Feature extraction of latent fault components of rolling bearing based on self-learned sparse atomics and frequency band entropy

Abstract

As the key rotating parts in machinery, it is crucial to extract the latent fault features of rolling bearing in machinery condition monitoring to avoid the occurrence of sudden accidents. Unfortunately, the latent fault features are hard to extract by using the traditional signal processing method such as envelope demodulation because the effect of envelope demodulation is influenced strongly by the degree of background noise. Sparse decomposition, as a new promising method being able of capturing the latent fault feature components buried in the vibration signal, has attracted a lot of attentions, especially the predefined dictionary-based sparse decomposition methods. However, the feature extraction effect of the predefined dictionary-based sparse decomposition depends on whether the prior knowledge of the analyzed signal is sufficient or not. To overcome the above problems, a feature extraction method of latent fault components of rolling bearing based on self-learned sparse atomics and frequency band entropy is proposed in the article. First, a self-learned sparse atomics method is applied on the early weak vibration signal of rolling bearing and several self-learned atomics are obtained. Then, the self-learned atomics owing bigger kurtosis values are selected and used to reconstruct the vibration signal to remove the other interference signals. Subsequently, the frequency band entropy method is used to analyze the reconstructed vibration signal, and the optimal parameter of band-pass filter could be calculated. At last, the reconstructed vibration signal is filtered using the optimal band-pass filter, envelope demodulation on the filtered signal is applied, and better fault feature is extracted. The feasibility and effectiveness of the proposed method are verified through the vibration data of the accelerated fatigue life test of rolling bearing. Besides, the analysis results of the same vibration data using Autogram and spectral kurtosis methods are also presented to highlight the superiority of the proposed method.

Keywords

Feature extraction latent fault rolling bearing sparse decomposition self-learned sparse atomics frequency band entropy

1. Introduction

As the commonly used supporting component in rotating machinery, it is significant to detect the fault feature of rolling bearing to avoid unplanned outage or disasters. The envelope demodulation spectral (EDS) is used for this purpose frequently. However, there exist two problems of EDS. First, the feature extraction effect of EDS would not be ideal if the fault feature signal of rolling bearing is interfered by strong background noise. Second, an optimal band-pass filter is needed to be constructed to filter the component for EDS to get better effect. Unfortunately, the parameters of the optimal band-pass filter such as center frequency are selected based on experience usually, and this will cause the contingency of feature extraction effect of EDS. To solve the above two problems, kinds of new signal processing methods have been arising in recent years (Antoni et al., 2004, 2006, 2007; Qiu et al., 2006; Selesnick., 2011; Wu and Huang, 2009), and most of research works on the first problem are based on orthogonal basis expansion which has the defect in interpreting the physical meaning of coefficients under these orthogonal bases. Besides, some other research works dealing with the vibration of shell structures have been arising in recent years (Zhu et al., 2017, 2020). Sparse decomposition (SD) is a relatively new signal processing method which could capture the essential feature of the analyzed signal without considering orthogonal basis expansion, and it has attracted more and more attention in mechanical fault diagnosis. Normally, the PDBSD method has the shortcomings of needing prior knowledge of the analyzed signal, and the self-learning dictionary SD method has the disadvantage of poor interference robustness. To address the same above problems, a parametric impulsive dictionary based on Laplace wavelets is designed for impulsive fault feature extraction of rolling bearing (Sun et al., 2019) which has better feature extraction performance. The traditional orthogonal matching pursuit (OMP) method was improved, and an adapted dictionary-free OMP (ADOMP) method was proposed in the article Huang et al. (2019) which has the advantages of flexibility and robustness of self-learned dictionary method, and the ADOMP method presented potential advantages in feature extracting bearing early weak fault. An improved variational mode decomposition (VMD) method was proposed by Li et al. (2019a) which overcomes the shortcomings of VMD in selecting the reasonable algorithm parameters. Besides, the improved VMD was combined with adaptive sparse code shrinkage denoising and used in fault diagnosis of rotating machinery. To overcome the difficulties in planet bearing fault identification due to the intricate kinematics and multiple modulation effects, a classification method based on dictionary learning was introduced by Zhao et al. (2018) which has the virtue of avoiding the feature design requirement in normal intelligent diagnosis method. To alleviate the drawback of pursuit efficiency of optimal atom in SD, a new SD method based on time-frequency spectrum segmentation was proposed by Yan et al. (2018), and the higher decomposition efficiency and better approximation precision of the proposed method were verified by conducting simulations and experiments. A new SD signal processing method based on multiscale chirplet was proposed in fault diagnosis of gearbox vibration signals (Peng et al., 2011), and the feasibility of the proposed method was validated by both simulations and experiments. A novel intelligent fault diagnosis method of rolling bearing was proposed by combing SD with the neighborhood preserving deep extreme learning machine (Li et al., 2019b). A supervised feature extraction method named supervised regularized sparse filtering was proposed in feature extraction of rotating machinery which provided a new way for optimizing the solving of sparsity (Qian et al., 2018).

