A Fast Signal Estimation Method Based on Probability Density Functions for Fault Feature Extraction of Rolling Bearings

applied sciences

Article A Fast Signal Estimation Method Based on Probability Density Functions for Fault Feature Extraction of Rolling Bearings

Shijun Li, Weiguo Huang *, Juanjuan Shi , Xingxing Jiang and Zhongkui Zhu

School of Rail Transportation, Soochow University, Suzhou 215131, China * Correspondence: [email protected]; Tel.: +86-136-4622-0530

Received: 14 August 2019; Accepted: 4 September 2019; Published: 9 September 2019

Abstract: Fault diagnosis of rolling bearings is essential to ensure the efficient and safe operation of mechanical equipment. The extraction of fault features of the repetitive transient component from noisy vibration signals is key to bearing fault diagnosis. However, the bearing fault-induced transients are often submerged by strong background noise and interference. To effectively detect such fault-related transient components, this paper proposes a probability- and statistics-based method. The maximum-a-posteriori (MAP) estimator combined with probability density functions (pdfs) of the repetitive transient component, which is modeled by a mixture of two Laplace pdfs and noise, were used to derive the fast estimation model of the transient component. Subsequently, the LapGauss pdf was adopted to model the noisy coefficients. The parameters of the model derived could then be estimated quickly using the iterative expectation–maximization (EM) algorithm. The main contributions of the proposed statistic-based method are that: (1) transients and their wavelet coefficients are modeled as mixed Laplace pdfs; (2) LapGauss pdf is used to model noisy signals and their wavelet coefficients, facilitating the computation of the proposed method; and (3) computational complexity changes linearly with the size of the dataset and thus contributing to the fast estimation, indicated by analysis of the computational performance of the proposed method. The simulation and experimental vibration signals of faulty bearings were applied to test the effectiveness of the proposed method for fast fault feature extraction. Comparisons of computational complexity between the proposed method and other transient extraction methods were also conducted, showing that the computational complexity of the proposed method is proportional to the size of the dataset, leading to a high computational efficiency.

Keywords: fault diagnosis; feature extraction; MAP estimator; probability density functions; EM algorithm

1. Introduction Rolling bearings are key parts of much of the modern mechanical equipment which plays an essential role in social production and living, such as engines, generators, machine tools, and so on. As rolling bearings usually operate under harsh working conditions and need to work long hours, they are prone to failures [1]. So, it is crucial to diagnose faults of the rolling bearings as soon as possible to avoid safety accidents and maintain good working efficiency. Vibration signals generated by faulty bearings directly reflect the dynamic characteristics of faulty bearings [2]. Therefore, it is effective and reasonable to extract the features of the repetitive transient components by analyzing the vibration signals of faulty bearings. However, the vibration signals measured are always drowned out by strong noise, which makes the features of the repetitive transient component difficult to extract [3]. Therefore, methods to effectively extract the features of the repetitive transient component from the noisy vibration signals are key to bearing fault diagnosis.

Appl. Sci. 2019, 9, 3768; doi:10.3390/app9183768 www.mdpi.com/journal/applsci Appl. Sci. 2019, 9, 3768 2 of 15

To deal with this issue, various effective methods, such as time–frequency analysis [4], spectral kurtosis [5], empirical mode decomposition [6], wavelet analysis [7], stochastic resonance [8], deep learning [9], sparse representation [10] and bispectrum analysis [11] have been proposed over the past decade. The time-frequency analysis method with high time-frequency aggregation has achieved good results in the extraction of frequency conversion curve and the extraction of fault feature frequency [12]. The empirical mode decomposition method has good time-frequency resolution and adaptability and is able to separate the noise and effective component by reasonably selecting the intrinsic mode functions without giving a given transform base [13]. The spectral kurtosis filtering method, based on the non-Gaussian property exhibited by the fault component, designs the filter to select the resonant frequency band of the fault component according to the principle of maximizing the statistic such as kurtosis, and then determines the fault feature frequency according to envelope spectrum analysis [14]. The deep learning method can adaptively extract features based on the model constructed, which provides a new idea for improving the more accurate and effective fault diagnosis of rolling bearings [15]. Sparse representation has been successfully applied to mechanical fault diagnosis and achieved significant results. As an optimization method, the sparse representation method needs to optimize an objective function, which consists of the fidelity term, penalty term and regularization parameter [16]. In the field of mechanical fault diagnosis, research on sparse representation has focused on the construction of penalty functions [17]. At the same time, in the sparse representation method an over-complete dictionary according to the characteristics of the components needs to be constructed to be estimated in advance [18]. Modulation signal bispectrum (MSB) is a representative of bispectrum analysis-based methods. The MSB method has inherent noise suppression capabilities and can help enhance the modulation effects of a bearing fault and achieve the goal of frequency band optimization for fault detection [19]. The aforementioned methods have achieved good results in fault feature extraction. However, most methods described above cannot satisfy the requirement of timeliness very well. From the point of view of the timeliness of fault feature extraction, a method to quickly estimate the transient components of noisy bearing fault signals and extract fault features is needed. Many methods based on probability and statistics have been proposed and successfully applied to many fields, including fault diagnosis and detection [20]. These methods can mainly be divided into three categories: maximum-likelihood (ML) [21], Bayesian [22], and maximum-a-posteriori (MAP) approaches [23]. The methods based on ML are common and use a simple approach, which needs to maximize a log-likelihood function. However, these approaches might not work for denoising problems [24]. The Bayesian-oriented methods make the optimal estimation based on the known probability distribution and observed data, which is often used to solve the optimization and approximation problems of nonlinear signal models or processing systems [25]. MAP approaches use an assumed prior distribution of the related signal to regularize the estimation process, and the basis of MAP is Bayesian formula. These methods usually have deduced probabilistic models and the pdfs of the related signals play an important role in these methods. Furthermore, most of these methods need to iteratively estimate the parameters of the models deduced. But, generally speaking, these methods have high computational efficiency. In this paper, we propose an effective wavelet-based method with low computational complexity to fast extract the fault feature components from noisy bearing fault signals. This method only needs one prior, the pdfs of related signals. The analysis of simulation and experimental noisy fault vibration signals prove that our proposed method can efficiently estimate the repetitive transient component and extract fault feature frequency. The proposed method uses the MAP estimator and derives the general model of estimating the transient components based on the pdfs of the transient components of faulty bearings and noise, which means that our method has a scientific explanation of probability theory. From the point of view of timeliness, the proposed method in this paper only requires a small amount of iterative parameter estimation of the model derived using the expectation-maximization (EM) algorithm. At the same time, the computational cost of the discrete wavelet transform is also very low. Therefore, compared with some other popular fault feature extraction methods, the computational Appl. Sci. 2019, 9, 3768 3 of 15 complexity of our proposed method is much lower and the running time also proves the high computation efficiency. Our method can not only get the result at a fast speed, but also ensure the accuracy of fault feature extraction. This paper is organized as follows. In Section2, the derivation of the model to estimate the transient components and the algorithm to estimate parameters are presented in detail. In Section3, the proposed method is used to analyze a simulated signal. In Section4, three experimental bearing fault signals are analyzed using the proposed method respectively. Lastly, Section5 presents the conclusions of the study.

