## 5 Experimental Verification and Analysis
### Case Study I: Fault Diagnosis of Rolling Bearing  
#### 5.1.1 Data Set Description  
The open vibration dataset of rolling bearings established by Paderborn University (Germany) was selected for the study [25]. The components of the vibration test rig used  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Method** & **RMSDE** & **MSE** & **MFE** & **MPE** \\ \hline Average calculation time & 0.267s & 1.874s & 2.058s & 0.091s \\ \hline \hline \end{tabular}
\end{table}
Table 1: Average calculation time comparison.  
Figure 8: Comparison of computational time consumption of different entropy methods.  
include the drive motor, measurement shaft, bearing test module, flywheel, load motor, and other components, as shown in Figure 9. Vibration data acquisition can be achieved by mounting rolling bearings with different prefabricated faults into the test module. For the prefabricated faults of the inner and outer rings of the bearings, three different machining methods are included: sharp trench by EDM, drilling and pitting by an electric engraver. An illustration of these three types of fault machining is shown in Figure 10.  
The first two methods mainly rely on mechanical processing, which is simpler, more accurate and easy to replicate but lacks some correlation with the actual bearing damage, while the third method is to form artificial damage by an electric engraving machine with irregular surface structure and depth, which can better simulate bearing pitting. The model of the tested bearing is a 6203 deep groove ball bearing. The specific structural parameters of the bearing are shown in Table 2. The test conditions in this data set contain a variety of different speed and load settings. In this section, the test conditions for the data selected for analysis are speed 1500 rpm, load torque 0.7 Nm, radial force 400 N and sample frequency 64 kHz. By combining the parameters in Table 2 with the theoretical calculation formula for bearing fault characteristic frequencies, the characteristic fault frequencies of the test bearing can be obtained. The specific calculation process can be referred to in reference [26]. Here, the characteristic frequency \(f_{0}=76.25\) Hz for the outer ring fault of the test bearing and \(f_{i}=123.3\) Hz for the inner ring fault can be calculated.  
Figure 10: Test bearings with artificial damage: (**a**) Sharp trench by EDM (**b**) Drilling (**c**) Pitting by electric engraver.  
Figure 9: Experimental platform for bearing datasets at Paderborn University.  
In this section, five different health states of bearing test data are selected for analysis, and the bearing codes are K001, K101, K103, KA01 and KA05, and the specific information is shown in Table 3. The extent of the damage describes the size of the damage in normalized levels, which are independent of the bearing size. The levels are based on the length of the damage, as this is a determining factor for the signal output. When the extent of damage is equal to 1, that means the damage size is less than 2 mm. The time domain waveforms of bearing vibration signals for the five health states are shown in Figure 11. The length of the bearing data for each health state is about 256,000 points, and by dividing each data into 50 samples, the data length of each sample is 4096 points, so the number of samples is constructed with 250 in total.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Type of bearing & 6203 \\ Pitch circle diameter & 28.55 mm \\ Diameter of inner raceway & 24.0 mm \\ Diameter of outer raceway & 33.1 mm \\ Diameter of rolling element & 6.75 mm \\ Number of rolling elements & 8 \\ Contact Angle & 0\({}^{\circ}\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: The specific structural parameters of the bearing.  
Figure 11: _Cont._