## 4 Vibration Characteristics for Bearing Clearance Monitoring
### Bearing Clearance Monitoring  
In condition monitoring practice, vibration responses in the time domain are often evaluated based on RMS value in many systems [34]. Study has shown that RMS value could serve as a leading indicator for early detection of faults in WT, giving WT owners time to schedule operations and maintenance [35]. RMS values of signals in the load and non-load directions under different clearance and load conditions are shown in Figure 13.  
In general, excepting the resonance peak in the low-loaded conditions, the RMS value showed an uptrend with the increase in clearance, which was more obvious in the load direction. However, it should be noted that the RMS value was not increasing monotonically with the increment of clearances and loads. In Figure 13a, there is a resonance peak in low-loaded conditions, i.e., 800 and 1600 N. This may have been caused by the differences of the load distribution under different clearance and load conditions. Due to fluctuations of the RMS values, it is not accurate enough to be used as an indicator to reflect the bearing clearance, especially when the load is relatively light. Thus, it is necessary to seek other indicators to monitor the variations of clearances.  
The spectrum exhibits the amplitude distribution and energy distribution of each frequency component. Envelope spectrum is an effective tool to reveal the amplitude modulation frequency components [36], which is a widely used method to diagnose bearing faults for mechanical machines. Through the envelope spectrum, as shown in Figure 14, BPFO and its harmonics were the primary  
Figure 12: Maximum of the dynamic force \(F_{r1}\) (**a**) under different clearances and loads and (**b**) increased proportion compared with 0 \(\upmu\)m.  
Figure 13: RMS value in (**a**) load and (**b**) non-load direction.  
modulation source in the bearing vibration signals. Note that \"800-0\" in Figure 14 means the load is 800 N and the clearance is 0\(\upmu\)m. Such expression is adopted in the following figures.  
Figure 15 depicts the spectrum of signal in the load and non-load directions, respectively. As can be seen, two peaked regions can be clearly distinguished from the spectrum in both directions, with one in the low frequency band from 500 to 1500 Hz and one in the high frequency band from 6000 to 9000 Hz. In addition, it can be observed that the low frequency band shifted right slightly with the increase in loads.  
Due to the difference between the low and high frequency band, spectral centroid [37] based on FFT spectrum was calculated according to Equation (16), as shown in Figure 16.  
\[\textit{Centroid}=\frac{\sum_{n=0}^{N-1}f(n)x(n)}{\sum_{n=0}^{N-1}x(n)} \tag{16}\]  
From the spectral centroid, signals of the low frequency band and the high frequency band in the load direction showed a relatively obvious uptrend with the increase in clearance. However, when under a relatively light load, especially 800 and 1600 N, the centroid line displayed large fluctuations under large clearances. In addition, in either the load direction or non-load direction, these lines showed much crossing, which still brought difficulty for monitoring bearing clearance. Fortunately, when under a widely used clearance range, from 0 to 60 \(\upmu\)m, these lines showed better separability, which showed a better performance over the RMS value.  
Figure 14: Envelope spectrum in (**a**) load and (**b**) non-load direction.  
Figure 15: FFT spectrum in (**a**) load and (**b**) non-load direction.  
In order to verify the effect of the spectral centroid in a real measured signal, the vibration signals of Section 3 were used to calculate the index values. In addition, RMS values of the low frequency band were calculated as a comparison to the spectral centroid, as shown in Figure 17. It was found that the spectral centroid showed a better performance in clearance monitoring.