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

Jia, Guangfei; Meng, Yanchao

doi:https://doi.org/10.1155/2023/5097144

Shock and Vibration

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Abstract Introduction Conclusion Abbreviations Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2023 | Article ID 5097144 | https://doi.org/10.1155/2023/5097144

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

Guangfei Jia¹and Yanchao Meng¹

Academic Editor: Xingxing Jiang

Received20 Apr 2023

Revised30 May 2023

Accepted31 May 2023

Published14 Jun 2023

Abstract

The vibration signal of rolling bearing fault is nonlinear and nonstationary under the interference of background noise, and it is difficult to extract fault features from it. When feature mode decomposition is used to analyze signals, prior parameter settings can easily affect the decomposition results. Therefore, a fault feature extraction method based on improved whale optimization algorithm is proposed to optimize feature modal decomposition parameters. The improved WOA integrates Lévy flight and adaptive weight, and envelope entropy is used as fitness function to optimize feature modal decomposition parameters. The feature mode decomposition of the original signal is performed using the optimal combination of parameters to obtain multiple IMF components. The optimal IMF component envelope demodulation analysis is selected according to the kurtosis value, and the fault feature is extracted through the envelope spectrum. Comparing the LMWOA method with PSO and WOA methods by simulated and experimental signals, the results show that the optimization speed of LMWOA is faster than that of other methods. Compared with CEEMD, VMD, and FMD methods, the improved WOA-FMD method has higher fault feature ratio and can accurately extract fault features under noise interference. This method can effectively solve the parameter adaptive ability and improve the accuracy of fault diagnosis, which has practical significance.

1. Introduction

As a standard component in industrial production field, bearing is widely applied in aerospace, energy, transportation, manufacturing, and other fields [1–3]. 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]. Therefore, it is necessary to monitor the running state of the bearing and diagnose the fault types in time. This is of great significance 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. The 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 [7–9].

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 effect of EMD in mechanical fault diagnosis is limited by mode overlap and boundary effect [12, 13]. Wu et al. [14] proposed EEMD, which decreases the effect of mode mixing by adding white noise, but it cannot effectively solve the problem of endpoint effect [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 filters, which adaptively decompose the signal into multiple narrowband signals with different frequencies. This 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 predefined 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 filter 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 different parameters have different effect on the final results, so it is not applicable. Yan et al. [25] proposed a PSO method to optimize the decomposition number of mode and filter length of FMD, which guarantees the parametric adaptability of feature mode decomposition. However, the influence 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 difficult 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 filter, the number of frequency bands will also affect the decomposition results. Therefore, 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. This leads to the failure to achieve the optimal decomposition effect. In order to overcome the above problems, this paper proposes an improved WOA-FMD algorithm. This method introduces adaptive weights and Lévy flight into WOA. The envelope entropy is used as the fitness function of the algorithm to optimize the FMD parameters. The kurtosis value is used as an index for Hilbert envelope demodulation analysis to extract the fault characteristic frequency of the signal.

2. Fundamental Algorithms

2.1. Feature Mode Decomposition

FMD is a new nonrecursive signal decomposition method, mainly including the steps of designing FIR filter bank, updating filter, calculating cycle, and calculating decomposition mode. By initializing the filter bank and iteratively updating its coefficients, different modes can be selected simultaneously and adaptively. Figure 1 displays the flowchart of FMD algorithm, and the algorithm is as follows:(1)Step 1. Read the collected raw signal and set FMD decomposition mode n, FIR filter bandwidth L, and iteration number I of FMD.(2)Step 2. The FIR filter bank is initialized using Hanning windows, and set start iteration .(3)Step 3. Use to obtain a signal to remove interference, that is, the IMF components obtained after decomposition, where , and represents the convolution operation.(4)Step 4. The filter coefficients are periodically updated based on the initial input signal , the number of decomposed modes , and the estimated fault frequency . is the time delay when the autocorrelation spectrum of reaches a local maximum after the first zero crossing.(5)Step 5. Determine whether the current number of iterations has reached the maximum number of iterations. If it is not satisfied, 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 . Select two adjacent modal components with the largest correlation coefficient 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 .(7)Step 7. Determine if the specified is the same as the number of modes . If not, go back to step 3. Otherwise, the iteration stops and the modal component of the output FMD is , which is the final decomposition result.

