Rolling bearing state assessment based on the composite multiscale weight slope entropy and hierarchical prototype-based approach

Abstract

To evaluate the state of rolling bearing more accurately, a new feature called composite multiscale weight slope entropy was proposed for the complexity measurement of vibration signals. On the basis of analyzing the fault signal structure, the new feature could consider the influence of the nonlinearity, the multiscale characteristics, the fluctuation of the amplitude, and the amplitude itself on the fault signals. In the following, different fault types and the corresponding damage degree of rolling bearings were identified with the hierarchical prototype-based approach. Compared with the results of different modified slope entropy, it is shown that composite multiscale weight slope entropy could significantly improve the identification accuracy. In the two designed testing schemes, ten and sixty state types of rolling bearings are respectively calculated, and the identification accuracy could reach up to 100% and 96.5% respectively, which illustrate the effectiveness and the validity of the proposed approach.

Introduction

Rolling bearings are critical components of complex devices such as motors. ¹ Timely diagnosis of their operation status and replacement could ensure well dynamic characteristics and motion accuracy of mechanical equipment, reduce the related failure of other components of equipment, so as to improve economic benefits and ensure the safety of operators. The operating state of the rolling bearing includes two aspects: fault type judgment and damage degree identification. The identification process is mainly divided into three parts: feature extraction, feature reduction, and fault classification.^2,3

In order to extract abundant feature information from vibration signals and reduce the noise interference in the process of feature extraction, many scholars have studied the features representing the operating state of rolling bearings, including time-domain features, frequency-domain features, and nonlinear features such as L-Z coefficient and entropy values.^4,5 Entropy is a metric of the degree of the data disorder and it has been widely investigated and applied for its good nonlinear expression of signals,^6,7 such as approximate entropy, ⁸ sample entropy, ⁹ fuzzy entropy, ¹⁰ hierarchical entropy, ¹¹ dispersion entropy, ¹² permutation entropy,^13,14 bubble entropy, ¹⁵ and gray entropy, ¹⁶ etc. The above types of entropy have made great achievements in improving the accuracy of rolling bearing fault diagnosis, but they still have further improvements, such as coarse-grained process of the signals to obtain entropy in multiple scales. ¹⁷ Fadlallah et al. ¹⁸ proposed weighted permutation entropy on the basis of permutation entropy, considering the influence of amplitude fluctuation. Based on the improvement of entropy, Zhang et al. ¹⁹ utilized the multiscale bubble entropy to identify the fault types and the damage degree of rolling bearings. Gan et al. ²⁰ investigated the influence of multi-scale, fluctuation of each scale and amplitude fluctuation of data, and then proposed composite multiscale weighted permutation entropy for bearing diagnosis. Li et al. ²¹ explored the application of refined composite multivariate multiscale dispersion entropy in rolling bearing fault diagnosis. Zhang et al. ²² achieved rolling bearing fault diagnosis with the refined composite generalized multiscale bubble entropy. Xi et al. ²³ presents refined composite multivariate multiscale fluctuation dispersion entropy to extract the recognition information of multi-channel signals with different scale factors. Sheng et al. ²⁴ constructed local mean decomposition Shannon entropy and Ma et al. ²⁵ employed wavelet packet-energy entropy respectively for bearing fault diagnosis with high accuracy.

In particular, data symbolization ²⁶ can generate low-resolution data from high-resolution data, so as to capture large-scale features and reduce the impact of dynamic noise and measurement noise on statistical algorithms. The symbolization of rolling bearing vibration data sequences could take the form of dichotomy-based symbolization, probability distribution-based symbolization and angular interval symbolization. ²⁷ At present, there are some problems, such as the lack of fine-grained symbolic entropy division, parameter selection based mainly on experience, failure to consider multiscale situations, and lack of adaptive capability for multiple fault types in rolling bearings, all of which limit the development of symbolic entropy analysis of complex systems. Slope entropy^28,29 is a measure developed from symbol entropy. Considering the amplitude, it can reduce the noise interference. Therefore, this paper proposes an improved entropy-composite multiscale weighted slope entropy to represent the operating state of rolling bearings, and combines the hierarchical prototype-based approach ³⁰ to realize the operating state identification of rolling bearings.

