Study on Fault Feature Extraction of Rolling Bearing Based on Improved WOA-FMD Algorithm

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Introduction
As a standard component in industrial production feld, bearing is widely applied in aerospace, energy, transportation, manufacturing, and other felds [1][2][3]. Because the bearings are used to support the rotating parts of machinery and bear various dynamic loads, they are prone to various faults during their operation [4,5]. Terefore, it is necessary to monitor the running state of the bearing and diagnose the fault types in time. Tis is of great signifcance to the safe and reliable operation of mechanical equipment and the reduction of equipment loss.
In the process of collecting vibration signals, the complex working environment makes the vibration signal mixed with a lot of noise. Te signal usually presents a nonlinear and nonstationary state [6]. In order to solve this problem, EMD, EWT, and VMD have been widely studied and applied in fault diagnosis [7][8][9].
Huang et al. [10] proposed a time signal decomposition method called EMD, which can adaptively decompose the signal into several intrinsic mode functions according to the characteristics of the data. Guo et al. [11] combined the shape controlling parameter based on cubic spline interpolation with EMD, and it has been applied to fault diagnosis of rolling bearing and achieved good results. However, the application efect of EMD in mechanical fault diagnosis is limited by mode overlap and boundary efect [12,13]. Wu et al. [14] proposed EEMD, which decreases the efect of mode mixing by adding white noise, but it cannot efectively solve the problem of endpoint efect [15]. Yeh et al. [16] proposed that CEEMD can decompose such nonstationary signals very well. Although the decomposition ability has been improved, a large number of noise residues still exist in the signal.
Gilles [17] proposed EWT based on wavelet transform. EWT is generated by splitting the Fourier spectrum, and binary band allocation may divide feature signals into different modes [18]. To solve the above problems, Dragomiretskiy and Zosso [19] proposed a VMD algorithm which can adaptively decompose through center frequency and bandwidth constraints. VMD essentially sets a set of Wiener flters, which adaptively decompose the signal into multiple narrowband signals with diferent frequencies. Tis avoids mode aliasing [20]. Jiang et al. [21] used the convergence trend phenomenon to quickly and adaptively determine the number of potential modes and the optimal initial center frequency of the signal. Based on this research, central frequency mode decomposition is proposed to better solve the problem of parameter adaptation [22]. Song et al. [23]overcame the problem of model parameter predefned in multivariate variational mode decomposition by using smart multichannel mode extraction and manifold learning methods.
Miao et al. [24] proposed a new decomposition mode-FMD, inspired by adaptive signal decomposition and deconvolution techniques. FMD is deduced from multiple FIR flter banks and calculating the maximum correlation peak deconvolution. It can consider the pulse property and periodicity of the signal at the same time and has good noise robustness. FMD algorithm requires manual input parameters, and diferent parameters have diferent efect on the fnal results, so it is not applicable. Yan et al. [25] proposed a PSO method to optimize the decomposition number of mode and flter length of FMD, which guarantees the parametric adaptability of feature mode decomposition. However, the infuence of the number of frequency bands on the decomposition results is neglected, and the iteration of PSO is prone to premature phenomenon and falls into local optimal solution [26,27]. Mirjalili and Lewis [28] proposed WOA to observe and simulate the predatory behavior of whales in 2016. WOA has the advantages of simple structure, fast searching speed, less parameters, and strong global convergence. In WOA, with the number of iterations increasing, the convergence speed of the algorithm gradually increases. It is easy to fall into a local optimal solution [29,30].
Under the interference of background noise, the EMD, CEEMD, and VMD algorithms are difcult to extract the fault characteristics of rolling bearings. When the signal is decomposed by the FMD method, in addition to the number of decomposed modes and the length of the flter, the number of frequency bands will also afect the decomposition results. Terefore, the number of frequency bands should also be included in the parameter optimization. Although whale optimization algorithm has advantages over particle swarm optimization algorithm in optimization speed and accuracy, it is easy to fall into local optimization in the process of optimization. Tis leads to the failure to achieve the optimal decomposition efect. In order to overcome the above problems, this paper proposes an improved WOA-FMD algorithm. Tis method introduces adaptive weights and Lévy fight into WOA. Te envelope entropy is used as the ftness function of the algorithm to optimize the FMD parameters. Te kurtosis value is used as an index for Hilbert envelope demodulation analysis to extract the fault characteristic frequency of the signal.

