#### Abstract

To promote the effect of variational mode decomposition (VMD) and further enhance the recognition performances of bearing fault signals, genetic algorithm (GA) is applied to optimize the combination of VMD parameters in this paper, and GA-VMD algorithm is put forward to improve the decomposition accuracy of VMD. In addition, combined with the center frequency, a feature extraction method based on GA-VMD and center frequency is proposed to ameliorate the difficulty of bearing fault feature extraction. Firstly, the bearing signal is decomposed into a series of intrinsic mode components (IMFs) by GA-VMD. Then, the Center Frequency of IMFs is extracted, and the recognition rate is calculated by k-nearest neighbor (KNN) algorithm. Simulation signal experiments state clearly that, compared with manual parameter setting-VMD algorithm and parameter optimization VMD algorithm based on particle swarm optimization (PSO), the decomposition result of GA-VMD has a smaller root mean square error and higher decomposition accuracy, which verifies the effectiveness of GA-VMD. The experimental results demonstrate that, by comparison with the feature extraction method based on envelope entropy, the feature extraction method based on center frequency has better inter class separability and higher mean recognition rate (the highest recognition rate of single feature is 94.5%, and in the case of multiple features, the recognition rate reaches 100% when four features are extracted) and can realize the accurate identification of different bearing fault signals.

#### 1. Introduction

As an essential part of various mechanical equipment, bearings have been widely used in civil, industrial, and military applications [1, 2]. However, due to the arduous working conditions, bearings are extremely susceptible to damage [3], which can cause economic losses and even affect personal safety. Simultaneously, the bearing signal in the fault state will produce some periodic transients, which contains the status information of the bearing during operation [4]. Therefore, fault diagnosis of bearings is of great significance to ensure the safe, stable, and reliable operation of mechanical equipment [5, 6].

In recent years, many scholars have put forward their methods for fault diagnosis [7, 8]. Generally speaking, there are two key steps in bearing signal fault diagnosis; the first step is the extracted features. Recently, there are many methods that can be extracted as features and used for fault detection of various types of signals, such as the Lempel-Ziv complexity (LZC) [9], the correlation dimension [10], the Lyapunov exponent [11]__,__ and the entropy-based method [12, 13]. Among which, LZC and the entropy-based method can effectively reflect the complexity of the time series and have better robustness and simplicity compared with other features [14, 15]. However, only relying on the separability of the extracted features will be susceptible to interference from noise signals.

Another critical step of bearing signal fault diagnosis is the signal processing method. In 1998, Huang et al. first proposed empirical mode decomposition (EMD) and used it for adaptive decomposition of target signals [16], which has received extensive attention. In 2009, Wu et al. proposed an ensemble empirical mode decomposition (EEMD) method on the basis of EMD, which can use the statistical properties of white noise to reduce modal aliasing [17, 18]. However, EEMD still belongs to recursive decomposition and is vulnerable to modal aliasing and endpoint effect, so there will be some decomposition errors [19, 20]. In order to solve these problems, Dragomiretskiy et al. proposed the variational modal decomposition (VMD) method in 2014 [21], which can achieve adaptive decomposition of the target signal by iteratively searching for the optimal solution of the variational model, so as to determine the center frequency and bandwidth of each mode. Compared with EMD and EEMD, VMD also has stronger robustness [22–24].

However, different from EMD and its improved algorithm, VMD is more dependent on the setting of parameters, including the number of decomposition layers and penalty factors. In recent years, many scholars have improved VMD [25–27]. Tang Guiji et al. used particle swarm optimization (PSO) algorithm to optimize combination parameters and established a fitness function based on minimum envelope entropy [28]; Mirjalili et al. proposed the whale optimization algorithm (WOA) algorithm in 2016 and proved that it has Strong local and global search ability [29]. In this paper, genetic algorithm (GA) [30, 31] is used as the optimization method, and envelope entropy is selected as the fitness function, and then combined with VMD for automatic optimization of parameters. Because the center frequency obtained by VMD does not need to be calculated like other features, a feature extraction method based on GA-VMD and center frequency is proposed and applied to bearing fault diagnosis. Compared with other methods, the method proposed in this paper has clear physical significance, and it is simpler and has higher recognition rate. After the feature extraction, we utilize K-Nearest Neighbor (KNN) [32] to classify all kinds of fault bearing signals, and both the simulated signal and the actual bearing signal experimental results show the effectiveness of the feature extraction method proposed in this paper.

The rest of this paper is structured as follows. Section 2 proposes GA-VMD algorithm and introduces the specific steps of GA-VMD algorithm. Section 3 introduces the main steps of the proposed fault feature extraction method. Section 4 carries out the simulation signal experiment and compares the GA-VMD algorithm and PSO-VMD algorithm. Section 5 carries out the experiments of bearing signal extraction and KNN classification based on center frequency and envelope entropy, respectively. Section 6 gives conclusion of this paper.

