# A Parameter Adaptive MOMEDA Method Based on Grasshopper Optimization Algorithm to Extract Fault Features

Authors: ChengJiang Zhou; Jun Ma; Jiande Wu; Zezhong Feng
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

The nonstationary components and noises contained in the bearing vibration signal make it particularly difficult to extract fault features, and minimum entropy deconvolution (MED), maximum correlated kurtosis deconvolution (MCKD), and fast spectral kurtosis (FSK) cannot achieve satisfactory results. However, the filter size and period range of multipoint optimal minimum entropy deconvolution adjusted (MOMEDA) need to be set in advance, so it is difficult to achieve satisfactory filtering results. Aiming at these problems, a parameter adaptive MOMEDA feature extraction method based on grasshopper optimization algorithm (GOA) is proposed. Firstly, the multipoint kurtosis (MKurt) of MOMEDA filtered signal is used as the optimization objective, and the optimal filter size and periodic initial value which matched with the vibration signal can be determined adaptively through multiple iterations of GOA. Secondly, the periodic impact contained in the vibration signal is extracted by the optimized MOMEDA, and the fault features in the impact signal are extracted by Hilbert envelope demodulation. Finally, the simulation signal and bearing signal are processed by the proposed approach. The results show that the introduction of GOA not only solves the problem of parameter selection in MOMEDA, but also achieves better performance compared with other optimization methods. Meanwhile, the feasibility and superiority of the approach are fully proved by comparing it with the three methods MED, MCKD, and FSK after parameter optimization. Therefore, this approach provides a novel way and solution for fault diagnosis of the rolling bearing, gear, and shaft.


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## Body

## 1. Introduction

The bearing is an important rotating part widely used in manufacturing, transportation, aerospace, and other fields, and one of the most vulnerable rotating components. Bearing failure may cause catastrophic damage to the rotating machinery and lead to serious loss of life and property. Therefore, bearing health monitoring and fault diagnosis are particularly important [1]. At present, the time-frequency analysis technology is the most popular feature extraction method. However, the bearing vibration signal contains complex background noise and interference components, and satisfactory feature extraction results cannot be obtained by using the traditional time-frequency analysis methods [2]. Therefore, it is of great significance to explore the simple and effective feature extraction methods.The fault impact and vibration source can be extracted from the vibration signal by Hilbert envelope demodulation [3]. However, this method is only applicable to the single AM-FM signal. Therefore, signal preprocessing is required before demodulation. Signal decomposition and mode selection are the most commonly used preprocessing methods. The empirical mode decomposition (EMD) [4] is the most classical method. In order to prevent mode mixing caused by the discontinuous signal in the screening process, ensemble empirical mode decomposition (EEMD) [5], complete ensemble empirical mode decomposition adaptive noise (CEEMDAN) [6], and other methods were proposed. Zhou et al. [7] first decomposed the bearing signal into several modes by EEMD, then extracted the weighted permutation entropy (PE) of the previous modes, and diagnosed the bearing faults through SVM ensemble classifier. Lv et al. [8] decomposed the bearing signal by CEEMDAN and calculated the improved multivariate multiscale sample entropy of the effective modes, and the method realizes the bearing early fault diagnosis. Inspired by EMD, Smith et al. proposed local mean decomposition (LMD) [9]. In order to improve its mode mixing problem, a variety of improved methods have been proposed, such as ensemble local mean decomposition (ELMD) [10], complete ensemble local mean decomposition (CELMD) [11], etc. Zhang et al. [12] proposed a weak fault diagnosis method based on mask signal and LMD, which can not only suppress the modal mixing of LMD but also effectively extract the fault features. Wang et al. [13] decomposed the bearing signal by ELMD and then selected the effective product functions (PF) according to kurtosis and finally extracted the fault pulse information by using fast spectral kurtosis (FSK). Recently, variational mode decomposition (VMD) [14], empirical wavelet transform (EWT) [15], and other methods have been used for signal decomposition. Wang et al. [16] proposed an improved VMD based on PE and white noise, which can adaptively select the mode number of VMD and is insensitive to noise. Zhang et al. [17] proposed an adaptive VMD based on grasshopper optimization algorithm (GOA), which can adaptively determine the parameters of VMD according to the signal characteristics, and the decomposition performance of the algorithm is improved. Wang et al. [18] proposed the sparsity guided EWT, which can automatically establish the Fourier segments required by empirical wavelet transform. Luo et al. [19] proposed the autoregressive EWT, which detects the boundaries of the autoregressive power spectrum by Burg algorithm. This method avoids the error boundary caused by white noise and nonstationary signal. Through these methods, the signal is decomposed into many subsignals in different frequency bands. Then, some indicators are often used to determine sensitive modes, including correlation coefficient, kurtosis, energy, and entropy. Furthermore, FSK [20], cyclostationary analysis (CA) [21], stochastic resonance (SR) [22], and other signal processing methods have also received increasing attention.The deconvolution method is another effective signal processing method, which can extract the impact components contained in the vibration signal. Minimum entropy deconvolution (MED) and maximum correlation kurtosis deconvolution (MCKD) are the most popular deconvolution methods. Endo et al. used MED to diagnose the fault of rotating machinery for the first time [23]. Li et al. [24] determined the filter size of MED by constructing the power spectral kurtosis index and then extracted the impact components in the filtered signal by time-delay feedback mono-stable stochastic resonance. Abboud et al. [25] enhanced the impact component of the signal through MED, then constructed the optimal frequency band by FSK, and finally extracted fault features by Hilbert envelope demodulation. Although mechanical faults can be detected by MED, the MED filter is not necessarily the optimal filter. In addition, the periodic impact signals contained in the vibration signal cannot be efficiently extracted by MED. Mcdonald et al. [26] proposed MCKD, which can extract a series of fault impacts from the vibration signal. Wan et al. [27] calculated the period of MCKD according to the estimated characteristic frequency and then extracted the early fault features contained in the filtered signal by FSK. The period and filter size of MCKD need to be set in advance, so the method lacks adaptability. In 2017, McDonald et al. [28] proposed the multipoint optimal minimum entropy deconvolution adjusted (MOMEDA). The method does not require the iterative solution in the filtering process, and it can extract the periodic fault impact according to the multipoint kurtosis (MKurt) in the period range [29]. Wang et al. [30] successfully extracted the multiple impact components from the simulation signal and gearbox vibration signal by MOMEDA. Zhu et al. [31] extracted the impact components from the bearing signal by MOMEDA and then extracted the fault features by Teager energy operator (TEO). Wang et al. [32] analyzed the variable speed signal by MOMEDA and resampling order analysis technology and achieved good results. Since the early fault features of the gearbox are often submerged by strong noise, Wang et al. [33] first used MED to suppress the noise and then extracted the composite fault features from the denoised signal by MOMEDA. Cai et al. [34] first used the ELMD and correlation coefficients to obtain the sensitive modes, and they used the combined product function method (CPF) to recombine the PF with the same period. Finally, periodic fault impacts in the combined signal were extracted by MOMEDA. Thus, the MOMEDA has been increasingly used in periodic impact extraction and fault diagnosis.In the deconvolution algorithm, a measure of a signal (also known as a norm) is first defined, and then a filter is constructed to maximize the norm of the filtered signal [28]. The norms of MED, MCKD, and MOMEDA are kurtosis, correlation kurtosis (CK), and multipoint kurtosis (MKurt), respectively. In the methods, MED and MCKD, the filter size and the number of iterations affect the norm of the filtered signal. In addition, the period is the main factor influencing the filtering effect of the MCKD. Therefore, the parameter optimization of deconvolution algorithms is particularly important, and the existing methods are mostly based on kurtosis. Cheng et al. [35] optimized the filter parameters of MED by particle swarm optimization (PSO) and then successfully extracted fault features by optimized MED. Instance et al. [36] optimized the filter size and the number of iterations of MED by shuffled frog leaping algorithm (SFLA), and the method avoids the subjectivity of artificially selected parameters. Miao et al. [37] estimated the period of MCKD by calculating the autocorrelation of envelope signals and improved the resampling process of MCKD. The above methods improve the filtering performance of MED and MCKD and provide the new idea and way for the parameter optimization problem of MOMEDA. The filter size and periodic initial value seriously affect the filtering results of MOMEDA. Meanwhile, the effects of these two parameters on the filtering results are nonlinear and complicated, and the satisfactory filtering results cannot be obtained by selecting random parameters. Therefore, the parameter optimization of MOMEDA is of great significance. Nowadays, many novel optimization methods are attracting attention, such as bat algorithm (BA), flower pollination algorithm (FPA), and cuckoo search algorithm (CS), etc. Saremi et al. [38] proposed the GOA in 2017 and compared it with a variety of optimization algorithms. The results show that GOA has outstanding advantages in optimizing the unimodal function, multimodal function, and compound function. Zhang et al. [17] optimized the mode number and penalty factor of VMD by GOA, which not only improved the performance of VMD but also improved the effects of feature extraction. Barman et al. [39] optimized the kernel function and penalty factor of support vector machine (SVM) by GOA and then used the model to predict the power load in the local climate.To solve the parameter selection problem of MOMEDA, a parameter adaptive MOMEDA method based on GOA is proposed, which combines the advantages of GOA and MOMEDA effectively. Firstly, the algorithm parameters are initialized according to the signal characteristics. Then, we optimize MOMEDA parameters by GOA algorithm. In each iteration, the impact component contained in the mixed signal is extracted by MOMEDA, and the opposite of the MKurt is served as the objective function to optimize the filter parameters. Finally, the optimized MOMEDA is used to extract the impact component from the signals, and the fault feature is extracted through Hilbert envelope demodulation. In order to prove the validity of the proposed approach, the method is used to analyze the simulation signal, Case Western Reserve University (CWRU) bearing dataset [40], and National Aeronautics and Space Administration (NASA) bearing dataset [41]. To highlight the advantages of GOA, the algorithm is compared with PSO, SFLA, and gray wolf optimization algorithm (GWO) [42], and then the proposed approach is compared with the MOMEDA methods with fixed parameters. Finally, the proposed approach is compared with the three methods, MED, MCKD, FSK after parameter optimization. The structure of the paper is shown in Figure 1(b).Figure 1
Parameter optimization process and the structure of the paper.
(a)
The implementation process of GOA (b)
Parameter optimization process of MOMEDAThe main contents of the paper are as follows. Theories of MOMEDA and GOA are introduced briefly in Section2. The process of optimizing MOMEDA filter parameters is introduced in Section 3. The validity of the proposed approach is verified by simulation signal in Section 4. In Section 5, the validity and reliability of the approach are validated by CWRU dataset and NASA dataset. In Section 6, the experimental process and preliminary conclusions are discussed. The conclusions of the experiment are summarized in Section 7.


## 2. A Brief Review of the Diagnostic Techniques

### 2.1. MOMEDA

The response signal can be expressed as follows [33, 34, 43]:(1)yn=hn∗xn+en,where y(n) is the vibration signal, x(n) is the impact component, h(n) is the system frequency response function, and e(n) is the background noise. In order to reconstruct the impact component, the deconvolution process is as follows:(2)xn=fn∗yn=∑l=1Lflyk+L-1.In (2), k=1,2,⋯,N-L, where N represents the signal length and L represents the filter size. In view of the characteristics of the periodic impact signal, the multipoint D-norm is introduced in this method, as shown below.(3)MultiD-Norm=MDNy→,t→=1t→t→Ty→y→.(4)MOMEDA=maxf→⁡MDNy→,t→=maxf→⁡t→Ty→y→,where t→ represents a constant vector and it is used to describe the position and weight of the target impact. The optimal filter can be obtained by solving the maximum of multipoint D-norm, and the deconvolution process also obtains the optimal solution at this time. In order to calculate the extreme value of (4), we first calculate the derivative of the filter coefficients f→=(f1,f2,⋯,fL).(5)ddf→t→Ty→y→=ddf→t1y1y→+ddf→t2y2y→+⋯+ddf→tN-LyN-Ly→.(6)ddf→tkyky→=y→-1tkM→k-y→-3tkykX0y→.(7)M→k=xx+L-1,xx+L-2,⋯,xkT.Therefore, (5) can be transformed into the following form.(8)ddf→t→Ty→y→=y→-1t1M→1+t2M→2+⋯tN-LM→N-L-y→-3t→Ty→X0y→.The equationt1M→1+t2M→2+⋯tN-LM→N-L=X0t→ is used to simplify the formula above, and (5) is equivalent to the following formula.(9)y→-1X0t→-y→-3t→Ty→X0y→=0→.Then, the equationt→Ty→/y→2X0y→=X0t→ can be obtained. Since y→=X0Tf→, assuming (X0X0T)-1 exists, the following formula can be obtained.(10)t→Ty→y→2f→=X0X0T-1X0t→.The special solution of (10) is a set of optimal filters, and the optimal filters can be expressed as follows.(11)f→=X0X0T-1X0t→.(12)X0=xLxL+1xL+2⋯xNxL-1xLxL+1⋯xN-1xL-2xL-1xL⋯xN-2⋮⋮⋮⋱⋮x1x2x3⋯xN-L+1LbyN-L+1.Then, the original impact signal can be reconstructed by the formulay→=X0Tf→.


