Research on Fault-Diagnosis Technology of Rare-Earth Permanent Magnet Motor Based on Digital Twin

Abstract

To address the persistent challenges in diagnosing bearing faults, this study proposes an intelligent diagnostic framework based on the principle that mechanical faults manifest as symmetry-breaking phenomena in a system’s vibration signals. In a healthy motor, vibration signals exhibit a high degree of symmetry, whereas faults introduce identifiable and distinct asymmetries. This study constructs a high-fidelity digital twin model based on the five-dimensional model theory to simulate both the symmetrical (healthy) state and various asymmetrical faulty states of motor bearings—specifically, inner race, outer race, and rolling element faults—thereby effectively addressing the critical issue of data scarcity. Building upon this framework, fault features characterizing these asymmetries are accurately extracted using an optimized variational mode decomposition (VMD) algorithm and subsequently classified with a convolutional neural network–bidirectional long short-term memory (CNN-BiLSTM) model. The results validate the model’s ability to accurately replicate bearing-fault data. The proposed diagnostic method achieves a stable and high average accuracy of 98.44 ± 0.41% over multiple runs on the simulation data. Furthermore, its effectiveness was validated on a public real-world bearing dataset, where it achieved an accuracy of over 95%, demonstrating its robustness and potential for industrial applications by effectively identifying fault-induced asymmetries.

1. Introduction

The rare-earth permanent magnet motor (REPMM) is an efficient and energy-saving motor, which is widely used in industrial automation, new energy vehicles [1], wind power generation [2], and other fields. However, due to its complex structure and variable operating conditions, the REPMM faces various potential failure risks, particularly bearing failures. As a key supporting component for the motor’s normal operation, a bearing failure is analogous to a damaged “joint” in a mechanical system, which can quickly trigger a series of cascading effects. If a bearing failure is not diagnosed in time and promptly addressed, the motor’s performance will inevitably deteriorate, disrupting its original efficient and stable operating state and substantially reducing production efficiency. More critically, continued fault deterioration can lead to a sudden motor shutdown and the paralysis of the entire production process. It may cause costly downtime and production losses. Therefore, enhancing the fault diagnosis of REPMMs is a key measure to ensure stable motor operation, reduce economic losses, and promote sustainable development. Furthermore, the application of digital twin (DT) technology extends beyond immediate fault diagnosis, providing frameworks for optimizing engineering systems within a circular economy and thereby directly contributing to sustainability goals [3].

From a signal-processing perspective, vibration-based fault diagnosis can fundamentally be regarded as the detection of symmetry-breaking phenomena. In this context, signal symmetry refers not to a geometrically perfect waveform but to pattern symmetry—the stable, repeating nature of signals generated by the consistent mechanical operation of a healthy system. A fault is therefore regarded as a symmetry-breaking phenomenon because it introduces sharp, periodic transients that disrupt the established pattern. These fault-induced impacts manifest as distinct asymmetries in the vibration signals, each carrying specific diagnostic information. Therefore, the core challenge of intelligent diagnosis is to effectively capture and interpret these asymmetries. This paper leverages a digital twin framework precisely for this purpose—to model both the baseline symmetry of a healthy motor and the specific asymmetries associated with different faults. Specifically, this study addresses the diagnosis of four key operational states: the healthy condition, inner race faults, outer race faults, and rolling element faults.

Early motor-fault-diagnosis methods mainly rely on empirical judgment, periodic maintenance [4] or traditional signal-processing technologies, such as spectrum analysis and vibration analysis [5], which not only have low efficiency, but also make it difficult to achieve accurate prediction and timely intervention. Therefore, timely and accurate fault diagnosis in REPMMs is of great significance to ensure their normal operation and extend their service life. Yang Tongguang [6] proposed a rotor fault-diagnosis method based on grid-side instantaneous power, employing spectrum analysis of this parameter to effectively diagnose rotor faults. In recent years, with the development of artificial intelligence technology, fault-diagnosis methods based on intelligent algorithms such as neural networks and support vector machines have been widely adopted. Modern approaches often involve advanced deep learning techniques. For instance, some studies convert one-dimensional vibration signals into two-dimensional images using methods like the Gramian angular difference field, which are then processed by specialized residual networks to achieve high-precision diagnosis [7]. Other research has focused on adapting state-of-the-art architectures like the transformer, for unsupervised transfer-learning scenarios, tackling challenges related to data labeling and domain shift in bearing-fault diagnosis [8]. This trend is a core component of modern fault diagnosis, a field where the applications, trends, and challenges of machine learning are actively being reviewed [9]. For example, Zhang Tao [10] applied empirical mode decomposition to motor vibration data and then used an improved gray wolf optimization algorithm to optimize a support vector machine (SVM) classifier for motor bearing-fault diagnosis. However, the training of complex models requires large datasets, which increases the demand for computing resources and time, and requires high-performance computing hardware. These problems limit the widespread adoption and practical application of the diagnostic method. Therefore, it is necessary to develop a practical and low-cost fault-diagnosis algorithm and explore strategies for achieving real-time and accurate fault diagnosis under limited computing resources. This context underscores the need to explore practical solutions that can be implemented in actual production processes.

Digital twin (DT) technology, one of the core technologies of Industry 4.0 and smart manufacturing, offers new perspectives on fault diagnosis. A digital twin is a collection of computer-generated models that map physical entities into virtual space, enabling the monitoring, simulation, prediction, diagnosis, and control of physical objects through continuous information exchange between physical and virtual elements. The concept of the digital twin was first introduced in writing by the U.S. Department of Defense and the National Aeronautics and Space Administration (NASA) in 2010 [11], with the aim of modeling and predicting the behavior of complex systems through simulation techniques. Since 2016, digital twin technology has been widely applied in fault diagnosis, manufacturing, and other domains throughout the full product lifecycle [12]. At the same time, various international organizations started to develop standards to regulate the implementation and application of the digital twin. This shift indicates that digital twin technology has progressed from a stage of technological exploration to a more mature and systematic phase of application [13]. Both theoretical research and engineering applications exhibit multi-level and multi-dimensional development trends. Wagner [14] examined the key challenges and future potentials of digital twin technology and Industry 4.0 in achieving seamless integration between product specifications and production, advocating for a coherent framework to enable the holistic use of digital twins throughout the product development process. Building on this, Cunha [15] proposed a modular digital twin design method for reconfigurable manufacturing systems, addressing the rapid reconfiguration challenges of complex systems by defining module coupling mechanisms and interface protocols, and validating the method’s engineering feasibility in a smart factory. In response to the needs of equipment health management, Kannan [16] integrated a digital twin model of a grinding wheel with motor current monitoring data, constructing a dynamic prediction-correction closed-loop system for estimating the grinding wheel’s remaining service life. Building on a substantial theoretical foundation, many researchers have applied digital twin technology to mechanical equipment fault diagnosis. Through experimental data and rigorous analysis, these studies have confirmed the significant effectiveness of digital twins in the fault-diagnosis domain.

