A method for constructing a digital twin model reflecting defect propagation of a rolling bearing

Abstract

The application discloses a kind of digital twin model construction methods reflecting the extension of rolling bearing defect, it is related to bearing health management and operation and maintenance field;The digital twin model presented, the vibration signal of rolling bearing is used to bridge physical space and virtual space, and the dynamic updating of virtual entity is realized using signal approximation method, without measuring the actual defect size of bearing, solve the difficult problem that the time-varying defect size of bearing under operating condition is difficult to obtain.The relationship between the root mean square value of measured vibration signal and the local defect length of virtual space is established using LSTM, the signal of physical space is mapped to virtual space, and the model can directly predict the local defect size of actual bearing according to the collected vibration signal under the same working condition after training with historical data, to help quickly update virtual entity, to realize the operation and maintenance and health monitoring of rolling bearing.

Description

Technical Field

[0001] This invention relates to the field of bearing health management and operation and maintenance, and in particular to a method for constructing a digital twin model that reflects the expansion of defects in rolling bearings. Background Technology

[0002] Rolling bearings are among the most commonly used components in rotating machinery, and their reliability directly affects the safety and operational stability of the equipment. Therefore, the operation and maintenance of rolling bearings are crucial. Many scholars have established local fault dynamic models to simulate the dynamic response of rolling bearings for the operation and maintenance of large machinery containing rolling bearings. However, given that the establishment of rolling bearing dynamic models involves varying degrees of idealization, the models cannot completely reproduce the motion behavior of the internal components of the rolling bearing, resulting in a difference between the simulated dynamic response and the actual dynamic response of the rolling bearing. Therefore, narrowing the gap between the simulated and actual dynamic responses of rolling bearings is a key issue in improving the accuracy of rolling bearing fault detection.

[0003] Digital twin technology can serve as a bridge between simulated dynamic responses and real dynamic responses. Most traditional digital twin models are based on real-time sensor data, and their modeling process often requires the assistance of simulation software such as ICEM CFD, MATLAB, ANSYS, or mathematical mechanics models. These continuously update the parameters of the virtual entity based on real-time sensor data to mimic the motion of the physical entity and generate simulated dynamic responses. Currently, this technology has been applied to bearing management and maintenance. Farhat et al. used digital twin technology to provide a dataset for a neural network classifying faulty bearings, and experiments showed that the dataset generated using digital twin technology can still meet the classification accuracy requirements. To accurately predict the remaining service life of rolling bearings during the degradation process, Liu et al. established a bearing digital twin model based on bearing vibration phenomena and used a Domain Adversarial Neural Network (DANN) to achieve domain adaptation between simulated and real data. Qin et al. proposed a data- and model-driven digital twin model construction method based on a dynamic model and historical bearing measurement data. The simulated vibration signal generated by this model, after being corrected by a CycleGAN, is highly consistent with the measured vibration signal. However, the information provided by sensors is limited, and some parameters cannot be obtained directly from sensors, such as the size of local defects in rolling bearings. This makes it impossible for traditional digital twin models to represent these physical quantities. Summary of the Invention

[0004] The technical problem to be solved by this invention is to provide a method for constructing a digital twin model that reflects the expansion of defects in rolling bearings, addressing the shortcomings of traditional digital twin models. Based on signal denoising methods, mathematical mechanism models, dynamic update methods, and deep learning technology, this method enables real-time prediction of the time-varying local defect size of rolling bearings and generates a simulated vibration signal that is highly consistent with the measured vibration signal.

