Additionally, a rapid increase in the running speed could lead to an increase in the ITB strain rate for both the pre-  and post-exhaustion  states. Conclusions: It should be noted that an exhaustion state could lead to an increase in the ITB strain rate. In addition, a rapid increase in running speed might cause a higher ITB strain rate, which is proposed to be the primary cause of ITBS. The risk of injury should also be considered due to the rapid increase in the training load involved. Running at a normal speed in a non-exhaustive state might be beneficial for the prevention and treatment of ITBS.Background: Iliotibial band syndrome (ITBS) is one of the most prevalent overuse injuries in runners. The strain rate in the iliotibial band (ITB) has been theorized to be the primary causative factor in the development of ITBS. Running speed and exhaustion might lead to an alteration in the biomechanics that influence the strain rate in the iliotibial band. Objectives: To identify how exhaustion states and running speeds affect the ITB strain and strain rate. Methods: A total of 26 healthy runners  ran at a normal preferred speed and a fast speed. Then, participants performed a 30 min exhaustive treadmill run at a self-selected speed. Afterward, participants were required to run at similar speeds to those of the pre-exhaustion state. Results: Both the exhaustion and running speeds were revealed to have significant influences on the ITB strain rate. After exhaustion, an increase of approximately 3% in the ITB strain rate was observed for both the normal speed ( Iliotibial band syndrome (ITBS) is one of the most prevalent overuse injuries in runners, which leads to lateral knee pain . The iliRunning speed is one of the main influential factors in gait. Compensatory strategies might be adopted when runners increase their speeds. One dominant strategy is for runners to shift to increase their stride cadence and push on the ground with a higher frequency, which consequently applies more force on the hip joint as the running speed increases . The incProlonged, exhaustive running has significant effects on the biomechanics of the lower limb joints, which are essential intrinsic factors of running-related injuries ,16. A siIt has also been implied that excessive mileage is associated with the development of ITBS . An alteTwenty-five participants  were recruited for this study. Details about the participants are listed in An eight-camera motion-capture system  was used to collect the motion data at a sampling frequency of 250 Hz. One force plate  was used to record the ground reaction force at a sample frequency rate of 1000 Hz, which recorded simultaneously with the Vicon system. The Vicon system was calibrated prior to each session to ensure that the image error was below 0.2 mm. Before the formal tests, a total of 34 markers were attached to the feet, legs, and torso of each participant. Reflective markers were placed on the anatomical landmarks: the forehead; acromioclavicular joints; posterior and anterior iliac spines; three skin-mounted in a triangle on the thigh; lateral and medial femoral epicondyles; three skin-mounted in a triangle on the shank; lateral and medial malleoli; the heads of the first and fifth metatarsals; and the tip of the toes. It was ensured that each segment possessed three markers that were used to track 3-dimensional motion, as shown in Each participant wore lab-provided sports shoes  of appropriate size. Participants were encouraged to become familiar with the shoes and warm up. Data of a static trial were collected for each participant standing on the force plate, which were used to scale the generic musculoskeletal model. Then, dynamic trials were performed using the normal running speed and a faster speed that was just over 10% of each participant’s normal speed. The two speed conditions were confirmed by the participants, with the speeds in a random order. A successful trial was confirmed by stepping on the force plate with the entire right foot. In addition, the speed tolerance of each trial of the same speed needed to be within ±5% to conduct the analysis. Five successful trials were collected for each condition. Afterward, each participant was asked to perform a treadmill run at their self-selected speed for 30 min, with each of them ultimately reaching an exhaustion state. Then, participants were asked to perform the dynamic overground running trials again. All participants performed running trials with a preferred speed and a fast speed during their pre- and post-treadmill runs, running at a self-selected speed for 30 min. The normal speed and the fast speed were maintained at approximately the same level. Real-time feedback on speed was provided for participants to confirm that the speed conditions were maintained similarly before and after the treadmill run. Before the formal tests, participants were required to conduct at least ten warm-up overground running trials to resume their natural gait. Notably, participants were not allowed to rest in order to maintain the exhaustive state. Their running speeds were confirmed using the average speed of the anterior–posterior speeds of the sacral marker, which could provide feedback immediately after each trial . DetailsThe musculoskeletal model of the ITB was established in OpenSim . The modt-tests were used to evaluate the group differences for different speed levels. The differences in the kinematics and kinetics of the hip and knee joints, and also in the strain and strain rate of the ITB between exhaustion (before and after) and speed levels, were examined using a two-way repeated-measures analysis of variance (ANOVA). Mauchly’s test was adopted to examine the sphericity assumption of the repeated-measures ANOVA. The results revealed that all of the variables passed the assumption. The moments were normalized according to the mass of the participants. Post-hoc tests were performed for significant ANOVA interactions using t-tests with Bonferroni corrections. The post-hoc tests with the adjustment of the Bonferroni method confirmed the presence of significant differences.SPSS software  was used to perform all of the statistical analyses. The significance level for all tests was set at 0.05. The mean variables were calculated by averaging the five successful trials, allowing for statistical analysis. The kinematics and kinetics of the knee and hip joints were obtained to perform the analysis. The strain and strain rate were calculated to conduct statistical analyses under different conditions. All variables were established to be normally distributed using the Shapiro–Wilk test. Independent paired p = 0.002) at the fast speed after the exhaustive run but showed no significant difference when the running speed changed. On the contrary, the hip internal rotation angle decreased significantly (p = 0.001) after an exhaustive run when the running speed increased. As for the knee joint, when the speed increased, the knee flexion angle revealed a significant increase under the same exhaustive state. Additionally, the angle was found to increase significantly after exhaustive running when the running speed remained similar.No significant interaction was identified for all variables of interest between the exhaustion states and running speeds. The kinematic parameters of the knee and hip joints were influenced by both speed and exhaustion states, as shown in p = 0.007) during the exhaustive state when the running speed increased. The peak moment of the knee extension revealed a significant increase (p = 0.002) at the normal speed after exhaustive running. However, the running speed showed no significant effects on the knee extension moment.The kinetics of the knee and hip joints were also influenced by the exhaustion states and running speeds, as shown in Peak ITB strains were similar when the running speed increased, as shown in The current study aimed to explore how exhaustion states (before and after an exhaustive run) and running speeds altered the behaviors of the ITB due to the changes in the kinematics and kinetics of knee and hip joints. The flexion angles of both the knee and hip joints revealed significant increases when the running speed improved. In addition, the angles increased significantly when participants experienced an exhaustive run. The peak angle of the hip adduction also increased significantly when comparing pre- and post-exhaustion states at a fast speed. The greater hip adduction was associated with the increase in the ITB strain . The runThe peak of the ITB strain remained similar when the running speed increased. Furthermore, it showed no significant changes after an exhaustive run. However, the peak of the ITB strain rate revealed a significant increase alongside the increase in running speeds for both the pre- and post-exhaustion states. Additionally, the ITB strain rate also showed a significant increase after an exhaustive run. The increase in the ITB strain rate has been proposed to be a major causative factor in the development of ITBS . The incThe current study found that the peak hip adduction angle showed a significant increase when participants ran at a fast speed before and after an exhaustive run, which is in agreement with previous findings . The peaWith respect to the influences of the running speeds, the peak angle of the hip internal rotation tended to decrease as the running speeds increased when participants were in the post-exhaustion state. During the pre-exhaustion state, a small decrease was also observed in the hip internal rotation angle. There were no significant differences in the similar speeds between pre- and post-exhaustion states, which was in agreement with a previous study . The incIn this study, a few limitations should be considered. Participants in this study were healthy runners, meaning the results might not be generalizable to injured populations. Additionally, gender differences were not a part of the investigation in the current study. The musculoskeletal model is not able to incorporate the complex anatomical structures of the ITB. Validation of the ITB model would be very difficult with non-invasive measures. The ITB was simulated to be completely passive and its length was represented by the length between the connected markers . All of In this study, a musculoskeletal model was used to examine the influences of the exhaustion states and running speeds on the ITB strain and strain rate. The hip adduction angle revealed an increase that corresponded with the exhaustion state. The peak of the hip internal rotation angle decreased due to the increase in running speed. Both the exhaustion states and running speeds showed a significant effect on the ITB strain rate. The results imply that a rapid increase in the training intensity might predispose healthy runners to sustain higher ITB strain rates, which might lead to injuries, such as ITBS.


