# A Fault Feature Extraction Method for Rolling Bearing Based on Intrinsic Time-Scale Decomposition and AR Minimum Entropy Deconvolution

Authors: Jiakai Ding; Liangpei Huang; Dongming Xiao; Lingli Jiang
Journal: Shock and Vibration
 (2021)

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## Abstract

It is very difficult to extract the feature frequency of the vibration signal of the rolling bearing early weak fault and in order to extract its feature frequency quickly and accurately. A method of extracting early weak fault vibration signal feature frequency of the rolling bearing by intrinsic time-scale decomposition (ITD) and autoregression (AR) minimum entropy deconvolution (MED) is proposed in this paper. Firstly, the original early weak fault vibration signal of the rolling bearing is decomposed by the ITD algorithm to proper rotations (PRs) with fault feature frequency. Then, the sample entropy value of each PR is calculated to find the largest PRs of the sample entropy. Finally, the AR-MED filtering algorithm is utilized to filter and reduce the noise of the largest PRs of the sample entropy value, and the early weak fault vibration signal feature frequency of the rolling bearing is accurately extracted. The results show that the ITD-AR-MED method can extract the early weak fault vibration signal feature frequency of the rolling bearing accurately.


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## Body

## 1. Introduction

Rolling bearings play a pivotal role in modern industry and their applications are very extensive, such as aircraft engines [1, 2], gas turbines [3], landing gear [4], flexible mechanism [5], and wind turbines [6–11]. However, the rolling bearing is very easy to cause damage in a strong vibration environment, and the rolling bearing is very prone to failure in most rotating machinery systems. The failure of the rolling bearing will even cause a catastrophic accident to the rotating machinery system, which will indirectly cause huge economic losses. Therefore, it is significant to accurately extract the early weak fault vibration signal feature frequency of the rolling bearing [12].At present, vibration signal [13–15] is mostly used for rolling bearing fault diagnosis analysis. Because the rolling bearing usually works in complex environmental noise, and the collected fault vibration signal feature frequency of the rolling bearing is usually weak [16, 17]. Many researchers use the signal decomposition algorithms such as empirical mode decomposition (EMD) [18, 19], wavelet packet decomposition (WPT) [20–22], and local mode decomposition (LMD) [23–25] to reduce the noise of the original signal. However, the wavelet basis function (WBF) is fixed before the WPT decomposition and the decomposition effect according to the selection of the WBF. Therefore, WPT has obvious defects in vibration signal decomposition. The EMD and LMD are prone to adverse factors such as endpoint effects and mode mixing [26–29]. Therefore, Frei and Osorio [30] proposed an intrinsic time-scale decomposition (ITD) adaptive signal decomposition algorithm. The ITD algorithm can effectively solve the unfavorable factors between EMD and LMD. Xing et al. [31] developed the ITD method combined with singular value decomposition (SVD) and support vector machine (SVM) method. The original vibration signal is decomposed by the ITD algorithm, and the proper rotations (PRs) with the greatest correlation was selected for subsequent analysis. An [32] utilized the ITD algorithm in fault diagnosis under four fault states collected. The first few PRs were selected for spectral analysis to find out their feature frequency. Feng et al. [33] utilized the ITD algorithm in fault diagnosis and feature extraction of the planetary gearbox.The fault vibration signal of the rolling bearing usually presents the characteristics of the impact waveform [34] when the rolling bearing fails. Wiggins [35] proposed a minimum entropy deconvolution (MED) filtering method to enhance the impact component in the fault vibration signal. This method can enhance the impact component and the fault feature in the signal. Sawalhi et al. [36] and Endo and Randall [37] utilized the MED to enhance the impact component of the rolling bearing and gear vibration signal, respectively, and the experiment results proved that this method could effectively enhance the feature frequency. Jiang et al. [38] proposed a condition monitoring method of the rolling bearing based on the MED and envelope spectrum. After the rolling bearing fault vibration signal was filtered by the MED, its fault feature frequency was enhanced, and then an envelope spectrum analysis was performed on it; thus, the rolling bearing fault feature frequency was separated accurately. Li et al. [39] developed an autoregression (AR) MED and variational mode decomposition (VMD) fault diagnosis method. Among them, the AR-MED method can eliminate the influence of background noise effectively, which can enhance the feature frequency in the fault signal.In summary, this paper studied the problem of extracting early weak fault vibration signal feature frequency of the rolling bearing under complex background noise. This paper proposes a method for extracting early weak fault vibration signal feature frequency of the rolling bearing based on ITD and AR-MED.