The rest of this article is organized as follows: the CMWSlE is derived in detail in Section “Composite multiscale weight slope entropy.” Section “Hierarchical prototype-based approach” briefly introduces HPA and a flowchart is presented in Section “Flowchart of rolling bearing state assessment.” Verification of the proposed method is performed with experiment data sets in Section “Experimental validation,” and the conclusions are drawn in section “Conclusion.”

Composite multiscale weight slope entropy

Slope entropy

Standardization of vibration sequences X is performed as:

$$\mathit{x}=\left(\mathit{X}-{\mu }_{\mathit{X}}\right)/{\sigma }_{\mathit{X}}$$

(1)

in which ${\mu }_{\mathit{X}}$ and ${\sigma }_{\mathit{X}}$ respectively are the mean and the standard deviation (Std.) of vibration sequences $\mathit{X}$ , and then a new time series $\mathit{x}=\{x_1,x_2,\cdots ,x_N\}$ is obtained according to equation (1) to ensure the following slope entropy computation.

The subsequence $X_i^{m,\tau }=\{x\left(i\right),x(i+\tau ),\cdots ,x(i+(m-1)\tau )\}$ , $i=1,2,\cdots ,N-(m-1)\tau$ with embedding dimension m and time delay τ is extracted from the vibration sequence x , and the difference in adjacent amplitudes is x_i - x_i-1. As shown in Figure 1, set the number of partitions q = 5 and the two threshold values as γ and δ. The procedure of data symbolization is as follows:

• (1) When $x_i>x_{i-1}+\gamma$ , the symbol is set as +2;

• (2) When $x_i>x_{i-1}+\delta$ and $x_i\le x_{i-1}+\gamma$ , $x_i$ is below the 45° angle and above the small region near 0, the symbol is set as +1;

• (3) When $|x_i-x_{i-1}|\le \delta$ , $x_i$ is in the small region around 0, the symbol is set as 0;

• When $x_i<x_{i-1}-\delta$ and $x_i\ge x_{i-1}-\gamma$ , $x_i$ is above −45° and below the small region near 0, the symbol is set as −1;

• When $x_i<x_{i-1}-\gamma$ , the symbol is set as −2.

Figure 1.

Slope entropy with three division levels.

When the embedding dimension is set as m = 4, the different types of patterns symbolized by the difference sequence are shown in Figure 2 according to the procedure of data symbolization above with specified γ and δ. Let n be the number of all pattern types, π_l be the lth pattern and $\Pi ={\left\{{\mathrm{\pi }}_l\right\}}_1^{q^m}$ , then the probability of each pattern is as follows:

$$p\left({\mathrm{\pi }}_l^{m,\tau }\right)=\frac{{\sum }_{i\le N-\left(m-1\right)\tau }{1}_{u:type\left(u\right)={\mathrm{\pi }}_l}\left(X_i^{m,\tau }\right)}{{\sum }_{i\le N-\left(m-1\right)\tau }{1}_{u:type\left(u\right)=\Pi }\left(X_i^{m,\tau }\right)}$$

(2)

Figure 2.

Illustration of the different patterns when m = 4.

in which $1_A\left(u\right)=\{\begin{array}{c}1,\mathrm{if}u\in A\hfill \\ \multicolumn{1}{c}{0,\mathrm{if}u\notin A}\end{array}$ is the indicator function of the set A. The slope entropy of the vibration sequence x with the specified embedding dimension m and time delay τ is obtained as:

$$\mathrm{SlE}(\mathit{x},m,\tau )=-\sum _{i:{\mathrm{\pi }}_l^{m,\tau }\in \Pi }p\left({\mathrm{\pi }}_l^{m,\tau }\right)\ln p\left({\mathrm{\pi }}_l^{m,\tau }\right)$$

(3)

Weighted slope entropy

Considering the fluctuation of the vibration sequence under the same structure, the literature Manis et al. ¹⁵ proposes an improved algorithm as shown in Figure 3. In this algorithm the same structure data is weighted according to its fluctuation, that is, equation (3) is replaced by:

$$p_w\left({\mathrm{\pi }}_l^{m,\tau }\right)=\frac{{\sum }_{i\le N-(m-1)\tau }{1}_{u:\mathrm{type}\left(u\right)={\mathrm{\pi }}_l}\left(X_i^{m,\tau }\right){\omega }_i}{{\sum }_{i\le N-(m-1)\tau }{1}_{u:\mathrm{type}\left(u\right)=\Pi }\left(X_i^{m,\tau }\right){\omega }_i}$$

(4)

Figure 3.