Although a large number of fault diagnosis methods for rolling bearing based on SD have been arising as above stated, most of them are PDBSD methods which have the drawback of lack of flexibility. Besides, most of the self-learned SD methods own the disadvantage of low computational efficiency. In the article, a feature extraction method of latent fault components of rolling bearing based on self-learned dictionary and frequency band entropy (FBE) is proposed. The self-learned dictionary method is based on the improved shift-invariant sparse coding (ISISC) which has higher computational efficiency compared with traditional shift-invariant sparse coding (SISC), and the FBE has the advantages of more simple theory basis and could calculate the center frequency of the optimal band-pass filter for better EDS effect much more effectively than spectral kurtosis and the other related methods. The main steps of the proposed method are as following: First, the early weak vibration signal of rolling bearing is analyzed by ISISC and the latent fault characteristic signal is obtained. Then apply FBE on the obtained signal and an optimal band-pass filter is constructed to filter the obtained signal for much better envelope demodulation effectiveness. At last, apply EDS on the filtered signal and satisfactory feature extraction results are obtained. The structure of the article is arranged as follows: Section 2 is dedicated to the processes of ISISC, and the related theory and calculation processes of FBE are given in Section 3. The flow chart of the proposed method is shown in Section 4, and Section 5 is the experiment to verify the effectiveness and feasibility of the proposed method. Besides, comparison result is given in Section 6 to show the advantage of the proposed method, and conclusions obtained from the above results are given in Section 7.

2. Theory of ISISC

The mathematical model of SISC could be referred to equation (1) (Blumensath et al., 2005; Lee et al., 2007)

x^{(i)} = \sum_{j} a^{(j)} * s^{(i, j)} + ε^{(i)}

(1)

where the basis function

a^{(j)} \in R^{q}, j = 1, \dots, n

is allowed to move arbitrarily in the time range of the signal, or every identical event in the signal can be expressed by the same basis function through any time shift.

s^{(i, j)}

in equation (1) is the sparse coefficient matrix corresponding to

a^{(j)}

, and

ε^{(i)}

represents the additive random white noise. The analyzed signal

x^{(i)} \in R^{P}, i = 1, \dots, m

could be expressed by the basis function through time shift. The convolution operator * in equation (1) is used so that each basis function

a^{(j)}

could appear at any position within the time domain of the signal. The difference between the conventional SD model and SISC model is that the dimension of basis function of SISC is allowed to be lower than the dimension of analyzed signal. Now, the solution of the sparsity coefficient

S

(

S

represents the vector group of

s

) and the construction of basis function

A

(

A

represents the vector group of

a

) could be reduced to equations (2) and (3).