2.2. Whale Optimization Algorithm

The WOA algorithm mainly includes three kinds of predator-prey behavior simulation methods: surrounding prey, bubble net predator, and prey search. The specific steps are as follows:(1)Step 1. When searching the prey position is unknown, the WOA assumes that the solution with the smallest fitness among individuals in the current population is the target or closest prey and updates the position of other search individuals after defining the optimal solution. When and , the behavior of surrounding prey is executed, and the update formula is where linearly decreases to 0 in the iteration process; is the current iteration number; and is the maximum iteration number; and are synergy coefficients; is a random number generated from the interval; is the distance between the optimal whale and the current whale; is the current optimal solution position; is the current solution position; and is the position of the whale after moving to the optimal solution position.(2)Step 2. When , the precontraction behavior of the bucket net predation of whales is simulated, and the position update formula is where is a constant and assumes a spiral shape and L is a random number that oscillates at .(3)Step 3. When and , WOA will enter the global search and randomly find the position of a search body in the population to replace the found optimal position. It no longer updates the position according to the found optimal position, which enhances the search energy of the algorithm. The formula is where is the random selection of the whale position.(4)Step 4. When , 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.

3. Feature Extraction Based on Improved WOA

3.1. Improved WOA

WOA is simpler and more effective 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. The Lévy flight process adopts the search method of long-term small steps and occasional long-distance jumps, which helps WOA to expand the search scope and enhance the global exploration and search ability. This can prevent the solution from producing local optima and accelerate the convergence speed [31]. The formula for Lévy flight 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 aswhere Is constant, and the value range is . Take ; ; . The formula for calculating is given by

After introducing Lévy flight 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. The 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. The calculation principle of iswhere is the current iteration number.

The 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. The 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.

3.2. Feature Extraction Process

The parameter decomposition mode , filter length , and frequency band number in FMD have great influence on signal decomposition results. For example, if the filter length is too short, it may lead to poor filtering results. If the filter 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. The larger the number of bands is, the heavier the computational burden will be, which affects the decomposition performance to a certain extent.

Therefore, 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 influence of human factors on the decomposition results. This can achieve the best decomposition effect. Its flowchart is shown in Figure 3. The LMWOA-FMD algorithm is implemented as follows:(1)Step 1. Input the collected rolling bearing signal and put up as the optimization parameters of the WOA. The optimization dimension is 4, the number of iterations , and the number of search population is 15. In order to avoid under-decomposition or over-decomposition, the search space of decomposition mode number n is . In order to achieve a sample number that includes two adjacent repetitive transients related to bearing faults, the search space for the filter length L is assumed to be , where is a rounding operation, and and are sampling frequency and bearing failure frequency, respectively; must be greater than for the mode to decompose, so the search space of is set as .(2)Step 2. Set the envelope entropy as the judgment value of the fitness 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 by using LMWOA to optimize the parameter combination.(4)Step 4. Input the optimal parameter combination into the FMD algorithm, disintegrate the signal into several IMF components, and use kurtosis value as the parameters of the target mode. Then, 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.(5)Step 5. Select the component with the highest kurtosis value as the target mode and calculate the envelope spectrum by Hilbert transformation to extract fault characteristics.

4. Case Study

4.1. Simulated Signal Analysis

The feasibility of the LMWOA-FMD method was verified by constructing the simulation signal of the inner ring fault of the rolling bearing, and the simulation signal expression iswhere is the amplitude, ; is frequency conversion, ; is the attenuation factor, ; is the resonance frequency, ; is a periodic shock component; represents the small fluctuation of the th shock with respect to period ; and fault frequency , . In the simulation signal, Gaussian white noise interference is added, the sampling points are selected as 10240, and the sampling frequency . The time-domain waveform diagram and sum spectrogram of the simulation signal after adding noise are shown in Figure 4.

The 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 fitness function. The value change of fitness function is shown in Figure 5. In the figure, LMWOA-FMD converges to obtain the best solution when the number of iterations reaches 5 and final fitness 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 verified that Lévy flight and adaptive weights can effectively avoid falling into local optimum and improve algorithm performance, which can save a lot of time for optimization.

After optimization by LMWOA, the final optimal parameter combination is as follows: decomposition mode , filter length , and frequency band number . The parameters are input into FMD, and the decomposition effect is shown in Figure 6. The 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.

The 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 figure, it can be seen clearly that the extracted fault frequency is . At the same time, the dual frequency of and the triple frequency of are also accurately extracted. The 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 verified that this method can accurately identify fault characteristics under background noise and has certain noise robustness.

The FMD of prior parameters was compared, where decomposition mode number , filter length , and frequency band number . In Figure 8, there is no mode aliasing during FMD decomposition. However, most components have passband ripple, which has a certain impact on the final decomposition result.