The extraction of the optimal mode of the bearing signal in the drive system of a corn harvester is a challenging task. In addition, the accuracy and robustness of the fault diagnosis model are low. Therefore, this paper proposes a fault diagnosis method that uses the optimal mode component as the input feature. The vibration signal is first decomposed by variational mode decomposition (VMD) based on the optimal parameters searched by the artificial bee colony (ABC). Moreover, the key components are screened using an evaluation function that is a fusion of the arrangement entropy, the signal-to-noise ratio, and the power spectral density weighting. The Stockwell transform is then used to convert the filtered modal components into time–frequency images. Finally, the MBConv quantity and activation function of the EfficientNet network are optimized, and the time–frequency pictures are imported into the optimized network model for fault diagnosis. The comparative experiments show that the proposed method accurately extracts the optimal modal component and has a fault classification accuracy greater than 98%. As a critical component of corn harvesters, the significance of the bearing lies in its crucial role in guaranteeing the safety and dependability of the equipment operation . In receThe recent studies on bearing fault diagnosis focused on signal extraction and processing. The parameter selection of the VMD method significantly affects the decomposition results, which limits its application range in bearing fault diagnosis. Wang et al.  used theBai et al.  proposedWith the increase in the application of neural network models in fault diagnosis, the large scale of network parameters often leads to overfitting in the training process. Therefore, it is necessary to adopt a lightweight model to perform fault diagnosis with high generalization ability.In summary, although VMD has improved the signal decomposition, it still has some problems, such as the number of decomposition layers and the penalty factor, which is challenging to select accurately. The EfficientNet classification model uses deep scalable convolutional blocks and lightweight feature extractors to provide the network with a strong representation with few parameters. However, the convergence process and classification accuracy of the model still have much room for improvement.To this end, this paper proposes a bearing fault diagnosis method based on ABC-ANC and optimizes the EfficientNet. The main contributions of this study are summarized as follows.(1) The artificial bee colony (ABC) algorithm is used to optimize the decomposition number K and the variational mode decomposition penalty factor. The permutation entropy, power spectral density, and signal-to-noise ratio are used to construct an evaluation function in order to select the best decomposition signal.(2) The EfficientNet model optimized for the count of MBConv modules and activation function is proposed to improve the stability of the training process and the classification accuracy of the model.x(t) into k amplitude-FM sub-signals. To determine ku and kω, the VMD can be written as a constrained variational problem described by the following equations:The VMD non-recursively decomposes a real-valued signal α and the Lagrange multiplier λ. The augmented Lagrange quantity function is given by:α is the equilibrium parameter of the data fidelity constraint.The constrained variational problem is solved using the quadratic penalty term The solution of the equation consists of updating each mode function and neutral frequency using the alternating direction multiplier method. The KTH amplitude-FM sub-signal obtained from the spectral domain solution is written as:The central frequency of each eigenmode component is expressed as :(5)ωkn+1xi  is randomly generated as:lb denotes the lower bound of the target space, ub represents the upper bound, φi represents a vector consisting of random values uniformly distributed between  and ix = T.The ABC algorithm should first initialize the colony. That is, the SN d-dimensional initial solution ijx is the JTH element of the ith solution, j and k are randomly selected within their value ranges.Note that after initialization, the three bees search cyclically until reaching the maximum number of iterations. In the search phase, new food sources are randomly selected by the bees as follows:According to Equation (7), the employed bee randomly selects a new food source, which fitness is then calculated as:The employed bee compares the fit value of the new scheme with the original scheme and chooses the best scheme. After the search, the employed bee gives the onlooker bee the latest information about food sources. The selection probability of each food source is then calculated according to Equation (9), and the watching bees select their food source scheme based on a roulette wheel style selection scheme.During the bystander bee search phase, a random value is generated in the range of . If the random value is less than the probabilistic value, the relevant onlooker bee will find a new solution based on Equation (7). The onlooker bee then chooses a better solution, just like the employed bee. If the solution is not updated after a specified number, in an iteration known as limiting, in which employed bees give up their food source and become scout bees. This prevents the ABC from falling into a locally optimal solution, which is then randomized by the scout bee to choose a new food source. The traditional VMD decomposition usually leads to the problem of over-decomposition due to the improper choice of the decomposition mode (K) and penalty factor, which results in the deterioration of the input quality of the fault diagnosis model. Therefore, the artificial bee colony algorithm is used to optimize the selection of K and penalty factors in order to enhance the effectiveness and precision of the VMD decomposition.Step 1: The parameter boundaries are determined, and the boundaries of K and α are initialized as  and , respectively .Step 2: The parameters of the artificial colony algorithm, including the number of food sources (SN), the ABC parameters , the maximum number of iterations, and the limit value, are set. K and α are then randomly searched to decompose the input signal VMD, while the fitness value of the bee number of the ABC is calculated in each iteration.Step 3: For the iteration termination conditions, a maximum number of iterations is set, and the best fitness value over several consecutive iterations below a set threshold is improved. If the iteration termination condition has not been met, the iterative process is repeated by updating the food sources K and α.Step 4: The optimal K and α values based on the best fitness value are determined, and the original signal is decomposed by VMD using the optimal parameters.In general, the bearing fault dataset cannot reach the million-level dataset required by EfficientNet to achieve the best classification effect. Too many MBConv modules in EfficientNet will result in high model complexity and computational workload. Therefore, this study adjusts and reduces the number of MBConv modules . AdjustmThe primary role of the activation function is to provide nonlinear expressivity to the neural network. The first-order derivative of the LeakyRelu activation function is constant, while the second and first-order derivatives of the Swish activation function are not constants, and the derivation process is more complex . AlthougIn this section, the precise diagnostic procedure shown in The bearing fault diagnosis classification model based on the ABC-VMD and optimized EfficientNet mainly comprises five steps.Step 1: The vibration signals of bearings under normal operation, ball fault, inner ring fault, and outer ring fault state are collected and optimized through the test bench, and vibration characteristics under different fault states are obtained. The irrelevant signals are eliminated by signal filtering using several techniques, such as thresholding to filter out the data that are not within the expected range and by replacement for irrelevant or missing data using interpolation or other methods.Step 2: ABC-VMD is used to find the optimal penalty factor and decomposition number. The best parameters for the modal decomposition of the vibration signal collected in Step 1 are determined.Step 3: The permutation entropy , power sStep 4: The selected signal components are converted to time–frequency images using the Stockwell transform . The latStep 5: The generated time–frequency images are fed into the optimized EfficientNet model for training and validation.The use of VMD as the signal processing means of the proposed method has the following advantages .(1) Adaptive decomposition: In contrast to the traditional fixed-base decomposition methods, VMD can adaptively select the decomposition mode to better match the characteristics of the signal.(2) Nonlinear and non-stationary signal analysis: VMD is suitable for analyzing nonlinear and non-stationary signals. It can adapt to the local characteristics of the signal, which allows it to provide a more precise analysis.(3) No prior information required: In contrast to the other methods, VMD does not require prior information about the signal or noise.(4) Sparsity: VMD can usually produce relatively sparse solutions, with each mode capturing only one specific signal component. This helps to better understand and visualize the structure of the signal.(5) Fewer pseudo components and mode mixing: VMD tries to reduce the generation of spurious components and the mixing between different modes, which provides a cleaner decomposition.