner race fault. Thus, it can be concluded that the local bearing fault is positioned on the inner race of the bearing. It is worth mentioning that, according to the calculated ratio, the bottomed IF ridge in the TFR of lower frequency band signal is the second harmonic of shaft IF, rather than the shaft IF. The top IF ridge is the fourth harmonic of the shaft IF.