At the same time, the CEEMD and VMD methods are also used to compare and verify. The number of mode decomposition of VMD is 5, and the penalty factor is 1500. The component with the maximum kurtosis value is continued to be used for envelope demodulation. The envelope spectra of the CEEMD, VMD, and FMD are shown in Figure 9. In the figure, 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 effect is not ideal. Further processing is required if the fault frequency is to be accurately extracted. The envelope spectrum of VMD method can accurately extract the fault frequency. However, the effect is not obvious due to the influence of the interference spectral line at the position of the second and third harmonic, and the robustness to noise is poor.

(a)

(b)

(c)

In order to further compare the advantages of these four methods, FFR is used as the evaluation index of the four methods. The larger the FFR value is, the more the periodic impact components are included in the IMF component [34]. The FFR is defined aswhere is the sum of the amplitude of the envelope spectrum; is fault characteristic frequency; and 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. The FFR value of LMWOA-FMD method is 0.059, which is higher than that of CEEMD, VMD, and FMD. This proves the superiority of this method.

4.2. 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 verification. The 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. The bearing model was SKF6205-2RS deep groove ball bearing, and the sampling frequency was 12 kHz. The 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. The bearing rotational frequency is 29.95 Hz, and the outer ring fault frequency is 107.3 Hz according to its parameters. The time-domain waveform and spectrum diagram with Gaussian white noise are shown in Figure 11.

(a)

(b)

The PSO, WOA, and LMWOA methods are, respectively, used to optimize FMD. The minimum envelope entropy is used as the fitness function to find the optimal parameter combination. The change of fitness function value is shown in Figure 12. The LMWOA-FMD in the figure converges to the optimal solution when the number of iterations reaches 3. The final fitness 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.

The decomposition mode number , filter length , and frequency band number are optimized by LMWOA. The parameters are input into FMD, and the decomposition effect is shown in Figure 13. The 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. The 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 figure that the extracted fault frequency is , which is close to the theoretical value of . The frequency doubling characteristics can be clearly seen in the figure, 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 influence of frequency conversion signal and noise on the fault signal and has certain robustness to noise.

The FMD of prior parameters was compared, where decomposition mode number , filter length , and frequency band number . The result is shown in Figure 15. In the figure, IMF2 and IMF5 have passband ripple and IMF3 and IMF4 have mode mixing. The 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. The number of mode decomposition of VMD is 5, and the penalty factor is 1500. The envelope spectra of the CEEMD, VMD, and FMD are shown in Figure 16. In the figure, the envelope spectra of the three methods accurately extract the fault frequency. However, the amplitude effects of the second and third harmonic of the envelope spectrum are not significant. Compared with LMWOA-FMD, the overall extraction performance of the three methods is not ideal.

(a)

(b)

(c)

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.

The LMWOA-FMD method is used to decompose the inner ring signal of rolling bearing with Gaussian white noise. The optimized decomposition results are the number of decomposition modes , the filter length , and the number of frequency bands . The 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. The 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 influence of noise in the figure, the extracted fault frequency is prominent in the envelope spectrum, which is only different from the theoretical value of , 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 decomposition. This will affect the final result of fault feature extraction.

Also, input the signal to CEEMD and VMD, where the number of mode decomposition of VMD is 5 and the penalty factor is 1500. The envelope spectra of the three methods are shown in Figure 20. The fault frequencies of the envelope spectra of three methods in the figure 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.

(a)

(b)

(c)

Table 7 shows the FFR values of four different 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.

5. Conclusion

The improved WOA-FMD algorithm is proposed for fault feature extraction of rolling bearing under noise in this paper. The main contributions of the paper are as follows:(1)The improved WOA with Lévy flight and adaptive weight can find the optimal value faster and more accurately than PSO and WOA in the optimization process of test function. It can find the optimal solution before 5 iterations in FMD optimization, which has stronger search ability and avoids falling into local optimal solution.(2)The LMWOA-FMD algorithm successfully decomposes the original signal into multiple IMF components without mode aliasing and passband ripple. This 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)Through 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. This 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.

Abbreviations

LMWOA:	Lévy flight-based modified whale optimization algorithm
FMD:	Feature mode decomposition
IMF:	Intrinsic mode function
EMD:	Empirical mode decomposition
EWT:	Empirical wavelet transform
VMD:	Variational mode decomposition
EEMD:	Ensemble empirical mode decomposition
CEEMD:	Complementary ensemble empirical mode decomposition
WOA:	Whale optimization algorithm
FIR:	Finite-impulse response
CC:	Correlation coefficient
PSO:	Particle swarm optimization
FFR:	Fault feature ratio
CWRU:	Case Western Reserve University
EDM:	Electric discharge machining.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This research was supported by the Natural Science Foundation of Hebei Province (no. E2021208004).

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Copyright © 2023 Guangfei Jia and Yanchao Meng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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