#### 2. Basic Principles

##### 2.1. Variational Mode Decomposition

Variational mode decomposition (VMD) is an adaptive signal decomposition algorithm. It can decompose a signal into multiple components, and its essence and core idea is the construction and solution of the variational problems. The solution process is as follows:

Firstly, set the variational model:where is partial derivative operation, is the Dirichlet distribution function, represents convolution, represents *K* IMFs obtained by VMD, represents the center frequency of each IMF, *f* represents the original signal, and *K* is the number of IMFs.

Then, (1) is transformed into an unconstrained problem by using penalty factor and multiplication operator; the augmented Lagrange function is obtained as follows:where is the Lagrange constant, and is the penalty factor.

Lastly, for the sake of solving the optimal solution of the variational problem, it is necessary to update variable by alternating direction multiplier of method (ADMM). Updated after *n* + 1 cycles, update the formulas for , , and as follows:where the noise tolerance, and , , and correspond to the Fourier transform of , , *f* () and , respectively.

The main iterative solution process of VMD algorithm is as follows:(1)Initialize , , (2)*n* = *n* + 1, enter the cycle, where *n* is the number of iterations(3)Update , , (4)Repeat steps (2) through (5) until the iteration stop condition is satisfied: where is a constant and .

##### 2.2. Optimization of VMD by Genetic Algorithm

As a signal decomposition algorithm, the variational modal decomposition (VMD) algorithm is greatly affected by the number of modal components and penalty factors, so it is necessary to use the optimization algorithm to optimize the parameters. Compared with the particle swarm optimization (PSO) algorithm, the genetic algorithm (GA) has higher global performance and a more mature convergence analysis method. The optimal parameters of VMD algorithm are searched by optimization algorithm. The algorithm steps of GA-VMD are shown in Figure 1, and the algorithm steps of GA-VMD are as follows:(1)The parameters of GA algorithm are determined, and the optimization objective function can be expressed as Constraints can be expressed as where *K* is the optimal number of IMFs, *i* is the IMF serial number (*i* = 1,2,3, .....) obtained from the decomposition of the original signal, represents the envelope entropy of the i-th IMF, C is the objective function, is the penalty factor of the i-th IMF, is the optimal number of IMF for the i-th IMF, and N is natural number.(2)Initialize the population, namely, VMD parameter [K, *α*].(3)Input the population into VMD to obtain the IMFs, calculate the average envelope entropy of all IMFs as the fitness function, and save the minimum average envelope entropy and its corresponding VMD parameters [K, *α*], and the envelope entropy can be expressed as where is the envelope signal obtained from the signal IMF_{i}(*j*) after Hilbert mediation, and is the normalized form of the signal .(4)Evaluate the fitness of all individuals in the population, that is, the parameter combination, and eliminate unsatisfactory individuals.(5)According to the principle that the higher the fitness is, the greater the selection probability is, two individuals are selected from the population as the parent and mother, and the offspring are generated through binary crossover and mutation;(6)Repeat steps (4) and (5) and update the population;(7)Cycle steps (3) ∼ (6) until the maximum number of iterations is completed;(8)The minimum value of the local minimum average envelope entropy in each iteration obtained in step (7) is selected as the global minimum value of this optimization, and its corresponding parameter combination [K, *α*] is the best parameter combination obtained in this optimization.

#### 3. Fault Feature Extraction Method

The bearing fault feature extraction flow chart is shown in Figure 2, and the specific steps are as follows.(1)The bearing signals under four different states are obtained, including normal signal, inner ring fault signal, outer ring fault signal, and rolling element fault signal.(2)GA-VMD algorithm is used to carry out signal adaptive decomposition of the four types of bearing signals, and the IMF combinations are obtained. The effectiveness of GA-VMD algorithm is convinced through simulation signal analysis in Section 4.(3)The central frequency of bearing signal IMF is extracted, and a comparison experiment with the envelope entropy is carried out.(4)Recognition results of bearing signals are obtained through KNN.

#### 4. Simulation Signal Analysis

In order to verify the effectiveness of GA-VMD algorithm, the VMD decomposition results based on particle swarm optimization algorithm are compared with the proposed method in this paper. Set up a set of simulation signals for analysis, and its specific expression iswhere *X* is the original signal, , , , are the subsignals, , , , are the subfrequencies of , , , , and their values are, respectively, 170 Hz, 120 Hz, 50 Hz, and 10 Hz.

Set the number of analysis points *N* as 1024 and the sampling frequency as 1024 kHz. Figure 3 shows the time-domain waveform of the simulation signal and its original component.