## 2. ITD Signal Decomposition Method

Suppose that the original signalXt to be decomposed is a real value discrete signal, ξ is defined as the baseline extraction factor of Xt, Ht=Xt−ξXt is defined as the proper rotations (PRs), where ξXt is the mean curve of the signal, denoted as Lt=ξXt; then, the primary decomposition of the original signal can be expressed as(1)Xt=Lt+Ht.Then, the steps of the ITD algorithm are as follows:(1)
Extract the extreme pointXk of Xt and the corresponding time point τk. Among them, k=1,2,…,N and definition τ0=0, so definition Lk+1 is(2)Lk+1=αXk+τk+1−τkτk+2−τkXk+2−Xk+1−αXk+1,whereα∈0,1, α=0.5, and k=1,2,…,N−2.(2)
The piecewise linear baseline extraction factor of signalXt is defined as(3)Lt=ξXt=Lk+Lk+1−LkXk+1−XkXt−Xk.(3)
Repeat steps (1)-(2) with baseline signalLt as the original signal; then, the original signal Xt is decomposed into(4)Xt=ξXt+HXt=H∑k=0p−1ξk+ξpXt,where HξkXt is the k+1 component of proper rotations (PRs) and ξpXt is the monotone trend signal. According to the above equation, the first PR component Ht1=Xt−Lt1 can be obtained. Then, Lt1 is decomposed until the new signal is generated, Ltp is monotone function or constant function, and then the decomposition is stopped. The time domain and frequency domain waveform of fault signals of the rolling bearing is shown in Figure 1.Figure 1
Time domain and frequency domain waveform of rolling bearing vibration signal: (a) inner race (IR) fault of rolling bearing vibration signal and (b) outer race (OR) fault of rolling bearing vibration signal.
(a)(b)It easy can be seen from the above ITD decomposition process that the baseline signalLt is obtained through the linear transformation of the signal, so the calculation process of ITD can be greatly simplified and the decomposition speed greatly improved.


## 3. AR-MED Filter

### 3.1. AR Filter

AR filtering is a parametric modeling method based on the rational transfer function. Because the system parameters of the AR filter are very sensitive to the state change of the system, it is of great value to apply it to signal analysis.When the data sequence slides along the signal sequence, the adaptive AR spectrum of the signal is formed. The parameter estimation of signal sequence AR model is to select appropriate parameters so that the residual of the AR modelεk is white noise. The commonly used methods are time series theory method and optimization theory estimation method. If there is a time series xn, there is a positive integer p, which makes(5)Γpxn=ap,n>p,where Γ is the Wold decomposition operator; p is analysis order; ap is AR model parameter.The AR model of signal sequencexn can be expressed as(6)Yk=∑i=1paixi+k+εk,where εk is a white noise process, and its variance represents the estimation error of the AR filter.The process of linear predictive filtering using the AR filter is shown in Figure2. After the original signal passes through p-order AR linear filter for predictive filtering, a new signal Yn is obtained. Due to the convolution operation in the filtering process, Yn is also intercepted to obtain a signal Yn−p.Figure 2
AR filtering process diagram.


### 3.2. MED Method

The purpose of MED is to find the source of impact in the signal. The principle is to select an appropriate filter coefficient to construct an inverse filter so that the output signal is restored to the input after the inverse filter. The minimum entropy of the signal is used as the determination condition, which can significantly improve the signal-to-noise ratio (SNR) of pulse impact signals, which is very suitable for bearing crack fault diagnosis. The fault vibration signal generated by the bearing in the event of failure usually has impact characteristics. The MED can be applied to the noise reduction treatment of impact fault of the rolling bearing to better highlight the impact pulse. It is assumed that when impact failure occurs to the rolling bearing, the collected discrete signals can be expressed as follows:(7)yt=gt+nt∗ht,where gt is the signal of rolling bearing fault impact source after AR filtering; nt is noise interference; ∗ is the convolution operator; and ht is the influence of the transfer path.The purpose of deconvolution is to find the filter coefficientf of the inverse filter so that the output signal yt is restored to the input signal gt; that is,(8)et=fn∗yt=∑l=1Lflyn−l,where et is the output signal of the inverse filter and the input signal gt can be recovered. L is the filter length.In this paper, the objective function method is selected to realize the MED, and the optimal inverse filter coefficient is found so that the objective function is maximized. The objective function can be defined as(9)O4fl=∑n=1Net2∑n=1Net22.Take the derivative of the objective function and set it equal to 0 to obtain(10)∂O4fl∂fl=0.Since∂et/∂fl=yn−l, we can get(11)∑n=1Net2∑n=1Net4∑n=1Net3yn−l=∑p=1Lfp∑n=1Nyn−lyn−p.Equation (11) can be written as matrix(12)f=A−1b,where b is the cross-correlation matrix of input signal gt and output signal et of the inverse filter; A is Toeplitz autocorrelation matrix of inverse filter input signal gt; and f is the filtering coefficient of the inverse filter. The solution process of MED is as follows:(1)
Toeplitz autocorrelation matrixA is calculated, and the filtering coefficient f0 of the inverse filter is initialized, which is usually set as the delayed pulse.(2)
According to the initial filter coefficientf0 and the inverse filter input signal g0, the output signal e0 is calculated.(3)
Vectorb1 is calculated based on the output signal e0 and the input signal g0.(4)
According to equation (12), f1=A−1b0.(5)
Calculate the iteration error:(13)δ=f1−μf0μf0,μ=Ef02Ef121/2.When the expected value of the iteration error is greater than the set error threshold, that is,Eδ>τ, f0=f1, recalculate step (2), repeat the calculation until Eδ≤τ or Eδ diverges, stop the calculation, and get the MED FIR filter parameters.