Volatility of data with the same structure.

in which the weighted value ${\omega }_i$ is the variance of the corresponding data sequence $X_i^{m,\tau }$ _. When ${\omega }_i=C,\forall i\le N$ and $C>0$ , the above equation degenerates to equation (2). Substitute the statistics of each pattern obtained in equation (4) into equation (3), and then the weighted slope entropy is obtained as:

$$\mathrm{WSlE}(\mathit{x},m,\tau )=-\sum _{i:{\mathrm{\pi }}_l^{m,\tau }\in \Pi }{p}_w\left({\mathrm{\pi }}_l^{m,\tau }\right)\ln p_w\left({\mathrm{\pi }}_l^{m,\tau }\right).$$

(5)

Composite multiscale weighted slope entropy

For a given s_max as the maximum of scale factor, the coarse-grained sequence of $\mathit{x}=\{x_1,x_2,\cdots ,x_N\}$ is:

$$y_j^{\left(s\right)}=\frac{1}{s}\sum _{i=(j-1)s+1}^{\mathrm{js}}{x}_i,1\le j\le \frac{N}{s},s=1,2,\cdots ,s_{max}$$

(6)

in which s is a positive integer termed scale factor, and a procedure of the coarse-grained process is illustrated in Figure 4. With the scale s, the multiscale weighted slope entropy values are calculated as:

$$\mathrm{MWSlE}(\mathit{x},m,\tau ,s)=\mathrm{WSlE}(y^{\left(s\right)},m,\tau ),$$

(7)

Figure 4.

The coarse-grained process of the vibration data sequence.

In further, the composite coarse-grained sequence of $\mathit{x}=\{x_1,x_2,\cdots ,x_N\}$ is constructed as:

$$y_{k,j}^{\left(s\right)}=\frac{1}{s}\sum _{i=(j-1)s+k}^{\mathrm{js}+k-1}{x}_i,1\le j\le \frac{N}{s},1\le k\le s$$

(8)

MWSlE values of s coarse-grained sequences $y_k^{\left(s\right)}(k=1,2,\cdots ,s)$ are calculated and the mean value is taken as the composite multiscale weighted slope entropy:

$$\mathrm{CMWSlE}(\mathit{x},m,\tau ,s)=\frac{1}{s}\sum _{k=1}^s\mathrm{WSlE}(y_k^{\left(s\right)},m,\tau )$$

(9)

For the CMWSlE, it degenerates to MSlE if no weighting and no averaging on each scale are performed; if just weighting on each scale is taken, it degenerates to MWSlE; if just averaging is performed, it degenerates to CMSlE. Alternatively, signal hierarchical decomposition is performed by the method proposed in the literature Richman and Moorman ⁹ and combined with weighted slope entropy to obtain HWSlE.

Hierarchical prototype-based approach

Let $\left\{\mathbf{\Theta }\right\}=\{{\mathit{\theta }}_1,{\mathit{\theta }}_2,\cdots ,{\mathit{\theta }}_k,\cdots ,{\mathit{\theta }}_K,\cdots \}$ consist of C different rolling bearing states, where ${\mathit{\theta }}_k={[{\theta }_{k,1},{\theta }_{k,2},\cdots ,\cdots {\theta }_{k,N}]}^T\in {\mathbf{R}}^N$ denotes the N-dimensional vectors of the k-th rolling bearing state. Suppose the set of samples of the i-th class of rolling bearing states be ${\left\{\mathit{\theta }\right\}}_K^i=\{{\mathit{\theta }}_1^i,{\mathit{\theta }}_2^i,\cdots ,{\mathit{\theta }}_{K^i}^i\}$ , $1\le i\le C$ , and the total number of samples be $K=\sum _{i=1}^C{K}^i$ .

General framework

The general structure of the hierarchical prototype-based approach is shown in Figure 5. Let ${\mathit{p}}_{l,j}^i$ denote the j-th prototype of the l-th layer concerning the i-th class fault, λ_i is the i-th class of faults, $1\le i\le C$ , $1\le l\le L$ , $1\le j\le M_l^i$ , and L is the number of classifier layers, $M_l^i$ is the number of prototypes of the l-th layer of the i-th class fault, and satisfies $M_1^i\le M_2^i\le \cdots \le M_{L-1}^i\le M_L^i$ .

Figure 5.

