Research on gearbox bearing fault diagnosis based on SSA-VMD-CNN algorithms

Research on gearbox bearing fault diagnosis based on SSA-VMD-CNN algorithms | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Article Research on gearbox bearing fault diagnosis based on SSA-VMD-CNN algorithms Jianrui Zhang, Jinchang Guo, Baolong Geng, Haoyang Song, Yue Zhang, and 1 more This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-4137819/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract Gearbox bearings are crucial components in numerous mechanical systems. These gearboxes typically operate in environments characterized by significant noise, causing their fault signals to be obscured by background interference, vibrations, and signals from other mechanical parts. This interference complicates the accurate extraction and diagnosis of fault characteristics from complex data. To address this challenge, we propose a novel bearing fault diagnosis model that integrates Variational Mode Decomposition (VMD), Convolutional Neural Network (CNN), and advanced optimization algorithms. Initially, the Squirrel Search Algorithm (SSA) is employed to automatically optimize VMD parameters, enabling efficient extraction of denoised signal features. VMD decomposes vibration signals into multiple Intrinsic Mode Functions (IMFs), which are then analyzed and reconstructed using kurtosis and cross-correlation criteria. Subsequently, these processed signals serve as input feature vectors for the CNN model, facilitating both training and testing phases. The model is designed to construct a singular value vector matrix that reflects the current fault state based on the position of each submatrix. Simulation verification of our model demonstrates an accuracy exceeding 95% in bearing fault diagnosis, a substantial improvement over traditional methods. This advancement offers a new perspective for the health monitoring and maintenance of critical mechanical equipment, such as gearboxes. It holds significant potential for application in intelligent manufacturing and automated monitoring systems in the future. Physical sciences/Engineering Physical sciences/Engineering/Mechanical engineering Fault diagnosis variational mode decomposition convolutional neural network vibration signal bearing Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 Figure 10 Figure 11 Figure 12 Figure 13 Figure 14 Figure 15 Figure 16 Figure 17 Figure 18 Figure 19 Figure 20 Introduction Gearbox bearing fault detection technology plays a critical role across a broad spectrum of industries, including automotive, aerospace, energy, manufacturing, and agricultural machinery [ 1 ]. The inherent complexity of operating environments, characterized by high levels of noise and interference, complicates the accurate extraction and diagnosis of fault signals from gearboxes. Such complexities necessitate the advancement of fault diagnosis methodologies to meet these challenges [ 2 ]. Bearings are essential components of mechanical equipment, and their malfunction can significantly degrade equipment performance or cause complete operational failures [ 3 ]. Timely fault diagnosis is imperative for maintaining safe operation, particularly in critical applications like train traction motors. These motors, essential for power drive and brake power storage during train operation, are prone to various failures due to fluctuating speeds and loads [ 4 – 6 ]. Bearing faults, including those in the inner ring, outer ring, and rolling elements, can have severe implications for train safety and operational integrity [ 7 , 8 ]. The operational environment of rolling bearings in mechanical systems is inherently complex, often involving high speeds and heavy loads. Unaddressed bearing failures can escalate over time, leading to serious consequences and substantial losses [ 9 ]. Consequently, the development of advanced and intelligent bearing fault diagnosis methods is of paramount engineering significance, especially with the increasing modernization and automation of industrial production [ 10 ]. Traditional fault diagnosis methods, such as vibration analysis and acoustic emission technology, often fall short under complex signal and noise conditions [ 11 ]. This limitation underscores the necessity for more effective and accurate fault diagnosis techniques. Variational Mode Decomposition (VMD) has emerged as a prominent solution in mechanical fault diagnosis, effectively decomposing vibration signals into band-limited intrinsic mode functions (BLIMFs) with specific sparsity in the time-frequency domain [12,213]. Unlike empirical mode decomposition (EMD), VMD does not require preset parameters, offering greater flexibility and higher frequency resolution. It efficiently identifies fault characteristics within noisy backgrounds and is particularly adept at processing nonlinear and non-stationary signals [ 14 ]. Recent studies have integrated VMD with machine learning algorithms, including deep learning and transfer learning, to enhance diagnostic accuracy and efficiency [ 15 ]. Convolutional Neural Networks (CNNs) have demonstrated exceptional capability in extracting meaningful features from complex data, such as the nonlinear and non-stationary vibration signals of gearboxes under varying conditions [ 16 ]. The application of CNNs in fault diagnosis, combined with their rapid response capabilities, makes them ideally suited for real-time monitoring systems [ 17 ]. The structure of this paper is organized as follows: Chap. 2 delineates the theoretical framework underpinning the proposed method. Chapter 3 details the methodology employed for the collection of test data. In Chap. 4, we conduct an analysis to assess the efficacy of the method. Finally, Chap. 5 provides a comprehensive summary of the research presented in this paper. Theoretical Model 1.1 VMD algorithm principle VMD is a non-recursive variational mode decomposition method proposed by Dragomiretskiy and Zosso in 2014. Its core idea is to decompose the signal into a discrete number of band-limited intrinsic mode functions (BLIMF) with specific sparsity [ 18 ]. The VMD decomposition method decomposes complex vibration signals into intrinsic mode functions with physical meaning by constructing variational problems and iterative solutions [ 19 ]. Specific steps are as follows: (1) Problem of Structural VariationThe challenge of structural variation entails the decomposition of the original input signal \(x\left(t\right)\) into K Intrinsic Mode Function (IMF) components, denoted as \(u\left(t\right)\) . Subsequently, Hilbert transform is applied to demodulate each component u(t) to derive its envelope signal. The comparison is then made with the estimated center frequency \({\omega }_{k}\) mixing. This process adheres to the constraint that the sum of each \(u\left(t\right)\) component is equivalent to the original signal \(x\left(t\right)\) . The structural variation problem is thus formulated as follows: $$\left\{\begin{array}{c}\underset{\left\{{u}_{k}\right\},\left\{{\omega }_{k}\right\}}{{min}}\left\{\left.\sum _{k=1}^{K}{‖{\partial }_{t}\left[\left(\delta \left(t\right)+\frac{j}{\pi t}\right)*{u}_{k}\left(t\right)\right]{e}^{-j{\omega }_{k}t}‖}_{2}^{2}\right\}\right.\\ s.t. \sum _{k=1}^{K}{u}_{k}=x\left(t\right)\end{array} \left(1\right)\right.$$ Where, \({u}_{k}\) is each mode, \({\omega }_{k}\) is the center frequency of each mode, K is the number of decomposition layers, \({\partial }_{t}\) is the partial derivative of t, and \(\delta \left(t\right)\) is the impact function. (2) Solve variational problems The Lagrangian multiplication operator \(\lambda \left(t\right)\) and the quadratic penalty factor α are added to the constrained variation problem to transform it into an unconstrained variation problem, and its expression is: $$L\left(\left\{{u}_{k}\right\},\left\{{\omega }_{k}\right\},\lambda \right)=\alpha \sum _{k}{‖{\partial }_{t}\left[\left(\delta \left(t\right)+\frac{j}{\pi t}\right)*{u}_{k}\left(t\right)\right]{e}^{-j{\omega }_{k}t}‖}_{2}^{2}+{‖x\left(t\right)-\sum _{k=1}^{K}{u}_{k}\left(t\right)‖}_{2}^{2}+⟨\lambda \left(t\right),x\left(t\right)-\sum _{k=1}^{K}{u}_{k}\left(t\right)⟩ \left(2\right)$$ The saddle point in Eq. (2), which is the optimal solution of Eq. (1), is obtained through the Alternating Direction Method of Multipliers (ADMM). The solution steps are: Step 1: Initialize \(\left\{{\widehat{u}}_{k}^{1}\right\}\) , \(\left\{{\widehat{\omega }}_{k}^{1}\right\}\) , \({\widehat{\lambda }}^{1}\) , n; Step 2: Write only loop n = n + 1; Step 3: For all \(\omega >0\) , update \({\widehat{u}}_{k}\) , \(k\in \left\{\text{1,2},\cdots ,K\right\}\) ; $$\begin{array}{c}{\widehat{u}}_{k}^{n+1}\left(\omega \right)\leftarrow \frac{\widehat{x}\left(\omega \right)-\sum _{i<k}{\widehat{u}}_{i}^{n+1}\left(\omega \right) -\sum _{i<k}{\widehat{u}}_{i}^{n}\left(\omega \right)+\frac{{\widehat{\lambda }}^{n}\left(\omega \right)}{2}}{1+2\alpha {\left(\omega -{\omega }_{k}^{n}\right)}^{2}} \left(3\right)\end{array}$$ Step 4: Update \({\omega }_{k}\) $$\begin{array}{c}{\omega }_{k}^{n+1}\leftarrow \frac{{\int }_{0}^{\infty }\omega {\left|{\widehat{u}}_{k}^{n+1}\left(\omega \right)\right|}^{2}d\omega }{{\int }_{0}^{\infty }{\left|{\widehat{u}}_{k}^{n+1}\left(\omega \right)\right|}^{2}d\omega },k\in \left\{1,K\right\} \left(4\right)\end{array}$$ Step 5: Update \(\lambda\) $$\begin{array}{c}{\widehat{\lambda }}^{n+1}\left(\omega \right)\leftarrow {\widehat{\lambda }}^{n}\left(\omega \right)+\tau \left[\widehat{x}\left(\omega \right)-\sum _{k}{\widehat{u}}_{k}^{n+1}\left(\omega \right)\right] \left(5\right)\end{array}$$ Step 6: Repeat steps 2 to 5 until formula (6) is satisfied and the iteration is stopped, that is, K IMF components are obtained. $$\begin{array}{c}\raisebox{1ex}{$\sum _{k}{‖{\widehat{u}}_{k}^{n+1}-{\widehat{u}}_{k}^{n}‖}_{2}^{2}$}\!\left/ \!\raisebox{-1ex}{${‖{\widehat{u}}_{k}^{n}‖}_{2}^{2}$}\right.<\epsilon \left(6\right)\end{array}$$ The primary merit of Variational Mode Decomposition (VMD) lies in its adaptive signal decomposition capability without necessitating prior knowledge [ 20 ]. The algorithm automatically estimates both the number (K) of modes and their respective center frequencies. Remarkably robust against noise, VMD yields Band-Limited Intrinsic Mode Functions (BLIMF) with compact support in the spectral domain, effectively capturing the inherent natural oscillation modes within the signal. Unlike the Empirical Mode Decomposition (EMD) method, VMD obviates the need for complex envelope estimation and extreme value tracking, rendering its calculations more straightforward [ 21 ]. In essence, the VMD algorithm tackles constrained variation problems through an iterative process grounded in Alternating Direction Multipliers (ADMM). Key operations within each iteration encompass updating the BLIMF via a Wiener filter with an offset frequency, adjusting the Lagrange multiplier, verifying convergence, and updating the estimated center frequency. Iterations persist until the residuals converge to a monotonic function. Ultimately, the signal undergoes decomposition into a series of BLIMFs and a residual. 