Fault Detection in Active Magnetic Bearings Using Digital Twin Technology

Abstract

Active magnetic bearings (AMBs) are widely used in different industries to offer non-contact and high-velocity rotational support. The AMB is prone to failures, which may result in system instability and decreased performance. The efficacy and reliability of magnetic bearings can be significantly affected by failures in the sensor and control systems, leading to system imbalance and possible damage. A digital twin is an advanced technology that has been increasingly used in different industrial fields. It allows for the creation and real-time monitoring of virtual replicas of physical systems. This paper proposes a novel method for fault detection of Active Magnetic Bearings (AMBs) using digital twin technology and a neural network. The digital twin model serves as a virtual representation that accurately replicates the actual AMB system’s efficiency and features, allowing continuous real-time monitoring and detection of faults. The conventional neural network (CNN) is used as the primary tool for identifying faults in the Active Magnetic Bearing (AMB) within a digital twin model. Experiments proved the effectiveness and robustness of the suggested approach method to fault detection in the AMB.

1. Introduction

Active magnetic bearings (AMBs) are a form of bearing technology that levitates and supports spinning equipment without any physical contact using magnetic fields and advanced control systems [1]. AMBs provide many benefits over conventional mechanical bearings due to eliminating physical contact, including less wear and friction, increased dependability, and the capacity to function at high speeds and in challenging conditions [2]. Equipment safety has commanded increasing attention recently, and magnetic bearing defect diagnostics is a popular study area. Many industrial applications, such as turbomachinery, compressors, generators, and high-speed trains, need magnetic bearings [3]. Because of the natural instability of AMBs, a stabilizing feedback control is required for their efficient operation. The AMB system has many components, such as a controller, position sensor, and actuators. The controller is an integral part of the AMB, and choosing it is very influential to the system. PID (Proportional–Integral–Derivative) control is widely used in AMB systems because of its low power consumption and ability to provide adaptability, precision, efficiency, and cost-effectiveness [4]. In an active magnetic bearing (AMB) system, the position sensor plays a role because it presents accurate and real-time monitoring on the rotor’s position with respect to the stator. The simulation tool is used by Basso, M. et al. to help researchers compare several position sensor options and choose the best one for their particular AMB rotor-supported system [5]. Understanding the possibility of failure mechanisms in an AMB is essential to developing a reliable and safe AMB for various applications [6]. Therefore, many studies introduce methods to detect many types of failures in the magnetic bearing. Anand S. Reddy et al. [7] developed simulation data-driven principal component analysis (PCA) based fault detection and diagnosis (FDD) with statistical analysis for fault detection and diagnosis in position sensors and actuators. They correctly detected a variety of defect categories, including bias, multiplicative, and noise addition. Silva and Pederiva [8] have presented a failure detection methodology that relies on a mathematical model and the correlation between determined states associated with the rotor and the control. Lee, X.Y. et al. [9] provided a collection of three convolution kernel-based techniques and demonstrate their effectiveness in detecting faults in vibration signals obtained from planetary gearboxes in a test rig. They surpass other methods and achieve an accuracy above 98.8%. Artificial neural networks were used to establish mappings for unmeasurable conditions, while the identification of errors was accomplished by the comparison of output correlations across neural networks. GOUWS, Rupert [10] proposed a method for managing energy in active magnetic bearing (AMB) systems by using fault conditions to enhance system performance. GOUWS, Rupert [11] used Cepstrum analysis, Wigner-Ville Distributions (WVD), and enveloped Equi-Sampled Discrete Fourier transforms (ESDFT) to monitor the internal components of a radial active magnetic bearing (AMB) system. In order to identify instances of malfunction within the AMB system, it is customary to conduct a comparative analysis between the observed behaviors of the system and its expected performance under normal operating circumstances [8]. An alternative approach for identifying problems in active magnetic bearings involves conducting a simulation of the model before its physical construction. The conventional approaches to failure diagnosis involve analyzing signals in both the time and frequency domains [12].

