Weak fault detection based on amplification-compression function and sparse spike deconvolution

Abstract

Weak fault detection is still a hotspot in these years. The key of this work is to enhance the ratio of signal to noise (SNR). Then a new extraction method of weak fault impact based on amplification algorithm is proposed. Since the fault impact is related to bigger derivative, the amplification function combined with the derivative of the vibration signal is used to amplify the amplitude of the impact. Then Sparse Spike Deconvolution (SSD) which has been widely used in earthquake for impact detection is able to extract small impact from the signal processed by amplification function. Then, in order to enhance the periodicity of fault impacts, the compression function is used to compress the amplitude of bigger impacts according to different scales. At last, the experiment result shows that the proposed method is more effective than the common envelope analysis.

Keywords

Weak fault impact derivative compression function

Introduction

Rolling elements Bearing is an important part of mechanical machine. Once damage happens on the surface of the bearing track or rollers, large damage may be occurred in the whole machine. Thus bearing diagnosis should be taken into full consideration for the normal machine operation. So far, many different fault detection methods have been applied for the engineering problem. For example, resonance demodulation which is a classic diagnosis has been used for several decades and developed into many new methods in these years. Spectral kurtosis¹ is one of the most successful modified envelope analysis methods, which is widely used in practical engineering problem. However, this method cannot find the most optimum frequency band with high SNR for band-pass filtering which is an important part of the envelope analysis. Thus, many new methods which intend to find optimal band are proposed. Gu² applied Morlet wavelet filter to find the best envelope parameters containing filtering band and filter construction. Randall³ used GA to find the optimal band with high SNR for the extraction of envelope spectrum. However, since the weak fault energy is very low, it is not easy to obtain effective envelope spectrum only by searching optimal band.

Then some other fault detection methods based on intelligent algorithms are proposed. Guo⁴ used network to extract weak fault feature as the health indicator which can tell the damage degree. And the support vector^5-8 is trained for weak fault recognition. Markov model^9-12 is also a popular tool for weak fault detection, which can decrease the effect of the random noise. However, these methods based on intelligent algorithm cannot precisely describe the real damage degree of the bearing, which just use the basic data characteristics to extract the fault feature without relating to the physical principle. When the damage occurs on the surface of the race order, the fault impact amplitude can indicate the real damage degree. Thus fault impact recognition and extraction is still an important way to detect the weak fault.

Then SSD^13-15 which is widely used in earthquake impact detection can be used here for fault extraction. Shulin Pan^16-18 has contributed a lot to the field of SSD and made a great progress on it in these years. His research proves that this algorithm is sensitive to impact based on sparsity assumption. Then a new weak fault impact extraction method based on amplification-compression function and SSD is proposed. Firstly, the amplification function is used to amplify the amplitude of the weak impact. Then the amplified signal is decomposed into impact train using sparse spike deconvolution which can decrease the influence of the gaussian noise. Then the compression function is used to compress the extracted impact amplitude, which can reduce noise influence. In the end, the fault vibration data of rolling bearing is used to verify the effectiveness of the proposed method. And a common method is used as comparison. The contribution of this paper is to proposed a method to amplify the weak fault of the bearing by amplifying the peaks which may relate to the fault impact. And every peak amplification scale is set according to derivative of the peak. And the differences of the peak amplitudes after amplification are bigger. Then compression method is used to decrease the bigger peak amplitude so that the fault impact periodicity can be strengthened. And the SSD can be applied to extract the amplified-compressed impact.

The principle of weak fault impact detection

The principle of amplified function

Since the derivative of fault impact is bigger than the one of many other vibration components, an amplification function combined with the signal derivative is built to enhance the fault impact amplitude. And the amplification function is shown as follows

y (t) = x (t) . * \exp ({(x (t))}^{'} / σ)

(1)

x (t)

is the original vibration signal.

{(x (t))}^{'}

is the derivative function of

x (t)

σ

is the amplification scale parameter, which is corresponding to the magnification times. And

y (t)

is the amplified signal. Based on equation (1), we can see that the amplitude of every signal point is amplified according to their derivatives. Bigger derivative means bigger amplification. Since impact has bigger derivative, magnification times of impact amplitude is bigger than many other signal components. Then the impact with small amplitude can be outstanding after this transform. However, some other noise with big derivative is still in the signal, some of which is random noise. Then the SSD is used in the next step.