[0005] To achieve the above objectives, the present invention provides the following technical solution:

[0006] A method for constructing a digital twin model reflecting the propagation of defects in rolling bearings includes the following steps:

[0007] S1: Obtain the vibration signal of the rolling bearing throughout its entire life cycle, and use singular value decomposition (SVD) technology to perform noise reduction preprocessing on the collected vibration signal;

[0008] S2: Construct a dynamic model based on Hertzian contact theory to describe the local defect propagation behavior of rolling bearings;

[0009] S3: Based on the signal approximation method, update the dynamic model parameters with actual measurement data and estimate the time-varying defect size of the rolling bearing;

[0010] S4: Utilize a Long Short-Term Memory (LSTM) network to establish the relationship between the actual measured vibration signal and the time-varying defect, build a digital twin model, and realize real-time mapping from physical entities to virtual entities;

[0011] S5: Monitor the operating status and health of rolling bearings based on the extent of local defect expansion, estimate the current operating stage of rolling bearings in the entire life cycle, and make timely decisions and feedback before bearing failure.

[0012] The noise reduction preprocessing of the acquired vibration signal using singular value decomposition specifically involves: constructing an r×s order Hankel matrix based on phase space reconstruction theory, and setting the vibration signal acquired by the sensor as Y=[y1 y2…y n Then the Hankel matrix is ​​shown in equation (1):

[0013]

[0014] Where H is an r×s matrix, n is the total length of the signal, n=r+s-1, and r≤s, H can be decomposed into H=D+W; D is the matrix of the smooth signal in the reconstruction space, and W is the matrix of the noise interference signal. H, D, and W are all r×s matrices. Singular value decomposition of H yields:

[0015] H = USV T (2)

[0016] Among them, U and VT Let S be an r×r matrix and S be an s×s matrix, respectively, where S is an r×s diagonal matrix with λ as the main diagonal. i (i = 1, 2, ..., m), where m = min(r, s), that is:

[0017] S=diag(λ1,λ2,…,λ m (3)

[0018] Where λ1,λ2,…,λ m These are the singular values ​​of H, and λ1≥λ2≥…≥λ m U and V T Let H be the left and right singular matrices; after the inverse operation of singular value decomposition, we obtain the reconstructed matrix H. n , H n Re-unfold into a new vibration signal Y n The noise-reduced vibration signal is obtained.

[0019] Define the singular value difference as b i =λ i -λ i+1 (i = 1, 2, ..., m-1), after differencing, a singular value difference sequence B = (b1 + b2 + ... + b) is formed. m-1 ).

[0020] The dynamic model describing the local defect propagation behavior of rolling bearings based on Hertzian contact theory is as follows:

[0021] S2.1: Define the local defects of the rolling bearing as cuboid defects with a depth of H, a length of L, and a width equal to the width of the outer raceway of the bearing. Analyze the fault stage based on the contact between the rolling elements and the outer raceway.

[0022] S2.2: A two-degree-of-freedom dynamic model is established by using piecewise functions as displacement excitation.

[0023] The fault analysis stage based on the contact condition between the rolling element and the outer raceway includes:

[0024] Based on geometric relationships, calculate the maximum radial displacement H when the rolling element and both edges of the local defect are in contact simultaneously. d As shown in equation (4):

[0025]

[0026] Where r is the radius of the rolling element;

[0027] If H d If H < H, the rolling element only contacts the edge of the defect, which is the early stage of failure;

[0028] If H dIf the value is greater than H, then the rolling element is in contact with both the edge and bottom of the defect, which indicates the later stage of the fault.

[0029] The method of using a piecewise function as displacement excitation to establish a two-degree-of-freedom dynamic model specifically includes:

[0030] Introducing piecewise displacement excitation functions:

[0031] For the early stage of the fault, the displacement excitation function is as shown in equation (5):

[0032]

[0033] For the later stage of the fault, the displacement excitation function is as shown in equation (6):

[0034]

[0035] Where R is the outer raceway radius, β0 is the initial angular position of the local defect relative to the positive X-axis, β is the local defect range angle, and θ i Let β be the angle of the i-th rolling element relative to the positive X-axis at time t. q H is the arc length when the rolling element contacts the bottom edge of the local defect. max The maximum actual radial displacement of the rolling element when it passes through a local defect is shown in equation (7):

[0036]

[0037] The contact stiffness between the rolling elements and the inner and outer raceways of the bearing satisfies Hertz contact theory, and its load-deformation relationship is shown in equation (8):

[0038] F=Kδ n (8)

