# Adaptive Asymmetric Real Laplace Wavelet Filtering and Its Application on Rolling Bearing Early Fault Diagnosis

Authors: Shuting Wan; Bo Peng
Journal: Shock and Vibration
 (2019)

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## Abstract

The early fault of rolling bearing is weak and may not be readily detected. To overcome this issue, the present paper comes up with a rolling bearing fault-diagnosing approach based on adaptive asymmetric real Laplace wavelet (ARLW) filtering, which is on the strength of water cycle optimization algorithm (WCA). Firstly, ARLW is introduced to filter the initial vibration signal since its waveform has the same asymmetric structure as the fault impact. Secondly, the optimum center frequency and bandwidth of ARLW is found out adaptively by applying the WCA through the proposed square envelope fault energy ratio (SEFER). Finally, envelope analysis is conducted to the narrowband signal obtained by the optimum ARLW filtering, and its envelope spectrum presents the rolling bearing fault characteristic frequency apparently. The proposed approach and two existing approaches are all tested in four signal analysis cases. The results are analyzed, and the conclusion is that the approach proposed by the present paper can detect the early fault of rolling bearing more accurately. The present research is valuable for diagnosing the early fault of rolling bearing.


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## Body

## 1. Introduction

Rolling bearing is broadly used in rotating equipment, and its fault acts on the safe operation of the whole equipment [1–3]. At the beginning of the rolling bearing fault, the impact component of vibration signal collected by the sensor is weak and often submerged in strong background noise, bringing challenges to the diagnostic process [4]. Consequently, diagnosing the early fault of rolling bearing acts as the focus and difficulty among researchers and scholars.The periodic impact produced by partial defects on the roiling bearing surface can arouse the resonance between the rolling bearing and its adjacent parts [5]. Using resonance demodulation to extract impact response characteristics from vibration signal is a fast and simple approach of rolling bearing fault diagnosis, the pivotal step of which is to precisely identify the resonance frequency band that contains plentiful fault information. The traditional resonance demodulation approach has the limitation of the request of artificially presetting the parameters of the bandpass filter (center frequency and bandwidth).Antoni et al. creatively put forward the spectral kurtosis theory [6, 7] and the fast spectrum kurtosis (FSK) approach [8] that can automatically set the parameters of the bandpass filter, which firstly used short time Fourier transform (STFT) or finite impulse response (FIR) filters to divide frequency bands and then used the kurtosis of signal as an evaluating indicator to identify the frequency band containing the most fault information. However, there are two disadvantages in the FSK approach. The first one is that kurtosis as an evaluation index cannot distinguish the random impact and periodic impact of the signal, which easily leads to identifying the erroneous resonance band. The second one is that the optimal frequency band selected by FSK may not include the whole resonant frequency band region. For the first disadvantage, several new evaluating indicators are raised to improve the accuracy of selecting the resonant frequency band, such as correlated kurtosis [9], harmonic-to-noise ratio [10], Gini index [11], and spectral L2/L1 [12]. In addition, the noise reduction approach can be used as a preprocessing method of FSK to avoid interference, such as the improved FSK approach with the aid of EEMD [13], ICA [14], and ITD [15]. Although the above approach improves the robustness of FSK, more prior knowledge is needed, and the calculation process is complex. For the second shortcoming, many scholars have done useful work. Lei et al. used the wavelet packet transform (WPT) instead of the STFT or FIR filters in the FSK, which segments the frequency band more finely [16–18]. Comparing with WPT, continuous wavelet transform (CWT) can flexibly and accurately divide the frequency domain. CWT segments the signal through constructing a series of filters with the same property but different center frequency (CF) and bandwidth (BW) by translating and stretching the mother wavelet. Thus, designing wavelet parameters properly to obtain the resonance frequency band is key to diagnosing rolling bearing fault by CWT. Qiu et al. [19] combines the Shannon entropy and SVD theory to realize the optimal wavelet transform. Bozchalooi et al. [20] uses the smoothing index to select the CF and BW of Gabor wavelet. The variable step size of the CF is 50 Hz, and the search range is [200, 6000]. The variable step size of the BW is 0.01, and the search range is [0.01, 1]. That is to say, we need to calculate 11700 (117 × 110) times to determine the best solution. It is obvious that intelligent algorithms should be used to replace the huge computation process and to reduce the computation time and memory footprint. Su et al. [21] puts forward an approach based on the minimal entropy criterion, using genetic algorithm to obtain the optimum wavelet parameters. Wang et al. [22] obtains the parameters of Morlet wavelet using simulated annealing algorithm that adopted the maximum sparsity as the fitness function. Chen et al. [23] used a particle swarm optimization algorithm on the basis of the correlated kurtosis of squared envelope spectrum as fitness function to obtain the parameters of Morlet wavelet.At present, there are three main issues in using wavelet filtering to determine the resonance frequency band of rolling bearing: the selection of the wavelet basic function, the selection of evaluation index, and fast determination of wavelet parameters. Accordingly, this paper proposed an adaptive asymmetric real Laplace wavelet (ARLW) filtering approach based on water cycle optimization algorithm (WCA), which has the following improvements: ARLW is used as wavelet basis function; square envelope fault energy ratio (SEFER) is chosen as a new evaluation index; WCA is used to choose the optimum wavelet parameters quickly. Four signal analysis cases are conducted to testify the validity of the proposed approach, the results of which are compared to that of the two existing approach.The rest of the present paper is structured as follows: Section2 reviews the theoretical background of wavelet filtering. Section 3 discusses the wavelet parameter optimization process. Section 4 proposes the rolling bearing fault diagnosis approach on the basis of adaptive wavelet filtering. Section 5 verifies the proposed approach through four signal analysis cases. Section 6 summarizes the full text.


## 2. Wavelet Filtering

### 2.1. Continuous Wavelet Transform

The CWT of one-dimensional signalxt is defined as(1)Wa,b=a−1/2∫−∞+∞xtφ∗t−badt,where Wa,b represents the wavelet coefficient, a represents the scale parameter, b represents the shift parameter, φ(·) represents the wavelet basis function, and ∗ represents the conjugate. In the frequency domain, Equation (1) is expressed as(2)Wa,b=a−1/2IFTXfψ∗af,where Xf refers to the Fourier transform of x(t), ψ(f) refers to the Fourier transform of φ(t), and IFT refers to the inverse Fourier transform.


### 2.2. Asymmetric Real Laplace Wavelet

In the fault diagnosis of the mechanical system, more fault features can be extracted when the wavelet function matches the impact response in the dynamic signal. Among the existing wavelet basic functions, the time domain waveform (TW) of ARLW and Morlet wavelet is similar to that of impact response caused by local fault of rolling bearing. Figures1(a)–1(c) depict the TW of ARLW, Morlet wavelet, and the actual rolling bearing signal, respectively. Compared with the Morlet wavelet with symmetrical structure, the waveform of ARLW with asymmetrical structure is more similar to fault waveform, so ARLW is more suitable for extracting rolling bearing fault features. The ARLW φ(t) can be expressed as the following equation [24, 25]:(3)ϕt=exp−πσtsin2πγt.Figure 1
The TW of (a) ARLW; (b) Morlet wavelet; (c) fault impact; (d) FS of (a).
(a)
(b)
(c)
(d)Fourier transform ofφ(t) is displayed as(4)ψf=σπσ2+4f−γ2i−σπσ2+4f+γ2i,where γ represents the center frequency (CF) and σ represents the bandwidth (BW).The ARLW transform of one-dimensional vibration signalx(t) is expressed as(5)Wxγ,σ=IFTXfψ∗f,where Xf represents the Fourier transform of x(t), Wx(γ, σ) represents the result of ARLW filtering, and IFT represents the inverse Fourier transform. According to Equation (4), ARLW transform is the product of the analyzed signal and the ARLW in the frequency domain. Figure 1(d) illustrates that the frequency spectrum (FS) of ARLW has the window form. Therefore, the ARLW transform has the same function as the bandpass filter. Based on the above analysis, designing wavelet parameters properly to obtain the resonance frequency band is the key to diagnosing rolling bearing fault by ARLW filtering. In addition, it should be noted that γ and σ are not completely independent in the ARLW filtering process. If the two parameters are optimized separately, the result is not the optimal solution.


## 2.1. Continuous Wavelet Transform

The CWT of one-dimensional signalxt is defined as(1)Wa,b=a−1/2∫−∞+∞xtφ∗t−badt,where Wa,b represents the wavelet coefficient, a represents the scale parameter, b represents the shift parameter, φ(·) represents the wavelet basis function, and ∗ represents the conjugate. In the frequency domain, Equation (1) is expressed as(2)Wa,b=a−1/2IFTXfψ∗af,where Xf refers to the Fourier transform of x(t), ψ(f) refers to the Fourier transform of φ(t), and IFT refers to the inverse Fourier transform.


