# Fault Diagnosis Method for Bearing Based on Digital Twin

Authors: Xuyang Xie; Zichun Yang; Wenhao Wu; Lei Zhang; Xuefeng Wang; Guoqing Zeng; Guobing Chen
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

The bearing is an essential component of rotating machinery, as its reliability and running state have a direct impact on the machinery’s performance. Considering that deep learning-based fault diagnosis methods for bearing require a large amount of labelled sample data, a novel fault diagnosis framework based on digital twin is proposed. In the case of fault data available, self-organizing maps with minimum quantization error and support vector machine are employed to analyze the data. Where fault data is unavailable, a bearing digital twin model is first constructed to simulate the data, and the convolutional neural network combined with transfer learning is utilized to diagnose the bearing faults. Then, the law of bearing performance degradation is investigated. The effectiveness of the proposed method is verified using bearing vibration data.


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## Body

## 1. Introduction

Bearings are widely used in rotating machinery, and they are one of the most important components. After long-term running under complex working conditions, bearings are prone to faults such as wear, spalling, and crack [1]. If the bearing fault is not handled in time, the machinery will fail to work, thus leading to unexpected economic loss and even terrible accidents. Therefore, the accurate diagnosis of bearing faults is of great significance in improving the stability and security of machinery and reducing the operation and maintenance cost [2].Scholars have made substantial research on the fault diagnosis of rotating machinery, such as bearings. Numerous model-based methods [3, 4], signal-based methods [5, 6], and data driven methods [7, 8] have been proposed. The model-based methods establish precise mathematical models of bearings and compare the output values with actual values to identify faults. Yang et al. [9] constructed a nonlinear dynamic model for rolling bearing, which successfully simulated both single faults and compound faults. Liu et al. [10] established a bearing finite element model to generate fault samples, then they were input into the support vector machine (SVM) to detect bearing faults. The conventional fault diagnosis methods based on signal processing aim at extracting fault-related features from signals manually to identify the bearing faults. Chen et al. [11] proposed a fault diagnosis method for bearings based on integration of resonance-based sparse signal decomposition and wavelet transform and achieved good results. Zeng et al. [12] presented a group-based K-singular value decomposition denoising algorithm to extract bearing fault features, which raised the diagnosis accuracy. Kong et al. [13] used the discrete particle swarm optimization algorithm to achieve the optimal sensor placement, which improved the fault diagnosis efficiency. However, these methods heavily rely on the sophisticated signal processing and abundant expertise [14]. In recent years, deep learning, owing to its excellent performance in automatic feature extraction, has enjoyed wide applications in fault diagnosis [15, 16]. Liu et al. [17] proposed a deep adversarial domain adaptation model, which provided an unsupervised learning method to effectively extract bearing fault features. Li et al. [18] converted the time series into graph data by the weighted horizontal visibility graph, and then the graph convolution network was performed to achieve an accurate diagnosis of bearing faults. Sun and Li [19] improved the convolution neural network (CNN) and applied it to bearing fault diagnosis. Cai et al. [20] used a Bayesian network to detect the early faults of bearings in permanent magnet synchronous motor based on vibration and acoustic emission signals. However, although these deep learning-based methods simplify the feature extraction process and raise the diagnosis accuracy, the training of deep learning models requires a large amount of labelled sample data. Unfortunately, fault data are generally insufficient in practical applications, which make it difficult to achieve accurate fault diagnosis.Motived by this problem, some scholars combined simulation models with deep learning to detect faults. They established accurate models of rotating machinery, such as bearings, to simulate sufficient fault data, which can be used for the training of deep learning models [21, 22]. Wang et al. [23] developed the lumped parameter model of gears to generate samples. Then, they were combined with measured signals to train different deep-learning models, and the diagnosis accuracy reached almost 100%. Liu and Gryllias [24] constructed a bearing phenomenological model to simulate vibration signals, and a domain adversarial neural network combined with transfer learning was trained for rolling bearing fault diagnosis. Gao et al. [25] simulated bearing samples of different type faults through the finite element method and expanded them by a generative adversarial network. Then, artificial intelligence models were trained to detect bearing faults. However, these simulation models are offline models that cannot be updated in real time based on the states of physical entities. The simulation results of the models may appear deviations as bearings run, which will lead to inaccurate fault diagnosis. Digital twin, an emerging virtual and real mapping technology, has seen rapid development in recent years. It can establish virtual entities integrating multiple disciplines, physical quantities, scales, and probabilities, and it possesses properties of real-time mapping and self-convergence. Some scholars have attempted to apply it to fault diagnosis [26]. Deebak and Al-Turjman [27] constructed the digital twin model of the machine tool, and deep transfer learning was performed to achieve condition monitoring and fault diagnosis. Jain et al. [28] presented a digital twin-based fault diagnosis method for distributed photovoltaic system, which demonstrated higher fault sensitivity. Guo et al. [29] simulated a large amount of fault data of a production line through digital twin and trained a reliable fault diagnosis model based on an improved random forest. However, few works have been reported to apply digital twin to bearing fault diagnosis. Therefore, it is a feasible and effective way to construct bearing the digital twin model and inject faults into it to simulate sensor data, which can address the problem of lacking fault data.Based on the above discussion, a novel bearing fault diagnosis framework based on digital twin is proposed in this article. In the case of fault data available, self-organizing maps (SOMs) with the minimum quantization error (MQE) and SVM are utilized to diagnose the bearing faults. In the case of fault data unavailable, CNN combined with transfer learning is performed to address the problem of low classification accuracy with small sample data. In addition, the law of bearing performance degradation is analyzed.The remainder of this article is organized as follows: in Section 2, the construction and update process of the bearing digital twin model is explained. Section 3 shows the theory and implementation procedure of the fault diagnosis method. In Section 4, the bearing performance degradation is analyzed. Finally, conclusions are drawn in Section 5.