The principle of sparse spike deconvolution

Sparse spike deconvolution is a popular algorithm which is widely used to detect impact in earthquake. This model of the algorithm assumes that the earth structure can be represented adequately by a set of planar layers of constant impedance, which is shown as follows

s_{t} = w_{t} * r_{t} + {r a n d}_{t}, t = 1,2, \dots \dots, N

(2)

where

w_{t}

is the seismic wavelet and

{r a n d}_{t}

is the additive noise, and

r_{t}

is the impact sequences. In bearing diagnosis,

w_{t}

is the vibration transition characteristics parameters, and

r_{t}

is the fault impact sequences. The solution of this equation is to extract the fault impact from

s_{t}

, which is very difficult. Then regularization method is used here to figure out this problem. Firstly, the expectation should be calculated as follows

e_{t} = {\sum_{i} r_{i} w}_{t - i} - s_{t}, i = 1,2, \dots \dots, N

(3)

Then another important formula is used here as follows

J = J_{r} + μ J_{x}

(4)

Here,

J

is the objective function. The first term

J_{r}

is shown as follows

J_{r} = \sum_{t} \frac{1}{2} e_{t}^{2}

(5)

While the second term which is regularization term is shown as follows

J_{x} = \sum_{t} | r_{t} |

(6)

$μ$ is the regularization parameter. The purpose of these formulas is to find the optimal $r_{i}$ to minimize J, which would be realized by solving the follow formula

\frac{\partial J}{\partial r_{i}} = \sum_{t} e_{t} w_{t - i} + μ \sum_{i} \frac{| r_{i} |}{r_{i}} = 0

(7)

Based on the above formula, another one can be deduced as follows

\sum_{t} \sum_{i} [(w_{t - i} w_{t - i} + μ \frac{1}{| r_{i} |}) r_{i}] = \sum_{t} (w_{t - i} d_{t})

(8)

which can be transformed to matrix form as follows

(R + Q) x = g, Q = d i a g (μ \frac{1}{| r_{i} |})

(9)

where diag() means diagonal transformation of the matrix in the brackets, and

x

is the column vector of

r_{t}

. The methods to solve x are many. Here, we use least square method to solve the problem. The basic steps are shown as follows:

(1) Set the regularization parameter μ

(2) Calculate matrix R and g

(3) Use $x_{i n i t i a l}$ to calculate the initial matrix Q

Q = d i a g (μ \frac{1}{| x_{i n i t i a l} |})

(10)

(4) Then the iteration can be conducted to calculate x and Q by following formula

Q^{(k)} = d i a g (μ \frac{1}{| x^{(k)} |})

(11)

x^{(k)} = {(R + Q^{(k - 1)})}^{- 1} g

(12)

where k is the iteration number.

(5) The iteration doesn’t stop until k reached its predetermined value.

The principle of compression procession

After the above procession, small impacts in the signal are amplified. However, the amplitude sizes of initial fault impacts are obviously different. Some of them are bigger while others are small or even not detected. Then if original signal is just processed by equation (1), the amplitude differences of these fault impacts would become extremely larger, which would weaken fault periodicity. Moreover, some small noise with bigger derivative is also amplified, which can negatively affect the SNR. Then clear fault frequency with its harmonics may not be clearly shown in envelope spectrum. Thus, the amplitudes of bigger impacts should be compressed to strengthen fault periodicity. As the amplitude of fault impact and amplified noise is compressed down to the similar size, the total energy of all fault impacts in the signal will be more apparent than the noise energy. Then the scanning compression method is used to reduce the bigger impact amplitudes. First, we can set several thresholds which should be set as follows

t h r e s h o l d = \frac{1}{n} \times A_{m}, (n = 1,2,3 \dots \dots, N)

(13)

where N is the number of the thresholds, and n is the compression factor, and

A_{m}

is the maximum amplitude of the amplified signal. Then all the thresholds are used to compare with every amplitude of the signal one by one. Firstly, we choose one threshold to compare with signal amplitude. If the amplitude is bigger than the threshold, it is replaced by the threshold. Otherwise, the amplitude remains unchanged. Then the next threshold is used for the next comparison. The iteration doesn’t stop until all the thresholds finish comparison. At last the correlated kurtosis of these compressed signals is calculated, which can indicate the periodicity of the fault impact. The bigger correlated kurtosis means more obvious periodicity. Its formula is expressed as