## 2.2. Asymmetric Real Laplace Wavelet

In the fault diagnosis of the mechanical system, more fault features can be extracted when the wavelet function matches the impact response in the dynamic signal. Among the existing wavelet basic functions, the time domain waveform (TW) of ARLW and Morlet wavelet is similar to that of impact response caused by local fault of rolling bearing. Figures1(a)–1(c) depict the TW of ARLW, Morlet wavelet, and the actual rolling bearing signal, respectively. Compared with the Morlet wavelet with symmetrical structure, the waveform of ARLW with asymmetrical structure is more similar to fault waveform, so ARLW is more suitable for extracting rolling bearing fault features. The ARLW φ(t) can be expressed as the following equation [24, 25]:(3)ϕt=exp−πσtsin2πγt.Figure 1
The TW of (a) ARLW; (b) Morlet wavelet; (c) fault impact; (d) FS of (a).
(a)
(b)
(c)
(d)Fourier transform ofφ(t) is displayed as(4)ψf=σπσ2+4f−γ2i−σπσ2+4f+γ2i,where γ represents the center frequency (CF) and σ represents the bandwidth (BW).The ARLW transform of one-dimensional vibration signalx(t) is expressed as(5)Wxγ,σ=IFTXfψ∗f,where Xf represents the Fourier transform of x(t), Wx(γ, σ) represents the result of ARLW filtering, and IFT represents the inverse Fourier transform. According to Equation (4), ARLW transform is the product of the analyzed signal and the ARLW in the frequency domain. Figure 1(d) illustrates that the frequency spectrum (FS) of ARLW has the window form. Therefore, the ARLW transform has the same function as the bandpass filter. Based on the above analysis, designing wavelet parameters properly to obtain the resonance frequency band is the key to diagnosing rolling bearing fault by ARLW filtering. In addition, it should be noted that γ and σ are not completely independent in the ARLW filtering process. If the two parameters are optimized separately, the result is not the optimal solution.


## 3. Wavelet Parameter Adaptive Optimization

Water cycle optimization algorithm (WCA) proposed by Eskandar is a new embedded optimization approach [26, 27], which can optimize multiple parameters in parallel. Compared with genetic algorithm, simulated annealing algorithm, and other traditional optimization algorithms, WCA is more efficient and unlikely to fall into local optimal solution. Based on this, the optimal ARLW filter can be constructed adaptively with the help of WCA optimization algorithm. In the iterative process of WCA, the wavelet obtained by each iteration should be evaluated and determined whether it is the optimal solution. Therefore, it is particularly necessary to select an appropriate evaluation index.


### 3.1. The Selection of Evaluation Index

Aiming at improving the performance of envelope analysis in resonance demodulation, Randall [28] proposed a method, namely, square envelope analysis, which can suppress noise and highlight fault characteristics. Based on this advantage, the square envelope fault energy ratio (SEFER) is proposed, and the specific calculation process is as follows.Step 1.
Performing Hilbert transform on the signalx(t).(6)x′t=1π∫−∞+∞xtt−τdt.Step 2.
Constructing the analytic signalz(t) and its conjugate complex analytic signal z′t.(7)zt=xt+jxt,z′t=xt−jxt′.Step 3.
The square envelope signalyt obtained by multiplying the analytic signal with the conjugate analytic signal.(8)yt=ztzt′.Step 4.
Fourier transform is conducted on the square envelope signalyt.(9)SESf=FTyt,where SES refers to squared envelope spectrum sequence and FT refers to the Fourier transform.Step 5.
The SEFER value of different fault type is calculated by the follow equation:(10)SEFERf′=FESE=∑n=15SES2nf′∑SES2f,where FE and SE denote the fault energy and signal total energy, respectively. SES (f) refers to the amplitude of each frequency in squared envelope spectrum. SES (nf′) refers to the amplitude of the fault feature frequency and its frequency doubling in squared envelope spectrum. f′ represents the feature frequency of different fault types.
The periodic impact signal overlaying the Gaussian noise signal can be served to simulate the rolling bearing fault signal. The mathematical model of the periodic impact signal is defined as(11)x=∑1KA1e−2gπfnt0⋅sin2fn1−g2t0,where the amplitude is A1 = 0.5, the damping coefficient is g=0.05, the resonance frequency is fn = 2500 Hz, the single impact sampling time is t0 = 0.025 s, and the number of impact is K = 20. The sampling frequency and sampling points is 8096 Hz and 4096, respectively. Figure 2 displays the simulation signal with different SNRs, where the periodic impact characteristics (red waveform line) of signals become more prominent with the increase of SNR. The variation tendency of SEFER values for different SNR signals is presented in Figure 3, where the SEFER value increases monotonically with SNR increase. To sum up, we can say that the larger the SEFER value is, the more obvious impact characteristic the signal has.Figure 2
The periodic impact signal with different SNRs: (a)  −12 dB; (b)  −10 dB; (c)  −8 dB; (d)  −6 dB.
(a)
(b)
(c)
(d)Figure 3
The TFR variation tendency with different SNRs.


### 3.2. Adaptive Optimization Process

According to the analysis of Section3.1, the larger the TFR value is, the more fault information the signal has. By comparing the SEFER value, we can judge the matching degree between ARLW and impulse components of the signal and determine the optimal ARLW filter according to the maximal value criterion. It should be noted that WCA is an algorithm for solving the minimization problem, and the process of finding the maximum SEFER should be converted into the minimization problem. The detailed steps of the adaptive optimization process of ARLW parameters based on WCA algorithm are as follows:Step 1.
Setting the reciprocal of SEFER of the filtered signal obtained by ARLW filtering as the objective function of WCA as follows:(12)F=1SEFERf′,where f′ represents the feature frequency of different fault type.
Setting the constraint condition of theγ and σ of ARLW. The literature shows that the optimal bandwidth of ARLW filter should be 3 times greater than the rolling bearing fault feature frequency. In addition, according to the sampling theory, the high-pass cutoff frequency of the ARLW filter needs to be less than half the time of the sampling frequency. Thus, the constraint conditions of γ and σ are as follows:(13)3f′<σ<fs2,σmin2<γ<fs2−σmin2,where f′ refers to the feature frequency of different fault type, fs refers to the sampling frequency, and σmin refers to the minimal value in the range of σ.Step 2.
Initializing the parameters of WCA algorithm. The number of design variables isNvar = 2, the population size is Npop = 10, the number of rivers and sea is Nsr = 2, the evaporation condition constant is dmax = 1e − 5, and the maximal number of iterations is Max_it = 50.Step 3.
Generating the initial raindrops and determining the number of river and stream. The initial raindrops can be expressed as follows.(14)X=x1i,x2i,…,xNvari,i=1,2,…,Npop,where x = LB + rand × (UB − LB), LB and UB represent the upper and lower limitations of the variables, and the rand represents a random number evenly distributed between 0 and 1.
Use Equation (15) to calculate the fitness function (objective function) of each raindrop:(15)Costi=Fx1i,x2i,…,xNvari,i=1,2,…,Npop.
The best individual (the minimal fitness value) is chosen as the sea, and some better individuals are selected as the river. The number of sea (optimum solution) is 1, the number of rivers isNr = Nsr − 1 = 1, and the number of streams flowing to rivers or sea is determined by.(16)NSn=roundCostn∑i=1NsrCosti×Npop−Nsr,where n = 1, 2, 3, ... , Nsr.Step 4.
In the process of water confluence, that is, raindrops form streams, some streams flow into the river and the others flow into the sea. The algorithm assumes that all rivers and streams become merged in the sea eventually. If the fitness value of a stream is greater than that of the adjacent river, exchange the location of the river and stream (i.e., stream becomes river and river becomes stream); if the fitness value of a river is greater than that of the adjacent sea, exchange the location of the sea and river (i.e., river becomes sea and sea becomes river). The location updating of stream and river can be expressed as follows:(17)Xstreami+1=Xstreami+rand×C×Xriveri−Xstreami,Xriveri+1=Xriveri+rand×C×Xseai−Xriveri,where the rand definition is the same as above, C refers to numbers between 1 and 2. Xstreami, Xriveri, and Xriveri refer to the position of the stream, river, and sea at the i times’ iteration, respectively.Step 5.
The process of evaporation and precipitation. Evaporation condition can be expressed as follows:(18)Xseai−Xriveri<dmax,where dmax can be reduced adaptively, and its calculation formula is as follows:(19)dmaxi+1=dmaxi−dmaxiMax−it
When fulfilling the evaporation criterion, the new raindrops will form and then enter the process of water confluence again.Step 6.
Checking the terminating condition. When living up to the maximal iteration, stop the algorithm and output the optimal solution. Or else, go back to the Step 4 and continue to solve.