## 3.1. AR Filter

AR filtering is a parametric modeling method based on the rational transfer function. Because the system parameters of the AR filter are very sensitive to the state change of the system, it is of great value to apply it to signal analysis.When the data sequence slides along the signal sequence, the adaptive AR spectrum of the signal is formed. The parameter estimation of signal sequence AR model is to select appropriate parameters so that the residual of the AR modelεk is white noise. The commonly used methods are time series theory method and optimization theory estimation method. If there is a time series xn, there is a positive integer p, which makes(5)Γpxn=ap,n>p,where Γ is the Wold decomposition operator; p is analysis order; ap is AR model parameter.The AR model of signal sequencexn can be expressed as(6)Yk=∑i=1paixi+k+εk,where εk is a white noise process, and its variance represents the estimation error of the AR filter.The process of linear predictive filtering using the AR filter is shown in Figure2. After the original signal passes through p-order AR linear filter for predictive filtering, a new signal Yn is obtained. Due to the convolution operation in the filtering process, Yn is also intercepted to obtain a signal Yn−p.Figure 2
AR filtering process diagram.


## 3.2. MED Method

The purpose of MED is to find the source of impact in the signal. The principle is to select an appropriate filter coefficient to construct an inverse filter so that the output signal is restored to the input after the inverse filter. The minimum entropy of the signal is used as the determination condition, which can significantly improve the signal-to-noise ratio (SNR) of pulse impact signals, which is very suitable for bearing crack fault diagnosis. The fault vibration signal generated by the bearing in the event of failure usually has impact characteristics. The MED can be applied to the noise reduction treatment of impact fault of the rolling bearing to better highlight the impact pulse. It is assumed that when impact failure occurs to the rolling bearing, the collected discrete signals can be expressed as follows:(7)yt=gt+nt∗ht,where gt is the signal of rolling bearing fault impact source after AR filtering; nt is noise interference; ∗ is the convolution operator; and ht is the influence of the transfer path.The purpose of deconvolution is to find the filter coefficientf of the inverse filter so that the output signal yt is restored to the input signal gt; that is,(8)et=fn∗yt=∑l=1Lflyn−l,where et is the output signal of the inverse filter and the input signal gt can be recovered. L is the filter length.In this paper, the objective function method is selected to realize the MED, and the optimal inverse filter coefficient is found so that the objective function is maximized. The objective function can be defined as(9)O4fl=∑n=1Net2∑n=1Net22.Take the derivative of the objective function and set it equal to 0 to obtain(10)∂O4fl∂fl=0.Since∂et/∂fl=yn−l, we can get(11)∑n=1Net2∑n=1Net4∑n=1Net3yn−l=∑p=1Lfp∑n=1Nyn−lyn−p.Equation (11) can be written as matrix(12)f=A−1b,where b is the cross-correlation matrix of input signal gt and output signal et of the inverse filter; A is Toeplitz autocorrelation matrix of inverse filter input signal gt; and f is the filtering coefficient of the inverse filter. The solution process of MED is as follows:(1)
Toeplitz autocorrelation matrixA is calculated, and the filtering coefficient f0 of the inverse filter is initialized, which is usually set as the delayed pulse.(2)
According to the initial filter coefficientf0 and the inverse filter input signal g0, the output signal e0 is calculated.(3)
Vectorb1 is calculated based on the output signal e0 and the input signal g0.(4)
According to equation (12), f1=A−1b0.(5)
Calculate the iteration error:(13)δ=f1−μf0μf0,μ=Ef02Ef121/2.When the expected value of the iteration error is greater than the set error threshold, that is,Eδ>τ, f0=f1, recalculate step (2), repeat the calculation until Eδ≤τ or Eδ diverges, stop the calculation, and get the MED FIR filter parameters.