Feature Mode Decomposition.
FMD is a new nonrecursive signal decomposition method, mainly including the steps of designing FIR flter bank, updating flter, calculating cycle, and calculating decomposition mode. By initializing the flter bank and iteratively updating its coefcients, diferent modes can be selected simultaneously and adaptively. Figure 1 displays the fowchart of FMD algorithm, and the algorithm is as follows: (1) Step 1. Read the collected raw signal x and set FMD decomposition mode n, FIR flter bandwidth L, and iteration number I of FMD. (2) Step 2. Te FIR flter bank is initialized using K Hanning windows, and set start iteration i � 1.
Step 3. Use u i k � x * f i k to obtain a signal to remove interference, that is, the IMF components obtained after decomposition, where k � 1, 2, ......K, and * represents the convolution operation. (4) Step 4. Te flter coefcients are periodically updated based on the initial input signal x, the number of decomposed modes u i k , and the estimated fault frequency T i k . T i k is the time delay when the autocorrelation spectrum R i k of u i k reaches a local maximum after the frst zero crossing. (5) Step 5. Determine whether the current number of iterations has reached the maximum number of iterations. If it is not satisfed, go back to step 3 and repeat the iteration. Otherwise, proceed to step 6. (6) Step 6. Calculate the CC between two adjacent components and construct a correlation matrix with CC (K×K) . Select two adjacent modal components with the largest correlation coefcient and calculate their correlation kurtosis according to the estimated fault period . Finally, the modal component with large correlation kurtosis is selected as the modal component of FMD, set K � K − 1.
Step 7. Determine if the specifed n is the same as the number of modes K. If not, go back to step 3. Otherwise, the iteration stops and the modal component of the output FMD is n, which is the fnal decomposition result.

Whale Optimization Algorithm.
Te WOA algorithm mainly includes three kinds of predator-prey behavior simulation methods: surrounding prey, bubble net predator, and prey search. Te specifc steps are as follows: (1) Step 1. When searching the prey position is unknown, the WOA assumes that the solution with the smallest ftness among individuals in the current population is the target or closest prey and updates the position of other search individuals after defning the optimal solution. When p < 0.5 and |A| < 1, the behavior of surrounding prey is executed, and the update formula is 2 Shock and Vibration where a linearly decreases to 0 in the iteration process; T is the current iteration number; and T max is the maximum iteration number; A and C are synergy coefcients; rand() is a random number generated from the (0, 1) interval; D is the distance between the optimal whale and the current whale; X * (t) is the current optimal solution position; X(t) is the current solution position; and X(t + 1) is the position of the whale after moving to the optimal solution position. (2) Step 2. When p ≥ 0.5, the precontraction behavior of the bucket net predation of whales is simulated, and the position update formula is where b is a constant and assumes a spiral shape and L is a random number that oscillates at [− 1, 1]. according to the found optimal position, which enhances the search energy of the algorithm. Te formula is where X 1 is the random selection of the whale position. (4) Step 4. When T � T max , max reaches the maximum number of iterations, and the convergence factor a linearly decreases to 0, the best search agent is output. If not, return to continue iteration.

Feature Extraction Based on Improved WOA
3.1. Improved WOA. WOA is simpler and more efective than other algorithms, but it is easy to fall into local optimum in the optimization process, resulting in a decrease in the accuracy of the results. Te Lévy fight process adopts the search method of long-term small steps and occasional longdistance jumps, which helps WOA to expand the search scope and enhance the global exploration and search ability. Tis can prevent the solution from producing local optima and accelerate the convergence speed [31]. Te formula for Lévy fight random step is calculated as follows: Random numbers are solved using the Mantegna [32] method of normal fractions to generate a random step of the Lévy distribution, which is formulated as where β Is constant, and the value range is 0 . Te formula for calculating σ is given by After introducing Lévy fight strategy, the position update formula of whale optimization algorithm is In the WOA, the global search ability needs to be improved in the early stage, and the local search ability needs to be improved in the late stage. As the iteration time increases, the convergence speed of the algorithm increases. It will be prone to fail to jump out of the local optimal solution. In order to solve the problem, the inertia weight parameter, which can adjust the algorithm, is designed. Te larger weight can be used for global optimization in a wide range, and the smaller weight can be used for local optimization near the optimal solution. Te calculation principle of ω is where t is the current iteration number. Te improved whale optimization algorithm position update formula is In order to verify the performance of LMWOA, 10 basic standard test functions are used for experiments. Te results are shown in Table 1 and Figure 2. Comparing the minimum value of the test function with the optimization results of the three algorithms, the optimization results of LMWOA are closer. According to Figure 2, it can be seen that the optimization speed of LMWOA is faster than that of the other two algorithms. After improvement, WOA can jump out of the local optimum faster, ensuring the accuracy of the algorithm.