2. Theoretical Description

2.1. Model Derivation In the time domain, a bearing fault signal corrupted by white Gaussian noise can be represented as

g = x + nt, (1) where g RN denotes the observed noisy bearing fault signal, x RN denotes the noise-free transient ∈ N ∈ components and nt R denotes noise in time domain. It is necessary to estimate the transient ∈ components x as accurately as possible based on the noisy signal g and their characteristics. In order to make the most of the statistical distribution characteristics of related signals, a linear transformation is needed. Wavelet transform is an efficient linear transformation and it transforms a signal in time domain into time-frequency domain, which can better observe the local characteristics of the signal and simultaneously observe the time and frequency information of the signal [26]. At the same time, the computational complexity of wavelet transform is also low, which can ensure the fast extraction of fault features. When the wavelet transform with orthogonality is performed, the model can be represented as y = w + n, (2) where y denotes the wavelet coefficients of the observed noisy bearing fault signal g, w denotes the wavelet coefficients of the noise-free transient components x and n denotes noise. Through the analysis of bearing fault mechanism and statistical analysis of simulation signals, we found that the pdfs of the noise-free transient components and its wavelet coefficients are mixture of two Laplace pdfs [27]: p(w) = k1lap(σ1) + k2lap(σ2), (3) where √ 1 2 w lap(σ) = e− σ | |, (4) √2σ and k1 + k2 = 1. And noise follows Gaussian distribution and is independent of transient components [28]. In order to minimize the noise contained in the estimated components w, the MAP estimator is adopted to estimate w according to noisy wavelet coefficients y [23]. The MAP estimator for w in Equation (2) is defined as wˆ (y) = arg max p(w y). (5) w | Based on the Bayes’ theorem, the maximum a posteriori estimator can be expressed as

p(y w)p(w) wˆ (y) = arg max | . (6) w p(y) Appl. Sci. 2019, 9, 3768 4 of 15

As the pdf p(y) on the denominator is irrelevant with w, the MAP estimator of w is transformed into the following form wˆ (y) = arg max p(y w) p(w). (7) w | · The Equation (7) is equivalent to solving the following formula

wˆ (y) = arg max ln[p(y w) p(w)]. (8) w | · Consequently, the MAP estimate of w can be obtained by solving the following formula

∂ ∂ ln p(y w) + ln p(w) = 0. (9) ∂w | ∂w According to the form of the above formula, it is necessary to know the conditional pdf p(y w). | As for the model in Equation (2), p(w y) denotes the pdf of n when the parameter w is given |

(y w)2 1 − 2 p(y w) = e− 2σn , (10) | √2πσn

2 where σn represents the variance of the white zero-mean Gaussian noise. The standard variance of the independent noise σn also needs to be estimated. The parameter can be estimated through a frequently used method when implementing denoising in wavelet domain

MAD σ = , (11) n 0.6745 where MAD denotes median absolute deviation MAD = Median yn Median(yn) , (12) | − | and yn denotes the wavelet coefficients in the finest level [29]. In this paper, the estimation is carried out at the first level which which is often the finest level used for estimation in wavelet denoising problems and thus yn denotes the level 1 detail coefficients. The single Laplace model will be discussed firstly and then extended to the mixture model. If the distribution of the wavelet coefficients w follow single Laplacian pdf,

√ 1 2 w p(w) = e− σ | |. (13) √2σ

Plug the above equation into Equation (9), it can be found that

√2σ2 y = wˆ + n sign(wˆ ). (14) σ · The relationship between wˆ and y can be represented as   y + T, y T  ≤ − wˆ =  y T , (15)  0,  | | ≤ y T, y T − ≥

2 where T = √2σn/σ. The above function is the frequently used soft thresholding and the notation is often written as    √2σ2  wˆ (y) = softy, n . (16)  σ  Appl. Sci. 2019, 9, 3768 5 of 15

Then, if the pdf of w is a mixture of two Laplace pdfs as in Equation (3), the result is shown as

     √2σ2   √2σ2  ( ) = ( )  n  + ( )  n  wˆ y pk1 y softy,  pk2 y softy, , (17)  σ1   σ2  where k g (y) ( ) = i i = pki y , i 1, 2. (18) k1 g1(y) + k2 g2(y) As to the model y = w + n, since w and n are independent of each other, the pdf of y can be calculated as the convolution of w and n according to relevant knowledge of probability theory

y2 √2 1 σ y 1 − 2σ2 gi(y) = e− i | | e n , i = 1, 2. (19) √2σi ∗ √2πσn

Through the aforementioned deductions, the pdf of y is needed to improve the eﬃciency of calculations. That is, it is necessary to ﬁnd the pdf of y = w + n, where w is the noise-free value with a Laplace distribution and n is the noise with a Gaussian distribution. As w is a Laplace random variable and assume that the standard deviation of w is σw. The pdf of w is: √ 1 2 w p(w) = e− σw | |, (20) √2σw and n denotes zero-mean Gaussian noise, assume the standard deviation of n is σn. The pdf of n is:

n2 1 2 p(n) = e− 2σn . (21) √2πσn

According to Hansen [30] and after veriﬁcation, the pdf of y is:

y2 ! ! 1 2 σn y σn y p(y) = e− 2σn [erfcx | | + erfcx + | | ], (22) 2 √2σw · σw − √2σn σw √2σn

where 2 erfcx(x) = ex erfc(x), (23) · erfc(x) = 1 erf(x), (24) − Z x 2 t2 erf(x) = e− dt. (25) √π 0

The notation LapGauss(y, σw, σn) is used to represent the pdf in Equation (22).