(6) Ability of real-time analysis: Due to its mathematical properties, VMD can be quickly executed, which makes it suitable for real-time or near-real-time signal analysis applications.(7) Robustness: VMD has a certain robustness and resistance to noise and other disturbances.However, the traditional VMD decomposition usually leads to problems such as over-decomposition due to the improper selection of the decomposition mode number (K) and penalty factor, which results in decreasing the input quality of the fault diagnosis model. Therefore, although the following problems are faced when dealing with large-scale datasets , artific(1) Local searchability: ABC has a low local search ability, which leads to getting stuck when approaching the global optimal solution.(2) Adaptability to complex problems: The ABC algorithm may need to perform better for some very complex or high-dimensional optimization problems.(3) Parameter tuning: Although fewer parameters exist, their choice may affect the performance of the algorithm. For different problems, the parameters may require to be accurately tuned.(4) Possibility of falling into local optimum: In some cases, especially in complex multimodal function optimization problems, ABC may fall into the local optimum solution.Therefore, the artificial bee colony algorithm has higher performance on the small dataset of bearings collected in this paper. More precisely, it has the following characteristics.(1) Global search ability: ABC can perform a global search in the solution space, and it can find the global optimal solution.(2) Robustness: Compared with many other optimization methods, ABC is less sensitive to the selection of initial parameters, and it has a higher robustness.(3) Ease of implementation: The rules of ABC are straightforward, and it can be easily implemented.(4) Existence of a small number of control parameters: ABC only requires a small number of control parameters that can be easily adjusted and optimized.(5) Parallelism: The natural parallel nature of the algorithm allows it to be effectively implemented in a parallel computing environment.The Stockwell transform in step 4 can provide a joint representation of the time and frequency of the signal. It then preserves the phase information of the signal, which allows a more accurate time–frequency and multi-resolution analysis. It has higher time resolution at low frequencies and higher frequency resolution at high frequencies . TherefoThe bearing failure dataset used in this paper is derived from the bearing dataset provided by Case Western Reserve University (CWRU). The experimental setup shown in In the experiment, the SKF6205 bearing type was divided into four states: normal, ball defects, inner ring failure, and outer ring failure. The failure under the load was 0, 1, 2, and 3 hp, where the damage degree of each failure diameter size was 0.007, 0.014, and 0.021 inches, leading to a total of 10 conditions of the bearing data. The dataset consists of 9660 samples, with 6720 and 2940 samples allocated to the training set and testing set, respectively. The details of the dataset are shown in In order to determine the valuable signal components, a comprehensive evaluation function is established, which fully considers the permutation entropy, PSD mean, and signal-to-noise ratio. The evaluation function is given by:The magnitude of change in each indicator is the basis for the selection of the weighting parameters. More precisely, the magnitude of the components of the permutation entropy transformation is relatively small, and thus, the components of the permutation entropy are set with smaller weights. The evaluation function is characterized by the combined permutation entropy, PSD mean, and signal-to-noise ratio in a weighted manner. The permutation entropy, PSD mean, and signal-to-noise ratio reflect the complexity, energy distribution, and intensity of the signal relative to the noise, respectively.The process of parameter optimization based on ABC-VMD is summarized as follows.(1) Initialization: The kurtosis is defined as the objective function for assessing the quality of the VMD, and the artificial bee colony is initialized. Each bee represents a possible solution .(2) Evaluation: For each solution in the bee colony, VMD decomposition is performed, and the obtained result is evaluated using the defined objective function.(3) Selection: Based on the assessment of the objective function, the best solutions for the worker bees to continue their search are selected while leaving the rest as scout bees to explore new potential solutions.(4) Search Optimization: The worker bees employ localized search to improve their solutions, while the scout bees use random search strategies to find entirely new solutions.(5) Iterative Process: The aforementioned steps are repeated until certain stopping criteria are met, such as reaching a maximum number of iterations or the solution’s improvement falls below a specific threshold.(6) Result Extraction: The optimal solution is extracted from the bee colony, identified by the best objective function value.The final optimal component number (K) is set to 6. The calculation process of the evaluation function based on the data in The magnitude of the IMF4 evaluation function is significantly reduced in In order to verify the outperformance of the evaluation function score, the PE, SNR, and PSD are compared with the evaluation function while considering the same inputs and the same training model to avoid the effect of random errors. The loss function of the signals extracted by different indicators in the network training process is shown in It can be seen that in the first ten rounds of iterations, the four models have a downward trend. However, when the iteration period gradually increases, the gap between the signal extracted by the PE, SNR, and PSD and the evaluation function score becomes gradually noticeable, which indicates that the modal features extracted by the indicators in this paper are more conducive to model training. After the tenth iteration, the loss value of the evaluation function score is significantly lower than the other three, showing a good model training process.To further demonstrate the outperformance of the proposed method in terms of fault classification, a confusion matrix comparison is performed using EfficientNet and a high-parameter number network. The obtained results are shown in VGGNet , DenseNeIn order to more intuitively show the feature extraction ability of the proposed model, the t-SNE  dimensioIn the actual working process, there are differences in the working environment of the bearings and the mode of operation, leading to deviations in the fault signals. The low robustness and generalization ability of the model affects the learning ability and the degree of fitting, which reduces its fault diagnosis accuracy. Therefore, the bearing fault signal of the bearing of the corn harvester under the working state is collected on the spot. The acquisition device is shown in The labels of the collected fault datasets are shown in In order to compare the recognition effects of the proposed model for different bearing datasets, the confusion matrix is introduced for EfficientNet, DenseNet, and VGGNet to verify the recognition accuracy of the training model in different fault samples in the test set. In order to intuitively compare the classification abilities of the six models, t-SNE  is used It can be seen from To further demonstrate the high accuracy of the proposed method, it is compared with existing state-of-the-art approaches. The obtained results are shown in (1) The approach presented in  is a bea(2) The authors of  introduc(3) The method presented in  uses a gThe proposed optimization of EfficientNet not only retains the advantages of the original model but also reduces the complexity and avoids overfitting, leading to an average accuracy rate of 98.7%.Fault characteristic signals are often submerged due to the complex working environment of corn harvester bearings. The accuracy and stability of the fault classification model depend on the characteristic signal processing and the convergence ability of the diagnosis model.The conclusions of this study are summarized as follows.(1) A fault signal decomposition method based on ABC-VMD is proposed. It relies on the ABC algorithm to search for the optimal decomposition number and penalty factor in order to improve the accuracy of fault feature extraction.(2) By combining the complexity of the signal, the ratio between the signal quality and the noise level, and the energy distribution in the frequency domain, the evaluation function based on the permutation entropy, signal-to-noise ratio, and power spectral density is constructed, which can fully reflect the fault signal characteristics.(3) An optimization approach for EfficientNet is proposed to solve the problem of high complexity and computational workload of the high parametric quantity models. The convergence ability of the model is effectively improved. The number of MBConv modules is optimized. In addition, LeakyRelu is used to reduce the calculation workload and optimize the iterative process of the fault classification model.Although the proposed method outperforms other existing models, after optimizing EfficientNet, the number of modules is reduced, which is mainly suitable for small datasets and can lead to model underfitting in the case of large amounts of data. Therefore, in future work, we aim to avoid the impact of different sample volumes on the recognition accuracy of EfficientNet.