Feature Extraction Process.
Te parameter decomposition mode n, flter length L, and frequency band number K in FMD have great infuence on signal decomposition results. For example, if the flter length is too short, it may lead to poor fltering results. If the flter length is too long, it may lead to distortion and further increase the computational burden. If there are too many decomposed modes, the result may be redundant modes. Te larger the number of bands is, the heavier the computational burden will be, which afects the decomposition performance to a certain extent.
Terefore, this paper uses LMWOA to optimize the parameters of FMD algorithm in order to get rid of the dependence of FMD on the prior knowledge of the fault cycle of the original signal and avoid the infuence of human factors on the decomposition results. Tis can achieve the best decomposition efect. Its fowchart is shown in Figure 3. Te LMWOA-FMD algorithm is implemented as follows: (1) Step 1. Input the collected rolling bearing signal and put up (n, L, K) as the optimization parameters of the WOA. Te optimization dimension is 4, the number of iterations T max � 20, and the number of search population is 15. In order to avoid underdecomposition or over-decomposition, the search space of decomposition mode number n is [3,7]. In order to achieve a sample number that includes two adjacent repetitive transients related to bearing faults, the search space for the flter length L is assumed to be [10, rand(f s /f g )], where rand() is a rounding operation, and f s and f g are sampling frequency and bearing failure frequency,  10,100,4) [− 50, 50] 0 0.103 0.023 0.016   respectively; K must be greater than n for the mode to decompose, so the search space of K is set as [4,10]. (2) Step 2. Set the envelope entropy as the judgment value of the ftness function. When the complexity of vibration signal is low and the noise interference is small, the envelope entropy is also small. On the contrary, when the feature information is small, the noise interference is larger and the envelope entropy is larger [33]. (3) Step 3. Obtain the optimal parameter combination (n, L, K) by using LMWOA to optimize the parameter combination. (4) Step 4. Input the optimal parameter combination (n, L, K) into the FMD algorithm, disintegrate the signal into several IMF components, and use kurtosis value as the parameters of the target mode. Ten, the kurtosis value of each component is calculated separately. When a bearing fault occurs, the vibration signal will deviate from the normal distribution under the action of fault impact, and the larger the kurtosis value, the richer the impact component and fault information.