## 2. Construction and Update of the Bearing Digital Twin Model

To construct an accurate digital twin model for bearing, the parameters, structure, working principle, and running mechanism of bearing as well as mathematical and physical knowledge are required. Currently, the mature modelling software related to digital twin mainly includes ANSYS Twin Builder, MATLAB Simulink, and Modelica-based MWorks. According to the requirements, digital twin models with different complexities can be built, including a single component, multiple components, and a system. A three-dimensional model of bearing is first built in SolidWorks in this article, and then the parameters are optimized. Finally, the multidomain unified modelling is conducted in Simulink to obtain an accurate model. The whole process of digital twin model construction is shown in Figure1.Figure 1
Flow chart of digital twin model construction.A bearing digital twin model is constructed and optimized based on a bearing test bench, as shown in Figure2. The rated current and power of the motor are 2.5 A and 250 W, respectively. It can achieve stepless speed regulation in the range of 0∼10000 rpm. The rotational speed is measured by a photoelectric sensor, while the vibration signals can be obtained through two MT3 speed sensors. The measured signals are transmitted to the computer via the ZXP-F8N data acquisition system.Figure 2
Bearing test bench.The outer race of bearing is connected with the bearing seat through interference fit, and the inner race is assembled with the shaft. So, only the inner race rotates, while the outer race remains fixed, which can support the shaft and lower the friction. Located at both ends of the shaft, the bearing seats are used for fixing and supporting the bearings, and the sealing device can reduce pollution. The assembly model of the whole device in SolidWorks is shown in Figure3.Figure 3
Device assembly diagram.The bearing faults are easy to mathematically model according to the differential equations of motion so as to obtain their time-domain waveforms. The bearing digital twin model is established in Simulink, and the relevant equations are as follows [30, 31]:(1)djt=xcosφjt+ysinφjt,where dj (t) represents the displacement of the groove bottom of outer race at φj (t) at time t, and x and y represent the horizontal and vertical displacements of the center of the inner race, respectively. The position angle φj (t) is given by(2)φjt=2πj−1/Z+wct,where Z represents the number of rollers, and wc is the angular velocity. The elastic deformation of the roller and raceway at φj (t) is(3)δj=djt−u.Assuming that the position angles of the defects on the outer race and inner race areφo and φi, respectively. Hence, φo is in the range of(4)ζo1=φo−WDe,ζo2=φo+WDe.φi is in the range of(5)ζi1=φo−WDi,ζi2=φo+WDi.When a roller is within the position angle range, the height of its center turns to(6)h=D21−cosWD.The parameterβ is introduced. When β = 1, the roller is in the defect; when β = 0, the roller is not in the defect.