## 3.1. The Selection of Evaluation Index

Aiming at improving the performance of envelope analysis in resonance demodulation, Randall [28] proposed a method, namely, square envelope analysis, which can suppress noise and highlight fault characteristics. Based on this advantage, the square envelope fault energy ratio (SEFER) is proposed, and the specific calculation process is as follows.Step 1.
Performing Hilbert transform on the signalx(t).(6)x′t=1π∫−∞+∞xtt−τdt.Step 2.
Constructing the analytic signalz(t) and its conjugate complex analytic signal z′t.(7)zt=xt+jxt,z′t=xt−jxt′.Step 3.
The square envelope signalyt obtained by multiplying the analytic signal with the conjugate analytic signal.(8)yt=ztzt′.Step 4.
Fourier transform is conducted on the square envelope signalyt.(9)SESf=FTyt,where SES refers to squared envelope spectrum sequence and FT refers to the Fourier transform.Step 5.
The SEFER value of different fault type is calculated by the follow equation:(10)SEFERf′=FESE=∑n=15SES2nf′∑SES2f,where FE and SE denote the fault energy and signal total energy, respectively. SES (f) refers to the amplitude of each frequency in squared envelope spectrum. SES (nf′) refers to the amplitude of the fault feature frequency and its frequency doubling in squared envelope spectrum. f′ represents the feature frequency of different fault types.
The periodic impact signal overlaying the Gaussian noise signal can be served to simulate the rolling bearing fault signal. The mathematical model of the periodic impact signal is defined as(11)x=∑1KA1e−2gπfnt0⋅sin2fn1−g2t0,where the amplitude is A1 = 0.5, the damping coefficient is g=0.05, the resonance frequency is fn = 2500 Hz, the single impact sampling time is t0 = 0.025 s, and the number of impact is K = 20. The sampling frequency and sampling points is 8096 Hz and 4096, respectively. Figure 2 displays the simulation signal with different SNRs, where the periodic impact characteristics (red waveform line) of signals become more prominent with the increase of SNR. The variation tendency of SEFER values for different SNR signals is presented in Figure 3, where the SEFER value increases monotonically with SNR increase. To sum up, we can say that the larger the SEFER value is, the more obvious impact characteristic the signal has.Figure 2
The periodic impact signal with different SNRs: (a)  −12 dB; (b)  −10 dB; (c)  −8 dB; (d)  −6 dB.
(a)
(b)
(c)
(d)Figure 3
The TFR variation tendency with different SNRs.


## 3.2. Adaptive Optimization Process

According to the analysis of Section3.1, the larger the TFR value is, the more fault information the signal has. By comparing the SEFER value, we can judge the matching degree between ARLW and impulse components of the signal and determine the optimal ARLW filter according to the maximal value criterion. It should be noted that WCA is an algorithm for solving the minimization problem, and the process of finding the maximum SEFER should be converted into the minimization problem. The detailed steps of the adaptive optimization process of ARLW parameters based on WCA algorithm are as follows:Step 1.
Setting the reciprocal of SEFER of the filtered signal obtained by ARLW filtering as the objective function of WCA as follows:(12)F=1SEFERf′,where f′ represents the feature frequency of different fault type.
Setting the constraint condition of theγ and σ of ARLW. The literature shows that the optimal bandwidth of ARLW filter should be 3 times greater than the rolling bearing fault feature frequency. In addition, according to the sampling theory, the high-pass cutoff frequency of the ARLW filter needs to be less than half the time of the sampling frequency. Thus, the constraint conditions of γ and σ are as follows:(13)3f′<σ<fs2,σmin2<γ<fs2−σmin2,where f′ refers to the feature frequency of different fault type, fs refers to the sampling frequency, and σmin refers to the minimal value in the range of σ.Step 2.
Initializing the parameters of WCA algorithm. The number of design variables isNvar = 2, the population size is Npop = 10, the number of rivers and sea is Nsr = 2, the evaporation condition constant is dmax = 1e − 5, and the maximal number of iterations is Max_it = 50.Step 3.
Generating the initial raindrops and determining the number of river and stream. The initial raindrops can be expressed as follows.(14)X=x1i,x2i,…,xNvari,i=1,2,…,Npop,where x = LB + rand × (UB − LB), LB and UB represent the upper and lower limitations of the variables, and the rand represents a random number evenly distributed between 0 and 1.
Use Equation (15) to calculate the fitness function (objective function) of each raindrop:(15)Costi=Fx1i,x2i,…,xNvari,i=1,2,…,Npop.
The best individual (the minimal fitness value) is chosen as the sea, and some better individuals are selected as the river. The number of sea (optimum solution) is 1, the number of rivers isNr = Nsr − 1 = 1, and the number of streams flowing to rivers or sea is determined by.(16)NSn=roundCostn∑i=1NsrCosti×Npop−Nsr,where n = 1, 2, 3, ... , Nsr.Step 4.
In the process of water confluence, that is, raindrops form streams, some streams flow into the river and the others flow into the sea. The algorithm assumes that all rivers and streams become merged in the sea eventually. If the fitness value of a stream is greater than that of the adjacent river, exchange the location of the river and stream (i.e., stream becomes river and river becomes stream); if the fitness value of a river is greater than that of the adjacent sea, exchange the location of the sea and river (i.e., river becomes sea and sea becomes river). The location updating of stream and river can be expressed as follows:(17)Xstreami+1=Xstreami+rand×C×Xriveri−Xstreami,Xriveri+1=Xriveri+rand×C×Xseai−Xriveri,where the rand definition is the same as above, C refers to numbers between 1 and 2. Xstreami, Xriveri, and Xriveri refer to the position of the stream, river, and sea at the i times’ iteration, respectively.Step 5.
The process of evaporation and precipitation. Evaporation condition can be expressed as follows:(18)Xseai−Xriveri<dmax,where dmax can be reduced adaptively, and its calculation formula is as follows:(19)dmaxi+1=dmaxi−dmaxiMax−it
When fulfilling the evaporation criterion, the new raindrops will form and then enter the process of water confluence again.Step 6.
Checking the terminating condition. When living up to the maximal iteration, stop the algorithm and output the optimal solution. Or else, go back to the Step 4 and continue to solve.


## 4. The Diagnosis Approach on the Basis of Adaptive Wavelet Filtering

Rolling bearing fault can be diagnosed quickly and effectively through the demodulation analysis of the resonance frequency band. Based on this idea, this paper came up with a rolling bearing fault diagnosis approach on the basis of adaptive ARLW filtering through WCA. Figure4 describes the diagnosis process, and the detailed steps are as follows:Figure 4
The diagnosis process of the proposed approach.Step 1.
Loading the original signal and calculating the feature frequency of different fault type. The mathematical formulas are listed below [29].(20)fi=Z21+dDcosαn60,fo=Z21−dDcosαn60,fe=D2d1−dDcosα2,fc=121−dDcosαn60,where d and D represent the diameter of the balls and the pitch, respectively; α represents the contact angle between the ball and the raceway; Z represents the number of rolling element; and n represents the rotating speed. fi refers to the fault frequency of the inner ring (IRFF), fo refers to the fault frequency of the outer ring (ORFF), fe refers to the fault frequency of the rolling element (REFF), and fc refers to the fault frequency of the cage (CFF).Step 2.
Constructing the optimal bandpass filter according to the result of WCA optimization. The corresponding optimizing process can be seen in Section3.2.Step 3.
Filtering the original signal with the constructed bandpass filter.Step 4.
Performing envelope demodulation on the filtered signal obtained by the optimal bandpass and comparing these frequencies corresponding to the spectral lines with larger amplitude in envelope spectrum with the fault characteristic frequency to judge the fault kind.


## 5. Validations for the Proposed Approach

In the following content, the proposed approach, the improved fast spectral kurtogram based on EEMD (EEMD-FSK) [13], and the improved kurtogram based on WPT (WPT-FSK) [16] are used to analyze a simulation signal, an artificial single fault signal, an artificial compound fault signal, and a life cycle fault signal, respectively.