Given the digitalization trends within the field of engineering, we propose a practical approach to engineering digitization. This method is established based on a physical sandbox model, camera equipment and simulation technology. We propose an image processing modeling method to establish high-precision continuous mathematical models of transmission towers. The calculation of the wind field is realized by using wind speed calculations, a load-wind-direction-time algorithm and the Continuum-Discontinuum Element Method (CDEM). The sensitivity analysis of displacement- and acceleration-controlled transmission tower loads under two different wind direction conditions is conducted. The results show that the digital model exhibits a proportional relationship with the physical dimensions of the transmission tower model. The error between the numerical simulation results and the experimental results falls within a reasonable range. Nodes at higher positions of the transmission tower experience significantly higher forces compared to those at lower positions, and the structural forms with larger windward projected areas yield similar simulation results. The proposed digital twin system can help monitor the performance of structural bodies and assess the disaster degree in extreme conditions. It can guide specific maintenance and repair tasks. The reliability and safety of power grid infrastructure have emerged as pivotal factors impacting both the country’s energy security and economic development, making power grid engineering a cornerstone industry within the national economy. Currently, the inadequate levels of informatization, delayed data collection and a lack of intelligent decision-support tools have constrained the progress and quality of power grid infrastructure development. Digital twinning technology offers a solution to these challenges by amalgamating physical systems with numerical models. Digital twinning could realize real-time data interaction between physical systems and dynamic simulation and modeling by constructing a digital mirror image of actual systems, providing precise information and decision support for the operation and management of tangible systems. Integrating digital twinning into power grid development can revitalize the industry, as this technology enables the remote monitoring and operation of power grid equipment, improving grid efficiency and safety. Digital twinning has substantial potential to address grid challenges and provide vital support for the nation’s energy supply and economic progress ,2,3.The application of digital twinning technology in civil engineering has become a hot topic recently ,5. DigitThe 3D numerical model is established through the algorithmic processing of two-dimensional images and the creation of corresponding digital three-dimensional scenes. The accuracy of the model is verified by comparing the results from physical and digital models.Within the framework of CDEM, we simulated the mechanical characteristics of transmission towers under unidirectional and variable wind conditions. We analyzed the impact of loads on structural displacement and acceleration.We converted historical on-site wind speeds into loads, accounting for wind direction effects. We coupled multiple factors through coding, such as load, direction and time, for transmission towers. We confirmed the feasibility and high accuracy of the model, algorithms and simulation methods by comparing the experimental data with the numerical simulation results.This paper presents an implementation approach to the entire lifecycle of a physical model under a digital twin technology framework. The framework is composed of the transmission tower sandbox model, camera monitoring equipment and digital simulation software. The sandbox model is as the physical object of study, the camera equipment captures the model’s geometric dimensions to create a virtual model and the digital simulation software can simulate the entire lifecycle process of the virtual model. This approach allows for the establishment of a high-precision continuous mathematical model, enabling a holographic mapping from the real space to the virtual digital space. The main content of this paper is as follows:The CDEM ,18 methoL represents the energy of the Lagrangian system and Qi represents the work done by non-conservative forces. The governing equation of the CDEM method is expressed as [u, The equation is shown as ,20:(1)ddessed as ,20:(2)MuThe power transmission tower is a spatial truss structure, where some members are connected through welding and bolting to transfer forces . In this work, the numerical model is reconstructed using the “pile” element. Each “pile” structural element is defined by its geometric form, material parameters and coupling-spring properties. It provides beam-like structural properties for determining the ultimate plastic moment capacity. Additionally, it can simulate frictional effects in both normal and shear directions.x-axis, (2) the x-axis points from node 1 to node 2, (3) the y-axis is aligned with the projection of Y onto the cross-sectional plane . These members can undergo axial yielding, and the yield strength depends on the inherent strength of the materials.Every “pile” element has its own coordinate system. This system is used to define cross-sectional moments of inertia and applied distributed loads. It also sets the conventions for force and moment distributions within each element, as illustrated in In three-dimensional image processing, two three-dimensional coordinate systems and two two-dimensional coordinate systems need to be considered . The fouIn The transformation relationships between three coordinate systems can be described using the extrinsic matrix and the intrinsic matrix. The extrinsic matrix represents the position and orientation of the camera relative to the reference point, and it is used to describe the transformation relationship between the world coordinate system and the camera coordinate system. On the other hand, the intrinsic matrix represents the fixed parameters of the camera, specifically the internal parameters that project objects from the camera coordinate system to the image plane. The intrinsic matrix is commonly used to describe the relationship between the camera coordinate system and the image coordinate system. By performing coordinate system transformations, the relationship between the two-dimensional image coordinate system and the physical world coordinate system could be determined :(3)Zcuv1In this transformation relationship, the intrinsic matrix can be accurately obtained through camera calibration. The intrinsic matrix is assumed to remain unchanged when the camera application environment remains static. However, the extrinsic matrix changes when the camera position changes. In this study, we utilize coordinate transformations to handle captured image data and reconstruct a three-dimensional virtual model that corresponds to the real-world scenes captured in the two-dimensional images.x- and y-axes for the wind at each angle. These load components are subsequently applied to different nodes of the structure, allowing for an accurate simulation of the wind load on the transmission tower.To simulate wind pressure in real-world scenarios, it is important to consider the dynamic characteristics of wind, in which both direction and magnitude change with time. In this paper, a wind direction simplification method is employed, as shown in z and The wind load values for the tower are computed in accordance with the regulations specified for overhead transmission line loads. The formula for calculating the standard wind load on the tower is as follows :(4)Ws=W02) and The value of lated as :(5)W0= VThis study developed a method that can consider the multivariate effects of load, direction and time during the simulation process. The GDEM software used in the study did not provide complex load application functions, which limited the study of the mechanical response of transmission towers in different scenarios. To address this limitation, the specific numerical values of wind loads are computed and stored in a text (txt) file. Afterward, the algorithm reads and stores these values in arrays. These values are then applied to the nodes of the respective component groups. The position of each component in the model is determined based on its group identifier and the load values from the specific arrays are applied to the nodes of these components. This iterative process continues until all load values in the array are computed. Through this approach, the fluctuations in wind force magnitude and direction at various locations within the model could be considered. Furthermore, adaptations to load application timing and computation time were achieved by modifying the time step for calculations, with the specific steps outlined in This paper compares the velocity curves of the physical model of the transmission tower with that obtained from the numerical model under operating conditions to validate the proposed method. The velocity curve of the physical model is monitored in real-time using sensor equipment. Sensor devices were used to measure and monitor various physical quantities and experimental parameters of the test structure. To ensure the validity of the experiments, the material parameters of the numerical model were consistent with the actual materials of the physical model, and the geometric dimensions were kept consistent with the physical