(7)βi=1,ζi1≤φjt≤ζi2,0,Beyond the range,βo=1,ζo1≤φjt≤ζo2,0,Beyond the range.The deformationδj between the roller and the raceway can be rewritten as(8)δj=djt−u−βihi−βoho,where u represents the radial clearance. Based on the Hertz equation, the contact load between the roller and the raceway can be obtained as follows:(9)Qjt=Kδjn,∀δj≥0,0,∀δj<0,where K is the deformation coefficient. When the roller is a sphere, n = 1.5; when the roller is a cylinder, n = 10/9. The resultant force of all the rollers acting on the inner race is decomposed into the forces in x and y directions, which can be expressed as follows:(10)QxQy=∑j=1ZQjcosφjsinφj.The kinematic differential equations of bearing are given by the Lagrangian equation(11)mx¨+Cx˙+Qx=Fr+mew2coswitmy¨+Cy˙+Qx=mew2sinwit,where m is the combined mass of bearing the inner race and shaft. C is the damping, and Fr represents the radial load. wi is the rotational speed of the inner race. e represents the eccentricity distance, and e = 0 in a balanced state. According to (12), The Simulink block diagram of bearing dynamic equations can be obtained, as shown in Figure 4.Figure 4
Simulink block diagram of bearing dynamic equations.The parameters of bearing are set according to [32]. The ode4 solver is used in Simulink, and the simulation results of a normal bearing are displayed in Figure 5. The relevant waveforms in [32] are shown in Figure 6. As can be seen, after the running state gradually turns to balance, the displacement and velocity waveforms in x and y directions are roughly the same, proving the validity of the bearing model in Simulink.Figure 5
Simulated displacement and velocity waveforms inx and y directions.
(a)(b)(c)(d)Figure 6
Displacement and velocity waveforms inx and y directions in [32].
(a)(b)(c)(d)After establishing the accurate digital twin model of bearing, the real-time running data of the bearing entity need to be collected and input into the model to continuously update and optimize it, thus raising the output accuracy of the model. To reduce the amount of data transmission and guarantee real-time performance, several simple state signals of bearing, such as temperature signals, are used to update the digital twin model, while other complex signals, such as vibration signals, are simulated by the model. The updating process of the bearing digital twin model is shown in Figure7. The consistency and synchronization between the digital twin model and the physical entity are achieved by comparing and validating the simulation results with collected data, which improves the model’s accuracy and confidence.Figure 7
Updating process of the bearing digital twin model.


## 3. Fault Diagnosis Method for Bearing Based on Digital Twin

### 3.1. Fault Diagnosis Scheme

It is impractical to get sufficient sample data for each possible bearing fault, which will lead to the reduction of diagnosis accuracy. To address this problem, a real-time updating digital twin model of bearing with faults injected is built to simulate the necessary data for the fault diagnosis method. The fault diagnosis process based on digital twin is displayed in Figure8.Figure 8
Fault diagnosis scheme based on digital twin.


### 3.2. Methods

#### 3.2.1. SOM-MQE

SOM [33] can get low-dimensional and discrete mapping after learning input layer data, which endows it with the characteristics of some dimensionality reduction algorithms. The training set in health data is used to train the SOM model, and the training set in fault data is applied to calculate the MQE value [34]. The MQE value can quantify the deviation of fault data between health data. The larger the MQE value, the more serious the fault. Figure 9 shows the process of state recognition based on SOM-MQE.Figure 9
State recognition process based on SOM-MQE.