### 5.1. Simulation Signal Case

The rolling bearing fault model [30, 31] is conducted for simulating the impact produced by the inner ring defect, and the strong white noise is overlaid for simulating the early fault. The mathematical model of simulation signal is as follows:(21)xt=st+nt=∑tAiht−iT−τi+nt,ht=exp−Ctsin2πfnt,Ai=A0sin2πfrt,where s(t) refers to the periodic impact component, n(t) refers to the Gaussian white noise, the amplitude refers to A0 = 0.5, τi = 0 refers to the slight fluctuation of the i-th time impact relative to periodic T, the attenuation coefficient refers to C = 1000, the resonance frequency refers to fn = 3 kHz, the shaft rotation frequency refers to fr = 20 Hz, the fault feature frequency refers to fi = 1/T = 130 Hz, the random fluctuation obeys the zero-mean normal distribution, the standard deviation is 0.5% of the shaft rotation frequency, the SNR of the simulation signal is −12 dB, the sampling frequency refers to fs = 8192 Hz, and the sampling point number refers to N = 4096. Figure 5(a) presents the TW of the simulation signal, where the periodic impact components (red waveform lines) are masked by strong noise and thus difficult to observe fault information. Envelope demodulation is conducted on the simulation signal. Figure 5(b) illustrates the ES, where the IRFF fi is detected, but its amplitude is not prominent comparing with noise.Figure 5
The simulation signal of inner ring defect: (a) TW; (b) ES.
(a)
(b)The proposed approach is conducted to analyze the simulation signal. The optimization range of BW is set asσ ∈ [0, 390], and the optimization range of CF is set as γ ∈ [180, 3916]. WCA algorithm is initialized and 50 iterations are performed. Figure 6(a) shows the optimization curve of function value with evolution algebra. It can be seen that the minimum value is 29.75, obtained after the 8 times’ iteration calculation. The parameter combination consisting of CF and BW corresponding to the minimal value is chosen to construct the optimal ARLW filter. Figure 6(b) illustrates the FS of the optimal ARLW filter, where the CF = 3046 Hz and BW = 1349 Hz. The TW of the narrowband signal acquired from the optimal ARLW filter is displayed as Figure 6(c). Figures 6(d) and 6(e) depict the FS and ES of the filtered signal, respectively. Figure 6(d) shows that the CF of the optimal resonance frequency band obtained by adaptive optimization is basically consistent with the resonance frequency of the simulation signal, and the BW is 3 times significantly greater than fault feature frequency. As shown in Figure 6(e), the IRFF fi and its frequency doubling 2fi−3fi are detected productively.Figure 6
Diagnosis results of the simulation signal by the proposed approach: (a) iteration curves for determining the optimal ARLW parameters; (b) FS of the optimal ARLW for detecting the inner ring defect information; (c) TW of the filtered signal obtained by (b); (d) FS of (c); (e) ES of (d).
(a)
(b)
(c)
(d)
(e)The EEMD-FSK approach is conducted to analyze the simulation signal. According to the diagnostic process in [13], the simulation signal is firstly decomposed into a group of IMFs by EEMD. The two IMFs with obvious impact characteristics are chosen by the measure factor to reconstruct signal. Figure 7(a) presents the reconstruct signal, and Figure 7(b) presents its ES. As seen, no useful information is detected. Then, the reconstruct signal is analyzed by FSK, and the kurtogram is displayed as Figure 7(c). The color block of kurtogram represents the kurtosis of the narrowband signal obtained by different bandpass filters. As seen, the maximal kurtosis is 0.2, corresponding to the bandpass filter (CF = 384 Hz, level = 6), which is circled by the red dotted line. Figure 7(d) depicts the filtered signal acquired by FSK, where the impact component in the filtered signal has no obvious periodicity. Envelope demodulation is conducted on the filtered signal. Figure 7(e) indicates the ES of the filtered signal, where the IRFF fi cannot be detected. The impact feature shown in Figure 7(d) is not caused by rolling bearing fault.Figure 7
Diagnosis results of the simulation signal by EEMD-FSK: (a) TW of the reconstruction signal obtained by EEMD; (b) ES of (a); (c) kurtogram; (d) TW of the filtered signal obtained by FSK; (e) ES of (d).
(a)
(b)
(c)
(d)
(e)The WPT-FSK approach is conducted to analyze the simulation signal. According to the diagnostic process in [16], a number of distinct frequency band signals are obtained after the initial vibration signal is processed by WPT algorithm. The kurtosis value of different frequency band signals are calculated and then presented in the kurtgram, where the lateral and vertical axes represent the frequency and the decomposition level of WPT, respectively. Each node of kurtgram represents the narrowband signal obtained by WPT decomposition, and these narrowband signals are in the different frequency bands. The signal corresponding to the node with the maximal kurtosis is chosen for envelope demodulation. Figure 8(a) presents that node (4, 13) has the maximal kurtosis. Figures 8(b) and 8(c) depict the TW and ES of the frequency band signal corresponding to node (4, 13). As seen, although the impact feature is seen in TW, there is no IRFF fi in ES. The impact feature shown in Figure 8(b) is not caused by rolling bearing fault.Figure 8
Diagnosis results of the simulation signal by WPT-FSK: (a) kurtogram; (b) TW of the filtered signal with the maximal kurtosis; (c) ES of (b).
(a)
(b)
(c)In the simulation signal case above, the proposed approach can efficiently detect the fault information and accurately determine that the rolling bearing is under inner ring defect, while the EEMD-FSK approach and the WPT-FSK approach cannot realize such functionality.


### 5.2. Artificial Single Fault Signal Case

Case Western Reserve University discloses the rolling bearing vibration signal under different operation conditions to the outside world [32]. Figure 9 depicts the overall appearance of the test platform. The experiment objective is SKF 6203-2RS rolling bearing, and Table 1 illustrates its structural parameters. Aiming at simulating the early fault signal of the rolling element, a single-point dent is machined on the surface of the rolling element by adopting electron discharge machining. The damaged diameters are divided into 0.1778 mm, 0.3556 mm, and 0.5334 mm. To show the superiority of the proposed approach, the vibration data corresponding to the least degree (0.1778 mm) of fault are selected for analysis. The shaft rotary speed refers to n = 1478 r/min, acceleration sensors collect the experimental data, and the sampling frequency refers to fs = 12000 Hz. The parameters shown in Table 1 and the shaft rotary speed are introduced into Equation (20), where the REFF fe ≈ 118 Hz can be obtained.Figure 9
Test platform of Case Western Reserve University.Table 1
Structural parameters of SKF6023-2RS.
Ball diameter
Pitch diameter
Ball number
Contact angle
6.7 mm
28.5 mm
8
0°The TW of the single fault experimental signal is displayed as Figure10(a), where no obvious periodic impact characteristics can be seen. A further envelope demodulation is conducted on of the analyzed signal. There are no frequency components related to rolling element fault in the ES (Figure 10(b)).Figure 10
The single fault experimental signal: (a) TW; (b) ES.
(a)
(b)The proposed approach is conducted to analyze the single fault experimental signal. The optimization range of BW is set asσ ∈ [357, 12000], and the optimization range of CF is set as γ ∈ [178.5, 11821.5]. WCA is initialized and 50 iterations are performed. Figure 11(a) shows the optimization curve of the function value with evolution algebra. It can be seen that the minimal value is 118.2 obtained after the 8 times’ iteration calculation. The parameter combination consisting of CF and BW corresponding to the minimal value is chosen to construct the optimal ARLW filter. Figure 11(b) illustrates the FS of the optimal ARLW filter, where the CF = 4390 Hz and BW = 911 Hz. The TW of the filtered signal acquired from the optimum ARLW filter is displayed as Figure 11(c). As seen, the filtered signal has and obvious periodic impact feature. The FS and the ES of the filtered are presented in Figures 11(d) and 11(e), respectively. As shown in Figure 11(e), the REFF fe and its frequency doubling 2fe are detected productively.Figure 11
Diagnosis results of the single fault experimental signal by the proposed approach: (a) iteration curve for determining the optimal ARLW parameters; (b) FS of the optimal ARLW for detecting the rolling element defect information; (c) TW of the filtered signal obtained by (b); (d) FS of (c); (e) ES of (d).
(a)
(b)
(c)
(d)
(e)Aiming at validating the superiority of the proposed approach, the EEMD-FSK approach and the WPT-FSK approach are conducted to analyze the single fault experimental signal. The reconstruct signal obtained after EEMD processing is displayed as Figure12(a), and its ES is displayed as Figure 12(b). As seen, no useful information is detected. Figure 12(c) presents the kurtogram of the reconstruct signal. As seen, the maximum kurtosis is 0.1, corresponding to the bandpass filter (CF = 2062.5 Hz, level = 7) that is circled by the red dotted line. Figure 12(d) depicts the filtered signal with the maximal kurtosis, where the impact component in the filtered signal has no obvious periodicity. Envelope demodulation is conducted on the filtered signal. Figure 12(e) illustrates the ES of the filtered signal, where the REFF fe cannot be detected. The impact feature shown in Figure 12(d) may be caused by random noise or background noise. Figure 13(a) presents the kurtogram generated after performing WPT-FSK on the original signal, where node (4, 5) has the maximal kurtosis. Figures 13(b) and 13(c) depict the TW and ES of the frequency band signal corresponding to node (4, 5), respectively. As seen, although the impact feature can be seen in TW, there is no REFF fe in ES. The impact feature shown in Figure 13(b) also may be caused by random noise or background noise.Figure 12
Diagnosis results of the single fault experimental signal by EEMD-FSK: (a) TW of the reconstruction signal obtained by EEMD; (b) ES of (a); (c) kurtogram; (d) TW of the filtered signal obtained by FSK; (e) ES of (d).
(a)
(b)
(c)
(d)
(e)Figure 13
Diagnosis results of the single fault experimental signal by WPT-FSK: (a) kurtogram; (b) TW of the filtered signal with the maximal kurtosis; (c) ES of (b).
(a)
(b)
(c)In the artificial single fault signal case above, the proposed approach can efficiently detect the fault information and accurately determine that the rolling bearing is under the rolling element defect, while the EEMD-FSK approach and the WPT-FSK approach cannot realize such functionality.The analysis results of Section5.1 and Section 5.2 illustrate the following conclusions: (1) SEFER index can overcome the influence of random impulse noise, thus it can accurately identify the periodic impact in the signal. Combining ARLW filter with WCA, the optimal resonance frequency band of rolling bearing can be determined fast and accurately. (2) Kurtosis index is easily disturbed by random noise. Although the noise reduction method is the preprocessing of FSK method, the resonance frequency band cannot be accurately found because the random noise is unavoidable.