Digital twins may be useful in this case. A digital twin is a virtual representation of a physical system that is continually updated to reflect the state of the system based on information from sensors and other sources. A digital twin can identify possible problems, forecast when failures might happen, and suggest steps to prevent or reduce them by doing real-time data analysis.

Digital twin modeling is one of the leading technologies propelling the development of AMBs [13]. The purpose of a digital twin is to imitate the behavior and performance of a physical asset or system under various operating situations [14]. A digital twin may be used to simulate the behavior of the rotating equipment, the control system, and the magnetic field in the context of AMBs [15]. This makes it possible to enhance the system’s performance and design before it is developed.

AMB system behavior may be predicted via digital twin modeling under various operating situations, such as loads, speeds, and vibration levels [16]. This shortens the time and expense of development by enabling the identification of possible problems and improving the system before it is developed. Digital twin modeling may be used to monitor system performance in real-time and improve the design of AMB systems [17]. The system’s behavior may be tracked and modified to improve its performance by connecting sensors and control algorithms with the digital twin. This might increase productivity, save downtime, and enhance system lifetime. Wang, Jinjiang, et al. [18] presented a Digital Twin reference model for rotating machinery fault diagnosis. They designed a model to be adaptable and updatable, and the researchers propose a model updating scheme based on parameter sensitivity analysis to improve its accuracy.

AMBs and digital twin modeling together provide a powerful new technique for enhancing the efficiency and dependability of rotating equipment. AMB systems may be created and refined to operate at their peak in various operating situations by using modern control systems and virtual modeling. However, this study aims to investigate and present a new methodology for identifying malfunctions in magnetic bearings through the utilization of digital twin technology. The objective of this study is to examine the necessity for efficient failure detection techniques in magnetic bearings in order to minimize system malfunctions, enhance reliability, and reduce downtime.

2. Materials and Methods

2.1. Structure of AMB

The stator and rotor form the AMB actuator’s structure. Figure 1 shows the cross section of the AMB actuator made of eight poles.

The stator, which is the stationary element of the magnetic bearing, is typically built from a number of electromagnets poles. Around the revolving shaft, these magnets are arranged in a circular or annular configuration. The revolving component of a magnetic bearing, the rotor, is fastened to the machine’s shaft. In order to create magnetic levitation and regulate the position of the shaft, the rotor has a collection of electromagnets that interact with the stator magnets. Position sensors are used to gauge the rotor’s position and displacement with respect to the stator. Hall effect, inductive, and optical sensors are frequently utilized as position sensors in active magnetic bearings. These sensors give the control system feedback, allowing it to adjust the magnetic forces and keep the shaft in the appropriate position. The control system is in charge of keeping track of the signals from the position sensors and generating the proper control signals to modify the magnetic fields. It calculates the necessary magnetic forces to oppose any outside forces or disturbances operating on the shaft, such as vibrations or axial loads, using feedback control algorithms. To improve the stability and performance of the bearing, the control system can additionally offer active dampening and vibration control capabilities. The control signals produced by the control system are amplified by power amplifiers, which then supply the required current to the stator electromagnets. The magnetic fields produced by the stator magnets must be powerful enough to support the weight of the rotor and maintain the proper position, which is what the power amplifiers make sure of.

2.2. Mathematical Model of AMB

To analyze and design a controller, one must first determine the system’s mathematical model [19]. An active magnetic bearing’s (AMB) mathematical model includes a description of the system’s forces and dynamics. The magnetic force on the rotor can be represented by the following equation [20]:

$${\mathrm{F}}_{\mathrm{x}}={\mathrm{k}}_{\mathrm{i}}{\mathrm{i}}_{\mathrm{x}}+{\mathrm{k}}_{\mathrm{s}}\mathrm{x} \mathrm{in} \mathrm{x} \mathrm{direction}$$