### 2.2. GOA

GOA is a new bionic algorithm, which simulates the foraging behavior of the grasshoppers [38]. The behavior of grasshoppers has the following characteristics: the nymphs move slowly and the step size of movement is small, while the adults move fast in a wide range of parameter space. Therefore, the search mechanism of the GOA algorithm consists of two tendencies: exploration and exploitation. The algorithm optimizes the objective function by judging the repulsion region, comfort zone, and attraction region formed between the grasshoppers. GOA has the advantages of few parameters and high convergence accuracy. The mathematical model of GOA [38] is as follows.(1) The Initialization of the Population. The grasshoppers random behavior can be represented by the following model.(13)Xi=Si+Gi+Ai,(14)Si=∑j=1,j≠iNsdijdij^Gi=-geg^Ai=uew^,where Xi represents the location of the i-th grasshopper, Si refers to the social interaction, Gi represents the gravity force of the i-th grasshopper, and Ai represents the wind advection. In (14), N represents the number of grasshoppers, dij represents the distance between the i-th and the j-th individual, and dij=xj-xi, dij^=(xj-xi)/dij represents a unit vector. The s function is shown in (15), which represents the social forces. g represents the gravitational constant, and eg^ represents a unit vector pointing to the earth. u is a constant drift, and ew^ represents a unit vector towards the wind direction. (15)sr=fe-r/l-e-r,where f represents the intensity of attraction and l represents the attractive length scale.(2) Location Update. Substituting Si, Gi, and Ai in (13), the specific position of the i-th grasshopper is as follows.(16)Xi=∑j=1,j≠iNsxj-xixj-xidij-geg^+uew^.Equation (16) shows that the grasshoppers move quickly to the comfort zone, at which point the swarm has not found a specified point. Therefore, the model needs to be improved before solving optimization problems. Actually, the model is considered in free space, and the gravity need not be considered (no Gi component). At the same time, we assume that the wind direction is the same as the target direction (Td^). An improved version of (16) is as follows:(17)Xid=c∑j=1,j≠iNcubd-lbd2sxjd-xidxj-xidij+Td^,where ubd and lbd are the upper and lower boundaries of the D-dimensional search space, respectively. Td^ represents the optimal solution found so far in the target direction. The outer c adjusts the exploration and exploitation of the swarm near the target. Inner c reduces the interaction between grasshoppers as the iteration number increases.(3) Balance of Search Ability. The parameter c is used to balance the exploration and exploitation of the swarm, which decreases linearly with the increase of iteration. This mechanism reduces the comfort zone and thus enhances the exploitation ability.(18)c=cmax-mcmax-cminM,where cmax and cmin are the upper and lower boundaries of c, respectively, and they take 1 and 0.00004 in this paper. m represents the current iteration, and M represents the maximum number of iterations.


## 2.1. MOMEDA

The response signal can be expressed as follows [33, 34, 43]:(1)yn=hn∗xn+en,where y(n) is the vibration signal, x(n) is the impact component, h(n) is the system frequency response function, and e(n) is the background noise. In order to reconstruct the impact component, the deconvolution process is as follows:(2)xn=fn∗yn=∑l=1Lflyk+L-1.In (2), k=1,2,⋯,N-L, where N represents the signal length and L represents the filter size. In view of the characteristics of the periodic impact signal, the multipoint D-norm is introduced in this method, as shown below.(3)MultiD-Norm=MDNy→,t→=1t→t→Ty→y→.(4)MOMEDA=maxf→⁡MDNy→,t→=maxf→⁡t→Ty→y→,where t→ represents a constant vector and it is used to describe the position and weight of the target impact. The optimal filter can be obtained by solving the maximum of multipoint D-norm, and the deconvolution process also obtains the optimal solution at this time. In order to calculate the extreme value of (4), we first calculate the derivative of the filter coefficients f→=(f1,f2,⋯,fL).(5)ddf→t→Ty→y→=ddf→t1y1y→+ddf→t2y2y→+⋯+ddf→tN-LyN-Ly→.(6)ddf→tkyky→=y→-1tkM→k-y→-3tkykX0y→.(7)M→k=xx+L-1,xx+L-2,⋯,xkT.Therefore, (5) can be transformed into the following form.(8)ddf→t→Ty→y→=y→-1t1M→1+t2M→2+⋯tN-LM→N-L-y→-3t→Ty→X0y→.The equationt1M→1+t2M→2+⋯tN-LM→N-L=X0t→ is used to simplify the formula above, and (5) is equivalent to the following formula.(9)y→-1X0t→-y→-3t→Ty→X0y→=0→.Then, the equationt→Ty→/y→2X0y→=X0t→ can be obtained. Since y→=X0Tf→, assuming (X0X0T)-1 exists, the following formula can be obtained.(10)t→Ty→y→2f→=X0X0T-1X0t→.The special solution of (10) is a set of optimal filters, and the optimal filters can be expressed as follows.(11)f→=X0X0T-1X0t→.(12)X0=xLxL+1xL+2⋯xNxL-1xLxL+1⋯xN-1xL-2xL-1xL⋯xN-2⋮⋮⋮⋱⋮x1x2x3⋯xN-L+1LbyN-L+1.Then, the original impact signal can be reconstructed by the formulay→=X0Tf→.


## 2.2. GOA

GOA is a new bionic algorithm, which simulates the foraging behavior of the grasshoppers [38]. The behavior of grasshoppers has the following characteristics: the nymphs move slowly and the step size of movement is small, while the adults move fast in a wide range of parameter space. Therefore, the search mechanism of the GOA algorithm consists of two tendencies: exploration and exploitation. The algorithm optimizes the objective function by judging the repulsion region, comfort zone, and attraction region formed between the grasshoppers. GOA has the advantages of few parameters and high convergence accuracy. The mathematical model of GOA [38] is as follows.(1) The Initialization of the Population. The grasshoppers random behavior can be represented by the following model.(13)Xi=Si+Gi+Ai,(14)Si=∑j=1,j≠iNsdijdij^Gi=-geg^Ai=uew^,where Xi represents the location of the i-th grasshopper, Si refers to the social interaction, Gi represents the gravity force of the i-th grasshopper, and Ai represents the wind advection. In (14), N represents the number of grasshoppers, dij represents the distance between the i-th and the j-th individual, and dij=xj-xi, dij^=(xj-xi)/dij represents a unit vector. The s function is shown in (15), which represents the social forces. g represents the gravitational constant, and eg^ represents a unit vector pointing to the earth. u is a constant drift, and ew^ represents a unit vector towards the wind direction. (15)sr=fe-r/l-e-r,where f represents the intensity of attraction and l represents the attractive length scale.(2) Location Update. Substituting Si, Gi, and Ai in (13), the specific position of the i-th grasshopper is as follows.(16)Xi=∑j=1,j≠iNsxj-xixj-xidij-geg^+uew^.Equation (16) shows that the grasshoppers move quickly to the comfort zone, at which point the swarm has not found a specified point. Therefore, the model needs to be improved before solving optimization problems. Actually, the model is considered in free space, and the gravity need not be considered (no Gi component). At the same time, we assume that the wind direction is the same as the target direction (Td^). An improved version of (16) is as follows:(17)Xid=c∑j=1,j≠iNcubd-lbd2sxjd-xidxj-xidij+Td^,where ubd and lbd are the upper and lower boundaries of the D-dimensional search space, respectively. Td^ represents the optimal solution found so far in the target direction. The outer c adjusts the exploration and exploitation of the swarm near the target. Inner c reduces the interaction between grasshoppers as the iteration number increases.(3) Balance of Search Ability. The parameter c is used to balance the exploration and exploitation of the swarm, which decreases linearly with the increase of iteration. This mechanism reduces the comfort zone and thus enhances the exploitation ability.(18)c=cmax-mcmax-cminM,where cmax and cmin are the upper and lower boundaries of c, respectively, and they take 1 and 0.00004 in this paper. m represents the current iteration, and M represents the maximum number of iterations.


## 3. Parameter Adaptive MOMEDA Method Based on GOA

Kurtosis can characterize the distribution density of the impact signal, and it has been widely used in signal processing. MKurt is based on kurtosis; it not only inherits the characteristics of the kurtosis but also extends its definition to multiple pulses at controlled locations [28, 33]. Therefore, MKurt can not only provide location information of fault, but also indicate the energy and period of the fault signal [29]. Meanwhile, a larger MKurt characterizes a larger proportion of the periodic impacts [33]. (19)MKurt=∑n=1N-Ltn22∑n=1N-Ltnyn4∑n=1N-Ltn8∑n=1N-Lyn22.So we expect to get the maximum of MKurt and the parameters of MOMEDA when the maximum is reached in this paper. Since the intention of GOA is to search for the minimum of the objective function [17], the opposite of MKurt is used as the objective function in this paper.(20)fitness=minγ=L,Ti-MKurtL∈1,500,Ti∈50,150,where fitness is the objective function and γ=(L,Ti) is a set of parameters that need to be optimized. L is a positive integer within range L∈[1,500], and Ti takes a value in range Ti∈[50,150]. The steps of the approach are as follows:( 1 ) Input the signal x(t), and initialize the ranges of the MOMEDA parameters. Meanwhile, initialize the GOA parameters and grasshoppers swarm.( 2 ) Extract the periodic impact signal by using MOMEDA, and calculate the fitness of the impact signal. Save the best fitness and corresponding position Td^ for each iteration of GOA.( 3 ) If the current iteration m is less than the maximum M, the decreasing coefficient c is updated by (18).( 4 ) For each swarm, normalize the distances between individuals in [1, 4], and update the location of the search agent by (17).( 5 ) If there is a better solution, update the best fitness and corresponding position Td^. Then, determine if the stop criterion is reached (m≥M). If yes, the iteration is terminated. Else, let m=m+1, and the iteration will be continued.( 6 ) Obtain the best fitness and the corresponding parameters of MOMEDA.( 7 ) Extract the periodic impact signal by using MOMEDA with the optimized parameters.( 8 ) Extract fault features from impact signal by using Hilbert envelope demodulation.


## 4. Simulations and Comparisons

To verify the feasibility of the approach, we analyzed the bearing simulation signal [44] by using the proposed method. Meanwhile, all the experiments were completed by using MATLAB R2014a under the environment of Intel(R) Core(TM) i5-5200U CPU @ 2.20GHz 2.19GHz, 8GRAM.(21)xt=y0e-2πfnξt∗sin⁡2πfn∗1-ξ2t,where the natural frequency fn=3000Hz, damping coefficient ξ=0.1, shift constant y0=2.5, impact period T=0.00625s (characteristic frequency theoretical value fchthe=160Hz), data length N=4096, and sampling frequency fs=12KHz. Moreover, the 5 dB noise is mixed into the original signal.The waveforms of the simulation signal and noisy signals are given in Figure2. In Figure 2(b), the frequency fchthe=160Hz and its harmonics are especially obvious. In Figures 2(c) and 2(d), the characteristic frequency of the noisy signal and harmonics are submerged by noise. Although the noisy signal is similar to the actual bearing fault signal, the waveforms of the noisy signal are more complex [44].Figure 2
Noisy signals and their frequency domain.
(a) (b) (c) (d) (e) (f)Then, we analyze the noisy signal by the proposed approach. Here, the search agentsSAn=100, the maximum number of iterations M=10, and the ranges of filter size and the periodic initial value are L∈[1,500] and Ti∈[50,150], respectively. As shown in Figure 3, along with the increasing of iterations, the grasshopper swarms tend to converge towards the best locations, and the convergent curve gradually converges to the minimum. When the number of iterations m=9, the MKurt (MKurt=0.31328) reaches the maximum, and the optimal parameters γ=(L,Ti)=(386,75.00464) of MOMEDA are obtained. Then, the impact signal in the noisy signal is extracted by the optimized MOMEDA, and the envelope of the impact component is given in Figure 4. The actual value of the characteristic frequency fch=161.1Hz is approximate to fchthe=160Hz, and its harmonics are particularly obvious. Therefore, the proposed approach is feasible in extracting the features from the noisy signal.Figure 3
Optimization results of MOMEDA.
(a)
The search history of grasshoppers (b)
Convergence curve of GOAFigure 4
The envelope of the impact signal.In order to explore the effect of nonoptimal parameters on the results, we compared four sets of MOMEDA methods with fixed parameters. As shown in Figures5(a) and 5(b), when the filter size L of MOMEDA is the optimal value L=386 obtained by GOA and periodic initial value Ti=50, the frequency 240.2Hz and its harmonics can be observed clearly. Meanwhile, when Ti=90, the frequency 134.8Hz and its harmonics are obvious. However, the frequency 240.2 Hz is far greater than the theoretical value of the characteristic frequency (fchthe=160Hz), and the frequency 134.8 Hz is far less than the theoretical value. Therefore, the improper selection of periodic initial value could lead to incorrect diagnostic results. As shown in Figures 5(c) and 5(d), when Ti of MOMEDA is set to the optimal value Ti=75.00464, the frequency 158.2 Hz and its harmonics can be observed clearly, and these frequencies are approximate to fchthe=160Hz and its harmonics. Meanwhile, the spectral peaks of the envelope spectrum are more obvious when L=900.Figure 5
The effect of nonoptimal parameters on the results.
(a) (b) (c) (d)Figure5 shows that the selection of filter size and periodic initial value has a great impact on the feature extraction results. Therefore, it is important to choose an effective optimization method. Next, we discuss the optimization performance of PSO, SFLA, GWO, and GOA. For comparison purposes, the maximum number of iterations and population numbers of all algorithms are set to 50 and 20, and the other parameters are the default values. The -6db noise is mixed into the original signal, and the waveforms of the noisy signal are shown in Figures 2(e) and 2(f).As shown in Figure6 and Table 1, the optimal fitness (-0.075004) obtained by PSO is greater than that obtained by GOA when the number of iterations is 29, and the results show that the optimization precision of PSO is not high. Moreover, the convergence speed of PSO is slow and the algorithm is time-consuming. The convergence speed of the SFLA is fast, and the iterations take little time. However, the optimal fitness (-0.0746) obtained by SFLA is greater than that of other algorithms, which indicates that the optimization precision of the SFLA is lower than that of other methods. The GWO has high convergence accuracy, but its convergence speed is very slow and the optimal fitness is obtained through 49 iterations. Compared with the above methods, the convergence speed of GOA is very fast. The optimal parameters are obtained when the number of iterations is 25. Moreover, the optimal fitness obtained by GOA is minimal. It can be concluded that GOA is not only simple in calculation, but also high in convergent speed and accuracy. The GOA has outstanding advantages compared with other optimization algorithms, so we optimized the parameters of the MOMEDA by using GOA in this paper.Table 1
The optimization results of algorithms.
Index PSO SFLA GWO GOA Fitness -0.075004 -0.07468 -0.07616 -0.076541 Time (s) 356.8335 294.9474 369.3333 321.8644 Convergence 29 27 49 25 BestL 500 478 494 494 BestTi 75.0050 72.2673 74.7739 74.99123Figure 6
Convergence curves of optimization algorithms.To highlight the advantage of the approach, the method is compared with the methods with the optimal parameters, such as MED, MCKD, and FSK. Since the norms of MED, MCKD, and MOMEDA are based on kurtosis, the optimal filter sizeL=386 obtained by GOA is used as the filter size of MED and MCKD, and the optimal periodic initial value Ti=75.00464 is taken as the period of MCKD. The number of iterations of MED and MCKD is both set to 30. Compared with the method in the literature [27], the method has good adaptability. According to the characteristic of the noisy signal, the decomposition level of FSK is 4. In Figures 7(a) and 7(b), the characteristic frequencies obtained by MED are submerged in irrelevant frequency components. Although the frequency 158.2Hz and its harmonics obtained by MCKD can be observed, these frequencies are ambiguous due to the interference of irrelevant frequencies. Similarly, the frequency 159.2Hz and its harmonics can be obtained by FSK, but many interference frequencies exist in the envelope spectrum. From the comparison between Figures 4 and 7, the proposed approach has an outstanding advantage in the feature extraction of the noisy signal.Figure 7
Comparison of feature extraction methods for the noisy signal.
(a)
Kurtogram analysis of the noisy signal (b)
The envelope analysis of impact signals obtained by MED, MCKD, and FSK


