*4.2. Decomposition of the Cylinder Vibration Signal*

First, we present the typical working conditions corresponding to the three load conditions considered in this analysis: working condition 1 (a fill rate of 10%, a powder-to-ball ratio of 0.4, and a grinding concentration of 0.5), working condition 2 (a fill rate of 30%, a powder-to-ball ratio of 0.6, and a grinding concentration of 0.5), and working condition 3 (a fill rate of 50%, a powder-to-ball ratio of 0.8, and a grinding concentration of 0.5). The waveforms of the cylinder vibration signals recorded under these three working conditions are shown in Figure 8.

**Figure 8.** Waveforms of the original cylinder vibration signals: (**a**) working condition 1; (**b**) working condition 2; (**c**) working condition 3.

As Figure 8 shows, there is a large amount of noise in the vibration signal from the mill cylinder in all three load states, which makes it difficult to effectively extract feature information. To extract the characteristics of the vibration signal of the cylinder, the original signal must be preprocessed. The preprocessing steps are as follows.


**Figure 9.** Relationship between the correlation coefficient and the sequence number of the amplitude modulation–frequency modulation (AM-FM) component.

As shown in Figure 9, the correlation coefficients between the AM-FM1, AM-FM2, and AM-FM5 components and the original signal for working condition 1 were greater than the threshold value of 0.21437. Thus, these components were identified as sensitive AM-FM components that characterize the vibration signal of the cylinder. For working condition 2, the AM-FM2, AM-FM4, and AM-FM6 components, with correlation coefficients greater than the threshold value of 0.23872, were selected as the sensitive mode components. For working condition 3, the AM-FM1, AM-FM3, and AM-FM4 components, with correlation coefficients greater than the threshold value of 0.23872, were selected as the sensitive mode components. All AM-FM components with correlation coefficients smaller than the corresponding threshold were removed.

3. The selected sensitive modal components are reconstructed, and the results are shown in Figure 10.

**Figure 10.** *Cont.*

**Figure 10.** Waveforms of the reconstructed cylinder vibration signals: (**a**) working condition 1; (**b**) working condition 2; (**c**) working condition 3.

Based on a comparison of Figures 8 and 10, the trend of the reconstructed signal waveform is basically the same as that of the original signal. Compared with the original signal, the impact profile of the reconstructed signal curve is obviously distinct, but it preserves the characteristic information of the original signal while effectively denoising the signal. To further quantitatively highlight the preprocessing effect in this paper, the EMD algorithm, EWT algorithm, and improved EWT algorithm are used to decompose the original signals of the three working conditions, and the sensitive components are reconstructed by the correlation coefficient method. Additionally, the SNR is introduced into the comparative analysis before and after processing to qualitatively analyze the comparison results, and the results are shown in Table 1.


**Table 1.** Signal-to-noise ratio (SNR) before and after signal processing.

As Table 1 shows, compared with the original signal, the SNR of the reconstructed signal processed by the improved EWT algorithm increases by 13.32 dB, 12.78 dB, and 17.52 dB under three typical working conditions, which indicates that the noise is considerably reduced after applying the improved EWT algorithm. Compared with those of the EMD algorithm and the EWT algorithm, the SNR of the reconstructed signal processed by the improved EWT algorithm increases the most. Therefore, the preprocessing effect of the improved EWT algorithm is best.

#### *4.3. Decomposition of the Cylinder Vibration Signal*

The FE of the reconstructed signal is calculated, and five groups of samples are assessed for each type of ball mill load state. The average value of the FE of the three-state data is calculated, as shown in Table 2.