model. Furthermore, the simulation monitoring points for the numerical model were positioned at the same locations as those on the physical model, as depicted in The experimental monitoring results have a good agreement with the numerical simulation results. As shown in The image-based modeling method provides an efficacious means for the visual examination and surveillance of transmission tower structures by generating detailed 3D models from visual data. This approach assures safety and dependability while also resulting in substantial time and resource economization.An image-based modeling approach was employed in this study to establish a digital model of the sandbox system, achieving the virtualization and digitization of the physical entity. The geometric shape and dimensions of the physical entity were transferred onto virtual objects through software calculations. This served as the basis for constructing a computable three-dimensional numerical model.The sandbox system is a scaled-down mountain model constructed based on the contour lines of a mountain surface, with simplified treatment on the remaining areas. The model includes many features such as trees, transmission towers, solar panels and power plants. The transmission tower model was fabricated using 3D printing technology and is depicted in A three-dimensional model is created by capturing a complete rotation of the object with a camera. A point cloud model is established by segmenting the video frames and then identifying and extracting key points from the images . ColorizThe digital model of the transmission tower was created based on the actual physical object. The geometric dimensions were measured to be 43 cm in height and printed using Acrylonitrile Butadiene Styrene (ABS) material. The physical and mechanical parameters of the model are listed in The transmission tower model is composed of 639 connected elements. The model has a height of 43 cm and a cross-arm length of 24 cm. To monitor the acceleration and displacement of nodes, four monitoring points are established. The model is divided into three groups from top to bottom. The distribution of the model and the monitoring points is illustrated in This paper analyzes the characteristics of transmission towers under variable-direction and unidirectional wind conditions by examining the acceleration and displacement at different locations of tower nodes.Initial cases: In the calculation for variable-direction wind conditions, meteorological wind speed data published by the weather station are selected. According to the tower wind load standard values as outlined in In the case of unidirectional wind conditions, a set of wind force data at a 45° wind direction angle are established. To study the structural stability and mechanical response of the transmission tower under fixed wind forces, the simulation lasted for 15 s at multiples of 100%, 130%, 150% and 200% of the wind force magnitude. Monitoring points were positioned at the top of each tower and within each group, which were systematically numbered from the tower base to the top.Boundary cases: The velocity at the bottom nodes of the transmission tower model is constrained to 0, thereby ensuring the stability of the bottom node positions.Transmission towers, as a type of highly flexible structure, have vibration characteristics like simply supported beams. Therefore, for simplification purposes, transmission towers are treated as beam element models in calculations. In the CDEM, the members are modeled using the “pile” model, which considers the structural behavior of the beam, including the ability to specify the plastic moment capacity. Additionally, the axial force  between members is calculated, and the load-carrying effect at the top of the tower is evaluated. In this paper, the members are modeled with a “failure” model to assess and analyze the stress state of individual transmission towers.The phenomenon known as whip-end effect occurs when a flexible structure is subjected to external excitations or loads, resulting in nonlinear vibration. Disturbance or excitation at one end of the structure leads to the propagation of vibration waves along the structure and the creation of reflections at the endpoint. These reflected waves interact with the propagating waves, which results in an increased amplitude and instability in the vibrations at the endpoint. As observed in When designing and optimizing transmission towers, it is crucial to pay special attention to the characteristics of the top structure. Rational design and optimization measures can mitigate the adverse effects of the whip-end phenomenon on the structure, thereby enhancing its stability and vibration suppression capabilities.As observed from Simultaneously, the displacements progressively decrease from the top to the bottom of the tower due to the lower tower components being loosely arranged and having a smaller windward area compared to the upper sections. The maximum displacement at the tower top corresponds to the peak values on the curves. The maximum displacement is 8 mm in the x-direction and 7 mm in the y-direction, resulting in a 1 mm difference between the two directions. This discrepancy is attributed to the varying structural configurations on the windward side of the transmission tower, where form factors differ, ultimately leading to greater displacements induced by the applied forces in the x-direction.The mechanical characteristics of a transmission tower in a complex wind field were investigated by applying the wind loads from As is shown in This paper presents an implementation approach to digital twins in a transmission tower sandbox model, considering the extensive application of digital twin technology in the engineering industry. The approach combines image recognition modeling, sensor monitoring and the CDEM. The feasibility of establishing a three-dimensional numerical model is investigated based on two-dimensional images and simulation studies under wind loading conditions.Image-based modeling methods enable the rapid identification of geometric dimensions and material properties based on actual structures. This novel modeling method streamlines the modeling process for large-scale structures. However, the details of this method still require manual refinements. It indicates the potential for extensive optimization and further research in the future.The implementation approach of a digital twin system for transmission towers in a sandbox is proven to be effective and reliable. It involves the utilization of image recognition modeling, camera sensor equipment and the CDEM. The transfer of information data from the physical model to the virtual model is achieved through the camera equipment. The numerical models, created from processed two-dimensional images, exhibit geometric dimensions consistent with the physical model. The dimensions have a 5 mm error, demonstrating the accuracy of the system.The functionality developed in Java language successfully achieved the application of multi-variable loads on the structure, including load, time and direction. The load application algorithm was shown to be effective and accurate, with the simulation results demonstrating an error of less than 5% when compared to the experimental data.Under unidirectional wind conditions, the displacement of the transmission tower exhibits a distinct “step-like” variation, with a simple linear relationship between the displacement and load magnitude as the load magnitude increases. However, under variable-direction wind loads, the variations in amplitude peaks are caused by differences in projected force areas resulting from different structural forms. The maximum acceleration at each monitoring point is approximately twice that of its lower position. With increasing height, both the displacement and acceleration of the transmission tower also increase accordingly.The main conclusions are as follows:


This present paper aims to examine various epidemiological aspects of the monkeypox viral infection using a fractional-order mathematical model. Initially, the model is formulated using integer-order nonlinear differential equations. The imperfect vaccination is considered for human population in the model formulation. The proposed model is then reformulated using a fractional order derivative with power law to gain a deeper understanding of disease dynamics. The values of the model parameters are determined from the cumulative reported monkeypox cases in the United States during the period from May 10th to October 10th, 2022. Besides this, some of the demographic parameters are evaluated from the population of the literature. We establish sufficient conditions to ensure the existence and uniqueness of the model’s solution in the fractional case. Furthermore, the stability of the endemic equilibrium of the fractional monkeypox model is presented. The Lyapunov function approach is used to demonstrate the global stability of the model equilibria. Moreover, the fractional order model is numerically solved using an efficient numerical technique known as the fractional Adams-Bashforth-Moulton method. The numerical simulations are conducted using estimated parameters, considering various values of the fractional order of the Caputo derivative. The finding of this study reveals the impact of various model parameters and fractional order values on the dynamics and control of monkeypox. The symptoms of monkeypox virus infection in humans are comparable to smallpox signs but less