1.2 Squirrel optimization algorithm model (SSA Model) In the Variational Mode Decomposition (VMD) computation, the decomposition number (K) and the penalty factor (α) are pivotal parameters influencing the performance and effectiveness of the decomposition [ 22 ]. The value of K directly dictates the quantity of decomposed modal components, whereas the value of α affects their bandwidth, significantly impacting decomposition performance. Currently, there is a lack of objective criteria for setting these parameters. To address this, the present study introduces the Squirrel Search Algorithm (SSA) for the adaptive optimization of VMD parameters [ 23 ]. The Squirrel Search Algorithm (SSA) exhibits notable proficiency in search capability and accuracy when addressing complex problems within a search space. Inspired by the gliding behavior of squirrels, which enables them to evade predators despite their inability to fly, the SSA algorithm simulates this adaptive strategy. The algorithm's search process mimics squirrels' foraging behavior, wherein they locate food by moving between various trees. This method effectively facilitates exploration across different areas of a metaphorical forest, represented by the varying positions of the squirrels within the search space [ 24 ]. Assuming the number of squirrels is n , the position the squirrel moves is determined by a vector, and its position is randomly initialized within the boundary range. $$\begin{array}{c}FS=\left[\begin{array}{cccc}F{S}_{\text{1,1}}& F{S}_{\text{1,2}}& \cdots & F{S}_{1,D}\\ F{S}_{\text{2,1}}& F{S}_{\text{2,2}}& \cdots & F{S}_{1,d}\\ ⋮& ⋮& ⋮& ⋮\\ F{S}_{n,1}& F{S}_{n,2}& \cdots & F{S}_{n,d}\end{array}\right] \left(7\right)\end{array}$$ \(F{S}_{n,d}\) is the value of the n -th mouse in the d -dimension, and the initial position of the squirrel in the forest is: $$\begin{array}{c}F{S}_{i}=F{S}_{L}+U\left(\text{0,1}\right)\times \left(F{S}_{U}-F{S}_{L}\right) \left(8\right)\end{array}$$ \(F{S}_{U}\) and \(F{S}_{L}\) are the upper and lower bounds of squirrel movement, and \(U\left(\text{0,1}\right)\) is a random number [0,1]. The grade of food source is represented by the fitness of each squirrel location, and the fitness value is calculated and classified in ascending order. The position with the least fitness is: the best food source ① pecan, the next three are normal food source ② oak, and the other locations have no food source ③ common tree [ 25 ]. Depending on the probability of the presence of its natural enemies, \({P}_{dp}\) squirrels update the location of their movement. Glide Path 1: ②→① $$\begin{array}{c}F{S}_{at}^{t+1}\left\{\begin{array}{c}F{S}_{at}^{t}+{d}_{g}\times {G}_{c}\times \left(F{S}_{ht}^{t}-F{S}_{at}^{t}\right),{R}_{1}\ge {P}_{dp}\\ \text{R}\text{a}\text{n}\text{d}\text{o}\text{m}\text{l}\text{o}\text{c}\text{a}\text{t}\text{i}\text{o}\text{n},{ R}_{1}<{P}_{dp}\end{array}\right. \left(9\right)\end{array}$$ Glide Path 2:③→② $$\begin{array}{c}F{S}_{nt}^{t+1}\left\{\begin{array}{c}F{S}_{nt}^{t}+{d}_{g}\times {G}_{c}\times \left(F{S}_{at}^{t}-F{S}_{nt}^{t}\right),{R}_{2}\ge {P}_{dp}\\ \text{R}\text{a}\text{n}\text{d}\text{o}\text{m}\text{l}\text{o}\text{c}\text{a}\text{t}\text{i}\text{o}\text{n},{ R}_{2}<{P}_{dp}\end{array}\right. \left(10\right)\end{array}$$ Glide Path 3:③→① $$\begin{array}{c}F{S}_{nt}^{t+1}\left\{\begin{array}{c}F{S}_{nt}^{t}+{d}_{g}\times {G}_{c}\times \left(F{S}_{ht}^{t}-F{S}_{nt}^{t}\right),{R}_{3}\ge {P}_{dp}\\ \text{R}\text{a}\text{n}\text{d}\text{o}\text{m}\text{l}\text{o}\text{c}\text{a}\text{t}\text{i}\text{o}\text{n},{ R}_{1}<{P}_{dp}\end{array}\right. \left(11\right)\end{array}$$ \({d}_{g}\) is the random glide distance, \({R}_{1}{R}_{2}{R}_{3}\) is the random number in the range of [0,1], \(F{S}_{at}^{t}\) is the squirrel's position on the oak, \(F{S}_{ht}^{t}\) is the squirrel's position on the mountain tree, \(F{S}_{nt}^{t}\) is the squirrel's position on the common tree, and \({G}_{c}\) is the slip coefficient. Seasonal changes affect squirrels' foraging activities, and seasonal changes are used to prevent algorithms from falling into local optimality. $$\begin{array}{c}{S}_{c}^{t}=\sqrt{{\sum }_{z=1}^{3}{\sum }_{k=1}^{d}{\left(F{S}_{at,k}^{t,z}-F{S}_{ht,k}\right)}^{2}} \left(12\right)\end{array}$$ $$\begin{array}{c}{S}_{min}=\frac{10E-6}{{365}^{2.5t/{t}_{m}}} \left(13\right)\end{array}$$ \(t\) and \({t}_{m}\) are the current value and the maximum iteration value respectively, \({S}_{min}\) is the minimum value of the seasonal constant, \({S}_{c}^{t}\) is the seasonal constant, and the seasonal change condition is \({S}_{c}^{t}<{S}_{min}\) . If this condition is met, the position of common squirrels changes randomly. $$\begin{array}{c}F{S}_{nt,i}^{t+1}=F{S}_{i,L}+Levy\left(F{S}_{i,U}-F{S}_{i,L}\right) \left(14\right)\end{array}$$ \(F{S}_{i,U}\) and \(F{S}_{i,L}\) are the upper and lower bounds of squirrel movement, and Levy is the Levi distribution, effectively searching globally to find a new location that is optimal from the current location. The process of optimizing VMD parameters based on squirrel search algorithm is as follows [ 26 ]. Step 1: Set SSA parameters, including population size, number of iterations and upper and lower bounds of the optimization range, and initialize the population position using Eq. (7). Step 2: VMD decomposition of the power sequence was performed according to the position of each squirrel (K,a), and the fitness of each individual was calculated and sorted according to Eq. (8). The squirrels were assigned to hickory trees (optimal solution), oak trees (suboptimal solution), and ordinary trees (general solution) in order to preserve the locations of the most individual squirrels. Step 3: Update individual squirrel locations. Step 4: Update the seasonal constant, when S_c^t < S_min, move the individuals in the ordinary tree randomly. Step 5: Repeat Step 2 for the newly generated location to update the optimal solution. Step 6: Repeat steps 3 to 5 until the maximum number of iterations is reached to stop the optimization and output the optimal parameters and fitness values. 1.3 CNN algorithm principle CNN is a feedforward neural network with a deep structure, including input layer, convolutional layer, pooling layer, fully connected layer and output layer [ 27 ]. The convolution layer extracts local features from the fault data, and the feature output of the convolution layer is obtained by activation function after convolution calculation. The feature expression is: $$\begin{array}{c}{x}_{j}^{l}=f\left(\sum _{i\in {M}_{j}}{x}_{i}^{l-1}*{k}_{ij}^{l}+{b}_{j}^{i}\right) \left(15\right)\end{array}$$ Where: \({k}_{ij}^{l}\) is the convolution kernel weight matrix; \(f\) is a nonlinear activation function; \(l\) is the l -layer in the network; \({M}_{j}\) is the input feature map. The pooling layer is divided into maximum pooling and average pooling, which can retain the main features while reducing the network parameters and calculation amount to avoid the occurrence of overfitting [ 28 ]. The fully connected layer further extracts the output features of the pooled layer and inputs them into the classifier for classification. Data Acquisition 2.1. Gearbox structure The gearbox fault diagnosis platform is a device used to detect and diagnose gearbox faults. This platform can simulate the operating status of the gearbox and monitor the operating status of the gearbox in real time by monitoring various indicators such as vibration and temperature of the gearbox, so as to detect faults and perform diagnosis in a timely manner [ 29 ]. The platform can collect various indicator data such as vibration, temperature, and current of the gearbox to facilitate fault diagnosis. The experimental platform consists of: a 2-horsepower drive motor, a torque transmission device in the middle and a tested gearbox [ 30 ]. Acceleration sensors are installed at the base, fan end, and drive end of the experimental motor respectively to obtain base acceleration data, fan end acceleration data, and drive end acceleration data. Figure 1 shows the overall design of the fault detection platform. 2.2. Data description To leverage neural network algorithms effectively, particularly convolutional neural networks (CNNs), it is imperative to amass a substantial corpus of images for image recognition tasks. A quintessential example is the MNIST dataset, which comprises 60,000 training images and 10,000 test images of handwritten digits. For comprehensive training of the CNN model, we curated an extensive collection of training samples derived from accelerometer data in a motor-driven mechanical system, recorded at a 12 kHz sampling frequency by the Case Western Reserve University (CWRU) Axis Data Center [ 31 ]. This dataset encapsulates four bearing fault types: normal, ball, inner ring, and outer ring faults, each manifesting in three distinct fault diameters: 0.007 inches, 0.014 inches, and 0.021 inches. This configuration results in a total of 10 unique fault scenarios. Each sample within the experiment comprises 2048 data points, facilitating the straightforward implementation of the Fast Fourier Transform (FFT) in the baseline algorithm. The datasets, labeled A, B, and C, each encompass 6,600 training samples and 250 test samples, distributed across 10 failure conditions at load capacities of 1, 2, and 3 horsepower, respectively. Dataset D extends this with 19,800 training and 750 test samples, covering the same three load conditions. To augment the training dataset, we employed overlapping samples, ensuring, however, that no overlap occurred within the test samples. Table 1 provides a comprehensive overview of these datasets. Table 1 Description of rolling element bearing datasets. Fault Location None Ball Inner Race Outer Race Load Category Labels 1 2 3 4 5 6 7 8 9 10 Fault diameter (inch) 0 0.007 0.014 0.021 0.007 0.014 0.021 0.007 0.014 0.021 Dataset A no. Train Test 660 25 660 25 660 25 660 25 660 25 660 25 660 25 660 25 660 25 660 25 1 Dataset B no. Train test 660 25 660 25 660 25 660 25 660 25 660 25 660 25 660 25 660 25 660 25 2 Dataset C no. train test 660 25 660 25 660 25 660 25 660 25 660 25 660 25 660 25 660 25 660 25 3 Dataset D no. Train Test 1980 75 1980 75 1980 75 1980 75 1980 75 1980 75 1980 75 1980 75 1980 75 1980 75 1,2,3 Fault Diagnosis Instance 3. Fault Diagnosis Instance 3.1. Fault detection and noise reduction based on VMD After data processing, two input signals were selected for display. Figure 2 is a gear without fault, and Fig. 3 is a faulty input signal. It can be clearly seen that although the first signal has no special pattern, it is relatively stable. The second signal is faulty and has significant noise that needs to be dealt with. Figure 4 is the signal after VMD noise reduction. The blue color in the figure is the input signal, and the red color is the signal output after the noise reduction is successful. The noise reduction effect is good. Figure 5 shows the frequency domain analysis of the four modes after VMD decomposition. The black dotted line is the signal added after the four decompositions. The result is relatively stable. Figure 6 shows the decomposed four signals. Figure 7 shows the frequency domain representation of the input signal. Figure 8 is a comparison, the horizontal axis is the number of training rounds, and the vertical axis is the accuracy rate. The red is the training set data, and the blue is the test set data. In the figure, the data accuracy of the training set is significantly higher than that of the test set, and the overfitting phenomenon is obvious. However, after VMD noise reduction, the fitting phenomenon is greatly alleviated. Figure 8 and Fig. 9 show the classification accuracy and loss function graphs of VMD denoising and VMD(called SSA-VMD) denoising after VMD and SSA parameter optimization respectively. The accuracy of VMD at 20 iterations is 80%, while the accuracy of SSA-VMD at 20 iterations is 87%, and SSA-VMD achieves higher accuracy with fewer iterations. Figure 9 shows the number of training rounds on the horizontal axis and the loss function on the vertical axis. 3.2. Gearbox fault diagnosis based on VMD and CNN 3.2.1 Optimization of CNN model Figure 10 , Fig. 11 and Fig. 12 shows the number of training rounds on the horizontal axis and the accuracy rate on the vertical axis. Red is the training set data, blue is the test set data. Figure 10 is the original CNN model without any optimization, Fig. 11 is the data after noise reduction only with SSA-VMD, and Fig. 12 is the final model after optimization with SSA-VMD and CNN (named SSA-SVM-CNN). It can be seen that with continuous optimization, not only the overfitting problem is solved, but also the accuracy rate is improved, as shown in Table 2 . Table 2 Correct rate data table Method CNN SSA-VMD SSA-VMD-CNN Accuracy 81% 87% 95% 3.2.2 Result visualization and analysis To elucidate the functionality of the proposed SSA-VMD-CNN network model, this study presents graphical representations of various model parameters—including accuracy rate, loss function, and confusion matrix—over 200 operational iterations. Precision, defined as the ratio of true positive samples to all predicted positive samples, is crucial for evaluating model performance. Recall, or the proportion of actual positive samples correctly identified, is equally significant. The F1-score, representing a harmonious mean of precision and recall, proves particularly useful in balancing these metrics, especially in datasets exhibiting imbalance. Figure 13 delineates the precision, recall, and F1-scores for each fault category within the network model. Notably, categories 0, 2, and 7 exhibit lower accuracy. Category 0 corresponds to rolling element faults with a diameter of 1.1778 mm, Category 2 to rolling element faults of 0.5334 mm, and Category 7 to outer ring gear faults, also of 0.5334 mm. This trend suggests a reduced efficacy of the model in identifying rolling element failures, indicating a potential improvement by augmenting rolling element data within the training set. Conversely, the model demonstrates exceptional accuracy in other categories, achieving a 100% recognition rate in four fault categories. This high accuracy offers the prospect of data reduction, thereby conserving resources and reducing operational time. Figure 14 shows the loss function of the model in this paper. The final loss function is around 0.25. If the training continues, the loss function can be reduced and the accuracy can be increased. In order to show the classification effect more clearly, the confusion matrix of diagnosis results is shown in Fig. 15 . The horizontal axis is the type recognized by the computer, and the vertical axis is the real category. It is obvious that group 1 has more errors in judgment. The other groups have very few errors. 3.3. Comparison of CNN Convolutional neural network with other models In this paper, the CNN model is compared with some classical algorithms. All models are denoised by SSA-VMD to ensure the fairness of the comparison. 3.3.1 Comparison between CNN and GRU algorithm GRU stands for Gated Recurrent Unit, which is a recurrent neural network (RNN) architecture introduced by Kyunghyun Cho et al. in 2014. GRU is similar to long short-term memory (LSTM) with forget gates, but it has fewer parameters than LSTM because it lacks an output gate. GRU uses a gating mechanism to control and manage the flow of information through the network. It consists of only three gates, namely reset gate, update gate and new memory gate. The reset gate determines how much past knowledge needs to be transferred to the future, while the update gate determines how much new information needs to be stored in memory. GRU shows similar performance to LSTM on some tasks, such as chord music modeling, speech signal modeling, and natural language processing. Figure 16 serves as a comparative examination, where the horizontal axis signifies the number of training rounds, and the vertical axis denotes the accuracy rate. The red curve represents data trained with specific parameters, while the blue curve encapsulates parameters affiliated with a test set unexposed to the model. Evidently, the accuracy of the training set, as delineated in the figure, markedly surpasses that of the test set, elucidating a conspicuous manifestation of overfitting. Positioned on the left is the Convolutional Neural Network (CNN) model, while on the right resides the Gated Recurrent Unit (GRU) model. Notably, the CNN model exhibits superior performance. The CNN model not only demonstrates enhanced model fitting but also maintains a stable efficacy. Despite the approximately 90% accuracy of the GRU on the right, discernible fluctuations and fitting issues persist within the GRU model. As depicted in Fig. 17 , the loss function for the Convolutional Neural Network (CNN) stands at 0.5, whereas the Gated Recurrent Unit (GRU) registers a loss function approximating 0. The discernible divergence in these loss functions implies that the CNN model retains significant room for further optimization within the 0.5 space, whereas the GRU model exhibits limited optimization potential. In summation, the superiority of the CNN model is evident. 3.3.2 Comparison between CNN and LSTM algorithms LSTM, an acronym signifying Long Short-Term Memory, represents an artificial neural network employed within the realms of artificial intelligence and deep learning. Operating as a subtype of recurrent neural networks (RNNs), LSTM specializes in discerning sequential dependencies pertinent to sequence prediction tasks. Diverging from conventional feedforward neural networks, LSTMs incorporate feedback connections. Notably, LSTMs excel in processing extensive sequential data, endowing RNNs with a durable short-term memory capable of spanning thousands of time steps, hence the nomenclature "long short-term memory." Widespread adoption of LSTM is evident in natural language processing and diverse domains, solidifying its status as a cornerstone in classical machine learning models. Figure 18 delineates an accuracy comparison, with the horizontal axis denoting the number of training rounds and the vertical axis representing accuracy. The red trace corresponds to the training data and the associated model parameters, while the blue trace pertains to parameters from the test set. Notably, while Long Short-Term Memory (LSTM) effectively mitigates the issue of overfitting, its accuracy lags approximately 20% behind that of Convolutional Neural Network (CNN). This performance discrepancy underscores that, in the context of accuracy, LSTM proves less apt for gearbox fault detection, advocating a preference for CNN-based approaches in this application domain. 3.3.3 Comparison between CNN and SVM algorithms The Support Vector Machine (SVM) stands as a supervised machine learning algorithm proficient in classification and regression tasks. SVM discerns a hyperplane within the training data, optimizing the separation of distinct classes. Rooted in the statistical learning framework and VC theory, SVM serves as a robust predictive method for both two-group and multi-class classification challenges. Functioning as a non-probabilistic binary linear classifier, SVM necessitates labeled data for effective training. Its applicability extends across diverse domains, encompassing text classification, image classification, and bioinformatics. Precision, denoting the proportion of correctly identified positive samples among predicted positives, and recall, representing the ratio of correctly predicted positive samples to the total actual positive samples, are pivotal metrics. The F1-score, a harmonic mean of precision and recall, provides a comprehensive evaluation of their balance. Particularly advantageous in managing imbalanced datasets, F1-score contributes nuanced insights. While accuracy remains a crucial metric, the F1-score's incorporation of precision and recall renders it a more insightful indicator, especially in scenarios involving imbalanced datasets. Figures 19 and 20 illustrate the three key indicators—Precision, Recall, and F1-score—depicting the classification performance of Convolutional Neural Network (CNN) and Support Vector Machine (SVM) across ten distinct fault conditions. Notably, with the exception of fault conditions in groups 0 and 3, where CNN exhibits a lower accuracy compared to SVM, CNN consistently outperforms SVM across other data categories. Particularly noteworthy is the second data type, where SVM's accuracy is alarmingly null, contrasting sharply with CNN's commendable accuracy of 87%. This underscores CNN's superior efficacy in diverse scenarios. In summation, our proposed CNN model, detailed in this chapter, emerges as the preeminent performer, boasting the highest accuracy in the test set and superior effectiveness. Following closely is the Gated Recurrent Unit (GRU) model, surpassing the performance of traditional models like SVM and Random Forest (RF). Through the above comparative analysis, the fault diagnosis accuracy of various methods is shown in Table 3 . The combination of parameter optimized VMD and CNN algorithm has significant advantages for the fault diagnosis of gearbox. Table 3 Fault diagnosis accuracy of CNN and other algorithms Method SSA-VMD-CNN SSA-VMD-GRU SSA-VMD-LSTM SSA-VMD-SVM Accuracy 95% 91% 70% 65% Conclusion In this study, we introduce a novel bearing fault diagnosis model that effectively detects faults in gearboxes operating amidst complex noise environments. The integration of Variational Mode Decomposition (VMD), Convolutional Neural Network (CNN), and Squirrel Search Algorithm (SSA) within this model enables the efficient extraction of pivotal signal features from noise, thereby facilitating accurate fault diagnosis. The employment of VMD in the model allows the decomposition of vibration signals into multiple Intrinsic Mode Functions (IMFs). These IMFs are subsequently analyzed and reconstructed based on kurtosis and cross-correlation criteria, yielding a more refined and precise signal foundation for fault detection. Optimizing VMD parameters using SSA further amplifies the model’s capabilities in noise reduction and feature extraction. The refined signals are then input into a CNN for training and testing, showcasing the model’s robustness and precision in handling intricate data sets. The model's efficacy is corroborated through simulations under various fault conditions, achieving an impressive accuracy rate exceeding 93%, markedly surpassing traditional fault diagnosis methodologies. This proposed bearing fault diagnosis model not only elevates the accuracy and efficiency of fault detection but also pioneers a novel approach for the health monitoring and maintenance of critical mechanical systems, such as gearboxes.5. Acknowledgments This work was supported in part by the Crossing Research Project of Longdong University, HXZK2318, partly by Natural Science Foundation of Gansu Province, 23JRRM737, partly by the Qingyang City Science and Technology Planning Project, QY-STK-2022B-151 and partly by the Longdong University Doctoral Fund, XYBYZK2301. Declarations 6. Data Availability All data generated or analysed during this study are included in this published article. Author Contribution Conceptualization, Jianrui Zhang and Yue Zhang; software, Jinchang Guo and Baolong Geng; validation, Haoyang Song; writing—original draft preparation, Rong Zhao. All the data used in the simulation analysis in this paper are available. All authors have read and agreed to the published version of the manuscript.All authors reviewed the manuscript. Data Availability The original experiments data was obtained from the accelerometers of the motor driving mechanical system at a sampling frequency of 12 kHz from the Case Western Reserve University (CWRU) Bearing Data Center. Lou, X.; Loparo, K.A. Bearing fault diagnosis based on wavelet transform and fuzzy inference. Mech. Syst. Signal Process. 2004, 18, 1077–1095. References Li, Y.; Xu, M.; Wei, Y.; Huang, W. A new rolling bearing fault diagnosis method based on multiscale permutation entropy and improved support vector machine based binary tree. Measurement 2016, 77, 80–94. Zhang X, Han P, Xu L, et al. Research on bearing fault diagnosis of wind turbine gearbox based on 1DCNN-PSO-SVM[J]. IEEE Access, 2020, 8: 192248–192258. Yu X, Li Z, He Q, et al. Gearbox fault diagnosis based on bearing dynamic force identification[J]. Journal of Sound and Vibration, 2021, 511: 116360. Zhou L, Duan F, Corsar M, et al. A study on helicopter main gearbox planetary bearing fault diagnosis[J]. 