# Bearing Fault Diagnosis Approach under Data Quality Issues
## 4 Results and Discussion  
### Fractal Characteristics of Vibration Signals  
Using the DFA, the fractal characteristics of vibration signals can be inferred by calculating scaling exponent. The bearing vibration data were collected as part of an experiment conducted by the Case Western Reserve University. At a sampling rate of 48,000 samples/s, Figure 4 shows 1-second vibration measurements from bearings under four different health conditions: no fault (normal), rolling element (ball), inner race, and outer race faults. Based on the measurements, the vibration signals in faulted bearings had higher deviation compared to the normal ones. In the normal bearing, the vibration amplitude varied between \(-0.2\) m/s\({}^{2}\) and \(0.2\) m/s\({}^{2}\). However, the outer race-faulted bearing had more severe deviations with vibration amplitude up to \(6\) m/s\({}^{2}\). In the faulted bearings, the ball fault had lower vibration amplitude compared to the other two faults (inner race and outer race). Figure 4: Vibration signals from bearings under different health conditions: (**a**) normal, (**b**) ball fault, (**c**) inner race fault, (**d**) outer race fault. Figure 3: Crossover phenomenon in correlated time series: (**a**) correlated time series (scaling exponent \((\alpha)=1.2\)); (**b**) sinusoidal trend (time period \((T)=~{}1256\)); (**c**) the correlated time series with a sinusoidal trend; (**d**) the fluctuation function of the correlated series with a sinusoidal trend. We applied the DFA method on the vibration signals from 28 bearings under three health conditions, no fault (normal), minor fault (0.18 mm), and severe fault (0.53 mm). In Figure 5, all the fluctuation functions show two scaling regions resulting from the crossover phenomenon. At the short-time scale (\(3<n<17\)), the average scaling exponents of vibration signals from bearings under normal state, minor faults, and severe faults were 1.39, 1.76, and 1.85, respectively. Therefore, the calculated scaling exponents (\(\alpha_{1}\)) indicated non-stationarity (\(\alpha>1\)) and long-range correlation (\(\alpha>0.5\)) at the short-time scale. The scaling exponents (\(\alpha_{1}\)) fell between 1 and 2 and they increased as we moved from normal toward severely faulted bearings. On the other hand, the second linear region was present at the long-time scale between the windows (n), 7000 and 200,000. At the long-time scale, the average scaling exponents of vibration signals from bearings under normal state, minor faults, and severe faults were 1.07, 1.01, and 1.02, respectively. The scaling exponents (\(\alpha_{2}\)) of the vibration signals were approximately distributed around 1 for all the bearings under the different health conditions. That means the vibration signals possessed long-range correlation and behaved similarly to white noise with scaling exponent close to 1. The fractal characteristics of the vibration signals could unravel the hidden features associated with the different bearing faults. By studying these characteristics, we provided a novel method to diagnose the health condition of the bearing based on its scaling exponents. As shown in Figure 6, the scaling exponents (\(\alpha_{1}\) and \(\alpha_{2}\)) of the bearings were plotted on the x- and y-axes. The range of the scaling exponents (\(\alpha_{2}\)) at the long-time scale was the same among normal and faulted bearings. That means the minor and severe faults posed no effect on the vibration dynamics at the long-time scale. On the other hand, the faulted bearings possessed higher scaling exponents (\(\alpha_{1}\)) at the short-time scale compared to the normal ones. The normal bearings had smaller scaling exponents (\(\alpha_{1}\)) distributed between 1.2 and 1.6. Moreover, the scaling exponents increased gradually as the severity of the fault increased.