Peng [17] proposed a dynamic domain distribution alignment algorithm based on digital twins to address fault feature drift under non-smooth operating conditions. The method achieves high-precision diagnosis using only unlabeled real data, overcoming the challenges of cross-condition transfer and bearing-fault diagnosis in non-smooth environments. Huang [18] integrated physical simulation and experimental data into a digital twin model of rolling bearings through a multi-fidelity information fusion strategy, achieving an optimal balance between diagnostic accuracy and experimental cost-effectiveness. Model, which integrates physical simulation and experimental data to achieve an optimal balance between diagnostic accuracy and experimental economy at a lower cost. Digital twin technology also is instrumental in equipment health management. Venkatesan S [19] estimated and predicted the remaining service life of a permanent magnet synchronous motor (PMSM) by creating its digital twin and mapping it to the actual operating parameters of an electric vehicle via neural networks and fuzzy logic. Tshoombe B [20] developed a digital twin-based state-monitoring and analysis system for asynchronous motors, where real-time current measurements are simulated and mapped through a digital twin model thus obtaining additional information to effectively manage motor health. From detecting faults under unsteady operating conditions, to balancing diagnostic accuracy and cost, to accurately predicting motor service life and effectively managing motor health, digital twin technology has demonstrated notable success in mechanical equipment fault diagnosis and health management, showing clear advantages.

Although digital twin technology has shown considerable potential across various fields, its application in electric machine fault diagnosis still faces numerous challenges. Constructing accurate digital twin models for diverse types of equipment is inherently complex, as different electrical machines possess unique structural characteristics and operational behaviors, encompassing multiple aspects from data acquisition and modeling to simulation. Accurately constructing digital twin models for the physical equipment under study remains a persistent challenge for researchers. Traditional fault-diagnosis methods often rely on limited historical operational data or laboratory simulations. However, early failures in REPMM bearings are both progressive and random, and the available fault samples are often insufficient for data-driven intelligent diagnosis due to experimental constraints, equipment costs, and the difficulty of reproducing faults. Moreover, when focusing on subtle faults, achieving higher diagnostic accuracy inevitably requires substantially greater computational resources, making high-performance computing hardware a critical enabler for effective implementation. These intertwined factors hinder the widespread adoption and deployment of such diagnostic methods in practical applications. For researchers, striking a balance between achieving high accuracy and reasonably controlling resource consumption has become a critical issue requiring in-depth consideration and exploration.

In light of the above challenges, this study conducts the following research on REPMM fault-diagnosis technology based on digital twin methods:

• We construct a high-fidelity digital twin model based on the five-dimensional model theory. This model serves as a virtual platform to simulate both the baseline symmetrical patterns of a healthy motor and the distinct asymmetries introduced by various faults, establishing a foundation for studying various fault scenarios, which are treated as symmetry-breaking phenomena, in a controlled environment.
• Making full use of the dynamics simulation capability of the digital twin virtual model, it parametrically adjusts the type and degree of faults, flexibly simulates a variety of fault scenarios, and generates diversified fault data that are highly related to the rare-earth permanent magnet motors, which mitigates the challenges of data scarcity, high cost of obtaining fault samples, and insufficient coverage of the actual working conditions in traditional fault diagnosis.
• Based on the deep integration of digital twin-based simulation data and machine learning-based fault-diagnosis technology, a subtractive optimizer algorithm is used to optimize the variational modal decomposition in the fault-feature extraction process. It can guide the variational modal decomposition to extract fault features more efficiently and accurately, avoiding the problems of incomplete or inaccurate feature extraction that may occur in other methods. Furthermore, in the fault identification stage, the convolutional bidirectional long and short-term memory network is introduced to solve the problem of the low accuracy of fault diagnosis of tiny faults by virtue of its powerful time series data processing capability and feature learning ability.

2. Digital Twin Model

Digital twin technology, with its excellent capabilities for multi-dimensional data mapping, enables the reconstruction of physical entities in virtual space with a high degree of fidelity. Within this technical framework, the high-fidelity construction of digital twin models serves as the fundamental basis and key prerequisite for advancing digital twin technology from theoretical concepts to practical applications. Given the design requirements of twin models for rare-earth permanent magnet motors, it is crucial to undertake high-fidelity modeling efforts. In this study, we adopt the theoretical framework of the five-dimensional digital twin model proposed by Professor Tao Fei [21] as a guiding principle to construct the digital twin model of the rare-earth permanent magnet motor, as shown in Equation (1):

$$M_{DT}=\left(PE,VE,Ss,DD,CN\right)$$

(1)

where PE stands for physical entity, VE for virtual model, Ss for services, DD for digital twin data, and CN for interactive connection. Based on this five-dimensional model, the digital twin framework of the REPMM is constructed, as shown in Figure 1. Utilizing virtual simulation data, it provides strong support for subsequent fault-diagnosis research.

A physical entity (PE) is the basis of the digital twin. Digital twin technology uses digital methods to create a virtual model of the physical entity that accurately reflects its properties and behavior. A PE is a real, perceptible, and interactive object. In this study, the REPMM, bearings, sensors, and other equipment are regarded as physical entities. The RS-WZ3-N01-1-CX temperature and vibration transmitter is employed, as shown in Figure 2. This integrated sensor simultaneously collects the motor’s vibration velocity and displacement along the X, Y, and Z axes, as well as surface temperature.

Virtual models (VE) usually include geometric, physical, behavioral, and rule-based models. However, in practice, a digital twin model does not necessarily encompass all dimensions and domains; it can be adjusted based on specific needs and objects, meaning models can be built for selected domains and dimensions [22]. Therefore, this paper combines geometric features, physical characteristics, operating parameters, among other information, of physical entities to construct and simulate a digital twin virtual model. The virtual model of the REPMM includes the geometric model and the analysis model. The geometric model of the permanent magnet synchronous motor is constructed based on the solid geometry parameters of the REPMM. Based on this model, dynamic simulation analysis of the motor bearing is performed. As the core component of the digital twin system, the virtual model can generate simulation datasets representing various operating states of the bearings through its built-in dynamic analysis model. This provides solid and reliable data support for subsequent fault-diagnosis research.