\min_{A,S} \sum_{i = 1}^{m} {‖ x^{(i)} - \sum_{j = 1}^{n} a^{(j)} * s^{(i, j)} ‖}_{2}^{2} + β \sum_{i, j} {‖ s^{(i, j)} ‖}_{1}

(2)

s . t . {‖ a^{(j)} ‖}_{2}^{2} \leq c, 1 \leq j \leq n

(3)

The constraint in equation (3) is to ensure that $a^{(j)}$ does not become too large. Although objective function (2) is not a convex optimization problem with respect to $A$ and $S$ at the same time, objective function (2) is a convex optimization problem with respect to the basis functions when the coefficient matrix is invariant and vice versa. That is to say, objective function (2) is also a convex optimization problem with respect to the sparsity coefficients when the basis function is invariant. Therefore, the optimal results of $A$ and $S$ could be obtained by setting one of them.

The direct solution of convolution problem in equation (2) makes the optimization problem very hard. Besides, the sparsity coefficients are coupled with each other. Considering the special structure of the formula, this section will introduce an ISISC method.

The solution of the sparsity coefficient $S$ is a $L_{1}$ regularized least squares problem by fixing the basis function $A$ , and it could be resolved by transferring into an unconstrained quadratic optimization problem as shown in equation (4) using feature-sign search algorithm (FSS) (Blumensath et al., 2005; Lee et al., 2007). The genetic algorithm (Zhi and Liu, 2019) is introduced into FSS to improve its efficiency, and the flow chart of ISISC is shown in Figure 1.

min_{A, S} {\sum_{i = 1}^{m} ‖ x^{(i)} - \sum_{j = 1}^{n} a^{(j)} * s^{(i, j)} ‖}_{2}^{2}

(4)

s . t . {‖ a^{(j)} ‖}_{2}^{2} \leq c, 1 \leq j \leq n

(5)

Figure 1.

Flow chart of improved shift-invariant sparse coding.

The direct expansion of the convolution in equation (4) will make the optimization problem to become very large. Besides, the sparsity coefficients are coupled with each other. We could transform equations (4) and (5) into frequency domain as shown in equations (6) and (7) by considering that the special structure of the formula and the convolution in the time domain can be replaced by multiplication in the frequency domain

\min_{\hat{A}} {\sum_{i = 1}^{m} ‖ \hat{x^{(i)}} - \sum_{j = 1}^{n} \hat{a^{(j)}} \hat{s^{(i, j)}} ‖}_{2}^{2}

(6)

s . t . {‖ \hat{a^{(j)}} ‖}_{2}^{2} \leq \hat{c} = c K, 1 \leq j \leq n

(7)

The discrete Fourier transforms of basis function $A = {\hat{a^{(1)}}, \hat{a^{(2)}}, \dots, \hat{a^{(n)}}}$ , the input signal $x^{(i)}$ , and the sparse coefficient $s^{(i, j)}$ are represented by $\hat{A} = {\hat{a^{(1)}}, \hat{a^{(2)}}, \dots, \hat{a^{(n)}}}$ , $\hat{x^{(i)}}$ , and $\hat{s^{(i, j)}}$ , respectively as shown in equation (6).

The Parseval’s theorem (Zhu et al., 2015) guarantees that problem equations (4) and (5) could be transformed into problem equations (6) and (7). We can use the Lagrangian method to solve this problem, and its Lagrangian function can be decomposed into the sum of quadratic terms, and each of them depends on a single frequency component $t$

L (\hat{A}, λ) = \sum_{t} ({‖ \hat{x_{t}} - \hat{s_{t}} \hat{a_{t}} ‖}_{2}^{2} + \hat{a_{t}^{*}} \land \hat{a_{t}}) - {\hat{c 1}}^{T} λ