2.2. Parameter Estimation

After the model is established, parameters k1, k2, σ1 and σ2 need to be estimated according to the wavelet coeﬃcients of the noisy bearing fault signals. At present, there are many mature iterative optimization algorithms to estimate the parameters of the model, such as gradient descent method, Gibbs sampling method and EM algorithm. The expectation-maximization (EM) algorithm has been successfully applied to the optimization of the sparse representation model and parameter estimation of mixture models in many ﬁelds [31]. Usually, the computational complexity of the EM algorithm for the one-dimensional signal is not very high. So, the EM algorithm is adopted in this paper to estimate the four parameters. Appl. Sci. 2019, 9, 3768 6 of 15

EM algorithm needs to set initial values for each parameter. In the proposed method, k1, k2, σ1 and σ2 is needed to be initialized. And then a series of expectation and maximization steps will alternate. The E-step updates the responsibility factors

kiLapGauss(yn, σ1, σn) qi(n) , i = 1, 2. (26) ← k1LapGauss(yn, σ1, σn) + k2LapGauss(yn, σ2, σn)

Next the updated parameters k1, k2, σ1 and σ2 are estimated in the M-step. k1 and k2 are estimated by N 1 X k q (n), i = 1, 2. (27) i ← N i n=1 Then σ2 , i = 1, 2 are estimated as weighted sum of the data values [32]. As to σ2 , the weight y(i) y(i) for yn is the probability that the data point yn is generated by LapGauss(y, σw, σn)

N P 2 qi(n)yn 2 n=1 σ ( ) , i = 1, 2. (28) y i ← PN qi(n) n=1

Finally, as the wavelet coeﬃcients w and n are independent, the standard deviation σ1 and σ2 can be calculated by σ2 = σ2 σ2, i = 1, 2. (29) i y(i) − n 2 2 And it is possible that the estimated variance σ1 and σ2 will be negative because these parameters are all estimated. So, in order to let the value under the radical sign not to be negative and the denominator not to be zero, σ1 and σ2 are estimated as [32]:

s ! σ2 σ = max σ2 σ2, n , i = 1, 2. (30) i y(i) − n 250

2.3. Method Summary The flowchart of the proposed method is shown in Figure1. Given an observed signal g, the corresponding wavelet coefficients w are obtained by wavelet transform firstly. Next, the standard deviation of the independent Gaussian white noise σn is estimated by Equation (11). Then the parameters k1, k2, σ1, σ2 are estimated by Equation (27) to Equation (30), and the new estimated wavelet coefficients wˆ are calculated by the Equation (17). Finally, the new wavelet coefficients wˆ are reconstructed to obtain the estimated signal xˆ, and the fault feature can be obtained through envelope spectrum analysis. Appl. Sci. 2019, 9, 3768 7 of 15 Appl. Sci. 2019, 9, x FOR PEER REVIEW 7 of 15

Original observed signal g

Wavelet transform

Wavelet coefficients y

Estimate noise standard deviation  n using Eq. (11), ie

 n  MAD/0.6745

Estimate parameters ab ,,,  12 using EM algorithm

Calculate w ˆ using Eq. (17), ie 22    22nn wˆ  pk  ysoft y ,  pk  y soft  y ,  12    12   

Reconstruction

Reconstructed signal xˆ

Envelope spectrum analysis

Figure 1. FlowchartFlowchart of of the the proposed proposed method. method. MAD; MAD; median median a absolutebsolute deviation. deviation.

3.3. Simulation Simulation Analysis Analysis InIn this this section, section, a a simulated simulated bearing bearing fault fault signal signal is is analyzed analyzed using using the the proposed proposed method, method, to to prove prove itsits effectiveness.effectiveness. Through Through the the analysis analysis of of the the failure failure mechanism, mechanism, the the bearing bearing fault fault signal signal can can be be constructedconstructed as " ζ # X − 2π f (t τ KT) √ 2 y(t) = Ae 1 ζ  − − 2π f (t τ KT) , (31) − 2f ( t KT ) × − − K 1 2 y( t ) Ae  2 f ( t   KT ) , (31)  where A denotes amplitude, ξ denotesK  damping ratio, f denotes frequency, τ denotes time index, K denotes the number of impulses and T denotes cyclic period [27]. The noisy signal is created by whereadding GaussianA denotes noise amplitude, to the simulated  denotes signal. damping We used ratio, the discrete f denotes Daubechies frequency, 5 wavelet,  denotes which timeis an index,orthogonal K waveletdenotes transform,the number with of threeimpulses decomposition and T denotes levels, cyclic and detail period coe ffi[27].cients The at noisy each level signal were is createdused for by the adding estimation Gaussian of parameters noise to thek1 simulated, k2, σ1 and signal.σ2 utilizing We used the the EM discrete algorithm. Daubechies The corresponding 5 wavelet, whichresults is are an shown orthogonal in Figure wavelet2. transform, with three decomposition levels, and detail coefficients at The waveform of the simulated bearing fault signal in the time domain is shown in Figure2a, each level were used for the estimation of parameters k1 , k2 , 1 and  2 utilizing the EM where amplitude A = 1.5, frequency f = 1500 Hz, damping ratio ξ = 0.080, cyclic period T = 0.007 s algorithm. The corresponding results are shown in Figure 2. and the corresponding theoretical fault feature frequency was f = 135.1 Hz. The length of this signal The waveform of the simulated bearing fault signal in thes time domain is shown in Figure 2a, was 51,200 and the sampling frequency was set at 51.2 kHz. The waveform of the noisy signal in the where amplitude = 1.5, frequency = 1500 Hz, damping ratio = 0.080, cyclic period = time domain is shown in Figure2b. The standard deviation of the additive Gaussian white noise was 0.007 s and the corresponding theoretical fault feature frequency was f = 135.1 Hz. The length of σn = 2.0. s this signalThe signal-to-noise was 51,200 and ratio the (SNR)sampling is introducedfrequency was to describe set at 51.2 the kHz. influence The waveform of noisy components,of the noisy signalwhich in is definedthe time as domain is shown in Figure 2b. The standard deviation of the additive Gaussian x 2 white noise was  n = 2.0. 2 SNR = 10 log10 k k . (32) The signal-to-noise ratio (SNR) is introduced to describen 2 the influence of noisy components, k k2 whichThe is defined SNR of as the simulated signal was 16.082 dB. The envelope spectrum analysis of the noisy − bearing fault signal is shown in Figure2c. It is obvious that2 the fault feature frequency cannot be x obtained directly. The time domain waveform of the estimated2 signal is shown in Figure2d. The SNR SNR 10log10 2 . (32) n 2

