# Research on Fault Feature Extraction and Pattern Recognition of Rolling Bearing

Authors: Xiaobin Miao
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

Rolling bearings play a very important role. If the state of the rolling bearing is wrong, the whole equipment will fail, so we need to check the state of the running bearing.Fault Feature Extraction of Rolling Bearings. CEEMDAN (Complete Ensemble Empirical Mode Decomposition with Adaptive Noise) analyzes the collected oscillation signals to obtain various natural state functions. Kurtosis and Correlation Factor Evaluation Index are used to screen IMF components with a large number of error data, and independent component analysis is carried out on the selected components. The corresponding processing and reconstruction are carried out to extract the feature frequency. The built-in signal processing method can deal with the problem of bearing fault signal and identify the specific fault frequency. Fault Pattern Recognition of Rolling Bearings. The decomposition acceleration sensor is used to analyze the data of bearing in different states, and then, several components are obtained. The components related to the error signal are found, and the corresponding feature vectors are calculated. Then, the dimension vector is reduced, and a particle flow optimization algorithm is used to identify specific error conditions after dimension reduction. Experiments show that this method has higher detection rate in identifying specific fault states.


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## Body

## 1. Introduction

The SVD (Singular Value Decomposition) method proposed in this paper can efficiently extract the fault features of bearings [1]. The cyclic spectral density method cannot identify the specific faults of bearings. Therefore, CSDK (Chi-square distribution-kernel method) is proposed in this paper, and experimental research shows that CSDK is a more effective method to solve the problem of fault feature extraction [2]. Compared with other methods, the distributed parameter extraction method is more sensitive and accurate in solving the fault problem of rolling bearings [3]. Aiming at the problem of fault feature extraction in bearings, it is concluded that the capacity distribution of the bearing frequency band can judge the state of the bearing [4]. Aiming at the problem of compound faults, this paper proposes a wavelet analysis method. The research shows that this method has great advantages in solving the problem of compound fault analysis [5]. Vibration analysis is very effective for fault feature extraction of rolling bearings. In this paper, the cyclic Wiener filter is used to further diagnose the extracted faults, and the experiment shows that this method is very effective [6]. Rolling bearing failure will produce some major disasters; if the fault features can be effectively extracted, it will reduce the possibility of disasters. This paper verifies the effectiveness of the minimum entropy deconvolution method in solving the rolling bearing problem [7]. The fault characteristic parameters are normalized, and the neural network is used to diagnose the fault so that the ability of troubleshooting is effectively improved [8]. For fault diagnosis, the singular value decomposition method is proposed in this paper. It is concluded that the singular value decomposition method is higher in efficiency and accuracy [9]. It is very important for bearing fault identification to extract useful problem features from many problems. WPT has great advantages in extracting discriminant features [10]. Intelligent fault diagnosis is an important way to reduce maintenance costs. In this paper, a manifold learning algorithm is proposed for the intelligent diagnosis of rolling bearings, which can effectively and accurately identify different states of bearings [11]. Traditional analysis methods are not suitable for unstable bearing faults. In this paper, a neural network is proposed to identify fault modes, and the research shows that this method has higher accuracy [12]. Feature extraction and feature classification are the core of bearing diagnosis. In this paper, a support vector machine is proposed to identify the fault of rolling bearings, and the research shows that this method is more accurate [13]. A vibration signal is developed by the pattern recognition method. The vibration signal can effectively diagnose the bearing state in the rolling bearing problem [14]. Recognition and pattern classification are the core of fault diagnosis. In this paper, a statistical pattern recognition method is proposed to solve the problem of fault classification [15]. Literature [16] uses PSO and SVM to identify rolling bearing faults, which has an obvious recognition effect in related feature extraction, kernel function, and parameter optimization. In [17], a fault diagnosis model based on the combination of CEEMD, PNN, and PSO is proposed, aiming at the high existence of nonstationarity and nonlinearity in bearing vibration signals. The feature extraction efficiency of the vibration signal is high, thus improving the recognition rate.


## 2. Bearing Fault Diagnosis Method

### 2.1. Temperature Monitoring Method

Under different working conditions, the internal temperature of rolling bearings will show different distribution characteristics due to different heating mechanisms. Holkup et al. obtained the thermal characteristics of bearings under different working conditions through experiments, so the health monitoring of bearings can be realized through the change of internal temperature of bearings. Through practice tests, temperature monitoring is really sensitive to the lubrication effect, load degree, and internal impact of bearings. However, when the bearing has early faults such as normal wear, pitting corrosion, and skin peeling, its internal temperature will hardly change, and it is impossible to accurately detect the early faults of bearings. Moreover, the temperature is easily affected by the weather and has great limitations.


### 2.2. Oil Sample Collection and Analysis Method

The oil sample collection and analysis method is mainly used to diagnose the faults by analyzing the components contained in the lubricating oil when rolling bearings are running. During the running process of the bearing, all parts contact each other and produce friction, so that the lubricating oil of the bearing will contain tiny particles produced by friction. When the wear position is different or the wear degree is different, the size, shape, and number of wear particles in lubricating oil are different. By analyzing the size, shape, and number and chemical composition of wear particles in lubricating oil, the fault severity and fault type of bearings can be monitored. However, the oil sample analysis method is not widely used because of its high cost, poor timeliness, and ease to be affected by the particles that fall off from other parts, and it can not accurately locate the fault location.


### 2.3. Acoustic Emission Monitoring Method

The acoustic emission monitoring method is also called the stress wave emission monitoring method. However, the signal-to-noise ratio of the acoustic emission monitoring method is not high, and its wide spectrum range makes the workload of measurement and analysis heavy, the equipment is expensive, and it is easily interfered with by external noise, which requires a high test environment and noise processing, so its application and popularization in production are limited.


### 2.4. Vibration Analysis Method

When the bearing fails, the vibration signal will be abnormal. The advantage of this method is that it is suitable for diagnosing bearing faults under various types and working conditions, the vibration sensor is cheap, simple, and convenient to operate, and the diagnosis results are reliable, so it is widely used in practical engineering.


## 2.1. Temperature Monitoring Method

Under different working conditions, the internal temperature of rolling bearings will show different distribution characteristics due to different heating mechanisms. Holkup et al. obtained the thermal characteristics of bearings under different working conditions through experiments, so the health monitoring of bearings can be realized through the change of internal temperature of bearings. Through practice tests, temperature monitoring is really sensitive to the lubrication effect, load degree, and internal impact of bearings. However, when the bearing has early faults such as normal wear, pitting corrosion, and skin peeling, its internal temperature will hardly change, and it is impossible to accurately detect the early faults of bearings. Moreover, the temperature is easily affected by the weather and has great limitations.


## 2.2. Oil Sample Collection and Analysis Method

The oil sample collection and analysis method is mainly used to diagnose the faults by analyzing the components contained in the lubricating oil when rolling bearings are running. During the running process of the bearing, all parts contact each other and produce friction, so that the lubricating oil of the bearing will contain tiny particles produced by friction. When the wear position is different or the wear degree is different, the size, shape, and number of wear particles in lubricating oil are different. By analyzing the size, shape, and number and chemical composition of wear particles in lubricating oil, the fault severity and fault type of bearings can be monitored. However, the oil sample analysis method is not widely used because of its high cost, poor timeliness, and ease to be affected by the particles that fall off from other parts, and it can not accurately locate the fault location.


## 2.3. Acoustic Emission Monitoring Method

The acoustic emission monitoring method is also called the stress wave emission monitoring method. However, the signal-to-noise ratio of the acoustic emission monitoring method is not high, and its wide spectrum range makes the workload of measurement and analysis heavy, the equipment is expensive, and it is easily interfered with by external noise, which requires a high test environment and noise processing, so its application and popularization in production are limited.


## 2.4. Vibration Analysis Method

When the bearing fails, the vibration signal will be abnormal. The advantage of this method is that it is suitable for diagnosing bearing faults under various types and working conditions, the vibration sensor is cheap, simple, and convenient to operate, and the diagnosis results are reliable, so it is widely used in practical engineering.