(1)

$${\mathrm{F}}_{\mathrm{y}}={\mathrm{k}}_{\mathrm{i}}{\mathrm{i}}_{\mathrm{y}}+{\mathrm{k}}_{\mathrm{s}}\mathrm{y} \mathrm{in} \mathrm{y} \mathrm{direction}$$

(2)

where, F_x,y is the magnetic force in x,y direction, i_x,y control current in x,y direction, x,y is displacement of the rotor form its original position in x,y direction, k_i is current gain, and k_s is position stiffness. The current gain and position stiffness can be calculated by:

$${\mathrm{k}}_{\mathrm{i}}=\frac{{\mathsf{\mu }}_0{\mathrm{N}}^2\mathrm{A}{\mathrm{i}}_0}{{\mathrm{g}}_0^2}$$

(3)

$${\mathrm{k}}_{\mathrm{s}}=\frac{{\mathsf{\mu }}_0{\mathrm{N}}^2\mathrm{A}{\mathrm{i}}_0^2}{{\mathrm{g}}_0^3}$$

(4)

where, ${\mathsf{\mu }}_0$ is the permeability of free space, N is the number of turns, A the cross-sectional area of the pole, ${\mathrm{i}}_0$ is the current acting on all AMB, and ${\mathrm{g}}_0$ air-gap between rotor and the stator. Figure 2 shows a block diagram of an AMB closed-loop system.

A control system is often used to maintain stability and regulate the position of the rotor. In AMB systems, a proportional-integral-derivative (PID) controller is often used [21]. The control system model can be represented by the following equation:

u = Kp * e + Ki * ∫ e dt + Kd * de/dt

(5)

where, u represents the control signal applied to the electromagnets, Kp is the proportional gain, Ki is the integral gain, Kd is the derivative gain, and e is the error between the desired and actual rotor position. The dynamics of the rotor may be mathematically described by a second-order differential equation that takes into consideration many variables such as the mass, inertia, and damping of the rotor [22]. As can see in the following equation:

M * d²x/dt² + C * dx/dt + K * x = F

(6)

where, x represents the displacement of the rotor, M is the mass of the rotor, C is the damping coefficient, K is the stiffness coefficient, and F is the total force acting on the rotor. The behavior of the electromagnets in the AMB system can be represented by an electrical circuit equation [23]. The current in the electromagnets can be described using the following equation:

V = L * di/dt + R * i

(7)

where, V represents the voltage applied to the electromagnets, L is the inductance of the electromagnets, R is the resistance of the electromagnets, and di/dt is the derivative of the current. A comprehensive mathematical model of an active magnetic bearing system may be developed by combining Equations (1), (2) and (5)–(7) and analyzing the relationships among the magnetic forces, control system, rotor dynamics, and electromagnet behavior. This model serves as a basic structure for the examination, development of control design, and simulation of AMB systems. All dimensions used in this study are listed in

Table 1. Different parameters used.

Parameter	Value
Height of the stack	71.42 mm
Rotor diameter	50 mm
Operating air gap	0.5 mm (radial)
The outer diameter of the AMB	140 mm
The inner diameter of the AMB	120 mm
Number of turns	80

.

2.3. Finite Element Model of the AMBs

A nonlinear model of the AMBs can be set up in ANSYS based on the 2-D finite element strategy, using ANSYS Maxwell (V.2022 R2) [24]. The software enables the execution of many steps, including geometry definition, physics application, meshing, equation solving, and result visualization [25]. Using Maxwell’s formulas for a steady magnetic field:

∇ × H = J

(8)

Here, J represents the steady current density (A/m²).

The material equation may be expressed using the reluctivity v:

H = νB

(9)

The variable v represents the reciprocal of the material-dependent (1/μ).

The magnetic vector potential A (Wb/m) is a measure of the magnetic flux density as:

B = ∇ × A

(10)

The basic equation of the vector potential formulation for the magnetic field can be obtained by substituting Equations (9) and (10) into Equation (8).