## 4. Early Weak Fault Feature Frequency Enhance Method of Rolling Bearing Based on ITD-AR-MED

In order to enhance the feature frequency in the complex background noise, a fault feature extraction method of the rolling bearing based on the ITD-AR-MED is proposed in this paper. The flow chart of fault feature frequency enhance method of rolling bearing based on ITD-AR-MED algorithm is shown in Figure3.Figure 3
The flow chart of fault feature frequency enhance method of rolling bearing based on the ITD-AR-MED algorithm.The steps of the ITD-AR-MED algorithm are as follows:(1)
Firstly, the early weak fault vibration signal of rolling bearing, which decomposed by the ITD algorithm into several PRs.(2)
Then, the sample entropy value of each PR is solved, and the PRs with the maximum entropy value is selected for subsequent processing.(3)
Finally, the selected PRs are filtered by the AR-MED method to obtain the final signal.The ITD-AR-MED algorithm decomposition advantages are compared by the fast kurtogram algorithm. The simulation signals are decomposed by the ITD-AR-MED algorithm and fast kurtogram algorithm, respectively. Its simulation signal is(14)x1t=0.5cos2πω1t·cos2πω2t+sin15πt2,x2t=0.25cos2πω3t·e−2t2,nt=noise,xt=x1t+x2t+nt,where ω1=6, ω2=50, ω3=20, nt is Gaussian white noise signal, t is time, x1t,x2t is simulation fault signal. The time domain and frequency domain waveform of the simulation fault signal xt is shown in Figure 4. The feature frequency of the fault simulation signal xt is f1=ω2−ω1=50−6=44Hz, f2=ω2+ω1=50+6=56Hz, and f3=ω3=20Hz. Time domain waveform of simulation signal xt and its components is shown in Figure 5. Time domain waveform of ITD decomposition of simulation signals xt and its PRs.Figure 4
Time domain and frequency domain waveform of the simulation signal.
(a)(b)Figure 5
Time domain waveform of simulation signalxt and its components.It can be seen from Figures5 and 6 that the ITD algorithm can roughly decompose the component signal of the simulation signal xt.Figure 6
Time domain waveform of ITD decomposition of simulation signalsxt and its PRs.