### 5.3. Artificial Compound Fault Signal Case

The rolling bearing compound fault test is conducted on the QPZZ-II rotating machinery fault test-bed. Figure14(a) depicts the overall appearance of the test-bed. The rotating shaft of the test-bed is connected with the driving motor through a coupling and a pulley. A pressure-loading device and two disks are fixed on the rotating shaft. Normal bearing is fixed on the middle bearing pedestal, and the defective bearing is fixed on the right bearing pedestal. The experiment objective is SKF 6203 rolling bearing, and Table 2 illustrates its structural parameters. The grooves with a width of 0.2 mm and a depth of 1.5 mm are, respectively, machined on the inner and outer ring by wire cutting technology to simulate the compound fault of rolling bearing. The PCB piezoelectric acceleration sensor is fixed on the right bearing pedestal to the collected vibration signal. Figures 14(b) and 14(c) present the damage rolling bearing and the sensor installation location, respectively. During experiment, the driving motor speed refers to n = 1466 r/min and the sampling frequency refers to fs = 12800 Hz. The parameters shown in Table 2 and the shaft rotary speed are introduced into Equation (20); the fault characteristic frequencies of bearing inner ring fi ≈ 132 Hz and outer ring fo ≈ 88 Hz are calculated, respectively.Figure 14
(a) QPZZ test platform; (b) bearing with inner and outer ring defects; (c) acceleration sensor location.
(a)
(b)
(c)Table 2
SKF6205 bearing parameters.
Ball diameter
Pitch diameter
Ball number
Contact angle
7.9 mm
39 mm
9
0°Figures15(a) and 15(b) illustrate the TW and the ES of the experiment signal. As seen, the ORFF fo and its frequency doubling 2fo−5fo are detected effectively, whereas the IRFF fi cannot be extracted.Figure 15
The compound fault experimental signal: (a) TW; (b) ES.
(a)
(b)The proposed approach is conducted to analyze the compound fault experiment signal. Aiming at detecting the inner ring defect, the optimization range of BW is set asσ ∈ [396, 6400], and the optimization range of CF is set as γ ∈ [198, 6202]. Aiming at detecting the outer ring defect, the optimization range of BW is set as σ ∈ [264, 6400], and the optimization range of CF is set as γ ∈ [132, 6268]. WCA algorithm is initialized, and 50 iterations are performed. Figures 16(a) and 16(b) show the convergent curve of the function value with evolution algebra. It can be seen that the minimum value is 201.4 and 4.64 which are obtained after the 16 times’ and 9 times’ iteration calculation, respectively. Figure 16(c) depicts the FS of the optimal ARLW filter for detecting the inner race defect information, where the CF = 5114 Hz and BW = 619 Hz. Figure 16(d) depicts the FS of the optimal ARLW filter for detecting the outer race defect information, where the CF = 1297 Hz and BW = 833 Hz. The TW of the filtered signals containing the most inner and outer ring defect information are presented as Figures 16(e) and 16(f). The FS and the ES of the above two filtered signals are illustrated as Figures 16(g)–16(j). As seen from Figure 16(i), the IRFF fi and its frequency doubling 2fi are identified productively. As seen from Figure 16(j), the ORFF fo and its frequency doubling 2fo−3fo are identified productively.Figure 16
Diagnosis results of the compound fault experimental signal using the approach proposed in the paper: (a), (b) iteration curves for determining the optimal ARLW parameters; (c), (d) FS of the optimal ARLW for detecting the inner and outer ring defect information; (e), (f) TW of the filtered signal obtained by (c) and (d); (g), (h) FS of (c) and (d); (i), (j) ES of (c) and (d).
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)Aiming at validating the superiority of the proposed approach, the EEMD-FSK approach and WPT-FSK approach are conducted to analyze the compound fault experimental signal. The reconstruct signal obtained after EEMD processing is displayed as Figure17(a), and its ES is displayed as Figure 17(b). As seen, the ORFF fo and its frequency doubling 2fo−5fo are detected effectively, whereas the IRFF fi cannot be extracted. Figure 17(c) illustrates the kurtogram of the reconstruct signal, where the maximal kurtosis is 1.4, corresponding to the bandpass filter (CF = 2000 Hz, level = 4.5) which is circled by the red dotted line. The filtered signal with the maximal kurtosis is shown in Figure 17(d), where the impact characteristics are more prominent. Envelope demodulation is conducted on the filtered signal. Figure 17(e) presents the ES of the filtered signal, where the ORFF fo and its frequency doubling 2fo−4fo are identified, whereas the IRFF fi cannot be detected. The spectral line of Figure 17(e) is more concise than that of Figures 15(b) and 17(b). Figure 18(a) displays the kurtogram generated after performing WPT-FSK on the original signal, where node (4, 7) has the maximal kurtosis. The TW and the ES of the frequency band signal corresponding to node (4, 7) are depicted as Figures 18(b) and 18(c), respectively. As seen, the ORFF fo and its frequency doubling 2fo−3fo are detected effectively, whereas the IRFF fi cannot be extracted. According to the above, the filtered signal obtained by the EEMD-FSK or WPT-FSK approach can only increase the SNR, while it still cannot express the periodic impact feature of the weaker fault.Figure 17
Diagnosis results of the compound fault experimental signal by EEMD-FSK: (a) TW of the reconstruction signal obtained by EEMD; (b) ES of (a); (c) kurtogram; (d) TW of the filtered signal obtained by FSK; (e) ES of (d).
(a)
(b)
(c)
(d)
(e)Figure 18
Diagnosis results of the compound fault experimental signal by WPT-FSK: (a) kurtogram; (b) TW of the filtered signal with the maximal kurtosis; (c) ES of (b).
(a)
(b)
(c)In the artificial compound fault signal case above, the proposed approach can efficiently detect the fault information and accurately determine that the rolling bearing is under inner ring and outer ring defect, while the EEMD-FSK approach and the WPT-FSK approach can only determine that the rolling bearing is under the outer ring defect.The analysis result of this section illustrates that the SEFER index with directivity can identify resonance frequency bands caused by different fault types according to different fault characteristic frequencies. However, in the absence of random impact interference, the kurtosis index only can identify the resonance frequency band caused by the periodic impact corresponding to the strong fault in the signal.