severe3. Typically, the infection begins with a fever, headache, muscle pain, and tiredness3. The primary distinction between the symptoms of monkeypox and smallpox is that the monkeypox infection results in lymphadenopathy, characterized by the swelling of lymph nodes3. The incubation period for this infection usually ranges from 7 to 14 days, although it can vary between 5 and 21 days3. Symptoms of monkeypox are usually mild, and the majority of patients naturally recover within a few weeks. Those with weakened immune systems may exhibit severe symptoms5. Smallpox and human monkeypox are clinically related due to their difficult-to-distinguish characteristics2. Typically, monkeypox is spread to humans through contact with animal blood or bites from rodents, pets, and primates3. Gambian pouched rats (Cricetomys Gambians), dormice (Graphiurus sp.), and African squirrels (Heliosciurus and Funisciurus) have also been found to be infected4. Currently, no approved and safe treatment mechanism specifically for monkeypox virus infection. Nevertheless, some particular controlling measures can be used to mitigate the spread of monkeypox. Vaccines, immune globulin, and antiviral medications introduced for smallpox control can be employed to control the transmission. However, that the smallpox vaccine is currently unavailable as smallpox has been eradicated worldwide7.Monkeypox is a viral disease caused by the monkeypox virus (MPXV), which belongs to the Orthopoxvirus genus. It primarily affects animals, particularly monkeys, squirrels, and rodents. It is a zoonotic disease that can be transmitted to humans through animals. Human-to-human transmission can also occur via close contact with infected individuals, particularly through respiratory droplets or contact with skin lesions or other body fluids. Besides these, other possible modes of transmission include contact with contaminated objects or surfaces known as environmental transmission. Although, infected cases are found in numerous countries in the world, but Africa is the main place where it is most commonly found. The first monkeypox case was identified in 1958 following the occurrence of outbreaks in a group of monkeys that were being studied. Globally, there have been a total of 88,122 reported cases of infection, with 30,555 of these cases occurring in the United States8. Their findings revealed that isolating infected people from other populations reduces disease incidence. In9 the authors developed a system of nonlinear differential equations to analyze different modes of monkeypox transmission. The numerical simulation indicates that the immunological status of an individual affects how well they recover from an orthopoxvirus infection. To learn more about the mechanisms of transmission and various controlling methods, many epidemic models on infectious disease have been investigated13. In14 authors developed a mathematical model to better comprehend the dynamics of the monkeypox. Their results suggest the monkeypox outbreak can be eradicated through the proper implementation of non-pharmaceutical interventions. Additionally, in the study of15, the numerical simulation of the model demonstrated that the treatment will result in the elimination of infected individuals from human and non-human primate populations throughout the course of the investigation. Recently, in17 the authors introduce a mathematical model with environmental transmission to study the dynamics and some controlling measures of the monkeypox 2022 outbreak.Monkeypox has received little attention in the past, making it difficult to understand its epidemiology. Despite this, numerous mathematical models have been recently developed to analyze the dynamics of the monkeypox. A novel mathematical model based on deterministic approach for the monkeypox outbreak was developed and examined in20. The memory effect is one of the remarkable characteristics of fractional-order models that cannot be developed in classical models because of the numerous properties of fractional operators22. Many researchers have recently employed fractional differential equations to represent a variety of infectious and non-infectious diseases. One of these study subjects that has received great attention and produced insightful results is the COVID-19 infection. Mathematical models for COVID-19 based on the fractional derivatives are taken into account in27. The dynamics of monkeypox with cross-infection via a novel stochastic model was studied in28. A novel modeling approach based on fractal-fractional operators demonstrating monkeying dynamics with animal-to-human transmission was proposed in29.Mathematical modeling with fractional differential equations has recently attracted the attention of researchers in a wide range of scientific and professional domains, particularly in epidemiologyThis study analyzes the dynamics of monkeypox under some controlling measures using fractional-order calculus. We present a comprehensive theoretical and numerical analysis of the fractional order epidemic model. Most of the model parameters are estimated from the actual data set of the monkeypox 2022 outbreak. The description of the manuscript’s main sections is as follows: The section titled “31.Fractional differential operators in mathematical modeling are well-known and it has been successful in the fields like science, engineering, and epidemiology. We give a few fundamental definitions along with some proprieties that be will used in the rest of the studyThe left and right Caputo fractional derivative operator of the function For Let, The total human population The susceptible human class is initiated through recruited rate 22. The proposed model for transmission of monkeypox model is described by the following systemThis section presents the extension of the integer order monkeypox model to the fractional case. The well-known Caputo derivative having order 3. While some of the demographic parameters  are estimated from the USA population32. The estimation of parameters from the real disease data is significant for carrying out the numerical simulations more realistically. The estimation process of the proposed model  between the model predictions and the corresponding actual data points.Define or set the model parameters’ initial values that need to be estimated. These values can be based on prior knowledge or initial guesses.Utilizing the initial parameter values set in step 2, the epidemic model is simulated in order to generate model predictions.Using a ’lsqcurvefit’ optimization technique, for the minimization of objective function.Set a convergence criterion to stop the iterative estimation scheme. This criterion is based on a maximum number of iterations, a minimal change in parameter estimates, or attaining a predefined threshold for the corresponding objective function.If the estimated parameter values do not adequately fit the real data curve or do not meet the convergence criterion, reset the initial parameter estimates and re-execute steps 3–5 until a reasonable agreement between the model simulation and the actual data is obtained.By iteratively minimizing the objective function via a standard nonlinear least squares technique, the parameter estimates are refined, which leads to improved agreement between the model-predicted and observed data. The fitted and estimated values of the monkeypox epidemic model’s parameters are listed in Table This procedure involves the following main steps: In this section, we address some of the basic and necessary mathematical analysis of the fractional monkeypox epidemic model . We procFor the positive initial values Let Next to we prove the boundedness of the fractional model solution. The total human individuals is given byT be a positive real number and consider J by F is continuous function defined as followsLet e system  as folloproperty  in propoF) described in  is  is 7) is33 we compute the basic reproduction number The model disease free equilibrium (DFE) is given byThe endemic equilibrium (EE) of the fractional monkeypox model is given byMoreover,In the system  \documenIf >1, then  at EE \dEE is GAS in Consider the following nonlinear Lyapunov function given by 20\documeIn this section, we analyze the impact of some model parameters on entclass1pt{minimaentclass1pt{minimaentclass1pt{minima34, an effective iterative method is used to solve the model numerically. The simulation for different model parameters and different values of In this section, we first investigate the numerical solution of the Caputo fractional monkeypox model. The generalized fractional Adams-Bash forth-Moulton approachmentclass2pt{minim34, a uniform grid on A concise numerical method for the approximate solution of the monkeypox transmission model in the Caputo sense is shown in this subsection. For this purpose, the fractional Adams-Bashforth-Moulton is used. The system  can be rIn this section, we applied the iterative technique developed in the preceding section in order to simulate the monkeypox Caputo epidemic model . The simIn the first case of simulation, we illustrate the impact of only fractional order In this subpart, we analyzed the impact of variation in recovery rate This section describes the influence of the infection transmission rates mentclass2pt{minimmentclasspt{minimamentclasspt{minimaIn this manuscript, we analyzed the dynamics of a novel monkeypox infection with the case study of the recently reported outbreak. The model is first formulated using an integer-order nonlinear system of ten differential equations. The human population was divided into six distinct subgroups, while the animal population was divided into four classes. To estimate the model parameters, we fit the model to the actual cases of the 2022 monkeypox outbreak in the USA. Moreover, keeping the importance of the fractional modeling approach, the integer case model is extended to fractional order via the well-known Caputo operator. In the initial stage, we presented a comprehensive theoretical analysis of the fractional monkeypox model, including the existence and uniqueness of the solutions. The existence and stability analysis of the model equilibria are provided. Furthermore, the fractional model is solved numerically using the fractional Adams-Bashforth-Moulton approach. The simulation results illustrate the impact of disease incidence graphically, considering both the baseline values of the parameters and different values of the fractional order of the Caputo operator. Moreover, the model is simulated by increasing the parameter