The general architecture of the hierarchical prototype-based classifier: (a) System structure and (b) Structure of the ith prototype-based hierarchy.

Learning process

For each sample ${\mathit{\theta }}_k^i$ , the following normalization process is first performed

$${\mathit{\theta }}_k^i\leftarrow \frac{{\mathit{\theta }}_k^i}{\Vert {\mathit{\theta }}_k^i\Vert }$$

(10)

in which $\Vert {\mathit{\theta }}_k^i\Vert =\sqrt{\sum _{j=1}^N{\left({\theta }_{k,j}^i\right)}^2}$ . The transformation is able to convert the Eulerian distance between two samples into a distance metric based on cosine dissimilarity, thus enhancing the ability of the hierarchical prototype-based approach in dealing with high-dimensional samples.

During the learning process, the first observation sample ${\mathit{\theta }}_{K^i}^i(K^i=1)$ of class i is used to initialize each fault type and treated as the first prototype of each layer

$$M_l^i\leftarrow 1;{\mathit{p}}_{l,M_l^i}^i\leftarrow {\mathit{x}}_{K^i}^i;S_{l,M_l^i}^i\leftarrow 1$$

(11)

in which $S_{l,M_l^i}^i$ is the number of data samples associated with ${\mathit{p}}_{l,M_l^i}^i$ .

For the top-level prototypes, as shown in Figure 6, they are defined as

$${\mathit{L}}_0^i\leftarrow \left\{{\mathit{p}}_{1,M_1^i}^i\right\}$$

(12)

Figure 6.

Illustration of the three-layer prototype-based hierarchy approach.

For the l-th level of ${\mathit{p}}_{l,M_l^i}^i(l=1,2,\cdots ,L-1)$ , they are defined as

$${\mathit{L}}_{l,M_l^i}^i\leftarrow \left\{{\mathit{p}}_{l+1,M_{l+1}^i}^i\right\}$$

(13)

By means of the above equation, the fault chain of the i-th class is established starting with ${\mathit{p}}_{1,M_1^i}^i$ .

When a new sample ${\mathit{\theta }}_{K^i}^i(K^i\leftarrow K^i+1)$ is available, the system will be updated from top to bottom. First, the prototype nearest to ${\mathit{\theta }}_{K^i}^i$ in the l-th layer is identified using the following equation

$$n_l^{*}=\{\begin{array}{c}\underset{\mathit{p}\in {\mathit{L}}_0^i}{\arg min}(\Vert {\mathit{\theta }}_{K^i}^i-\mathit{p}\Vert ),\mathrm{if}l=1\hfill \\ \multicolumn{1}{c}{\underset{\mathit{p}\in {\mathit{L}}_{l-1,n_{l-1}^{*}}^i}{\arg min}(\Vert {\mathit{\theta }}_{K^i}^i-\mathit{p}\Vert ),\mathrm{if}l=2,3,\cdots ,L}\end{array}$$

(14)

The above equation reduces the scope of the search from the entire data space to a small number of neighboring prototypes, avoiding the time wastage caused by searching for a large number of prototypes far away ${\mathit{\theta }}_{K^i}^i$ in the data space.

Decision process

As shown in Figure 7, for the testing sample ${\mathit{\theta }}_K$ , the nearest prototype of the l-th level of each class is found directly and the score is calculated

$${\lambda }^i\left({\mathit{\theta }}_K\right)=\underset{\mathit{p}\in {\left\{\mathit{p}\right\}}_l^i}{max}\left(e^{-{\Vert \mathit{p}-{\mathit{\theta }}_K\Vert }^2}\right)$$

(15)

Figure 7.

Illustration of rolling bearing state classification.

where ${\left\{\mathit{p}\right\}}_l^i=\{{\mathit{p}}_{l,1}^i,{\mathit{p}}_{l,2}^i,\cdots ,{\mathit{p}}_{l,M_l^i}^i\}$ denotes the prototype training data of the l-th layer of the i-th class fault. The fault type of the sample to be tested can be obtained by the nearest distance evaluation according to the equation (15).

Flowchart of rolling bearing state assessment

The flowchart of rolling bearing state assessment based on the composite multiscale weighted slope entropy and the hierarchical prototype-based approach is as follows:

• Step 1. Collect signals of different fault types with corresponding damage degrees of rolling bearings. Set the data length n = 2048 for the prepare of CMWSlE features extraction, as well as the entropy features used for comparison: MSlE, MWSlE, CMSlE, and HWSlE.

