## 5 Verification  
### Experiment Verification  
The rolling bearing life cycle experiment is conducted in the section and the signal collected at the early weak failure stage is analyzed using the combined two-stage method. The accelerated bearing life test (ABLT-1A) is provided by Hangzhou Bearing Test & Research Center. It simultaneously hosts four rolling element bearings on one shaft driven by an AC motor and coupled by rubber belts. The test rig is shown in Figure 6. A new bearing will be installed if one fails. Select one of the same test bearings as [45] as the study object. The parameters and the fault characteristic frequencies of the test rolling bearing are shown in Tables 1 and 2, respectively. Figure 5: Flow chart of the proposed method. The whole life cycle of the selected test bearing is 2469 min. The vibration data corresponding to the 2297 th minute are used as the analyzed object, whose time-domain diagram is presented in Figure 7a. It is found that failure arises on the inner ring of the test bearing by disassembling it after experiment and the FCF of the inner race is 245 Hz. The transient characteristics of the faulty bearing are obscured for the effect of strong interferences and the judgement of the fault location occurring on the inner race would not be realized based on the periodic interval between the two adjacent impacts in the waveform as presented in Figure 7a. Figure 7b is the full-band ES analysis result and only the spectral energy distribution characteristics between 0 Hz and 800 Hz are displayed for convenient analysis. Unfortunately, any useful fault characteristic information could not be obtained based on Figure 7b by applying ES analysis over the full frequency band and the main reason is still due to the strong inferences. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Type**} & **Ball** & **Ball** & **Pitch** & **Contact** & **Motor** & **Load** \\  & **Number** & **Diameter** & **Diameter** & **Angle** & **Speed** & **(rpm)** & **(kN)** \\ \hline
6307 & 8 & 13.494 & 58.5 & 0 & 3000 & 12.744 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The parameters of the test rolling bearing. Figure 6: The test rig. Figure 7: _Cont_. Based on the described method in Figure 5, AMF is used to preprocess the original signal initially and the de-noised signal is shown in Figure 7c. By comparing Figure 7c with Figure 7a, the transient characteristics of the former are enhanced significantly and the kurtosis index of the de-noised signal is about 5.2 times that of the original signal through statistical calculation, which proves the enhancement effect of AMF on the transient characteristics of the faulty bearing. To verify the necessity of further processing on the de-noised signal, full-band ES analysis is carried out on the signal as shown in Figure 7c,d, which show the corresponding result: although the FCF of inner race is extracted, its harmonics still could not be identified effectively due to the inference of strong interferences, and further FBS is needed to handle de-noised signal. Apply FBS on the de-noised signal and Figure 7e presents the analysis result, based on which the key parameters (center frequency \(f_{c}\) and bandwidth \(B_{w}\)) of the selected optimal frequency band with the biggest SI (SI = 5.78) for ES analysis are as follows: \(f_{c}\) = 3200 Hz and \(B_{w}\) = 6400 Hz. According to the above two obtained parameters, a band-pass filter is constructed to further filter the signal shown in Figure 7c and the envelope result of the further de-noised signal is displayed in Figure 7f. Apply FFT on the envelope signal shown in Figure 7f and the last fault features extraction effect is shown in Figure 7g, based on which both the FCF of the inner race and its harmonics are extracted perfectly. Figure 7: Experiment verification of the proposed method. (**a**) The original vibration signal of the experimental bearing at its early weak fault stage. (**b**) Envelope spectral of the signal shown in (**a**).