## 5. Case Studies

### 5.1. Case1: Vibration Signal Analysis of CWRU Bearings

The data are derived from CWRU bearing dataset [40]. The bearing test rig is given in Figure 8 [44], and it is composed of four parts: a 2 hp motor, a decoder or a torque sensor, a dynamometer, and a controller. The sensors are fixed on the motor, and the motor speed is 1797rpm (the rotational frequency fr=1797/60Hz=29.95Hz), the 6205-2RSJEMSKF bearing parameters are listed in Table 2. The sampling frequency fs=12KHz, and the experimental data length N=2048. We use the inner ring, outer ring, and rolling element signals to prove the feasibility of the proposed approach. The fault diameter of these three bearing components is 0.007 inches (0.01778 cm). Based on the parameters listed in Table 2 and the formulas in literature [45], we can calculate the corresponding characteristic frequencies and they are listed in Table 3.Table 2
The SKF 6205 bearing parameters.
Ball diameter (d) Number of balls (N) Pitch diameter (D) Contact angle (α) 7.938mm 9 39mm 0Table 3
Characteristic frequencies.
Inner ring Outer ring Rolling element 162.1852Hz 107.3648Hz 141.1693HzFigure 8
Bearing test rig.


#### 5.1.1. Parameter Initialization

The parameters of MOMEDA include filter sizeL, window function Window, periodic initial value Ti, and periodic final value Tf. Next, we discuss the effect of different parameters on the MKurt of the filtered signal. The effect of window function on MKurt is not discussed in this paper, and we use a rectangular window in all the tests [28].The approximate range of MOMEDA parameters can be determined according to the signal characteristics. Firstly, the FIR filters with different filter sizeL are constructed, the range of filter size is L∈[1,800], and the period interval is T=[60:0.1:300] (Ti=60,Tf=300). Then, the periodic impacts are extracted from the different fault signals by these filters. Finally, the MKurt of these impact signals is obtained, and the results are shown in Figure 9.Figure 9
The relationship between MKurt and filter size.The following conclusions can be drawn:(1) The MKurt of the filtered signals increases with the increase of the filter size L. However, none of the four MKurt curves is strictly monotone increasing function, and the local feature of these MKurt curves is irregular. Therefore, the filter with a larger filter size is not necessarily the optimal filter. (2) The MKurt of filtered signals in different fault states is different. Therefore, MKurt can characterize the characteristics of different fault signals. (3) When the range of filter size is L≤500, the three health statuses of bearings can be effectively distinguished by calculating the MKurt of the filtered signal. If the filter size is too large, some information of the periodic impact signal may be removed and the length of the filtered signal (N-L+1) becomes shorter. In addition, the deconvolution efficiency of MOMEDA is reduced, and the efficiency of parameter optimization is also reduced. Therefore, the range of filter sizes is L∈[1,500] in this experiment.Then, we analyze the effect of the periodic initial valueTi on the MKurt of the filtered signal. Firstly, the FIR filters with different Ti are constructed, the range of Ti is Ti∈[5,200], the periodic final value is Tf=400, and the filter size is L=400. Then, the periodic impacts are extracted from the different fault signals by these filters. Finally, the MKurt of these impact signals is obtained, and the results are shown in Figure 10.Figure 10
The relationship between MKurt and periodic initial value.The following conclusions can be drawn:(1) With the increase of Ti, the change of MKurt is irregular. (2) The MKurt of the filtered signal is large when Ti is small, the reason is that the high-frequency noise and high harmonics of the fault frequencies are mixed into the filtered signal. It can be seen that a filter with a large Ti can remove the high-frequency noise. Therefore, the range of Ti is Ti∈[50,150] in this experiment.Then, we analyze the effect of the periodic final valueTf on the MKurt of the filtered signal. Firstly, the FIR filters with different Tf are constructed, the range of Tf is Tf∈[200,600], the periodic initial value is Ti=60, and the filter size is L=400. Then, the periodic impacts are extracted from the different fault signals by these filters. Finally, the MKurt of these impact signals is obtained, and the results are shown in Figure 11.Figure 11
The relationship between MKurt and periodic final value.The results show that the periodic final valueTf does not affect the filtering results. Therefore, the optimization of the Tf is not necessary, and the final value Tf=400 is used for all experiments. In summary, the number of search agents SAn=300, the maximum number of iterations M=10, the range of filter size L∈[1,500], the range of periodic initial value Ti∈[50,150], and periodic final value Tf=400.


#### 5.1.2. Inner Ring Signal Analysis of CWRU Bearings

As shown in Figure12, the waveforms of the inner ring signal have a certain regularity, which indicates that the rolling bearing has failed [44]. However, background noise and interference frequencies make it difficult to extract the inner ring fault characteristics directly from the frequency domain [2]. Therefore, we extract the inner ring fault features by using the proposed approach. Firstly, the parameters of the MOMEDA are optimized by GOA algorithm and the results are shown in Figure 13.Figure 12
Inner ring signal and its frequency domain.Figure 13
Optimization results of MOMEDA obtained by GOA.
(a)
The search history of grasshoppers (b)
Average fitness curve and convergence curve of GOAAs shown in Figure13, along with the increasing of iteration times, the grasshopper swarms tend to converge towards the best location, and the convergent curve gradually converges to the minimum. Meanwhile, a large number of particles are gathered at the right edge of Figure 13(a), and the results indicate that the optimal target may exist outside the range of the parameters. In order to ensure the efficiency of the algorithm and the filtering effect, we only consider the optimal solution of the algorithm when the filter size is less than or equal to 500. When the number of iterations m=9, the MKurt (MKurt=0.2869) is the maximum, and the optimal parameters γ=(L,Ti)=(499,74.15955) of MOMEDA are obtained.Subsequently, the impact component in the inner ring signal is extracted by the optimized MOMEDA, and the envelope of the impact signal is given in Figure14. When the frequencies are 164.1Hz and its multiples, several spectral peaks can be observed clearly. Moreover, these peak frequencies are approximate to the fault frequency (162.1852Hz) of inner ring and its harmonics. Due to the interference of transmission path and interference noise, there are some slight deviations between the measured values and the theoretical values. However, the proposed approach can effectively diagnose the inner ring fault.Figure 14
The envelope of the inner ring impact signal.In order to explore the effect of nonoptimal parameters on the results, we compared four sets of MOMEDA methods with fixed parameters. As shown in Figures15(a) and 15(b), when the filter size L of MOMEDA is set to the optimal value L=499 obtained by GOA and Ti=50, the frequency 240.2Hz and its harmonics can be observed clearly. Meanwhile, when Ti=90, the frequency 134.8Hz and its harmonics are obvious. However, the frequency 240.2 Hz is far greater than the characteristic frequency (162.1852Hz) of inner ring, and 134.8 Hz is far less than the frequency. Therefore, the values of these two sets of parameters are not appropriate. As shown in Figure 15(c), when Ti of MOMEDA is set to the optimal value Ti=74.15955 and L=20, the frequency 164.1Hz and its second harmonic can be observed clearly. However, the envelope spectrum is disturbed by noise, and its high-order harmonics are not obvious. Therefore, if the filter size L of MOMEDA is too small, the noise in the signal cannot be removed effectively, and the satisfactory effect cannot be obtained. In Figure 15(d), the frequency 164.1Hz and its harmonics can be observed clearly when L=900, and there is little interference around these harmonics. It can be seen that a better filtering effect can be obtained when the filter size L is larger. However, some information of the periodic impact signal may be removed when L is too large and the optimization process is time-consuming. We can draw a preliminary conclusion that the periodic initial value affects the size of the filtered signal characteristic frequency, and the filter size affects the noise reduction effect.Figure 15
The effect of nonoptimal parameters on the results.
(a) (b) (c) (d)To demonstrate the preponderance of the approach, the method is compared with the methods with the optimal parameters, such as MED, MCKD, and FSK. Since the norms of MED, MCKD, and MOMEDA are based on kurtosis, the optimal filter sizeL=499 obtained by GOA is used as the filter size of MED and MCKD, and the optimal periodic initial value Ti=74.15955 is taken as the period of MCKD. The number of iterations of MED and MCKD is both set to 30. In view of the characteristic of the signal, the decomposition level of FSK is set to 3. As shown in Figure 16, the three methods MED, MCKD, and FSK can extract the inner ring fault frequency and its harmonics. However, the envelope spectra obtained by these three methods are disturbed by the irrelevant frequency components, and their harmonic envelopes are very blurred. Therefore, the proposed approach has outstanding performance in the feature extraction of the inner ring.Figure 16
Comparison of feature extraction methods for the inner ring signal.
(a)
Kurtogram analysis of the inner signal (b)
The envelope analysis of impact signals obtained by MED, MCKD, and FSK


#### 5.1.3. Outer Ring Signal Analysis of CWRU Bearings

As shown in Figure17, the waveforms of the outer ring fault signal have a certain regularity, which indicates that the rolling bearing has failed. However, background noise and irrelevant components make it difficult to extract the outer ring fault characteristics directly from the frequency domain [2]. Therefore, we extract the outer ring fault features by using the proposed approach. Firstly, the parameters of the MOMEDA are optimized by GOA algorithm and the results are shown in Figure 18.Figure 17
Outer ring signal and its frequency domain.Figure 18
Optimization results of MOMEDA obtained by GOA.
(a)
The search history of grasshoppers (b)
Average fitness curve and convergence curve of GOAAs shown in Figures18(a) and 18(b), along with the increasing of iteration times, the grasshopper swarms tend to converge towards the best location, and the convergent curve gradually converges to the minimum. When the number of iterations m=7, the MKurt (MKurt=0.5128) is the maximum, and the optimal parameters γ=(L,Ti)=(459,111.4268) of MOMEDA are obtained.Then, the impact component in the outer ring signal is extracted by the optimized MOMEDA, and the envelope of the impact signal is given in Figure19. When the frequencies are 105.5Hz and its multiples, several spectral peaks can be observed clearly. Moreover, these peak frequencies are approximate to the characteristic frequency (107.3648Hz) of outer ring and its harmonics. Due to the influence of interference noise, there are some slight deviations between the measured values and the theoretical values. However, the proposed approach can effectively diagnose the outer ring fault of the bearing.Figure 19
The envelope of the outer ring impact signal.To explore the impact of nonoptimal parameters on the results, we compared four sets of MOMEDA methods with fixed parameters. As shown in Figures20(a) and 20(b), when the filter size L of MOMEDA is set to the optimal value L=459 obtained by GOA, Ti=50 or Ti=90, the frequencies 240.2Hz, 134.8Hz, and their harmonics can be observed clearly. However, these frequencies are far greater than the characteristic frequency (107.3648Hz) of outer ring and its harmonics. Therefore, the values of these two sets of parameters are not appropriate. As shown in Figure 20(c), when Ti of MOMEDA is set to the optimal value Ti=111.4268 and L=20, the frequency 105.5Hz can be observed clearly. However, the envelope spectrum is disturbed by noise, and its harmonics are submerged by interference frequency components. Therefore, the feature extraction method will fail when the filter size is too small. In Figure 20(d), 105.5Hz and its harmonics can be observed clearly when L=900, and there is little interference frequency around these harmonics. It can be seen that a better filtering effect can be obtained when the filter size is larger. However, some information of the periodic impact signal may be removed when L is too large and the optimization process is time-consuming. Meanwhile, the larger the periodic initial value, the smaller the characteristic frequency of the filtered signal obtained by MOMEDA.Figure 20
The effect of nonoptimal parameters on the results.
(a) (b) (c) (d)To demonstrate the superiority of the approach, the method is compared with the methods MED, MCKD, and FSK with the optimal parameters. The optimal filter sizeL=459 obtained by GOA is used as the filter size of MED and MCKD, and the optimal periodic initial value Ti=111.4268 is taken as the period of MCKD. The number of iterations of MED and MCKD is both set to 30. According to the characteristic of the outer ring signal, the decomposition level of FSK is set to 4. As shown in Figure 21, the three methods MED, MCKD, and FSK can extract the characteristic frequency of outer ring and its harmonics. However, the envelope obtained by MED contains many interference frequencies. For the outer ring, MCKD, FSK, and the proposed approach can achieve good results.Figure 21
Comparison of feature extraction methods for the outer ring signal.
(a)
Kurtogram analysis of the outer ring signal (b)
The envelope analysis of impact signals obtained by MED, MCKD, and FSK


