# Fault Diagnosis for a Bearing Rolling Element Using Improved VMD and HT
## 5 Analysis of Vibration Signals of the Motor Bearing Rolling Element  
### Effectiveness Analysis of the Center Frequency Method of the Multi-Threshold  
The motor bearing data came from the electrical engineering laboratory of Case Western Reserve University [82]. The deep groove ball bearing of 6205-2 RS 6 JEM SKF[M1] (SKF company, Goteborg, Sweden) was used in this experiment. The experiment platform is shown in Figure 3. It used electro-discharge technology to machine a single point of damage on the bearing. The fault diameter was 0.1778 mm, the rotating speed was 1797 r/min, the sampling frequency was 12 kHz, and the signal length was 2048 points. The bearing information of the 6205-2 RS JEM SKF deep groove ball bearing is described in Table 1. \begin{table}
\begin{tabular}{c c c c c c c} \hline
**Inside** & **Outside** & **Thickness** & **Ball** & **Pitch** & **Roller** & **Rotating** \\
**Diameter** & **Diameter** & & **Diameter** & **Diameter** & **Number** & **Speed** \\ \hline
25 mm & 52 mm & 15 mm & 8.182 mm & 44.2 mm & 9 & 1797 r/min \\ \hline \end{tabular}
\end{table} 
Table 1: The bearing information of the 6205-2 RS JEM SKF deep groove ball bearing. Figure 3: The experiment platform [79]. Figure 2: The flow of the fault diagnosis method. The fault characteristic frequency of the outer ring, \(f_{o}\), inner ring, \(f_{i}\), and the rolling element, \(f_{b}\), can been calculated by the theoretical formula as follows:  
\[f_{o}=\frac{z}{2}f_{r}(1-\frac{B_{d}}{P_{d}}\cos\phi) \tag{17}\]  
\[f_{i}=\frac{z}{2}f_{r}(1+\frac{d}{D}\cos\alpha) \tag{18}\]  
\[f_{b}=\frac{D}{2d}f_{r}\left[1-\left(\frac{d}{D}\cos\alpha\right)^{2}\right] \tag{19}\]  
where \(f_{r}\) is the axis revolution frequency (Hz), \(z\) is the number of rolling elements, \(B_{d}\) is the diameter of the rolling elements, \(P_{d}\) is the pitch diameter of the rolling elements, and \(\phi\) is the pressure angle of the rolling elements. The theoretical calculation values of the fault characteristic frequency were calculated and are shown in Table 2.  
The time domain waveform and frequency domain waveform of the motor bearing rolling element fault signal under no-load are shown in Figures 4 and 5. The improved VMD decomposition method was used to decompose the fault signal of a motor bearing rolling element under no-load, and the center frequency of each IMF component under different scales is shown in Figure 6.  
The center frequency method of the multi-threshold was used to decompose the rolling element data in order to determine the modal number of the VMD decomposition. When \(K\) = 2, 3, 4, 5, the difference between the maximum value of the center frequency value of the corresponding mode under different scales and the maximum value of the center frequency of all modes is less than \(T_{1}(\Delta cf\_max_{IMF_{K}}\leq T_{1})\), which is not met. When there is \(K=6,7\), the difference between the value of the center frequency of the \(L-th\) IMF component and the value of the center frequency of the \(L-1-th\) IMF component under the decomposition scale, \(K\) (\(K=L\)), is less than or equal to \(T_{2}(\Delta cf_{IMF_{K}}\leq T_{2})\), which is not met, so the maximum value of all modal center frequencies was \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Inner Ring** & **Outer Ring** & **Rolling Element** & **Switching Frequency** \\ \hline
162.1852 (Hz) & 107.2931 (Hz) & 141.0751 (Hz) & 29.2 (Hz) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Fault characteristic frequencies. Figure 4: Time domain waveform. recalculated. It can be seen from Figure 3 that when \(K=7\), the center frequency of the seventh IMF component obtains the maximum value of all modal center frequencies, so the center frequency of the sixth IMF component is determined as the maximum value of all modal center frequencies, and the threshold was recalculated \(T_{1}=a*cf\_max_{IMF}(a=8\%)\), \(T_{2}=b*cf\_max_{IMF}\) (\(b=15\%\)), and the threshold set to \(T_{3}=c*cf\_max_{IMF}\).