(8)

with dual variables

λ \in R^{n}

, unit vector

1 \in R^{n}

, and

\hat{a_{t}} = (\begin{matrix} \hat{a_{t}^{(1)}} \\ \dots \\ \hat{a_{t}^{(n)}} \end{matrix}), \hat{x_{t}} = (\begin{matrix} \hat{x_{t}^{(1)}} \\ \dots \\ \hat{x_{t}^{(n)}} \end{matrix}), \land = diag (λ), \hat{s_{t}} = (\begin{matrix} \hat{s_{t}^{(11)}} & \hat{s_{t}^{(12)}} & \dots \\ \hat{s_{t}^{(21)}} & \hat{s_{t}^{(22)}} & \dots \\ \dots & \dots & \dots \end{matrix})

Although the Lagrange function in equation (8) is a complex function, it could be expressed as a real function by dividing $\hat{A}$ into real part and imaginary part. The obtaining of $\hat{A^{\min}}$ can be realized by optimizing over $Re (\hat{A})$ and $Im (\hat{A})$

\hat{A_{t}^{\min}} = {(\hat{s_{t}^{*}} \hat{s_{t}} + \land)}^{- 1} \hat{s_{t}^{*}} \hat{x_{t}}

(9)

The specific optimization processes could be referred to the works of Lee et al., (2007) and Blumensath et al., (2005).

3. FBE

The definition of entropy: Suppose the information source given by a discrete random variable $X = {x_{1}, x_{2}, \dots x_{N}}$ , and the occurrence probability of each element is $p_{i} = P (x_{i}) (i = 1,2, \dots, N)$ . Besides, there exists $\sum_{i = 1}^{N} p_{i} = 1$ . Then, the information entropy of the information source $X = {x_{1}, x_{2}, \dots x_{N}}$ could be calculated using the following equation

H (x) = - \sum_{i = 1}^{N} p_{i} \ln p_{i}

(10)

where

\ln 0 = 0

It could be observed in equation (10) that when the probability distribution of each variable in the information source is more uniform, the information entropy increases. Conversely, when the probability distribution of each variable in the information source is more uneven, the information entropy decreases. So, the information entropy could be used to measure the degree of state chaos. Here, the concept of entropy is introduced into the fault diagnosis of rolling bearings: when the rolling bearings are running normally, the frequency information distribution is uniform and the information entropy value is relatively large. The frequency information concentrates on the fault characteristic frequency mainly when fault occurs, and the corresponding information entropy value is becoming relatively smaller.

Amplitude spectrum analysis is the most commonly used frequency-domain analysis method. This study combines it with information entropy, and amplitude spectrum entropy is proposed. The normalized amplitude spectral entropy (Liu et al., 2014) is calculated as follows

{\begin{cases} H_{s} = - (\sum_{i = 1}^{N} P_{i} \cdot \ln (P_{i})) / \ln (N) \\ P_{i} = X (i) / \sum_{j = 1}^{N} X (j) \\ \sum_{i = 1}^{N} P_{i} = 1 \end{cases}

(11)

where

X (i)

is the amplitude spectral of the series

{x_{i}}, i = 1, 2, \dots, N

and

H_{s}

is the corresponding amplitude spectral entropy.

Amplitude spectral entropy can represent the complexity of the frequency structure over a period of time, but it could not reveal the change of frequency components over time. To reveal the complexity of the frequency components changing with time, the study combines the time-frequency analysis with the amplitude spectral entropy and the FBE method is obtained. The calculation processes of FBE are as follows:

1. Apply short-time Fourier transform on the FFT result of the original signal ${x_{i}}, i = 0, 1, \dots, N - 1$ and the corresponding time-frequency distribution matrix is obtained

T F R = [\begin{matrix} r_{1, 1} & \dots & r_{1, C} \\ \dots & \dots & \dots \\ r_{M, 1} & \dots & r_{M, C} \end{matrix}]

(12)

where the rows of the matrix represent the time distribution and the columns represent the frequency distribution.

M

is the number of frequency points.

C = N / L

and

L

represents the step of window function moving along the time axis.