J = ∇ × (ν∇ × A)

(11)

The structural design of AMB includes a rotor and stator, both equipped with coils. The chosen material for the rotor and stator components is JFE_Steel_50JN1000, whereas copper is selected as the material for the coils. Figure 1 illustrates the heterpolar architecture of the AMB, which has eight poles spaced 45 degrees away from one another. Table 1 lists the AMB’s parameters. The parameters mentioned are used to design AMB by implementing the finite element technique, explicitly using the ANSYS software. Figure 3 illustrates the mesh configuration of AMB with eight poles. Figure 4 shows the magnetic flux density of an eight-pole Active Magnetic Bearing (AMB).

3. Failures in AMB

To effectively develop a reliable and safe AMB system for different purposes, it is imperative to have a comprehensive understanding of the potential failure mechanisms associated with AMBs. The failure of any component impacts the performance of an AMB. AMB faults must be found at the early stages of expression. There are three essential stages for dealing with faults:

• Determining the fault’s timing.
• Determining the component containing a defect.
• Finding the type of the fault.

Tsai, N. C., King, et al. [26] classified faults into multiplicative, bias, and noise addition. They explicitly classified faults in the sensors and actuators, as shown in Figure 5.

Every failure has a degree of importance to the system’s performance that depends on the cause. Lijesh, K.P. and Hirani, H. [27] used the failure mode and effect analysis (FMEA) methodology to figure out the possible causes of failure for different passive magnetic bearing (PMB) configurations, as well as the resultant impacts of these failures on the overall performance of the bearings. The AMB often experiences problems in the electric systems represented by amplifiers and in the sensors, which are typically brought on by rising temperatures, wear and tear, or short circuits. A solid-state amplifier is typically utilized to supply power to each magnetic coil [28]. An amplifier is an important part of the AMB. The amplifiers are set up to operate in either voltage control or current control mode. Amplifier malfunctions can happen as a result of environmental factors, internal thermal effects, or high external load requirements [27]. All AMB systems fail due to the amplifier failures, causing no output power. Initially, a linear power amplifier was implemented in AMBs, but because linear power devices utilize a large amount of energy, they have been replaced with switching power amplifiers (SPAs) [29]. Failure in the position sensor results in errors in the signal feedback to the control system, which causes the rotor to become unstable and finally leads to failure in the AMB system. Failure in the position sensor may be related to the power supply that feeds the sensor and relative damage due to physical contact with parts of the AMB. Shishir Bisht, et al. [30] studied different control techniques and failure modes to increase AMB life, performance, and reliability. They used Failure Mode and Effects Analysis (FMEA) to identify and prioritize system failures in AMB components. This method involves evaluating the severity, occurrence, and detection of possible failures, which results in the identification of important failure modes [31]. The Risk Priority Number (RPN) value was determined for every single cause of failure. The significance of the analysis of causes (RPN) in relation to the failure of the position sensor is as follows:

• A value of 20 is given to the failure of an electric circuit of the sensor.
• A value of 18 is given to the physical contact between the sensor and the rotor.
• A value of 8 is given to the risk of damage to the shaft.
• A value of 3 is given to the presence of debris.

Moreover, these failures result in a decrease in speed, causing the rotor to become levitated from its original position.

4. Digital Twin Fundamentals and Application in AMB

A digital twin represents the physical object, process, or system in virtual representation [32]. It includes both the physical and digital domains through duplicated real-time data from the physical system with a simulation model to enhance its physical counterpart’s behavior, characteristics, and performance [33]. It works as a link between the real and simulated model to allow real-time monitoring, analysis, and optimization of the system [34]. The digital twin updated by capturing the data from the sensors of the physical system continuously. By implementing machine learning algorithms and simulation models, digital twins can achieve comprehensive insights, decision-making, and optimized performance by simulating and predicting the physical system’s behavior [35]. The digital twin can play an essential role in monitoring and optimizing, diagnosing failure, and enabling predictive maintenance of the AMB. The system behavior, including rotor position, magnetic force, and control signals, can be covered by the digital twin through integrating real-time data from the position sensor, amplifier, and other related sources for the AMB. This makes it possible to identify faults earlier, analyze performance, and simulate “what-if” scenarios to determine the effects of faults and strategies for optimization. The digital twin offers a simulation model for applying a control algorithm, optimizing control parameters, and testing system performance under various operating conditions. Also, it provides system monitoring and fault detection, permitting preventative measures to reduce failures, improve the system’s reliability, and reduce downtime.