#### 5.1.4. Rolling Element Signal Analysis of CWRU Bearings

According to the theory of Kumar et al. [46], when the rolling element fails, the frequency spectrum contains the second harmonic of the spin frequency. The spin frequency of the rolling element is generated by impacting the inner ring or outer ring. In general, the rolling element rotates once and it produces two impacts. Therefore, the characteristic frequency of the rolling element usually submerged in interference frequencies. Ma et al. [44] also proved that the proposed demodulation approach can obtain good results in the feature extraction of the inner ring and outer ring, but it has a poor effect in the feature extraction of the rolling element.As shown in Figure22, the regularity and periodicity of the waveforms are not obvious, and the frequencies contained in frequency domain waveform are more complex. In addition, the interference frequencies and noise make it difficult to extract fault features directly from frequency domain. Therefore, we extract the fault features by using the proposed approach. Firstly, the parameters of MOMEDA are optimized by GOA algorithm and the results are shown in Figure 23.Figure 22
Rolling element signal and frequency domain.Figure 23
Optimization results of MOMEDA obtained by GOA.
(a)
The search history of grasshoppers (b)
Average fitness curve and convergence curve of GOAAs shown in Figure23, along with the increasing of iteration times, the grasshopper swarms tend to converge towards the best location, and the convergent curve gradually converges to the minimum. When the number of iterations m=9, the MKurt (MKurt=0.18179) is the maximum, and the optimal parameters γ=(L,Ti)=(494,85.5723) of MOMEDA are obtained. Then, the impact component in the rolling element signal is extracted by the optimized MOMEDA, and the envelope of the impact signal is given in Figure 24. When the frequencies are 140.6Hz and its multiples, several spectral peaks can be observed clearly. Moreover, these peak frequencies are close to the characteristic frequency (141.1693Hz) of rolling element and its harmonics. Due to the effect of interference noise, there are some slight deviations between the measured values and the theoretical values. However, the rolling element fault of the bearing can be identified accurately by the proposed approach.Figure 24
The envelope of the rolling element impact signal.To explore the impact of nonoptimal parameters on the results, we compared four sets of MOMEDA methods with fixed parameters. As shown in Figures25(a) and 25(b), when the filter size L of MOMEDA is the optimal value L=494 obtained by GOA and Ti=50, the frequency 240.2Hz and its harmonics can be observed clearly. Meanwhile, when Ti=90, the frequency 134.8Hz and its harmonics are obvious. However, these two frequencies and their harmonics are independent of the fault characteristic frequency. Therefore, the values of these two sets of parameters are not appropriate. As shown in Figure 25(c), when Ti of MOMEDA is set to the optimal value Ti=85.5723 and L=20, the effective frequency is submerged by the interference frequencies and the fault features cannot be extracted. So, if the filter size is too small, the noise in the signal cannot be effectively removed, and the feature extraction method will completely fail. In Figure 25(d), 140.6Hz and its harmonics can be observed clearly when L=900. To sum up, the feature extraction method will fail when the periodic initial value Ti is not appropriate. In general, the larger Ti, the smaller the characteristic frequency of the filtered signal. The filter size L affects the noise reduction effect; if L is too small, the interference frequencies and noise in the signal cannot be removed effectively.Figure 25
The effect of nonoptimal parameters on the results.
(a) (b) (c) (d)To demonstrate the superiority of the approach, the method is compared with the methods MED, MCKD, and FSK with the optimal parameters. The optimal filter sizeL=494 obtained by GOA is used as the filter size of MED and MCKD, and the optimal periodic initial value Ti=85.5723 is taken as the period of MCKD. The number of iterations of MED and MCKD is both set to 30. According to the characteristics of the rolling element signal, the decomposition level of FSK is set to 4. As shown in Figure 26, the envelope spectra obtained by MED, MCKD, and FSK all contain a large number of unrelated frequency components, and the effective frequencies are all submerged in the interference frequencies. For the rolling element signal, the above three feature extraction methods cannot achieve satisfactory results. Therefore, it is particularly difficult to extract the rolling element characteristic frequency compared with the inner ring and outer ring. However, the proposed method still achieves good results in the feature extraction of the rolling element.Figure 26
Comparison of feature extraction methods for the rolling element signal.
(a)
Kurtogram analysis of the rolling element signal (b)
The envelope analysis of impact signals obtained by MED, MCKD, and FSK


### 5.2. Case2: Vibration Signal Analysis of NASA Bearings

To verify the universality and reliability of the proposed approach, we analyze the NASA bearing dataset [41] by using the proposed approach. Figure 27 shows the test platform [47] of NASA, and four lubricated Rexnord ZA-2115 bearings are fixed on the spindle of the test rig. The spindle rotates under the drive of the motor, and its speed is 2000 rpm. Therefore, the rotating frequency of the bearing is 33.33 Hz (fr = 2000/60 = 33.33 Hz). The PCB353B33 acceleration sensors are fixed on the bearing housing, and the signals are collected by NI DAQ Card 6062E data acquisition card. The data length N=20480 and the sampling frequency fs=20KHz. In this experiment, the inner ring signal comes from Bearing 3 of No. 1 dataset, the outer ring signal comes from Bearing 1 of No. 2 dataset, the rolling element signal comes from Bearing 3 of No. 1 dataset, and the data length is 2048. The parameters of Rexnord ZA-2115 bearing are listed in Table 4. Based on the parameters listed in Table 4 and the formulas in literature [45], we can calculate the corresponding characteristic frequencies and they are listed in Table 5.Table 4
The Rexnord ZA-2115 bearing parameters.
Ball diameter (d) Number of balls (Z) Pitch diameter (D) Contact angle (α) 8.407 mm 16 71.501 mm 15.17Table 5
Characteristic frequencies.
Inner ring Outer ring Rolling element 296.91Hz 236.4Hz 139.9HzFigure 27
Bearing experiment platform.The structure and operation mode of bearings share some similarities, and the bearings vibration signals have a corresponding periodicity. Therefore, the initialization of parameters in this section is the same as that in Section5.1.1.


#### 5.2.1. Inner Ring Signal Analysis of NASA Bearings

As shown in Figure28, due to the interference of interference frequencies and noise, it is difficult to extract the inner ring fault features. Therefore, we extract the inner ring fault features by using the proposed method. Firstly, the MOMEDA parameters are optimized by GOA algorithm.Figure 28
Inner ring signal and its frequency domain.As shown in Figure29, along with the increasing of iteration times, the convergent curve gradually converges to the minimum. When the number of iterations m=3, the MKurt (MKurt=0.2168) is the maximum, and the optimal parameters γ=(L,Ti)=(498,67.7462) of MOMEDA are obtained. Then, the impact component in the inner ring signal is extracted by the optimized MOMEDA, and the envelope of the impact signal is given in Figure 30(d). When the frequencies are 293 Hz and its multiples, several spectral peaks can be observed clearly. Moreover, these peak frequencies are approximate to the characteristic frequency (296.91Hz) of inner ring and its harmonics. Due to the interference of background noise and transmission path, there are some slight deviations between the measured values and the theoretical values. However, the proposed approach can effectively diagnose the inner ring fault of the bearing.Figure 29
Optimization results of MOMEDA obtained by GOA.Figure 30
The envelope analysis of impact signals obtained by MED, MCKD, FSK, and MOMEDA.To demonstrate the superiority of the approach, the method is compared with the methods with the optimal parameters, such as MED, MCKD, and FSK. Since the norms of MED, MCKD, and MOMEDA are based on kurtosis, the optimal filter sizeL=498 obtained by GOA is used as the filter size of MED and MCKD, and the optimal periodic initial value Ti=67.7462 is taken as the period of MCKD. The number of iterations of MED and MCKD is both set to 30. In view of the characteristic of the signal, the decomposition level of FSK is set to 3. The kurtosis of the filtered signal obtained by FSK is maximum when the decomposition level is 2. At this point, the central frequency of the filtered signal is 8750Hz, the bandwidth is 2500Hz, and the envelope of the filtered signal is given in Figure 30(c). As shown in Figures 30(a) and 30(c), the envelope spectra obtained by MED and FSK are disturbed by many irrelevant frequencies, and it is unrealistic to extract the fault characteristic frequency. In Figure 30(b), although MCKD can extract the inner ring fault features, the frequency and its harmonics are disturbed by many irrelevant frequencies. Therefore, the proposed approach has outstanding advantages in the feature extraction of the inner ring.


#### 5.2.2. Outer Ring Signal Analysis of NASA Bearings

As shown in Figure31, although the waveforms have a certain regularity, it is difficult to extract fault features directly from the frequency domain. Therefore, we extract the outer ring fault features by using the proposed method. Firstly, the parameters of MOMEDA are optimized by GOA algorithm and the results are shown in Figure 32.Figure 31
Outer ring signal and its frequency domain.Figure 32
Optimization results of MOMEDA obtained by GOA.When the number of iterationsm=9, the MKurt (Mkurt=0.2385) is the maximum, and the optimal parameters γ=(L,Ti)=(496,84.7148) of MOMEDA are obtained. Then, the impact component in the outer ring signal is extracted by the optimized MOMEDA, and the envelope of the impact component is given in Figure 33(d). When the frequencies are 234.4Hz and its multiples, several spectral peaks can be observed clearly. Moreover, these peak frequencies are approximate to the outer ring characteristic frequency (236.4Hz) and its harmonics. Due to the interference of interference noise, there are slight deviations between the measured values and the theoretical values. However, the proposed approach can effectively diagnose the outer ring fault.Figure 33
The envelope analysis of impact signals obtained by MED, MCKD, FSK, and MOMEDA.Then, we compared the proposed approach with the methods MED, MCKD, and FSK with the optimal parameters. The optimal filter sizeL=496 obtained by GOA is used as the filter size of MED and MCKD, and the optimal periodic initial value Ti=84.7148 is taken as the period of MCKD. The number of iterations of MED and MCKD is both set to 30, and the decomposition level of the FSK is set to 3. The kurtosis of the filtered signal obtained by FSK is maximum when the decomposition level is 1.5. At this point, the central frequency of the filtered signal is 1666.666Hz, the bandwidth is 3333.33Hz, and the envelope analysis of the filtered signal is shown in Figure 33(c). In Figures 33(a) and 33(c), the characteristic frequencies and their harmonics can be observed. However, these frequencies are disturbed by many interference frequencies. Therefore, it is difficult to identify the outer ring characteristic frequency through MED and FSK. In Figure 33(b), although MCKD can extract the outer ring characteristic frequency and its harmonics, the amplitudes of these harmonics are small and the spectral peaks are not obvious. Therefore, the proposed approach has outstanding performance in the feature extraction of the outer ring.