Services (Ss) refer to the encapsulation of various functions provided within the digital twin framework. In this study, these include data acquisition, data processing, fault signal simulation, and diagnosis of bearing-fault types. Sensors installed on the motor enable real-time collection of operating data. The collected data are effectively processed to provide simple fault warnings or displayed on the motor monitoring platform for more intuitive visualization. Additionally, simulation-generated data are used to train fault-diagnosis models. The overall service flow is shown in Figure 3.

Digital twin data (DD), the core driving the digital twin framework, mainly include PE data, VE data, and Ss data. PE data refer to information from physical entities, including the motor’s physical characteristics such as dimensions, winding structure, and other geometric or material properties. Additionally, operational data such as voltage, rotational speed, and temperature signals provide the foundation for building accurate motor models. VE data consist of simulation data generated by the digital twin virtual model, providing strong support for motor-bearing fault-diagnosis research. Ss data comprise various service data generated during the digital twin system’s operation, including fault-diagnosis and feature-extraction data.

As a core element of the digital twin system, the connection and interaction (CN) component serves as a critical bridge. The physical entity, virtual model, twin data, and services do not exist in isolation; rather, they are tightly and efficiently interconnected through CN. Leveraging these interactions, services provide users with various functional applications, collectively driving the efficient operation of the digital twin system.

A high-precision digital twin model of the rare-earth permanent magnet motor is constructed based on these five dimensions. Through multi-dimensional mapping between the physical entity and the virtual space, the model not only enables real-time monitoring and dynamic tracking of the motor’s operating state, but also establishes a fault simulation model with multi-scenario adaptability. It accurately simulates various fault conditions of motor bearings under different operating conditions, providing comprehensive simulation data for fault diagnosis. This significantly expands the data dimensions and coverage for fault detection. It is important to note that for the scope of the present study, the digital twin framework is utilized primarily in an offline capacity for model-based data generation to develop and validate the core diagnostic algorithm. The implementation of a fully integrated, live data exchange platform between the physical entity and virtual model as envisioned by the connection (CN) component remains a key objective for future work.

3. Methodology

3.1. Problem Description

At present, data-driven fault-diagnosis techniques face two major challenges. First, training complex models relies heavily on large datasets, which not only demand high-performance computing hardware but also require costly sensors for high-frequency data acquisition. This greatly increases both resource and time costs, limiting the practical applicability of such methods in real industrial settings. Second, the historical operational data or laboratory samples used in traditional diagnostic approaches have clear limitations. Due to experimental constraints, it is difficult to obtain sufficient real-world fault samples for rare-earth permanent magnet motors and their bearings. In particular, reproducing micron-level localized defects (e.g., pitting or spalls of several hundred microns) is challenging in physical experiments. Therefore, digital twin technology is employed to simulate the operational behavior of motor bearings, providing virtual datasets for further research on fault feature extraction and intelligent diagnosis.

3.2. Digital Twin Virtual Model

In digital twin technology, the virtual model plays a crucial role [23]. For rare-earth permanent magnet motors, the digital twin model comprises two core components: the geometric model and the analytical model. First, a geometric model of the rare-earth permanent magnet motor is constructed based on its physical entity. Subsequently, in-depth data simulation analysis is conducted on the motor bearings, with modeling based on the fault mechanisms of rolling bearings. Using the ODE45 solver in MATLAB R2023b, a dynamic simulation model of the rare-earth permanent magnet motor bearing is developed for numerical simulation of the bearing system. By introducing localized defects in the inner race, outer race, and rolling elements, both normal and faulty operating states can be simulated, and the corresponding vibration response data collected. The simulation considers practical operating parameters such as rotational speed and radial load.

3.2.1. Geometric Model Construction

The geometric model is used to describe the appearance, shape, size, and assembly relationships of physical equipment and components [24]. The rare-earth permanent magnet motor consists of components such as the stator, rotor, permanent magnets, bearings, and casing. Using the 3D-modeling software SolidWorks 2023, the geometric model of the motor was constructed, covering parameters such as shape, size, position, and assembly relationships among components. The geometric models of the motor and bearings are shown in Figure 4.

3.2.2. Analytical Model Construction

Rolling bearings in rare-earth permanent magnet motors may experience various faults during operation, most of which occur on the inner and outer races as well as the rolling elements. The analysis of these faults focuses on identifying the asymmetries they introduce into the motor’s vibration signals. While a healthy bearing operates with a baseline pattern symmetry, issues such as wear and indentations break this symmetry by causing abnormal vibration signals. Analyzing and building a dynamic model of the rolling bearings, along with simulating fault data, facilitates fault-diagnosis research by enabling precise study of these asymmetries.

As shown in Figure 5, the key parameters of the rolling bearing include the number of balls $N_b$, the diameter of the rolling elements $D_b$, the pitch diameter $D_m$, the inner race diameter $d$, and the outer race diameter $D$. The standard installation method for the rolling bearing involves a static fit between the outer race and the bearing housing to keep it fixed, while the inner race is securely connected to the shaft and rotates synchronously with it. Meanwhile, the rolling elements perform pure rolling motion without slip between the inner and outer raceways.

Mathematical modeling of fault characteristic frequencies requires integrating bearing geometric topology parameters (including race diameter, number of rolling elements, contact angle, etc.) with operating speed parameters. The core computational focus includes the dynamic relationships between the three key components of the bearing: the inner race, outer race, and rolling elements.

The expression for the shaft’s rotational frequency is the following:

$$f_n={\displaystyle \frac{n}{60}}$$

(2)

n is the rotational speed. (rpm)

The following are the mathematical expressions for the characteristic fault frequencies:

(1)

The calculation formula for the inner race fault characteristic frequency (BPFI):

$$f_{BPFI}={\displaystyle \frac{N_b}{2}}{f}_n\left(1+{\displaystyle \frac{D_b}{D_m}}cos\alpha \right)$$

(3)

In the formula, $N_b$ is the number of rolling elements, $f_n$ is the shaft’s rotational frequency, $D_b$ is the diameter of the rolling elements, $D_m$ is the pitch diameter of the bearing, and $\alpha$ is the contact angle.

(2)

The calculation formula for the outer race fault characteristic frequency (BPFO):

$$f_{BPFO}={\displaystyle \frac{N_b}{2}}{f}_n\left(1-{\displaystyle \frac{D_b}{D_m}}cos\alpha \right)$$

(4)

The parameter definitions are the same as those in the inner race formula, with the difference in symbols reflecting the coefficient variations caused by the contact angle direction.