5. Modeling of Digital Twin for AMB

The position signals, control signals, and current signals from sensors, controllers, and actuators, respectively, are used in this paper. Figure 6 shows the description of available electrical signals.

Based on the AMB working states, the physical and virtual models have been established. The physical entity’s state’s position, control, and current data are collected in the physical model to update the digital twin model and combined with the sensors, controllers, and actuators to synchronize with the virtual model. The sensor, controller, and actuator failures will be predicted using the synchronization data, preventing the AMB system from fault using the digital twin. The efficiency of the suggested digital twin model is validated by simulation and experimentation utilizing an AMB system. Figure 7 shows the experimental setup. In order to achieve stable levitation with four degrees of freedom (4-DOF), a pair of radial magnetic bearings are positioned at either end of the rotor. The rotor is mechanically linked to a 1.2-kilowatt motor through a flexible connection. The AMB control system utilizes a digital control modeled on the quick-control prototype dSPACE DS1007. To monitor the position of the rotor in the x and y axes, four eddy current position sensors are fixed adjacent to two radial bearings. The amplifier supports the AMB at 50 v, while the switching frequency is set at 20 kHz. The Tektronix A622 current probe is used to acquire the coil current signal. The AMB configuration utilized in the simulation and experiments’ physical and control parameters are shown in

Table 2. The AMB configuration parameters used in experiments.

Symbol	Parameter	Value	Unit
m	Rotor mass	3.34	kg
Ip	Momentum of inertia	0.00187	kg⋅m2
ki	Current stiffness	99	N/A
kx	Displacement stiffness	473	N/mm
ks	Gain of displacement sensor	5	V/mm
KP	Proportional coefficient of PID	1.4
KI	Integral coefficient of PID	30
KD	Differential coefficient of PID	0.001
kAD	Gain of A/D conversion	4096/5
kDA	Gain of D/A conversion	5/4096

.

The Matlab-Simulink software(V.R2022b) was implemented to represent the virtual model. The mathematical model was built to accurately represent the magnetic bearing system, including the essential equations and parameters. The selection of MATLAB as the simulation platform was based on its flexibility and comprehensive range of tools for numerical calculations and system modeling. As stated in Section 2.2, a mathematical model based on electromagnetic force theory and control theory was used to represent the active magnetic bearing system. The model was designed to include pertinent variables, such as position, control, and current signals, to effectively describe the system’s dynamic behavior. The parameters listed in Table 2 were used to configure the MATLAB simulation. The data used in the MATLAB simulation was obtained from the actual active magnetic bearing system being investigated. The system’s installed sensors’ position, control, and current signals were captured immediately throughout the data-collecting procedure. The raw data obtained from the physical system may potentially include many sources of noise and artifacts. In order to tackle this issue, signal processing methodologies were used, including the implementation of filtering methods and algorithms for noise reduction. The purpose of these preprocessing operations was to enhance the quality and reliability of the input signals, hence improving the accuracy of the MATLAB simulation outputs. The MATLAB simulation was enhanced by using actual input data obtained from the physical system, resulting in a more precise simulation of the behavior of the active magnetic bearing. This methodology facilitated the evaluation of the digital twin’s ability to identify failures in a manner that closely corresponds to the actual operating circumstances in the real system. The data collection process included the identification of various sensors used in the physical system, such as position sensors, control signal sensors, and current sensors. The suitable dSPACE (DS1007) hardware platform was chosen for integrating sensors with the simulation model as show in Figure 5. The hardware was set up for sensor signal collecting using dSPACE software tools, namely ControlDesk (V. 7.6.1.1). To ease data communication in real-time between the sensors and the simulation model executed in MATLAB, the implementation used dSPACE software modules, namely TargetLink (V.5.0). The synchronization between the simulation model and the hardware sampling rates was carefully implemented to mimic the real-time simulation capabilities. The integration enabled the interaction between the simulation model and the physical system using the collected sensor data. Figure 8 shows the design of the digital twin model for the AMB.