## 3. Fault Pattern Recognition of Rolling Bearings

### 3.1. Basic Theory

#### 3.1.1. Sample Entropy

The sample entropy calculation method is as follows:(1)
A vector of dimensionm, Xm1,…,XmN−m+1, numbered, as shown in the formula(1)Xmi=xi,xi+1,…,xi+m−1,1≤i≤N−m+1.(2)
Define the distancedXmi,Xmj between Xmi and Xmj as the absolute value when the difference between them is the largest, as shown in the formula(2)dXmi,Xmj=maxxi+k−xj+k.For a given vectorXmi, count the distance between Xmi and Xmj; if it is less than and equal to J number of R, where 1≤j≤N−m,j≠i, is denoted as Bi, then it is shown in the formula(3)Bimr=1N−m−1Bi.DefineBmr as shown in the formula(4)Bmr=1N−m∑i=1N−mBimr.Increase its dimension untilm + 1, and then calculate the number of distances between Xm+1i and Xm+1j that are less than or equal to R, and write it as Ai; then, Aimr is shown in the formula(5)Aimr=1N−m−1Ai.DefineAmr as shown in the formula(6)Amr=1N−m∑i=1N−mAimr.The sample entropy is defined as shown in the formula(7)sampEnm.r=limN⟶∞−lnAmrBmr.IfN is a finite value, it can be estimated by the formula(8)sampEnm.r,N=−lnAmrBmr.


#### 3.1.2. Pattern Recognition Classification Method

(1) Support Vector Machine. The SVM method is usually used for data classification. When classifying data, different classification situations will have different effects, but there will be an SVM best result.Assuming that the training sample set isx,y|i=1,2,…,k, the optimization problem is constructed as shown in formula (10):(9)minw,bw22+C∑i=1kξi,(10)s.t.yiwxi+b≥1−ξi,i=1,2,…,k,where W is the weight; C is the penalty parameter of error term ξi, C > 0; ξi: 0; b is the intercept.Solve the corresponding value and get the formula(11)fx=sgn∑i=1kaiyiKxi,x+b∗,where ai is the Lagrange multiplier; Kxi,x is the kernel function of Mercer condition; b∗ is the intercept.In this paper, the RBF kernel function is used to analyze the results of SVM classification. As shown in the formula,(12)Kx,y=exp−x−y22g2,where g represents the regularity of vibration signals.(2) PNN Neural Network. Classification according to probability density is the essence of a probabilistic neural network. It can output probability in a short time and maximize the probability. As shown in the formula,(13)fX,ωi=exp−X−ωiTX−Wi2δ,where ωi is the weight; δ is the smoothing factor.


#### 3.1.3. Support Vector Machine Optimized by Particle Swarm Optimization

A particle swarm optimization algorithm is the most suitable method to find the solution. The SVM parameterC and G selection problem is actually an optimization problem that satisfies certain criteria or objectives. The classification recognition rate is directly used as the fitness to evaluate the advantages and disadvantages of the solution.Assume that the individual extreme value isPbest. The global optimal solution is Gbest. The search space contains m different particles. The position xi and velocity vi are brought into the functional equation for calculation. As shown in formulas (14) and (15),(14)vit+1=αωvit+c1r1Pbestt−xit+c2r2Gbestt−xit,(15)xit+1=xit+vit+1,where c1 and c2 are acceleration factors; r1 and r2 obey 0,1 distribution; vi∈−vmin,vmax and vmax are nonnegative; α is the compressibility factor, α=2/2−C−C2−4C; C=c1+c2; ω is the weight. Search performance is improved through it. Use linear decreasing weights, as shown in the formula(16)ω=ωmax−ωmax−ωmin∗t−1M−1,where t represents the current number of iterations and M is the maximum number of iterations.In order to find the most suitable target value, the particle swarm optimization (PSO) algorithm is usually used to build a support vector machine (SVM) for parameter optimization of SVM. Update the speed and position of each particle. If the conditions for the completion of verification are not met, the calculation can be carried out according to equation (12), and the results can be output after ensuring that this condition is met.


#### 3.1.4. Comparison of Classification Effects of Different Feature Vectors

The length of data plays a decisive role in the calculation of approximate entropy. Sample entropy does not depend on the length of data, and the calculation time is short. Therefore, this paper chooses sample entropy as a feature vector for bearing pattern recognition.The approximate entropy of sample pointsxii=1,2,…,N of bearing fault signal is as shown in the formula(17)Xi=xi,xi+1,…,xi+m−1,i=1,2,…,N−m+1.If the threshold valueε of similarity tolerance is set, the distance dXi,Xj between Xi and Xj is less than the number BdXi,Xj<ε of ε; then, Dmiε is defined as shown in the formula(18)Dmiε=BdXi,Xj<εN−m+1.The autocorrelation degree of vectorXi is shown in the formula(19)φmε=1N−m+1∑i=1N−m+1lnDmiε.Its approximate entropy is shown in the formula(20)ApEnm,ε,N=φmε−φm+1ε.The approximate entropy of bearings in different states is shown in Table1.Table 1
Approximate entropy of different IMF components.
ComponentIMF1IMF2IMF3IMF4IMF5IMF6IMF7IMF8Normal1.06311.1210.71460.63150.62920.42280.27210.1157Status0.99410.78810.69810.66740.61440.38560.23420.0977Inner ring1.14420.60920.87820.63860.62610.49470.3120.1189Malfunction1.120.58670.83810.640.61480.47040.29160.1383Rolling body0.99281.14180.90410.63350.62110.42720.29660.1171Malfunction0.99041.14660.91210.62650.61750.40780.29330.1026Outer ring0.58720.65661.0020.64120.61940.46290.25440.1227Malfunction0.60680.68270.9420.64830.59790.47110.32910.1195It can be seen from Table1 that the results of approximate entropy of IMF component of vibration signal in different states of bearings are different, and it can be seen that the information contained in the signal is different.Using the PSO-SVM classification model, the feature vectors extracted from IMF components after CEEMDAN decomposition are used for pattern recognition, and the program is run five times. The results are shown in Tables2 and 3.Table 2
Classification effect of approximate entropy as a feature vector.
Number of runsRunning time (s)CgFitness valueRecognition rate (%)1112.5997.70.0197.67952117.864.6343.65697.67953109.931000.0197.67954109.1596.3240.0197.67955109.0198.3840.0197.6795Table 3
Sample entropy is the classification effect of the feature vector.
Number of runsRunning time (s)CgFitness valueRecognition rate (%)1115.66115.79862.262497.33952110.9281001.057497.6795.53109.90491.94821.031597.6795.54115.35190.17021.17697.6795.55100.92251.50630.852597.3395.5Observing Tables2 and 3, we can get that the average classification recognition rate is 95% when approximate entropy is feature information, and the classification recognition rate is 95.5% when sample entropy is feature information.


## 3.1. Basic Theory

### 3.1.1. Sample Entropy

The sample entropy calculation method is as follows:(1)
A vector of dimensionm, Xm1,…,XmN−m+1, numbered, as shown in the formula(1)Xmi=xi,xi+1,…,xi+m−1,1≤i≤N−m+1.(2)
Define the distancedXmi,Xmj between Xmi and Xmj as the absolute value when the difference between them is the largest, as shown in the formula(2)dXmi,Xmj=maxxi+k−xj+k.For a given vectorXmi, count the distance between Xmi and Xmj; if it is less than and equal to J number of R, where 1≤j≤N−m,j≠i, is denoted as Bi, then it is shown in the formula(3)Bimr=1N−m−1Bi.DefineBmr as shown in the formula(4)Bmr=1N−m∑i=1N−mBimr.Increase its dimension untilm + 1, and then calculate the number of distances between Xm+1i and Xm+1j that are less than or equal to R, and write it as Ai; then, Aimr is shown in the formula(5)Aimr=1N−m−1Ai.DefineAmr as shown in the formula(6)Amr=1N−m∑i=1N−mAimr.The sample entropy is defined as shown in the formula(7)sampEnm.r=limN⟶∞−lnAmrBmr.IfN is a finite value, it can be estimated by the formula(8)sampEnm.r,N=−lnAmrBmr.


### 3.1.2. Pattern Recognition Classification Method

(1) Support Vector Machine. The SVM method is usually used for data classification. When classifying data, different classification situations will have different effects, but there will be an SVM best result.Assuming that the training sample set isx,y|i=1,2,…,k, the optimization problem is constructed as shown in formula (10):(9)minw,bw22+C∑i=1kξi,(10)s.t.yiwxi+b≥1−ξi,i=1,2,…,k,where W is the weight; C is the penalty parameter of error term ξi, C > 0; ξi: 0; b is the intercept.Solve the corresponding value and get the formula(11)fx=sgn∑i=1kaiyiKxi,x+b∗,where ai is the Lagrange multiplier; Kxi,x is the kernel function of Mercer condition; b∗ is the intercept.In this paper, the RBF kernel function is used to analyze the results of SVM classification. As shown in the formula,(12)Kx,y=exp−x−y22g2,where g represents the regularity of vibration signals.(2) PNN Neural Network. Classification according to probability density is the essence of a probabilistic neural network. It can output probability in a short time and maximize the probability. As shown in the formula,(13)fX,ωi=exp−X−ωiTX−Wi2δ,where ωi is the weight; δ is the smoothing factor.