The SNR of the simulated signal was −16.082 dB. The envelope spectrum analysis of the noisy bearing fault signal is shown in Figure 2c. It is obvious that the fault feature frequency cannot be obtained directly. The time domain waveform of the estimated signal is shown in Figure 2d. The SNR Appl.Appl. Sci. Sci. 20192019, ,99, ,x 3768 FOR PEER REVIEW 88 of of 15 15 of the estimated signal was −4.526 dB. We can see that the SNR improved significantly. The root mean of the estimated signal was 4.526 dB. We can see that the SNR improved signiﬁcantly. The root mean square error (RMSE) was also− used to evaluate our method. The RMSE is defined as square error (RMSE) was also used to evaluate our method. The RMSE is deﬁned as 1 2 RMSE=r xx ˆ , (33) N1 2 RMSE = x xˆ 2, (33) N k − k2 where x denotes the noise-free simulated signal, xˆ denotes the estimated signal and N denotes thewhere lengthx denotes of the the signal. noise-free The RMSE simulated of the signal, estimatedxˆ denotes signal the is estimated 0.5328. The signal noise and levelN denotes is reduced the successfully.length of the signal. The RMSE of the estimated signal is 0.5328. The noise level is reduced successfully.

(a) Noise-free simulation signal.

(b) Noisy signal.

(d) Estimated signal.

fs 135.1Hz

(e) Envelope spectrum analysis of estimated signal.

FigureFigure 2. 2. ResultsResults of of simulated simulated signal. signal.

TheThe envelope envelope spectrum spectrum analysis analysis of of the the estimated estimated bearing bearing fault fault signal signal is is shown shown in in Figure Figure 2e.2e. f ThroughThrough the the envelope analysis of the estimatedestimated signal,signal, thethe faultfault featurefeature frequencyfrequency was was s =fs135.1 = 135.1 Hz, Hz,which which is equal is equal to theto the simulated simulated value value (135.1 (135.1 Hz). Hz). The The comparisoncomparison ofof FigureFigure2 2c,ec,eshows shows thatthat thethe proposedproposed method method can can effectively eﬀectively extract extract the the fault fault feature feature frequency frequency from from the the noisy noisy signal, signal, even even if if the the transienttransient component component is is drowned drowned by by heavy heavy noise noise.. TheThe verification veriﬁcation of of the the distribution distribution of of transient transient components components and and noisy noisy bearing bearing fault fault signals signals is is shownshown in in F Figureigure 33.. The distribution of transient components data is shown in Figure 3 3a.a. ItIt isis shownshown thatthat the pdf pdf of of noise-free noise-free transient transient component component was was a mixture a mixt ofure two of Laplace two Laplace pdfs, where pdfs, the where estimated the parameters were k1 = 0.416, k2 = 0.584, σ1 = 0.488 and σ2 = 0.049. The distribution of noisy data estimated parameters were k1  0.416 , k2  0.584 , 1  0.488 and  2  0.049 . The distribution of noisy data value is shown in Figure 3b. It can be found that the pdf of noisy value was a mixture of Appl. Sci. 2019, 9, 3768 9 of 15 Appl.Appl. Sci.Sci. 20192019,, 99,, xx FORFOR PEERPEER REVIEWREVIEW 99 ofof 1515 two LapGauss pdfs, where the estimated parameters k  0.250 , k  0.750 ,   0.447 and valuetwo LapGauss is shown in pdfs, Figure where3b. It can the be estimated found that parameters the pdf ofnoisy k11  value0.250 was, k22 a mixture0.750 , of11 two 0.447 LapGauss and pdfs,  0.186 where. the estimated parameters k1 = 0.250, k2 = 0.750, σ1 = 0.447 and σ2 = 0.186. 22  0.186 .

histogramhistogram histogramhistogram 9 9 pphistogram1histogram1 histogramhistogrampp11 99 ppp2121 00..22 pp11pp22 88 ppp(2(x2x)) 00.2.2 pp22pp((yy)) 88 pp((xx)) pp((yy)) 77 77 00..1515 66 00.15.15

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((aa)) HistogramHistogram ofof thethe noisenoise--freefree signal.signal. ((bb)) HistogramHistogram ofof thethe noisynoisy signals.signals. Figure 3. Histogram of related signals. Figure 3. Histogram of related signals.

