## 2 Vibration Signal Analysis
### Signal Analysis  
The testing and validation of the proposed wear state estimation system was focused towards the first operating condition, with speeds of 1800 rpm and an applied force of 4000 N, for seven bearing test cases. The duration of each test case varied in length with the longest test case reaching almost 8 h and the shortest lasting only 2 h and 25 min as illustrated in Table 1 and Figure 2. Since the run-to-failure test was conducted with no artificial mechanical tampering of the constituent bearings, it can be expected that a spread of different fault types would have occurred in the bearings, involving the rolling elements, the cage and the inner-race and outer-race parts. Figure 3 demonstrates examples of typical time domain vibration signals where different fault types occur. The signal depicted in the three top panels show a very gradual increase in the vibration amplitude for bearing S. 01 before the fault occurs. The fault in bearing S. 04, depicted in the three lower panels, manifests itself as a sudden change in the vibration amplitude about three quarters through the lifetime. This indicates the occurrence of a very different, abrupt fault mode, such as the rapid formation of a catastrophic crack, a part snapping or sudden deformation due to heat induced by friction. These two examples, that of the slowly evolved degradation mode and the rapid formation fault mode, go some way to demonstrate the inherent degree of complexity and \begin{table}
\begin{tabular}{c c} \hline \hline
**Sample Bearing (S.)** & **Lifetime Duration (Hour \(|\) Min \(|\) Sec)** \\ \hline S. 01 & 7 h 47 m 00 s \\ S. 02 & 2 h 25 m 00 s \\ S. 03 & 5 h 00 m 10 s \\ S. 04 & 3 h 09 m 40 s \\ S. 05 & 6 h 23 m 30 s \\ S. 06 & 6 h 23 m 29 s \\ S. 07 & 4 h 10 m 11 s \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Timing characteristics of datasets. Figure 2: Moving average (MA) plots of 7 vibration signals are shown in the form of acceleration level (dB) across time (s), for condition 1 of the bearing dataset [62]. The MA interval is 512 points and a reference level of 1 \(\upmu\)m/s\({}^{2}\) was used, as per [66]. challenges that exist in developing robust condition-monitoring systems using machine-learning methods. It is difficult to ascertain trends and patterns in the time domain signals for the vibration amplitudes alone. Hence, a conversion to the frequency domain is necessary to observe spectral signatures and trends throughout the duration of the test to failure. Accordingly, a Short-Time Fourier Transform (STFT) process was applied to each bearing test case with the parameters set to produce a multivariate spectral description of the data. Table 2 details the STFT algorithm parameters used in this work. The concept of STFT analysis is fundamental for describing any quasi-stationary (slowly time varying) signal. In general, one can define the STFT in terms of the output of an arbitrary bank of filters. The amplitude spectrum of each frequency component of the signal was converted to a decibel scale. Figure 2 illustrates the variance of mean STFT spectral component energies for each time sample for all 7 of the bearing test cases, using the first 50 time samples of each as their reference baseline value. The natural degradation trend of a bearing does not represent a gradual, linear pattern as represented in Figure 2.  
All seven bearings test cases vary in the number of time samples as no two experiments were equal in duration, however, the avalanche-like pattern of degradation is apparent in all cases. This sudden effect makes the RUL estimation more difficult as the wear states leading to this stage share a great deal of the same feature values, and some do not vary from their initial baseline values until roughly 75% of their usable lifetime has passed. Figure 3: Plots (**a,d**) illustrate the time series vibration signals of bearings S. 01 and S. 04, respectively. Panes (**b,e**) show the variance of the total frequency amplitude values of each signal acquisition over time.