• Step 2. Normalize the data of each segment, and then extract the CMWSlE features and the entropy features for comparison. The parameters are set as: embedding dimension m = 4, scale factor s = 20, time delay τ = 1, predefined thresholds γ = 1, δ = 0.001.

• Step 3. Design two test schemes to verify the effectiveness of this method in combination with the needs of the practical project. The two schemes respectively test the fault types of rolling bearings under different working conditions and the same damage degree, and the fault types with corresponding damage degrees of rolling bearings under the same working conditions. There are 50 samples in each rolling bearing state, of which 10 samples are used to train the HPA, and the other 40 samples are used to verify the method proposed in this paper. The accuracy of the proposed method is:

$$Acc=\frac{n_R}{n_T}\times 100\%$$

(16)

in which n_R and n_T respectively are the total number of correctly classified samples and the total number of test samples.

Experimental validation

The test schemes including test group A and test group B are designed to verify the proposed approach as shown in Table 1. The two test groups have certain practical values in engineering. The experimental device of CWRU for artificial fault bearings is composed of drive motor, loading motor, torque sensor, etc. The test bearing is the type of deep groove ball bearing with model SKF6205, and it is installed at the drive end, as shown in Figure 8. Single point damage on bearings are made by electric spark and there are three types of failure types: REF, IRF and ORF. The diameter of single point damage of each fault type is shown in Table 1, which can indicate the degree of damage. From the fault signals of each component of the rolling bearing in Figure 9 and the vibration signals of the rolling elements under different loads when the damage degree is 0.007 in in Figure 10, it could be seen that the fault signals of different components are easy to distinguish, while the vibration signals of the slightly worn rolling balls under different working conditions are difficulty to distinguish. The reason is that the rolling balls are always in the complex motion state during the working process. In addition, the vibration signals of ORF at 6:00 are selected.

Table 1.

Design of test schemes regarding bearings with artificial faults.

Testing groups	Working condition		Fault types	Damage diameter (inch)	Number of training samples	Number of testing samples	Labels
	Load (HP)	Speed(rpm)
A	2	1750	Normal	0	10	40	1
	2	1750	REF	0.007	10	40	2
	2	1750	REF	0.014	10	40	3
	2	1750	REF	0.021	10	40	4
	2	1750	IRF	0.007	10	40	5
	2	1750	IRF	0.014	10	40	6
	2	1750	IRF	0.021	10	40	7
	2	1750	ORF	0.007	10	40	8
	2	1750	ORF	0.014	10	40	9
	2	1750	ORF	0.021	10	40	10
B	0	1797	Normal	0.007	10	40	1
	1	1772	Normal	0.007	10	40	2
	2	1750	Normal	0.007	10	40	3
	3	1730	Normal	0.007	10	40	4
	0	1797	REF	0.007	10	40	5
	1	1772	REF	0.007	10	40	6
	2	1750	REF	0.007	10	40	7
	3	1730	REF	0.007	10	40	8
	0	1797	IRF	0.007	10	40	9
	1	1772	IRF	0.007	10	40	10
	2	1750	IRF	0.007	10	40	11
	3	1730	IRF	0.007	10	40	12
	0	1797	ORF	0.007	10	40	13
	1	1772	ORF	0.007	10	40	14
	2	1750	ORF	0.007	10	40	15
	3	1730	ORF	0.007	10	40	16

Figure 8.

Bearing fault test equipment of CWRU. ³¹

Figure 9.

Vibration signals at the damage level 0.007 in for different fault types of bearings on the condition of load 2 HP and rotating speed 1750 r/min in testing group A.

Figure 10.

Vibration signals of the rolling elements at the damage level 0.007 in for different loads in testing group B.

According to the parameter settings in Section 4, the CMWSlE features are extracted from test group A and test group B, as well as the entropy features used for comparison: MSLE, MWSlE, HWSlE, and CMSlE, as shown in Figures 11 and 12 respectively. In HWSlE calculation, three layers are considered for the hierarchical structure and other settings are the same. It could be seen from Figures 11 and 12 that compared with the statistical values under 50 samples, CMWSlE has smaller Std. than MSLE, MWSlE, HWSlE, and CMSlE, indicating the stability of CMWSlE in rolling bearing fault diagnosis. The trends of CMWSlE and CMSLE at each scale are basically the same, and the standard deviation of CMWSlE is relatively large, but it contains more abundant information at each scale.