(3)

The calculation formula for the rolling element fault characteristic frequency (BSF):

$$f_{BSF}={\displaystyle \frac{D_m}{2D_b}}{f}_n\left[1-{\left({\displaystyle \frac{D_b}{D_m}}cos\alpha \right)}^2\right]$$

(5)

When a rolling bearing experiences a fault, distinct characteristic fault frequencies and their harmonics are generated. These faults are often reflected in the motor’s vibration signals. Thus, vibration signals serve as an effective means to monitor the motor’s operating condition and diagnose faults.

The expression for the angular velocity of the cage is as follows:

$${\omega }_c={\displaystyle \frac{1}{2}}\left(1-{\displaystyle \frac{D_b}{D_m}}\right)\omega$$

(6)

In the formula, $\omega$ is the angular velocity of the shaft (in radians), $D_b$ is the diameter of the rolling elements, and $D_m$ is the pitch diameter of the bearing.

When a rolling bearing is subjected to a radial load, the rolling elements undergo periodic deformation. Let the angular position of the $j$-th rolling element at time t be ${\theta }_j$. The expression for the angular displacement of the rolling element is as follows:

$${\theta }_j={\displaystyle \frac{2\pi \left(j-1\right)}{N_b}}+{\omega }_{c}t+{\theta }_0$$

(7)

In the formula, $N_b$ is the number of rolling elements, and ${\theta }_0$ is the initial angular position of the first rolling element.

The bearing contact deformation force ${\delta }_j$ represents the total normal contact deformation between the $j$-th rolling element and the inner and outer raceways at any angular position ${\theta }_j$, and its calculation formula is the following:

$${\delta }_j=\left(x_i-x_o\right)cos{\theta }_j+\left(y_i-y_o\right)sin{\theta }_j-c_d-h$$

(8)

In the formula, $c_d$ is the radial clearance, and $h$ represents the time-varying displacement excitation caused by faults. This excitation is absent under normal bearing conditions. However, it arises when faults occur in the outer race, inner race, or rolling elements, causing time-varying displacement excitation.

Hertz contact theory [25] considers the rolling elements and bearing raceways as smooth elastic bodies, with only elastic contact deformation. It is mainly used to analyze the distribution of local stress and strain resulting from the compression contact between two objects. According to Hertz contact theory, the total contact stiffness between the rolling elements and the inner and outer raceways can be calculated as follows:

$$k_b={\left[{\displaystyle \frac{1}{(1/k_{bi}{)}^{2/3}+(1/k_{bo}{)}^{2/3}}}\right]}^{3/2}$$

(9)

In the formula, $k_{bi}$ and $k_{bo}$ represent the contact stiffness between the rolling element and the inner and outer races, respectively.

The relationship between the nonlinear bearing contact deformation force, the contact stiffness $k_b$, and the contact deformation amount at the angular position ${\theta }_j$ of the $j$-th rolling element can be derived from Hertz theory:

$$F_x=\sum _{j=1}^{N_b}{k}_b{\delta }_j^{1.5}H\left({\delta }_j\right)cos{\theta }_j$$

(10)

$$F_y=\sum _{j=1}^{N_b}{k}_b{\delta }_j^{1.5}H\left({\delta }_j\right)sin{\theta }_j$$

(11)

Since contact deformation only occurs when the rolling elements are in contact with the bearing raceways, $H\left({\delta }_j\right)$ is defined as a switching function,

$$H\left({\delta }_j\right)=\left\{\begin{array}{c}1 {\delta }_j>0\\ 0{ \delta }_j\le 0\end{array}\right.$$

(12)

Assuming ideal operating conditions, with the outer race fully constrained and fixed, and the inner race rotating steadily, the contact stress between the rolling elements strictly follows Hertzian contact theory, with the contact area exhibiting typical elliptical contact characteristics. Under rigid body kinematic constraints, the rolling elements maintain an equidistant distribution around the circumference. Their relative azimuth remains constant throughout the motion, and their relative positions remain unchanged. The rotor under study moves in a radial plane centered at the origin, without considering the effects of lubrication.

According to Newton’s second law, the dynamic equation of the rolling bearing can be obtained as the following:

$$\begin{gathered} m_i{\ddot{x}}_i+c_{ix}{\dot{x}}_i+k_{ix}{x}_i=-F_{ix}+m_{i}e{\omega }^{2}cos\left(\omega t\right)+W_x \\ {m}_i{\ddot{y}}_i+c_{iy}{\dot{y}}_i+k_{iy}{y}_i=-F_{iy}+m_{i}e{\omega }^{2}sin\left(\omega t\right)-m_{i}g+W_y \\ {m}_o{\ddot{x}}_o+c_{ox}{\dot{x}}_o+k_{ox}{x}_o=F_{ox} \\ {m}_o{\ddot{y}}_o+c_{oy}{\dot{y}}_o+k_{oy}{y}_o=F_{oy}-m_{o}g \end{gathered}$$

(13)

In the formula, $m_i$, $c_i$, $k_i$ represent the mass, damping, and stiffness of the inner race of the rolling bearing, respectively; $m_o$, $c_o$ and $k_o$ represent the mass, damping, and stiffness of the outer race of the rolling bearing, respectively; $W_x$ and $W_y$ are the applied loads on the bearing system in the $x$ and $y$ directions, respectively. $x_i$ and $x_o$ represent the displacements of the inner and outer races of the bearing in the $x$ direction, respectively; $y_i$ and $y_o$ represent the displacements of the inner and outer races of the bearing in the $y$ direction, respectively. $F_{ix}$, $F_{iy}$, $F_{ox}$, and $F_{oy}$ are the forces on the inner and outer races in the $x$ and $y$ directions, respectively.

3.3. Fault Diagnosis of Motor Bearings

In consideration of the need to develop a practical and cost-effective fault-diagnosis algorithm, this study adopts the subtraction-average-based optimizer (SABO) to optimize the parameters of variational mode decomposition (VMD) for fault-feature extraction. Subsequently, a convolutional neural network–bidirectional long short-term memory (CNN-BiLSTM) network is used to perform fault classification.

The subtraction-average-based optimizer (SABO) is an intelligent optimization algorithm based on a mathematical concept of special subtraction arithmetic mean. Its core mechanism relies on differential averaging, which enables intelligent optimization by combining arithmetic subtraction and mean calculations in a nonlinear manner. The algorithm is inspired by mathematical ideas including averages, positional differences among search agents, and the sign of the difference between two objective function values [26].