#### 3.2.2. SVM

SVM [35] is a supervised learning algorithm that is designed to find a hyperplane to completely separate two types of data and make them the farthest from this hyperplane. The conventional binary SVM has gradually developed to multiclass SVMs, including one-to-one, one-to-multiple, directed acyclic graph, and binary tree SVMs. The hyperplane can be expressed as(12)wTx+b=0,where w is the weight vector and b is the bias.The training set isT = {(x1, y1), (x2, y2),…, (xN, yN)}, where xi ∈ Rn, yi∈{+1, −1}, i = 1, 2, …, N. xi is the ith feature vector. yi is the class label of xi, and each xi corresponds to yi. The training sample which is the closest to the hyperplane is represented by the support vector, and the distance to the optimal hyperplane is(13)d=wTx+bw,where w=w12+w22+...+wn2. The distances between other points and the hyperplane are larger than d, so(14)wTx+bwd≥1,y=1,wTx+bwd≥1,y=−1..(14) can be rewritten into(15)d=ywTx+bw.For the sake of simplification, we sety (wTx + b) = 1 to get the interval between two heterogeneous support vectors(16)γ=2w2.The goal of SVM is to maximizeγ, that is, to minimize 1/2w2 . Therefore, it becomes an optimization problem(17)min12w2,s.t.yiwTxi+b≥1,i=1,2,…,N.Since the constraints involve inequality, it is a convex quadratic programming problem. To solve the problem more efficiently, the Lagrange multiplier method is employed to obtain the dual problem so that the optimization problem can be written as(18)minα12∑i=1N∑j=1Nαiαjyiyjxixj−∑i=1Nαis.t.∑i=1Nαiyi=0αi≥0,i=1,2,…,N,where αi is the Lagrange multiplier.The analysis data are not generally linearly separable in practical applications. To deal with the problem, it is necessary to make some points not satisfy the constraintyi (wTxi + b) ≥ 1. The hinge loss can be applied to rewrite the optimization problem as(19)minα12∑i=1N∑j=1Nαiαjyiyjxixj−∑i=1Nαis.t.∑i=1Nαiyi=00≤αi≤C,i=1,2,…,N,where C > 0 is the penalty function. After obtaining the optimal solution αi∗=α1∗,α2∗,…,αN∗T , we calculate w∗=∑i=1Nαi∗yixi and b∗=yi−∑i=1Nαi∗yixixj , so the classification decision function is obtained according to the hyperplane equation w∗x + b∗ = 0(20)fx=signw∗x+b∗.The state recognition process based on SVM is displayed in Figure10. Similarly, the historical running data of bearings are divided into the training set and the test set. Then, the feature parameters are selected, and the corresponding feature matrix is obtained. The data in the feature matrix need to be normalized, which is more suitable for a comprehensive comparison.Figure 10
State recognition process based on SVM.


#### 3.2.3. CNN Combined with Digital Twin

For the data that are difficult to obtain and label in the experiment, it can be simulated by the digital twin model after injecting the fault. The simulated data are used for the training of the Inception V3 model, and the parameters of trained model are transferred. Then, the real-time running data of bearing are input into the transfer model to recognize its state, which effectively addresses the problems of lacking fault data and computing resource. The implementation process is shown in Figure11.Figure 11
State recognition process based on CNN and digital twin.


#### 3.2.4. Implementation Procedure of the Proposed Method

In the case of sufficient fault data, the SOM-MQE model is trained to identify the running state of bearing. The SVM model is employed when there is only a small amount of sample data. With little fault data, the digital twin model is performed to simulate the corresponding data to train the Inception V3 model combined with transfer learning [36], which reduces the requirement for training samples and time. The complete process of the proposed method is shown in Figure 12.Figure 12
Flowchart of the proposed method.


## 3.1. Fault Diagnosis Scheme

It is impractical to get sufficient sample data for each possible bearing fault, which will lead to the reduction of diagnosis accuracy. To address this problem, a real-time updating digital twin model of bearing with faults injected is built to simulate the necessary data for the fault diagnosis method. The fault diagnosis process based on digital twin is displayed in Figure8.Figure 8
Fault diagnosis scheme based on digital twin.