2. Calculate the amplitude spectrum entropy of the time distribution ${r_{n, i}}, i = 1, 2, \dots, C$ at frequency $f_{j}$

{\begin{cases} H_{s, j} = - (\sum_{i = 1}^{C} P_{j, i} \cdot \ln (P_{j, i}) / ln (C)) \\ P_{j, i} = | \sum_{n = 0}^{C - 1} r_{j, n + 1} W_{C}^{n (i - 1)} | / \sum_{k = 1}^{C} | \sum_{n = 0}^{C - 1} r_{j, n + 1} W_{C}^{n (k - 1)} | \\ \sum_{i = 1}^{C} P_{j, i} = 1 \end{cases}

(13)

where

| \sum_{n = 0}^{C - 1} r_{j, n + 1} W_{C}^{n (i - 1)} |

is the amplitude spectrum of

{r_{j, i}}, i = 1, 2, \dots, C

and

W_{C}^{n i} = e^{- j (2 π / C) n i}

is the Fourier transform factor.

H_{s, j}

is the amplitude spectrum entropy at frequency

f_{j}

, and it is also named FBE.

4. Proposed method

The flow chart of the proposed method is shown in Figure 2, and its specific steps are as follows:

Figure 2.

Flow chart of the proposed method.

Step 1

Segment the original signal into several groups and the basis functions of each group are learned by using the ISISC method.

Step 2

Calculate the kurtosis index of each basis function and the basis functions with bigger kurtosis index are selected to reconstruct the signal, and the reason is that bigger kurtosis index means much more impulsive characteristic components being contained in the basis function.

Step 3

Apply FFT on the reconstructed signal first, then apply short-time Fourier transform on the FFT result of the reconstructed signal and the corresponding time–frequency distribution matrix is obtained.

Step 4

Amplitude spectral entropy is calculated on the analysis results of Step 3.

Step 5

The FBE analysis is applied on the analysis results of Step 4 and optimal parameters of the band-pass filter are obtained.

Step 6

The optimal band-pass filter is constructed to filter the reconstructed signal, and then apply EDS analysis on the filtered signal and satisfactory fault features are extracted at last.

5. Experiment

It is a complex process from the installation and use of rolling bearings to their complete failure. It is of great economic and safety significance to study the method of extracting early weak fault features of rolling bearings to avoid sudden accidents. In the article, the analysis data are collected from the accelerated bearing life test and the accelerated bearing life test (ABLT-1A) is provided by the Hangzhou Bearing Test and Research Center. The concrete experiment processes and the parameters of the test bearings could be referred to the work Wang et al., (2014). In the experiment, one group of 20,480 points is collected per minute with sampling rate 25.6 kHz and two of the test bearings, namely, B12 and B14 are selected to verify the effectiveness of the proposed method. Because of the external radial load P, the test bearings all occur inner race fault.

5.1. B12

The disassembly diagram of B12 fatigue failure could be referred to the work Wang et al., (2014) which shows that fault arises in the inner race of rolling bearing. The kurtosis value curve of the whole life cycle of the experimental bearing B12 could be referred to work Wang et al., (2014). The reason is that the kurtosis value is more sensitive to the impulse signal characteristics when rolling bearing fails. It can be seen from the kurtosis value curve of the whole life cycle of the experimental bearing B12 that the kurtosis increases abruptly at 2306th minute, which indicates that the rolling bearing failed completely at this moment. The data before 2306th minute could be regarded as the early weak fault data of rolling bearings. Same as the work Ming et al., (2011), the data at 2297th minute are used to verify the effectiveness of the proposed method in the article. Time-domain waveform and the corresponding EDS analysis of the data at 2297th minute could be referred to work Ming et al., (2011), and both of them could not reflect the inner race fault feature of test bearing. This proves the invalidity of the envelope demodulation analysis method in extracting weak fault features of rolling bearings and the reasons could be attributed to the following: First, the impact characteristic energy of rolling bearing in the early weak fault stage is very weaker compared with other signal components such as rotating frequency with its harmonics. Second, the early weak fault signal is also influenced by the strong background noise and other signal sources. The following is the analysis processes of the proposed method.