4.4. ExperimentalExperimental ResultsResults InIn this this section, section, thethe cylindricalcylindrical rollerroller bearingbearing installedinstalled installed onon on thethe the testtest test rigrig rig isis is takentaken taken asas as thethe the testtest test object.object. object. TheThe test test rig rig is is shownshown in in FigureFigure 4 4.4.. ThreeThreeThree NJ208NJ208NJ208 (TMB)(TMB)(TMB) typetypetype cylindricalcylindricalcylindrical rollerrollerroller elementelementelement bearingbearingbearing werewerewere usedused for for testing. testing. TheThe The vibration vibration vibration sensor sensor sensor model model model was was was a a a PCBPCB PCBM621B40M621M621B40B40 and and the the sensor sensor had had a a reference reference sensitivitysensitivity of of 10 10 m mvmvv/g//gg and and t thethehe frequency frequency range range o ofoff measurable measurable signals signals was was 1.6 1.6 to to 30,000 30,000 Hz. Hz. The The data data acquisitionacquisition instrumentinstrument modelmodel waswas anan INV3018C.INV3018C. TheThe testtest waswas carriedcarried outout under under the the conditions conditions of of settingsetting fault: fault: aa a throughthrough crackcrack faultfault withwith widthwidth andand depthdepth ofof of 0.20.2 mmmm waswas setset onon thethe outerouter race race,race,, inner inner racerace and and ballball ofofof the. the.the. rollingrollingrolling bearing bearingbearing respectively respectivelyrespectively by byby using usingusing the thethe wire-cut wirewire--cutcut method. methodmethod The.. TheThe load loadload applied appliedapplied to the toto thetestingthe testingtesting bearings bearingsbearings was waswas about aboutabout 1 kN. 11 ThekN.kN. samplingTheThe samplingsampling frequency frequencyfrequency was 51.2 waswas kHz 51.251.2 and kHzkHz the andand Shaft thethe rotatingShaftShaft rotatingrotating speed speedwasspeed 1496 waswas r /14961496min. r/min.r/min. The theoretical TheThe theoreticaltheoretical fault feature faultfault featurefeature frequencies frequenciesfrequencies of the ofof outer thethe outerouter race, innerrace,race, innerinner race and racerace ball andand were ballball were142.8were 142.8142.8 Hz, 206.2 Hz,Hz, 206.2206.2 Hz and HzHz 132.6 andand 132.6132.6 Hz respectively. HzHz respectively.respectively. And AndAnd the corresponding thethe correspondingcorresponding periods periodsperiods were werewere 7.00 7.007.00 ms, ms,ms, 4.85 4.854.85 ms, ms,andms, andand 7.54 7.547.54 ms, ms, respectively.ms, respectively.respectively. As in AsAs the inin simulation, thethe simulation,simulation, we used wewe usedused the discrete thethe discretediscrete Daubechies DaubechiesDaubechies 5 wavelet 55 waveletwavelet with threewithwith threedecompositionthree decomposition decomposition levels, andlevels,levels, detail and and coe detail detailﬃcients coefficientscoefficients at each level atat each eachwere levelused level for were were the usedused estimation for for the the of estimationparameters estimation ofk of1 , parametersk2, σ1 and σ 2 kutilizing, k ,  the EMand algorithm. utilizing The the results EM algorithm. for the outer The race, results inner for race the and outer ball race, are shown inner parameters k11 , k22 , 11 and 22 utilizing the EM algorithm. The results for the outer race, inner raceinrace Figures andand ballball5–7 are arerespectively. shownshown inin FiguresFigures 55––77 respectively.respectively.

MeasuredMeasured BearingBearing CCushioningushioning CouplingCoupling DynamometerDynamometer DDeviceevice

SensorSensor

MotorMotor

AdjustingAdjusting Loading Device Loading Device DDeviceevice

FigureFigure 4. 4. TestTest rig rig for for bearing bearing signals. signals.

4.1.4.1. OuterOuter RaceRace InIn this this section,section, the the bearing bearing outer outer race race faultfault signalsignal measuredmeasured from from the the test test rig rig is is analyzed analyzed and and thethe correspondingcorresponding results results are are shown shown in in F FigureFigigureure5 5.5.. Appl. Sci. 2019, 9, 3768 10 of 15 Appl. Sci. 2019, 9, x FOR PEER REVIEW 10 of 15

TheThe time time domain domain waveforms waveforms of the noisynoisy outerouter racerace fault fault signal signal measured measured are are shown shown in in Figure Figure5a. 5a.The The length length of thisof this signal signal was was 76,800. 76,800. The The envelope envelope spectrum spectrum analysis analysis of the of noisy the noisy bearing bearing fault signalfault signalis shown is shown in Figure in Figure5b. Also, 5b. theAlso, fault the featurefault feature frequency frequency cannot cannot be extracted be extracted directly. directly. The results The results of the ofestimated the estimated transient transient components components in the time in the domain time domain is shown is in shown Figure in5c. Figure The envelope 5c. The spectrum envelope spectrumanalysis ofanalysis the estimated of the estimated transient componentstransient components is shown in is Figureshown5 d.in InFigure Figure 5d.5d, In it Figure can be 5d, seen it thatcan be seen that the fault feature frequency of the outer race was f  142.8 Hz, which is equal to the the fault feature frequency of the outer race was f1 = 142.8 Hz, which1 is equal to the theoretical value theor(142.8etical Hz). value (142.8 Hz).

(a) Original noisy signal.

(b) Envelope spectrum analysis of noisy bearing signal.

f1 142.8Hz

(d) Envelope spectrum analysis of estimated signal.

FigureFigure 5. 5. ResultsResults of of experimental experimental outer outer race race fault fault signal. signal.

4.2.4.2. Inner Inner Race Race InIn this this section, section, the the bearing bearing inner inner race race fault fault signal signal measured measured from from the the test test rig rig is is analyzed analyzed and and the the correspondingcorresponding results results are are shown shown in in F Figureigure 66.. TheThe time time domain domain waveforms waveforms of the noisynoisy innerinner raceracefault fault signal signal measured measured are are shown shown in in Figure Figure6a. 6a.The The length length of thisof this signal signal was was 76,800. 76,800. The The envelope envelope spectrum spectrum analysis analysis of the of noisy the noisy bearing bearing fault signalfault signalis shown is shown in Figure in 6Figureb. The 6b. fault The feature fault frequency feature fr alsoequency could also not becould obtained not be directly. obtained The directly. result of The the resultestimated of the transient estimated components transient components in the time in domain the time is domain shown in is Figureshown6 inc. Figure The envelope 6c. The spectrumenvelope spectrumanalysis ofanalysis the estimated of the estimated transient transient components components is shown is in shown Figure in6d. Figure From 6d. Figure From6d, Figure it could 6d, be it obtained that the fault feature frequency of the inner race was f2 = 206 Hz, which is nearly equal to could be obtained that the fault feature frequency of the inner race was f2 = 206 Hz, which is nearly the theoretical value (206.2 Hz). Since there are always unavoidable errors in the measurement and equal to the theoretical value (206.2 Hz). Since there are always unavoidable errors in the calculation process, 206 Hz should be the correct required fault feature frequency. measurement and calculation process, 206 Hz should be the correct required fault feature frequency. 4.3. Ball 4.3. Ball In this section, the bearing ball fault signal measured from the test rig is analyzed and the In this section, the bearing ball fault signal measured from the test rig is analyzed and the corresponding results are shown in Figure7. corresponding results are shown in Figure 7. The time domain waveforms of the noisy ball fault signal measured are shown in Figure 7a. The length of this signal was 76,800. The envelope spectrum analysis of the noisy bearing fault signal is Appl. Sci. 2019, 9, 3768 11 of 15