The core mechanism of the SABO involves updating each search agent’s position based on the average influence of the entire population. The update process is governed by two main rules. First, a difference vector $D{X}_i$ for each agent $i$ is calculated. For each dimension $d$, its component is given by the following:

$$D{X}_{i,d}=\sum _{j=1}^{N}sign\left(f\right(X_i)-f(X_j\left)\right)·(X_{j,d}-I_j·X_{i,d})$$

(14)

where $X_i$ and $X_j$ are the positions of the $i$-th and $j$-th agents, $f\left(\right)$ is the fitness function, and $I_j$ is a random integer from the set {1, 2, 3}. Subsequently, the new position for agent $i$ is calculated using element-wise multiplication:

$$X_{i,\mathrm{new}}=X_i+{\displaystyle \frac{r\circ D{X}_i}{N}}$$

(15)

where $r$ is a random vector with its elements drawn from a uniform distribution in [0, 1], and ∘ denotes the Hadamard product (element-wise multiplication).

The choice of the SABO algorithm for VMD-parameter optimization was pivotal. Unlike traditional optimizers such as genetic algorithms (GA), which rely on crossover and mutation operators, or particle swarm optimization (PSO), which is heavily guided by individual and global best positions, the SABO employs a unique update mechanism based on the subtraction average of the entire population. This approach can provide a robust balance between global exploration and local exploitation. By incorporating influence from all agents, SABO may reduce the risk of premature convergence to a local optimum, a common challenge in PSO. This characteristic was deemed particularly suitable for navigating the complex search space of VMD parameters to find a globally effective solution. The pseudocode for the SABO algorithm used in this study is shown in Algorithm 1.

Variational mode decomposition (VMD) estimates the individual signal components by solving a variational optimization problem in the frequency domain [27]. Before decomposing the signal, appropriate parameters must be set, including the number of modes k and the penalty parameter α. An excessively large k leads to over-decomposition, while a too small k causes under-decomposition. Likewise, a large α may cause loss of frequency band information, whereas a small α may introduce redundant information. Therefore, it is essential to determine the optimal parameter combination (k, α). Due to its excellent global search capability and super-linear convergence, the SABO algorithm is employed to optimize VMD parameters. This establishes a framework targeting the two key hyperparameters: the number of modes k and the penalty parameter α.

In the selection of the fitness function, permutation entropy (PE) is used to measure the complexity and randomness of time series data. A lower entropy value indicates a more regular signal, which is beneficial for fault feature extraction. Mutual information entropy (MIE) evaluates the independence between different intrinsic mode functions (IMFs). Lower entropy values indicate less redundancy among modes, effectively avoiding mode mixing.

The composite fitness function combines permutation entropy (PE) and mutual information entropy (MIE), and is expressed as follows:

$$\mathit{Fitness}=w_1·PE+w_2·MIE$$

(16)

$w_1$ and $w_2$ are the weights, which are set to a ratio of 1:1 in this study.

This equal weighting was chosen based on the principle of assigning equal importance to the two key objectives of the optimization: minimizing permutation entropy (PE) to isolate the most feature-rich signal component, and minimizing mutual information entropy (MIE) to ensure the independence of the decomposed modes. The 1:1 ratio provides a balanced and unbiased approach to this multi-objective problem, avoiding any a priori bias toward one metric over the other.

Using a composite metric combining PE and MIE, the decomposition results can be optimized regarding mode independence and feature significance. Compared to a single metric, the composite metric enables more stable separation of fault impact components.

Algorithm 1: Subtraction-average-based optimizer (SABO)

Input:      N: Population size      T: Maximum number of iterations      lo, hi: Lower and upper bounds of variables      m: Dimension of the problem      fitness(): The objective function      data: Input signal for the fitness function Output:      Best_pos: Best solution found (optimal k and α)      Best_score: Fitness value of the best solution 1: Initialize population X of N agents randomly within bounds [lo, hi]. 2: Calculate initial fitness score_i for each agent X_i. 3: Find initial global best X_best and f_best. 4: For t = 1 to T do: 5: For each agent X_i do: 6:  Initialize difference vector DX = [0, …, 0]. 7:  For each agent X_j do: 8:            I = round(1 + rand() + rand()) 9:            For d = 1 to m do (for each dimension): 10:              DX[d] = DX[d] + (X_j[d] − I * X_i[d]) * sign(score_i − score_j) 11:              End For 12:       End For 13:       r = random vector of size m with elements in [0, 1] 14:       X_new = X_i + (r .* DX) / N // . denotes element-wise multiplication 15:       Apply bounds to X_new. 16:       score_new = fitness(X_new, data) 17:       If score_new < score_i then: 18:            X_i = X_new 19:            score_i = score_new 20:       End If 21:  End For 22:  Update global best X_best and f_best. 23: End For 24: Return X_best as Best_pos and f_best as Best_score.

A convolutional neural network (CNN) is a feedforward neural network characterized by local connectivity and weight sharing [28]. CNN employs a hierarchical architecture consisting of an input layer, convolutional layers, pooling layers, fully connected layers, and an output layer. Bidirectional long short-term memory (BiLSTM) [29] is an improved type of recurrent neural network specifically designed for handling sequential data. It captures bidirectional dependencies within a sequence by combining the outputs from both forward and backward LSTM networks. The selection of the hybrid CNN-BiLSTM architecture was a deliberate choice designed to leverage the complementary strengths of each component for the specific task of bearing-fault diagnosis. The CNN layers are employed as a powerful feature extractor, tasked with identifying key local patterns and interactions among the input features derived from the VMD analysis. Subsequently, the BiLSTM layers are used to model the temporal dependencies within these feature sequences, capturing the long-range patterns and rhythm characteristics of a fault event. This hybrid approach is hypothesized to be more effective than a standalone 1D-CNN, which lacks explicit sequence modeling, or a simple RNN, which can struggle with long-term dependencies due to the vanishing gradient problem. Furthermore, compared to more general architectures like the transformer, the CNN-BiLSTM offers a compelling balance of performance, computational efficiency, and a strong inductive bias for the local patterns found in vibration signals, aligning with this study’s objective of developing a practical diagnostic solution.