## 3.2. Methods

### 3.2.1. SOM-MQE

SOM [33] can get low-dimensional and discrete mapping after learning input layer data, which endows it with the characteristics of some dimensionality reduction algorithms. The training set in health data is used to train the SOM model, and the training set in fault data is applied to calculate the MQE value [34]. The MQE value can quantify the deviation of fault data between health data. The larger the MQE value, the more serious the fault. Figure 9 shows the process of state recognition based on SOM-MQE.Figure 9
State recognition process based on SOM-MQE.


### 3.2.2. SVM

SVM [35] is a supervised learning algorithm that is designed to find a hyperplane to completely separate two types of data and make them the farthest from this hyperplane. The conventional binary SVM has gradually developed to multiclass SVMs, including one-to-one, one-to-multiple, directed acyclic graph, and binary tree SVMs. The hyperplane can be expressed as(12)wTx+b=0,where w is the weight vector and b is the bias.The training set isT = {(x1, y1), (x2, y2),…, (xN, yN)}, where xi ∈ Rn, yi∈{+1, −1}, i = 1, 2, …, N. xi is the ith feature vector. yi is the class label of xi, and each xi corresponds to yi. The training sample which is the closest to the hyperplane is represented by the support vector, and the distance to the optimal hyperplane is(13)d=wTx+bw,where w=w12+w22+...+wn2. The distances between other points and the hyperplane are larger than d, so(14)wTx+bwd≥1,y=1,wTx+bwd≥1,y=−1..(14) can be rewritten into(15)d=ywTx+bw.For the sake of simplification, we sety (wTx + b) = 1 to get the interval between two heterogeneous support vectors(16)γ=2w2.The goal of SVM is to maximizeγ, that is, to minimize 1/2w2 . Therefore, it becomes an optimization problem(17)min12w2,s.t.yiwTxi+b≥1,i=1,2,…,N.Since the constraints involve inequality, it is a convex quadratic programming problem. To solve the problem more efficiently, the Lagrange multiplier method is employed to obtain the dual problem so that the optimization problem can be written as(18)minα12∑i=1N∑j=1Nαiαjyiyjxixj−∑i=1Nαis.t.∑i=1Nαiyi=0αi≥0,i=1,2,…,N,where αi is the Lagrange multiplier.The analysis data are not generally linearly separable in practical applications. To deal with the problem, it is necessary to make some points not satisfy the constraintyi (wTxi + b) ≥ 1. The hinge loss can be applied to rewrite the optimization problem as(19)minα12∑i=1N∑j=1Nαiαjyiyjxixj−∑i=1Nαis.t.∑i=1Nαiyi=00≤αi≤C,i=1,2,…,N,where C > 0 is the penalty function. After obtaining the optimal solution αi∗=α1∗,α2∗,…,αN∗T , we calculate w∗=∑i=1Nαi∗yixi and b∗=yi−∑i=1Nαi∗yixixj , so the classification decision function is obtained according to the hyperplane equation w∗x + b∗ = 0(20)fx=signw∗x+b∗.The state recognition process based on SVM is displayed in Figure10. Similarly, the historical running data of bearings are divided into the training set and the test set. Then, the feature parameters are selected, and the corresponding feature matrix is obtained. The data in the feature matrix need to be normalized, which is more suitable for a comprehensive comparison.Figure 10
State recognition process based on SVM.


### 3.2.3. CNN Combined with Digital Twin

For the data that are difficult to obtain and label in the experiment, it can be simulated by the digital twin model after injecting the fault. The simulated data are used for the training of the Inception V3 model, and the parameters of trained model are transferred. Then, the real-time running data of bearing are input into the transfer model to recognize its state, which effectively addresses the problems of lacking fault data and computing resource. The implementation process is shown in Figure11.Figure 11
State recognition process based on CNN and digital twin.


### 3.2.4. Implementation Procedure of the Proposed Method

In the case of sufficient fault data, the SOM-MQE model is trained to identify the running state of bearing. The SVM model is employed when there is only a small amount of sample data. With little fault data, the digital twin model is performed to simulate the corresponding data to train the Inception V3 model combined with transfer learning [36], which reduces the requirement for training samples and time. The complete process of the proposed method is shown in Figure 12.Figure 12
Flowchart of the proposed method.