Basalt fiber-reinforced polymer (BFRP) reinforced concrete is a new alternative to conventional steel-reinforced concrete due to its high tensile strength and corrosion resistance characteristics. However, as BFRP is a brittle material, unexpected failure of concrete structures reinforced with BFRP may occur. In this study, the damage initiation and progression of BFRP-reinforced concrete slabs were monitored using the acoustic emission (AE) method as a structural health monitoring (SHM) solution. Two simply supported slabs were instrumented with an array of AE sensors in addition to a high-resolution camera, strain, and displacement sensors and then loaded until failure. The dominant damage mechanism was concrete cracking due to the over-reinforced design and adequate BFRP bar-concrete bonding. The AE method was evaluated in terms of identifying the damage initiation, progression from tensile to shear cracks, and the evolution of crack width. Unsupervised machine learning was applied to the AE data obtained from the first slab testing to develop the clusters of the damage mechanisms. The cluster results were validated using the k-means supervised learning model applied to the data obtained from the second slab. The accuracy of the K-NN model trained on the first slab was 99.2% in predicting three clusters . Due to the limitation of a single indicator to characterize complex damage properties, a Statistical SHapley Additive exPlanation (SHAP) analysis was conducted to quantify the contribution of each AE feature to crack width. Based on the SHAP analysis, the AE duration had the highest correlation with the crack width. The cumulative duration of the AE sensor near the crack had close to 100% accuracy to track the crack width. It was concluded that the AE sensors positioned at the mid-span of slabs can be used as an effective SHM solution to monitor the initiation of tensile cracks, sudden changes in structural response due to major damage, damage evolution from tensile to shear cracks, and the progression of crack width. Reinforced concrete is one of the most common composite structural systems for constructing buildings, bridge girders, bridge decks, and dams. While concrete has exceptional durability compared to other materials, minor cracks may cause the penetration of water and the corrosion of conventional steel reinforcing bars. Fiber-reinforced polymer (FRP) bars are alternatives to steel reinforcing bars as they are corrosion resistant and lightweight. The alternative polymer materials are basalt, glass, carbon, and aramid. Basalt fibers are made from igneous volcanic rocks melted at 1400 °C using a technology similar to that used to produce glass and carbon fibers. The basalt fiber reinforced polymer (BFRP) bars have high tensile strength which is comparable to glass fiber-reinforced polymer (GFRP), while often being more cost-efficient than the carbon fiber-reinforced polymer (CFRP) ,3. BFRP SHM methods can be adapted to structures for identifying, locating, and sizing defects. Typical SHM methods applied to concrete structures are vibration ,7,8, strDue to the inherent heterogeneous characteristics of concrete, it is difficult to use a single AE feature to evaluate and predict concrete damage evolution. It is necessary to consider multiple AE features, although the relationship between AE features and damage is very complex. SHapley Additive exPlanations (SHAP) is an integrated framework for interpreting model prediction, expressing the importance of each feature for a specific prediction. Lee et al.  conducteWhile the AE method has been applied to both reinforced concrete ,28,29,30In this paper, the AE data collected from two large-scale concrete slabs with different BFRP reinforcing bar sizes were trained to detect damage evolution and characterization. The slab size was comparable to short-span bridge decks with dimensions of 3.05 m in length, 1.22 m in width, and 0.2 m in depth. Typical AE features adapted to reinforced concrete structures, including AE hit rate and amplitude, were evaluated. The damage initiation can easily be determined using the cumulative AE hits in the laboratory environment as the load gradually increases. In an actual application, the slab is under varying loading. A feature-based unsupervised machine learning algorithm was trained to classify each AE hit as representing friction emissions  and tensile/shear cracks using the data obtained from the first slab. The repeatability of the machine learning algorithm was tested using the data obtained from the second slab. SHAP analysis was conducted to investigate the sensitivity of each AE feature to the crack width as an effective AE feature differentiation tool compared to conventional graphical correlation plots. It was shown that the AE sensors positioned at the mid-span of slabs can be used as an effective SHM solution to monitor (i) the initiation of tensile cracks using the machine learning developed in this study, (ii) sudden changes in structural response due to major damage, shifting of clusters from tensile crack to shear crack, and (iii) the progression of crack width by tracking cumulative AE duration using the sensor at the vicinity of the crack. To the best knowledge of the authors, this is the first study in the literature to implement AE in two BFRP reinforced slabs in large scale and apply SHAP analysis to determine the most correlated AE feature with a physical quality .AE features are extracted from time history AE signals to monitor the damage initiation and correlate them with the AE sources. The damage modes in concrete are identified using cluster analyses of two or more AE features in the multivariant domain. Shear and tensile cracks are separated using RA (depending on the rise time and amplitude), and AF (average frequency) scatter plots. RA is calculated by the ratio of rise time and amplitude. AF is defined as the ratio of AE counts and duration. The tensile crack appears with a shorter rise time and a larger amplitude during the fracture process, such that a lower RA value is obtained. Shear crack has a longer rise time and lower amplitude. The data points above the pre-defined diagonal line indicate the tensile crack, whereas the part below represents the shear crack. RA-AF analysis is adaptable to small-scale experiments . HoweverTwo slabs with different BFRP bar sizes summarized in Sand-coated BFRP bars, composed of 75% fiber content set in vinyl ester resin, were utilized as primary reinforcement . The tenThe slabs were loaded from one point using hydraulic actuators as shown in The parameters for hit definition time, hit lockout time, and peak definition time were fixed at their default values, which are 1000 μs, 1000 μs, and 200 μs respectively. Hit definition time determines the end of the AE signal (no more threshold crossing within the window of the hit definition time). Hit lockout time is employed to restrict the system from recording new AE signals, thereby reducing the chance of classifying reflected AE signals as new hits. Peak definition time is utilized to identify the maximum amplitude of an AE signal. While the AE sensors were positioned for developing two-dimensional source localization, the heterogeneity of concrete and the formation of multiple simultaneous cracks caused a complex data set for applying the conventional location algorithms ,44. TherThe slabs were instrumented with three vertical LVDT sensors positioned in the mid-span to measure the deflection of the slab. The pi-shaped displacement transducers were placed to monitor the crack-width growth at the bottom surface of the concrete, illustrated as crack meters in The AE data were evaluated to understand the earliest stage of crack initiation, the identification of damage modes using machine learning methods, and the AE features representing the crack width using SHAP analysis. While these values  can be measured in the laboratory environment through strain and displacement sensors at discrete points providing local information, the AE method provides an alternative as a non-intrusive solution for field deployment as a global monitoring approach. During the loading process, each slab underwent damage. After the tensile stress exceeded the modulus of rupture of concrete, the tensile crack began to form at the positive moment region as shown in The time of the first crack development in the positive moment region was evaluated by the load value corresponding to the bending initiation in the