# Fault Prediction of Mechanical Equipment Based on Hilbert-Full-Vector Spectrum and TCDAN
## 4 Experiment Verification
### Application of Hilbert-Full-Vector Spectrum  
Taking the data collected at 2510 min of Bearing 3_1 as an example, the vibration data collected in the horizontal and vertical directions were processed using the ordinary Hilbert-envelope spectrum [36; 37] and Hilbert-full-vector spectrum technology, respectively, and the processing results are shown in Figures 9 and 10.  
As shown in Figure 9, the Hilbert-envelope spectrum in both directions has a certain prominent reflection at the outer-ring fault frequency of Bearing 3_1, which is consistent with the actual situation of bearing outer-ring fault, but the prominent characteristic frequency of the two is not completely consistent as a whole. According to Figure 9a, it can be seen that the horizontal Hilbert-envelope spectrum does not prominently reflect the 2 times frequency of the rotation frequency of Bearing 3_1 and the 2 times frequency of the outer-ring fault frequency. According to Figure 9b, it can be seen that the vertical Hilbert-envelope spectrum does not prominently reflect the 1 times frequency of the rotation  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Value** & **Parameter** & **Value** \\ \hline Diameter of inner raceway/mm & 29.30 & \begin{tabular}{c} Diameter of bearing rolling \\ element/mm \\ \end{tabular} & 7.92 \\ Diameter of outer raceway/mm & 39.80 & \
\begin{tabular}{c} The number of bearing \\ rolling element \\ \end{tabular} & 8 \\ The pitch diameter of the & 34.55 &
\begin{tabular}{c} contact angle/\({}^{\circ}\) \\ \end{tabular} & 0 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters of Bearing 2_1 and Bearing 3_1.  
frequency of Bearing 3_1. The Hilbert-envelope spectrum in either direction does not fully reflect the state information of Bearing 3_1.  
As shown in Figure 10, the Hilbert-full-vector spectrum is prominently reflected in the rotation frequency, outer-ring fault frequency, and the corresponding frequency doubling of Bearing 3_1, and the outer-ring characteristic frequency and the corresponding frequency doubling are even more prominent, which reflects the outer-ring fault information of Bearing 3_1. Compared with the traditional Hilbert-envelope spectrum, the Hilbert-full  
Figure 10: Hilbert-envelope full-vector spectrum.  
Figure 9: Hilbert-envelope spectrum: (**a**) Hilbert-envelope spectrum of horizontal direction; (**b**) Hilbert-envelope spectrum of vertical direction.  
vector spectrum contains more comprehensive information about Bearing 3_1 by fusing the vibration data in two directions, and can better reflect the running state of Bearing 3_1.  
The samples of the Bearing 3_1 health state, initial fault stage, middle fault stage, and late fault stage are, respectively, selected and processed based on the Hilbert-full-vector spectrum technique, as shown in Figure 11. The corresponding amplitudes of the rotation frequency, the outer-ring fault frequency, and the outer-ring fault frequency doubling of the bearing change obviously in different operating stages of Bearing 3_1. This change proves the effectiveness of using the corresponding amplitudes of different characteristic frequencies as equipment fault-prediction features.