## 4 Results and Observations  
### Fault Detection Based on Averaged Envelope Spectrum  
As the radial force of test bearing is imposed with a hydraulic loading from the horizontal direction, the raw vibration signals of the horizontal accelerometers are utilized for further analysis; they are shown in Figure 5.  
In order to identify faults in each bearing at an early stage, the proposed averaged envelope spectrum is calculated according to Section 3.1.2 for each acceleration signal during the experiment. Subsequently, a spectral inspection is carried out on each envelope spectrum to pinpoint the bearing's fault characteristic frequencies. This process is performed for every signal measurement throughout the bearing's lifespan which helps in identifying the initiation of the bearing fault and facilitates the generation of a precise detection time \(t_{FDT}\) for the further prediction of RUL. The results of the fault detection are summarized in Table 4, where \(t_{FDT}\) denotes the first detection time of faults and the bearing EOL time is further discussed in Section 4.2. For illustration, Figures 6-8 present the spectral changes during the initial detection of a bearing fault for Bearings 2, 4 and 6, marking the transition from the healthy state to the degradation stage. As shown in Figure 6, the vibration signals of Bearing 2 at the 58 th and the 59 th minute are both dominated by the frequency component at the shaft speed. However, the peaks at the BPFO and its second multiple become more prominent in the latter spectrum (Figure 6b). Despite the weaker energy compared to the rotation frequency ('1X'), the BPFO components indicate the onset of the outer-race fault which occurs after the 59 th minute during the run-to-failure experiment. Figure 4: The overall procedure of the EKF-based RUL prediction method. Similarly, the spectral variations at the fault initiation of Bearing 4 are presented in Figure 7. It is seen that the envelope spectrum of the 46 th minute is dominated by the shaft rotation frequency (Figure 7a), whereas the frequency components at the BPFO and its harmonic intensify and dominate the envelope spectrum at the 47 th minute. With further inspection of the full life-cycle data, it is verified that the outer-race fault has manifested in Bearing 4 since the 47 th minute. Likewise, Figure 8 shows the envelope spectra during the fault onset for Bearing 6. It is seen that the envelope spectrum in Figure 8a is governed by the shaft speed but also masked with random components. However, although dominated by the shaft rotation frequency, the envelope spectrum at the 1418 th minute exhibits evident frequency components at the BPFI and its second multiple. Furthermore, the presence of sidebands at the synchronous \begin{table}
\begin{tabular}{c c c c c} \hline
**Operating Condition** & **Bearing ID** & \
\begin{tabular}{c} **First Detection Time** \\ **of Fault** \(t_{\text{EDT}}\) \\ \end{tabular} &
\begin{tabular}{c} **End-of-Life Time** \\ **t\({}_{\text{EOL}}\)** \\ \end{tabular} & **Fault Type** \\ \hline C1: 35 Hz/12 kN & 1 & 35 min & 126 min & Outer-Race Fault \\  & 2 & 59 min & 151 min & Outer-Race Fault \\ \hline C2: 37.5 Hz/11 kN & 3 & 446 min & 490 min & Inner-Race Fault \\  & 4 & 47 min & 160 min & Outer-Race Fault \\ \hline C3: 40 Hz/10 kN & 5 & 327 min & 353 min & Inner-Race Fault \\  & 6 & 1418 min & 1478 min & Inner-Race Fault \\ \hline \end{tabular}
\end{table} 
Table 4: Results of bearing fault detection based on envelope spectrum inspection. Figure 5: The horizontal acceleration signals of the analyzed test bearings: (**a**–**f**): Bearings 1 to 6.  
speed ('1X') suggests a strong modulation of the BPFI by the shaft speed, which indicates the occurrence of the inner-race fault in Bearing 6 since the 1418 th minute. Figure 6: The averaged envelope spectra indicating the fault initiation of Bearing 2: (**a**) The envelope spectrum at the 58 th minute. (**b**) The envelope spectrum at the 59 th minute, where the outer-race fault is identified.