## 3.2.1. SOM-MQE

SOM [33] can get low-dimensional and discrete mapping after learning input layer data, which endows it with the characteristics of some dimensionality reduction algorithms. The training set in health data is used to train the SOM model, and the training set in fault data is applied to calculate the MQE value [34]. The MQE value can quantify the deviation of fault data between health data. The larger the MQE value, the more serious the fault. Figure 9 shows the process of state recognition based on SOM-MQE.Figure 9
State recognition process based on SOM-MQE.


## 3.2.2. SVM

SVM [35] is a supervised learning algorithm that is designed to find a hyperplane to completely separate two types of data and make them the farthest from this hyperplane. The conventional binary SVM has gradually developed to multiclass SVMs, including one-to-one, one-to-multiple, directed acyclic graph, and binary tree SVMs. The hyperplane can be expressed as(12)wTx+b=0,where w is the weight vector and b is the bias.The training set isT = {(x1, y1), (x2, y2),…, (xN, yN)}, where xi ∈ Rn, yi∈{+1, −1}, i = 1, 2, …, N. xi is the ith feature vector. yi is the class label of xi, and each xi corresponds to yi. The training sample which is the closest to the hyperplane is represented by the support vector, and the distance to the optimal hyperplane is(13)d=wTx+bw,where w=w12+w22+...+wn2. The distances between other points and the hyperplane are larger than d, so(14)wTx+bwd≥1,y=1,wTx+bwd≥1,y=−1..(14) can be rewritten into(15)d=ywTx+bw.For the sake of simplification, we sety (wTx + b) = 1 to get the interval between two heterogeneous support vectors(16)γ=2w2.The goal of SVM is to maximizeγ, that is, to minimize 1/2w2 . Therefore, it becomes an optimization problem(17)min12w2,s.t.yiwTxi+b≥1,i=1,2,…,N.Since the constraints involve inequality, it is a convex quadratic programming problem. To solve the problem more efficiently, the Lagrange multiplier method is employed to obtain the dual problem so that the optimization problem can be written as(18)minα12∑i=1N∑j=1Nαiαjyiyjxixj−∑i=1Nαis.t.∑i=1Nαiyi=0αi≥0,i=1,2,…,N,where αi is the Lagrange multiplier.The analysis data are not generally linearly separable in practical applications. To deal with the problem, it is necessary to make some points not satisfy the constraintyi (wTxi + b) ≥ 1. The hinge loss can be applied to rewrite the optimization problem as(19)minα12∑i=1N∑j=1Nαiαjyiyjxixj−∑i=1Nαis.t.∑i=1Nαiyi=00≤αi≤C,i=1,2,…,N,where C > 0 is the penalty function. After obtaining the optimal solution αi∗=α1∗,α2∗,…,αN∗T , we calculate w∗=∑i=1Nαi∗yixi and b∗=yi−∑i=1Nαi∗yixixj , so the classification decision function is obtained according to the hyperplane equation w∗x + b∗ = 0(20)fx=signw∗x+b∗.The state recognition process based on SVM is displayed in Figure10. Similarly, the historical running data of bearings are divided into the training set and the test set. Then, the feature parameters are selected, and the corresponding feature matrix is obtained. The data in the feature matrix need to be normalized, which is more suitable for a comprehensive comparison.Figure 10
State recognition process based on SVM.


## 3.2.3. CNN Combined with Digital Twin

For the data that are difficult to obtain and label in the experiment, it can be simulated by the digital twin model after injecting the fault. The simulated data are used for the training of the Inception V3 model, and the parameters of trained model are transferred. Then, the real-time running data of bearing are input into the transfer model to recognize its state, which effectively addresses the problems of lacking fault data and computing resource. The implementation process is shown in Figure11.Figure 11
State recognition process based on CNN and digital twin.