C K_{M} (T) = \frac{\sum_{i = 1}^{N} {(\prod_{m = 0}^{M} y_{i - m T})}^{2}}{{(\sum_{n = 1}^{N} y_{i}^{2})}^{M + 1}}

(14)

Here, $y_{i}$ stands for the compressed signal and T is the cycle of the fault impulse signal. M represents the number of cycles in the migration. CK takes both the impulsion and the cyclicity into consideration Figure 1.

Figure 1.

Flow chart of proposed method.

The flow chart of the whole proposed method is shown as follows:

Simulated validation

The simulated fault signal is used here to validate the effectiveness of the proposed method, whose model is shown as follow

{\begin{cases} x (t) = \sum_{i = 1}^{N} s (t - i T - τ_{i}) + B (t) + r a n d (t) \\ s (t) = e^{- 2 π f_{n} ξ t} \sin (2 π f_{n} t + ϕ_{w}) \\ B (t) = B_{0} \sin (2 π f_{m} t + π / 2) \end{cases}

(15)

Here,

f r

is the roller’s rotation frequency.

B (t)

is the background harmonic component.

f_{n}

stands for natural frequency of the system and

S (t)

is an exponential decay pulse. The fault impact interval is set as

T

τ_{i}

is the ith impulsion delay.

ξ

stands for system damping coefficient and

r a n d (t)

is white noise.

ϕ_{w}

is the initial phase.

B_{0}

is the amplitude of the sinusoidal signal. The parameters of the fault signal model are set as bellow:

$f r = 40, f n = 1600, T = 1 / 200, τ_{i} = 1, A_{0} = 2, B_{0} = 4, c_{A} = 0.5$ , $f_{m} = 2000$ . Sample frequency is set as 20,000 Hz. And the fault timewave and spectrum is shown in Figure 2.

Figure 2.

Simulated fault signal (a) Timewave of simulated Fault signal (b )frequency spectrum of simulated fault signal.

Then the proposed method and fast-kurtogram¹ are, respectively, used to process the simulated signal. The proposed method contains amplification-compression part and SSD part. The amplification procedure uses equation (1) to enlarge the impact amplitude, and SSD is used to extract impact from these amplified signals. Then several thresholds are set to compress the bigger impact amplitude for the reinforcement of fault periodicity. Here, the

σ

is set as 1.5.

μ

is set as 6. The iteration number k is set as 100. The compression factor n corresponding to thresholds in equation (13) is set as [1 2 3 4 5]. The correlated kurtosis of these compressed signals is calculated and shown in Table 1. The signals processed by different compression factors are shown in Figure 3, Figure 4, Figure 5, Figure 6, Figure 7. In Table 1, the biggest correlated kurtosis is 0.0037, which is corresponding to compression factor n = 4.

Table 1.

Different correlated kurtosis of vibration signal processed by different compression factor.

Compression factor	Correlated kurtosis
1	0.0011
2	0.0022
3	0.0025
4	0.0037
5	0.0028

Figure 3.

Timewave and the corresponding envelope spectrum (n = 1, $μ = 6$ , $σ = 1.5$ ) (a) Timewave (b) Envelope spectrum.

Figure 4.

Timewave and the corresponding envelope spectrum (n = 2, $μ = 6$ , $σ = 1.5$ ) (a) Timewave (b) Envelope spectrum.

Figure 5.

Timewave and the corresponding envelope spectrum (n = 3, $μ = 6$ , $σ = 1.5$ ) (a) Timewave (b) Envelope spectrum.

Figure 6.

Timewave and the corresponding envelope spectrum (n = 4, $μ = 6$ , $σ = 1.5$ ) (a) Timewave (b) Envelope spectrum.

Figure 7.

Timewave and the corresponding envelope spectrum (n = 5, $μ = 6$ , $σ = 1.5$ ) (a) Timewave (b) Envelope spectrum.