#### 5.2.3. Rolling Element Signal Analysis of NASA Bearings

As shown in Figure34, the regularity of the waveforms is not significant, and the frequencies contained in the frequency domain waveform are more complex. In addition, the interference frequencies and noise make it difficult to extract fault features directly from frequency domain. Therefore, we extract the fault features by the proposed approach. Firstly, the parameters of MOMEDA are optimized by GOA algorithm and the results are shown in Figure 35.Figure 34
Rolling element signal and its frequency domain.Figure 35
Optimization results of MOMEDA obtained by GOA.As shown in Figure35, when the number of iterations m=2, the MKurt (Mkurt=0.19684) is the maximum, and the optimal parameters γ=(L,Ti)=(500,143.0353) of MOMEDA are obtained. Then, the impact component is extracted by the optimized MOMEDA, and the envelope of the impact signal is given in Figure 36(d). When the frequencies are 136.7Hz and its multiples, several spectral peaks can be observed clearly. Moreover, these peak frequencies are approximate to the characteristic frequency (139.9Hz) of rolling element and its harmonics. Due to the interference of background noise and transmission path, there are some slight deviations between the measured values and the theoretical values. However, the proposed approach can effectively diagnose the rolling element fault.Figure 36
The envelope analysis of impact signals obtained by MED, MCKD, FSK, and MOMEDA.Then, the proposed method is compared with the methods MED, MCKD, and FSK with the optimal parameters. The optimal filter sizeL=500 obtained by GOA is used as the filter size of MED and MCKD, and the optimal period initial value Ti=143.0353 is taken as the period of MCKD. The number of iterations of MED and MCKD is both set to 30, and the decomposition level of FSK is set to 3. The kurtosis of the filtered signal obtained by FSK is maximum when the decomposition level is 0. At this point, the central frequency of the filtered signal is 5000Hz, the bandwidth is 10000Hz, and the envelope of the filtered signal is given in Figure 36(c). As shown in Figures 36(a) and 36(c), the envelope spectra obtained by MED and FSK are disturbed by many irrelevant frequencies, and it is difficult to extract the rolling element characteristic frequency through MED and FSK. In Figure 36(b), although MCKD can extract the characteristic frequency, the deviation between the peak frequency (146.5Hz) and fault characteristic frequency (136.7Hz) is large. Therefore, the proposed approach has excellent performance in the feature extraction of the rolling element.


## 5.1. Case1: Vibration Signal Analysis of CWRU Bearings

The data are derived from CWRU bearing dataset [40]. The bearing test rig is given in Figure 8 [44], and it is composed of four parts: a 2 hp motor, a decoder or a torque sensor, a dynamometer, and a controller. The sensors are fixed on the motor, and the motor speed is 1797rpm (the rotational frequency fr=1797/60Hz=29.95Hz), the 6205-2RSJEMSKF bearing parameters are listed in Table 2. The sampling frequency fs=12KHz, and the experimental data length N=2048. We use the inner ring, outer ring, and rolling element signals to prove the feasibility of the proposed approach. The fault diameter of these three bearing components is 0.007 inches (0.01778 cm). Based on the parameters listed in Table 2 and the formulas in literature [45], we can calculate the corresponding characteristic frequencies and they are listed in Table 3.Table 2
The SKF 6205 bearing parameters.
Ball diameter (d) Number of balls (N) Pitch diameter (D) Contact angle (α) 7.938mm 9 39mm 0Table 3
Characteristic frequencies.
Inner ring Outer ring Rolling element 162.1852Hz 107.3648Hz 141.1693HzFigure 8
Bearing test rig.


### 5.1.1. Parameter Initialization

The parameters of MOMEDA include filter sizeL, window function Window, periodic initial value Ti, and periodic final value Tf. Next, we discuss the effect of different parameters on the MKurt of the filtered signal. The effect of window function on MKurt is not discussed in this paper, and we use a rectangular window in all the tests [28].The approximate range of MOMEDA parameters can be determined according to the signal characteristics. Firstly, the FIR filters with different filter sizeL are constructed, the range of filter size is L∈[1,800], and the period interval is T=[60:0.1:300] (Ti=60,Tf=300). Then, the periodic impacts are extracted from the different fault signals by these filters. Finally, the MKurt of these impact signals is obtained, and the results are shown in Figure 9.Figure 9
The relationship between MKurt and filter size.The following conclusions can be drawn:(1) The MKurt of the filtered signals increases with the increase of the filter size L. However, none of the four MKurt curves is strictly monotone increasing function, and the local feature of these MKurt curves is irregular. Therefore, the filter with a larger filter size is not necessarily the optimal filter. (2) The MKurt of filtered signals in different fault states is different. Therefore, MKurt can characterize the characteristics of different fault signals. (3) When the range of filter size is L≤500, the three health statuses of bearings can be effectively distinguished by calculating the MKurt of the filtered signal. If the filter size is too large, some information of the periodic impact signal may be removed and the length of the filtered signal (N-L+1) becomes shorter. In addition, the deconvolution efficiency of MOMEDA is reduced, and the efficiency of parameter optimization is also reduced. Therefore, the range of filter sizes is L∈[1,500] in this experiment.Then, we analyze the effect of the periodic initial valueTi on the MKurt of the filtered signal. Firstly, the FIR filters with different Ti are constructed, the range of Ti is Ti∈[5,200], the periodic final value is Tf=400, and the filter size is L=400. Then, the periodic impacts are extracted from the different fault signals by these filters. Finally, the MKurt of these impact signals is obtained, and the results are shown in Figure 10.Figure 10
The relationship between MKurt and periodic initial value.The following conclusions can be drawn:(1) With the increase of Ti, the change of MKurt is irregular. (2) The MKurt of the filtered signal is large when Ti is small, the reason is that the high-frequency noise and high harmonics of the fault frequencies are mixed into the filtered signal. It can be seen that a filter with a large Ti can remove the high-frequency noise. Therefore, the range of Ti is Ti∈[50,150] in this experiment.Then, we analyze the effect of the periodic final valueTf on the MKurt of the filtered signal. Firstly, the FIR filters with different Tf are constructed, the range of Tf is Tf∈[200,600], the periodic initial value is Ti=60, and the filter size is L=400. Then, the periodic impacts are extracted from the different fault signals by these filters. Finally, the MKurt of these impact signals is obtained, and the results are shown in Figure 11.Figure 11
The relationship between MKurt and periodic final value.The results show that the periodic final valueTf does not affect the filtering results. Therefore, the optimization of the Tf is not necessary, and the final value Tf=400 is used for all experiments. In summary, the number of search agents SAn=300, the maximum number of iterations M=10, the range of filter size L∈[1,500], the range of periodic initial value Ti∈[50,150], and periodic final value Tf=400.


### 5.1.2. Inner Ring Signal Analysis of CWRU Bearings

As shown in Figure12, the waveforms of the inner ring signal have a certain regularity, which indicates that the rolling bearing has failed [44]. However, background noise and interference frequencies make it difficult to extract the inner ring fault characteristics directly from the frequency domain [2]. Therefore, we extract the inner ring fault features by using the proposed approach. Firstly, the parameters of the MOMEDA are optimized by GOA algorithm and the results are shown in Figure 13.Figure 12
Inner ring signal and its frequency domain.Figure 13
Optimization results of MOMEDA obtained by GOA.
(a)
The search history of grasshoppers (b)
Average fitness curve and convergence curve of GOAAs shown in Figure13, along with the increasing of iteration times, the grasshopper swarms tend to converge towards the best location, and the convergent curve gradually converges to the minimum. Meanwhile, a large number of particles are gathered at the right edge of Figure 13(a), and the results indicate that the optimal target may exist outside the range of the parameters. In order to ensure the efficiency of the algorithm and the filtering effect, we only consider the optimal solution of the algorithm when the filter size is less than or equal to 500. When the number of iterations m=9, the MKurt (MKurt=0.2869) is the maximum, and the optimal parameters γ=(L,Ti)=(499,74.15955) of MOMEDA are obtained.Subsequently, the impact component in the inner ring signal is extracted by the optimized MOMEDA, and the envelope of the impact signal is given in Figure14. When the frequencies are 164.1Hz and its multiples, several spectral peaks can be observed clearly. Moreover, these peak frequencies are approximate to the fault frequency (162.1852Hz) of inner ring and its harmonics. Due to the interference of transmission path and interference noise, there are some slight deviations between the measured values and the theoretical values. However, the proposed approach can effectively diagnose the inner ring fault.Figure 14
The envelope of the inner ring impact signal.In order to explore the effect of nonoptimal parameters on the results, we compared four sets of MOMEDA methods with fixed parameters. As shown in Figures15(a) and 15(b), when the filter size L of MOMEDA is set to the optimal value L=499 obtained by GOA and Ti=50, the frequency 240.2Hz and its harmonics can be observed clearly. Meanwhile, when Ti=90, the frequency 134.8Hz and its harmonics are obvious. However, the frequency 240.2 Hz is far greater than the characteristic frequency (162.1852Hz) of inner ring, and 134.8 Hz is far less than the frequency. Therefore, the values of these two sets of parameters are not appropriate. As shown in Figure 15(c), when Ti of MOMEDA is set to the optimal value Ti=74.15955 and L=20, the frequency 164.1Hz and its second harmonic can be observed clearly. However, the envelope spectrum is disturbed by noise, and its high-order harmonics are not obvious. Therefore, if the filter size L of MOMEDA is too small, the noise in the signal cannot be removed effectively, and the satisfactory effect cannot be obtained. In Figure 15(d), the frequency 164.1Hz and its harmonics can be observed clearly when L=900, and there is little interference around these harmonics. It can be seen that a better filtering effect can be obtained when the filter size L is larger. However, some information of the periodic impact signal may be removed when L is too large and the optimization process is time-consuming. We can draw a preliminary conclusion that the periodic initial value affects the size of the filtered signal characteristic frequency, and the filter size affects the noise reduction effect.Figure 15
The effect of nonoptimal parameters on the results.
(a) (b) (c) (d)To demonstrate the preponderance of the approach, the method is compared with the methods with the optimal parameters, such as MED, MCKD, and FSK. Since the norms of MED, MCKD, and MOMEDA are based on kurtosis, the optimal filter sizeL=499 obtained by GOA is used as the filter size of MED and MCKD, and the optimal periodic initial value Ti=74.15955 is taken as the period of MCKD. The number of iterations of MED and MCKD is both set to 30. In view of the characteristic of the signal, the decomposition level of FSK is set to 3. As shown in Figure 16, the three methods MED, MCKD, and FSK can extract the inner ring fault frequency and its harmonics. However, the envelope spectra obtained by these three methods are disturbed by the irrelevant frequency components, and their harmonic envelopes are very blurred. Therefore, the proposed approach has outstanding performance in the feature extraction of the inner ring.Figure 16
Comparison of feature extraction methods for the inner ring signal.
(a)
Kurtogram analysis of the inner signal (b)
The envelope analysis of impact signals obtained by MED, MCKD, and FSK


### 5.1.3. Outer Ring Signal Analysis of CWRU Bearings

As shown in Figure17, the waveforms of the outer ring fault signal have a certain regularity, which indicates that the rolling bearing has failed. However, background noise and irrelevant components make it difficult to extract the outer ring fault characteristics directly from the frequency domain [2]. Therefore, we extract the outer ring fault features by using the proposed approach. Firstly, the parameters of the MOMEDA are optimized by GOA algorithm and the results are shown in Figure 18.Figure 17
Outer ring signal and its frequency domain.Figure 18
Optimization results of MOMEDA obtained by GOA.
(a)
The search history of grasshoppers (b)
Average fitness curve and convergence curve of GOAAs shown in Figures18(a) and 18(b), along with the increasing of iteration times, the grasshopper swarms tend to converge towards the best location, and the convergent curve gradually converges to the minimum. When the number of iterations m=7, the MKurt (MKurt=0.5128) is the maximum, and the optimal parameters γ=(L,Ti)=(459,111.4268) of MOMEDA are obtained.Then, the impact component in the outer ring signal is extracted by the optimized MOMEDA, and the envelope of the impact signal is given in Figure19. When the frequencies are 105.5Hz and its multiples, several spectral peaks can be observed clearly. Moreover, these peak frequencies are approximate to the characteristic frequency (107.3648Hz) of outer ring and its harmonics. Due to the influence of interference noise, there are some slight deviations between the measured values and the theoretical values. However, the proposed approach can effectively diagnose the outer ring fault of the bearing.Figure 19
The envelope of the outer ring impact signal.To explore the impact of nonoptimal parameters on the results, we compared four sets of MOMEDA methods with fixed parameters. As shown in Figures20(a) and 20(b), when the filter size L of MOMEDA is set to the optimal value L=459 obtained by GOA, Ti=50 or Ti=90, the frequencies 240.2Hz, 134.8Hz, and their harmonics can be observed clearly. However, these frequencies are far greater than the characteristic frequency (107.3648Hz) of outer ring and its harmonics. Therefore, the values of these two sets of parameters are not appropriate. As shown in Figure 20(c), when Ti of MOMEDA is set to the optimal value Ti=111.4268 and L=20, the frequency 105.5Hz can be observed clearly. However, the envelope spectrum is disturbed by noise, and its harmonics are submerged by interference frequency components. Therefore, the feature extraction method will fail when the filter size is too small. In Figure 20(d), 105.5Hz and its harmonics can be observed clearly when L=900, and there is little interference frequency around these harmonics. It can be seen that a better filtering effect can be obtained when the filter size is larger. However, some information of the periodic impact signal may be removed when L is too large and the optimization process is time-consuming. Meanwhile, the larger the periodic initial value, the smaller the characteristic frequency of the filtered signal obtained by MOMEDA.Figure 20
The effect of nonoptimal parameters on the results.
(a) (b) (c) (d)To demonstrate the superiority of the approach, the method is compared with the methods MED, MCKD, and FSK with the optimal parameters. The optimal filter sizeL=459 obtained by GOA is used as the filter size of MED and MCKD, and the optimal periodic initial value Ti=111.4268 is taken as the period of MCKD. The number of iterations of MED and MCKD is both set to 30. According to the characteristic of the outer ring signal, the decomposition level of FSK is set to 4. As shown in Figure 21, the three methods MED, MCKD, and FSK can extract the characteristic frequency of outer ring and its harmonics. However, the envelope obtained by MED contains many interference frequencies. For the outer ring, MCKD, FSK, and the proposed approach can achieve good results.Figure 21
Comparison of feature extraction methods for the outer ring signal.
(a)
Kurtogram analysis of the outer ring signal (b)
The envelope analysis of impact signals obtained by MED, MCKD, and FSK