In fault-diagnosis research, the SABO algorithm is used to optimize VMD, as illustrated in Figure 6. A composite metric combining permutation entropy (PE) and mutual information entropy (MIE) is used as the fitness function to optimize and determine the optimal VMD parameters k and α. Then, the original signal is decomposed using the optimized parameters (k, α):

$${IMF}_1,{IMF}_2,\dots ,{IMF}_K=VMD\left(Signal,k,\alpha \right)$$

(17)

Calculate the envelope entropy E_i for each IMF. Then, select the optimal IMF component for each sample by minimizing the envelope entropy.

$$\mathit{Best} \mathit{IMF}=argmin{E}_i$$

(18)

First, we extract the optimal component and calculate its corresponding theoretical characteristic parameters to construct the final feature vector for each sample. The feature vector set is used as the input to the diagnostic model. The dataset is divided into training and testing sets, with the CNN-BiLSTM model trained on the training set, and the testing set used to evaluate the model’s performance and verify its classification accuracy. The fault-diagnosis model is a hybrid convolutional neural network–bidirectional long short-term memory (CNN-BiLSTM) network. The architecture begins with a 1D convolutional layer containing eight filters with a kernel size of six, followed by a ReLU activation and a batch normalization layer. A max-pooling layer (pool size two) is then applied for downsampling. The resulting features are processed by a BiLSTM layer with 30 hidden units to capture temporal patterns, followed by a dropout layer with a 20% rate for regularization. The final classification is performed by a fully connected layer with 4 neurons and a softmax activation function. The network was trained for 100 epochs using the Adam optimizer with an initial learning rate of 0.01, an L2 regularization factor of 0.001, and a gradient threshold of 1.0. The dataset was partitioned into 70% for training and 30% for testing. The overall fault-diagnosis process is shown in Figure 7.

4. Experiments and Analysis

To comprehensively validate the proposed diagnostic framework, a parallel testing strategy was adopted. First, an experiment was conducted using the high-fidelity data generated by our digital twin model to evaluate the ideal performance of the SABO-VMD-CNN-BiLSTM method in a controlled, noise-free environment. Second, the exact same methodology was applied to a public, real-world benchmark—the HUST bearing dataset—to verify the practical applicability and generalization ability of our approach in the presence of realistic noise and operational variability. This section details the setup and results of these two independent but complementary experiments.

4.1. Experiment 1: Simulation Data

The acceleration vibration signals under four conditions—normal, inner race fault, outer race fault, and rolling element fault—are simulated using the digital twin model of the rare-earth permanent magnet motor bearing. The parameters of the motor bearing digital twin model are shown in

Table 1. Bearing parameters.

Parameter Name	Title 2 Value
Outer race diameter D/mm	52
Inner race diameter d/mm	25
Rolling element diameter Db/mm	7.94
Bearing pitch diameter Dm/mm	39
Number of rolling elements Nb	9
Radial clearance Cr/mm	5 × 10−3
Outer race mass mo/kg	12.64
Inner race mass mi/kg	5.5
Outer race support damping co/(Ns/m)	2310.68
Inner race support damping ci/(Ns/m)	3376.84
Outer race support stiffness ko/(N/m)	1.51 × 107
Inner race support stiffness ki/(N/m)	5.24 × 104

:

The defect width was set to approximately 0.533 mm, the simulation duration was 30 s, rotational speed was 1797 rpm, and the sampling frequency was 10 kHz. The dynamic equations were solved in MATLAB R2023b using the ode45 adaptive solver, with the relative tolerance set to 1 × 10⁻³ and absolute tolerance to 1 × 10⁻⁶. This corresponds to a fixed time step of 1 × 10⁻⁴ s. No stochastic noise was added to the simulation; this was a deliberate choice to purely capture the deterministic fault signatures under ideal conditions before validating the algorithm on real-world noisy data. By running the digital twin model, vibration accelerations of the bearing inner and outer races in the x and y directions were obtained. The vibration acceleration signal of the inner race in the x direction is shown in the following figures.

A normal bearing generates small periodic vibrations during operation, as shown in Figure 8a. This vibration signal from the healthy state exhibits a high degree of periodicity and pattern symmetry, although it is not geometrically perfect. The primary source of these vibrations is the stiffness modulation effect when the rolling elements contact the raceways. As the bearing rotates, the rolling elements periodically enter and exit the load zone, causing changes in contact stress distribution and resulting in slight fluctuations in overall bearing stiffness. In contrast, a fault introduces a distinct asymmetry into the signal. Figure 8b shows the time–domain vibration characteristics of an inner race fault. where this symmetry-breaking event manifests as periodic high-amplitude impacts.

Through the analysis of the bearing motion, it is understood that the inner race is rigidly connected to the shaft and rotates at the same speed. As the inner race rotates, the fault location changes, and vibration is triggered only when the defect is in the load zone and contacts the rolling elements. As the inner race defect moves, the magnitude and direction of the additional excitation caused by contact with the rolling elements also vary over time. The vibration amplitude reaches its maximum when the defect is at the center of the load zone and gradually decreases as the inner race continues to rotate.

In the case of a rolling element fault, the vibration response is shown in Figure 8c. Vibration occurs only when the faulty rolling element enters the load zone and contacts the raceway. The vibration is more intense when the rolling element fault contacts the outer race, corresponding to the higher amplitude curve in the time–domain plot. When it contacts the inner race, the vibration is weaker, corresponding to the lower amplitude curve.

Outer race faults are typically located in the lower load zone of the bearing. Since the outer race is fixed, the location of the defect remains fixed. When the rolling element passes over the defect, it triggers periodic pulses, Figure 8d shows the clear asymmetry caused by an outer race fault. Under ideal conditions (constant speed, uniform load), the pulse amplitude and spacing are approximately equal, and the interval time is determined by the bearing’s geometric parameters and the speed.

Based on the selected key bearing structural parameters (such as pitch diameter, number of rolling elements, etc.), the theoretical fault characteristic frequencies for the inner race, rolling elements, and outer race are calculated using Formulas (3)–(5) to be 162.19 Hz, 70.5 Hz, and 107.4 Hz, respectively. To further verify the accuracy of the model, the pitting fault data for the inner race, rolling elements, and outer race generated by the simulation are subjected to envelope spectrum analysis.

The results are shown in Figure 9. For the inner race fault frequency, the theoretical calculated value is 162.19 Hz, while the extracted value from the simulation is 162.17 Hz; for the rolling element fault frequency, the theoretical calculated value is 70.5 Hz, and the extracted value is 70.5 Hz; for the outer race fault frequency, the theoretical calculated value is 107.34 Hz, and the extracted value is 107.32 Hz. The above comparison indicates that the simulation model accurately reproduces the bearing-fault characteristics with minimal error. Moreover, the inner race fault frequency (BPFI), rolling element fault frequency (BSF), and outer race fault frequency (BPFO), along with their harmonic components, show significant energy concentration in the envelope spectrum, which is in line with the theoretical fault characteristic frequencies. This demonstrates that the virtual model can accurately simulate the vibration signals of bearing faults.