### 3.1.3. Support Vector Machine Optimized by Particle Swarm Optimization

A particle swarm optimization algorithm is the most suitable method to find the solution. The SVM parameterC and G selection problem is actually an optimization problem that satisfies certain criteria or objectives. The classification recognition rate is directly used as the fitness to evaluate the advantages and disadvantages of the solution.Assume that the individual extreme value isPbest. The global optimal solution is Gbest. The search space contains m different particles. The position xi and velocity vi are brought into the functional equation for calculation. As shown in formulas (14) and (15),(14)vit+1=αωvit+c1r1Pbestt−xit+c2r2Gbestt−xit,(15)xit+1=xit+vit+1,where c1 and c2 are acceleration factors; r1 and r2 obey 0,1 distribution; vi∈−vmin,vmax and vmax are nonnegative; α is the compressibility factor, α=2/2−C−C2−4C; C=c1+c2; ω is the weight. Search performance is improved through it. Use linear decreasing weights, as shown in the formula(16)ω=ωmax−ωmax−ωmin∗t−1M−1,where t represents the current number of iterations and M is the maximum number of iterations.In order to find the most suitable target value, the particle swarm optimization (PSO) algorithm is usually used to build a support vector machine (SVM) for parameter optimization of SVM. Update the speed and position of each particle. If the conditions for the completion of verification are not met, the calculation can be carried out according to equation (12), and the results can be output after ensuring that this condition is met.


### 3.1.4. Comparison of Classification Effects of Different Feature Vectors

The length of data plays a decisive role in the calculation of approximate entropy. Sample entropy does not depend on the length of data, and the calculation time is short. Therefore, this paper chooses sample entropy as a feature vector for bearing pattern recognition.The approximate entropy of sample pointsxii=1,2,…,N of bearing fault signal is as shown in the formula(17)Xi=xi,xi+1,…,xi+m−1,i=1,2,…,N−m+1.If the threshold valueε of similarity tolerance is set, the distance dXi,Xj between Xi and Xj is less than the number BdXi,Xj<ε of ε; then, Dmiε is defined as shown in the formula(18)Dmiε=BdXi,Xj<εN−m+1.The autocorrelation degree of vectorXi is shown in the formula(19)φmε=1N−m+1∑i=1N−m+1lnDmiε.Its approximate entropy is shown in the formula(20)ApEnm,ε,N=φmε−φm+1ε.The approximate entropy of bearings in different states is shown in Table1.Table 1
Approximate entropy of different IMF components.
ComponentIMF1IMF2IMF3IMF4IMF5IMF6IMF7IMF8Normal1.06311.1210.71460.63150.62920.42280.27210.1157Status0.99410.78810.69810.66740.61440.38560.23420.0977Inner ring1.14420.60920.87820.63860.62610.49470.3120.1189Malfunction1.120.58670.83810.640.61480.47040.29160.1383Rolling body0.99281.14180.90410.63350.62110.42720.29660.1171Malfunction0.99041.14660.91210.62650.61750.40780.29330.1026Outer ring0.58720.65661.0020.64120.61940.46290.25440.1227Malfunction0.60680.68270.9420.64830.59790.47110.32910.1195It can be seen from Table1 that the results of approximate entropy of IMF component of vibration signal in different states of bearings are different, and it can be seen that the information contained in the signal is different.Using the PSO-SVM classification model, the feature vectors extracted from IMF components after CEEMDAN decomposition are used for pattern recognition, and the program is run five times. The results are shown in Tables2 and 3.Table 2
Classification effect of approximate entropy as a feature vector.
Number of runsRunning time (s)CgFitness valueRecognition rate (%)1112.5997.70.0197.67952117.864.6343.65697.67953109.931000.0197.67954109.1596.3240.0197.67955109.0198.3840.0197.6795Table 3
Sample entropy is the classification effect of the feature vector.
Number of runsRunning time (s)CgFitness valueRecognition rate (%)1115.66115.79862.262497.33952110.9281001.057497.6795.53109.90491.94821.031597.6795.54115.35190.17021.17697.6795.55100.92251.50630.852597.3395.5Observing Tables2 and 3, we can get that the average classification recognition rate is 95% when approximate entropy is feature information, and the classification recognition rate is 95.5% when sample entropy is feature information.


## 3.1.1. Sample Entropy

The sample entropy calculation method is as follows:(1)
A vector of dimensionm, Xm1,…,XmN−m+1, numbered, as shown in the formula(1)Xmi=xi,xi+1,…,xi+m−1,1≤i≤N−m+1.(2)
Define the distancedXmi,Xmj between Xmi and Xmj as the absolute value when the difference between them is the largest, as shown in the formula(2)dXmi,Xmj=maxxi+k−xj+k.For a given vectorXmi, count the distance between Xmi and Xmj; if it is less than and equal to J number of R, where 1≤j≤N−m,j≠i, is denoted as Bi, then it is shown in the formula(3)Bimr=1N−m−1Bi.DefineBmr as shown in the formula(4)Bmr=1N−m∑i=1N−mBimr.Increase its dimension untilm + 1, and then calculate the number of distances between Xm+1i and Xm+1j that are less than or equal to R, and write it as Ai; then, Aimr is shown in the formula(5)Aimr=1N−m−1Ai.DefineAmr as shown in the formula(6)Amr=1N−m∑i=1N−mAimr.The sample entropy is defined as shown in the formula(7)sampEnm.r=limN⟶∞−lnAmrBmr.IfN is a finite value, it can be estimated by the formula(8)sampEnm.r,N=−lnAmrBmr.


## 3.1.2. Pattern Recognition Classification Method

(1) Support Vector Machine. The SVM method is usually used for data classification. When classifying data, different classification situations will have different effects, but there will be an SVM best result.Assuming that the training sample set isx,y|i=1,2,…,k, the optimization problem is constructed as shown in formula (10):(9)minw,bw22+C∑i=1kξi,(10)s.t.yiwxi+b≥1−ξi,i=1,2,…,k,where W is the weight; C is the penalty parameter of error term ξi, C > 0; ξi: 0; b is the intercept.Solve the corresponding value and get the formula(11)fx=sgn∑i=1kaiyiKxi,x+b∗,where ai is the Lagrange multiplier; Kxi,x is the kernel function of Mercer condition; b∗ is the intercept.In this paper, the RBF kernel function is used to analyze the results of SVM classification. As shown in the formula,(12)Kx,y=exp−x−y22g2,where g represents the regularity of vibration signals.(2) PNN Neural Network. Classification according to probability density is the essence of a probabilistic neural network. It can output probability in a short time and maximize the probability. As shown in the formula,(13)fX,ωi=exp−X−ωiTX−Wi2δ,where ωi is the weight; δ is the smoothing factor.


## 3.1.3. Support Vector Machine Optimized by Particle Swarm Optimization

A particle swarm optimization algorithm is the most suitable method to find the solution. The SVM parameterC and G selection problem is actually an optimization problem that satisfies certain criteria or objectives. The classification recognition rate is directly used as the fitness to evaluate the advantages and disadvantages of the solution.Assume that the individual extreme value isPbest. The global optimal solution is Gbest. The search space contains m different particles. The position xi and velocity vi are brought into the functional equation for calculation. As shown in formulas (14) and (15),(14)vit+1=αωvit+c1r1Pbestt−xit+c2r2Gbestt−xit,(15)xit+1=xit+vit+1,where c1 and c2 are acceleration factors; r1 and r2 obey 0,1 distribution; vi∈−vmin,vmax and vmax are nonnegative; α is the compressibility factor, α=2/2−C−C2−4C; C=c1+c2; ω is the weight. Search performance is improved through it. Use linear decreasing weights, as shown in the formula(16)ω=ωmax−ωmax−ωmin∗t−1M−1,where t represents the current number of iterations and M is the maximum number of iterations.In order to find the most suitable target value, the particle swarm optimization (PSO) algorithm is usually used to build a support vector machine (SVM) for parameter optimization of SVM. Update the speed and position of each particle. If the conditions for the completion of verification are not met, the calculation can be carried out according to equation (12), and the results can be output after ensuring that this condition is met.