We can see that the impacts are enlarged and shown obviously in Figure 6(a). And we can also clearly see the preset fault frequency 200 Hz with its multiply order in Figure 6(b) which is the most optimal envelope spectrum in these figures. The timewave in Figure 3(a) with n = 1 shows several bigger impacts. However, the corresponding envelope spectrum can hardly show the fault frequency. That is because these several bigger impacts which are the main energy components of the signal are not periodic. Then the aperiodicity of these several bigger impacts covers up periodicity of the smaller fault impacts. Thus, the whole signal doesn’t show the fault impacts periodicity in envelope spectrum. Then as the compression factor become bigger, the fault frequency in the envelope spectrum becomes more obvious in Figure 4 and Figure 5. This is because bigger compression factor weakens these bigger impacts energy. And the SNR becomes high. Then CK reaches maximum value in Figure 6. Then as n become bigger, the fault frequency becomes unapparent in Figure 7. This is because the over compression reduces the amplitudes of the fault impacts sequence. And more small noise relatively becomes bigger, which reduce the SNR. Based on the above analysis, the optimal output envelope signal is Figure 6(a) relating to the biggest CK. The results show that the compression factor should not be too big or too small.

Then different regularization parameters $μ$ are also used to process the signal. And the other parameters are the same with the above ones. The output envelope signals with their corresponding parameters are shown in Figure 8, Figure 9, Figure 10, and Table 2. Based on Figure 8, we can see that different regularization parameters result in different effect envelope spectrums. Here, output envelope spectrum corresponding to $μ = 6$ is the best one in these figures. And the other two envelope spectrums are less effective. Thus the regularization parameter should not be too big or too small, neither.

Figure 8.

Timewave and the corresponding envelope spectrum ( $μ = 2$ ) (a) Timewave (b) Envelope spectrum.

Figure 9.

Timewave and the corresponding envelope spectrum ( $μ = 6$ ) (a) Timewave (b) Envelope spectrum.

Figure 10.

Timewave and the corresponding envelope spectrum ( $μ = 10$ ) (a) Timewave (b) Envelope spectrum.

Table 2.

Different biggest correlated kurtosis of vibration signal processed by different regularization parameter.

Regularization parameter	The biggest correlated kurtosis
2	0.0018
6	0.0037
10	0.0021

Figure 11 is the kurtogram obtained by fast-kurtogram method, which shows the optimal band in level 6.6 with its center frequency 8500. Then the corresponding envelope timewave and spectrum are, respectively, shown in Figure 11(b) and Figure 11(c). However, the timewave shown in Figure 11(b) is hardly to show the impacts. And the fault frequency cannot be seen in envelope spectrum.

Figure 11.

Fast-kurtogram method processing results (a) Fast-kurtogram (b) The optimal envelope timewave (c) The optimal envelope spectrum.

Therefore, fast-kurtogram method is less effective than proposed method.

Experiment validation

In this part, the experimental data is used for the modified envelope analysis. In order to acquire the complicated vibration signal, self-designed experimental device is built by simulating the motor output structure of the marine gas turbine. And the structure of the experimental device is shown in Figure 12 and Figure 13. The location of the bearing used for experiment is shown in Figure 14. From Figure 14, we can see that the vibration transfer path of the fault bearing is first to go through the cage, and then to pass the two flanges, and to go through the thick chassis, and at last to reach the position of sensor. Thus, this path is a complicated one, which means the fault signal may bury into the huge background noise. Therefore, the vibration collected from measure point can be used for the comparison filtering. The fault of bearing is made by wire-electrode cutting.

Figure 12.

The integral figure of the experimental device.

Figure 13.

Support structure for rolling elements bearing.

Figure 14.

Original vibration timewave and the corresponding frequency spectrum (7000r/min).

The bearing fault experiment is conducted at rotation speeds 7000r/min. The type of the device for vibration measurement is B&K3560 C. And the sample frequency is 131,072 Hz. The experimental bearing type is NSK6010, whose inner diameter and outer diameter, respectively, are 50 mm and 80 mm. The number of rolling elements is 13 with the element’s diameter 9 mm and $α = 0 °$ . The artificial fault is located on the surface of inner race, and the corresponding frequency is 874.1 Hz. The original vibration signal is shown in Figure 14 which contains the waveform in time domain and frequency spectrum. And the fast-kurtogram method is used as the comparison factor for the feature extraction.