Simulated Signal Analysis.
Te feasibility of the LMWOA-FMD method was verifed by constructing the simulation signal of the inner ring fault of the rolling bearing, and the simulation signal expression is h(t) � e − ct sin 2πf n t , where A 0 is the amplitude, A 0 � 0.5; f r is frequency conversion, f r � 20Hz; c is the attenuation factor, c � 800; f n is the resonance frequency, f n � 4000Hz; S(t) is a periodic shock component; τ i represents the small fuctuation of the ith shock with respect to period T; and fault frequency f i � 1/T, f i � 120Hz. In the simulation signal, 1dB Gaussian white noise interference is added, the sampling points are selected as 10240, and the sampling frequency f s � 12kHz. Te time-domain waveform diagram and sum spectrogram of the simulation signal after adding noise are shown in Figure 4. Te PSO, WOA, and LMWOA methods are, respectively, used to search the optimal parameter combination of FMD, and the minimum envelope entropy is used as the ftness function. Te value change of ftness function is shown in Figure 5. In the fgure, LMWOA-FMD converges to obtain the best solution when the number of iterations reaches 5 and fnal ftness function value is 9.08. Compared with PSO and WOA, LMWOA has better search ability, has faster convergence speed, and can seek the optimal FMD parameters faster. It is verifed that Lévy fight and adaptive weights can efectively avoid falling into local optimum and improve algorithm performance, which can save a lot of time for optimization.
After optimization by LMWOA, the fnal optimal parameter combination is as follows: decomposition mode n � 4, flter length L � 15, and frequency band number K � 5. Te parameters are input into FMD, and the decomposition efect is shown in Figure 6. Te frequency of each mode component in the decomposition result is uniformly distributed, which avoids the appearance of mode aliasing and avoids losing some important information.
Te kurtosis values of the four IMF components obtained after LMWOA-FMD treatment were calculated, respectively, and the results are shown in Table 2. After comparison of kurtosis values, IMF3 component is selected for Hilbert transform analysis of envelope demodulation, and its envelope spectrum is shown in Figure 7. According to the fgure, it can be seen clearly that the extracted fault frequency is 120Hz. At the same time, the dual frequency of 240Hz and the triple frequency of 360Hz are also accurately extracted. Te inner ring failure frequency extracted by this method is near to the theoretical value, which means that the rolling bearing has inner ring damage. It is verifed that this method can accurately identify fault characteristics under background noise and has certain noise robustness.
Te FMD of prior parameters was compared, where decomposition mode number n � 5, flter length L � 100, and frequency band number K � 6. In Figure 8, there is no mode aliasing during FMD decomposition. However, most components have passband ripple, which has a certain impact on the fnal decomposition result.
At the same time, the CEEMD and VMD methods are also used to compare and verify. Te number of mode decomposition of VMD is 5, and the penalty factor is 1500. Te component with the maximum kurtosis value is continued to be used for envelope demodulation. Te envelope spectra of the CEEMD, VMD, and FMD are shown in   Figure 9. In the fgure, the envelope spectrum of CEEMD and FMD can extract the fault frequency under the interference of noise. However, there are many interference lines and the efect is not ideal. Further processing is required if the fault frequency is to be accurately extracted. Te envelope spectrum of VMD method can accurately extract the fault frequency. However, the efect is not obvious due to the infuence of the interference spectral line at the position of the second and third harmonic, and the robustness to noise is poor. In order to further compare the advantages of these four methods, FFR is used as the evaluation index of the four methods. Te larger the FFR value is, the more the periodic impact components are included in the IMF component [34]. Te FFR is defned as where S is the sum of the amplitude of the envelope spectrum; f is fault characteristic frequency; and S(kf) is the amplitude of the envelope spectrum corresponding to each octave of the fault characteristic frequency. Table 3 shows the FFR values corresponding to the four methods. Te FFR value of LMWOA-FMD method is 0.059, which is higher than that of CEEMD, VMD, and FMD. Tis proves the superiority of this method.     Figure 7: Te analysis result by LMWOA-FMD method for the simulated inner ring fault signal.