### 5.4. Life Cycle Fault Signal Case

The experimental data are obtained from the life cycle acceleration test of rolling bearings in the center of the NSFI/UCR intelligent maintenance system [19]. Figure 19 presents the overall appearance of the test-bed. Four ZA2115 rolling bearings are installed on the rotating shaft of the test bench, and Table 3 illustrates the structural parameters. During experiment, the radial load is about 2671N, the shaft rotary speed refers to n = 1478 r/min, the experimental data are collected by ICP acceleration sensors, and the sampling frequency refers to fs = 20000 Hz. The duration of the experiment is 164 hours, 984 data files are collected, the sampling interval is 10 minutes, and each sampling is 1 second in length. After the test, obvious erosion phenomenon is found on the outer ring of the No. 1 bearing. The parameters shown in Table 3 and the shaft rotary speed are introduced into Equation (20), where the ORFF fo ≈ 148 Hz can be obtained.Figure 19
Schematic diagram of experiment platform.Table 3
Bearing structure factor of ZA2115.
Ball diameter
Pitch diameter
Ball number
Contact angle
8.4 mm
71.5 mm
16
15.17°Root mean square (RMS) is a useful index in the condition monitoring of mechanical equipment. The RMS variation tendency of the bearing life cycle (0∼9790 min) is shown in Figure20. As seen, after about 5100 minutes, the RMS began to increase, but the amplitude of fluctuation was not large. This stage is generally called the early stage of bearing fault. After about 7020 minutes, the RMS began to change significantly and reached the extreme value at 9790 min. In this stage, the bearing fault gradually aggravated until the limit of life.Figure 20
Bearing fault tendency chart.Aiming at proving verifying the validity of the proposed approach for the early fault of rolling bearing, the 530th data file (measured at 5300 minute) is selected for analysis. The life cycle of the bearing is about 7 days, and the collection time of the selected data file is 3 days earlier than that of the final test shutdown. The data points of the analyzed signal are 4096. Figures21(a) and 21(b) show the TW and ES of the analyzed signal, respectively, where no periodic impact and no fault feature can be seen.Figure 21
The life cycle fault signal: (a) TW; (b) ES.
(a)
(b)The proposed approach is conducted to analyze the life cycle fault signal. The optimization range of BW is set asσ ∈ [0, 200], and the optimization range of CF is set as γ ∈ [0, 100]. WCA algorithm is initialized, and 50 iterations are performed. Figure 22(a) shows the convergent curve of the function value versus the iterative number. It can be seen that the minimum value is 645.5 obtained after the 7 times’ iteration calculation. The parameter combination consisting of CF and BW corresponding to the minimal value is chosen to construct the optimal ARLW filter. Figure 22(b) illustrates the FS of the optimal ARLW filter, where the CF = 8370 Hz and BW = 2869 Hz. The TW of the filtered signal acquired from the optimal ARLW filter is displayed as Figure 22(c), where the periodic impact characteristics can be seen. Figures 22(d) and 22(e) display the FS and ES of the filtered signal, respectively. As shown in Figure 22(e), the ORFF fo and its frequency doubling 2fo−3fo are detected productively.Figure 22
Diagnosis results of the life cycle fault using the approach proposed in the paper: (a) iteration curves for determining the optimal ARLW parameters; (b) FS of the optimal ARLW for detecting the outer ring defect information; (c) TW of the filtered signal obtained by (b); (d) FS of (c); (e) ES of (d).
(a)
(b)
(c)
(d)
(e)Aiming at validating the superiority of the proposed approach, the EEMD-FSK approach and the WPT-FSK approach are conducted to analyze the life cycle fault signal. The reconstruct signal obtained after EEMD processing is displayed as Figure23(a), and its ES is presented as Figure 23(b). As seen, no useful information is detected. Figure 23(c) illustrates the kurtogram of the reconstruct signal. As seen, the maximal kurtosis is 0.2, corresponding to the bandpass filter (CF = 3125 Hz, level = 4.5), which is circled by the red dotted line. Figure 23(d) depicts the filtered signal with the maximal kurtosis, where the impact component in the filtered signal has no obvious periodicity. Envelope demodulation is conducted on the filtered signal. Figure 23(e) illustrates the ES of the filtered signal, where the ORFF fo cannot be detected. Figure 24(a) displays the kurtogram generated after WPT-FSK performing on the original signal, where node (4, 5) has the maximum kurtosis. The TW and ES of the frequency band signal corresponding to node (4, 5) are depicted as Figures 24(b) and 24(c), respectively. As seen, although the impact feature can be seen in TW, there is no ORFF fo in ES.Figure 23
Diagnosis results of the life cycle fault signal by EEMD-FSK: (a) TW of the reconstruction signal obtained by EEMD; (b) ES of (a); (c) kurtogram; (d) TW of the filtered signal obtained by FSK; (e) ES of (d).
(a)
(b)
(c)
(d)
(e)Figure 24
Diagnosis results of the life cycle fault signal by WPT-FSK: (a) kurtogram; (b) TW of the filtered signal with the maximal kurtosis; (c) ES of (b).
(a)
(b)
(c)In the life cycle fault signal case above, the proposed approach can efficiently detect the fault information and accurately determine that the rolling bearing is under outer ring defective, while the EEMD-FSK approach and the WPT-FSK approach cannot realize such functionality.Compared with the simulation fault signal and artificial fault signal, the life cycle fault signal can better reflect the evolution process of bearing fault occurrence and development in actual working conditions. Therefore, the analysis result of this section can further reflect the practicability of the proposed approach in the early fault detection of rolling bearing.


## 5.1. Simulation Signal Case

The rolling bearing fault model [30, 31] is conducted for simulating the impact produced by the inner ring defect, and the strong white noise is overlaid for simulating the early fault. The mathematical model of simulation signal is as follows:(21)xt=st+nt=∑tAiht−iT−τi+nt,ht=exp−Ctsin2πfnt,Ai=A0sin2πfrt,where s(t) refers to the periodic impact component, n(t) refers to the Gaussian white noise, the amplitude refers to A0 = 0.5, τi = 0 refers to the slight fluctuation of the i-th time impact relative to periodic T, the attenuation coefficient refers to C = 1000, the resonance frequency refers to fn = 3 kHz, the shaft rotation frequency refers to fr = 20 Hz, the fault feature frequency refers to fi = 1/T = 130 Hz, the random fluctuation obeys the zero-mean normal distribution, the standard deviation is 0.5% of the shaft rotation frequency, the SNR of the simulation signal is −12 dB, the sampling frequency refers to fs = 8192 Hz, and the sampling point number refers to N = 4096. Figure 5(a) presents the TW of the simulation signal, where the periodic impact components (red waveform lines) are masked by strong noise and thus difficult to observe fault information. Envelope demodulation is conducted on the simulation signal. Figure 5(b) illustrates the ES, where the IRFF fi is detected, but its amplitude is not prominent comparing with noise.Figure 5
The simulation signal of inner ring defect: (a) TW; (b) ES.
(a)
(b)The proposed approach is conducted to analyze the simulation signal. The optimization range of BW is set asσ ∈ [0, 390], and the optimization range of CF is set as γ ∈ [180, 3916]. WCA algorithm is initialized and 50 iterations are performed. Figure 6(a) shows the optimization curve of function value with evolution algebra. It can be seen that the minimum value is 29.75, obtained after the 8 times’ iteration calculation. The parameter combination consisting of CF and BW corresponding to the minimal value is chosen to construct the optimal ARLW filter. Figure 6(b) illustrates the FS of the optimal ARLW filter, where the CF = 3046 Hz and BW = 1349 Hz. The TW of the narrowband signal acquired from the optimal ARLW filter is displayed as Figure 6(c). Figures 6(d) and 6(e) depict the FS and ES of the filtered signal, respectively. Figure 6(d) shows that the CF of the optimal resonance frequency band obtained by adaptive optimization is basically consistent with the resonance frequency of the simulation signal, and the BW is 3 times significantly greater than fault feature frequency. As shown in Figure 6(e), the IRFF fi and its frequency doubling 2fi−3fi are detected productively.Figure 6
Diagnosis results of the simulation signal by the proposed approach: (a) iteration curves for determining the optimal ARLW parameters; (b) FS of the optimal ARLW for detecting the inner ring defect information; (c) TW of the filtered signal obtained by (b); (d) FS of (c); (e) ES of (d).
(a)
(b)
(c)
(d)
(e)The EEMD-FSK approach is conducted to analyze the simulation signal. According to the diagnostic process in [13], the simulation signal is firstly decomposed into a group of IMFs by EEMD. The two IMFs with obvious impact characteristics are chosen by the measure factor to reconstruct signal. Figure 7(a) presents the reconstruct signal, and Figure 7(b) presents its ES. As seen, no useful information is detected. Then, the reconstruct signal is analyzed by FSK, and the kurtogram is displayed as Figure 7(c). The color block of kurtogram represents the kurtosis of the narrowband signal obtained by different bandpass filters. As seen, the maximal kurtosis is 0.2, corresponding to the bandpass filter (CF = 384 Hz, level = 6), which is circled by the red dotted line. Figure 7(d) depicts the filtered signal acquired by FSK, where the impact component in the filtered signal has no obvious periodicity. Envelope demodulation is conducted on the filtered signal. Figure 7(e) indicates the ES of the filtered signal, where the IRFF fi cannot be detected. The impact feature shown in Figure 7(d) is not caused by rolling bearing fault.Figure 7
Diagnosis results of the simulation signal by EEMD-FSK: (a) TW of the reconstruction signal obtained by EEMD; (b) ES of (a); (c) kurtogram; (d) TW of the filtered signal obtained by FSK; (e) ES of (d).
(a)
(b)
(c)
(d)
(e)The WPT-FSK approach is conducted to analyze the simulation signal. According to the diagnostic process in [16], a number of distinct frequency band signals are obtained after the initial vibration signal is processed by WPT algorithm. The kurtosis value of different frequency band signals are calculated and then presented in the kurtgram, where the lateral and vertical axes represent the frequency and the decomposition level of WPT, respectively. Each node of kurtgram represents the narrowband signal obtained by WPT decomposition, and these narrowband signals are in the different frequency bands. The signal corresponding to the node with the maximal kurtosis is chosen for envelope demodulation. Figure 8(a) presents that node (4, 13) has the maximal kurtosis. Figures 8(b) and 8(c) depict the TW and ES of the frequency band signal corresponding to node (4, 13). As seen, although the impact feature is seen in TW, there is no IRFF fi in ES. The impact feature shown in Figure 8(b) is not caused by rolling bearing fault.Figure 8
Diagnosis results of the simulation signal by WPT-FSK: (a) kurtogram; (b) TW of the filtered signal with the maximal kurtosis; (c) ES of (b).
(a)
(b)
(c)In the simulation signal case above, the proposed approach can efficiently detect the fault information and accurately determine that the rolling bearing is under inner ring defect, while the EEMD-FSK approach and the WPT-FSK approach cannot realize such functionality.