Here, the $σ$ of the proposed method is set as 1.5. Compression factor n is also set as [1 2 3 4 5]. The regularization parameter is set as 6. And the correlated kurtosis of these processed signals is shown in Table 1. The processed signal with different compression factor n are shown in Figure 15, Figure 16, Figure 17, Figure 18, Figure 19. In Table 3, the biggest correlated kurtosis is 0.0065, which relates to timewave of Figure 17(a) with compression factor n = 3. And the impacts in the timewave of Figure 17(a) is enlarged obviously. And fault frequency 875 Hz with its multiply order is clearly shown in Figure 17(b), which is the most optimal envelope spectrum in these figures. Although in Figure 15 with n = 1, we can see several bigger impacts in the timewave, the corresponding envelope spectrum can hardly show the fault frequency. That means bigger differences among the impacts’ amplitudes can weaken the periodicity of the bearing fault. And the corresponding fault characteristics cannot be obviously presented in the envelope spectrum. Then as the compression factor become bigger, the fault frequency in the envelope spectrum becomes more obvious in Figure 16 and Figure 17. Then after the biggest CK, the fault frequency starts to become unapparent in Figure 18 and Figure 19. That means too much compression may relatively enhance the noise amplitude and decrease SNR. And the fault frequency with its harmonics is unobvious in the spectrum. Thus based on the above results, the derivative based amplification can effectively amplify the impact. By combining with the multiple-scale compression method, the optimal envelope spectrum can be outstanding.

Figure 15.

Timewave and the corresponding envelope spectrum (n = 1, $μ = 6$ , $σ = 1.5$ ) (a) Timewave (b) Envelope spectrum.

Figure 16.

Timewave and the corresponding envelope spectrum (n = 2, $μ = 6$ , $σ = 1.5$ ) (a) Timewave (b) Envelope spectrum.

Figure 17.

Timewave and the corresponding envelope spectrum (n = 3, $μ = 6$ , $σ = 1.5$ ) (a) Timewave (b) Envelope spectrum.

Figure 18.

Timewave and the corresponding envelope spectrum (n = 4, $μ = 6$ , $σ = 1.5$ ) (a) Timewave (b) Envelope spectrum.

Figure 19.

Timewave and the corresponding envelope spectrum (n = 5, $μ = 6$ , $σ = 1.5$ ) (a) Timewave (b) Envelope spectrum.

Table 3.

Different correlated kurtosis of vibration signal processed by different compression factor.

Compression factor	Correlated kurtosis
1	0.0053
2	0.0054
3	0.0065
4	0.0059
5	0.0049

Then different regularization parameters are also used to process the signal. And the corresponding results are shown in Figure 20 and Table 4. In these figures, we can see that the output envelope spectrum corresponding to $μ = 6$ is better. Though Figure 20 can show obvious fault frequency, it is less effective in comparison with Figure 21. And Figure 22 is less effective compared with other two pictures. Thus regularization parameter should be chosen appropriately.

Figure 20.

Timewave and the corresponding envelope spectrum ( $μ = 2$ ) (a) Timewave (b) Envelope spectrum.

Table 4.

Different biggest correlated kurtosis of vibration signal processed by different regularization parameter.

Regularization Parameter	The biggest correlated kurtosis
2	0.0051
6	0.0065
10	0.0061

Figure 21.

Timewave and the corresponding envelope spectrum ( $μ = 6$ ) (a) Timewave (b) Envelope spectrum.

Figure 22.

Timewave and the corresponding envelope spectrum ( $μ = 10$ ) (a) Timewave (b) Envelope spectrum.

The kurtogram of the signal processed by fast-kurtogram method is shown in Figure 23, which shows the optimal band in level 5.6 with its center frequency 49,720 Hz. Then the corresponding envelope timewave and spectrum are, respectively, shown in Figure 23(b) and Figure 23(c). The timewave shown in Figure 23(a) only show the first order of fault frequency. And other order of the fault frequency cannot be seen in band. Thus the fast-kurtogram method cannot well handle with the detection of weak fault within the huge background noise. One important reason that cause this result is the stubborn filtering band selection. And this kind of selection can find optimal band in a tinny possibility.