## 5. Experiment Analysis

### 5.1. Fault Diagnosis of the Rolling Bearing Experiment

The standard rolling bearing data set selected from Case Western Reserve University (CWRU) was tested by the ITD-AR-MED method for fault diagnosis of bearing outer and inner rings [40]. The test rig is shown in Figure 7. The test rig equipment consists of motor, power meter, torque sensor, acceleration sensor, and electronic control equipment. Among them, the fault bearing should be as close as possible to the motor. The single point of bearing fault is processed by Electric Discharge Machining (EDM) technology, with a diameter of 0.117 mm and a depth of 0.011 mm. The fault vibration signal generated by the test rig is collected by the acceleration sensor, which is installed directly above the motor drive end shell. The motor frequency is 30 Hz, and the 16-channel DAT data acquisition system is adopted to collect the vibration signal of the rolling bearing fault. The signal sampling frequency is 12 kHz, the spindle speed is 1797 r/min, and the rolling bearing model is 6205-2rs JEM SKF.Figure 7
Test rig of the CWRU.According to the empirical equations (15) and (16), the fault feature frequency is 107.3 Hz and 162.2 Hz of the inner race (IR) and the outer race (OR):(15)f0=Z21−dDcosαN60,(16)f1=Z21+dDcosαN60,where D is the section bearing diameter; N is the spindle speed; d is the diameter of the roll; and z is the number of the ball.It is necessary to conduct noise reduction decomposition base on Figure1’s results. The decomposition results of the ITD algorithm for the IR fault signal and OR fault signal of the rolling bearing are shown in Figures 8 and 9, respectively.Figure 8
IR fault vibration signal of rolling bearing is decomposed by ITD.
(a)(b)Figure 9
OR fault vibration signal of rolling bearing is decomposed by ITD.
(a)(b)It can be seen from Figures8 and 9 that the IR fault signal of rolling bearing has the widest frequency range of PR1 after ITD decomposition, so it covers the most characteristic information. The OR fault signal of rolling bearing has the widest frequency range of PR2 after ITD decomposition, so it covers the most characteristic information.The sample entropy value of each PR is obtained. Table1 shows the sample entropy values of each PR of the rolling bearing.Table 1
Sample entropy value of each PR of IR and OR of rolling bearing.
PR1PR2PR3PR4PR5IR fault1.85451.79071.21450.62860.2009OR fault0.90541.62890.91470.4052—According to the obtained sample entropy value of each PR component, the fault vibration signal of the IR is decomposed by the ITD, and the PR1 component is selected for subsequent analysis. After the fault vibration signal of the OR is decomposed by ITD, the PR2 component is selected for subsequent analysis.Firstly, the PR1 of the IR fault is filtered by AR, and Figure10 is obtained. Figure 10 shows the spectrum of the PR1 component filtered by ITD-AR. After AR filtering, the signal is processed by MED. Figure 11 shows the spectrum diagram of the PR1 component by the ITD-AR-MED algorithm.Figure 10
Spectrum diagram of PR1 component of fault vibration signal of the inner ring of rolling bearing filtered by ITD-AR.Figure 11
Spectrum diagram of PR1 component of fault vibration signal of IR of rolling bearing by ITD-AR-MED algorithm.Figures10 and 11 show that the feature frequency of the IR fault vibration signal is 162.2 Hz. The envelope spectrum of PR1 by the ITD-AR-MED algorithm is shown in Figure 12.Figure 12
Envelope spectrum of fault vibration signal PR1 of IR of rolling bearing by ITD-AR-MED algorithm.As shown in Figure12, the fault frequency of the PR1 can be accurately obtained, which is close to the theoretical value of 161.9 Hz. In addition, the envelope spectrum also accurately extracted the fault feature frequency at the doubling frequency of 323 Hz and the motor rotation frequency at the doubling frequency of 60.06 Hz.Then, the fault vibration signal PR2 component is filtered by AR. The spectrum of PR2 component filtered by ITD-AR is shown in Figure13. After AR filtering, the signal is processed by MED. Figure 14 shows the spectrum diagram of PR2 component filtered by ITD-AR.Figure 13
Spectrum diagram of PR2 component of fault vibration signal of OR of rolling bearing filtered by ITD-AR.Figure 14
Spectrum diagram of PR2 component of fault vibration signal of OR of rolling bearing by ITD-AR-MED algorithm.As shown in Figures13 and 14, the spectrum diagram shows that the fault feature frequency is not clearly displayed at 107 Hz, which covers a large amount of background noise frequency. Therefore, the spectrum diagram obtained by the MED is used for envelope spectrum analysis. The envelope spectrum of PR2 by the ITD-AR-MED algorithm is shown in Figure 15.Figure 15
Envelope spectrum of fault vibration signal PR2 of OR of rolling bearing by the ITD-AR-MED algorithm.As shown in Figure15, the fault frequency of the PR2 component can be accurately obtained, which is similar to the theoretical value 107.7 Hz. In addition, the envelope spectrum also accurately extracted the fault feature frequency, both 215.3 Hz and 323 Hz, which are two-time frequency and three-time frequency, as well as the motor rotation frequency 30 Hz. This proves the effectiveness of the proposed fault diagnosis method.In order to highlight the advantages of the ITD-AR-MED algorithm, the fast kurtogram algorithm is substituted for the ITD algorithm to obtain the envelope spectrum. The fault vibration signal is filtered and enhanced by the fast kurtogram algorithm. Figure16 shows the fast kurtosis spectrum of rolling bearing IR fault vibration signal. It can be seen from Figure 16 that the IR bearing fault signal is obtained by the fast kurtogram algorithm and the center frequency and bandwidth of the bandpass filter are 2250 Hz and 1500 Hz, respectively. Figure 17 shows the envelope spectrum of the rolling bearing IR fault vibration signal after the fast kurtogram algorithm. By comparing Figure 12 with Figure 17, the fast kurtogram algorithm can extract IR fault feature frequency, but its accuracy is slightly lower than that of the ITD-AR-MED algorithm.Figure 16
Fast kurtosis spectrum of rolling bearing IR fault vibration signal.Figure 17
Envelope spectrum of the rolling bearing IR fault vibration signal after the fast kurtogram algorithm, (a) original signal, (b) envelope of filtered signalBw = Fs/23, fc = 750 Hz, Kurt = 1.4, α = 0.1%, and (c) amplitude spectrum of the squared envelope.
(a)(b)(c)Figure18 shows the fast kurtosis spectrum of rolling bearing OR fault vibration signal. It can be seen from Figure 18 that the OR bearing fault signal is obtained by the fast kurtogram algorithm and the center frequency and bandwidth of the bandpass filter are 3000 Hz and 2000 Hz, respectively. Figure 19 shows the envelope spectrum of the rolling bearing OR fault vibration signal after the fast kurtogram algorithm. By comparing Figure 15 with Figure 19, the fast kurtogram algorithm can extract OR fault feature frequency, but its accuracy is slightly lower than that of the ITD-AR-MED algorithm.Figure 18
Fast kurtosis spectrum of rolling bearing OR fault vibration signal.Figure 19
Envelope spectrum of the rolling bearing OR fault vibration signal after the fast kurtogram algorithm, (a) original signal, (b) envelope of filtered signalBw = Fs/24, fc = 2625 Hz, Kurt = 5.6, α = 0.1%, and (c) amplitude spectrum of the squared envelope.
(a)(b)(c)