### 5.1.4. Rolling Element Signal Analysis of CWRU Bearings

According to the theory of Kumar et al. [46], when the rolling element fails, the frequency spectrum contains the second harmonic of the spin frequency. The spin frequency of the rolling element is generated by impacting the inner ring or outer ring. In general, the rolling element rotates once and it produces two impacts. Therefore, the characteristic frequency of the rolling element usually submerged in interference frequencies. Ma et al. [44] also proved that the proposed demodulation approach can obtain good results in the feature extraction of the inner ring and outer ring, but it has a poor effect in the feature extraction of the rolling element.As shown in Figure22, the regularity and periodicity of the waveforms are not obvious, and the frequencies contained in frequency domain waveform are more complex. In addition, the interference frequencies and noise make it difficult to extract fault features directly from frequency domain. Therefore, we extract the fault features by using the proposed approach. Firstly, the parameters of MOMEDA are optimized by GOA algorithm and the results are shown in Figure 23.Figure 22
Rolling element signal and frequency domain.Figure 23
Optimization results of MOMEDA obtained by GOA.
(a)
The search history of grasshoppers (b)
Average fitness curve and convergence curve of GOAAs shown in Figure23, along with the increasing of iteration times, the grasshopper swarms tend to converge towards the best location, and the convergent curve gradually converges to the minimum. When the number of iterations m=9, the MKurt (MKurt=0.18179) is the maximum, and the optimal parameters γ=(L,Ti)=(494,85.5723) of MOMEDA are obtained. Then, the impact component in the rolling element signal is extracted by the optimized MOMEDA, and the envelope of the impact signal is given in Figure 24. When the frequencies are 140.6Hz and its multiples, several spectral peaks can be observed clearly. Moreover, these peak frequencies are close to the characteristic frequency (141.1693Hz) of rolling element and its harmonics. Due to the effect of interference noise, there are some slight deviations between the measured values and the theoretical values. However, the rolling element fault of the bearing can be identified accurately by the proposed approach.Figure 24
The envelope of the rolling element impact signal.To explore the impact of nonoptimal parameters on the results, we compared four sets of MOMEDA methods with fixed parameters. As shown in Figures25(a) and 25(b), when the filter size L of MOMEDA is the optimal value L=494 obtained by GOA and Ti=50, the frequency 240.2Hz and its harmonics can be observed clearly. Meanwhile, when Ti=90, the frequency 134.8Hz and its harmonics are obvious. However, these two frequencies and their harmonics are independent of the fault characteristic frequency. Therefore, the values of these two sets of parameters are not appropriate. As shown in Figure 25(c), when Ti of MOMEDA is set to the optimal value Ti=85.5723 and L=20, the effective frequency is submerged by the interference frequencies and the fault features cannot be extracted. So, if the filter size is too small, the noise in the signal cannot be effectively removed, and the feature extraction method will completely fail. In Figure 25(d), 140.6Hz and its harmonics can be observed clearly when L=900. To sum up, the feature extraction method will fail when the periodic initial value Ti is not appropriate. In general, the larger Ti, the smaller the characteristic frequency of the filtered signal. The filter size L affects the noise reduction effect; if L is too small, the interference frequencies and noise in the signal cannot be removed effectively.Figure 25
The effect of nonoptimal parameters on the results.
(a) (b) (c) (d)To demonstrate the superiority of the approach, the method is compared with the methods MED, MCKD, and FSK with the optimal parameters. The optimal filter sizeL=494 obtained by GOA is used as the filter size of MED and MCKD, and the optimal periodic initial value Ti=85.5723 is taken as the period of MCKD. The number of iterations of MED and MCKD is both set to 30. According to the characteristics of the rolling element signal, the decomposition level of FSK is set to 4. As shown in Figure 26, the envelope spectra obtained by MED, MCKD, and FSK all contain a large number of unrelated frequency components, and the effective frequencies are all submerged in the interference frequencies. For the rolling element signal, the above three feature extraction methods cannot achieve satisfactory results. Therefore, it is particularly difficult to extract the rolling element characteristic frequency compared with the inner ring and outer ring. However, the proposed method still achieves good results in the feature extraction of the rolling element.Figure 26
Comparison of feature extraction methods for the rolling element signal.
(a)
Kurtogram analysis of the rolling element signal (b)
The envelope analysis of impact signals obtained by MED, MCKD, and FSK


## 5.1.1. Parameter Initialization

The parameters of MOMEDA include filter sizeL, window function Window, periodic initial value Ti, and periodic final value Tf. Next, we discuss the effect of different parameters on the MKurt of the filtered signal. The effect of window function on MKurt is not discussed in this paper, and we use a rectangular window in all the tests [28].The approximate range of MOMEDA parameters can be determined according to the signal characteristics. Firstly, the FIR filters with different filter sizeL are constructed, the range of filter size is L∈[1,800], and the period interval is T=[60:0.1:300] (Ti=60,Tf=300). Then, the periodic impacts are extracted from the different fault signals by these filters. Finally, the MKurt of these impact signals is obtained, and the results are shown in Figure 9.Figure 9
The relationship between MKurt and filter size.The following conclusions can be drawn:(1) The MKurt of the filtered signals increases with the increase of the filter size L. However, none of the four MKurt curves is strictly monotone increasing function, and the local feature of these MKurt curves is irregular. Therefore, the filter with a larger filter size is not necessarily the optimal filter. (2) The MKurt of filtered signals in different fault states is different. Therefore, MKurt can characterize the characteristics of different fault signals. (3) When the range of filter size is L≤500, the three health statuses of bearings can be effectively distinguished by calculating the MKurt of the filtered signal. If the filter size is too large, some information of the periodic impact signal may be removed and the length of the filtered signal (N-L+1) becomes shorter. In addition, the deconvolution efficiency of MOMEDA is reduced, and the efficiency of parameter optimization is also reduced. Therefore, the range of filter sizes is L∈[1,500] in this experiment.Then, we analyze the effect of the periodic initial valueTi on the MKurt of the filtered signal. Firstly, the FIR filters with different Ti are constructed, the range of Ti is Ti∈[5,200], the periodic final value is Tf=400, and the filter size is L=400. Then, the periodic impacts are extracted from the different fault signals by these filters. Finally, the MKurt of these impact signals is obtained, and the results are shown in Figure 10.Figure 10
The relationship between MKurt and periodic initial value.The following conclusions can be drawn:(1) With the increase of Ti, the change of MKurt is irregular. (2) The MKurt of the filtered signal is large when Ti is small, the reason is that the high-frequency noise and high harmonics of the fault frequencies are mixed into the filtered signal. It can be seen that a filter with a large Ti can remove the high-frequency noise. Therefore, the range of Ti is Ti∈[50,150] in this experiment.Then, we analyze the effect of the periodic final valueTf on the MKurt of the filtered signal. Firstly, the FIR filters with different Tf are constructed, the range of Tf is Tf∈[200,600], the periodic initial value is Ti=60, and the filter size is L=400. Then, the periodic impacts are extracted from the different fault signals by these filters. Finally, the MKurt of these impact signals is obtained, and the results are shown in Figure 11.Figure 11
The relationship between MKurt and periodic final value.The results show that the periodic final valueTf does not affect the filtering results. Therefore, the optimization of the Tf is not necessary, and the final value Tf=400 is used for all experiments. In summary, the number of search agents SAn=300, the maximum number of iterations M=10, the range of filter size L∈[1,500], the range of periodic initial value Ti∈[50,150], and periodic final value Tf=400.


## 5.1.2. Inner Ring Signal Analysis of CWRU Bearings

As shown in Figure12, the waveforms of the inner ring signal have a certain regularity, which indicates that the rolling bearing has failed [44]. However, background noise and interference frequencies make it difficult to extract the inner ring fault characteristics directly from the frequency domain [2]. Therefore, we extract the inner ring fault features by using the proposed approach. Firstly, the parameters of the MOMEDA are optimized by GOA algorithm and the results are shown in Figure 13.Figure 12
Inner ring signal and its frequency domain.Figure 13
Optimization results of MOMEDA obtained by GOA.
(a)
The search history of grasshoppers (b)
Average fitness curve and convergence curve of GOAAs shown in Figure13, along with the increasing of iteration times, the grasshopper swarms tend to converge towards the best location, and the convergent curve gradually converges to the minimum. Meanwhile, a large number of particles are gathered at the right edge of Figure 13(a), and the results indicate that the optimal target may exist outside the range of the parameters. In order to ensure the efficiency of the algorithm and the filtering effect, we only consider the optimal solution of the algorithm when the filter size is less than or equal to 500. When the number of iterations m=9, the MKurt (MKurt=0.2869) is the maximum, and the optimal parameters γ=(L,Ti)=(499,74.15955) of MOMEDA are obtained.Subsequently, the impact component in the inner ring signal is extracted by the optimized MOMEDA, and the envelope of the impact signal is given in Figure14. When the frequencies are 164.1Hz and its multiples, several spectral peaks can be observed clearly. Moreover, these peak frequencies are approximate to the fault frequency (162.1852Hz) of inner ring and its harmonics. Due to the interference of transmission path and interference noise, there are some slight deviations between the measured values and the theoretical values. However, the proposed approach can effectively diagnose the inner ring fault.Figure 14
The envelope of the inner ring impact signal.In order to explore the effect of nonoptimal parameters on the results, we compared four sets of MOMEDA methods with fixed parameters. As shown in Figures15(a) and 15(b), when the filter size L of MOMEDA is set to the optimal value L=499 obtained by GOA and Ti=50, the frequency 240.2Hz and its harmonics can be observed clearly. Meanwhile, when Ti=90, the frequency 134.8Hz and its harmonics are obvious. However, the frequency 240.2 Hz is far greater than the characteristic frequency (162.1852Hz) of inner ring, and 134.8 Hz is far less than the frequency. Therefore, the values of these two sets of parameters are not appropriate. As shown in Figure 15(c), when Ti of MOMEDA is set to the optimal value Ti=74.15955 and L=20, the frequency 164.1Hz and its second harmonic can be observed clearly. However, the envelope spectrum is disturbed by noise, and its high-order harmonics are not obvious. Therefore, if the filter size L of MOMEDA is too small, the noise in the signal cannot be removed effectively, and the satisfactory effect cannot be obtained. In Figure 15(d), the frequency 164.1Hz and its harmonics can be observed clearly when L=900, and there is little interference around these harmonics. It can be seen that a better filtering effect can be obtained when the filter size L is larger. However, some information of the periodic impact signal may be removed when L is too large and the optimization process is time-consuming. We can draw a preliminary conclusion that the periodic initial value affects the size of the filtered signal characteristic frequency, and the filter size affects the noise reduction effect.Figure 15
The effect of nonoptimal parameters on the results.
(a) (b) (c) (d)To demonstrate the preponderance of the approach, the method is compared with the methods with the optimal parameters, such as MED, MCKD, and FSK. Since the norms of MED, MCKD, and MOMEDA are based on kurtosis, the optimal filter sizeL=499 obtained by GOA is used as the filter size of MED and MCKD, and the optimal periodic initial value Ti=74.15955 is taken as the period of MCKD. The number of iterations of MED and MCKD is both set to 30. In view of the characteristic of the signal, the decomposition level of FSK is set to 3. As shown in Figure 16, the three methods MED, MCKD, and FSK can extract the inner ring fault frequency and its harmonics. However, the envelope spectra obtained by these three methods are disturbed by the irrelevant frequency components, and their harmonic envelopes are very blurred. Therefore, the proposed approach has outstanding performance in the feature extraction of the inner ring.Figure 16
Comparison of feature extraction methods for the inner ring signal.
(a)
Kurtogram analysis of the inner signal (b)
The envelope analysis of impact signals obtained by MED, MCKD, and FSK