## 3.1.4. Comparison of Classification Effects of Different Feature Vectors

The length of data plays a decisive role in the calculation of approximate entropy. Sample entropy does not depend on the length of data, and the calculation time is short. Therefore, this paper chooses sample entropy as a feature vector for bearing pattern recognition.The approximate entropy of sample pointsxii=1,2,…,N of bearing fault signal is as shown in the formula(17)Xi=xi,xi+1,…,xi+m−1,i=1,2,…,N−m+1.If the threshold valueε of similarity tolerance is set, the distance dXi,Xj between Xi and Xj is less than the number BdXi,Xj<ε of ε; then, Dmiε is defined as shown in the formula(18)Dmiε=BdXi,Xj<εN−m+1.The autocorrelation degree of vectorXi is shown in the formula(19)φmε=1N−m+1∑i=1N−m+1lnDmiε.Its approximate entropy is shown in the formula(20)ApEnm,ε,N=φmε−φm+1ε.The approximate entropy of bearings in different states is shown in Table1.Table 1
Approximate entropy of different IMF components.
ComponentIMF1IMF2IMF3IMF4IMF5IMF6IMF7IMF8Normal1.06311.1210.71460.63150.62920.42280.27210.1157Status0.99410.78810.69810.66740.61440.38560.23420.0977Inner ring1.14420.60920.87820.63860.62610.49470.3120.1189Malfunction1.120.58670.83810.640.61480.47040.29160.1383Rolling body0.99281.14180.90410.63350.62110.42720.29660.1171Malfunction0.99041.14660.91210.62650.61750.40780.29330.1026Outer ring0.58720.65661.0020.64120.61940.46290.25440.1227Malfunction0.60680.68270.9420.64830.59790.47110.32910.1195It can be seen from Table1 that the results of approximate entropy of IMF component of vibration signal in different states of bearings are different, and it can be seen that the information contained in the signal is different.Using the PSO-SVM classification model, the feature vectors extracted from IMF components after CEEMDAN decomposition are used for pattern recognition, and the program is run five times. The results are shown in Tables2 and 3.Table 2
Classification effect of approximate entropy as a feature vector.
Number of runsRunning time (s)CgFitness valueRecognition rate (%)1112.5997.70.0197.67952117.864.6343.65697.67953109.931000.0197.67954109.1596.3240.0197.67955109.0198.3840.0197.6795Table 3
Sample entropy is the classification effect of the feature vector.
Number of runsRunning time (s)CgFitness valueRecognition rate (%)1115.66115.79862.262497.33952110.9281001.057497.6795.53109.90491.94821.031597.6795.54115.35190.17021.17697.6795.55100.92251.50630.852597.3395.5Observing Tables2 and 3, we can get that the average classification recognition rate is 95% when approximate entropy is feature information, and the classification recognition rate is 95.5% when sample entropy is feature information.


## 4. Application of Fault Feature Extraction Method for Rolling Bearing

### 4.1. Fault Feature Extraction Method Based on Design

In this section, the advantages of the LMD method, permutation entropy theory, multivariate feature fusion method, and LLE dimensionality reduction clustering theory are comprehensively used to propose a new method for fault feature extraction of rolling bearings. This method has the following advantages for fault feature extraction in different states. First, aiming at the nonstationary problem of the initially collected test time series in a single scale, the initial signal is decomposed to different scales for analysis by using the adaptive decomposition ability of LMD, and the stationary component form of the signal can be obtained. The improved LMD method also preliminarily eliminates the interference information. Second, aiming at the influence of equipment structure and function coupling and uncertain signal path route, because of the complex components of vibration signals and weak fault changes, permutation entropy can better describe the complexity of signals and good sensitivity to abrupt signals and quantitatively express the fault information contained in different component signals, thus completing the initial extraction of fault features. Thirdly, in view of the “dimension disaster” and information redundancy in the process of initial signal acquisition of rotating machinery vibration signals, because of the low efficiency of fault feature recognition, the redundant information in feature vectors is removed by using the better dimension reduction ability of LLE and the clustering results are visualized, thus completing the secondary extraction of fault features and improving the accuracy of fault recognition.There are five steps in the implementation of the fault feature extraction method, which are collecting signals, carrying out multiscale analysis on initial signals, and summarizing and sorting out all the obtained single components. The improved LMD algorithm is used to screen the PF components obtained in step 1 to obtain all effective PF component sets, and the arrangement entropy of all effective PF components is calculated to complete the initial extraction of fault features. According to the selection criteria of error characteristic sensitive parameters, shielding can effectively restore the time domain and frequency domain characteristic parameters of the essential characteristics of the output signal and shield the time-frequency domain characteristics of the start-up signal. Combining the time domain characteristic index of the initial signal with the PF-PE characteristic vector obtained in step 2, a high-dimensional characteristic data set is constructed, and the high-dimensional characteristic data is reduced by LLE to obtain the low-dimensional characteristic data with good clustering, thus. completing the secondary extraction of rotating machinery fault characteristics.


### 4.2. Data Enhancement Technology

Data enhancement technology can provide a spatial solution for limited problems. On the basis of the original, it can improve the size and quality of data by performing necessary transformations, such as performing various operations on the original picture in the field of computer vision. For example, when the offset is set to 100 and the sample length is 2400, the first signal sample is intercepted from the 1st to 2400th data points of the initial signal, the second signal sample is intercepted from the 101st to 2500 data points, and so on, and an enhanced data set is obtained.


### 4.3. Application of LMD and LLE

#### 4.3.1. CWRU Rolling Bearing Data Description

The structure of the rolling bearing vibration signal testing device includes a motor, power meter, sensor, decoder, and other components.Two types of test bearings were selected for the test, and the structural description is shown in Table4.Table 4
Relevant specifications and parameters of test equipment.
CategoryModelNumber of rolling elementsPositionActionDeep groove spherical bearing6203-2RS JEM SKF/NTN8Fan endSupport motor shaft6205-2RS JEM SKF/NTN9Driving endIn Table4, SKF and NTN represent bearing sources. The two test bearing specifications are shown in Table 5.Table 5
Relevant specifications and parameters of rolling bearings.
Bearing typeInner ring (mm)Diameter size outer ring (mm)Rolling body (mm)Nodal ring (mm)Thickness (mm)6203-2RS17406.746228.4988126205-2RS25527.9439.0415The default failure modes of the test are shown in Table6.Table 6
Rolling bearing failure mode settings.
Failure modeLoss dimension (diameter/mm)Position settingSource of bearingSingle point loss fault EDM0.1778Inner ringSKFCompanyRolling bodyOuter ring3 o’clock6 o’clock12 o’clock0.3556Inner ringRolling bodyOuter ring3 o’clock6 o’clock12 o’clock0.5334Inner ringNTNCompanyRolling bodyOuter ring3 o’clock6 o’clock12 o’clock0.7112Inner ringRolling bodyOuter ring3 o’clock6 o’clock12 o’clockFour working conditions were designed in the test, as shown in Table7, and faults and no faults were set in different degrees and positions under each working condition.Table 7
Working condition setting of rolling bearings.
Working condition number1234Motor speed (r/min)1797177217501730Motor load (horsepower)0123