[0039] Where K is the total contact stiffness between the rolling element and the inner and outer rings, F is the Hertzian contact force, δ is the contact deformation, and n is the load-deformation index. For ball bearings, n = 1.5, K i and K o The contact stiffness between the rolling element and the inner and outer rings are respectively. The bearing system is treated as a concentrated spring-mass model, and a two-degree-of-freedom dynamic equation is established, as shown in equation (9):

[0040]

[0041] Where M is the sum of the masses of the inner ring and the shaft, C is the damping of the system, and K is the contact stiffness between the rolling element and the raceways of the inner and outer rings. These represent the vibration accelerations of the inner ring in the X and Y directions, respectively. F represents the vibration velocity of the inner ring in the X and Y directions, respectively. x Fy These are the load components in the horizontal and vertical directions, respectively, δ i The contact deformation between the i-th rolling element and the raceway is shown in equation (10):

[0042] δ i =xcosθ i +ysinθ i -cH′ (10)

[0043] Where x and y are the vibration displacements of the inner ring in the X and Y directions, respectively, c is the radial displacement of the rolling bearing, and H′ is the displacement excitation function. d When H < H, H′ = H1, when H d When H > H, H′ = H2, and λ represents the effective contact area parameter of the i-th rolling element, as shown in equation (11):

[0044]

[0045] The method of updating the dynamic model parameters with actual measurement data and estimating the time-varying defect size of the rolling bearing based on the signal approximation method specifically includes the following steps:

[0046] S3.1: Calculate the peak value pkm(i) of each group of target signals, i = 1, 2, ..., n, where n is the index of the last group of target signals; the target signals are the full lifecycle noise-reduced signals;

[0047] S3.2: Set the initial defect length L', step size Δ and stopping approximation error δ, solve the dynamic equation using the Runge-Kutta numerical integration method, and obtain the peak vibration signal pks corresponding to the current defect length. If pks < pkm, L' = L' + Δ, then compare pks and pkm again until pks ≥ pkm.

[0048] S3.3: Let L' = L' + Δ / 100, obtain pks corresponding to the current defect length. If pks ≥ pkm, continue L' = L' + Δ / 100 until pks - pkm ≤ δ.

[0049] S3.4: Apply S3.1-S3.3 to each group of target signals to obtain the defect length L(i) of the virtual entity throughout its entire life cycle and its corresponding vibration signal peak value pks(i);

[0050] S3.5: Substitute the obtained defect length L(i) of the virtual entity into the formula to obtain the simulated vibration signal for the entire life cycle.

[0051] The method of establishing the relationship between actual measured vibration signals and time-varying defects using a Long Short-Time Memory (LSTM) network to achieve real-time mapping from physical entities to virtual entities includes the following steps:

[0052] S4.1: Calculate the root mean square value rms(i) of the measured vibration signal, use rms(i) as the input of LSTM, and use the defect length L(i) of the virtual entity as the output to construct a sample;

[0053] S4.2: Divide the samples into training and validation sets according to the proportions to train the LSTM neural network;

[0054] S4.3: Input the vibration signal collected by the sensor into the trained LSTM neural network to obtain the estimated time-varying defect size.

[0055] Beneficial technical effects

[0056] 1. Compared with traditional digital twin models, the technical solution adopted in this invention not only makes full use of historical measurement data and dynamic models, but also dynamically updates the virtual entity using the proposed dynamic update method based on the relationship between the fault mechanism and the dynamic response of the rolling bearing. This allows the invention to not only generate simulated vibration signals consistent with the measured vibration signals, but also directly estimate the size of local defects in the rolling bearing, explore the expansion behavior of local defects in rolling bearings under continuous service conditions, and provide a new means for the operation and maintenance and health monitoring of rolling bearings.