## 5.1.3. Outer Ring Signal Analysis of CWRU Bearings

As shown in Figure17, the waveforms of the outer ring fault signal have a certain regularity, which indicates that the rolling bearing has failed. However, background noise and irrelevant components make it difficult to extract the outer ring fault characteristics directly from the frequency domain [2]. Therefore, we extract the outer ring fault features by using the proposed approach. Firstly, the parameters of the MOMEDA are optimized by GOA algorithm and the results are shown in Figure 18.Figure 17
Outer ring signal and its frequency domain.Figure 18
Optimization results of MOMEDA obtained by GOA.
(a)
The search history of grasshoppers (b)
Average fitness curve and convergence curve of GOAAs shown in Figures18(a) and 18(b), along with the increasing of iteration times, the grasshopper swarms tend to converge towards the best location, and the convergent curve gradually converges to the minimum. When the number of iterations m=7, the MKurt (MKurt=0.5128) is the maximum, and the optimal parameters γ=(L,Ti)=(459,111.4268) of MOMEDA are obtained.Then, the impact component in the outer ring signal is extracted by the optimized MOMEDA, and the envelope of the impact signal is given in Figure19. When the frequencies are 105.5Hz and its multiples, several spectral peaks can be observed clearly. Moreover, these peak frequencies are approximate to the characteristic frequency (107.3648Hz) of outer ring and its harmonics. Due to the influence of interference noise, there are some slight deviations between the measured values and the theoretical values. However, the proposed approach can effectively diagnose the outer ring fault of the bearing.Figure 19
The envelope of the outer ring impact signal.To explore the impact of nonoptimal parameters on the results, we compared four sets of MOMEDA methods with fixed parameters. As shown in Figures20(a) and 20(b), when the filter size L of MOMEDA is set to the optimal value L=459 obtained by GOA, Ti=50 or Ti=90, the frequencies 240.2Hz, 134.8Hz, and their harmonics can be observed clearly. However, these frequencies are far greater than the characteristic frequency (107.3648Hz) of outer ring and its harmonics. Therefore, the values of these two sets of parameters are not appropriate. As shown in Figure 20(c), when Ti of MOMEDA is set to the optimal value Ti=111.4268 and L=20, the frequency 105.5Hz can be observed clearly. However, the envelope spectrum is disturbed by noise, and its harmonics are submerged by interference frequency components. Therefore, the feature extraction method will fail when the filter size is too small. In Figure 20(d), 105.5Hz and its harmonics can be observed clearly when L=900, and there is little interference frequency around these harmonics. It can be seen that a better filtering effect can be obtained when the filter size is larger. However, some information of the periodic impact signal may be removed when L is too large and the optimization process is time-consuming. Meanwhile, the larger the periodic initial value, the smaller the characteristic frequency of the filtered signal obtained by MOMEDA.Figure 20
The effect of nonoptimal parameters on the results.
(a) (b) (c) (d)To demonstrate the superiority of the approach, the method is compared with the methods MED, MCKD, and FSK with the optimal parameters. The optimal filter sizeL=459 obtained by GOA is used as the filter size of MED and MCKD, and the optimal periodic initial value Ti=111.4268 is taken as the period of MCKD. The number of iterations of MED and MCKD is both set to 30. According to the characteristic of the outer ring signal, the decomposition level of FSK is set to 4. As shown in Figure 21, the three methods MED, MCKD, and FSK can extract the characteristic frequency of outer ring and its harmonics. However, the envelope obtained by MED contains many interference frequencies. For the outer ring, MCKD, FSK, and the proposed approach can achieve good results.Figure 21
Comparison of feature extraction methods for the outer ring signal.
(a)
Kurtogram analysis of the outer ring signal (b)
The envelope analysis of impact signals obtained by MED, MCKD, and FSK


## 5.1.4. Rolling Element Signal Analysis of CWRU Bearings

According to the theory of Kumar et al. [46], when the rolling element fails, the frequency spectrum contains the second harmonic of the spin frequency. The spin frequency of the rolling element is generated by impacting the inner ring or outer ring. In general, the rolling element rotates once and it produces two impacts. Therefore, the characteristic frequency of the rolling element usually submerged in interference frequencies. Ma et al. [44] also proved that the proposed demodulation approach can obtain good results in the feature extraction of the inner ring and outer ring, but it has a poor effect in the feature extraction of the rolling element.As shown in Figure22, the regularity and periodicity of the waveforms are not obvious, and the frequencies contained in frequency domain waveform are more complex. In addition, the interference frequencies and noise make it difficult to extract fault features directly from frequency domain. Therefore, we extract the fault features by using the proposed approach. Firstly, the parameters of MOMEDA are optimized by GOA algorithm and the results are shown in Figure 23.Figure 22
Rolling element signal and frequency domain.Figure 23
Optimization results of MOMEDA obtained by GOA.
(a)
The search history of grasshoppers (b)
Average fitness curve and convergence curve of GOAAs shown in Figure23, along with the increasing of iteration times, the grasshopper swarms tend to converge towards the best location, and the convergent curve gradually converges to the minimum. When the number of iterations m=9, the MKurt (MKurt=0.18179) is the maximum, and the optimal parameters γ=(L,Ti)=(494,85.5723) of MOMEDA are obtained. Then, the impact component in the rolling element signal is extracted by the optimized MOMEDA, and the envelope of the impact signal is given in Figure 24. When the frequencies are 140.6Hz and its multiples, several spectral peaks can be observed clearly. Moreover, these peak frequencies are close to the characteristic frequency (141.1693Hz) of rolling element and its harmonics. Due to the effect of interference noise, there are some slight deviations between the measured values and the theoretical values. However, the rolling element fault of the bearing can be identified accurately by the proposed approach.Figure 24
The envelope of the rolling element impact signal.To explore the impact of nonoptimal parameters on the results, we compared four sets of MOMEDA methods with fixed parameters. As shown in Figures25(a) and 25(b), when the filter size L of MOMEDA is the optimal value L=494 obtained by GOA and Ti=50, the frequency 240.2Hz and its harmonics can be observed clearly. Meanwhile, when Ti=90, the frequency 134.8Hz and its harmonics are obvious. However, these two frequencies and their harmonics are independent of the fault characteristic frequency. Therefore, the values of these two sets of parameters are not appropriate. As shown in Figure 25(c), when Ti of MOMEDA is set to the optimal value Ti=85.5723 and L=20, the effective frequency is submerged by the interference frequencies and the fault features cannot be extracted. So, if the filter size is too small, the noise in the signal cannot be effectively removed, and the feature extraction method will completely fail. In Figure 25(d), 140.6Hz and its harmonics can be observed clearly when L=900. To sum up, the feature extraction method will fail when the periodic initial value Ti is not appropriate. In general, the larger Ti, the smaller the characteristic frequency of the filtered signal. The filter size L affects the noise reduction effect; if L is too small, the interference frequencies and noise in the signal cannot be removed effectively.Figure 25
The effect of nonoptimal parameters on the results.
(a) (b) (c) (d)To demonstrate the superiority of the approach, the method is compared with the methods MED, MCKD, and FSK with the optimal parameters. The optimal filter sizeL=494 obtained by GOA is used as the filter size of MED and MCKD, and the optimal periodic initial value Ti=85.5723 is taken as the period of MCKD. The number of iterations of MED and MCKD is both set to 30. According to the characteristics of the rolling element signal, the decomposition level of FSK is set to 4. As shown in Figure 26, the envelope spectra obtained by MED, MCKD, and FSK all contain a large number of unrelated frequency components, and the effective frequencies are all submerged in the interference frequencies. For the rolling element signal, the above three feature extraction methods cannot achieve satisfactory results. Therefore, it is particularly difficult to extract the rolling element characteristic frequency compared with the inner ring and outer ring. However, the proposed method still achieves good results in the feature extraction of the rolling element.Figure 26
Comparison of feature extraction methods for the rolling element signal.
(a)
Kurtogram analysis of the rolling element signal (b)
The envelope analysis of impact signals obtained by MED, MCKD, and FSK


## 5.2. Case2: Vibration Signal Analysis of NASA Bearings

To verify the universality and reliability of the proposed approach, we analyze the NASA bearing dataset [41] by using the proposed approach. Figure 27 shows the test platform [47] of NASA, and four lubricated Rexnord ZA-2115 bearings are fixed on the spindle of the test rig. The spindle rotates under the drive of the motor, and its speed is 2000 rpm. Therefore, the rotating frequency of the bearing is 33.33 Hz (fr = 2000/60 = 33.33 Hz). The PCB353B33 acceleration sensors are fixed on the bearing housing, and the signals are collected by NI DAQ Card 6062E data acquisition card. The data length N=20480 and the sampling frequency fs=20KHz. In this experiment, the inner ring signal comes from Bearing 3 of No. 1 dataset, the outer ring signal comes from Bearing 1 of No. 2 dataset, the rolling element signal comes from Bearing 3 of No. 1 dataset, and the data length is 2048. The parameters of Rexnord ZA-2115 bearing are listed in Table 4. Based on the parameters listed in Table 4 and the formulas in literature [45], we can calculate the corresponding characteristic frequencies and they are listed in Table 5.Table 4
The Rexnord ZA-2115 bearing parameters.
Ball diameter (d) Number of balls (Z) Pitch diameter (D) Contact angle (α) 8.407 mm 16 71.501 mm 15.17Table 5
Characteristic frequencies.
Inner ring Outer ring Rolling element 296.91Hz 236.4Hz 139.9HzFigure 27
Bearing experiment platform.The structure and operation mode of bearings share some similarities, and the bearings vibration signals have a corresponding periodicity. Therefore, the initialization of parameters in this section is the same as that in Section5.1.1.


### 5.2.1. Inner Ring Signal Analysis of NASA Bearings

As shown in Figure28, due to the interference of interference frequencies and noise, it is difficult to extract the inner ring fault features. Therefore, we extract the inner ring fault features by using the proposed method. Firstly, the MOMEDA parameters are optimized by GOA algorithm.Figure 28
Inner ring signal and its frequency domain.As shown in Figure29, along with the increasing of iteration times, the convergent curve gradually converges to the minimum. When the number of iterations m=3, the MKurt (MKurt=0.2168) is the maximum, and the optimal parameters γ=(L,Ti)=(498,67.7462) of MOMEDA are obtained. Then, the impact component in the inner ring signal is extracted by the optimized MOMEDA, and the envelope of the impact signal is given in Figure 30(d). When the frequencies are 293 Hz and its multiples, several spectral peaks can be observed clearly. Moreover, these peak frequencies are approximate to the characteristic frequency (296.91Hz) of inner ring and its harmonics. Due to the interference of background noise and transmission path, there are some slight deviations between the measured values and the theoretical values. However, the proposed approach can effectively diagnose the inner ring fault of the bearing.Figure 29
Optimization results of MOMEDA obtained by GOA.Figure 30
The envelope analysis of impact signals obtained by MED, MCKD, FSK, and MOMEDA.To demonstrate the superiority of the approach, the method is compared with the methods with the optimal parameters, such as MED, MCKD, and FSK. Since the norms of MED, MCKD, and MOMEDA are based on kurtosis, the optimal filter sizeL=498 obtained by GOA is used as the filter size of MED and MCKD, and the optimal periodic initial value Ti=67.7462 is taken as the period of MCKD. The number of iterations of MED and MCKD is both set to 30. In view of the characteristic of the signal, the decomposition level of FSK is set to 3. The kurtosis of the filtered signal obtained by FSK is maximum when the decomposition level is 2. At this point, the central frequency of the filtered signal is 8750Hz, the bandwidth is 2500Hz, and the envelope of the filtered signal is given in Figure 30(c). As shown in Figures 30(a) and 30(c), the envelope spectra obtained by MED and FSK are disturbed by many irrelevant frequencies, and it is unrealistic to extract the fault characteristic frequency. In Figure 30(b), although MCKD can extract the inner ring fault features, the frequency and its harmonics are disturbed by many irrelevant frequencies. Therefore, the proposed approach has outstanding advantages in the feature extraction of the inner ring.


### 5.2.2. Outer Ring Signal Analysis of NASA Bearings

As shown in Figure31, although the waveforms have a certain regularity, it is difficult to extract fault features directly from the frequency domain. Therefore, we extract the outer ring fault features by using the proposed method. Firstly, the parameters of MOMEDA are optimized by GOA algorithm and the results are shown in Figure 32.Figure 31
Outer ring signal and its frequency domain.Figure 32
Optimization results of MOMEDA obtained by GOA.When the number of iterationsm=9, the MKurt (Mkurt=0.2385) is the maximum, and the optimal parameters γ=(L,Ti)=(496,84.7148) of MOMEDA are obtained. Then, the impact component in the outer ring signal is extracted by the optimized MOMEDA, and the envelope of the impact component is given in Figure 33(d). When the frequencies are 234.4Hz and its multiples, several spectral peaks can be observed clearly. Moreover, these peak frequencies are approximate to the outer ring characteristic frequency (236.4Hz) and its harmonics. Due to the interference of interference noise, there are slight deviations between the measured values and the theoretical values. However, the proposed approach can effectively diagnose the outer ring fault.Figure 33
The envelope analysis of impact signals obtained by MED, MCKD, FSK, and MOMEDA.Then, we compared the proposed approach with the methods MED, MCKD, and FSK with the optimal parameters. The optimal filter sizeL=496 obtained by GOA is used as the filter size of MED and MCKD, and the optimal periodic initial value Ti=84.7148 is taken as the period of MCKD. The number of iterations of MED and MCKD is both set to 30, and the decomposition level of the FSK is set to 3. The kurtosis of the filtered signal obtained by FSK is maximum when the decomposition level is 1.5. At this point, the central frequency of the filtered signal is 1666.666Hz, the bandwidth is 3333.33Hz, and the envelope analysis of the filtered signal is shown in Figure 33(c). In Figures 33(a) and 33(c), the characteristic frequencies and their harmonics can be observed. However, these frequencies are disturbed by many interference frequencies. Therefore, it is difficult to identify the outer ring characteristic frequency through MED and FSK. In Figure 33(b), although MCKD can extract the outer ring characteristic frequency and its harmonics, the amplitudes of these harmonics are small and the spectral peaks are not obvious. Therefore, the proposed approach has outstanding performance in the feature extraction of the outer ring.