Figure 23.

Experimental result obtained by fast-kurtogram method (a) fast-kurtogram (b) optimal envelope time wave (c) optimal envelope spectrum.

Another experiment with the same bearing is conducted, and the rotation speed is 8000r/min. The fault frequency is 986.67 Hz. The corresponding original vibration timewave and frequency spectrum are shown in Figure 24. The parameters of the previous experiment with rotation speed 7000 r/min are still used here, which means σ, Compression factor n, and the regularization parameter remain unchanged. Then the processed signal with different compression factor n are shown in Figure 25, Figure 26, Figure 27, Figure 28, Figure 29. Table 5 shows different correlated kurtosis of the signals processed by different compression factors. Table 6 shows different biggest correlated kurtosis of vibration signal processed by different regularization parameter. Figure 30, Figure 31, Figure 32 show that the effect different regularization parameters have on the envelope spectrum. Figure 33 is the optimal envelope timewave and spectrum obtained by fast-kurtogram method.

Figure 24.

Timewave and the corresponding frequency spectrum (8000r/min).

Figure 25.

Timewave and the corresponding envelope spectrum (n = 1, $μ = 6$ , $σ = 1.5$ ) (a) Timewave (b) Envelope spectrum.

Figure 26.

Timewave and the corresponding envelope spectrum (n = 2, $μ = 6$ , $σ = 1.5$ ) (a) Timewave (b) Envelope spectrum.

Figure 27.

Timewave and the corresponding envelope spectrum (n = 3, $μ = 6$ , $σ = 1.5$ ) (a) Timewave (b) Envelope spectrum.

Figure 28.

Timewave and the corresponding envelope spectrum (n = 4, $μ = 6$ , $σ = 1.5$ ) (a) Timewave (b) Envelope spectrum.

Figure 29.

Timewave and the corresponding envelope spectrum (n = 5, $μ = 6$ , $σ = 1.5$ ) (a) Timewave (b) Envelope spectrum.

Table 5.

Different correlated kurtosis of vibration signal processed by different compression factor.

Compression factor	Correlated kurtosis
1	0.0033
2	0.0037
3	0.0045
4	0.0031
5	0.0029

Table 6.

Different biggest correlated kurtosis of vibration signal processed by different regularization parameter.

Regularization Parameter	The biggest correlated kurtosis
2	0.0039
6	0.0045
10	0.0041

Figure 30.

Timewave and the corresponding envelope spectrum ( $μ = 2$ ) (a) Timewave (b) Envelope spectrum.

Figure 31.

Timewave and the corresponding envelope spectrum ( $μ = 6$ ) (a) Timewave (b) Envelope spectrum.

Figure 32.

Timewave and the corresponding envelope spectrum ( $μ = 10$ ) (a) Timewave (b) Envelope spectrum.

Figure 33.

Experimental result obtained by fast-kurtogram method (a) fast-kurtogram (b) optimal envelope time wave (c) optimal envelope spectrum.

In Figure 25, Figure 26, Figure 27, Figure 28, Figure 29, we can see the best envelope spectrum whose compression factor is 3. In Figure 30, Figure 31, Figure 32, the fault feature of the spectrum with $μ = 2$ is less obvious in comparison to the other two. These results also show that the different compression factor and regularization parameter can bring different results of fault feature extraction. And the relationship between these parameters and feature extraction result is not monotone increasing or decreasing. The compression factor has great affect on the envelope analysis result. We can see the amplitude of a few impacts in the signal of Figure 26 is too big, it will decrease the periodicity of the impact train. And the corresponding envelope spectrum cannot show obvious fault characteristics. However, if the signal amplitude is compressed too much, the noise becomes big to decrease SNR. And the fault feature in the corresponding envelope spectrum is not obvious, which can be shown in Figure 19. Then regularization parameter and compression factor should be chosen appropriately. In Figure 33, the optimal envelope spectrum can show the fault frequency with its harmonics. However, the noise which negatively influence the fault feature extraction result is huge in comparison with the proposed method.

Therefore, based on the above analysis, the proposed method can obviously amplify the weak fault impact. And the effect of the proposed method is better than the common method.