[0057] 2. This invention bridges the physical and virtual spaces using vibration signals from rolling bearings. It employs a signal approximation method to dynamically update the virtual entity without requiring measurement of the actual bearing defect size, thus solving the problem of obtaining time-varying defect sizes during bearing operation. By utilizing LSTM to establish the relationship between the root mean square value of the measured vibration signal and the length of local defects in the virtual space, the physical space signal is mapped to the virtual space. After training with historical data, the model can directly predict the actual local defect size of the bearing under the same operating conditions based on the collected vibration signals, facilitating rapid updates to the virtual entity. Attached Figure Description

[0058] Figure 1 This is a flowchart of a method for constructing a digital twin model reflecting the expansion of defects in rolling bearings, as proposed in an embodiment of the present invention.

[0059] Figure 2 This is a technical roadmap of the digital twin model construction method proposed in this embodiment of the invention;

[0060] Figure 3 This is a schematic diagram illustrating the local defect propagation of a rolling bearing according to an embodiment of the present invention.

[0061] Figure 4 This is a schematic diagram of the geometric relationship of the rolling bearing displacement excitation function proposed in an embodiment of the present invention;

[0062] Figure 5This is a flowchart of the signal approximation method proposed in an embodiment of the present invention. Detailed Implementation

[0063] The specific embodiments of the present invention will be described in further detail below with reference to the accompanying drawings and examples.

[0064] This invention proposes a method for constructing a digital twin model reflecting the expansion of defects in rolling bearings. Most traditional digital twin models continuously update the parameters of a virtual entity based on real-time sensor data to mimic the motion of the physical entity and generate a simulated dynamic response. However, the information provided by sensors is limited, and some parameters, such as the size of local defects in the rolling bearing, cannot be directly obtained through sensors. This prevents traditional digital twin models from displaying these physical quantities. To address the difficulty of online monitoring of local defect sizes during rolling bearing operation, this method is based on the five-dimensional digital twin model theory, incorporating signal processing, dynamic modeling, and deep learning techniques. It estimates the size of local defects in the rolling bearing and uses a signal approximation method to dynamically update the digital twin model, simulating the entire process of defect expansion. Finally, LSTM is used to establish the relationship between the measured vibration signal of the rolling bearing and the estimated time-varying defect size, achieving real-time mapping. The advantage of this model is that the model construction process does not require measuring the actual size of the local defects in the rolling bearing; changes in the virtual entity can represent changes in the physical entity. This method provides a method for constructing a digital twin model that dynamically updates and maps the local defect expansion behavior of rolling bearings in real time. Figure 1 As shown, it includes the following steps:

[0065] To achieve the above objectives, the present invention provides the following technical solution:

[0066] A method for constructing a digital twin model that dynamically updates and maps the local defect propagation behavior of rolling bearings in real time, such as... Figure 1 As shown, it includes the following steps:

[0067] S1: Obtain the vibration signal of the rolling bearing throughout its entire life cycle, and use singular value decomposition (SVD) technology to perform noise reduction preprocessing on the collected vibration signal;

[0068] The noise reduction preprocessing of the acquired vibration signal using singular value decomposition specifically involves: constructing an r×s order Hankel matrix based on phase space reconstruction theory, and setting the vibration signal acquired by the sensor as Y=[y1 y2…y n Then the Hankel matrix is ​​shown in equation (1):

[0069]

[0070] Where H is an r×s matrix, n is the total length of the signal, n=r+s-1, and r≤s, H can be decomposed into H=D+W; D is the matrix of the smooth signal in the reconstruction space, and W is the matrix of the noise interference signal. H, D, and W are all r×s matrices. Singular value decomposition of H yields:

[0071] H = USV T (2)

[0072] Among them, U and V T Let S be an r×r matrix and S be an s×s matrix, respectively, where S is an r×s diagonal matrix with λ as the main diagonal. i (i = 1, 2, ..., m), where m = min(r, s), that is:

[0073] S=diag(λ1,λ2,…,λ m (3)

[0074] Where λ1,λ2,…,λ m These are the singular values ​​of H, and λ1≥λ2≥…≥λ m U and V T Let H be the left and right singular matrices; after the inverse operation of singular value decomposition, we obtain the reconstructed matrix H. n , H n Re-unfold into a new vibration signal Y n The noise-reduced vibration signal is obtained.