**Table 2.** Fuzzy entropy values of three types of load state vibration signals.

Table 2 shows that the FE value of the vibration signal varies by load state and that the FE value of the vibration signal under the same load state fluctuates back and forth near the average value. By comparing the FE values of three different load vibration signals, the FE values of the underloaded state are found to be relatively large, which is due to the relatively small amount of steel ball and mineral material in the cylinder under this condition. Additionally, the collision frequency between the mineral material and the steel ball in grinding production increases with the movement of the cylinder body to a certain height under the action of friction, and the collision frequency with other steel balls, minerals, and the cylinder walls is high in the process of falling. Energy is mainly consumed in the collisions between the steel ball and the tube wall and between the steel ball and other steel balls; thus, the vibration signal is complex, and the signal is highly random. However, the FE value under overloaded conditions is relatively small because there are more steel balls and minerals in the cylinder under these conditions, causing the steel ball and minerals to undergo peristalsis in the grinding process. In this case, the randomness of the signal is small. Under a normal load, energy is mainly used for grinding the quantity of minerals, and so the complexity of generating a signal is relatively moderate. For underloaded conditions and a normal load state, the sample entropy values are similar, and individual overlap occurs, which results in a discriminating effect. Therefore, MSE is introduced into the analysis of the mill vibration signal. The MFE of the reconstructed signal that can characterize the characteristic information of the vibration signal under three different load conditions is calculated. To highlight the superiority of the feature extraction method used in this paper, four combination methods (EWT-MSE, EWT-MFE, improved EWT-MSE, and improved EWT-MFE) are used to analyze the vibration signals of the cylinder of the ball mill under three load conditions. The mean value and standard deviation curve of the three states (20 samples per group) are shown in Figure 11. The parameter selection process of the algorithm is as described above.

**Figure 11.** Reconstructed signal waveform under three working conditions: (**a**) EWT-MSE, (**b**) improved EWT-MSE, (**c**) EWT-MFE, and (**d**) improved EWT-MFE.

In Figure 11, it is evident that the order of the mean value of FE of the vibration signal of the ball mill cylinder under three working conditions displays the following order: underloaded > normal load > overloaded. Specifically, as the ball mill load increases, the amplitude of each component of the vibration signal in the spectrum obviously increases, which leads to a decrease in entropy. Although the variation trend of the FE of the cylinder vibration signal with the scale factor is the same in different load states, the fluctuation range of the entropy value varies, which indicates that FE can be effectively used to identify the load state. By comparing the four graphs, we see that there are obvious fluctuations and interval intersections between the EWT-MSE method and the EWT-MFE method. Although the entropy curve of the improved EWT-MSE method is smooth and the three states are distinguished to a certain extent, there are still overlap and intersection issues at small scales, which may lead to judgment errors. However, the MSE curves of the three load states obtained with the improved EWT-MFE method are smooth, and the fluctuation intervals have obvious limits. This finding indicates that the improved EWT-MFE method can effectively distinguish among the three load states of the ball mill.

#### *4.4. Training and Testing*

To verify the effectiveness of the proposed load identification model for a ball mill, 3 × 100 samples were randomly selected from each of the three classes of vibration signals, including 150 as training samples and 150 as test samples. The selected samples were first decomposed via the improved EWT method. The sensitive mode component signal with load state information was screened by the correlation coefficient method and reconstructed. Then, the MFE of the reconstructed signal was normalized as the input of the load state identification model of the ball mill based on AEPSO\_PNN, and the load state of the ball mill was output. To highlight the superiority of AEPSO\_PNN classification and identification, three clustering methods, namely, PNN classification, back propagation (BP) neural

network, and Bayes identification methods, were trained and tested with the abovementioned samples. Then, the identification results were compared with the AEPSO\_PNN identification results. For simplicity of description, the underloaded, normal load, and overloaded conditions are indicated by working condition numbers 1, 2 and 3, respectively. The identification effects of various classification methods are shown in Figure 12 and Table 3.

**Figure 12.** Recognition results of test samples for each classifier. (**a**) BP neural network; (**b**) Bayes identification method; (**c**) PNN; (**d**) AEPSO\_PNN.

**Table 3.** Singular value entropy results for the three working conditions. BP: back propagation. probabilistic neural network (PNN): probabilistic neural network.


Figure 12 and Table 3 show that the predicted load state of the AEPSO\_PNN model of ball mill load state recognition is largely consistent with the real state. Only four samples are misdiagnosed, and the overall recognition accuracy is 97.3%. Specifically, the recognition accuracy of AEPSO\_PNN classification for three different load states is 96%, 98%, and 98%, all of which are high recognition levels. The BP neural network, Bayes discriminant method, and PNN classification can also achieve

effective load identification. The highest accuracies of these methods are 89.3%, 92.0%, and 94.0%. Compared with the back propagation (BP) neural network, Bayes discriminant method, and PNN classification, AEPSO\_PNN classification increases the overall recognition rate by 8%, 5.3%, and 3.3%. The results show that the mill load identification method based on the improved EWT-MFE method and AEPSO\_PNN classification is effective, and the identification effect is excellent. Thus, this method provides a new approach for ball mill load identification.