## 3.2.4. Implementation Procedure of the Proposed Method

In the case of sufficient fault data, the SOM-MQE model is trained to identify the running state of bearing. The SVM model is employed when there is only a small amount of sample data. With little fault data, the digital twin model is performed to simulate the corresponding data to train the Inception V3 model combined with transfer learning [36], which reduces the requirement for training samples and time. The complete process of the proposed method is shown in Figure 12.Figure 12
Flowchart of the proposed method.


## 4. Bearing Performance Degradation Analysis Based on Digital Twin

The IMS bearing run-to-failure dataset is adopted for the analysis of performance degradation [37]. The bearing is installed on a shaft with 2000 rpm speed. The sampling interval is 10 min, and the sampling frequency is set to 20 kHz. 20480 points are sampled each time. The performance degradation curve is obtained in the following steps: (1)
Feature extraction. Figure13 illustrates the time domain waveform of the bearing run to failure signal. We can find that the bearing runs stably in the early stage, but it starts vibrating violently some time before the fault occurs. The spectrum kurtosis is extremely sensitive to the transient components in the signal, and the spectrum kurtosis corresponding to different time intervals and frequencies varies significantly, as displayed in Figure 14. Therefore, 11 time-domain features including mean, standard deviation, and skewness as well as four features of spectrum kurtosis such as mean, and skewness are used to extract fault characteristics.(2)
Feature selection. The previously mentioned 15 features have a different correlation with the degradation curve, so it is necessary to select the features with strong correlation. The 15 features are formed into a normalized feature matrix. Then, PCA is performed to screen the principal components of features, as shown in Figure15. We can observe that fusion features 9 to 15 outperform fusion features 1 to 8, so fusion features 9 to 15 are selected for further analysis.Figure16 displays the comparative scatter plots of fusion feature 9 with the other 6 fusion features. It can be found that fusion feature 9 is the most sensitive and changes the fastest, so it is selected as the health indicator.(3)
Degradation curve fitting. The health indicator represents the current degradation state of bearing, and its change with time is illustrated in Figure17. It is obvious that the health indicator fluctuates in a small range before 7000 min. After that, great fluctuations begin to appear, which demonstrates the deterioration of a bearing state. From about 9000 min, the state drops sharply, and the bearing exhibits a fault state at 9840 min.Figure 13
Time domain waveform of the bearing signal.Figure 14
Spectrum kurtosis of sample data. (a) Spectrum kurtosis of all sample data. (b) Spectrum kurtosis of partial sample data.
(a)(b)Figure 15
Feature selection based on PCA.Figure 16
Comparisons of fusion feature 9 with fusion features (a) 10, (b) 11, (c) 12, (d) 13, (e) 14, and (f) 15.
(a)(b)(c)(d)(e)(f)Figure 17
Scatter plot of the health indicator.A degradation curve is obtained by fitting the health indicator. The commonly used models include the exponential model, power model, and polynomial model. Figure18 displays the degradation curve fitted by the exponential model. We can observe that it is not accurate, failing to reflect the rapid changes of the health indicator.Figure 18
Degradation curve fitted by the exponential model.The degradation curve fitted by the combined power function [38] is shown in Figure 19. It exhibits a satisfactory fitting effect, which can provide a reference for the prediction of remaining useful life of the bearing.Figure 19
Degradation curve fitted by the combined power function.


## 5. Conclusions

To solve the problem of deep learning-based fault diagnosis methods requiring a large amount of labelled sample data, a novel fault diagnosis framework for bearings based on digital twin is proposed in this article. When fault data are available, SOM-MQE and SVM are employed to detect bearing faults. With little sample data, the bearing digital twin model is established to simulate the corresponding data needed by the CNN fault diagnosis model with transfer learning, which lowers the requirement of sample data. The law of bearing performance degradation is investigated, and the analysis results of bearing vibration data verify the effectiveness of the proposed method. Since there are generally insufficient fault data in actual circumstances, the proposed method has a great potential in engineering applications. Despite the advantages, the manually extracted fault features cannot guarantee high accuracy in performance degradation analysis. In future research, we will explore to automatically extract hidden features.



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*Source: 2982746-2022-11-17.xml*