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The root mean square error is used to calculate the amplitude difference between the original component signal and the decomposed component signal to verify the decomposition performance of the two algorithms. The root mean square error calculation formula iswhere *n* is the number of observations is the original component signal, and is decomposed component signal. The root mean square error reflects the difference of signal amplitude between the decomposed signal and the original signal component. The smaller the RMSE value is, the better the decomposition effect is.

The VMD algorithm based on genetic algorithm (GA-VMD) and the parameter optimization VMD algorithm based on particle swarm optimization (PSO-VMD) are used to decompose the simulation signal. The genetic algorithm parameters are set as follows: the initial population is 20, the number of iterations is 100, the crossover probability is 0.8, and the mutation probability is 0.4. The specific parameters of Particle swarm optimization are as follows: the inertia weight W is 1, the acceleration factors C1 and C2 are 1.5, the maximum optimization speed of K value is 1, the minimum optimization speed is −1, and the optimization range is 3 to 5. The maximum optimization speed of *α* value is 2000, the minimum optimization speed is 200, and the optimization range is from 200 to 2000. The optimal input parameters obtained by GA-VMD and PSO-VMD search are and , respectively. GA-VMD and PSO-VMD decomposition results are shown in Figures 4 and 5.

According to Figures 4 and 5, IMF1 corresponds to the original signal component , IMF2 to the original signal component , IMF3 to the original signal component , and IMF4 to the original signal component . Since there are two parameters in the parameter combination, modal component K and penalty factor , to prove the influence of the two parameters on the decomposition results, the values of K and are changed, respectively, for VMD decomposition. The four groups of VMD parameter combinations determined manually are as follows: when the value of change and the value of K remain unchanged, determine the parameter combination ; when the value of K changes and the value of remain unchanged, determine the parameter combination ;

To further compare the effectiveness of the two algorithms and manual parameter setting-VMD, the root mean square error of the two optimization algorithms and manual parameter setting-VMD are shown in Table 1.

It can be seen from Table 1 that the root mean square error of the optimal result of GA-VMD is 0.002 lower than that of PSO-VMD, and the IMF decomposed by GA-VMD is closer to the original signal component. In addition, The root mean square error of GA-VMD is lower than that of Manual parameter setting-VMD. Therefore, GA-VMD is used to analyze the bearing fault signal in this paper.

#### 5. Bearing Signal Feature Extraction

##### 5.1. Bearing Signals

Bearing signal feature extraction experiment adopts bearing data provided by the laboratory of Case Western Reserve University [33]. The diameter of rolling bearing failure is 0.1778 mm, and the speed is 1730r/min. Normal signal and the sampling frequency of the signal, which include acceleration data of inner ring fault, outer ring fault, and rolling element fault obtained from the rolling bearing fault, are 12 kHz. There are 100 samples for each type of bearing signal, and each sample contains 1200 sampling points. The time-domain waveform of rolling bearing signals under four states (one sample) is shown in Figure 6.

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##### 5.2. Bearing Signal Decomposition

The optimal number of IMF components *K* of the four bearing signals decomposed by GA-VMD algorithm is 9, and the penalty factors are 12040, 15480, 9880, and 15560, respectively. The decomposition results of the four bearing signals obtained are shown in Figure 7.

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In order to further verify the effectiveness of GA-VMD algorithm, comparative experiments with PSO-VMD algorithm are carried out. The bearing signal is decomposed by PSO-VMD to obtain the corresponding IMF, and then the root mean square error of the measured signal by the two optimization algorithms is shown in Table 2.

It can be seen from Table 2 that, for each bearing signal, the root mean square error of the optimization result of GA-VMD is lower than that of PSO-VMD, especially normal signal and rolling element fault signal; for normal signal, the root mean square error of GA-VMD is about 0.01 lower than that of PSO-VMD; for rolling element fault signal, the root mean square error of GA-VMD is about 0.0176 lower than that of PSO-VMD. Therefore, compared with PSO-VMD, GA-VMD has better optimization results.

##### 5.3. Feature Extraction of Bearing Signals

The characteristics of center frequency and envelope entropy of IMF of four kinds of bearing signals are extracted. The characteristic distribution of center frequency and envelope entropy of IMF of four kinds of bearing signals are shown in Figure 8 and 9.

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It can be seen from Figures 8 and 9 that the envelope entropy distribution in each IMF of the four kinds of bearing signals is chaotic, and there are many overlaps, while the center frequency distribution of each mode of the four types of bearing signals is more regular, and the overlaps are significantly reduced; the center frequency values of inner ring fault signal, outer ring fault signal, and rolling element fault in IMF4 and IMF7 are very close, while there are obvious differences from the center frequency of normal signal; the envelope entropy of outer ring fault signals in IMF4 and IMF5 is lower than that of the other three kinds of signals; the center frequency of normal signals in IMF7 is almost at the same level, and the frequency range of the center frequency of other three types of bearing signals also has obvious differences; compared with enveloping entropy, extracting center frequency features is more beneficial to distinguish four kinds of bearing signals.