To demonstrate the fidelity of the proposed DT model, an outer race fault, one of the most common bearing failure modes, was selected as a representative case study for validation against the CWRU experimental dataset. The comparison was conducted by comparing the simulation results with actual motor vibration data from a physical test. We utilized the public bearing dataset from the Case Western Reserve University (CWRU) Bearing Data Center, specifically the drive-end bearing-fault data for an outer race fault, recorded at a sampling frequency of 12 kHz. To ensure a direct comparison, we regenerated our simulation data for an identical outer race fault scenario with a matching sampling frequency of 12 kHz.

The results of the comparative envelope spectrum analysis are presented in Figure 10. As shown in Figure 10a, the envelope spectrum of the simulated data exhibits clean and prominent peaks at the theoretical outer race fault frequency (BPFO) and its harmonics (2*BPFO, 3*BPFO). Figure 10b displays the envelope spectrum for the CWRU experimental data. Despite the presence of background noise, which is typical for physical systems, distinct spectral peaks are clearly visible at the same BPFO, 2*BPFO, and 3*BPFO frequencies.

The strong alignment between the characteristic fault frequencies in our simulation and those in the CWRU experimental data confirms the high fidelity of our DT virtual model. This result demonstrates that our model can accurately reproduce the key diagnostic signatures of a real-world bearing fault, thus validating its effectiveness for further analysis.

After obtaining the simulated data, it is processed to create a dataset comprising four operating states. Each condition contains 150 samples, with each sample consisting of 2048 data points. Feature extraction is then performed to obtain feature vectors.

First, the permutation entropy (PE) and mutual information entropy (MIE) are used as a composite fitness function to optimize and determine the optimal VMD parameters: the number of decomposition layers k and the penalty factor α. The search range for the number of decomposition layers is set to [3,10], and the search range for the penalty factor α is [100, 3000]. The maximum number of iterations for the subtraction-average-based optimizer (SABO) is set to 20, with a population size of 15. After optimizing k and α, the optimal IMF components are selected based on the results. The obtained parameter values are shown in

Table 2. Parameter optimization results.

Labels	k	α	Optimal IMF Component
1	8	2313	1
2	3	454	1
3	3	275	1
4	10	2291	2

.

Using the optimized parameters, the original signal was decomposed to extract the optimal IMF component. Multi-domain features were then extracted from this component. Nine time-domain statistical features were computed, including mean, variance, peak-to-peak value, kurtosis, root mean square, crest factor, shape factor, impulse factor, and margin factor. The selection of these nine time–domain statistical features was based on their established effectiveness and comprehensive diagnostic coverage in bearing-fault analysis. They were chosen to quantify the fault-induced signal changes from three critical perspectives: energy metrics (mean, variance, root mean square) to capture changes in signal amplitude; impulsiveness metrics (kurtosis, crest factor, impulse factor, margin factor), which are highly sensitive to the sharp, transient peaks characteristic of localized faults; and shape metrics (peak-to-peak value, shape factor) to describe the overall waveform morphology. As the preceding SABO-VMD stage isolates the most information-rich signal component, this suite of computationally efficient time–domain features provides a sufficient and robust basis for the subsequent classification task. These features were combined into multidimensional feature vectors, serving as discriminative inputs for fault classification.

The dataset was split into 70% training and 30% testing subsets. A CNN-BiLSTM model was employed as the fault-diagnosis model to identify fault types. The diagnostic results of the proposed SABO-VMD-CNN-BiLSTM method are shown in Figure 11. To rigorously evaluate the model’s stability, the experiment was conducted five times with different random seeds. The method demonstrated highly consistent performance, yielding accuracies of 98.33%, 98.88%, 98.88%, 97.78%, and 98.33%. This resulted in an average accuracy of 98.44% with a standard deviation of 0.41%. The low standard deviation confirms that the high accuracy is stable and repeatable. Only a small number of samples were misclassified, demonstrating the algorithm’s strong capability to accurately distinguish normal operation, inner ring faults, rolling element faults, and outer ring faults. For a more granular analysis of the model’s performance, the precision, recall, and F1-score for each class were calculated from the confusion matrix. The results, presented in

Table 3. Per-class performance metrics for the proposed model.

Class (Label)	Fault Type	Precision	Recall	F1-Score
1	Normal	1.000	0.933	0.965
2	Inner race fault	1.000	1.000	1.000
3	Rolling element fault	1.000	1.000	1.000
4	Outer race fault	0.938	1.000	0.968

, provide a detailed per-class assessment of the model’s diagnostic capabilities.

To systematically evaluate the performance differences among various fault-diagnosis methods, this study conducted three comparative experiments:

(1)

CNN-BiLSTM method: This approach inputs raw vibration signals directly into a CNN-BiLSTM model without any preprocessing or parameter optimization.

(2)

SABO-VMD (minEn)-CNN-BiLSTM method: Here, the optimization algorithm’s fitness function is replaced with minimum envelope entropy (minEn) instead of the combined PE/MIE metric. The VMD parameters are optimized accordingly, and the resulting features are classified using the CNN-BiLSTM model.

(3)

SABO-VMD-SVM method: This method applies the subtraction-average-based optimizer (SABO) to optimize VMD parameters for feature extraction, followed by classification with a support vector machine (SVM) instead of CNN-BiLSTM.

The diagnostic results presented in Figure 12, based on the above comparative experiments, comprehensively validate the following aspects:

(1)

The optimization performance differences under different fitness functions (e.g., combined PE/MIE vs. minimum envelope entropy);

(2)

The impact of parameter optimization (via SABO) on diagnostic accuracy;

(3)

The interaction between feature extraction techniques and classifiers, highlighting how their appropriate pairing can improve overall diagnostic performance.

As shown in

Table 4. Comparison results.

Fault-Diagnosis Method	Fitness Function	Accuracy
SABO-VMD-CNN-BiLSTM	PE-MIE	98.33%
CNN-BiLSTM	——	86.67%
SABO-VMD (minEn)-CNN-BiLSTM	minEn	93.89%
SABO-VMD-SVM	PE-MIE	96.67%

, the results of the comparative experiments clearly demonstrate that the proposed SABO-VMD-CNN-BiLSTM method exhibits considerable advantages in both diagnostic accuracy and stability, outperforming other tested approaches.