### 5.2. Fault Diagnosis of the Desulfurization Fan Rolling Bearing Experiment

The desulfurization fan rolling bearing test rig is shown in Figure20. The test rig equipment consists of motor, bearing, hydraulic coupler, acceleration sensor, and fan. The signal sampling frequency is 25.6 kHz. The OR fault feature frequency is 60.31 Hz of the desulfurization fan rolling bearing.Figure 20
Test rig of the desulfurization fan rolling bearing.The decomposition results of the ITD algorithm for the OR fault signal of the desulfurization fan rolling bearing are shown in Figure21.Figure 21
OR fault vibration signal of the desulfurization fan rolling bearing is decomposed by ITD.
(a)(b)It can be seen from Figure21 that the OR fault signal of the desulfurization fan rolling bearing has the widest frequency range of PR2 after ITD decomposition, so it covers the most characteristic information.The sample entropy value of each PR is obtained. Table2 shows the sample entropy values of each PR of the desulfurization fan rolling bearing.Table 2
Sample entropy value of each PR OR fault signal of the desulfurization fan rolling bearing.
PR1PR2PR3PR4OR fault1.01091.20760.73480.2199According to the obtained sample entropy value of each PR component, the fault vibration signal of the OR is decomposed by the ITD, and PR2 component is selected for subsequent analysis.Firstly, the PR2 of the IR fault is filtered by AR, and Figure22 is obtained. Figure 22 shows the spectrum of the PR2 component filtered by ITD-AR. After AR filtering, the signal is processed by MED. Figure 23 shows the spectrum diagram of the PR2 component by the ITD-AR-MED algorithm.Figure 22
Spectrum diagram of the PR2 component of fault vibration signal of OR of the desulfurization fan rolling bearing filtered by ITD-AR.Figure 23
Spectrum diagram of the PR2 component of fault vibration signal of OR of the desulfurization fan rolling bearing by ITD-AR-MED algorithm.Figures22 and 23 show that the feature frequency of the OR fault vibration signal is 50 Hz. The envelope spectrum of PR1 by the ITD-AR-MED algorithm is shown in Figure 24.Figure 24
Envelope spectrum of fault vibration signal PR2 of OR of the desulfurization fan rolling bearing by ITD-AR-MED algorithm.As shown in Figure24, the fault frequency of the PR2 can be accurately obtained, which is close to the theoretical value of 60.31 Hz. Figure 25 shows the fast kurtosis spectrum of the desulfurization fan rolling bearing OR fault vibration signal. It can be seen from Figure 25 that the OR bearing fault signal is obtained by the fast kurtogram algorithm and the center frequency and bandwidth of the bandpass filter are 3000 Hz and 2000 Hz, respectively. Figure 26 shows the envelope spectrum of the rolling bearing OR fault vibration signal after the fast kurtogram algorithm. By comparing Figure 24 with Figure 26, the fast kurtogram algorithm can extract OR fault feature frequency, but its accuracy is slightly lower than that of the ITD-AR-MED algorithm.Figure 25
Fast kurtosis spectrum of the desulfurization fan rolling bearing OR fault vibration signal.Figure 26
Envelope spectrum of the desulfurization fan rolling bearing OR fault vibration signal after the fast kurtogram algorithm, (a) original signal, (b) envelope of filtered signalBw = Fs/26.585, fc = 12133.3333 Hz, Kurt = 17.8, α = 0.1%, and (c) amplitude spectrum of the squared envelope.
(a)(b)(c)