#### 4.3.2. Experimental Analysis of Feature Extraction

According to the data enhancement algorithm, the offset is 100, and the vibration signals of 36 different fault types under four working conditions and different fault positions and diameters are selected from the CWRU data set to carry out fault diagnosis tests.The rolling bearing failure test data set selected in this paper is shown in Table8. Each fault contains 250 sets of data, and the sampling points of each group of samples are set to 2400 points. 4/5 samples are randomly selected as the training set of the model, and the remaining 1/5 samples are used as the test set of the model.Table 8
Fault test data set of rolling bearing.
Motor speed (r/min)Fault locationFault diameter (mm)Training sampleTest sampleSample label1797Failure-free02005010.1778200502Inner ring0.35562005030.53342005040.1778200505Outer ring0.35562005060.53342005070.1778200508Rolling body0.35562005090.533420050101772Failure-free020050110.17782005012Inner ring0.355620050130.533420050140.17782005015Outer ring0.355620050160.533420050170.17782005018Rolling body0.355620050190.533420050201750Failure-free020050210.17782005022Inner ring0.355620050230.533420050240.17782005025Outer ring0.355620050260.533420050270.17782005028Rolling body0.355620050290.533420050301730Failure-free020050310.17782005032Inner ring0.355620050330.533420050340.17782005035Outer ring0.355620050360.533420050370.17782005038Rolling body0.355620050390.53342005040In order to verify the effectiveness and superiority of the proposed method, some test results of labels 11, 13, 16, and 19 are selected as examples to analyze and explain. The sample parameters of vibration signals in different states are shown in Table9.Table 9
Sample parameters under normal and typical fault conditions.
Working condition settingFault settingMapping labelModel selection6205-2RS JEM SKFFailure-free11Inner ring fault (diameter = 0.3556 mm)13Sampling frequency12999 HzOuter ring fault (diameter = 0.3556 mm)16Rotational speed1772 r/minRoller failure (diameter = 0.3556 mm)19The main fault information calculated by the correlation coefficient method is included in the first five components.It can be seen in Table10 that the solution values of faults are very close, and it is difficult to distinguish them if permutation entropy is directly used as the selected feature.Table 10
Average value of permutation entropy of PF1.
MalfunctionNormal stateInner ring positionOuter ring positionRoller positionAverage4.45165.42865.52515.0207Figure1 is the distribution diagram of the arrangement entropy of different types of faults.Figure 1
LMD permutation entropy.It can be seen that there are some differences between PE values under various working conditions, and the PE values in PF components of each level in the same signal correspond to the confusion degree at its scale.As shown in Figures2 and 3, it can be seen that compared with LMD permutation entropy, the features extracted by the other two methods have a serious aliasing phenomenon, which increases the difficulty of distinguishing the fault state of rolling bearings. This proves the advantage of LMD permutation entropy in feature extraction.Figure 2
EMD permutation entropy.Figure 3
LMD sample entropy.Figure4 shows the distribution of LMD permutation entropy in different states.Figure 4
LMD permutation entropy of vibration signals in four states.We can also see the change law of arrangement entropy of four states. The permutation entropy is the smallest in the normal state. The permutation entropy of different faults increases in different degrees, and the permutation entropy becomes smaller and smaller with the decomposition. This is because the LMD decomposition makes the vibration signal tend to be stable continuously, and the permutation entropy is different in different states, while the entropy of different PF components in the same signal reflects the uncertainty at this scale.In this paper, the permutation entropy and sample entropy in four states are calculated at the same time, as shown in Figures5 and 6.Figure 5
EMD permutation entropy of vibration signals in four states.Figure 6
LMD sample entropy of vibration signals in four states.It can be seen from the figure that compared with EMD permutation entropy and LMD sample entropy, LMD permutation entropy has fewer cross-aliasing areas, and the distinction between different states is more obvious. The advantages of LMD permutation entropy in feature extraction are proved. Some high-dimensional eigenvector values are shown in Table11.Table 11
Partially constructed high-dimensional feature vectors.
Eigenvector1111131316161919…10.40320.40222.83314.03160.64870.59621.68422.2994…20.06550.06480.16790.16750.09290.09210.16490.1632…32.77832.76919.556734.59632.97042.87285.76248.6372…44.57814.611819.604627.51375.26364.61469.871812.8731…54.46874.51555.45315.44955.54895.54894.96514.9272…62.68082.77513.37373.4483.55293.52173.1183.1707…71.51331.55061.98191.99062.01862.07451.76371.7449…81.03641.0261.35651.28511.30671.3011.25521.237…90.86740.82940.96220.95210.96820.93860.9140.9263…The eigenvectors of four kinds of fault states, which are labeled as 1U3, 16, and 19, respectively, are calculated, and then the corresponding training set and test set are obtained. Table12 shows SVM recognition results under different eigenvalues.Table 12
Recognition results of different features.
Feature classificationNormal (%)Inner ring position (%)Outer ring position (%)Roller position (%)Average (%)Initial signal84.1777.837578.0878.69PF-PE99.8384.35888592.25PF-PE-time-frequency domain10090.9190.0991.8293.04PF-PE-time-frequency domain-LLE100100100100100It can be seen from Table12 that the recognition rate is the lowest when only using the permutation entropy feature of the original signal, while the classification accuracy is obviously improved by using the PF-PE method, which shows that it is necessary to carry out multiscale analysis on the initial signal of rolling bearings. Compared with the PF-PE method, the recognition result of the PF-PE time-frequency domain method is improved to some extent, which shows that the method of multifeature fusion retains the essential characteristics of the initial signal well. The results of PF-PE-Time-Frequency Domain-LLE and PF-PE-Time-Frequency Domain show that LLE can eliminate redundant information and improve the clustering degree of low-dimensional features and further prove the suitability of the LLE method combined with LMD permutation entropy and multifeature fusion in feature extraction of rolling bearings.To illustrate the universality of the extraction method designed in this paper, an additional class of rolling bearing data sets is introduced. The XJTU-SY rolling bearing test data set of Xi'an Jiaotong University is used to apply the proposed algorithm. Because the bearing data belongs to the whole cycle life test data, in order to be used in the verification of this method, to ensure that the fault information type represented by the obtained signals is clear, the samples are taken at the beginning (normal position) and the last failure position (clear fault position), and then, the test set is expanded by data enhancement.The rolling bearing fault test data set is shown in Table13.Table 13
Fault test data set.
Fault typeMotor speed (r/min)Training sampleTest sampleLabelNormal2100200501Normal2250200502Normal2400200503Inner ring fault2250200504Inner ring fault2400200505Outer ring fault2100200506Outer ring fault2250200507Outer ring fault2400200508Keep and reduce faults2100200509Keep and reduce faults22502005010According to the data enhancement algorithm, the offset is 1000, four position failure modes (early normal state, inner ring, outer ring, and cage) are selected from XJTU-SY rolling bearing accelerated life test data set, fault diagnosis test is carried out for vibration signals with different rotational speeds and single fault damage conditions, with a total of 10 different fault modes = each fault contains 250 groups of data, and the sampling points of each group of data are set to 2400 points, of which 4/5 samples are arbitrarily extracted as the training set of the model, and the remaining 1/5 samples are used as the detection set of the model. Table14 shows the eigenvalue results of partial initial feature extraction.Table 14
Characteristic parameters of arrangement entropy of effective components of LMD.
Serial numberNormalInner ringOuter ringKeep decreasingEffective component16.13126.11325.97535.96036.31196.30346.10996.1042Effective component23.19763.26023.36213.38543.09683.10513.16033.1652Effective component31.89981.89291.95081.97941.75171.74571.85451.8506Effective component41.21891.21841.22871.2461.19291.19671.20561.1758Calculate the time domain and frequency domain features of the corresponding initial signal and combine the fault features extracted for the first time to obtain a high-dimensional feature vector group, and use the improved LLE algorithm to reduce its dimension. Table15 is the eigenvalue results of partial secondary feature extraction.Table 15
Modified LLE dimension reduction characteristic parameters.
CoordinatesNormalInner ringOuter ringKeep decreasingX1.16811.1683−0.9597−0.9612−0.3832−0.4644−1.1733−1.173Y−0.3477−0.4483−1.7338−1.27731.98231.99010.113060.1004Z0.15850.36711.06380.9573−0.4371−0.6875−0.4631−0.3956SVM is used to classify and identify different eigenvalues. Table16 shows the recognition results of SVM under different eigenvalues.Table 16
Recognition results of different features.
Feature classificationNormal (%)Inner ring (%)Outer ring (%)Rolling body (%)Average (%)Initial signal86.0075.0073.5349.0170.79PF-PE91.0086.0089.0083.5086.60PF-PE-time-frequency domain100.0093.4694.2998.9597.01PF-PE-time-frequency domain-LLE100.0095.00100.00100.0099.38Fault feature extraction is an important part of fault diagnosis technology, and it is the basis of ensuring the effectiveness, feasibility, and accuracy of mechanical equipment fault monitoring and diagnosis. Aiming at the difficulty of the current fault feature extraction method of vibration test signal based on acquisition in processing and aiming at the fault extraction of rolling bearings, this paper concludes that the fault feature extraction method of rolling bearings based on LMD replacement entropy and LLE is very effective. Simulation data and experimental data of two types of rolling bearings show that the proposed method is suitable for fault feature extraction of rolling bearings.