Figure 11.

Comparison of entropy values at the damage level 0.007 in for different fault types of bearings on the condition of load 2 HP and rotating speed 1750 r/min in testing group A.

Figure 12.

Comparison of entropy values at the damage level 0.007 in of rolling bearings at different working conditions in testing group B.

Support vector machine mainly solves the problem of two classification. For the classification of various fault states, it is necessary to construct appropriate classification algorithms, such as “one-to-one” algorithm and “one to many” algorithm. The classification strategies developed on the proposed algorithms include binary tree, DAG and so on. Among them, DAG-SVM structure ³² is faster and more accurate. Therefore, this paper compares it with HPA.

In test group A and test group B, each group carries out 10 times of random sample extraction according to Table 1 for the training of DAG-SVM and HPA, and the remaining samples are used for testing. In each calculation, the classification accuracy is counted, and the statistic results are shown in Figures 13 and 14. It could be seen that the average classification accuracy of the HPA is significantly better than DAG-SVM, and the standard deviation is less than DAG-SVM, which further illustrates the stability of the HPA in the multiple fault classification of rolling bearings. In Figure 13, CMSLE and CMWSLE respectively combined with HPA have 100% classification accuracy in 10 tests, so its standard deviation is 0. In Figure 14, compared with CMSLE, CMWSLE has higher classification accuracy and better stability for more rolling bearing fault states under more complex working conditions. As shown in Table 2, the results of recent research on artificial fault bearing data of CWRU are collected and compared with this paper. Compared with Zheng et al. and Gao et al.,^33,34 the proposed method can distinguish more rolling bearing fault states when the classification accuracy is the same as that of test group A; Compared with Tang et al., ³⁵ there is no need for preprocessing and feature selection of rolling bearing fault signals. Although the classification accuracy of test group B is slightly lower than that of literature,^36,37 the number of classification of fault states is significantly increased.

Figure 13.

Comparison of classification accuracy of testing group A.

Figure 14.

Comparison of classification accuracy of testing group B.

Table 2.

Comparisons of classification accuracy with approaches published in the literatures.

Pre-processing	Feature extraction	Feature selection	Classifier	Classification number	Classification accuracy	Reference
Null	MFE	Laplace score	VPMBCD	8	100%	Zheng et al. 33
STFT	Images	NMF	Bayesian classification	9	100%	Gao et al. 34
DTCWPT	IMPE	LLTST	ELM	10	100%	Tang et al. 35
Null	Entropy and other features	Null	Gray correlation	11	98.4%	Yang et al. 36
ALIF	MSE, MFE, MPE	Null	KSRC	12	99.4%	Zhang et al. 37
Null	CMWSlE	Null	HPA	10	100%	This paper
Null	CMWSlE	Null	HPA	16	96.5%	This paper

STFT: short time Fourier transform; NMF: nonnegative matrix factorization; DTCWPT: dual tree complex wavelet packet transform; IMPE: improved multiscale permutation entropy; LLTST: linear local tangent space transformation; ELM: extreme learn machine; ALIF: adaptive local iterative filtering; MSE: multiscale sample entropy; MFE: multiscale fuzzy entropy; MPE: multiscale permutation entropy; KSRC: Kernel sparse representation classification; VPMBCD: variable predictive model-based class discrimination.

Conclusions

In this paper, the composite multiscale weighted slope entropy combined with the hierarchical prototype-based approach is proposed to identify various fault states of rolling bearings. The main conclusions are as follows:

1. The CMWSlE features could consider the structure, volatility, multiscale characteristics, and amplitude effects of rolling bearing vibration data. Compared with other entropy values, the composite multiscale weighted slope entropy has less volatility under multiple sample statistics and contains more fault state information.

2. In the two test groups of different fault types with corresponding damage degree under the same working conditions, and different fault types with the same damage degree under different working conditions, the CMWSlE combined with the HPA could significantly improve the classification accuracy, which are 100% and 96.5% respectively. This shows the validity of the method proposed in this paper and the effectiveness of distinguishing more fault states of rolling bearings under complex working conditions.

3. In the two test groups, the method proposed in this paper reduces the preprocessing and feature selection of fault vibration signals on the basis of ensuring the accuracy of rolling bearing fault state identification, which makes the method more simple to use, and reduces the calculation time, laying a foundation for its application in complex mechanical equipment.