6. Vibration Image

The fault diagnosis process may be considered a problem of identification of patterns, where the extraction of features plays an important part in recognizing these patterns. The selection of the methodology used to characterize faults adequately significantly influences the precision of fault identification. The displacement sensors used by the AMB serve two primary purposes: first, to provide feedback about any deviations in the control loop, and second, to monitor the operational status of the AMB. Figure 9 illustrates the spatial locations of sensors denoted as X1, Y1, X2, and Y2. Both X1/Y1 and X2/Y2 are placed in orthogonal orientations. The points X1/X2 and Y1/Y2 are situated on the same axis line of the rotor. In order to get a vibration image, two channels of vibration signals (X1/Y1, X2/Y2) captured by different sensors are combined. Pairs of vibration signals, occurring simultaneously, are then represented as point pairings inside orthogonal coordinates. The trajectory is formed by collecting all point pairings, with the vertical and horizontal coordinates represented by both channels. The trajectory in the given coordinates is often referred to as a vibration image, while the specific places along this trajectory are denoted as vibration locations. This trajectory is called a shaft orbit. A shaft orbit is a useful tool for describing problems and is often used in monitoring since its form changes depending on the operational status of the rotating machinery. As seen in Figure 10, the vibration image exhibits different features than the conventional shaft orbit approach. In its normal condition, the conventional approach typically displays an approximately circular pattern, but the vibration image showcases a variety of patterns.

In cases when the number of signal channels exceeds two, the construction of the vibration image involves the arbitrary selection of two channels. Consequently, this process allows for the generation of many vibration images. Four sensors within the implemented AMB system are capable of generating six vibration images. Every vibration image is handled to get its features. Before the vibration image could be utilized in machine learning, it was necessary to convert it to a discretized representation and convert one dimension (1D) into a composite signal with two dimensions (2D). The region of the vibration image is split into L² grids. The active area, denoted by a value of 1, is defined as the region where the signal points interact. Conversely, the inactive area, represented by a value of 0, covers the remainder of the area. The representation of the vibration image can be shown as a binary image consisting of L² pixels, forming a matrix with values of 1 and 0. Figure 11 illustrates the mechanism by which the vibration image is formed.

As such, vibration images may be analyzed using a wide range of currently available image processing and description technologies. The vibration image shows the relationship between the two signal channels. Putting the two channels together makes it easier to show the phase and amplitude of the vibration image as well as their relationship, which is not possible when they are kept separate.

7. Classification Algorithm

The identification of faults in rotary equipment is a challenge in the field of pattern recognition. In addition to feature extraction, classification is another significant challenge in the field. There are many machine learning algorithms for fault diagnoses in rotary machines, such as Support Vector Machines (SVM), Artificial Neural Networks (ANN), Random Forest (RF), Hidden Markov Models (HMM), and Deep Learning Techniques (including Convolutional Neural Networks (CNNs), Recurrent Neural Networks (RNNs), and Long Short-Term Memory (LSTM) [36,37,38,39]. The use of deep learning models is essential in order to extract comprehensive, high-level semantic information, hence facilitating the execution of fault classification with enhanced effectiveness. The technique presented in this study incorporates CNN and its many adaptations to deal with the difficulty of multiple-feature fusion. In order to achieve end-to-end learning, the vibration image is integrated with an enhanced Convolutional Neural Network (CNN).