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Chen Z Q, Li C, Sanchez R V. Gearbox fault identification and classification with convolutional neural networks[J]. Shock and Vibration, 2015, 2015. Jin T, Yan C, Chen C, et al. Light neural network with fewer parameters based on CNN for fault diagnosis of rotating machinery[J]. Measurement, 2021, 181: 109639. Wang D, Guo Q, Song Y, et al. Application of multiscale learning neural network based on CNN in bearing fault diagnosis[J]. Journal of Signal Processing Systems, 2019, 91: 1205–1217. Wang X, Mao D, Li X. Bearing fault diagnosis based on vibro-acoustic data fusion and 1D-CNN network[J]. Measurement, 2021, 173: 108518. Deng H, Zhang W, Liang Z. Application of BP neural network and convolutional neural network (CNN) in bearing fault diagnosis[C]//IOP Conference Series: Materials Science and Engineering. IOP Publishing, 2021, 1043(4): 042026. Zheng X, Wu J, Ye Z. An end-to-end CNN-BiLSTM attention model for gearbox fault diagnosis[C]//2020 IEEE International Conference on Progress in Informatics and Computing (PIC). IEEE, 2020: 386–390. Pan H, He X, Tang S, et al. An improved bearing fault diagnosis method using one-dimensional CNN and LSTM[J]. Strojniski Vestnik/Journal of Mechanical Engineering, 2018, 64. Han S, Zhong X, Shao H, et al. Novel multi-scale dilated CNN-LSTM for fault diagnosis of planetary gearbox with unbalanced samples under noisy environment[J]. Measurement Science and Technology, 2021, 32(12): 124002. Jiang G, He H, Yan J, et al. Multiscale convolutional neural networks for fault diagnosis of wind turbine gearbox[J]. IEEE Transactions on Industrial Electronics, 2018, 66(4): 3196–3207. Chen Z, Gryllias K, Li W. Intelligent fault diagnosis for rotary machinery using transferable convolutional neural network[J]. IEEE Transactions on Industrial Informatics, 2019, 16(1): 339–349. Huang D, Zhang W A, Guo F, et al. Wavelet packet decomposition-based multiscale CNN for fault diagnosis of wind turbine gearbox[J]. IEEE Transactions on Cybernetics, 2021. Yao Y, Zhang S, Yang S, et al. Learning attention representation with a multi-scale CNN for gear fault diagnosis under different working conditions[J]. Sensors, 2020, 20(4): 1233. Peng D, Wang H, Liu Z, et al. Multibranch and multiscale CNN for fault diagnosis of wheelset bearings under strong noise and variable load condition[J]. IEEE Transactions on Industrial Informatics, 2020, 16(7): 4949–4960. Guo Z, Yang M, Huang X. Bearing fault diagnosis based on speed signal and CNN model[J]. Energy Reports, 2022, 8: 904–913. Chen Z, Liu C, Gryllias K, et al. Gearbox fault diagnosis using convolutional neural networks and support vector machines[C]//2019 27th European Signal Processing Conference (EUSIPCO). IEEE, 2019: 1–5. Chen Z, Gryllias K, Li W. Mechanical fault diagnosis using convolutional neural networks and extreme learning machine[J]. Mechanical systems and signal processing, 2019, 133: 106272. Jin T, Yan C, Chen C, et al. New domain adaptation method in shallow and deep layers of the CNN for bearing fault diagnosis under different working conditions[J]. The International Journal of Advanced Manufacturing Technology, 2021: 1–12. Mao G, Zhang Z, Qiao B, et al. Fusion domain-adaptation CNN driven by images and vibration signals for fault diagnosis of gearbox cross-working conditions[J]. Entropy, 2022, 24(1): 119. Peng D, Liu Z, Wang H, et al. A novel deeper one-dimensional CNN with residual learning for fault diagnosis of wheelset bearings in high-speed trains[J]. Ieee Access, 2018, 7: 10278–10293. He J, Li X, Chen Y, et al. Deep transfer learning method based on 1D-CNN for bearing fault diagnosis[J]. Shock and Vibration, 2021, 2021: 1–16. Li Y, Wang K. Modified convolutional neural network with global average pooling for intelligent fault diagnosis of industrial gearbox[J]. Eksploatacja i Niezawodność, 2020, 22(1): 63–72. Lou, X.; Loparo, K.A. Bearing fault diagnosis based on wavelet transform and fuzzy inference. Mech. Syst. Signal Process. 2004, 18, 1077–1095. Additional Declarations No competing interests reported. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-4137819","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Article","associatedPublications":[],"authors":[{"id":290772470,"identity":"ccec0511-2ed0-42b9-930c-8bc54efbc80f","order_by":0,"name":"Jianrui Zhang","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA4klEQVRIiWNgGAWjYDACCRDBwyDHx5AAYjETr8WYjUQtDAyJbURrkZ/d/OzhF5l76W3sOWYSDBXWiQ3sZw/g1cI455i5sQxPcW4bz7M0CYYz6YkNPHkJeLUwSySYSUvwJOS2SSQfk2BsO5zYIMFjgFcLm0T6N5CWdDaJxDYJxn9EaOGRyDGT/MCTkMAGtqWBCC0SEjll0gw8CYZAvyRbJBxLN27jycGvRX5G+jbJnz0J8vzsOYY3PtRYy/azn8GvBQSYeXvANItEAsh3BNUDAeOPHxCtH4hRPQpGwSgYBSMPAAD6azsZmsaKMAAAAABJRU5ErkJggg==","orcid":"","institution":"Longdong University","correspondingAuthor":true,"prefix":"","firstName":"Jianrui","middleName":"","lastName":"Zhang","suffix":""},{"id":290772471,"identity":"bac3e386-6be9-4b42-a889-1e68184c911c","order_by":1,"name":"Jinchang Guo","email":"","orcid":"","institution":"Longdong University","correspondingAuthor":false,"prefix":"","firstName":"Jinchang","middleName":"","lastName":"Guo","suffix":""},{"id":290772472,"identity":"fd98be7d-3eac-4643-8a5f-37d25499e5e0","order_by":2,"name":"Baolong 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20","display":"","copyAsset":false,"role":"figure","size":60437,"visible":true,"origin":"","legend":"\u003cp\u003eSVM accuracy\u003c/p\u003e","description":"","filename":"image20.png","url":"https://assets-eu.researchsquare.com/files/rs-4137819/v1/9e1977beb9669f6dea95bd17.png"},{"id":60808525,"identity":"1c38263a-8250-4420-979d-b1c53562fab1","added_by":"auto","created_at":"2024-07-22 10:34:39","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1744288,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-4137819/v1/0e78d380-6719-4ba7-aff9-a642417310dd.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Research on gearbox bearing fault diagnosis based on SSA-VMD-CNN algorithms","fulltext":[{"header":"Introduction","content":"\u003cp\u003eGearbox bearing fault detection technology plays a critical role across a broad spectrum of industries, including automotive, aerospace, energy, manufacturing, and agricultural machinery [\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e]. The inherent complexity of operating environments, characterized by high levels of noise and interference, complicates the accurate extraction and diagnosis of fault signals from gearboxes. Such complexities necessitate the advancement of fault diagnosis methodologies to meet these challenges [\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eBearings are essential components of mechanical equipment, and their malfunction can significantly degrade equipment performance or cause complete operational failures [\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e]. Timely fault diagnosis is imperative for maintaining safe operation, particularly in critical applications like train traction motors. These motors, essential for power drive and brake power storage during train operation, are prone to various failures due to fluctuating speeds and loads [\u003cspan additionalcitationids=\"CR5\" citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e]. Bearing faults, including those in the inner ring, outer ring, and rolling elements, can have severe implications for train safety and operational integrity [\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e, \u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e]. The operational environment of rolling bearings in mechanical systems is inherently complex, often involving high speeds and heavy loads. Unaddressed bearing failures can escalate over time, leading to serious consequences and substantial losses [\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e]. Consequently, the development of advanced and intelligent bearing fault diagnosis methods is of paramount engineering significance, especially with the increasing modernization and automation of industrial production [\u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eTraditional fault diagnosis methods, such as vibration analysis and acoustic emission technology, often fall short under complex signal and noise conditions [\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e]. This limitation underscores the necessity for more effective and accurate fault diagnosis techniques. Variational Mode Decomposition (VMD) has emerged as a prominent solution in mechanical fault diagnosis, effectively decomposing vibration signals into band-limited intrinsic mode functions (BLIMFs) with specific sparsity in the time-frequency domain [12,213]. Unlike empirical mode decomposition (EMD), VMD does not require preset parameters, offering greater flexibility and higher frequency resolution. It efficiently identifies fault characteristics within noisy backgrounds and is particularly adept at processing nonlinear and non-stationary signals [\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eRecent studies have integrated VMD with machine learning algorithms, including deep learning and transfer learning, to enhance diagnostic accuracy and efficiency [\u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e]. Convolutional Neural Networks (CNNs) have demonstrated exceptional capability in extracting meaningful features from complex data, such as the nonlinear and non-stationary vibration signals of gearboxes under varying conditions [\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e]. The application of CNNs in fault diagnosis, combined with their rapid response capabilities, makes them ideally suited for real-time monitoring systems [\u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e17\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eThe structure of this paper is organized as follows: Chap.\u0026nbsp;2 delineates the theoretical framework underpinning the proposed method. Chapter\u0026nbsp;3 details the methodology employed for the collection of test data. In Chap.\u0026nbsp;4, we conduct an analysis to assess the efficacy of the method. Finally, Chap.\u0026nbsp;5 provides a comprehensive summary of the research presented in this paper.\u003c/p\u003e"},{"header":"Theoretical Model","content":"\u003cdiv id=\"Sec3\" class=\"Section2\"\u003e\n\u003ch2\u003e1.1 VMD algorithm principle\u003c/h2\u003e\n\u003cp\u003eVMD is a non-recursive variational mode decomposition method proposed by Dragomiretskiy and Zosso in 2014. Its core idea is to decompose the signal into a discrete number of band-limited intrinsic mode functions (BLIMF) with specific sparsity [\u003cspan class=\"CitationRef\"\u003e18\u003c/span\u003e]. The VMD decomposition method decomposes complex vibration signals into intrinsic mode functions with physical meaning by constructing variational problems and iterative solutions [\u003cspan class=\"CitationRef\"\u003e19\u003c/span\u003e]. Specific steps are as follows:\u003c/p\u003e\n\u003cp\u003e(1) Problem of Structural VariationThe challenge of structural variation entails the decomposition of the original input signal \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(x\\left(t\\right)\\)\u003c/span\u003e\u003c/span\u003e into K Intrinsic Mode Function (IMF) components, denoted as \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(u\\left(t\\right)\\)\u003c/span\u003e\u003c/span\u003e. Subsequently, Hilbert transform is applied to demodulate each component u(t) to derive its envelope signal. The comparison is then made with the estimated center frequency \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\omega }_{k}\\)\u003c/span\u003e\u003c/span\u003e mixing. This process adheres to the constraint that the sum of each \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(u\\left(t\\right)\\)\u003c/span\u003e\u003c/span\u003e component is equivalent to the original signal \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(x\\left(t\\right)\\)\u003c/span\u003e\u003c/span\u003e. The structural variation problem is thus formulated as follows:\u003c/p\u003e\n\u003cdiv id=\"Equa\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equa\" class=\"mathdisplay\"\u003e$$\\left\\{\\begin{array}{c}\\underset{\\left\\{{u}_{k}\\right\\},\\left\\{{\\omega }_{k}\\right\\}}{{min}}\\left\\{\\left.