[0075] The singular value difference (SVD) reconstruction components were determined using the singular value difference spectral method. The difference value reflects the degree of difference between two singular values; a larger difference value indicates a greater difference between the singular values. The appearance of these large difference values ​​is due to the high degree of uncorrelation between the effective signal and the noise signal. The singular value difference spectral sequence can describe the changes and trends of the singular value sequence.

[0076] Define the singular value difference as b i =λ i -λ i+1 (i = 1, 2, ..., m-1), after differencing, a singular value difference sequence B = (b1 + b2 + ... + b) is formed. m-1 );

[0077] In this embodiment, the purpose of noise reduction preprocessing on the measured vibration signal is to reduce the influence of white noise on the amplitude of the vibration signal, thereby making the subsequent dynamic update steps more accurate and the estimated size of the local defect in the rolling bearing more accurate.

[0078] S2: Construct a dynamic model based on Hertzian contact theory to describe the local defect propagation behavior of rolling bearings;

[0079] like Figure 2As shown, a method for constructing a digital twin model that dynamically updates and maps the local defect expansion behavior of rolling bearings in real time is based on the five-dimensional digital twin model theory, integrating physical space, virtual space, twin space, services, and data connectivity. The physical space of the model mainly includes the construction of a physical platform, the arrangement of sensors, and data uploading. The virtual space mainly includes a geometric model, a physical model, a behavioral model, and a rule model. The geometric model describes the geometric parameters of the physical entity, such as shape and size. The physical model describes the physical properties, stress conditions, and constraints of the physical entity. In this framework, the behavioral model mainly represents the process of the local defect expansion of the rolling bearing over time. The rule model includes regular rules based on historical correlation data and experience summarized based on implicit knowledge, which in this framework corresponds to using LSTM to establish the relationship between measured vibration signals and virtual entities.

[0080] The aim is to simulate the internal motion logic of rolling bearings using a mathematical mechanism model, presenting the impact of local defect propagation on the dynamic response of rolling bearings from both mechanistic and mathematical perspectives; such as Figure 3 As shown, simulation data similar to actual measurement data is generated, including the following steps:

[0081] S2.1: Define the local defects of the rolling bearing as cuboid defects with a depth of H, a length of L, and a width equal to the width of the outer raceway of the bearing. Analyze the fault stage based on the contact between the rolling elements and the outer raceway.

[0082] The fault analysis stage based on the contact condition between the rolling element and the outer raceway includes:

[0083] Based on geometric relationships, calculate the maximum radial displacement H when the rolling element and both edges of the local defect are in contact simultaneously. d As shown in equation (4):

[0084]

[0085] Where r is the radius of the rolling element;

[0086] If H d If H < H, the rolling element only contacts the edge of the defect, which is the early stage of failure;

[0087] If H d If the value is greater than H, then the rolling element will be in contact with both the edge and bottom of the defect, which indicates the later stage of the failure.

[0088] S2.2: A two-degree-of-freedom dynamic model is established using piecewise functions as displacement excitation;

[0089] The method of using a piecewise function as displacement excitation to establish a two-degree-of-freedom dynamic model specifically includes:

[0090] like Figure 4 As shown, a piecewise displacement excitation function is introduced:

[0091] For the early stage of the fault, the displacement excitation function is as shown in equation (5):

[0092]

[0093] For the later stage of the fault, the displacement excitation function is as shown in equation (6):

[0094]

[0095] Where R is the outer raceway radius, β0 is the initial angular position of the local defect relative to the positive X-axis, β is the local defect range angle, and θ i Let β be the angle of the i-th rolling element relative to the positive X-axis at time t. q H is the arc length when the rolling element contacts the bottom edge of the local defect. max The maximum actual radial displacement of the rolling element when it passes through a local defect is shown in equation (7):

[0096]

[0097] The contact stiffness between the rolling elements and the inner and outer raceways of the bearing satisfies Hertz contact theory, and its load-deformation relationship is shown in equation (8):