## 5.2. Artificial Single Fault Signal Case

Case Western Reserve University discloses the rolling bearing vibration signal under different operation conditions to the outside world [32]. Figure 9 depicts the overall appearance of the test platform. The experiment objective is SKF 6203-2RS rolling bearing, and Table 1 illustrates its structural parameters. Aiming at simulating the early fault signal of the rolling element, a single-point dent is machined on the surface of the rolling element by adopting electron discharge machining. The damaged diameters are divided into 0.1778 mm, 0.3556 mm, and 0.5334 mm. To show the superiority of the proposed approach, the vibration data corresponding to the least degree (0.1778 mm) of fault are selected for analysis. The shaft rotary speed refers to n = 1478 r/min, acceleration sensors collect the experimental data, and the sampling frequency refers to fs = 12000 Hz. The parameters shown in Table 1 and the shaft rotary speed are introduced into Equation (20), where the REFF fe ≈ 118 Hz can be obtained.Figure 9
Test platform of Case Western Reserve University.Table 1
Structural parameters of SKF6023-2RS.
Ball diameter
Pitch diameter
Ball number
Contact angle
6.7 mm
28.5 mm
8
0°The TW of the single fault experimental signal is displayed as Figure10(a), where no obvious periodic impact characteristics can be seen. A further envelope demodulation is conducted on of the analyzed signal. There are no frequency components related to rolling element fault in the ES (Figure 10(b)).Figure 10
The single fault experimental signal: (a) TW; (b) ES.
(a)
(b)The proposed approach is conducted to analyze the single fault experimental signal. The optimization range of BW is set asσ ∈ [357, 12000], and the optimization range of CF is set as γ ∈ [178.5, 11821.5]. WCA is initialized and 50 iterations are performed. Figure 11(a) shows the optimization curve of the function value with evolution algebra. It can be seen that the minimal value is 118.2 obtained after the 8 times’ iteration calculation. The parameter combination consisting of CF and BW corresponding to the minimal value is chosen to construct the optimal ARLW filter. Figure 11(b) illustrates the FS of the optimal ARLW filter, where the CF = 4390 Hz and BW = 911 Hz. The TW of the filtered signal acquired from the optimum ARLW filter is displayed as Figure 11(c). As seen, the filtered signal has and obvious periodic impact feature. The FS and the ES of the filtered are presented in Figures 11(d) and 11(e), respectively. As shown in Figure 11(e), the REFF fe and its frequency doubling 2fe are detected productively.Figure 11
Diagnosis results of the single fault experimental signal by the proposed approach: (a) iteration curve for determining the optimal ARLW parameters; (b) FS of the optimal ARLW for detecting the rolling element defect information; (c) TW of the filtered signal obtained by (b); (d) FS of (c); (e) ES of (d).
(a)
(b)
(c)
(d)
(e)Aiming at validating the superiority of the proposed approach, the EEMD-FSK approach and the WPT-FSK approach are conducted to analyze the single fault experimental signal. The reconstruct signal obtained after EEMD processing is displayed as Figure12(a), and its ES is displayed as Figure 12(b). As seen, no useful information is detected. Figure 12(c) presents the kurtogram of the reconstruct signal. As seen, the maximum kurtosis is 0.1, corresponding to the bandpass filter (CF = 2062.5 Hz, level = 7) that is circled by the red dotted line. Figure 12(d) depicts the filtered signal with the maximal kurtosis, where the impact component in the filtered signal has no obvious periodicity. Envelope demodulation is conducted on the filtered signal. Figure 12(e) illustrates the ES of the filtered signal, where the REFF fe cannot be detected. The impact feature shown in Figure 12(d) may be caused by random noise or background noise. Figure 13(a) presents the kurtogram generated after performing WPT-FSK on the original signal, where node (4, 5) has the maximal kurtosis. Figures 13(b) and 13(c) depict the TW and ES of the frequency band signal corresponding to node (4, 5), respectively. As seen, although the impact feature can be seen in TW, there is no REFF fe in ES. The impact feature shown in Figure 13(b) also may be caused by random noise or background noise.Figure 12
Diagnosis results of the single fault experimental signal by EEMD-FSK: (a) TW of the reconstruction signal obtained by EEMD; (b) ES of (a); (c) kurtogram; (d) TW of the filtered signal obtained by FSK; (e) ES of (d).
(a)
(b)
(c)
(d)
(e)Figure 13
Diagnosis results of the single fault experimental signal by WPT-FSK: (a) kurtogram; (b) TW of the filtered signal with the maximal kurtosis; (c) ES of (b).
(a)
(b)
(c)In the artificial single fault signal case above, the proposed approach can efficiently detect the fault information and accurately determine that the rolling bearing is under the rolling element defect, while the EEMD-FSK approach and the WPT-FSK approach cannot realize such functionality.The analysis results of Section5.1 and Section 5.2 illustrate the following conclusions: (1) SEFER index can overcome the influence of random impulse noise, thus it can accurately identify the periodic impact in the signal. Combining ARLW filter with WCA, the optimal resonance frequency band of rolling bearing can be determined fast and accurately. (2) Kurtosis index is easily disturbed by random noise. Although the noise reduction method is the preprocessing of FSK method, the resonance frequency band cannot be accurately found because the random noise is unavoidable.


## 5.3. Artificial Compound Fault Signal Case

The rolling bearing compound fault test is conducted on the QPZZ-II rotating machinery fault test-bed. Figure14(a) depicts the overall appearance of the test-bed. The rotating shaft of the test-bed is connected with the driving motor through a coupling and a pulley. A pressure-loading device and two disks are fixed on the rotating shaft. Normal bearing is fixed on the middle bearing pedestal, and the defective bearing is fixed on the right bearing pedestal. The experiment objective is SKF 6203 rolling bearing, and Table 2 illustrates its structural parameters. The grooves with a width of 0.2 mm and a depth of 1.5 mm are, respectively, machined on the inner and outer ring by wire cutting technology to simulate the compound fault of rolling bearing. The PCB piezoelectric acceleration sensor is fixed on the right bearing pedestal to the collected vibration signal. Figures 14(b) and 14(c) present the damage rolling bearing and the sensor installation location, respectively. During experiment, the driving motor speed refers to n = 1466 r/min and the sampling frequency refers to fs = 12800 Hz. The parameters shown in Table 2 and the shaft rotary speed are introduced into Equation (20); the fault characteristic frequencies of bearing inner ring fi ≈ 132 Hz and outer ring fo ≈ 88 Hz are calculated, respectively.Figure 14
(a) QPZZ test platform; (b) bearing with inner and outer ring defects; (c) acceleration sensor location.
(a)
(b)
(c)Table 2
SKF6205 bearing parameters.
Ball diameter
Pitch diameter
Ball number
Contact angle
7.9 mm
39 mm
9
0°Figures15(a) and 15(b) illustrate the TW and the ES of the experiment signal. As seen, the ORFF fo and its frequency doubling 2fo−5fo are detected effectively, whereas the IRFF fi cannot be extracted.Figure 15
The compound fault experimental signal: (a) TW; (b) ES.
(a)
(b)The proposed approach is conducted to analyze the compound fault experiment signal. Aiming at detecting the inner ring defect, the optimization range of BW is set asσ ∈ [396, 6400], and the optimization range of CF is set as γ ∈ [198, 6202]. Aiming at detecting the outer ring defect, the optimization range of BW is set as σ ∈ [264, 6400], and the optimization range of CF is set as γ ∈ [132, 6268]. WCA algorithm is initialized, and 50 iterations are performed. Figures 16(a) and 16(b) show the convergent curve of the function value with evolution algebra. It can be seen that the minimum value is 201.4 and 4.64 which are obtained after the 16 times’ and 9 times’ iteration calculation, respectively. Figure 16(c) depicts the FS of the optimal ARLW filter for detecting the inner race defect information, where the CF = 5114 Hz and BW = 619 Hz. Figure 16(d) depicts the FS of the optimal ARLW filter for detecting the outer race defect information, where the CF = 1297 Hz and BW = 833 Hz. The TW of the filtered signals containing the most inner and outer ring defect information are presented as Figures 16(e) and 16(f). The FS and the ES of the above two filtered signals are illustrated as Figures 16(g)–16(j). As seen from Figure 16(i), the IRFF fi and its frequency doubling 2fi are identified productively. As seen from Figure 16(j), the ORFF fo and its frequency doubling 2fo−3fo are identified productively.Figure 16
Diagnosis results of the compound fault experimental signal using the approach proposed in the paper: (a), (b) iteration curves for determining the optimal ARLW parameters; (c), (d) FS of the optimal ARLW for detecting the inner and outer ring defect information; (e), (f) TW of the filtered signal obtained by (c) and (d); (g), (h) FS of (c) and (d); (i), (j) ES of (c) and (d).
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)Aiming at validating the superiority of the proposed approach, the EEMD-FSK approach and WPT-FSK approach are conducted to analyze the compound fault experimental signal. The reconstruct signal obtained after EEMD processing is displayed as Figure17(a), and its ES is displayed as Figure 17(b). As seen, the ORFF fo and its frequency doubling 2fo−5fo are detected effectively, whereas the IRFF fi cannot be extracted. Figure 17(c) illustrates the kurtogram of the reconstruct signal, where the maximal kurtosis is 1.4, corresponding to the bandpass filter (CF = 2000 Hz, level = 4.5) which is circled by the red dotted line. The filtered signal with the maximal kurtosis is shown in Figure 17(d), where the impact characteristics are more prominent. Envelope demodulation is conducted on the filtered signal. Figure 17(e) presents the ES of the filtered signal, where the ORFF fo and its frequency doubling 2fo−4fo are identified, whereas the IRFF fi cannot be detected. The spectral line of Figure 17(e) is more concise than that of Figures 15(b) and 17(b). Figure 18(a) displays the kurtogram generated after performing WPT-FSK on the original signal, where node (4, 7) has the maximal kurtosis. The TW and the ES of the frequency band signal corresponding to node (4, 7) are depicted as Figures 18(b) and 18(c), respectively. As seen, the ORFF fo and its frequency doubling 2fo−3fo are detected effectively, whereas the IRFF fi cannot be extracted. According to the above, the filtered signal obtained by the EEMD-FSK or WPT-FSK approach can only increase the SNR, while it still cannot express the periodic impact feature of the weaker fault.Figure 17
Diagnosis results of the compound fault experimental signal by EEMD-FSK: (a) TW of the reconstruction signal obtained by EEMD; (b) ES of (a); (c) kurtogram; (d) TW of the filtered signal obtained by FSK; (e) ES of (d).
(a)
(b)
(c)
(d)
(e)Figure 18
Diagnosis results of the compound fault experimental signal by WPT-FSK: (a) kurtogram; (b) TW of the filtered signal with the maximal kurtosis; (c) ES of (b).
(a)
(b)
(c)In the artificial compound fault signal case above, the proposed approach can efficiently detect the fault information and accurately determine that the rolling bearing is under inner ring and outer ring defect, while the EEMD-FSK approach and the WPT-FSK approach can only determine that the rolling bearing is under the outer ring defect.The analysis result of this section illustrates that the SEFER index with directivity can identify resonance frequency bands caused by different fault types according to different fault characteristic frequencies. However, in the absence of random impact interference, the kurtosis index only can identify the resonance frequency band caused by the periodic impact corresponding to the strong fault in the signal.