Two key problems are needed to be solved before applying the ISISC method on the analyzed signal: the first one is the selection of the data length of the basis functions. Ideally, the length of the basis function should be greater than the length of single impact of the signal to be analyzed and less than the length between two consecutive impacts. So the length of the basis functions could be estimated as following: it may be assumed that a single shock contains 10 attenuation oscillation periods, and the resonance frequency is 2 kHz, and the duration of a single shock is 0.005 s. The length of a single shock is calculated equal to 128 points based on the above assumption and the sampling frequency (25.6 kHz), so the length of basis functions could be selected as 256 points in the study.

The second one is determination of the number of the basis functions. The number of basis functions is generally based on the number of potential components in the analyzed signal. Of course, it is difficult to predict the characteristics of different mechanisms in the signal beforehand. Because this study mainly considers the single fault type, the number of basis functions is as small as possible under the premise of guaranteeing the effect. It is not only to reduce the calculation time but also to prevent the same fault feature from matching to multiple basis functions. In the study Tang et al. (2014), the number of the basis functions is selected as 8. To ensure sufficient redundancy of the constructed dictionary, the number of the basis functions is selected as 10 in the study. Apply the ISISC method on the original vibration signal of B12 at 2297th minute, and the learned basis functions are presented in Figure 3. Then, the kurtosis values of the learned basis functions are calculated and their corresponding values are given in Table.1. As mentioned earlier, the basis functions with bigger kurtosis values are selected to reconstruct the signal because bigger kurtosis value means that more typical impact features of the basis functions to pursuit the fault signals of rolling bearing much more effectively. The first six basis functions with larger kurtosis values are selected to reconstruct the signal, and the reconstructed signal is presented in Figure 4. It is evident that the impulse characteristics of reconstructed signals are significantly enhanced by comparing Figure 4 with the original vibration signal of B12 at 2297th minute, and it is verified that the ISISC method could capture the weak fault feature of rolling bearing effectively. The full-band EDS of the signal being shown in Figure 4 is presented in Figure 5 from which satisfactory fault feature extraction results could not be obtained. Usually, a band-pass filter should be created to filter the original signal, and then the envelope demodulation analysis will be carried out subsequently to obtain better feature extraction effect. However, the key parameters such as center frequency of the band-pass filter often require expert experience which results in the contingency of the band-pass filter effect, so the FBE is used in the study to solve the above problem. The FBE curve of the signal shown in Figure 4 is presented in Figure 6 from which the center frequency of the optimal band-pass filter could be selected as 4300 Hz. Then, the optimal band-pass filter is constructed by using the selected center frequency to filter the signal as shown in Figure 4. At last, apply envelope demodulation analysis on the filtered signal, and the last obtained results are shown in Figure 7 from which satisfactory result could be observed.

Figure 3.

The learned basis functions with their corresponding time-domain waveforms of B12 vibration signal at 2297th minute.

Table 1.

Kurtosis values of the learned basis functions.

Basis function	1	2	3	4	5	6	7	8	9	10
Kurtosis value	4.207	2.465	3.416	4.072	3.741	5.083	2.813	3.247	2.435	2.286

Figure 4.

The reconstructed signal using the selected basic functions.

Figure 5.

The full-band envelope demodulation spectral of the signal shown in Figure 4.

Figure 6.

The frequency band entropy analysis result of the signal shown in Figure 4.

Figure 7.

The envelope demodulation spectral of the filtered signal B12 using the optimal band-pass filter.

5.2. B14

The disassembly diagram of B14 fatigue failure is shown in Figure 8 which shows that fault also arises in the inner race. The kurtosis value curve of the whole life cycle of B14 is presented in Figure 9, and it is evident that the kurtosis increases abruptly at 980th minute, which indicates that the rolling bearing failed completely at this moment. The vibration data of B12 at 978th minute are taken as the early weak fault occurring moment and are analyzed to further verify the effectiveness of the proposed method, and the analysis processes and selection of relative parameters are same as B12. The analysis details of B14 are omitted because of the article length limit, and the last envelope analysis result using the proposed method is shown in Figure 10 from which the inner race fault characteristic frequency is extracted successfully. The effectiveness of the proposed method is verified further.