[0098] F=Kδ n (8)

[0099] Where K is the total contact stiffness between the rolling element and the inner and outer rings, F is the Hertzian contact force, δ is the contact deformation, and n is the load-deformation index. For ball bearings, n = 1.5, K i and K o The contact stiffness between the rolling element and the inner and outer rings are respectively. The bearing system is treated as a concentrated spring-mass model, and a two-degree-of-freedom dynamic equation is established, as shown in equation (9):

[0100]

[0101] Where M is the sum of the masses of the inner ring and the shaft, C is the damping of the system, and K is the contact stiffness between the rolling element and the raceways of the inner and outer rings. These represent the vibration accelerations of the inner ring in the X and Y directions, respectively. F represents the vibration velocity of the inner ring in the X and Y directions, respectively. x F y These are the load components in the horizontal and vertical directions, respectively, δ i The contact deformation between the i-th rolling element and the raceway is shown in equation (10):

[0102] δ i=xcosθ i +ysinθ i -cH′ (10)

[0103] Where x and y are the vibration displacements of the inner ring in the X and Y directions, respectively, c is the radial displacement of the rolling bearing, and H′ is the displacement excitation function. d When H < H, H′ = H1, when H d When H > H, H′ = H2, and λ represents the effective contact area parameter of the i-th rolling element, as shown in equation (11):

[0104]

[0105] S3: Based on the signal approximation method, update the dynamic model parameters with actual measurement data and estimate the time-varying defect size of the rolling bearing; such as Figure 5 As shown, establishing the relationship between virtual space simulation signals and physical space measured signals allows the simulation data generated in the virtual space to approximate the physical space measured data as closely as possible. This enables the virtual entity parameters to be continuously updated based on the physical space data. Furthermore, the virtual entity data is used to characterize the process of defect expansion in the physical entity, thus achieving digital twinning.

[0106] In this embodiment, as Figure 5 As shown, since it is impossible to measure the size of time-varying defects in rolling bearings during operation, the defect propagation of bearings can only be analyzed through vibration signals collected by sensors. Therefore, a signal approximation method is proposed to establish the relationship between the simulated signal in virtual space and the measured signal in physical space. The purpose of the signal approximation method is to approximate the simulated data generated in virtual space to the measured data in physical space as closely as possible, so that the virtual entity parameters are continuously updated according to the physical space data, and the virtual entity data is used to characterize the process of defect propagation in physical entities, thus realizing digital twins. Since the target signal for signal approximation is the measured signal after SVD noise reduction preprocessing, its amplitude changes smoothly and its impact characteristics are relatively obvious. Therefore, the peak value, which has a relatively simple approximation algorithm, is selected as the indicator for evaluating the approximation effect. The specific steps include:

[0107] S3.1: Calculate the peak value pkm(i) of each group of target signals, i = 1, 2, ..., n, where n is the index of the last group of target signals; the target signals are the full lifecycle noise-reduced signals;

[0108] S3.2: Set the initial defect length L', step size Δ and stopping approximation error δ, solve the dynamic equation using the Runge-Kutta numerical integration method, and obtain the peak vibration signal pks corresponding to the current defect length. If pks < pkm, L' = L' + Δ, then compare pks and pkm again until pks ≥ pkm.

[0109] S3.3: Let L' = L' + Δ / 100, obtain pks corresponding to the current defect length. If pks ≥ pkm, continue L' = L' + Δ / 100 until pks - pkm ≤ δ.

[0110] S3.4: Apply S3.1-S3.3 to each group of target signals to obtain the defect length L(i) of the virtual entity throughout its entire life cycle and its corresponding vibration signal peak value pks(i);

[0111] S3.5: Substitute the obtained defect length L(i) of the virtual entity into the formula to obtain the simulated vibration signal of the entire life cycle;

[0112] S4: Utilize a Long Short-Term Memory (LSTM) network to establish the relationship between the actual measured vibration signal and the time-varying defect, build a digital twin model, and realize real-time mapping from physical entities to virtual entities;

[0113] The LSTM neural network establishes the relationship between physically measured signals and virtual entities. The specific steps are as follows:

[0114] S4.1: Calculate the root mean square value rms(i) of the measured vibration signal, use rms(i) as the input of LSTM, and use the defect length L(i) of the virtual entity as the output to construct a sample;

[0115] S4.2: Divide the samples into training set and validation set according to the proportion to train the LSTM neural network; in this embodiment, 70% is used as training set and 30% is used as validation set.

