## 2 Fault Diagnosis of Rolling Bearing Based on VMD and ANN  
### Data Preprocessing  
The raw vibration dataset of rolling bearings are from the Case Western Reserve University Bearing Data Center (CWRU). The dataset contains 10 rolling bearing fault states of nine fault types and one normal state. There are three kinds of fault locations in the inner race, outer race and ball. The diameters of the faults were 0.007 inches, 0.014 inches, 0.021 inches. Table 1 shows the details of the fault dataset in the bearing fault experiment. The sampling frequency of the rolling bearing vibration signal was 48 kHz.  
For one type of bearing fault status, the random step increment slip method [28] was used to intercept and divide each vibration signal into 700 groups of training samples and 100 groups of testing samples, as shown in Figure 1. A new sample was obtained by a random start point, and the next random start point slips with a certain distance increment. For each raw vibration signal of the 10 fault types, 1024 points were collected as the training data. For the 10 types of fault types, we obtained a shape of 8000 \(\times\) 1024 training data and a shape of 8000 \(\times\) 10 labels of the fault types represented by a one-hot code. Moreover, the min-max normalization method was adopted to map the vibration signals to interval [0, 1]. Finally, the raw vibration signal was divided into 500 groups of training samples, 200 groups of validation samples and 100 groups of testing samples for each fault type.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**No.**} & **Fault** & \multirow{2}{*}{_d (mils)_} & **Training** & **Validation** & \multirow{2}{*}{**Testing Set**} & **Target Label** \\  & **Location** & & **Set** & **Set** & & \\ \hline
1 & Inner Race & 0.007 & 500 & 200 & 100 & [1 0 0 0 0 0 0 0 0 0 ] \\
2 & Inner Race & 0.014 & 500 & 200 & 100 & [0 1 0 0 0 0 0 0 0 ] \\
3 & Inner Race & 0.021 & 500 & 200 & 100 & [0 0 1 0 0 0 0 0 0 0 ] \\
4 & Outer Race & 0.007 & 500 & 200 & 100 & [0 0 0 1 0 0 0 0 0 0 0 ] \\
5 & Outer Race & 0.014 & 500 & 200 & 100 & [0 0 0 0 0 0 0 0 0 0 ] \\
6 & Outer Race & 0.021 & 500 & 200 & 100 & [0 0 0 0 0 1 0 0 0 ] \\
7 & Ball & 0.007 & 500 & 200 & 100 & [0 0 0 0 0 0 1 0 0 ] \\
8 & Ball & 0.014 & 500 & 200 & 100 & [0 0 0 0 0 0 0 1 0 ] \\
9 & Ball & 0.021 & 500 & 200 & 100 & [0 0 0 0 0 0 0 1 0 ] \\
10 & Normal & Normal & 500 & 200 & 100 & [0 0 0 0 0 0 0 0 1 ] \\ \hline \hline \end{tabular} Here, \(d\) denotes the fault diameter.  
\end{table}
Table 1: Details of the dataset in the bearing fault experiment.  
To study the influences of the sample quantity and sampling period on the prediction accuracy and computation time of the proposed method, we processed the vibration data of rolling bearings from two aspects. On the one hand, to study the sample quantity's influences on the model's performance, it extracted samples with the number ranging from 5000 to 30,000 as the input for the model to explore the sample number's influences on the prediction accuracy and computation time of the model. As shown by the research results in Section 3.3, the optimal sample number for fault diagnosis based on VMD and ANN was 8000. On the other hand, the impact of the sampling period on the fault classification accuracy and computation time was studied. As shown in Figure 1, the length of every sampling point for the data on bearing faults was 1024. The resampling of all fault data was carried out to attain the sampling data of different sampling periods, with the specific procedure shown in Figure 2. Though our investigation, it was found that when the sampling period was 1/6 ms, the classification accuracy of the bearing faults would be maximized and the computation time would be minimized. Therefore, the optimal sampling period in this paper is defined as 1/6 ms.