The time domain waveforms of the noisy ball fault signal measured are shown in Figure7a. The lengthAppl. Sci. of2019 this,, 9,, xx signal FORFOR PEERPEER was REVIEWREVIEW 76,800. The envelope spectrum analysis of the noisy bearing11 fault of 15 signal is shownshown in in Figure Figure7 b.7b. The The faultfault featurefeature frequency frequency also also cannot cannot be beobtained obtained directly. directly. The Theresult result of the of the estimatedestimated transient transient components components in in thethe timetime domain is is shown shown in inFigure Figure 7c.7 Thec. The envelope envelope spectrum spectrum analysisanalysis of theof the estimated estimated transient transient components components is is shown shown inin Figure Figure 7d.7 d. From From Figure Figure 7d, 7 itd, can it can be be = obtainedobtained that that the the fault fault feature feature frequencyfrequency of of the the ball ball was was ff33 f132.73 132.7 Hz, whichHz, whichis nearly is equal nearly to the equal to the theoreticaltheoreticaltheoretical value value value (132.6 (132.6 (132.6 Hz). Hz). Hz). Since Since Since there there there are are are alway alway alwayss unavoidable unavoidable errors errors in the in measurement the measurement and and calculationcalculation process, process, 132.7 132.7 Hz Hz should should be be thethe correctcorrect required required fault fault feature feature frequency. frequency.

((a)) Original noisy signal.

((b)) EnvelopeEnvelope spectrumspectrum analysisanalysis ofof noisy bearing signal.

((c)) EstimatedEstimated signal.signal.

ff22  206Hz

((d)) EnvelopeEnvelope spectrumspectrum analysisanalysis ofof estimatedestimated signal.signal.

FigureFigure 6. 6.Results Results ofof experimental inner inner race race fault fault signal. signal.

((a)) Original noisy signal.

((b)) EnvelopeEnvelope spectrumspectrum analysisanalysis ofof noisynoisy bearingbearing signal.signal.

((c)) EstimatedEstimated signal.signal.

Figure 7. Cont. Appl. Sci. 2019, 9, 3768 12 of 15 Appl. Sci. 2019, 9, x FOR PEER REVIEW 12 of 15

f3 132.7Hz

(d) Envelope spectrum analysis of estimated signal.

FigureFigure 7. 7.Results Results of experimental experimental ball ball fault fault signal. signal.

Therefore,Therefore, the the proposed proposed method method can can effectivelyeﬀectively extract extract the the fault fault feature feature frequencies frequencies of three of three diﬀerentdifferent types types of faultsof faults from from the the corresponding corresponding noisy experimental experimental signal signal.. The The aforementioned aforementioned simulatedsimulated and and engineer engineer tests tests were were all implemented all implemented in MATLAB2016b in MATLAB2016b on a on computer a computer whose whose operating operating system was Windows 10. The CPU parameter of the computer was Intel(R) Core(TM) i7- system was Windows 10. The CPU parameter of the computer was Intel(R) Core(TM) i7-4790 at 4790 at 3.60 GHz and the RAM was 12.0 GB, and the execution time of both simulated and 3.60 GHz and the RAM was 12.0 GB, and the execution time of both simulated and engineering signals engineering signals were less than 1 s. It can be seen that the time consumption of the proposed weremethod less than is very 1 s. low It and can stable. be seen that the time consumption of the proposed method is very low and stable. 4.4. Computational Complexity Comparison 4.4. Computational Complexity Comparison Comparisons of the computational complexity between the proposed method and some other populaComparisonsr fault feature of the extraction computational methods complexityare shown in between Table 1. the proposed method and some other popular fault feature extraction methods are shown in Table1. Table 1. Computational complexity comparison.

Table 1. Computational complexityTime comparison.- The Spectral Empirical Mode Sparse Methods Frequency Proposed SpectralKurtosis EmpiricalDecomposition Mode Time-Frequency RepresentationSparse The Proposed Methods Analysis Method Kurtosis Decomposition Analysis Representation Method Computational Computational ONN( log ) O( rN log N ) ON() complexity O(N log N) O(N log N) O(N log N) O(rN log N) O(N) complexity Usually, the computational complexity of the spectral kurtosis method and empirical mode decompositionUsually, the computational method are ONN( complexity log ) . The of frequently the spectral used kurtosis time-frequency method analysis and empirical methods mode decompositioninclude the method short-time are FourierO(N log transform,N). The continuous frequently wavelet used time-frequency transform and Hilbert analysis-Huang methods includetransform. the short-time The computational Fourier transform, complexity continuous of these wavelet methods transform is also and Hilbert-Huang. The sparse transform. The computationalrepresentation method complexity requires of these iterative methods optimization is also O to( N solvelog N many). The sub sparse-problems representation of the whole method requiresmodel. iterative At present, optimization many mature to iterative solve many optimization sub-problems algorithms of thehave whole been applied model. to At solve present, sparse many maturerepresentation iterative optimization models, such as algorithms the alternating have direction been applied method toof multipliers solve sparse (ADMM) representation [33], the split models, suchaugmented as the alternating Lagrangian direction shrinkage method algorithm of multipliers (SALSA) [34], (ADMM) majorization [33],- theminimization split augmented (MM) [35] Lagrangian and the forward-backward splitting (FBS) algorithm [36]. However, these iterative optimizations of shrinkage algorithm (SALSA) [34], majorization-minimization (MM) [35] and the forward-backward sparse representation always involve a matrix-vector. This operation has O( rN log N ) cost and r splitting (FBS) algorithm [36]. However, these iterative optimizations of sparse representation always is the redundancy factor [10]. Furthermore, calculating the inverse matrix may be involved involve a matrix-vector. This operation has O(rN log N) cost and r is the redundancy factor [10]. sometimes. So, the computational complexity of the sparse representation method, if the construction Furthermore,of the sparse calculating dictionary is the not inverse taken into matrix account, may is at be least involved sometimes., which obviously So, the computational limits the complexityexecution of efficiency the sparse of representation the algorithm. method, if the construction of the sparse dictionary is not taken into account,The EM is at algorithm least O( rNusedlog inN the), whichproposed obviously method limitshas computational the execution complexity efficiency ON of() the when algorithm. estimatingThe EM algorithm model parameters used in iteratively. the proposed Besides, method the computational has computational complexity complexity of the discreteO(N) when estimatingwavelet model transform parameters is also iteratively.. So, the Besides, computational the computational complexity of complexity the whole proposed of the discrete method wavelet transformis is alsoand Othe(N fault). So, feature the computational extraction results complexity can be obtained of the quickly. whole proposed method is O(N) and the fault feature extraction results can be obtained quickly. 5. Discussion 5. Discussion The proposed method is expected to be applied to fault diagnosis of the rolling bearings used in modernThe proposed mechanical method equipment. is expected In this paper, to be applieda simulated to signal fault diagnosisand three experimental of the rolling signals bearings with used in moderndifferent mechanical fault types are equipment. analyzed, Inand this the paper,envelope a simulated spectrum of signal the estimated and three signal experimental is compared signals with different fault types are analyzed, and the envelope spectrum of the estimated signal is compared with the envelope spectrum of the original noisy signal to determine whether the fault features can be effectively extracted. The results indicate that the proposed method can extract fault features accurately, although there is still some noise in the estimated signal. Once the fault feature is obtained, Appl. Sci. 2019, 9, 3768 13 of 15 we can determine the type of failure in practical use. The signal data used in the experiments were collected from cylindrical roller bearings mounted on the test rig, which could effectively simulate the bearings working in mechanical equipment. Therefore, using these data for analysis is reliable and can be extended to practical bearing fault diagnosis after some improvements and testing. Furthermore, the proposed method has a great advantage, that is, the time required to extract the fault feature is very short (about 0.8 s), which is very important for quickly diagnosing bearing faults in practical applications. However, the proposed method has some inaccuracy in estimating the standard deviations of the mixture Laplace model, which may affect the noise level of the estimated bearing fault signal in the time domain. So, future research will focus on the application of the proposed method to actual mechanical equipment and finding a way to more accurately estimate the parameters of the mixture Laplace model while preserving the extraction time.