##### 5.4. Classification of Bearing Signals

###### 5.4.1. Single Feature Classification

For purpose of further identifying the features, KNN classifier is used to classify the signals. The center frequency and envelope entropy of a single IMF in 200 sample signals in each of the four states are classified as characteristic parameters.

KNN is one of the most basic and simplest machine learning algorithms. KNN is classified by measuring the distance between different eigenvalues. Its core idea is as follows: a sample is most similar to the K samples in the dataset. If most of the K samples belong to category A, the sample also belongs to category A. The method only determines the category of the samples to be divided according to the category of the nearest one or several samples.

In KNN, the number of neighbors *K* is 1. Besides, in this experiment, for each type of bearing signal, 200 samples are randomly selected; the first 50 sample signals are used as training samples, while the other 150 sample signals are used as test samples. The characteristic classification results of the center frequency and envelope entropy of each IMF are shown in Figures 10 and 11.

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It can be seen from Figures 10 and 11 that, compared with the envelope entropy feature, under the same mode, fewer samples are identified incorrectly by using the method characterized by center frequency; for the center frequencies of different modes, IMF2 has the best result of feature classification. Classification recognition rates of single IMF component center frequency and envelope entropy are shown in Table 3 and 4 respectively.

It can be seen from Tables 3 and 4 that the center frequency feature extraction and recognition effect of a single IMF component is better than that of the single component feature extraction and recognition of IMF component envelope entropy; the average recognition rate of envelope entropy feature of single IMF component is generally lower than 60%, some even lower than 40%, and the maximum recognition rate is 61%; the recognition effect is poor, so it is impossible to identify and diagnose the four bearing signals; except for a few components, the accuracy of extraction and classification of central frequency features of most IMF components reached more than 75%, and the accuracy of IMF2 component even reached 94.5%; in the recognition of normal signals, the recognition accuracy of center frequency of IMF5, IMF6, IMF7, and IMF9 components all reached 100%; the recognition rate of IMF2 and IMF3 component center frequency in inner ring fault recognition is also 100%.

###### 5.4.2. Multifeature Classification

Aiming to improve the signal recognition rate, a multicomponent feature recognition method is adopted, that is, the method of simultaneously recognizing the center frequency or envelope entropy of multiple IMF components, to classify and recognize the four bearing signals.

KNN classifier is used to extract and classify the component features of the center frequency and envelope entropy of the four kinds of bearing signals. Classification recognition rates of center frequency and envelope entropy are shown in Tables 5 and 6, where IMF_{1-2} means to simultaneously identify the features of IMF component 1 and 2 signals, IMF_{1–3} means to simultaneously identify the features of IMF component 1, 2 and 3 signals, and so on.

It can be seen from Tables 4 and 5 that, under the same number of components, the classification effect of multicomponent central frequency feature extraction is significantly better than that of multicomponent envelope entropy feature extraction; when the number of IMFs is greater than 2, the average recognition rate of envelope entropy feature extraction increases with the increase of the number of components, up to 98.5%; the average recognition rate of multicomponent center frequency feature extraction is not less than 98.5%, which is higher than the highest value of recognition rate of envelope entropy feature extraction; when the number of components is 9, the envelops entropy feature extraction effect is optimal, and the average recognition rate is 98.5%; when the number of components is 4, the recognition accuracy of multifeature center frequency extraction has reached 100%; therefore, compared with feature extraction based on envelope entropy, feature extraction based on center frequency can achieve higher recognition rate in the case of a small number of components and can realize effective recognition of bearing faults.

#### 6. Conclusion

In this paper, a bearing fault feature extraction method based on GA-VMD and center frequency is proposed, and the feasibility of the proposed method is verified by feature extraction and classification recognition experiments on four kinds of measured bearing signals. The main conclusions are as follows:(1)Compared with PSO-VMD algorithm, the root mean square error of GA-VMD algorithm is smaller, and the decomposed IMF is closer to the corresponding signal component(2)For single feature extraction method, the highest recognition rate of feature extraction method based on center frequency is 94.5%, which is 33.5% higher than that of feature extraction method based on envelope entropy(3)Compared with the single feature extraction method, the average recognition rate of multifeature extraction method is 17% (center frequency) and 32% (envelope entropy) higher than that of single feature extraction method; compared with the multifeature center frequency extraction method, the average recognition rate of multifeature center frequency feature extraction method is 19.5% higher than that of multifeature envelope entropy extraction method

#### Data Availability

Data are available upon request to the authors.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.