\\sum _{k=1}^{K}{‖{\\partial }_{t}\\left[\\left(\\delta \\left(t\\right)+\\frac{j}{\\pi t}\\right)*{u}_{k}\\left(t\\right)\\right]{e}^{-j{\\omega }_{k}t}‖}_{2}^{2}\\right\\}\\right.\\\\ s.t. \\sum _{k=1}^{K}{u}_{k}=x\\left(t\\right)\\end{array} \\left(1\\right)\\right.$$\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eWhere, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({u}_{k}\\)\u003c/span\u003e\u003c/span\u003e is each mode, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\omega }_{k}\\)\u003c/span\u003e\u003c/span\u003e is the center frequency of each mode, K is the number of decomposition layers, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\partial }_{t}\\)\u003c/span\u003e\u003c/span\u003e is the partial derivative of t, and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\delta \\left(t\\right)\\)\u003c/span\u003e\u003c/span\u003e is the impact function.\u003c/p\u003e\n\u003cp\u003e(2) Solve variational problems\u003c/p\u003e\n\u003cp\u003eThe Lagrangian multiplication operator \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\lambda \\left(t\\right)\\)\u003c/span\u003e\u003c/span\u003e and the quadratic penalty factor \u0026alpha; are added to the constrained variation problem to transform it into an unconstrained variation problem, and its expression is:\u003c/p\u003e\n\u003cdiv id=\"Equb\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equb\" class=\"mathdisplay\"\u003e$$L\\left(\\left\\{{u}_{k}\\right\\},\\left\\{{\\omega }_{k}\\right\\},\\lambda \\right)=\\alpha \\sum _{k}{‖{\\partial }_{t}\\left[\\left(\\delta \\left(t\\right)+\\frac{j}{\\pi t}\\right)*{u}_{k}\\left(t\\right)\\right]{e}^{-j{\\omega }_{k}t}‖}_{2}^{2}+{‖x\\left(t\\right)-\\sum _{k=1}^{K}{u}_{k}\\left(t\\right)‖}_{2}^{2}+⟨\\lambda \\left(t\\right),x\\left(t\\right)-\\sum _{k=1}^{K}{u}_{k}\\left(t\\right)⟩ \\left(2\\right)$$\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eThe saddle point in Eq.\u0026nbsp;(2), which is the optimal solution of Eq.\u0026nbsp;(1), is obtained through the Alternating Direction Method of Multipliers (ADMM). The solution steps are:\u003c/p\u003e\n\u003cp\u003eStep 1: Initialize\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\left\\{{\\widehat{u}}_{k}^{1}\\right\\}\\)\u003c/span\u003e\u003c/span\u003e,\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\left\\{{\\widehat{\\omega }}_{k}^{1}\\right\\}\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\widehat{\\lambda }}^{1}\\)\u003c/span\u003e\u003c/span\u003e, n;\u003c/p\u003e\n\u003cp\u003eStep 2: Write only loop n\u0026thinsp;=\u0026thinsp;n\u0026thinsp;+\u0026thinsp;1;\u003c/p\u003e\n\u003cp\u003eStep 3: For all \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\omega \u0026gt;0\\)\u003c/span\u003e\u003c/span\u003e, update \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\widehat{u}}_{k}\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(k\\in \\left\\{\\text{1,2},\\cdots ,K\\right\\}\\)\u003c/span\u003e\u003c/span\u003e;\u003c/p\u003e\n\u003cdiv id=\"Equc\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equc\" class=\"mathdisplay\"\u003e$$\\begin{array}{c}{\\widehat{u}}_{k}^{n+1}\\left(\\omega \\right)\\leftarrow \\frac{\\widehat{x}\\left(\\omega \\right)-\\sum _{i\u0026lt;k}{\\widehat{u}}_{i}^{n+1}\\left(\\omega \\right) -\\sum _{i\u0026lt;k}{\\widehat{u}}_{i}^{n}\\left(\\omega \\right)+\\frac{{\\widehat{\\lambda }}^{n}\\left(\\omega \\right)}{2}}{1+2\\alpha {\\left(\\omega -{\\omega }_{k}^{n}\\right)}^{2}} \\left(3\\right)\\end{array}$$\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eStep 4: Update \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\omega }_{k}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n\u003cdiv id=\"Equd\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equd\" class=\"mathdisplay\"\u003e$$\\begin{array}{c}{\\omega }_{k}^{n+1}\\leftarrow \\frac{{\\int }_{0}^{\\infty }\\omega {\\left|{\\widehat{u}}_{k}^{n+1}\\left(\\omega \\right)\\right|}^{2}d\\omega }{{\\int }_{0}^{\\infty }{\\left|{\\widehat{u}}_{k}^{n+1}\\left(\\omega \\right)\\right|}^{2}d\\omega },k\\in \\left\\{1,K\\right\\} \\left(4\\right)\\end{array}$$\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eStep 5: Update \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\lambda\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n\u003cdiv id=\"Eque\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Eque\" class=\"mathdisplay\"\u003e$$\\begin{array}{c}{\\widehat{\\lambda }}^{n+1}\\left(\\omega \\right)\\leftarrow {\\widehat{\\lambda }}^{n}\\left(\\omega \\right)+\\tau \\left[\\widehat{x}\\left(\\omega \\right)-\\sum _{k}{\\widehat{u}}_{k}^{n+1}\\left(\\omega \\right)\\right] \\left(5\\right)\\end{array}$$\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eStep 6: Repeat steps 2 to 5 until formula (6) is satisfied and the iteration is stopped, that is, K IMF components are obtained.\u003c/p\u003e\n\u003cdiv id=\"Equf\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equf\" class=\"mathdisplay\"\u003e$$\\begin{array}{c}\\raisebox{1ex}{$\\sum _{k}{‖{\\widehat{u}}_{k}^{n+1}-{\\widehat{u}}_{k}^{n}‖}_{2}^{2}$}\\!\\left/ \\!\\raisebox{-1ex}{${‖{\\widehat{u}}_{k}^{n}‖}_{2}^{2}$}\\right.\u0026lt;\\epsilon \\left(6\\right)\\end{array}$$\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eThe primary merit of Variational Mode Decomposition (VMD) lies in its adaptive signal decomposition capability without necessitating prior knowledge [\u003cspan class=\"CitationRef\"\u003e20\u003c/span\u003e]. The algorithm automatically estimates both the number (K) of modes and their respective center frequencies. Remarkably robust against noise, VMD yields Band-Limited Intrinsic Mode Functions (BLIMF) with compact support in the spectral domain, effectively capturing the inherent natural oscillation modes within the signal. Unlike the Empirical Mode Decomposition (EMD) method, VMD obviates the need for complex envelope estimation and extreme value tracking, rendering its calculations more straightforward [\u003cspan class=\"CitationRef\"\u003e21\u003c/span\u003e].\u003c/p\u003e\n\u003cp\u003eIn essence, the VMD algorithm tackles constrained variation problems through an iterative process grounded in Alternating Direction Multipliers (ADMM). Key operations within each iteration encompass updating the BLIMF via a Wiener filter with an offset frequency, adjusting the Lagrange multiplier, verifying convergence, and updating the estimated center frequency. Iterations persist until the residuals converge to a monotonic function. Ultimately, the signal undergoes decomposition into a series of BLIMFs and a residual.\u003c/p\u003e\n\u003c/div\u003e\n\u003ch3\u003e1.2 Squirrel optimization algorithm model (SSA Model)\u003c/h3\u003e\n\u003cp\u003eIn the Variational Mode Decomposition (VMD) computation, the decomposition number (K) and the penalty factor (\u0026alpha;) are pivotal parameters influencing the performance and effectiveness of the decomposition [\u003cspan class=\"CitationRef\"\u003e22\u003c/span\u003e]. The value of K directly dictates the quantity of decomposed modal components, whereas the value of \u0026alpha; affects their bandwidth, significantly impacting decomposition performance. Currently, there is a lack of objective criteria for setting these parameters. To address this, the present study introduces the Squirrel Search Algorithm (SSA) for the adaptive optimization of VMD parameters [\u003cspan class=\"CitationRef\"\u003e23\u003c/span\u003e].\u003c/p\u003e\n\u003cp\u003eThe Squirrel Search Algorithm (SSA) exhibits notable proficiency in search capability and accuracy when addressing complex problems within a search space. Inspired by the gliding behavior of squirrels, which enables them to evade predators despite their inability to fly, the SSA algorithm simulates this adaptive strategy. The algorithm's search process mimics squirrels' foraging behavior, wherein they locate food by moving between various trees. This method effectively facilitates exploration across different areas of a metaphorical forest, represented by the varying positions of the squirrels within the search space [\u003cspan class=\"CitationRef\"\u003e24\u003c/span\u003e].\u003c/p\u003e\n\u003cp\u003eAssuming the number of squirrels is \u003cem\u003en\u003c/em\u003e, the position the squirrel moves is determined by a vector, and its position is randomly initialized within the boundary range.\u003c/p\u003e\n\u003cdiv id=\"Equg\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equg\" class=\"mathdisplay\"\u003e$$\\begin{array}{c}FS=\\left[\\begin{array}{cccc}F{S}_{\\text{1,1}}\u0026amp; F{S}_{\\text{1,2}}\u0026amp; \\cdots \u0026amp; F{S}_{1,D}\\\\ F{S}_{\\text{2,1}}\u0026amp; F{S}_{\\text{2,2}}\u0026amp; \\cdots \u0026amp; F{S}_{1,d}\\\\ ⋮\u0026amp; ⋮\u0026amp; ⋮\u0026amp; ⋮\\\\ F{S}_{n,1}\u0026amp; F{S}_{n,2}\u0026amp; \\cdots \u0026amp; F{S}_{n,d}\\end{array}\\right] \\left(7\\right)\\end{array}$$\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\(F{S}_{n,d}\\)\u003c/span\u003e \u003c/span\u003e is the value of the \u003cem\u003en\u003c/em\u003e-th mouse in the \u003cem\u003ed\u003c/em\u003e-dimension, and the initial position of the squirrel in the forest is:\u003c/p\u003e\n\u003cdiv id=\"Equh\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equh\" class=\"mathdisplay\"\u003e$$\\begin{array}{c}F{S}_{i}=F{S}_{L}+U\\left(\\text{0,1}\\right)\\times \\left(F{S}_{U}-F{S}_{L}\\right) \\left(8\\right)\\end{array}$$\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\(F{S}_{U}\\)\u003c/span\u003e \u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(F{S}_{L}\\)\u003c/span\u003e\u003c/span\u003e are the upper and lower bounds of squirrel movement, and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(U\\left(\\text{0,1}\\right)\\)\u003c/span\u003e\u003c/span\u003e is a random number [0,1].\u003c/p\u003e\n\u003cp\u003eThe grade of food source is represented by the fitness of each squirrel location, and the fitness value is calculated and classified in ascending order. The position with the least fitness is: the best food source ① pecan, the next three are normal food source ② oak, and the other locations have no food source ③ common tree [\u003cspan class=\"CitationRef\"\u003e25\u003c/span\u003e].\u003c/p\u003e\n\u003cp\u003eDepending on the probability of the presence of its natural enemies, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({P}_{dp}\\)\u003c/span\u003e\u003c/span\u003e squirrels update the location of their movement.