6. Conclusions In this paper, an efficient method to quickly reduce noise level and extract fault feature frequencies of noisy bearing fault signals was proposed. This proposed method is based on the MAP estimator and pdfs of bearing fault source signal and noise, thereby having a reliable probabilistic interpretation. The proposed method first performed discrete wavelet transform on the noisy signal to obtain wavelet coefficients. Based on the wavelet coefficients obtained, the EM algorithm was adopted for parameter estimation of the derived model, and the newly estimated wavelet coefficients were then calculated using this model. Finally, the estimated transient components could be obtained after reconstruction. The computational complexity of this method was lower than some popular fault feature extraction methods and the fault feature extraction results could be obtained quickly. To validate the efficiency and accuracy of the proposed method, we analyzed a simulated bearing fault signal and three experimental signals with different types of faults using this method. The results indicate that this method can effectively and efficiently achieve the goal of noise level reduction and fault feature extraction in rolling element bearings.

Author Contributions: Conceptualization, S.L. and W.H.; methodology, S.L. and W.H.; validation, J.S. and X.J.; formal analysis, S.L. and W.H.; investigation, S.L. and W.H.; resources, W.H. and Z.Z.; writing—original draft preparation, S.L. and W.H.; writing—review and editing, J.S. and X.J.; funding acquisition, J.S., X.J. and Z.Z. Funding: This research was funded by the National Natural Science Foundation of China, grant number 51405320, 51875376 and 51605319, China Postdoctoral Science Foundation, grant number 2017M611896 and 2017M621811 and Natural Science Foundation of Jiangsu Province, grant number BK20160318. Conﬂicts of Interest: The authors declare no conﬂict of interest.

References

1. Wang, J.; Peng, Y.; Qiao, W. Current-Aided Order Tracking of Vibration Signals for Bearing Fault Diagnosis of Direct-Drive Wind Turbines. IEEE Trans. Ind. Electron. 2016, 63, 6336–6346. [CrossRef] 2. Qin, Y. A New Family of Model-Based Impulsive Wavelets and Their Sparse Representation for Rolling Bearing Fault Diagnosis. IEEE Trans. Ind. Electron. 2018, 65, 2716–2726. [CrossRef] 3. Liu, Y.; He, B.; Liu, F.; Lu, S.; Zhao, Y. Feature Fusion Using Kernel Joint Approximate Diagonalization of Eigen-Matrices for Rolling Bearing Fault Identiﬁcation. J. Sound Vib. 2016, 385, 389–401. [CrossRef] 4. Wang, S.; Chen, X.; Cai, G.; Chen, B.; Li, X.; He, Z. Matching Demodulation Transform and Synchrosqueezing in Time-Frequency Analysis. IEEE Trans. Signal Process. 2014, 62, 69–84. [CrossRef] 5. Tian, J.; Morillo, C.; Azarian, M.H.; Pecht, M. Motor Bearing Fault Detection Using Spectral Kurtosis-Based Feature Extraction Coupled with K-Nearest Neighbor Distance Analysis. IEEE Trans. Ind. Electron. 2016, 63, 1793–1803. [CrossRef] 6. Komaty, A.; Boudraa, A.-O.; Augier, B.; Dare-Emzivat, D. Emd-Based Filtering Using Similarity Measure between Probability Density Functions of Imfs. IEEE Trans. Instrum. Meas. 2014, 63, 27–34. [CrossRef] 7. Yan, R.; Gao, R.X.; Chen, X. Wavelets for Fault Diagnosis of Rotary Machines: A Review with Applications. Signal Process. 2014, 96, 1–15. [CrossRef] Appl. Sci. 2019, 9, 3768 14 of 15