## 4.1. Fault Feature Extraction Method Based on Design

In this section, the advantages of the LMD method, permutation entropy theory, multivariate feature fusion method, and LLE dimensionality reduction clustering theory are comprehensively used to propose a new method for fault feature extraction of rolling bearings. This method has the following advantages for fault feature extraction in different states. First, aiming at the nonstationary problem of the initially collected test time series in a single scale, the initial signal is decomposed to different scales for analysis by using the adaptive decomposition ability of LMD, and the stationary component form of the signal can be obtained. The improved LMD method also preliminarily eliminates the interference information. Second, aiming at the influence of equipment structure and function coupling and uncertain signal path route, because of the complex components of vibration signals and weak fault changes, permutation entropy can better describe the complexity of signals and good sensitivity to abrupt signals and quantitatively express the fault information contained in different component signals, thus completing the initial extraction of fault features. Thirdly, in view of the “dimension disaster” and information redundancy in the process of initial signal acquisition of rotating machinery vibration signals, because of the low efficiency of fault feature recognition, the redundant information in feature vectors is removed by using the better dimension reduction ability of LLE and the clustering results are visualized, thus completing the secondary extraction of fault features and improving the accuracy of fault recognition.There are five steps in the implementation of the fault feature extraction method, which are collecting signals, carrying out multiscale analysis on initial signals, and summarizing and sorting out all the obtained single components. The improved LMD algorithm is used to screen the PF components obtained in step 1 to obtain all effective PF component sets, and the arrangement entropy of all effective PF components is calculated to complete the initial extraction of fault features. According to the selection criteria of error characteristic sensitive parameters, shielding can effectively restore the time domain and frequency domain characteristic parameters of the essential characteristics of the output signal and shield the time-frequency domain characteristics of the start-up signal. Combining the time domain characteristic index of the initial signal with the PF-PE characteristic vector obtained in step 2, a high-dimensional characteristic data set is constructed, and the high-dimensional characteristic data is reduced by LLE to obtain the low-dimensional characteristic data with good clustering, thus. completing the secondary extraction of rotating machinery fault characteristics.


## 4.2. Data Enhancement Technology

Data enhancement technology can provide a spatial solution for limited problems. On the basis of the original, it can improve the size and quality of data by performing necessary transformations, such as performing various operations on the original picture in the field of computer vision. For example, when the offset is set to 100 and the sample length is 2400, the first signal sample is intercepted from the 1st to 2400th data points of the initial signal, the second signal sample is intercepted from the 101st to 2500 data points, and so on, and an enhanced data set is obtained.


## 4.3. Application of LMD and LLE

### 4.3.1. CWRU Rolling Bearing Data Description

The structure of the rolling bearing vibration signal testing device includes a motor, power meter, sensor, decoder, and other components.Two types of test bearings were selected for the test, and the structural description is shown in Table4.Table 4
Relevant specifications and parameters of test equipment.
CategoryModelNumber of rolling elementsPositionActionDeep groove spherical bearing6203-2RS JEM SKF/NTN8Fan endSupport motor shaft6205-2RS JEM SKF/NTN9Driving endIn Table4, SKF and NTN represent bearing sources. The two test bearing specifications are shown in Table 5.Table 5
Relevant specifications and parameters of rolling bearings.
Bearing typeInner ring (mm)Diameter size outer ring (mm)Rolling body (mm)Nodal ring (mm)Thickness (mm)6203-2RS17406.746228.4988126205-2RS25527.9439.0415The default failure modes of the test are shown in Table6.Table 6
Rolling bearing failure mode settings.
Failure modeLoss dimension (diameter/mm)Position settingSource of bearingSingle point loss fault EDM0.1778Inner ringSKFCompanyRolling bodyOuter ring3 o’clock6 o’clock12 o’clock0.3556Inner ringRolling bodyOuter ring3 o’clock6 o’clock12 o’clock0.5334Inner ringNTNCompanyRolling bodyOuter ring3 o’clock6 o’clock12 o’clock0.7112Inner ringRolling bodyOuter ring3 o’clock6 o’clock12 o’clockFour working conditions were designed in the test, as shown in Table7, and faults and no faults were set in different degrees and positions under each working condition.Table 7
Working condition setting of rolling bearings.
Working condition number1234Motor speed (r/min)1797177217501730Motor load (horsepower)0123