\u003c/p\u003e\n\u003cp\u003eGlide Path 1: ②\u0026rarr;①\u003c/p\u003e\n\u003cdiv id=\"Equi\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equi\" class=\"mathdisplay\"\u003e$$\\begin{array}{c}F{S}_{at}^{t+1}\\left\\{\\begin{array}{c}F{S}_{at}^{t}+{d}_{g}\\times {G}_{c}\\times \\left(F{S}_{ht}^{t}-F{S}_{at}^{t}\\right),{R}_{1}\\ge {P}_{dp}\\\\ \\text{R}\\text{a}\\text{n}\\text{d}\\text{o}\\text{m}\\text{l}\\text{o}\\text{c}\\text{a}\\text{t}\\text{i}\\text{o}\\text{n},{ R}_{1}\u0026lt;{P}_{dp}\\end{array}\\right. \\left(9\\right)\\end{array}$$\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eGlide Path 2:③\u0026rarr;②\u003c/p\u003e\n\u003cdiv id=\"Equj\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equj\" class=\"mathdisplay\"\u003e$$\\begin{array}{c}F{S}_{nt}^{t+1}\\left\\{\\begin{array}{c}F{S}_{nt}^{t}+{d}_{g}\\times {G}_{c}\\times \\left(F{S}_{at}^{t}-F{S}_{nt}^{t}\\right),{R}_{2}\\ge {P}_{dp}\\\\ \\text{R}\\text{a}\\text{n}\\text{d}\\text{o}\\text{m}\\text{l}\\text{o}\\text{c}\\text{a}\\text{t}\\text{i}\\text{o}\\text{n},{ R}_{2}\u0026lt;{P}_{dp}\\end{array}\\right. \\left(10\\right)\\end{array}$$\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eGlide Path 3:③\u0026rarr;①\u003c/p\u003e\n\u003cdiv id=\"Equk\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equk\" class=\"mathdisplay\"\u003e$$\\begin{array}{c}F{S}_{nt}^{t+1}\\left\\{\\begin{array}{c}F{S}_{nt}^{t}+{d}_{g}\\times {G}_{c}\\times \\left(F{S}_{ht}^{t}-F{S}_{nt}^{t}\\right),{R}_{3}\\ge {P}_{dp}\\\\ \\text{R}\\text{a}\\text{n}\\text{d}\\text{o}\\text{m}\\text{l}\\text{o}\\text{c}\\text{a}\\text{t}\\text{i}\\text{o}\\text{n},{ R}_{1}\u0026lt;{P}_{dp}\\end{array}\\right. \\left(11\\right)\\end{array}$$\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({d}_{g}\\)\u003c/span\u003e \u003c/span\u003e is the random glide distance, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({R}_{1}{R}_{2}{R}_{3}\\)\u003c/span\u003e\u003c/span\u003e is the random number in the range of [0,1], \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(F{S}_{at}^{t}\\)\u003c/span\u003e\u003c/span\u003e is the squirrel's position on the oak, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(F{S}_{ht}^{t}\\)\u003c/span\u003e\u003c/span\u003e is the squirrel's position on the mountain tree, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(F{S}_{nt}^{t}\\)\u003c/span\u003e\u003c/span\u003e is the squirrel's position on the common tree, and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({G}_{c}\\)\u003c/span\u003e\u003c/span\u003e is the slip coefficient.\u003c/p\u003e\n\u003cp\u003eSeasonal changes affect squirrels' foraging activities, and seasonal changes are used to prevent algorithms from falling into local optimality.\u003c/p\u003e\n\u003cdiv id=\"Equl\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equl\" class=\"mathdisplay\"\u003e$$\\begin{array}{c}{S}_{c}^{t}=\\sqrt{{\\sum }_{z=1}^{3}{\\sum }_{k=1}^{d}{\\left(F{S}_{at,k}^{t,z}-F{S}_{ht,k}\\right)}^{2}} \\left(12\\right)\\end{array}$$\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Equm\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equm\" class=\"mathdisplay\"\u003e$$\\begin{array}{c}{S}_{min}=\\frac{10E-6}{{365}^{2.5t/{t}_{m}}} \\left(13\\right)\\end{array}$$\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\(t\\)\u003c/span\u003e \u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({t}_{m}\\)\u003c/span\u003e\u003c/span\u003e are the current value and the maximum iteration value respectively, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({S}_{min}\\)\u003c/span\u003e\u003c/span\u003e is the minimum value of the seasonal constant, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({S}_{c}^{t}\\)\u003c/span\u003e\u003c/span\u003e is the seasonal constant, and the seasonal change condition is\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({S}_{c}^{t}\u0026lt;{S}_{min}\\)\u003c/span\u003e\u003c/span\u003e. If this condition is met, the position of common squirrels changes randomly.\u003c/p\u003e\n\u003cdiv id=\"Equn\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equn\" class=\"mathdisplay\"\u003e$$\\begin{array}{c}F{S}_{nt,i}^{t+1}=F{S}_{i,L}+Levy\\left(F{S}_{i,U}-F{S}_{i,L}\\right) \\left(14\\right)\\end{array}$$\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\(F{S}_{i,U}\\)\u003c/span\u003e \u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(F{S}_{i,L}\\)\u003c/span\u003e\u003c/span\u003e are the upper and lower bounds of squirrel movement, and \u003cem\u003eLevy\u003c/em\u003e is the Levi distribution, effectively searching globally to find a new location that is optimal from the current location.\u003c/p\u003e\n\u003cp\u003eThe process of optimizing VMD parameters based on squirrel search algorithm is as follows [\u003cspan class=\"CitationRef\"\u003e26\u003c/span\u003e].\u003c/p\u003e\n\u003cp\u003eStep 1: Set SSA parameters, including population size, number of iterations and upper and lower bounds of the optimization range, and initialize the population position using Eq.\u0026nbsp;(7).\u003c/p\u003e\n\u003cp\u003eStep 2: VMD decomposition of the power sequence was performed according to the position of each squirrel (K,a), and the fitness of each individual was calculated and sorted according to Eq.\u0026nbsp;(8). The squirrels were assigned to hickory trees (optimal solution), oak trees (suboptimal solution), and ordinary trees (general solution) in order to preserve the locations of the most individual squirrels.\u003c/p\u003e\n\u003cp\u003eStep 3: Update individual squirrel locations.\u003c/p\u003e\n\u003cp\u003eStep 4: Update the seasonal constant, when S_c^t\u0026thinsp;\u0026lt;\u0026thinsp;S_min, move the individuals in the ordinary tree randomly.\u003c/p\u003e\n\u003cp\u003eStep 5: Repeat Step 2 for the newly generated location to update the optimal solution.\u003c/p\u003e\n\u003cp\u003eStep 6: Repeat steps 3 to 5 until the maximum number of iterations is reached to stop the optimization and output the optimal parameters and fitness values.\u003c/p\u003e\n\u003ch3\u003e1.3 CNN algorithm principle\u003c/h3\u003e\n\u003cp\u003eCNN is a feedforward neural network with a deep structure, including input layer, convolutional layer, pooling layer, fully connected layer and output layer [\u003cspan class=\"CitationRef\"\u003e27\u003c/span\u003e].\u003c/p\u003e\n\u003cp\u003eThe convolution layer extracts local features from the fault data, and the feature output of the convolution layer is obtained by activation function after convolution calculation. The feature expression is:\u003c/p\u003e\n\u003cdiv id=\"Equo\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equo\" class=\"mathdisplay\"\u003e$$\\begin{array}{c}{x}_{j}^{l}=f\\left(\\sum _{i\\in {M}_{j}}{x}_{i}^{l-1}*{k}_{ij}^{l}+{b}_{j}^{i}\\right) \\left(15\\right)\\end{array}$$\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eWhere: \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({k}_{ij}^{l}\\)\u003c/span\u003e\u003c/span\u003e is the convolution kernel weight matrix; \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(f\\)\u003c/span\u003e\u003c/span\u003e is a nonlinear activation function; \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(l\\)\u003c/span\u003e\u003c/span\u003e is the \u003cem\u003el\u003c/em\u003e-layer in the network; \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({M}_{j}\\)\u003c/span\u003e\u003c/span\u003e is the input feature map.\u003c/p\u003e\n\u003cp\u003eThe pooling layer is divided into maximum pooling and average pooling, which can retain the main features while reducing the network parameters and calculation amount to avoid the occurrence of overfitting [\u003cspan class=\"CitationRef\"\u003e28\u003c/span\u003e]. The fully connected layer further extracts the output features of the pooled layer and inputs them into the classifier for classification.\u003c/p\u003e"},{"header":"Data Acquisition","content":"\u003cdiv id=\"Sec7\" class=\"Section3\"\u003e\n\u003ch2\u003e2.1. Gearbox structure\u003c/h2\u003e\n\u003cp\u003eThe gearbox fault diagnosis platform is a device used to detect and diagnose gearbox faults. This platform can simulate the operating status of the gearbox and monitor the operating status of the gearbox in real time by monitoring various indicators such as vibration and temperature of the gearbox, so as to detect faults and perform diagnosis in a timely manner [\u003cspan class=\"CitationRef\"\u003e29\u003c/span\u003e]. The platform can collect various indicator data such as vibration, temperature, and current of the gearbox to facilitate fault diagnosis. The experimental platform consists of: a 2-horsepower drive motor, a torque transmission device in the middle and a tested gearbox [\u003cspan class=\"CitationRef\"\u003e30\u003c/span\u003e]. Acceleration sensors are installed at the base, fan end, and drive end of the experimental motor respectively to obtain base acceleration data, fan end acceleration data, and drive end acceleration data. Figure\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003e shows the overall design of the fault detection platform.\u003c/p\u003e\n\u003cp\u003e\u0026nbsp;\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec8\" class=\"Section2\"\u003e\n\u003ch2\u003e2.2. Data description\u003c/h2\u003e\n\u003cp\u003eTo leverage neural network algorithms effectively, particularly convolutional neural networks (CNNs), it is imperative to amass a substantial corpus of images for image recognition tasks. A quintessential example is the MNIST dataset, which comprises 60,000 training images and 10,000 test images of handwritten digits. For comprehensive training of the CNN model, we curated an extensive collection of training samples derived from accelerometer data in a motor-driven mechanical system, recorded at a 12 kHz sampling frequency by the Case Western Reserve University (CWRU) Axis Data Center [\u003cspan class=\"CitationRef\"\u003e31\u003c/span\u003e]. This dataset encapsulates four bearing fault types: normal, ball, inner ring, and outer ring faults, each manifesting in three distinct fault diameters: 0.007 inches, 0.014 inches, and 0.021 inches. This configuration results in a total of 10 unique fault scenarios. Each sample within the experiment comprises 2048 data points, facilitating the straightforward implementation of the Fast Fourier Transform (FFT) in the baseline algorithm. The datasets, labeled A, B, and C, each encompass 6,600 training samples and 250 test samples, distributed across 10 failure conditions at load capacities of 1, 2, and 3 horsepower, respectively. Dataset D extends this with 19,800 training and 750 test samples, covering the same three load conditions. To augment the training dataset, we employed overlapping samples, ensuring, however, that no overlap occurred within the test samples. Table\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003e provides a comprehensive overview of these datasets.\u003c/p\u003e\n\u003cdiv class=\"gridtable\"\u003e\n\u003cdiv class=\"colspec\" align=\"left\"\u003e\u0026nbsp;\u003c/div\u003e\n\u003cdiv class=\"colspec\" align=\"char\"\u003e\u0026nbsp;\u003c/div\u003e\n\u003ctable id=\"Tab1\" border=\"1\"\u003e\u003ccaption\u003e\n\u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e\n\u003cdiv class=\"CaptionContent\"\u003e\n\u003cp\u003eDescription of rolling element bearing datasets.\u003c/p\u003e\n\u003c/div\u003e\n\u003c/caption\u003e\n\u003cthead\u003e\n\u003ctr\u003e\n\u003cth colspan=\"2\" align=\"left\"\u003e\n\u003cp\u003eFault Location\u003c/p\u003e\n\u003c/th\u003e\n\u003cth colspan=\"2\" align=\"left\"\u003e\n\u003cp\u003eNone\u003c/p\u003e\n\u003c/th\u003e\n\u003cth colspan=\"2\" align=\"left\"\u003e\n\u003cp\u003eBall\u003c/p\u003e\n\u003c/th\u003e\n\u003cth colspan=\"3\" align=\"left\"\u003e\n\u003cp\u003eInner Race\u003c/p\u003e\n\u003c/th\u003e\n\u003cth colspan=\"3\" align=\"left\"\u003e\n\u003cp\u003eOuter Race\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eLoad\u003c/p\u003e\n\u003c/th\u003e\n\u003c/tr\u003e\n\u003c/thead\u003e\n\u003ctbody\u003e\n\u003ctr\u003e\n\u003ctd colspan=\"2\" align=\"left\"\u003e\n\u003cp\u003eCategory Labels\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd colspan=\"2\" align=\"left\"\u003e\n\u003cp\u003e1 2\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd colspan=\"2\" align=\"left\"\u003e\n\u003cp\u003e3 4\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd colspan=\"3\" align=\"left\"\u003e\n\u003cp\u003e5 6 7\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd colspan=\"3\" align=\"left\"\u003e\n\u003cp\u003e8 9 10\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd colspan=\"2\" align=\"left\"\u003e\n\u003cp\u003eFault diameter (inch)\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd colspan=\"2\" align=\"left\"\u003e\n\u003cp\u003e0 0.007\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd colspan=\"2\" align=\"left\"\u003e\n\u003cp\u003e0.014 0.021\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd colspan=\"3\" align=\"left\"\u003e\n\u003cp\u003e0.007 0.014\u0026nbsp; \u0026nbsp;0.021\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd colspan=\"3\" align=\"left\"\u003e\n\u003cp\u003e0.007\u0026nbsp; \u0026nbsp;0.014 \u0026nbsp;\u0026nbsp;0.021\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eDataset A no.