# Research on Rolling Bearing Fault Diagnosis Method Based on ECA-MRANet
## 2 Fault Feature Extraction of Rolling Bearings
### Feature Extraction Method for Rolling Bearings Based on SSA Optimization VMD
#### 2.2.3 Establishing a SSA-VMD Algorithm Framework  
In this paper, the SSA algorithm will be used to find the optimum of [\(\alpha\),K]; however, the key to optimizing the VMD parameters by the use of an SSA lies in choosing the appropriate fitness function to guide the global optimum. Envelope entropy [20] is a metric used to characterize signal complexity and irregularity. In bearing fault diagnosis, a smaller envelope entropy implies a higher smoothness of the signal; this can provide a clearer signal characterization, which helps in the detection of bearing anomalies and faults. Kurtosis [21] is a metric used to characterize the statistical properties of a signal. Greater kurtosis indicates sharper peaks in the signal, with more outliers relative to the normal distribution. In bearing fault diagnosis, the fault usually causes nonlinear changes and increased irregularities in the signal, which leads to outliers in the signal. Larger kurtosis can help detect these anomalies and help to detect bearing failures. Therefore, in this paper, the combination of envelope entropy and kurtosis is selected as the fitness function of the SSA for the optimization of the VMD parameters.  
After the original signal is decomposed by the VMD algorithm, the envelope entropy and kurtosis of each IMF produced can be expressed as:  
\[p_{i,n}=h_{i}(n)/\sum_{n=1}^{N}h_{i}(n) \tag{9}\]  
\[E_{i}=-\sum_{n=1}^{N}p_{i,n}lgp_{i,n} \tag{10}\]  
\[K_{i}=\frac{E(u_{i}(n)-\mu)^{4}}{\sigma^{4}} \tag{11}\]  
where \(u_{i}(n)\) is the is the decomposed IMF, \(h_{i}(n)\) is the envelope signal of \(u_{i}(n)\) demodulated by [PERSON], \(p_{i,n}\) is the probability of the nth point of \(u_{i}(n)\), \(E_{i}\) is the envelope entropy of \(u_{i}(n)\), \(\mu\) is the mean of \(u_{i}(n)\), \(\sigma\) is \(u_{i}(n)\) standard deviation, and \(K_{i}\) is the kurtosis of \(u_{i}(n)\).  
The fitness function is shown in Equation (12):  
\[fit=E_{i}+1/K_{i} \tag{12}\]  
Based on a study of the literature, it is found that the range of K values is [3; 10], the range of \(\alpha\) values is [100, 3000], and the best combination of [\(\alpha\),K] is sought from adaptation, as shown in Figure 3.  
Based on the above method, the population size of the SSA algorithm is set to 20, and the number of iterations is set to 20. The fault signal of the inner ring of the bearing shown in Figure 1 is used as an example for analysis. Figure 4 shows the fitness function curve of SSA-optimized VMD, and the fitness value reaches the minimum value of 9.7936 after 11 iterations, and the corresponding [\(\alpha\),K] is [142,4]. This parameter is used to decompose the VMD of the inner ring of the bearing, and Figure 5 shows the time and frequency domain diagrams of each mode.  
Figure 4: Iterative curve of the fitness function of the fault signal of the inner ring of the bearing.  
Figure 3: Flowchart of SSA-optimized VMD.  
As can be seen from Figure 5, the center frequency of each IMF is evenly distributed, and there is no modal aliasing, indicating that the decomposition effect is good. The frequency spectrum is also consistent with the time-frequency diagram of the continuous wavelet variation, which verifies the correctness of the above method. In order to screen out the IMF components with more obvious fault characteristics, the envelope entropy, kurtosis, and fitness function values of each IMF were calculated separately, as shown in Table 2.  
In Table 2, the IMF3 modal component has the smallest value of the fitness function, which indicates that the fault characteristics of this component are more obvious relative to other components. Therefore, in this paper, the modal component with the smallest fitness function value after VMD decomposition is selected for fault detection and analysis.