Figure 8.

The inner race fault of B14.

Figure 9.

The kurtosis values over B14’ whole life.

Figure 10.

The envelope demodulation spectral of the filtered signal of B14 using the proposed method.

6. Comparison

The analysis results using related method such as Autogram (Moshrefzadeh et al., 2018) method are presented in the section to verify the advantages of the proposed method. The Autogram method is based on cyclic autocorrelation variance by using the second-order cyclostationary characteristic of rolling bearing fault. Its steps are mainly as follows. First, the frequency band of fault signal is divided by maximum overlapping discrete wavelet packet transform. Second, the second-order cycle of rolling bearing is used. The periodic characteristic of autocorrelation variance function of stationary fault signal is calculated, and the unbiased autocorrelation function of envelope result of fault signal corresponding to each division frequency band is calculated. Then, the kurtosis value corresponding to unbiased autocorrelation function is calculated, and the spectrum kurtosis map based on cyclic autocorrelation is obtained. According to the spectrum, the key parameter of the optimal band-pass filter corresponding to envelope demodulation is selected as center frequency. According to the key parameters, the optimal band-pass filter is constructed to filter the fault signal, and the filtered signal is analyzed by envelope demodulation to extract features. To verify the advantage of the proposed method, the corresponding vibration data of B12 are analyzed for the purpose of comparison, and the corresponding analysis results are given as following under the guidance of the above stated: The unbiased autocorrelation function spectral of the signal shown in Figure 4 handled by maximal overlap discrete wavelet packet transform is given in Figure 11, and the corresponding Autogram spectrum analysis result is shown in Figure 12. The optimal band-pass filter is constructed based on the optimal parameter obtained by Figure 12 to filter the signal shown in Figure 4, then apply EDS on the filtered signal and the last obtained result is shown in Figure 13. It is evident that no satisfactory fault feature results are extracted by comparing with the proposed method.

Figure 11.

The un-biased auto-correlation function spectral of the signal shown in Figure 4 handled by maximal overlap discrete wavelet packet transform.

Figure 12.

The Autogram spectrum analysis of the signal shown in Figure 4.

Figure 13.

Envelope demodulation spectral of the filtered signal by the optimal band-pass filter being constructed using the analysis results of Figure 12.

In the section, the spectral kurtosis is also used for comparison, and the spectral kurtosis analysis result of the signal shown in Figure 4 is presented in Figure 14. Through the comparison of Figure 14 with 7, the better fault feature extraction result of the proposed method over spectral kurtosis is further verified.

Figure 14.

The spectral kurtosis analysis result of the signal as shown in Figure 4 and the corresponding envelope analysis result.

7. Conclusion

A feature extraction method of latent fault components of rolling bearing based on self-learned sparse atomics and FBE is proposed in the article. The genetic algorithm is introduced into the theoretical model of self-learning dictionary method traditional SISC to improve its computational efficiency. Besides, the problems of determining the number and length of basis functions in the proposed ISISC method are also solved. To improve the effect of envelope demodulation of the reconstructed signal using the self-learned basis functions and to solve the problem of selecting the optimal center frequency in envelope demodulation analysis, the FBE method is used in the envelope demodulation analysis. Besides, the accelerated bearing life test is carried out, and the design and selection of the experiment are mainly focusing on the early fault detection and the fault detection under strong interference. The experimental results using the proposed method show that the proposed method has a good ability to extract weak features and can be used to detect early weak fault of rolling bearing.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The research is supported by the National Natural Science Foundation (approved grant: U1804141) and the Key Science and Technology Research Project of the Henan Province (approved grant: 192102210105).

ORCID iD

HongChao Wang

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