8. Lu, S.; He, Q.; Kong, F. Stochastic Resonance with Woods-Saxon Potential for Rolling Element Bearing Fault Diagnosis. Mech. Syst. Signal Process. 2014, 45, 488–503. [CrossRef] 9. Zhao, M.; Kang, M.; Tang, B.; Pecht, M. Deep Residual Networks with Dynamically Weighted Wavelet Coefficients for Fault Diagnosis of Planetary Gearboxes. IEEE Trans. Ind. Electron. 2018, 65, 4290–4300. [CrossRef] 10. Zhang, H.; Chen, X.; Du, Z.; Yan, R. Kurtosis Based Weighted Sparse Model with Convex Optimization Technique for Bearing Fault Diagnosis. Mech. Syst. Signal Process. 2016, 80, 349–376. [CrossRef] 11. Gu, F.; Shao, Y.; Hu, N.; Naid, A.; Ball, A.D. Electrical motor current signal analysis using a modified bispectrum for fault diagnosis of downstream mechanical equipment. Mech. Syst. Signal Process. 2011, 25, 360–372. [CrossRef] 12. Wang,L.; Liu, Z.; Miao, Q.; Zhang, X. Time-Frequency Analysis Based on Ensemble Local Mean Decomposition and Fast Kurtogram for Rotating Machinery Fault Diagnosis. Mech. Syst. Signal Process. 2018, 103, 60–75. [CrossRef] 13. Chen, D.; Lin, J.; Li, Y. Modified Complementary Ensemble Empirical Mode Decomposition and Intrinsic Mode Functions Evaluation Index for High-Speed Train Gearbox Fault Diagnosis. J. Sound Vib. 2018, 424, 192–207. [CrossRef] 14. Liu, H.; Huang, W.; Wang, S.; Zhu, Z. Adaptive Spectral Kurtosis Filtering Based on Morlet Wavelet and Its Application for Signal Transients Detection. Signal Process. 2014, 96, 118–124. [CrossRef] 15. Zhang, Z.; Li, S.; Wang, J.; Xin, Y.; An, Z. General Normalized Sparse Filtering: A Novel Unsupervised Learning Method for Rotating Machinery Fault Diagnosis. Mech. Syst. Signal Process. 2019, 124, 596–612. [CrossRef] 16. Elad, M.; Matalon, B.; Zibulevsky, M. Coordinate and Subspace Optimization Methods for Linear Least Squares with Non-Quadratic Regularization. Appl. Comput. Harmon. Anal. 2007, 23, 346–367. [CrossRef] 17. Sun, C.; Wang, P.; Yan, R.; Gao, R.X.; Chen, X. Machine Health Monitoring Based on Locally Linear Embedding with Kernel Sparse Representation for Neighborhood Optimization. Mech. Syst. Signal Process. 2019, 114, 25–34. [CrossRef] 18. Cui, L.; Wang, J.; Lee, S. Matching Pursuit of an Adaptive Impulse Dictionary for Bearing Fault Diagnosis. J. Sound Vib. 2014, 333, 2840–2862. [CrossRef] 19. Tian, X.; Gu, J.X.; Rehab, I.; Abdalla, G.M.; Gu, F.; Ball, A.D. A robust detector for rolling element bearing condition monitoring based on the modulation signal bispectrum and its performance evaluation against the Kurtogram. Mech. Syst. Signal Process. 2018, 100, 167–187. [CrossRef] 20. Bahadorinejad, A.; Imani, M.; Braga-Neto, U. Adaptive Particle Filtering for Fault Detection in Partially-Observed Boolean Dynamical Systems. IEEE/ACM Trans. Comput. Biol. 2018.[CrossRef] 21. Johansen, A.M.; Doucet, A.; Davy, M. Particle methods for maximum likelihood estimation in latent variable models. Stat. Comput. 2008, 18, 47–57. [CrossRef] 22. Imani, M.; Ghoreishi, S.F.; Allaire, D.; Braga-Neto, U.M. MFBO-SSM: Multi-fidelity Bayesian optimization for fast inference in state-space models. In Proceedings of the AAAI Conference on Artificial Intelligence, Honolulu, HI, USA, 27 January–1 February 2019; pp. 7858–7865. 23. Zou, X.; Jancovic, P.; Liu, J.; Kokuer, M. Speech Signal Enhancement Based on Map Algorithm in the Ica Space. IEEE Trans. Signal Process. 2008, 56, 1812–1820. [CrossRef] 24. Elad, M.; Milanfar, P.; Rubinstein, R. Analysis versus synthesis in signal priors. Inverse Probl. 2007, 23, 947–968. [CrossRef] 25. Imani, M.; Ghoreishi, S.F.; Braga-Neto, U.M. Bayesian control of large MDPs with unknown dynamics in data-poor environments. In Proceedings of the Advances in Neural Information Processing Systems, Montreal, QC, Canada, 2–8 December 2018; pp. 8146–8156. 26. Daubechies, I. The Wavelet Transform, Time-Frequency Localization and Signal Analysis. IEEE Trans. Inf. Theory 1990, 36, 961–1005. [CrossRef] 27. Wang, L.; Cai, G.; Wang, J.; Jiang, X.; Zhu, Z. Dual-Enhanced Sparse Decomposition for Wind Turbine Gearbox Fault Diagnosis. IEEE Trans. Instrum. Meas. 2019, 68, 450–461. [CrossRef] 28. Wang, L.; Cai, G.; You, W.; Huang, W.; Zhu, Z. Transients Extraction Based on Averaged Random Orthogonal Matching Pursuit Algorithm for Machinery Fault Diagnosis. IEEE Trans. Instrum. Meas. 2017, 66, 3237–3248. [CrossRef] Appl. Sci. 2019, 9, 3768 15 of 15

29. Donoho, D.L.; Johnstone, I.M. Ideal Spatial Adaptation by Wavelet Shrinkage. Biometrika 1994, 81, 425–455. [CrossRef] 30. Hansen, M.; Yu, B. Wavelet Thresholding Via Mdl for Natural Images. IEEE Trans. Inf. Theory 2000, 46, 1778–1788. [CrossRef] 31. Jiang, B.; Xiang, J.; Wang, Y. Rolling Bearing Fault Diagnosis Approach Using Probabilistic Principal Component Analysis Denoising and Cyclic Bispectrum. J. Vib. Control 2016, 22, 2420–2433. [CrossRef] 32. Rabbani, H.; Nezafat, R.; Gazor, S. Wavelet-Domain Medical Image Denoising Using Bivariate Laplacian Mixture Model. IEEE Trans. Biomed. Eng. 2009, 56, 2826–2837. [CrossRef] 33. Yi, C.; Lv, Y.; Dang, Z. A Fault Diagnosis Scheme for Rolling Bearing Based on Particle Swarm Optimization in Variational Mode Decomposition. Shock Vib. 2016.[CrossRef] 34. Ding, Y.; He, W.; Chen, B.; Zi, Y.; Selesnick, I.W. Detection of Faults in Rotating Machinery Using Periodic Time-Frequency Sparsity. J. Sound Vib. 2016, 382, 357–378. [CrossRef] 35. He, W.; Ding, Y.; Zi, Y.; Selesnick, I.W. Sparsity-Based Algorithm for Detecting Faults in Rotating Machines. Mech. Syst. Signal Process. 2016, 72–73, 46–64. [CrossRef] 36. Cai, G.; Selesnick, I.W.; Wang, S.; Dai, W.; Zhu, Z. Sparsity-Enhanced Signal Decomposition Via Generalized Minimax-Concave Penalty for Gearbox Fault Diagnosis. J. Sound Vib. 2018, 432, 213–234. [CrossRef]

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