### 5.2.3. Rolling Element Signal Analysis of NASA Bearings

As shown in Figure34, the regularity of the waveforms is not significant, and the frequencies contained in the frequency domain waveform are more complex. In addition, the interference frequencies and noise make it difficult to extract fault features directly from frequency domain. Therefore, we extract the fault features by the proposed approach. Firstly, the parameters of MOMEDA are optimized by GOA algorithm and the results are shown in Figure 35.Figure 34
Rolling element signal and its frequency domain.Figure 35
Optimization results of MOMEDA obtained by GOA.As shown in Figure35, when the number of iterations m=2, the MKurt (Mkurt=0.19684) is the maximum, and the optimal parameters γ=(L,Ti)=(500,143.0353) of MOMEDA are obtained. Then, the impact component is extracted by the optimized MOMEDA, and the envelope of the impact signal is given in Figure 36(d). When the frequencies are 136.7Hz and its multiples, several spectral peaks can be observed clearly. Moreover, these peak frequencies are approximate to the characteristic frequency (139.9Hz) of rolling element and its harmonics. Due to the interference of background noise and transmission path, there are some slight deviations between the measured values and the theoretical values. However, the proposed approach can effectively diagnose the rolling element fault.Figure 36
The envelope analysis of impact signals obtained by MED, MCKD, FSK, and MOMEDA.Then, the proposed method is compared with the methods MED, MCKD, and FSK with the optimal parameters. The optimal filter sizeL=500 obtained by GOA is used as the filter size of MED and MCKD, and the optimal period initial value Ti=143.0353 is taken as the period of MCKD. The number of iterations of MED and MCKD is both set to 30, and the decomposition level of FSK is set to 3. The kurtosis of the filtered signal obtained by FSK is maximum when the decomposition level is 0. At this point, the central frequency of the filtered signal is 5000Hz, the bandwidth is 10000Hz, and the envelope of the filtered signal is given in Figure 36(c). As shown in Figures 36(a) and 36(c), the envelope spectra obtained by MED and FSK are disturbed by many irrelevant frequencies, and it is difficult to extract the rolling element characteristic frequency through MED and FSK. In Figure 36(b), although MCKD can extract the characteristic frequency, the deviation between the peak frequency (146.5Hz) and fault characteristic frequency (136.7Hz) is large. Therefore, the proposed approach has excellent performance in the feature extraction of the rolling element.


## 5.2.1. Inner Ring Signal Analysis of NASA Bearings

As shown in Figure28, due to the interference of interference frequencies and noise, it is difficult to extract the inner ring fault features. Therefore, we extract the inner ring fault features by using the proposed method. Firstly, the MOMEDA parameters are optimized by GOA algorithm.Figure 28
Inner ring signal and its frequency domain.As shown in Figure29, along with the increasing of iteration times, the convergent curve gradually converges to the minimum. When the number of iterations m=3, the MKurt (MKurt=0.2168) is the maximum, and the optimal parameters γ=(L,Ti)=(498,67.7462) of MOMEDA are obtained. Then, the impact component in the inner ring signal is extracted by the optimized MOMEDA, and the envelope of the impact signal is given in Figure 30(d). When the frequencies are 293 Hz and its multiples, several spectral peaks can be observed clearly. Moreover, these peak frequencies are approximate to the characteristic frequency (296.91Hz) of inner ring and its harmonics. Due to the interference of background noise and transmission path, there are some slight deviations between the measured values and the theoretical values. However, the proposed approach can effectively diagnose the inner ring fault of the bearing.Figure 29
Optimization results of MOMEDA obtained by GOA.Figure 30
The envelope analysis of impact signals obtained by MED, MCKD, FSK, and MOMEDA.To demonstrate the superiority of the approach, the method is compared with the methods with the optimal parameters, such as MED, MCKD, and FSK. Since the norms of MED, MCKD, and MOMEDA are based on kurtosis, the optimal filter sizeL=498 obtained by GOA is used as the filter size of MED and MCKD, and the optimal periodic initial value Ti=67.7462 is taken as the period of MCKD. The number of iterations of MED and MCKD is both set to 30. In view of the characteristic of the signal, the decomposition level of FSK is set to 3. The kurtosis of the filtered signal obtained by FSK is maximum when the decomposition level is 2. At this point, the central frequency of the filtered signal is 8750Hz, the bandwidth is 2500Hz, and the envelope of the filtered signal is given in Figure 30(c). As shown in Figures 30(a) and 30(c), the envelope spectra obtained by MED and FSK are disturbed by many irrelevant frequencies, and it is unrealistic to extract the fault characteristic frequency. In Figure 30(b), although MCKD can extract the inner ring fault features, the frequency and its harmonics are disturbed by many irrelevant frequencies. Therefore, the proposed approach has outstanding advantages in the feature extraction of the inner ring.


## 5.2.2. Outer Ring Signal Analysis of NASA Bearings

As shown in Figure31, although the waveforms have a certain regularity, it is difficult to extract fault features directly from the frequency domain. Therefore, we extract the outer ring fault features by using the proposed method. Firstly, the parameters of MOMEDA are optimized by GOA algorithm and the results are shown in Figure 32.Figure 31
Outer ring signal and its frequency domain.Figure 32
Optimization results of MOMEDA obtained by GOA.When the number of iterationsm=9, the MKurt (Mkurt=0.2385) is the maximum, and the optimal parameters γ=(L,Ti)=(496,84.7148) of MOMEDA are obtained. Then, the impact component in the outer ring signal is extracted by the optimized MOMEDA, and the envelope of the impact component is given in Figure 33(d). When the frequencies are 234.4Hz and its multiples, several spectral peaks can be observed clearly. Moreover, these peak frequencies are approximate to the outer ring characteristic frequency (236.4Hz) and its harmonics. Due to the interference of interference noise, there are slight deviations between the measured values and the theoretical values. However, the proposed approach can effectively diagnose the outer ring fault.Figure 33
The envelope analysis of impact signals obtained by MED, MCKD, FSK, and MOMEDA.Then, we compared the proposed approach with the methods MED, MCKD, and FSK with the optimal parameters. The optimal filter sizeL=496 obtained by GOA is used as the filter size of MED and MCKD, and the optimal periodic initial value Ti=84.7148 is taken as the period of MCKD. The number of iterations of MED and MCKD is both set to 30, and the decomposition level of the FSK is set to 3. The kurtosis of the filtered signal obtained by FSK is maximum when the decomposition level is 1.5. At this point, the central frequency of the filtered signal is 1666.666Hz, the bandwidth is 3333.33Hz, and the envelope analysis of the filtered signal is shown in Figure 33(c). In Figures 33(a) and 33(c), the characteristic frequencies and their harmonics can be observed. However, these frequencies are disturbed by many interference frequencies. Therefore, it is difficult to identify the outer ring characteristic frequency through MED and FSK. In Figure 33(b), although MCKD can extract the outer ring characteristic frequency and its harmonics, the amplitudes of these harmonics are small and the spectral peaks are not obvious. Therefore, the proposed approach has outstanding performance in the feature extraction of the outer ring.


## 5.2.3. Rolling Element Signal Analysis of NASA Bearings

As shown in Figure34, the regularity of the waveforms is not significant, and the frequencies contained in the frequency domain waveform are more complex. In addition, the interference frequencies and noise make it difficult to extract fault features directly from frequency domain. Therefore, we extract the fault features by the proposed approach. Firstly, the parameters of MOMEDA are optimized by GOA algorithm and the results are shown in Figure 35.Figure 34
Rolling element signal and its frequency domain.Figure 35
Optimization results of MOMEDA obtained by GOA.As shown in Figure35, when the number of iterations m=2, the MKurt (Mkurt=0.19684) is the maximum, and the optimal parameters γ=(L,Ti)=(500,143.0353) of MOMEDA are obtained. Then, the impact component is extracted by the optimized MOMEDA, and the envelope of the impact signal is given in Figure 36(d). When the frequencies are 136.7Hz and its multiples, several spectral peaks can be observed clearly. Moreover, these peak frequencies are approximate to the characteristic frequency (139.9Hz) of rolling element and its harmonics. Due to the interference of background noise and transmission path, there are some slight deviations between the measured values and the theoretical values. However, the proposed approach can effectively diagnose the rolling element fault.Figure 36
The envelope analysis of impact signals obtained by MED, MCKD, FSK, and MOMEDA.Then, the proposed method is compared with the methods MED, MCKD, and FSK with the optimal parameters. The optimal filter sizeL=500 obtained by GOA is used as the filter size of MED and MCKD, and the optimal period initial value Ti=143.0353 is taken as the period of MCKD. The number of iterations of MED and MCKD is both set to 30, and the decomposition level of FSK is set to 3. The kurtosis of the filtered signal obtained by FSK is maximum when the decomposition level is 0. At this point, the central frequency of the filtered signal is 5000Hz, the bandwidth is 10000Hz, and the envelope of the filtered signal is given in Figure 36(c). As shown in Figures 36(a) and 36(c), the envelope spectra obtained by MED and FSK are disturbed by many irrelevant frequencies, and it is difficult to extract the rolling element characteristic frequency through MED and FSK. In Figure 36(b), although MCKD can extract the characteristic frequency, the deviation between the peak frequency (146.5Hz) and fault characteristic frequency (136.7Hz) is large. Therefore, the proposed approach has excellent performance in the feature extraction of the rolling element.


## 6. Discussions

Through the comparison and analysis of several sets of experiments, the effectiveness and significant advantages of the proposed method are proved. The experimental process and preliminary conclusions of this paper are discussed below.( 1 ) It can be seen from the analysis of the simulation signal that MED, MCKD, and FSK cannot achieve satisfactory results in noisy environments. However, the characteristic frequencies contained in the noisy signal can be extracted effectively by the proposed approach. By analyzing the bearing signals come from two different experimental platforms, the results show that the fault features of the inner ring and outer ring are obvious, and the fault features are easy to be extracted. The rolling element with local damage periodically hits the inner ring and outer ring, which leads to the coupling effect between the inner and outer ring [44]. In addition, the rolling element vibration signal is affected by background noise and interference components. Therefore, the feature extraction of the rolling element is difficult and the characteristic frequency is not obvious. Nevertheless, the proposed method still achieves the desired effect, and the validity and reliability of the method are verified.( 2 ) It is known from the parameter initialization that the filter size is positively correlated with the MKurt of the filtered signal, but the MKurt curve is not a strictly monotone increasing curve. Therefore, there is a local optimal solution for the filter size. In addition, the change of MKurt is irregular when the periodic initial value changes, so the parameter optimization is very difficult. It is known from the experiments with fixed parameters that a large amount of noise and interference is removed when the filter size is large, and the envelope spectra are clear. However, the efficiency of the algorithm is very low when the filter size is too large, and the fault impact signal may be removed. The larger the periodic initial value is, the smaller the characteristic frequency extracted from the filtered signal is. Comparison experiments of optimization algorithms show that GOA method is not only simple in calculation, but also high in convergent speed and accuracy. From the optimization process of GOA, it is known that the grasshoppers are widely distributed in the whole search space at the beginning of iteration, which ensures the global search scope. At the end of iteration, the grasshopper swarm approaches the optimal position and moves slowly, which ensures the local search accuracy. Therefore, the GOA algorithm guarantees the reliability of parameter optimization.( 3 ) The optimal parameters of MOMEDA obtained by GOA are used in MED and MCKD because these three deconvolution methods are all based on kurtosis norm. The experimental results demonstrate that MED is only suitable for single impact signal. Meanwhile, the envelope spectrum obtained by MED contains a large number of interference frequencies. Although MCKD can extract a series of impact components from the noisy signal, the higher harmonics obtained by this method are often disturbed by irrelevant frequencies. In the process of processing the weak noise signal, FSK can achieve satisfactory results. However, when the signal contains strong background noise or interference frequencies, it is sometimes difficult to extract fault features through FSK. Therefore, for the above three methods, signal preprocessing is necessary before feature extraction. All signals have not been preprocessed in this paper, and the proposed method still achieves satisfactory results.


## 7. Conclusions

A parameter adaptive MOMEDA feature extraction method based on GOA is proposed for solving the problem of the bearing fault diagnosis. The GOA is introduced to determine the optimal parameters of MOMEDA. The algorithm has its own unique advantages, and it updates the position of individual search agent by involving all search agents. In addition, GOA can adaptively determine the optimal filter size and periodic initial value of MOMEDA based on the signal characteristics and effectively avoid the randomness of artificial adoption. The simulation signal and bearing signals from two different experimental platforms are analyzed by the proposed method, satisfactory feature extraction results can be obtained, and the validity and reliability of the method are verified. Therefore, several meaningful contributions of this paper are as follows: (a) the main parameters affecting the filtering results and feature extraction results are evaluated; (b) a new optimization model containing these influence parameters is constructed; (c) GOA is introduced to determine the appropriate MOMEDA parameters while extracting fault features. Through several case studies, some beneficial conclusions are gained.(a) The effect of feature extraction is affected by the filter size and the periodic initial value, and the influences of these two parameters on the results are different. Therefore, these influencing factors must be considered comprehensively in the process of parameter optimization.(b) The feature extraction effect of the proposed method is compared with that of MOMEDA with fixed parameters. The results show that the filter size affects the noise reduction effect, and the periodic initial value affects the size of the characteristic frequency.(c) The proposed approach is compared with the three methods MED, MCKD, and FSK with optimal parameters. We found that the feature extraction effect of the proposed approach is better than that of the other three methods in all cases, and it meets the requirement of feature extraction for all fault types.However, the damage problem of the bearing is complicated in the actual industrial production process, the fault impact signals are often submerged by noises, and the composite fault may occur. Therefore, we will collect bearing data from the industrial environment and analyze these data by the approach. In addition, the composite fault experiment and its analysis will be implemented step by step, and the proposed method is expected to address these complex issues.



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*Source: 7182539-2019-03-05.xml*