A Convolutional Neural Network (CNN) is a computational structure consisting of numerous layers that work together to analyze and extract important features from input data. The CNN architecture consists of many layers, including convolutional layers, activation functions, pooling layers, and fully connected layers. It is designed to process input data in the form of either 1D signals or 2D images. Each layer within the network performs distinct functions in the overall operations of the CNN. To execute convolutions, the input data is passed through a number of filters or kernels known as convolutional layers. Every filter is designed to extract certain characteristics by identifying and analyzing patterns and spatial data such as edges and textures in the images. Each filter produces a feature map as its output. An activation function called the Rectified Linear Unit (ReLU), which is applied after each convolutional layer, provides non-linearity to the network.

Pooling layers are commonly employed following convolutional layers as a method of down sampling. Their primary purpose is to decrease the dimensionality of feature maps and spatial dimensions. This reduction in dimensionality reduces the chance of overfitting and enhances the model’s robustness. Additionally, pooling layers aid in summarizing and extracting important information from the feature maps while simultaneously reducing the computational complexity of the network. There are two basic techniques for pooling layers, namely average pooling and maximal pooling. The approach most frequently applied in the suggested algorithm is maximum pooling.

After the development of conventional and pooling layers, the input data’s features are acquired in the form of a one-dimensional vector. The vector is then inputted into fully connected layers, which facilitate the network’s capacity for advanced reasoning and allow it to provide predictions or classifications by employing the extracted features.

The last layer of Convolutional Neural Networks (CNNs) is often referred to as the output layer, responsible for producing the network’s predictions. The activation function used for this layer in the classification task is always softmax. The difference between the actual truth labels and the expected outputs is measured using the loss function. Cross-entropy is often used as the basis for the loss function. Stochastic gradient descent (SGD) or its derivatives, such as the Adam optimizer, are optimization techniques that are used to minimize the loss function throughout CNN training [40]. This procedure, often referred to as backpropagation, is used to modify the weights and biases of the neural network with the aim of enhancing its overall performance. Figure 12 illustrates the CNN structure.

In order to implement the fault diagnostic procedure utilizing the suggested algorithm, the signals are acquired under various fault conditions during the operation of the Active Magnetic Bearing (AMB) system. These signals are then separated into two distinct sets: a training set and a testing set. The calculation of the vibration image is based on the signals present in each individual sample. The input for training the network parameters of the CNN consists of vibration images, while the output corresponds to the fault category of each sample. Subsequently, the vibration images obtained from the testing samples are fed into the convolutional neural network (CNN) that has been previously trained, resulting in the determination of fault categories. The AMB system includes four sensors that have the capability of collecting six vibration images. The integration of the data from these six vibration images allows for a comprehensive representation of the fail-related information. It is necessary to develop a new network structure and integrate the data obtained from the vibration images. The redesigned topology of the CNN contains many branches and combinations. Every branch relies on other branches and includes multiple conventional layers and pooling layers. In total, there exist six branches. The structure of these branches consists of a sequence of densely connected layers. Figure 13 illustrates the structure of the multi-branch network. The integration of many branches results in a significant amount of information derived from vibration images, which is then collected by merging the signals from all sensors using a sequence of fully connected layers. The fault representation acquired has a greater amount of information and possesses a higher level of discriminability.

In cases when obtaining and labeling experimental data is challenging, the digital twin model may be used to replicate the data after introducing a defect. The data that has been simulated is used to train the model, and the variables of the model that was trained are transmitted. Subsequently, the transfer model is fed with the real-time operating data of the AMB in order to identify its condition. This approach successfully resolves the issues of insufficient defect data and computational resources. Figure 14 illustrates the implementation method.

8. Results

To determine how to figure out the efficacy of the suggestion of the digital twin model, a sequence of tests was conducted on an AMB system (which was explained in Section 6). The system was tested at different rotation speeds (3000, 6000, 9000, and 12,000 rpm) under normal conditions. Figure 15 illustrates examples of vibration images under normal conditions at different rotation speeds.

Figure 16a illustrates the difference between the simulated signal response and the actual measured signals of the AMB system. By feeding the real-time inputs from the sensors into the simulation digital twin model, the error was minimized to achieve the most accurate update of the digital twin model, as seen in Figure 16b.