[0116] S4.3: Input the vibration signal collected by the sensor into the trained LSTM neural network to obtain the estimated time-varying defect size;

[0117] like Figure 5 As shown, the amplitude of the denoised signal is generally smaller than that of the noisy signal. After denoising, the amplitude of the signal is smoother. This denoised signal is more suitable as the objective function for signal approximation, laying the foundation for updating the virtual entity using the signal approximation method in the next stage.

[0118] S5: Monitor the operating status and health of rolling bearings based on the extent of local defect expansion, estimate the current operating stage of rolling bearings in the entire life cycle, and make timely decisions and feedback before bearing failure.

Claims

1. A method for constructing a digital twin model reflecting the propagation of defects in rolling bearings, characterized in that, Includes the following steps: S1: Obtain the vibration signal of the rolling bearing throughout its entire life cycle, and use singular value decomposition technology to perform noise reduction preprocessing on the collected vibration signal; S2: Construct a dynamic model based on Hertzian contact theory to describe the local defect propagation behavior of rolling bearings; S3: Based on the signal approximation method, update the dynamic model parameters with actual measurement data and estimate the time-varying defect size of the rolling bearing; S4: Utilize a long short-term memory network to establish the relationship between the actual measured vibration signal and the time-varying defect size, build a digital twin model, and realize real-time mapping from physical entities to virtual entities; S5: Monitor the operating status and health of rolling bearings based on the extent of local defect propagation, estimate the current operating stage of rolling bearings in the entire life cycle, and make timely decisions and feedback before bearing failure. The dynamic model describing the local defect propagation behavior of rolling bearings based on Hertzian contact theory is as follows: S2.1: Define the local defects of the rolling bearing as cuboid defects with a depth of H, a length of L, and a width equal to the width of the outer raceway of the bearing. Analyze the fault stage based on the contact between the rolling elements and the outer raceway. S2.2: A two-degree-of-freedom dynamic model is established using piecewise functions as displacement excitation; The fault analysis stage based on the contact condition between the rolling element and the outer raceway includes: Based on geometric relationships, calculate the maximum radial displacement H when the rolling element and both edges of the local defect are in contact simultaneously. d As shown in equation (4): Where r is the radius of the rolling element; If H d If H < H, the rolling element only contacts the edge of the defect, which is the early stage of failure; If H d If the value is greater than H, then the rolling element is in contact with both the edge and bottom of the defect, which indicates the later stage of the failure. The method of using a piecewise function as displacement excitation to establish a two-degree-of-freedom dynamic model specifically includes: Introducing piecewise displacement excitation functions: For the early stage of the fault, the displacement excitation function is as shown in equation (5): For the later stage of the fault, the displacement excitation function is as shown in equation (6): Where β0 is the initial angular position of the local defect relative to the positive X-axis, β is the local defect range angle, and θ i Let β be the angle of the i-th rolling element relative to the positive X-axis at time t. q H is the arc length when the rolling element contacts the bottom edge of the local defect. max The maximum actual radial displacement of the rolling element when it passes through a local defect is shown in equation (7): The contact stiffness between the rolling elements and the inner and outer raceways of the bearing satisfies Hertz contact theory, and its load-deformation relationship is shown in equation (8): F=Kδ n (8) Where K is the total contact stiffness between the rolling element and the inner and outer rings, F is the Hertzian contact force, δ is the contact deformation, and n is the load-deformation index. For ball bearings, n = 1.

5. The bearing system is treated as a concentrated spring-mass model, and a two-degree-of-freedom dynamic equation is established.

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