### 4.3.2. Experimental Analysis of Feature Extraction

According to the data enhancement algorithm, the offset is 100, and the vibration signals of 36 different fault types under four working conditions and different fault positions and diameters are selected from the CWRU data set to carry out fault diagnosis tests.The rolling bearing failure test data set selected in this paper is shown in Table8. Each fault contains 250 sets of data, and the sampling points of each group of samples are set to 2400 points. 4/5 samples are randomly selected as the training set of the model, and the remaining 1/5 samples are used as the test set of the model.Table 8
Fault test data set of rolling bearing.
Motor speed (r/min)Fault locationFault diameter (mm)Training sampleTest sampleSample label1797Failure-free02005010.1778200502Inner ring0.35562005030.53342005040.1778200505Outer ring0.35562005060.53342005070.1778200508Rolling body0.35562005090.533420050101772Failure-free020050110.17782005012Inner ring0.355620050130.533420050140.17782005015Outer ring0.355620050160.533420050170.17782005018Rolling body0.355620050190.533420050201750Failure-free020050210.17782005022Inner ring0.355620050230.533420050240.17782005025Outer ring0.355620050260.533420050270.17782005028Rolling body0.355620050290.533420050301730Failure-free020050310.17782005032Inner ring0.355620050330.533420050340.17782005035Outer ring0.355620050360.533420050370.17782005038Rolling body0.355620050390.53342005040In order to verify the effectiveness and superiority of the proposed method, some test results of labels 11, 13, 16, and 19 are selected as examples to analyze and explain. The sample parameters of vibration signals in different states are shown in Table9.Table 9
Sample parameters under normal and typical fault conditions.
Working condition settingFault settingMapping labelModel selection6205-2RS JEM SKFFailure-free11Inner ring fault (diameter = 0.3556 mm)13Sampling frequency12999 HzOuter ring fault (diameter = 0.3556 mm)16Rotational speed1772 r/minRoller failure (diameter = 0.3556 mm)19The main fault information calculated by the correlation coefficient method is included in the first five components.It can be seen in Table10 that the solution values of faults are very close, and it is difficult to distinguish them if permutation entropy is directly used as the selected feature.Table 10
Average value of permutation entropy of PF1.
MalfunctionNormal stateInner ring positionOuter ring positionRoller positionAverage4.45165.42865.52515.0207Figure1 is the distribution diagram of the arrangement entropy of different types of faults.Figure 1
LMD permutation entropy.It can be seen that there are some differences between PE values under various working conditions, and the PE values in PF components of each level in the same signal correspond to the confusion degree at its scale.As shown in Figures2 and 3, it can be seen that compared with LMD permutation entropy, the features extracted by the other two methods have a serious aliasing phenomenon, which increases the difficulty of distinguishing the fault state of rolling bearings. This proves the advantage of LMD permutation entropy in feature extraction.Figure 2
EMD permutation entropy.Figure 3
LMD sample entropy.Figure4 shows the distribution of LMD permutation entropy in different states.Figure 4
LMD permutation entropy of vibration signals in four states.We can also see the change law of arrangement entropy of four states. The permutation entropy is the smallest in the normal state. The permutation entropy of different faults increases in different degrees, and the permutation entropy becomes smaller and smaller with the decomposition. This is because the LMD decomposition makes the vibration signal tend to be stable continuously, and the permutation entropy is different in different states, while the entropy of different PF components in the same signal reflects the uncertainty at this scale.In this paper, the permutation entropy and sample entropy in four states are calculated at the same time, as shown in Figures5 and 6.Figure 5
EMD permutation entropy of vibration signals in four states.Figure 6
LMD sample entropy of vibration signals in four states.It can be seen from the figure that compared with EMD permutation entropy and LMD sample entropy, LMD permutation entropy has fewer cross-aliasing areas, and the distinction between different states is more obvious. The advantages of LMD permutation entropy in feature extraction are proved. Some high-dimensional eigenvector values are shown in Table11.Table 11
Partially constructed high-dimensional feature vectors.
Eigenvector1111131316161919…10.40320.40222.83314.03160.64870.59621.68422.2994…20.06550.06480.16790.16750.09290.09210.16490.1632…32.77832.76919.556734.59632.97042.87285.76248.6372…44.57814.611819.604627.51375.26364.61469.871812.8731…54.46874.51555.45315.44955.54895.54894.96514.9272…62.68082.77513.37373.4483.55293.52173.1183.1707…71.51331.55061.98191.99062.01862.07451.76371.7449…81.03641.0261.35651.28511.30671.3011.25521.237…90.86740.82940.96220.95210.96820.93860.9140.9263…The eigenvectors of four kinds of fault states, which are labeled as 1U3, 16, and 19, respectively, are calculated, and then the corresponding training set and test set are obtained. Table12 shows SVM recognition results under different eigenvalues.Table 12
Recognition results of different features.
Feature classificationNormal (%)Inner ring position (%)Outer ring position (%)Roller position (%)Average (%)Initial signal84.1777.837578.0878.69PF-PE99.8384.35888592.25PF-PE-time-frequency domain10090.9190.0991.8293.04PF-PE-time-frequency domain-LLE100100100100100It can be seen from Table12 that the recognition rate is the lowest when only using the permutation entropy feature of the original signal, while the classification accuracy is obviously improved by using the PF-PE method, which shows that it is necessary to carry out multiscale analysis on the initial signal of rolling bearings. Compared with the PF-PE method, the recognition result of the PF-PE time-frequency domain method is improved to some extent, which shows that the method of multifeature fusion retains the essential characteristics of the initial signal well. The results of PF-PE-Time-Frequency Domain-LLE and PF-PE-Time-Frequency Domain show that LLE can eliminate redundant information and improve the clustering degree of low-dimensional features and further prove the suitability of the LLE method combined with LMD permutation entropy and multifeature fusion in feature extraction of rolling bearings.To illustrate the universality of the extraction method designed in this paper, an additional class of rolling bearing data sets is introduced. The XJTU-SY rolling bearing test data set of Xi'an Jiaotong University is used to apply the proposed algorithm. Because the bearing data belongs to the whole cycle life test data, in order to be used in the verification of this method, to ensure that the fault information type represented by the obtained signals is clear, the samples are taken at the beginning (normal position) and the last failure position (clear fault position), and then, the test set is expanded by data enhancement.The rolling bearing fault test data set is shown in Table13.Table 13
Fault test data set.
Fault typeMotor speed (r/min)Training sampleTest sampleLabelNormal2100200501Normal2250200502Normal2400200503Inner ring fault2250200504Inner ring fault2400200505Outer ring fault2100200506Outer ring fault2250200507Outer ring fault2400200508Keep and reduce faults2100200509Keep and reduce faults22502005010According to the data enhancement algorithm, the offset is 1000, four position failure modes (early normal state, inner ring, outer ring, and cage) are selected from XJTU-SY rolling bearing accelerated life test data set, fault diagnosis test is carried out for vibration signals with different rotational speeds and single fault damage conditions, with a total of 10 different fault modes = each fault contains 250 groups of data, and the sampling points of each group of data are set to 2400 points, of which 4/5 samples are arbitrarily extracted as the training set of the model, and the remaining 1/5 samples are used as the detection set of the model. Table14 shows the eigenvalue results of partial initial feature extraction.Table 14
Characteristic parameters of arrangement entropy of effective components of LMD.
Serial numberNormalInner ringOuter ringKeep decreasingEffective component16.13126.11325.97535.96036.31196.30346.10996.1042Effective component23.19763.26023.36213.38543.09683.10513.16033.1652Effective component31.89981.89291.95081.97941.75171.74571.85451.8506Effective component41.21891.21841.22871.2461.19291.19671.20561.1758Calculate the time domain and frequency domain features of the corresponding initial signal and combine the fault features extracted for the first time to obtain a high-dimensional feature vector group, and use the improved LLE algorithm to reduce its dimension. Table15 is the eigenvalue results of partial secondary feature extraction.Table 15
Modified LLE dimension reduction characteristic parameters.
CoordinatesNormalInner ringOuter ringKeep decreasingX1.16811.1683−0.9597−0.9612−0.3832−0.4644−1.1733−1.173Y−0.3477−0.4483−1.7338−1.27731.98231.99010.113060.1004Z0.15850.36711.06380.9573−0.4371−0.6875−0.4631−0.3956SVM is used to classify and identify different eigenvalues. Table16 shows the recognition results of SVM under different eigenvalues.Table 16
Recognition results of different features.
Feature classificationNormal (%)Inner ring (%)Outer ring (%)Rolling body (%)Average (%)Initial signal86.0075.0073.5349.0170.79PF-PE91.0086.0089.0083.5086.60PF-PE-time-frequency domain100.0093.4694.2998.9597.01PF-PE-time-frequency domain-LLE100.0095.00100.00100.0099.38Fault feature extraction is an important part of fault diagnosis technology, and it is the basis of ensuring the effectiveness, feasibility, and accuracy of mechanical equipment fault monitoring and diagnosis. Aiming at the difficulty of the current fault feature extraction method of vibration test signal based on acquisition in processing and aiming at the fault extraction of rolling bearings, this paper concludes that the fault feature extraction method of rolling bearings based on LMD replacement entropy and LLE is very effective. Simulation data and experimental data of two types of rolling bearings show that the proposed method is suitable for fault feature extraction of rolling bearings.


## 4.3.1. CWRU Rolling Bearing Data Description

The structure of the rolling bearing vibration signal testing device includes a motor, power meter, sensor, decoder, and other components.Two types of test bearings were selected for the test, and the structural description is shown in Table4.Table 4
Relevant specifications and parameters of test equipment.
CategoryModelNumber of rolling elementsPositionActionDeep groove spherical bearing6203-2RS JEM SKF/NTN8Fan endSupport motor shaft6205-2RS JEM SKF/NTN9Driving endIn Table4, SKF and NTN represent bearing sources. The two test bearing specifications are shown in Table 5.Table 5
Relevant specifications and parameters of rolling bearings.
Bearing typeInner ring (mm)Diameter size outer ring (mm)Rolling body (mm)Nodal ring (mm)Thickness (mm)6203-2RS17406.746228.4988126205-2RS25527.9439.0415The default failure modes of the test are shown in Table6.Table 6
Rolling bearing failure mode settings.
Failure modeLoss dimension (diameter/mm)Position settingSource of bearingSingle point loss fault EDM0.1778Inner ringSKFCompanyRolling bodyOuter ring3 o’clock6 o’clock12 o’clock0.3556Inner ringRolling bodyOuter ring3 o’clock6 o’clock12 o’clock0.5334Inner ringNTNCompanyRolling bodyOuter ring3 o’clock6 o’clock12 o’clock0.7112Inner ringRolling bodyOuter ring3 o’clock6 o’clock12 o’clockFour working conditions were designed in the test, as shown in Table7, and faults and no faults were set in different degrees and positions under each working condition.Table 7
Working condition setting of rolling bearings.
Working condition number1234Motor speed (r/min)1797177217501730Motor load (horsepower)0123