Two categories of data failures, including imbalance and misalignment, were acquired. A total of 1000 samples was randomly chosen from each category for training purposes, while the remaining samples were used for testing (as show in

Table 3. The AMB data used in the experiments.

Rotation Speed (rpm)	Category	Number of Samples	Number of Training Samples	Number of Testing Samples
3000	Normal	11,251	1000	10,251
	Imbalance	5012	1000	4012
	Misalignment	4120	1000	3120
6000	Normal	12,001	1000	11,001
	Imbalance	5743	1000	4743
	Misalignment	4490	1000	3490
9000	Normal	12,256	1000	11,256
	Imbalance	6277	1000	5277
	Misalignment	5295	1000	4295
12,000	Normal	9501	1000	8501
	Imbalance	3420	1000	2420
	Misalignment	2832	1000	1832

). Examples of normal and imbalanced signals are presented in Figure 17.

When testing a proposed CNN algorithm, it was observed that the precision of the test data was around 94%, although the precision of the training data was approximately 96.5%, as shown in Figure 18. The result indicates the model acts at a high level for detecting faults, proving it to be robust, efficient, and trusted. In Figure 19, the confusion matrix is shown. The expected faults are shown on the vertical axis of the matrix, the actual faults are shown on the horizontal axis, and the percentage of the prediction value is shown on the diagonal axis. The accuracy of fault detection in the AMB is greatly influenced by the decision to gather data from multiple sensors. The results of testing the digital twin model using different two-sensor combinations (x1, y1), (x1, x2), (x1, y2), (y1, x2), (y1, y2), and (x2, y2) are shown in Figure 20.

9. Discussion

The study emphasizes the significance of fault detection in Active Magnetic Bearing (AMB) systems, given how important they are in various industries like aerospace, power generation, and manufacturing. The paper proposes using a digital twin of the AMB system to monitor and analyze its performance in real-time. The digital twin acts as a standard to detect failures by simulating the system’s behavior. The research proposes to use a neural network as the primary tool for digital twin fault identification. The neural network is trained using real-time sensor data (including normal condition, imbalance and misalignment) from the digital twin. Real-time sensor data from the digital twin model is supplied into the neural network once it has been trained. This allows the network to analyze, realize, and classify various types of AMB system failures.

The experimental results were presented to validate the methodology. Because of the implementation of digital twin technology, the model proved to have a high fault detection accuracy compared with the literature studies, as shown in

Table 4. How the digital twin–CNN accuracy compares with other algorithms.

Algorithms	Accuracy
Digital twin–CNN	99%
SVI+MCNN [41]	98.6%
AdaBoos [42]	95%
Fault dictionary [12]	93%
CNN-A [43]	92.9%
CNN-B [44]	96.3%
sdAE [45]	95.01%
LSTM-WDCNN [46]	96.2%
CNN-HMMS [47]	98%
ANN [48]	95.14%
DBN [45]	94.07%
LSTM [49]	100%

The results reflect a high level of accuracy in identifying and classifying several types of faults, such as imbalance and misalignment.

.

10. Conclusions

In this paper, the concept of a digital twin was presented to reflect the behavior and characteristics of the physical system in the virtual model. The digital twin model is a reference for real-time signals to monitor and detect faults in the AMB system.

The neural network is employed for fault detection with a digital twin model. The various types of faults, like imbalance and misalignment, are detected by training the CNN using real-time signals. The experiment validated the efficiency of the suggested technique by the neural network using real-time signal data from the digital twin model.

Using digital twin technology and neural networks, we observed several advantages, such as real-time monitoring, early detection of faults, improving system robustness, and optimizing maintenance methods.

Finally, integrating the digital twin technology with CNN algorithms presents a promising strategy for accurate and rapid fault classification, facilitating early repair and reducing operational downtime. This study established the way for further research in this area and presented new opportunities for improving the performance and dependability of AMB systems.