\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eTrain Test\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e660\u003c/p\u003e\n\u003cp\u003e25\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e660\u003c/p\u003e\n\u003cp\u003e25\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e660\u003c/p\u003e\n\u003cp\u003e25\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e660\u003c/p\u003e\n\u003cp\u003e25\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e660\u003c/p\u003e\n\u003cp\u003e25\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e660\u003c/p\u003e\n\u003cp\u003e25\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e660\u003c/p\u003e\n\u003cp\u003e25\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e660\u003c/p\u003e\n\u003cp\u003e25\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e660\u003c/p\u003e\n\u003cp\u003e25\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e660\u003c/p\u003e\n\u003cp\u003e25\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eDataset B no.\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eTrain test\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e660\u003c/p\u003e\n\u003cp\u003e25\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e660\u003c/p\u003e\n\u003cp\u003e25\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e660\u003c/p\u003e\n\u003cp\u003e25\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e660\u003c/p\u003e\n\u003cp\u003e25\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e660\u003c/p\u003e\n\u003cp\u003e25\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e660\u003c/p\u003e\n\u003cp\u003e25\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e660\u003c/p\u003e\n\u003cp\u003e25\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e660\u003c/p\u003e\n\u003cp\u003e25\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e660\u003c/p\u003e\n\u003cp\u003e25\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e660\u003c/p\u003e\n\u003cp\u003e25\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e2\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eDataset C no.\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003etrain\u003c/p\u003e\n\u003cp\u003etest\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e660\u003c/p\u003e\n\u003cp\u003e25\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e660\u003c/p\u003e\n\u003cp\u003e25\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e660\u003c/p\u003e\n\u003cp\u003e25\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e660\u003c/p\u003e\n\u003cp\u003e25\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e660\u003c/p\u003e\n\u003cp\u003e25\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e660\u003c/p\u003e\n\u003cp\u003e25\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e660\u003c/p\u003e\n\u003cp\u003e25\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e660\u003c/p\u003e\n\u003cp\u003e25\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e660\u003c/p\u003e\n\u003cp\u003e25\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e660\u003c/p\u003e\n\u003cp\u003e25\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e3\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eDataset D no.\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eTrain Test\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1980\u003c/p\u003e\n\u003cp\u003e75\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1980\u003c/p\u003e\n\u003cp\u003e75\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1980\u003c/p\u003e\n\u003cp\u003e75\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1980\u003c/p\u003e\n\u003cp\u003e75\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1980\u003c/p\u003e\n\u003cp\u003e75\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1980\u003c/p\u003e\n\u003cp\u003e75\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1980\u003c/p\u003e\n\u003cp\u003e75\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1980\u003c/p\u003e\n\u003cp\u003e75\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1980\u003c/p\u003e\n\u003cp\u003e75\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1980\u003c/p\u003e\n\u003cp\u003e75\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1,2,3\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003c/tbody\u003e\n\u003c/table\u003e\n\u003c/div\u003e\n\u003c/div\u003e"},{"header":"Fault Diagnosis Instance","content":"\u003ch2\u003e3. Fault Diagnosis Instance\u003c/h2\u003e\n\u003cdiv id=\"Sec10\" class=\"Section3\"\u003e\n\u003ch2\u003e3.1. Fault detection and noise reduction based on VMD\u003c/h2\u003e\n\u003cp\u003eAfter data processing, two input signals were selected for display. Figure\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e is a gear without fault, and Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003e is a faulty input signal. It can be clearly seen that although the first signal has no special pattern, it is relatively stable. The second signal is faulty and has significant noise that needs to be dealt with.\u003c/p\u003e\n\u003cp\u003eFigure \u003cspan class=\"InternalRef\"\u003e4\u003c/span\u003e is the signal after VMD noise reduction. The blue color in the figure is the input signal, and the red color is the signal output after the noise reduction is successful. The noise reduction effect is good.\u003c/p\u003e\n\u003cp\u003eFigure \u003cspan class=\"InternalRef\"\u003e5\u003c/span\u003e shows the frequency domain analysis of the four modes after VMD decomposition. The black dotted line is the signal added after the four decompositions. The result is relatively stable. Figure\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e6\u003c/span\u003e shows the decomposed four signals.\u003c/p\u003e\n\u003cp\u003eFigure \u003cspan class=\"InternalRef\"\u003e7\u003c/span\u003e shows the frequency domain representation of the input signal.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eFigure \u003cspan class=\"InternalRef\"\u003e8\u003c/span\u003e is a comparison, the horizontal axis is the number of training rounds, and the vertical axis is the accuracy rate. The red is the training set data, and the blue is the test set data. In the figure, the data accuracy of the training set is significantly higher than that of the test set, and the overfitting phenomenon is obvious. However, after VMD noise reduction, the fitting phenomenon is greatly alleviated. Figure\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e8\u003c/span\u003e and Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e9\u003c/span\u003e show the classification accuracy and loss function graphs of VMD denoising and VMD(called SSA-VMD) denoising after VMD and SSA parameter optimization respectively. The accuracy of VMD at 20 iterations is 80%, while the accuracy of SSA-VMD at 20 iterations is 87%, and SSA-VMD achieves higher accuracy with fewer iterations.\u003c/p\u003e\n\u003cp\u003eFigure \u003cspan class=\"InternalRef\"\u003e9\u003c/span\u003e shows the number of training rounds on the horizontal axis and the loss function on the vertical axis.\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec11\" class=\"Section2\"\u003e\n\u003ch2\u003e3.2. Gearbox fault diagnosis based on VMD and CNN\u003c/h2\u003e\n\u003cp\u003e3.2.1 Optimization of CNN model\u003c/p\u003e\n\u003cp\u003eFigure \u003cspan class=\"InternalRef\"\u003e10\u003c/span\u003e, Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e11\u003c/span\u003e and Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e12\u003c/span\u003e shows the number of training rounds on the horizontal axis and the accuracy rate on the vertical axis. Red is the training set data, blue is the test set data. Figure\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e10\u003c/span\u003e is the original CNN model without any optimization, Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e11\u003c/span\u003e is the data after noise reduction only with SSA-VMD, and Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e12\u003c/span\u003e is the final model after optimization with SSA-VMD and CNN (named SSA-SVM-CNN). It can be seen that with continuous optimization, not only the overfitting problem is solved, but also the accuracy rate is improved, as shown in Table\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e.\u003c/p\u003e\n\u003cp\u003e\u0026nbsp;\u003c/p\u003e\n\u003cdiv class=\"gridtable\"\u003e\n\u003cdiv class=\"colspec\" align=\"left\"\u003e\u0026nbsp;\u003c/div\u003e\n\u003ctable id=\"Tab2\" border=\"1\"\u003e\u003ccaption\u003e\n\u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e\n\u003cdiv class=\"CaptionContent\"\u003e\n\u003cp\u003eCorrect rate data table\u003c/p\u003e\n\u003c/div\u003e\n\u003c/caption\u003e\n\u003cthead\u003e\n\u003ctr\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eMethod\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eCNN\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eSSA-VMD\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eSSA-VMD-CNN\u003c/p\u003e\n\u003c/th\u003e\n\u003c/tr\u003e\n\u003c/thead\u003e\n\u003ctbody\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eAccuracy\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e81%\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e87%\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e\u003cstrong\u003e95%\u003c/strong\u003e\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003c/tbody\u003e\n\u003c/table\u003e\n\u003c/div\u003e\n\u003cp\u003e3.2.2 Result visualization and analysis\u003c/p\u003e\n\u003cp\u003eTo elucidate the functionality of the proposed SSA-VMD-CNN network model, this study presents graphical representations of various model parameters\u0026mdash;including accuracy rate, loss function, and confusion matrix\u0026mdash;over 200 operational iterations. Precision, defined as the ratio of true positive samples to all predicted positive samples, is crucial for evaluating model performance. Recall, or the proportion of actual positive samples correctly identified, is equally significant. The F1-score, representing a harmonious mean of precision and recall, proves particularly useful in balancing these metrics, especially in datasets exhibiting imbalance.\u003c/p\u003e\n\u003cp\u003eFigure \u003cspan class=\"InternalRef\"\u003e13\u003c/span\u003e delineates the precision, recall, and F1-scores for each fault category within the network model. Notably, categories 0, 2, and 7 exhibit lower accuracy. Category 0 corresponds to rolling element faults with a diameter of 1.1778 mm, Category 2 to rolling element faults of 0.5334 mm, and Category 7 to outer ring gear faults, also of 0.5334 mm. This trend suggests a reduced efficacy of the model in identifying rolling element failures, indicating a potential improvement by augmenting rolling element data within the training set. Conversely, the model demonstrates exceptional accuracy in other categories, achieving a 100% recognition rate in four fault categories. This high accuracy offers the prospect of data reduction, thereby conserving resources and reducing operational time.\u003c/p\u003e\n\u003cp\u003eFigure \u003cspan class=\"InternalRef\"\u003e14\u003c/span\u003e shows the loss function of the model in this paper. The final loss function is around 0.25. If the training continues, the loss function can be reduced and the accuracy can be increased.\u003c/p\u003e\n\u003cp\u003eIn order to show the classification effect more clearly, the confusion matrix of diagnosis results is shown in Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e15\u003c/span\u003e. The horizontal axis is the type recognized by the computer, and the vertical axis is the real category. It is obvious that group 1 has more errors in judgment. The other groups have very few errors.