# Vibration Energy at Damage-Based Statistical Approach to Detect Multiple Damages in Roller Bearings
## 2 Vibration Energy at Damage-Based Statistical Approach
\[BPFO=\frac{n}{2}\bigg{[}1-\frac{D_{B}}{D_{P}}\cos\theta\bigg{]} \tag{6}\]  
\[BPFI=\frac{n}{2}\bigg{[}1+\frac{D_{B}}{D_{P}}\cos\theta\bigg{]} \tag{7}\]  
\[BSF=\frac{D_{P}}{2D_{B}}\bigg{[}1-\bigg{(}\frac{D_{B}}{D_{P}}\bigg{)}^{2}\cos^ {2}\theta\bigg{]} \tag{8}\]  
\[FTF=\frac{1}{2}\bigg{[}1-\frac{D_{B}}{D_{P}}\cos\theta\bigg{]} \tag{9}\]  
where, \(D_{B}\) is the ball diameter, \(D_{P}\) is the pitch diameter, \(n\) is the number of balls, and \(\theta\) is the angle of contact. The ER-16K faulty rolling element bearings, with ball diameter \(D_{B}=8\) mm and pitch diameter \(D_{P}=38\) mm, number of balls \(n=9\), and angle of contact \(\theta=9.08^{\circ}\), are used in this study. By putting the values of \(D_{B}\), \(D_{P}\), \(n\), and \(\theta\) in the above Equations (6)-(9), the characteristic fault frequencies of the roller bearings are calculated and shown in Table 1.  
The characteristics fault frequencies are obtained by multiplying the operating speed by the multipliers calculated in the second row of \[|x|=X\quad,\ \left|\dot{x}\right|=\omega X\quad,\ \left|\ddot{x}\right|= \omega^{2}X \tag{2}\]0 The peaks at these characteristics fault frequencies and their respective harmonics are analyzed, and the bearing is often regarded as a damaged bearing if the vibrations are high at these characteristics fault frequencies (BPFO, BPFI, BSF, and FTF). However, in reality, these peaks are submerged by the noise, and it is not an easy task to recognize the peaks at these characteristic fault \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Operating Speed RPM** & \begin{tabular}{c} **Ball Pass Frequency** \\ **Outer (BPFO)** \\ \end{tabular} & \begin{tabular}{c} **Ball Pass Frequency** \\ **Inner (BPFI)** \\ \end{tabular} & \
\begin{tabular}{c} **Ball Spin Frequency** \\ **(BSF)** \\ \end{tabular} &
\begin{tabular}{c} **Fundamental Train** \\ **Frequency (FTF)** \\ \end{tabular} \\ \hline  & 3.572 & 5.43 & 2.322 & 0.402 \\ \hline
1002 & 3579.144 & 5440.86 & 2326.644 & 402.804 \\ \hline
1500 & 5358 & 8145 & 3483 & 603 \\ \hline
2400 & 8572.8 & 13,032 & 5572.8 & 964.8 \\ \hline
3000 & 10,716 & 16,290 & 6966 & 1206 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The characteristic fault frequency multipliers of roller bearings used in this study. frequencies. Thus, a novel method is proposed in this study, based on the vibration energy at damage. The histograms, along with the kurtosis values, were used to analyze the vibration data of an intact and damaged bearing. The kurtosis is a measurement of the frequency of extreme values, or peakedness of distribution. The kurtosis \((K)\) can be calculated from the following Equation (10):  
\[K=\ \frac{m_{4}}{{m_{2}}^{2}}=\ \frac{m_{4}}{{(\sigma^{2})}^{2}} \tag{10}\]  
where \(m_{4}\) is the fourth moment, \(m_{2}\) is the second moment, and \(\sigma\) is the variance. First, the deviation from the mean is calculated by using Equation (11), then the second moment \((m_{2})\), variance \((\sigma)\), and the fourth moment \((m_{4})\) can be calculated from Equations (12)-(14), respectively. The averaged value of kurtosis is used in this study so that the accuracy of measurement can be improved. The averaged value of kurtosis is the mathematical average of the numeric (absolute) value of all data blocks. \[\textit{deviation}\ \textit{from}\ \textit{the}\ \textit{mean}=X_{i}-X_{avg} \tag{11}\]  
\[m_{2}=\frac{\sum_{i=1}^{n}(X_{i}-X_{avg})^{2}}{n} \tag{12}\]  
\[\sigma^{2}=\frac{\sum_{i=1}^{n}(X_{i}-X_{avg})^{2}}{n} \tag{13}\]  
\[m_{4}=\frac{\sum_{i=1}^{n}(X_{i}-X_{avg})^{4}}{n} \tag{14}\]  
where, \(X_{i}\) is the \(i\)th observation, \(X_{avg}\) is the arithmetic average of all values, and \(n\) is the number of values. The formula to calculate the second moment \((m_{2})\) and variance \((\sigma)\) is the same.