## 5.1. Fault Diagnosis of the Rolling Bearing Experiment

The standard rolling bearing data set selected from Case Western Reserve University (CWRU) was tested by the ITD-AR-MED method for fault diagnosis of bearing outer and inner rings [40]. The test rig is shown in Figure 7. The test rig equipment consists of motor, power meter, torque sensor, acceleration sensor, and electronic control equipment. Among them, the fault bearing should be as close as possible to the motor. The single point of bearing fault is processed by Electric Discharge Machining (EDM) technology, with a diameter of 0.117 mm and a depth of 0.011 mm. The fault vibration signal generated by the test rig is collected by the acceleration sensor, which is installed directly above the motor drive end shell. The motor frequency is 30 Hz, and the 16-channel DAT data acquisition system is adopted to collect the vibration signal of the rolling bearing fault. The signal sampling frequency is 12 kHz, the spindle speed is 1797 r/min, and the rolling bearing model is 6205-2rs JEM SKF.Figure 7
Test rig of the CWRU.According to the empirical equations (15) and (16), the fault feature frequency is 107.3 Hz and 162.2 Hz of the inner race (IR) and the outer race (OR):(15)f0=Z21−dDcosαN60,(16)f1=Z21+dDcosαN60,where D is the section bearing diameter; N is the spindle speed; d is the diameter of the roll; and z is the number of the ball.It is necessary to conduct noise reduction decomposition base on Figure1’s results. The decomposition results of the ITD algorithm for the IR fault signal and OR fault signal of the rolling bearing are shown in Figures 8 and 9, respectively.Figure 8
IR fault vibration signal of rolling bearing is decomposed by ITD.
(a)(b)Figure 9
OR fault vibration signal of rolling bearing is decomposed by ITD.
(a)(b)It can be seen from Figures8 and 9 that the IR fault signal of rolling bearing has the widest frequency range of PR1 after ITD decomposition, so it covers the most characteristic information. The OR fault signal of rolling bearing has the widest frequency range of PR2 after ITD decomposition, so it covers the most characteristic information.The sample entropy value of each PR is obtained. Table1 shows the sample entropy values of each PR of the rolling bearing.Table 1
Sample entropy value of each PR of IR and OR of rolling bearing.
PR1PR2PR3PR4PR5IR fault1.85451.79071.21450.62860.2009OR fault0.90541.62890.91470.4052—According to the obtained sample entropy value of each PR component, the fault vibration signal of the IR is decomposed by the ITD, and the PR1 component is selected for subsequent analysis. After the fault vibration signal of the OR is decomposed by ITD, the PR2 component is selected for subsequent analysis.Firstly, the PR1 of the IR fault is filtered by AR, and Figure10 is obtained. Figure 10 shows the spectrum of the PR1 component filtered by ITD-AR. After AR filtering, the signal is processed by MED. Figure 11 shows the spectrum diagram of the PR1 component by the ITD-AR-MED algorithm.Figure 10
Spectrum diagram of PR1 component of fault vibration signal of the inner ring of rolling bearing filtered by ITD-AR.Figure 11
Spectrum diagram of PR1 component of fault vibration signal of IR of rolling bearing by ITD-AR-MED algorithm.Figures10 and 11 show that the feature frequency of the IR fault vibration signal is 162.2 Hz. The envelope spectrum of PR1 by the ITD-AR-MED algorithm is shown in Figure 12.Figure 12
Envelope spectrum of fault vibration signal PR1 of IR of rolling bearing by ITD-AR-MED algorithm.As shown in Figure12, the fault frequency of the PR1 can be accurately obtained, which is close to the theoretical value of 161.9 Hz. In addition, the envelope spectrum also accurately extracted the fault feature frequency at the doubling frequency of 323 Hz and the motor rotation frequency at the doubling frequency of 60.06 Hz.Then, the fault vibration signal PR2 component is filtered by AR. The spectrum of PR2 component filtered by ITD-AR is shown in Figure13. After AR filtering, the signal is processed by MED. Figure 14 shows the spectrum diagram of PR2 component filtered by ITD-AR.Figure 13
Spectrum diagram of PR2 component of fault vibration signal of OR of rolling bearing filtered by ITD-AR.Figure 14
Spectrum diagram of PR2 component of fault vibration signal of OR of rolling bearing by ITD-AR-MED algorithm.As shown in Figures13 and 14, the spectrum diagram shows that the fault feature frequency is not clearly displayed at 107 Hz, which covers a large amount of background noise frequency. Therefore, the spectrum diagram obtained by the MED is used for envelope spectrum analysis. The envelope spectrum of PR2 by the ITD-AR-MED algorithm is shown in Figure 15.Figure 15
Envelope spectrum of fault vibration signal PR2 of OR of rolling bearing by the ITD-AR-MED algorithm.As shown in Figure15, the fault frequency of the PR2 component can be accurately obtained, which is similar to the theoretical value 107.7 Hz. In addition, the envelope spectrum also accurately extracted the fault feature frequency, both 215.3 Hz and 323 Hz, which are two-time frequency and three-time frequency, as well as the motor rotation frequency 30 Hz. This proves the effectiveness of the proposed fault diagnosis method.In order to highlight the advantages of the ITD-AR-MED algorithm, the fast kurtogram algorithm is substituted for the ITD algorithm to obtain the envelope spectrum. The fault vibration signal is filtered and enhanced by the fast kurtogram algorithm. Figure16 shows the fast kurtosis spectrum of rolling bearing IR fault vibration signal. It can be seen from Figure 16 that the IR bearing fault signal is obtained by the fast kurtogram algorithm and the center frequency and bandwidth of the bandpass filter are 2250 Hz and 1500 Hz, respectively. Figure 17 shows the envelope spectrum of the rolling bearing IR fault vibration signal after the fast kurtogram algorithm. By comparing Figure 12 with Figure 17, the fast kurtogram algorithm can extract IR fault feature frequency, but its accuracy is slightly lower than that of the ITD-AR-MED algorithm.Figure 16
Fast kurtosis spectrum of rolling bearing IR fault vibration signal.Figure 17
Envelope spectrum of the rolling bearing IR fault vibration signal after the fast kurtogram algorithm, (a) original signal, (b) envelope of filtered signalBw = Fs/23, fc = 750 Hz, Kurt = 1.4, α = 0.1%, and (c) amplitude spectrum of the squared envelope.
(a)(b)(c)Figure18 shows the fast kurtosis spectrum of rolling bearing OR fault vibration signal. It can be seen from Figure 18 that the OR bearing fault signal is obtained by the fast kurtogram algorithm and the center frequency and bandwidth of the bandpass filter are 3000 Hz and 2000 Hz, respectively. Figure 19 shows the envelope spectrum of the rolling bearing OR fault vibration signal after the fast kurtogram algorithm. By comparing Figure 15 with Figure 19, the fast kurtogram algorithm can extract OR fault feature frequency, but its accuracy is slightly lower than that of the ITD-AR-MED algorithm.Figure 18
Fast kurtosis spectrum of rolling bearing OR fault vibration signal.Figure 19
Envelope spectrum of the rolling bearing OR fault vibration signal after the fast kurtogram algorithm, (a) original signal, (b) envelope of filtered signalBw = Fs/24, fc = 2625 Hz, Kurt = 5.6, α = 0.1%, and (c) amplitude spectrum of the squared envelope.
(a)(b)(c)