Experimental Signal Analysis.
For the sake of further verifying the performance of the LMWOA-FMD method in practical application, the bearing dataset provided by CWRU [35] was used for experimental verifcation. Te CWRU rolling bearing test stand is shown in Figure 10.
In the test bench, the driving end bearing of the induction motor with rated power of 1.5 kW and speed of 1797 r/min was selected as the research object. Te bearing model was SKF6205-2RS deep groove ball bearing, and the sampling frequency was 12 kHz. Te inner ring, outer ring, and roller components of the bearing were processed by the EDM method to produce tiny pits with the size of 0.117 mm to simulate the fault of the bearing. Te bearing rotational frequency is 29.95 Hz, and the outer ring fault frequency is 107.3 Hz according to its parameters. Te time-domain waveform and spectrum diagram with Gaussian white noise are shown in Figure 11.
Te PSO, WOA, and LMWOA methods are, respectively, used to optimize FMD. Te minimum envelope entropy is used as the ftness function to fnd the optimal parameter combination. Te change of ftness function value is shown in Figure 12. Te LMWOA-FMD in the fgure converges to the optimal solution when the number of iterations reaches 3. Te fnal ftness function value was 9.05. It is proved that the method avoids falling into local optimum and can accurately search the optimal solution. Compared with PSO and WOA, it has stronger searching ability, faster convergence speed, and faster searching for FMD optimal parameters.
Te decomposition mode number n � 4, flter length L � 34, and frequency band number K � 5 are optimized by LMWOA. Te parameters are input into FMD, and the decomposition efect is shown in Figure 13. Te frequency belt of each modal component in the decomposition result is narrow, avoiding the phenomenon of mode overlap.
After carrying out LMWOA-FMD, four IMF components were obtained to calculate their kurtosis values, respectively, and the results are shown in Table 4. Te component IMF3 with the largest kurtosis value is selected        to perform Hilbert transform for envelope demodulation analysis, and its envelope spectrum is shown in Figure 14. It can be clearly seen in the fgure that the extracted fault frequency is 107.8Hz, which is close to the theoretical value of 107.3Hz. Te frequency doubling characteristics can be clearly seen in the fgure, and the spectral lines evenly distributed on both sides of the frequency doubling are low. It can be seen that this method can accurately extract the fault frequency under the noise while suppressing the infuence of frequency conversion signal and noise on the fault signal and has certain robustness to noise. Te FMD of prior parameters was compared, where decomposition mode number n � 5, flter length L � 100, and frequency band number K � 6. Te result is shown in Figure 15. In the fgure, IMF2 and IMF5 have passband ripple and IMF3 and IMF4 have mode mixing. Te signal is not well decomposed, and redundant information appears.
Furthermore, CEEMD and VMD methods are used to compare and verify the advantages of this method. Te number of mode decomposition of VMD is 5, and the penalty factor is 1500. Te envelope spectra of the CEEMD, VMD, and FMD are shown in Figure 16. In the fgure, the envelope spectra of the three methods accurately extract the fault frequency. However, the amplitude efects of the second and third harmonic of the envelope spectrum are not signifcant. Compared with LMWOA-FMD, the overall extraction performance of the three methods is not ideal.
In order to further compare the advantages of these four methods, FFR is used as the evaluation index of the four methods. Table 5 shows the FFR values corresponding to other three methods. Among the four methods, the FFR value of the LMWOA-FMD algorithm is 0.122, which is greater than that of other three methods. It has been proven that this method has better noise resistance than other three methods.
Te LMWOA-FMD method is used to decompose the inner ring signal of rolling bearing with Gaussian white noise. Te optimized decomposition results are the number of decomposition modes n � 4, the flter length L � 20, and the number of frequency bands K � 7. Te parameters are input into the FMD, and the decomposition result is shown in Figure 17. LMWOA-FMD divides the signal into four parts, avoiding redundant information and mode aliasing and avoiding the loss of important information.
After carrying out LMWOA-FMD, four IMF components were obtained to calculate their kurtosis values, respectively, and the results are shown in Table 6. Te component IMF3 with the largest kurtosis value is selected to perform Hilbert transform for envelope demodulation analysis, and its envelope spectrum is shown in Figure 18. Although the spectrum line is cluttered under the infuence of noise in the fgure, the extracted fault frequency is prominent in the envelope spectrum, which is only 0.5Hz diferent from the theoretical value of 162.2Hz, and twice the frequency of the fault feature can be extracted. It can be seen that the method can accurately extract the fault frequency under noise and has certain noise robustness.
After the same FMD parameters are imputed manually, the results are as shown in Figure 19. Passband ripple occurs in IMF2 and IMF5 components, which may cause a large amount of interference information in the signal after  Also, input the signal to CEEMD and VMD, where the number of mode decomposition of VMD is 5 and the penalty factor is 1500. Te envelope spectra of the three methods are shown in Figure 20. Te fault frequencies of the envelope spectra of three methods in the fgure are completely submerged by noise frequency and there are many interference lines, which make it impossible to extract fault features accurately. Compared with LMWOA-FMD, the two methods have poor noise resistance and cannot be applied in bearing fault diagnosis under noise background. Table 7 shows the FFR values of four diferent methods. Among the four methods, the FFR value of the LMWOA-FMD algorithm is 0.022, which is greater than that of other three methods. It has been proven that the LMWOA-FMD method has better noise resistance than other three methods.

Conclusion
Te improved WOA-FMD algorithm is proposed for fault feature extraction of rolling bearing under noise in this paper. Te main contributions of the paper are as follows: (1) Te improved WOA with Lévy fight and adaptive weight can fnd the optimal value faster and more accurately than PSO and WOA in the optimization process of test function. It can fnd the optimal solution before 5 iterations in FMD optimization, which has stronger search ability and avoids falling into local optimal solution. (2) Te LMWOA-FMD algorithm successfully decomposes the original signal into multiple IMF components without mode aliasing and passband ripple. Tis overcomes the problem of input parameters by prior values and may realize the parameter adaptation of FMD. It can extract fault features better and improve the accuracy of signal decomposition. (3) Trough the analysis of simulated and experimental signals, the fault feature ratio extracted by this method in the background noise reaches 0.059 and 0.122. Tis is larger than the fault feature ratio of CEEMD, VMD, and FMD methods. It is proved that this method has strong noise robustness and can extract fault features more accurately.

Data Availability
Te data used to support the fndings of this study are included within the article.

Conflicts of Interest
Te authors declare that they have no conficts of interest.