Conclusion

In this study, amplification-compression function and SSD algorithm are used to extract weak fault impact. Most of the common methods focus on the elimination of the noise. This paper tries another way to amplify the fault impact itself. And it is shown that the amplification-compression method can effectively amplify the small impact in the signal by remaining strong periodicity of the fault impact. Then the amplified impact can be extracted by SSD. The results show that amplification-compression procession can enhance the SNR with appropriate regularization parameter and compression factor. Therefore, the advantage of this method is able to extract weak impact under background noise with high amplitude. And this derivative feature is sensitive to the impact wave, which can apparently distinguish the transient component. Thus, it is more stable and applicable in the real-world application in comparison to the fault detection model based on data-training. Moreover, the extracted impact is closely related to the damage severity which can indicate the health condition. However, the selection of the most optimal regularization parameter or compression factor should be analyzed in depth using optimal algorithms, which can be studied in further research.

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: This work was supported by the National Natural Science Foundation of China (grant 51579242).

ORCID iD

Shuai Zhang

References

Antoni

. Fast computation of the kurtogram for the detection of transient faults. Mech Syst Signal Process 2007; 21(1): 108–124.

Yang

Liu

, et al. Compound faults detection of the rolling element bearing based on the optimal complex Morlet wavelet filter. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 2018; 232(10): 1786–1801.

Zhang

Randall

. Rolling element bearing fault diagnosis based on the combination of genetic algorithms and fast kurtogram. Mechanical Systems and Signal Processing 2009; 23(5): 1509–1517.

GuoLiJia

LNF

Lei

Lin

. A recurrent neural network based health indicator for remaining useful life prediction of bearings. Neurocomputing 2017; 240(5): 98–109.

TanXieLiu

HSR

. Bearing fault identification based on stacking modified composite multiscale dispersion entropy and optimised support vector machine. [J]. Measurement 2021; 186(4): 110180.

Chen

XiaoliWang

ChuangchuangWu

, et al. Fault diagnosis of rolling bearing using marine predators algorithm-based support vector machine and topology learning and out-of-sample embedding. J]. Measurement 2021; 176(1): 88–97.

Tan

Cho

Lee

. Human mini-brain models. Nat Biomed Eng 2021; 5(4): 11–25.

Van

Hoang

D T

Kang

H J

. Bearing fault diagnosis using a particle swarm optimization-least squares wavelet support vector machine classifier[J]. Sensors (Basel, Switzerland) 2020; 20(12): 15–23.

Rehab

Ali

Gomaa

, et al. Bearings Fault detection using hidden markov models and principal component analysis enhanced features. 2021.

10.

WangXiangZhong

SJY

Zhou

. Convolutional neural network-based hidden Markov models for rolling element bearing fault identification. Knowl Based Syst 2018; 144(mar.15): 65–76.

11.

Lopez

Naranjo

, et al. Hidden Markov Model based Stochastic Resonance and its Application to Bearing Fault Diagnosis[J]. Journal of Sound and Vibration 2022; 5: 52–68.

12.

LiFang

Huang

Wei

, et al. Data-driven bearing fault identification using improved hidden Markov model and self-organizing map. Computers and Industrial Engineering 2018; 116(2): 37–46.

13.

Velis

D R

. Stochastic sparse-spike deconvolution. Geophysics 2008; 73(1): R1–R9.

14.

Sui

. Blind sparse spike deconvolution with thin layers and structure[J]. Geophysics 2020; 85(6): 1–92.

15.

Chai

Tang

Lin

, et al. Deep learning for multitrace sparse-spike deconvolution[J]. Geophysics: Journal of the Society of Exploration Geophysicists 2021; 3: 86–96.

16.

Pan

Yan

Lan

, et al. Adaptive step-size fast Iterative shrinkage-thresholding algorithm and sparse-spike deconvolution. Computers and Geosciences 2020; 134: 104343.

17.

Pan

Yan

Lan

, et al. A Bregman adaptive sparse-spike deconvolution method in the frequency domain. Appl Geophys 2019; 16(4): 463–472.

18.

Pan

Yan

Lan

, et al. A sparse spike deconvolution algorithm based on a recurrent neural network and the iterative shrinkage-thresholding algorithm. Energies 2020; 13: 3074.