## 5.2. Fault Diagnosis of the Desulfurization Fan Rolling Bearing Experiment

The desulfurization fan rolling bearing test rig is shown in Figure20. The test rig equipment consists of motor, bearing, hydraulic coupler, acceleration sensor, and fan. The signal sampling frequency is 25.6 kHz. The OR fault feature frequency is 60.31 Hz of the desulfurization fan rolling bearing.Figure 20
Test rig of the desulfurization fan rolling bearing.The decomposition results of the ITD algorithm for the OR fault signal of the desulfurization fan rolling bearing are shown in Figure21.Figure 21
OR fault vibration signal of the desulfurization fan rolling bearing is decomposed by ITD.
(a)(b)It can be seen from Figure21 that the OR fault signal of the desulfurization fan rolling bearing has the widest frequency range of PR2 after ITD decomposition, so it covers the most characteristic information.The sample entropy value of each PR is obtained. Table2 shows the sample entropy values of each PR of the desulfurization fan rolling bearing.Table 2
Sample entropy value of each PR OR fault signal of the desulfurization fan rolling bearing.
PR1PR2PR3PR4OR fault1.01091.20760.73480.2199According to the obtained sample entropy value of each PR component, the fault vibration signal of the OR is decomposed by the ITD, and PR2 component is selected for subsequent analysis.Firstly, the PR2 of the IR fault is filtered by AR, and Figure22 is obtained. Figure 22 shows the spectrum of the PR2 component filtered by ITD-AR. After AR filtering, the signal is processed by MED. Figure 23 shows the spectrum diagram of the PR2 component by the ITD-AR-MED algorithm.Figure 22
Spectrum diagram of the PR2 component of fault vibration signal of OR of the desulfurization fan rolling bearing filtered by ITD-AR.Figure 23
Spectrum diagram of the PR2 component of fault vibration signal of OR of the desulfurization fan rolling bearing by ITD-AR-MED algorithm.Figures22 and 23 show that the feature frequency of the OR fault vibration signal is 50 Hz. The envelope spectrum of PR1 by the ITD-AR-MED algorithm is shown in Figure 24.Figure 24
Envelope spectrum of fault vibration signal PR2 of OR of the desulfurization fan rolling bearing by ITD-AR-MED algorithm.As shown in Figure24, the fault frequency of the PR2 can be accurately obtained, which is close to the theoretical value of 60.31 Hz. Figure 25 shows the fast kurtosis spectrum of the desulfurization fan rolling bearing OR fault vibration signal. It can be seen from Figure 25 that the OR bearing fault signal is obtained by the fast kurtogram algorithm and the center frequency and bandwidth of the bandpass filter are 3000 Hz and 2000 Hz, respectively. Figure 26 shows the envelope spectrum of the rolling bearing OR fault vibration signal after the fast kurtogram algorithm. By comparing Figure 24 with Figure 26, the fast kurtogram algorithm can extract OR fault feature frequency, but its accuracy is slightly lower than that of the ITD-AR-MED algorithm.Figure 25
Fast kurtosis spectrum of the desulfurization fan rolling bearing OR fault vibration signal.Figure 26
Envelope spectrum of the desulfurization fan rolling bearing OR fault vibration signal after the fast kurtogram algorithm, (a) original signal, (b) envelope of filtered signalBw = Fs/26.585, fc = 12133.3333 Hz, Kurt = 17.8, α = 0.1%, and (c) amplitude spectrum of the squared envelope.
(a)(b)(c)


## 6. Conclusions

Aiming at the difficulty of extracting the early weak fault feature frequency, the characteristics of the shock waveform of the IR and OR of rolling bearing are combined. In this paper, the ITD-AR-MED method is used for weak fault feature extraction, and the following conclusions are as follows.(1)
The ITD algorithm can decompose the early weak fault vibration signal effectively, retaining the fault impact characteristics of the rolling bearing.(2)
The MED algorithm can be used to separate the noise from the original signal. The envelope spectrum can extract the early weak fault feature frequency of the rolling bearing, which verifies the effectiveness and accuracy of the method.



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*Source: 6673965-2021-01-13.xml*