## 5.4. Life Cycle Fault Signal Case

The experimental data are obtained from the life cycle acceleration test of rolling bearings in the center of the NSFI/UCR intelligent maintenance system [19]. Figure 19 presents the overall appearance of the test-bed. Four ZA2115 rolling bearings are installed on the rotating shaft of the test bench, and Table 3 illustrates the structural parameters. During experiment, the radial load is about 2671N, the shaft rotary speed refers to n = 1478 r/min, the experimental data are collected by ICP acceleration sensors, and the sampling frequency refers to fs = 20000 Hz. The duration of the experiment is 164 hours, 984 data files are collected, the sampling interval is 10 minutes, and each sampling is 1 second in length. After the test, obvious erosion phenomenon is found on the outer ring of the No. 1 bearing. The parameters shown in Table 3 and the shaft rotary speed are introduced into Equation (20), where the ORFF fo ≈ 148 Hz can be obtained.Figure 19
Schematic diagram of experiment platform.Table 3
Bearing structure factor of ZA2115.
Ball diameter
Pitch diameter
Ball number
Contact angle
8.4 mm
71.5 mm
16
15.17°Root mean square (RMS) is a useful index in the condition monitoring of mechanical equipment. The RMS variation tendency of the bearing life cycle (0∼9790 min) is shown in Figure20. As seen, after about 5100 minutes, the RMS began to increase, but the amplitude of fluctuation was not large. This stage is generally called the early stage of bearing fault. After about 7020 minutes, the RMS began to change significantly and reached the extreme value at 9790 min. In this stage, the bearing fault gradually aggravated until the limit of life.Figure 20
Bearing fault tendency chart.Aiming at proving verifying the validity of the proposed approach for the early fault of rolling bearing, the 530th data file (measured at 5300 minute) is selected for analysis. The life cycle of the bearing is about 7 days, and the collection time of the selected data file is 3 days earlier than that of the final test shutdown. The data points of the analyzed signal are 4096. Figures21(a) and 21(b) show the TW and ES of the analyzed signal, respectively, where no periodic impact and no fault feature can be seen.Figure 21
The life cycle fault signal: (a) TW; (b) ES.
(a)
(b)The proposed approach is conducted to analyze the life cycle fault signal. The optimization range of BW is set asσ ∈ [0, 200], and the optimization range of CF is set as γ ∈ [0, 100]. WCA algorithm is initialized, and 50 iterations are performed. Figure 22(a) shows the convergent curve of the function value versus the iterative number. It can be seen that the minimum value is 645.5 obtained after the 7 times’ iteration calculation. The parameter combination consisting of CF and BW corresponding to the minimal value is chosen to construct the optimal ARLW filter. Figure 22(b) illustrates the FS of the optimal ARLW filter, where the CF = 8370 Hz and BW = 2869 Hz. The TW of the filtered signal acquired from the optimal ARLW filter is displayed as Figure 22(c), where the periodic impact characteristics can be seen. Figures 22(d) and 22(e) display the FS and ES of the filtered signal, respectively. As shown in Figure 22(e), the ORFF fo and its frequency doubling 2fo−3fo are detected productively.Figure 22
Diagnosis results of the life cycle fault using the approach proposed in the paper: (a) iteration curves for determining the optimal ARLW parameters; (b) FS of the optimal ARLW for detecting the outer ring defect information; (c) TW of the filtered signal obtained by (b); (d) FS of (c); (e) ES of (d).
(a)
(b)
(c)
(d)
(e)Aiming at validating the superiority of the proposed approach, the EEMD-FSK approach and the WPT-FSK approach are conducted to analyze the life cycle fault signal. The reconstruct signal obtained after EEMD processing is displayed as Figure23(a), and its ES is presented as Figure 23(b). As seen, no useful information is detected. Figure 23(c) illustrates the kurtogram of the reconstruct signal. As seen, the maximal kurtosis is 0.2, corresponding to the bandpass filter (CF = 3125 Hz, level = 4.5), which is circled by the red dotted line. Figure 23(d) depicts the filtered signal with the maximal kurtosis, where the impact component in the filtered signal has no obvious periodicity. Envelope demodulation is conducted on the filtered signal. Figure 23(e) illustrates the ES of the filtered signal, where the ORFF fo cannot be detected. Figure 24(a) displays the kurtogram generated after WPT-FSK performing on the original signal, where node (4, 5) has the maximum kurtosis. The TW and ES of the frequency band signal corresponding to node (4, 5) are depicted as Figures 24(b) and 24(c), respectively. As seen, although the impact feature can be seen in TW, there is no ORFF fo in ES.Figure 23
Diagnosis results of the life cycle fault signal by EEMD-FSK: (a) TW of the reconstruction signal obtained by EEMD; (b) ES of (a); (c) kurtogram; (d) TW of the filtered signal obtained by FSK; (e) ES of (d).
(a)
(b)
(c)
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(e)Figure 24
Diagnosis results of the life cycle fault signal by WPT-FSK: (a) kurtogram; (b) TW of the filtered signal with the maximal kurtosis; (c) ES of (b).
(a)
(b)
(c)In the life cycle fault signal case above, the proposed approach can efficiently detect the fault information and accurately determine that the rolling bearing is under outer ring defective, while the EEMD-FSK approach and the WPT-FSK approach cannot realize such functionality.Compared with the simulation fault signal and artificial fault signal, the life cycle fault signal can better reflect the evolution process of bearing fault occurrence and development in actual working conditions. Therefore, the analysis result of this section can further reflect the practicability of the proposed approach in the early fault detection of rolling bearing.


## 6. Conclusions

This paper came up with a rolling bearing diagnosis approach based on ARLW filtering with the help of WCA, which can detect the fault information by demodulation analysis of the determined resonance frequency band. The proposed approach has three improvements: (1) ARLW is used as a bandpass filter to process the initial signal, which can filter out noise efficiently and extract more fault information; (2) WCA is employed to optimize the ARLW parameters in parallel adaptively, which can avoid artificial interference and improve robustness; (3) The proposed SEFER index can evaluate the quality of the bandpass filters constructed for different fault types.The proposed approach and the traditional resonance demodulation methods, such as EEMD-FSK approach and WPT-FSK approach, are all conducted to analyze the simulation signal, the artificial single fault signal, the artificial compound fault signal, and the life cycle fault signal. The analysis results of the single fault signal show that the proposed approach can efficiently detect the early fault information, while the EEMD-FSK approach and the WPT-FSK approach cannot realize such functionality. The analysis result of the compound fault signal illustrates that the proposed approach can detect both strong and weak fault information, while the EEMD-FSK approach and the WPT-FSK approach can only detect the strong fault information. The analysis result of the life cycle fault signal further reflects the advantages of the proposed approach in the early fault detection of rolling bearing.In summary, the proposed approach can efficiently detect the fault feature information and accurately determine the rolling bearing fault type, which is valuable in the engineering industry.



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*Source: 7475868-2019-01-14.xml*