load–deflection curve obtained by LVDT as shown in The correlation between AE features and local strain response offers an alternative means of monitoring local damage when direct strain data are not available. In practical scenarios where strain sensing is absent, identifying the damage status using AE becomes valuable. By investigating the correlation between AE features and strain, the relationship between AE signals and the underlying damage in concrete can be leveraged. Using the data recorded by one sensor placed near the support and at the mid-span of both slabs with varying reinforcement ratios, a comparative study was carried out on the variance of AE hit with respect to strain level throughout the loading testing until concrete failed, as shown in As shown in Differentiating AE sources with distinct characteristics, such as rise time, frequency centroid, and peak frequency is achieved by the clustering method, which is based on separating the AE sources by grouping the AE signals with similar characteristics. The most common clustering method is k-mean cluster analysis, that divides a dataset into k distinct, non-overlapping subsets. It iteratively reassigns the data points to the nearest centroid and recalculates the centroid, ensuring the convergence to a solution. The k-mean method is particularly suitable for large datasets, taking advantage of its computation efficiency and simplicity. Herein, k-mean cluster analysis was implemented to fingerprint the AE clusters  following the training and the testing of data collected from slab 1 and slab 2, respectively. As the AE features depend on the source–sensor distance, the algorithm was designed for one sensor (sensor 5) as the training set. The dataset was selectively chosen in order to minimize the impact of a single cluster dominating the majority of the signals. The first 180 s of data (noted as 20 s after the tensile crack appears observed in the load–deflection curve) and the last 20 s of data (mainly the shear crack) were selected to apply unsupervised machine learning. The algorithm was further validated by utilizing the sensor 6 data, which was also placed close to mid-span. Before conducting the cluster analysis, three preprocessing steps were applied. First, nine AE features, namely, counts, energy, absolute energy, rise time, initiation frequency, peak frequency, frequency centroid, amplitude, and duration, were selected to represent each AE signal. Then, the AE features were normalized with unit variance to reduce the influence of feature units in the cluster analysis. The unit variance normalization addresses the heterogeneities of units in the data and aligns with best practices for handling diverse units in the principal component analysis. The final preprocessing step, as depicted in The last step before cluster analysis was to determine the ideal number of clusters within the data set by the silhouette value, which is employed to evaluate the agreement of an object with the other elements in its own cluster compared to those in other clusters ,46,47. Tx-axis, and PCA-1 and PCA-3 are selected as the other axes to achieve the best separation and optimal visualization. The clusters possess varying characteristics regarding both time and frequency features, as indicated in terature ,49,50. TTo better understand the relationship between each cluster and its corresponding AE source and evaluate the robustness of this clustering model, a supervised machine learning method using the k-nearest neighbor (K-NN) method  was applThe machine learning model was validated using sensor 6 data, positioned at the first slab’s mid-span. The machine learning model was further applied to the slab 2 data recorded by sensor 5. As shown in As a summary, When the crack width reaches 0.3 mm, it is classified as a structural crack that leads to a decrease in the structural capacity . In thisSHAP analysis was implemented to quantify the contributions of the selected AE features to the crack width. The SHAP value is based on the game’s theoretically optimal Shapley values. The SHAP method explains the prediction of an instance by computing each feature to the prediction. To quantify the contribution of individual features, the SHAP value is calculated by first examining all possible combinations of features. It then measures how much the prediction changes when the feature of interest is included versus when it is excluded. The model ensures the feature contributions are fairly allocated, adding up to the total change in prediction. This helps to understand the distinct role each feature plays in the prediction. A detailed conceptual explanation of SHAP can be found in . Herein,The linear relationships between all eight cumulative AE features and crack width data were quantified using the Pearson correlation coefficient, which is given by the following equation : (1)rfeaThe SHM applications to transfer from laboratory to field can be influenced by environmental factors and traffic noise. Since the AE method relies on high-frequency elastic waves, it is generally not affected by temperature changes. Concrete structures, being relatively more rigid than steel structures, tend to exhibit lower friction emissions. In a typical monitoring application, it is possible to determine the background noise characteristics during the initial stages of the measurement. When using the same type of AE sensor and similar sensor distance to the expected crack regions as reported in this study, the AE characteristics of cracks are expected to be similar. Therefore, the cluster features representing tensile and shear cracks can be transferred to the field. The secondary emissions representing noise cluster need to be identified at the initial stages of the field measurement. The AE method can be utilized for both global and local monitoring purposes. In the case of global monitoring, sensors are strategically placed across the structure to encompass the entire system. On the other hand, local monitoring is typically employed to track the progression of known defects or monitor critical regions. The relationship between AE duration and crack width proposed in this study specifically focuses on local monitoring.The load–deflection and load–strain curves were good indicators of damage initiation, as corroborated by visual inspection. Prior to the crack initiation, the load–deflection behaviors of the two slabs exhibited similar patterns, however, once the tensile cracks formed, the behaviors varied with respect to the reinforcement ratio. The AE behavior changed accordingly. The slab with the higher reinforcement ratio released fewer AE hits due to stiffer behavior. The reinforcement ratio plays an important role when the cumulative AE hits or energy curves are used to differentiate the damage progression. The relationship between AE features and the local strain was found to provide damage information in the absence of strain data. Mid-span sensor locations exhibited higher AE activity than support regions, as expected, due to their proximity to the tensile crack source.A methodology integrating cluster analysis and the k-nearest neighbor (K-NN) algorithm was presented for predicting the required principal component axes and clusters within data sets. The machine learning model’s validity was evaluated using multiple data sets to detect cracking types and progression toward failure. A distinct characteristic between tensile and shear cracks was shown to detect the failure progression. In a realistic environment where the load is not gradually increased to form damage, the machine learning model developed in this study can differentiate AE sources and identify the progression of cracking from tensile to shear. The accuracy of the K-NN model trained on the first slab was 99.2% in predicting three clusters .SHAP (SHapley Additive exPlanations) analysis was shown as an effective signal processing tool to identify the most sensitive AE feature to the physical variable. The AE duration was determined as the feature related to the crack width. This is explained by the source function behavior that longer crack growth has a longer duration of source function that contributes to the duration of the AE signal as the AE signal is the convolution of source and medium transfer functions. The Pearson correlation coefficient of AE duration related to crack width achieves 0.9899. The approach is applicable for local monitoring to track the progression of known defects or to evaluate the damage state of critical regions.In this study, the AE method was demonstrated as a viable SHM solution for detecting the initiation and progression of damage in BFPR-reinforced slabs. The main conclusions of this study are as follows:In summary, the output of AE sensors at the mid-span of BFRP slabs could be used to monitor the initiation of tensile cracks, sudden changes in structural response due to major damage, and the progression of crack width (with AE duration). It is important to note that the absolute and cumulative AE values are influenced by the reinforcement ratio in addition to the AE sensor type, distance to AE source, and data acquisition variables. However, the machine learning model developed in this study reduces their influences by normalizing data and measuring the distances between cluster centers as a relative variable. Future investigation would involve analyzing the effectiveness and reliability of these AE time-frequency features in detecting and evaluating the damage in BFRP-reinforced concrete ,56.