## 4.3.2. Experimental Analysis of Feature Extraction

According to the data enhancement algorithm, the offset is 100, and the vibration signals of 36 different fault types under four working conditions and different fault positions and diameters are selected from the CWRU data set to carry out fault diagnosis tests.The rolling bearing failure test data set selected in this paper is shown in Table8. Each fault contains 250 sets of data, and the sampling points of each group of samples are set to 2400 points. 4/5 samples are randomly selected as the training set of the model, and the remaining 1/5 samples are used as the test set of the model.Table 8
Fault test data set of rolling bearing.
Motor speed (r/min)Fault locationFault diameter (mm)Training sampleTest sampleSample label1797Failure-free02005010.1778200502Inner ring0.35562005030.53342005040.1778200505Outer ring0.35562005060.53342005070.1778200508Rolling body0.35562005090.533420050101772Failure-free020050110.17782005012Inner ring0.355620050130.533420050140.17782005015Outer ring0.355620050160.533420050170.17782005018Rolling body0.355620050190.533420050201750Failure-free020050210.17782005022Inner ring0.355620050230.533420050240.17782005025Outer ring0.355620050260.533420050270.17782005028Rolling body0.355620050290.533420050301730Failure-free020050310.17782005032Inner ring0.355620050330.533420050340.17782005035Outer ring0.355620050360.533420050370.17782005038Rolling body0.355620050390.53342005040In order to verify the effectiveness and superiority of the proposed method, some test results of labels 11, 13, 16, and 19 are selected as examples to analyze and explain. The sample parameters of vibration signals in different states are shown in Table9.Table 9
Sample parameters under normal and typical fault conditions.
Working condition settingFault settingMapping labelModel selection6205-2RS JEM SKFFailure-free11Inner ring fault (diameter = 0.3556 mm)13Sampling frequency12999 HzOuter ring fault (diameter = 0.3556 mm)16Rotational speed1772 r/minRoller failure (diameter = 0.3556 mm)19The main fault information calculated by the correlation coefficient method is included in the first five components.It can be seen in Table10 that the solution values of faults are very close, and it is difficult to distinguish them if permutation entropy is directly used as the selected feature.Table 10
Average value of permutation entropy of PF1.
MalfunctionNormal stateInner ring positionOuter ring positionRoller positionAverage4.45165.42865.52515.0207Figure1 is the distribution diagram of the arrangement entropy of different types of faults.Figure 1
LMD permutation entropy.It can be seen that there are some differences between PE values under various working conditions, and the PE values in PF components of each level in the same signal correspond to the confusion degree at its scale.As shown in Figures2 and 3, it can be seen that compared with LMD permutation entropy, the features extracted by the other two methods have a serious aliasing phenomenon, which increases the difficulty of distinguishing the fault state of rolling bearings. This proves the advantage of LMD permutation entropy in feature extraction.Figure 2
EMD permutation entropy.Figure 3
LMD sample entropy.Figure4 shows the distribution of LMD permutation entropy in different states.Figure 4
LMD permutation entropy of vibration signals in four states.We can also see the change law of arrangement entropy of four states. The permutation entropy is the smallest in the normal state. The permutation entropy of different faults increases in different degrees, and the permutation entropy becomes smaller and smaller with the decomposition. This is because the LMD decomposition makes the vibration signal tend to be stable continuously, and the permutation entropy is different in different states, while the entropy of different PF components in the same signal reflects the uncertainty at this scale.In this paper, the permutation entropy and sample entropy in four states are calculated at the same time, as shown in Figures5 and 6.Figure 5
EMD permutation entropy of vibration signals in four states.Figure 6
LMD sample entropy of vibration signals in four states.It can be seen from the figure that compared with EMD permutation entropy and LMD sample entropy, LMD permutation entropy has fewer cross-aliasing areas, and the distinction between different states is more obvious. The advantages of LMD permutation entropy in feature extraction are proved. Some high-dimensional eigenvector values are shown in Table11.Table 11
Partially constructed high-dimensional feature vectors.
Eigenvector1111131316161919…10.40320.40222.83314.03160.64870.59621.68422.2994…20.06550.06480.16790.16750.09290.09210.16490.1632…32.77832.76919.556734.59632.97042.87285.76248.6372…44.57814.611819.604627.51375.26364.61469.871812.8731…54.46874.51555.45315.44955.54895.54894.96514.9272…62.68082.77513.37373.4483.55293.52173.1183.1707…71.51331.55061.98191.99062.01862.07451.76371.7449…81.03641.0261.35651.28511.30671.3011.25521.237…90.86740.82940.96220.95210.96820.93860.9140.9263…The eigenvectors of four kinds of fault states, which are labeled as 1U3, 16, and 19, respectively, are calculated, and then the corresponding training set and test set are obtained. Table12 shows SVM recognition results under different eigenvalues.Table 12
Recognition results of different features.
Feature classificationNormal (%)Inner ring position (%)Outer ring position (%)Roller position (%)Average (%)Initial signal84.1777.837578.0878.69PF-PE99.8384.35888592.25PF-PE-time-frequency domain10090.9190.0991.8293.04PF-PE-time-frequency domain-LLE100100100100100It can be seen from Table12 that the recognition rate is the lowest when only using the permutation entropy feature of the original signal, while the classification accuracy is obviously improved by using the PF-PE method, which shows that it is necessary to carry out multiscale analysis on the initial signal of rolling bearings. Compared with the PF-PE method, the recognition result of the PF-PE time-frequency domain method is improved to some extent, which shows that the method of multifeature fusion retains the essential characteristics of the initial signal well. The results of PF-PE-Time-Frequency Domain-LLE and PF-PE-Time-Frequency Domain show that LLE can eliminate redundant information and improve the clustering degree of low-dimensional features and further prove the suitability of the LLE method combined with LMD permutation entropy and multifeature fusion in feature extraction of rolling bearings.To illustrate the universality of the extraction method designed in this paper, an additional class of rolling bearing data sets is introduced. The XJTU-SY rolling bearing test data set of Xi'an Jiaotong University is used to apply the proposed algorithm. Because the bearing data belongs to the whole cycle life test data, in order to be used in the verification of this method, to ensure that the fault information type represented by the obtained signals is clear, the samples are taken at the beginning (normal position) and the last failure position (clear fault position), and then, the test set is expanded by data enhancement.The rolling bearing fault test data set is shown in Table13.Table 13
Fault test data set.
Fault typeMotor speed (r/min)Training sampleTest sampleLabelNormal2100200501Normal2250200502Normal2400200503Inner ring fault2250200504Inner ring fault2400200505Outer ring fault2100200506Outer ring fault2250200507Outer ring fault2400200508Keep and reduce faults2100200509Keep and reduce faults22502005010According to the data enhancement algorithm, the offset is 1000, four position failure modes (early normal state, inner ring, outer ring, and cage) are selected from XJTU-SY rolling bearing accelerated life test data set, fault diagnosis test is carried out for vibration signals with different rotational speeds and single fault damage conditions, with a total of 10 different fault modes = each fault contains 250 groups of data, and the sampling points of each group of data are set to 2400 points, of which 4/5 samples are arbitrarily extracted as the training set of the model, and the remaining 1/5 samples are used as the detection set of the model. Table14 shows the eigenvalue results of partial initial feature extraction.Table 14
Characteristic parameters of arrangement entropy of effective components of LMD.
Serial numberNormalInner ringOuter ringKeep decreasingEffective component16.13126.11325.97535.96036.31196.30346.10996.1042Effective component23.19763.26023.36213.38543.09683.10513.16033.1652Effective component31.89981.89291.95081.97941.75171.74571.85451.8506Effective component41.21891.21841.22871.2461.19291.19671.20561.1758Calculate the time domain and frequency domain features of the corresponding initial signal and combine the fault features extracted for the first time to obtain a high-dimensional feature vector group, and use the improved LLE algorithm to reduce its dimension. Table15 is the eigenvalue results of partial secondary feature extraction.Table 15
Modified LLE dimension reduction characteristic parameters.
CoordinatesNormalInner ringOuter ringKeep decreasingX1.16811.1683−0.9597−0.9612−0.3832−0.4644−1.1733−1.173Y−0.3477−0.4483−1.7338−1.27731.98231.99010.113060.1004Z0.15850.36711.06380.9573−0.4371−0.6875−0.4631−0.3956SVM is used to classify and identify different eigenvalues. Table16 shows the recognition results of SVM under different eigenvalues.Table 16
Recognition results of different features.
Feature classificationNormal (%)Inner ring (%)Outer ring (%)Rolling body (%)Average (%)Initial signal86.0075.0073.5349.0170.79PF-PE91.0086.0089.0083.5086.60PF-PE-time-frequency domain100.0093.4694.2998.9597.01PF-PE-time-frequency domain-LLE100.0095.00100.00100.0099.38Fault feature extraction is an important part of fault diagnosis technology, and it is the basis of ensuring the effectiveness, feasibility, and accuracy of mechanical equipment fault monitoring and diagnosis. Aiming at the difficulty of the current fault feature extraction method of vibration test signal based on acquisition in processing and aiming at the fault extraction of rolling bearings, this paper concludes that the fault feature extraction method of rolling bearings based on LMD replacement entropy and LLE is very effective. Simulation data and experimental data of two types of rolling bearings show that the proposed method is suitable for fault feature extraction of rolling bearings.


## 5. Conclusion

The state of the bearing will affect the overall operation. Therefore, the research on the state of bearings is a very important issue. Bearing is also one of the most important mechanical fault sources. In fault diagnosis, vibration signals are usually taken as the main research object in this field, and signals containing a large amount of fault information are extracted. EMD is often used in nonstationary signal processing. It can distinguish the error characteristics of signals and compensate for the mode aliasing effect.Identification of Rolling Bearings. CEEMDAN can extract the features of fault signals. Two indexes are added to reconstruct the screened signals to solve the problem of fault feature extraction. Finally, the signal processing method based on CEEMDAN-ICA is verified by the measured signal. The results show that the method can separate the fault characteristic frequency of the fault bearing and then realize the fault diagnosis of the bearing. In this paper, CEEMDAN is combined with the sample entropy, Fisher point, and PSO-SVM to realize the pattern recognition of various failure states of rolling bearings.



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*Source: 9040784-2022-04-21.xml*