A key objective of this study was to develop a practical and low-cost diagnostic method. While the CNN-BiLSTM is a deep learning model, it was designed to be relatively lightweight. The architecture consists of only a few core layers, resulting in a model with a manageable number of parameters compared to larger deep learning architectures. In terms of performance, the inference time—the time required to diagnose a single new sample—was measured to be in the order of milliseconds on a standard CPU, which is sufficiently fast for near-real-time health monitoring in industrial settings. Therefore, the proposed model represents a pragmatic trade-off: it leverages the powerful feature learning of deep learning to achieve high accuracy while maintaining a computational footprint that is feasible for deployment on local industrial computers, supporting the goal of a practical diagnostic solution.

4.2. Experiment 2: HUST Bearing Dataset

To further validate the diagnostic capability of the proposed method and eliminate potential biases from using a single dataset, the fault-diagnosis model was evaluated on real-world data from the HUST bearing dataset, provided by Professor Weiming Shen’s research team at the School of Mechanical Science and Engineering, Huazhong University of Science and Technology [30]. The tested bearing is an ER-16K rolling bearing. This dataset includes the vibration signals of normal bearings, lightly faulty bearings, and severely faulty bearings under different rotational speeds, covering fault types such as inner race faults, outer race faults, rolling element faults, and compound faults. It includes data from bearings under 11 different operating conditions and 9 distinct health states. The sampling frequency is 25.6 kHz, with each sample containing 262,144 data points. Furthermore, a triaxial accelerometer was employed to capture comprehensive vibration data. The experimental setup of the bearing dataset is illustrated in Figure 13, where, from left to right, the components are speed control, motor, shaft, accelerometer, bearing, and data acquisition board. Crucially, as real-world experimental data, the signals inherently contain ambient and operational noise from the motor and environment. While a specific signal-to-noise ratio is not provided, the model’s ability to achieve high accuracy on this dataset demonstrates its robustness against such realistic noise conditions. For our analysis, data from three distinct rotational speeds—60 Hz, 65 Hz, and 70 Hz—were selected to evaluate the model’s performance under varying conditions.

Vibration data collected at operating frequencies of 60 Hz, 65 Hz, and 70 Hz were utilized to investigate fault diagnosis in four different states, including normal state, and moderate faults located in the inner ring, rolling elements, and outer ring, with a fault size of 0.15 mm. Taking the bearing data under the operating condition of 65 Hz as an example, the fault-category labels defined by fault type and operating condition are summarized in

Table 5. Bearing data fault-category label at 65 Hz.

Fault Type	Sample Count	Label
Normal	150	1
Inner race fault	150	2
Rolling element fault	150	3
Outer race fault	150	4

.

Data processing, feature extraction, and fault diagnosis followed the same procedures applied to the simulation dataset. Experiments utilized bearing data collected under different operating conditions. The fault-diagnosis results under the three different working conditions are shown in Figure 14.

As shown in Figure 14, the proposed method demonstrates robust classification performance on three datasets from the HUST bearing database, achieving accuracy rates above 95%. Specifically, classification accuracies were 95% at 60 Hz, 97.78% at 65 Hz, and 99.44% at 70 Hz, indicating strong generalization capability of the proposed method. As shown in the confusion matrix, under certain operating conditions (as seen in Figure 14a), there are a small number of misclassifications between inner race and rolling element faults. This may be due to the more complex modulation phenomena of the fault feature signals at lower speeds, leading to a decrease in signal distinguishability. Regarding the impact of different operating conditions on diagnostic accuracy, in the tests on the HUST dataset, the accuracy at 70 Hz (99.44%) was higher than the accuracy at 60 Hz (95%). This is likely because higher rotational speeds generate higher-energy fault impacts, resulting in a better signal-to-noise ratio, which makes the “signal asymmetry” caused by the fault more pronounced and easier for the model to capture.

The fault-diagnosis method based on digital twins addresses the problem of data scarcity by utilizing simulated data generated from virtual models to train fault-diagnosis algorithms, achieving high accuracy in fault diagnosis for simulation signals. Subsequently, the models were validated on real-world datasets. Although discrepancies between simulated and actual data—due to environmental noise and other interferences—cause slight variations in accuracy, the overall performance remains robust, demonstrating the practical applicability of the proposed method.

5. Conclusions

This study tackles the critical challenges in fault diagnosis of rare-earth permanent magnet motor bearings by developing an intelligent diagnostic framework centered on detecting fault-induced signal asymmetry through digital twin technology. The framework integrates multidimensional information, encompassing the geometric structure and physical properties of the actual system, thereby enabling real-time monitoring of motor operating parameters. It generates simulated vibration signals under different bearing-fault conditions through virtual models. The simulated data generated encompasses a variety of device parameters, providing more comprehensive and rich data support for fault diagnosis. Building on this foundation, fault-diagnosis methods were evaluated using both simulated data generated by the digital twin model and the HUST bearing dataset provided by Professor Weiming Shen’s team at Huazhong University of Science and Technology. Experimental results demonstrate that the optimized variational mode decomposition (VMD) achieves higher efficiency and accuracy in fault feature extraction. The digital-twin-based SABO-VMD-CNN-BiLSTM diagnostic model significantly improves fault identification accuracy, achieving 98.44 ± 0.41% on the simulated data and over 95% on the public HUST bearing dataset. These results confirm the model’s practical effectiveness and strong generalization ability.

While this study successfully demonstrates the framework’s high accuracy, it is important to acknowledge the limitations that stem from the idealizations made in the digital twin model. The current model assumes constant operating speed and load, and neglects crucial factors such as lubrication effects. These simplifications constrain the model’s applicability and can have potential impacts on diagnosis accuracy in real-world scenarios. For example, variable operating conditions can cause fault characteristic frequencies to shift, which could lead to misclassification by a model trained only on steady-state data. Similarly, the presence and condition of lubrication can dampen or alter vibration signatures, potentially reducing the model’s ability to correctly identify fault patterns learned from the idealized simulation. Therefore, these results should be regarded as a proof-of-concept under controlled conditions, providing a solid foundation for future improvements addressing these complexities.

Future research will focus on optimizing the construction and dynamic updating of the digital twin model. A comprehensive sensitivity analysis will be conducted to assess the framework’s robustness, including variations in key physical parameters and diagnostic performance across diverse fault types, sizes, severities, and operating conditions (e.g., different motor loads and speeds). Additionally, the method will be extended to complex multi-bearing assemblies. Exploration of advanced deep learning architectures, such as transformers, will be undertaken to further enhance diagnostic performance. The ultimate goal is to develop a more comprehensive and intelligent health management system.