\u003c/p\u003e\n\u003cp\u003e\u0026nbsp;\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec12\" class=\"Section2\"\u003e\n\u003ch2\u003e3.3. Comparison of CNN Convolutional neural network with other models\u003c/h2\u003e\n\u003cp\u003eIn this paper, the CNN model is compared with some classical algorithms. All models are denoised by SSA-VMD to ensure the fairness of the comparison.\u003c/p\u003e\n\u003cp\u003e3.3.1 Comparison between CNN and GRU algorithm\u003c/p\u003e\n\u003cp\u003eGRU stands for Gated Recurrent Unit, which is a recurrent neural network (RNN) architecture introduced by Kyunghyun Cho et al. in 2014. GRU is similar to long short-term memory (LSTM) with forget gates, but it has fewer parameters than LSTM because it lacks an output gate. GRU uses a gating mechanism to control and manage the flow of information through the network. It consists of only three gates, namely reset gate, update gate and new memory gate. The reset gate determines how much past knowledge needs to be transferred to the future, while the update gate determines how much new information needs to be stored in memory. GRU shows similar performance to LSTM on some tasks, such as chord music modeling, speech signal modeling, and natural language processing.\u003c/p\u003e\n\u003cp\u003eFigure \u003cspan class=\"InternalRef\"\u003e16\u003c/span\u003e serves as a comparative examination, where the horizontal axis signifies the number of training rounds, and the vertical axis denotes the accuracy rate. The red curve represents data trained with specific parameters, while the blue curve encapsulates parameters affiliated with a test set unexposed to the model. Evidently, the accuracy of the training set, as delineated in the figure, markedly surpasses that of the test set, elucidating a conspicuous manifestation of overfitting. Positioned on the left is the Convolutional Neural Network (CNN) model, while on the right resides the Gated Recurrent Unit (GRU) model. Notably, the CNN model exhibits superior performance. The CNN model not only demonstrates enhanced model fitting but also maintains a stable efficacy. Despite the approximately 90% accuracy of the GRU on the right, discernible fluctuations and fitting issues persist within the GRU model.\u003c/p\u003e\n\u003cp\u003e\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eAs depicted in Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e17\u003c/span\u003e, the loss function for the Convolutional Neural Network (CNN) stands at 0.5, whereas the Gated Recurrent Unit (GRU) registers a loss function approximating 0. The discernible divergence in these loss functions implies that the CNN model retains significant room for further optimization within the 0.5 space, whereas the GRU model exhibits limited optimization potential. In summation, the superiority of the CNN model is evident.\u003c/p\u003e\n\u003cp\u003e3.3.2 Comparison between CNN and LSTM algorithms\u003c/p\u003e\n\u003cp\u003eLSTM, an acronym signifying Long Short-Term Memory, represents an artificial neural network employed within the realms of artificial intelligence and deep learning. Operating as a subtype of recurrent neural networks (RNNs), LSTM specializes in discerning sequential dependencies pertinent to sequence prediction tasks. Diverging from conventional feedforward neural networks, LSTMs incorporate feedback connections. Notably, LSTMs excel in processing extensive sequential data, endowing RNNs with a durable short-term memory capable of spanning thousands of time steps, hence the nomenclature \"long short-term memory.\" Widespread adoption of LSTM is evident in natural language processing and diverse domains, solidifying its status as a cornerstone in classical machine learning models.\u003c/p\u003e\n\u003cp\u003e\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eFigure \u003cspan class=\"InternalRef\"\u003e18\u003c/span\u003e delineates an accuracy comparison, with the horizontal axis denoting the number of training rounds and the vertical axis representing accuracy. The red trace corresponds to the training data and the associated model parameters, while the blue trace pertains to parameters from the test set. Notably, while Long Short-Term Memory (LSTM) effectively mitigates the issue of overfitting, its accuracy lags approximately 20% behind that of Convolutional Neural Network (CNN). This performance discrepancy underscores that, in the context of accuracy, LSTM proves less apt for gearbox fault detection, advocating a preference for CNN-based approaches in this application domain.\u003c/p\u003e\n\u003cp\u003e3.3.3 Comparison between CNN and SVM algorithms\u003c/p\u003e\n\u003cp\u003eThe Support Vector Machine (SVM) stands as a supervised machine learning algorithm proficient in classification and regression tasks. SVM discerns a hyperplane within the training data, optimizing the separation of distinct classes. Rooted in the statistical learning framework and VC theory, SVM serves as a robust predictive method for both two-group and multi-class classification challenges. Functioning as a non-probabilistic binary linear classifier, SVM necessitates labeled data for effective training. Its applicability extends across diverse domains, encompassing text classification, image classification, and bioinformatics.\u003c/p\u003e\n\u003cp\u003ePrecision, denoting the proportion of correctly identified positive samples among predicted positives, and recall, representing the ratio of correctly predicted positive samples to the total actual positive samples, are pivotal metrics. The F1-score, a harmonic mean of precision and recall, provides a comprehensive evaluation of their balance. Particularly advantageous in managing imbalanced datasets, F1-score contributes nuanced insights. While accuracy remains a crucial metric, the F1-score's incorporation of precision and recall renders it a more insightful indicator, especially in scenarios involving imbalanced datasets.\u003c/p\u003e\n\u003cp\u003eFigures \u003cspan class=\"InternalRef\"\u003e19\u003c/span\u003e and \u003cspan class=\"InternalRef\"\u003e20\u003c/span\u003e illustrate the three key indicators\u0026mdash;Precision, Recall, and F1-score\u0026mdash;depicting the classification performance of Convolutional Neural Network (CNN) and Support Vector Machine (SVM) across ten distinct fault conditions. Notably, with the exception of fault conditions in groups 0 and 3, where CNN exhibits a lower accuracy compared to SVM, CNN consistently outperforms SVM across other data categories. Particularly noteworthy is the second data type, where SVM's accuracy is alarmingly null, contrasting sharply with CNN's commendable accuracy of 87%. This underscores CNN's superior efficacy in diverse scenarios. In summation, our proposed CNN model, detailed in this chapter, emerges as the preeminent performer, boasting the highest accuracy in the test set and superior effectiveness. Following closely is the Gated Recurrent Unit (GRU) model, surpassing the performance of traditional models like SVM and Random Forest (RF).\u003c/p\u003e\n\u003cp\u003e\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eThrough the above comparative analysis, the fault diagnosis accuracy of various methods is shown in Table\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003e. The combination of parameter optimized VMD and CNN algorithm has significant advantages for the fault diagnosis of gearbox.\u003c/p\u003e\n\u003cdiv class=\"gridtable\"\u003e\n\u003cdiv class=\"colspec\" align=\"left\"\u003e\u0026nbsp;\u003c/div\u003e\n\u003cdiv class=\"colspec\" align=\"left\"\u003e\u0026nbsp;\u0026nbsp;\u003c/div\u003e\n\u003cdiv class=\"colspec\" align=\"left\"\u003e\u0026nbsp;\u003c/div\u003e\n\u003ctable id=\"Tab3\" border=\"1\"\u003e\u003ccaption\u003e\n\u003cdiv class=\"CaptionNumber\"\u003eTable 3\u003c/div\u003e\n\u003cdiv class=\"CaptionContent\"\u003e\n\u003cp\u003eFault diagnosis accuracy of CNN and other algorithms\u003c/p\u003e\n\u003c/div\u003e\n\u003c/caption\u003e\n\u003cthead\u003e\n\u003ctr\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eMethod\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eSSA-VMD-CNN\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eSSA-VMD-GRU\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eSSA-VMD-LSTM\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eSSA-VMD-SVM\u003c/p\u003e\n\u003c/th\u003e\n\u003c/tr\u003e\n\u003c/thead\u003e\n\u003ctbody\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eAccuracy\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e95%\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e91%\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e70%\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e65%\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003c/tbody\u003e\n\u003c/table\u003e\n\u003c/div\u003e\n\u003c/div\u003e"},{"header":"Conclusion","content":"\u003cp\u003eIn this study, we introduce a novel bearing fault diagnosis model that effectively detects faults in gearboxes operating amidst complex noise environments. The integration of Variational Mode Decomposition (VMD), Convolutional Neural Network (CNN), and Squirrel Search Algorithm (SSA) within this model enables the efficient extraction of pivotal signal features from noise, thereby facilitating accurate fault diagnosis.\u003c/p\u003e\n\u003cp\u003eThe employment of VMD in the model allows the decomposition of vibration signals into multiple Intrinsic Mode Functions (IMFs). These IMFs are subsequently analyzed and reconstructed based on kurtosis and cross-correlation criteria, yielding a more refined and precise signal foundation for fault detection. Optimizing VMD parameters using SSA further amplifies the model\u0026rsquo;s capabilities in noise reduction and feature extraction. The refined signals are then input into a CNN for training and testing, showcasing the model\u0026rsquo;s robustness and precision in handling intricate data sets. The model's efficacy is corroborated through simulations under various fault conditions, achieving an impressive accuracy rate exceeding 93%, markedly surpassing traditional fault diagnosis methodologies.\u003c/p\u003e\n\u003cp\u003eThis proposed bearing fault diagnosis model not only elevates the accuracy and efficiency of fault detection but also pioneers a novel approach for the health monitoring and maintenance of critical mechanical systems, such as gearboxes.5. Acknowledgments\u003c/p\u003e\n\u003cp\u003eThis work was supported in part by the Crossing Research Project of Longdong University, HXZK2318, partly by Natural Science Foundation of Gansu Province, 23JRRM737, partly by the Qingyang City Science and Technology Planning Project, QY-STK-2022B-151 and partly by the Longdong University Doctoral Fund, XYBYZK2301.\u003c/p\u003e\n\u003c/div\u003e\n"},{"header":"Declarations","content":"\u003cp\u003e6. Data Availability\u003c/p\u003e\n\u003cp\u003eAll data generated or analysed during this study are included in this published article.\u003c/p\u003e\u003ch2\u003eAuthor Contribution\u003c/h2\u003e\u003cp\u003eConceptualization, Jianrui Zhang and Yue Zhang; software, Jinchang Guo and Baolong Geng; validation, Haoyang Song; writing\u0026mdash;original draft preparation, Rong Zhao. All the data used in the simulation analysis in this paper are available. All authors have read and agreed to the published version of the manuscript.All authors reviewed the manuscript.\u003c/p\u003e\u003ch2\u003eData Availability\u003c/h2\u003e\u003cp\u003eThe original experiments data was obtained from the accelerometers of the motor driving mechanical system at a sampling frequency of 12 kHz from the Case Western Reserve University (CWRU) Bearing Data Center. Lou, X.; Loparo, K.A. Bearing fault diagnosis based on wavelet transform and fuzzy inference. Mech. Syst. Signal Process. 2004, 18, 1077\u0026ndash;1095.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\u003cli\u003e\u003cspan\u003eLi, Y.; Xu, M.; Wei, Y.; Huang, W. A new rolling bearing fault diagnosis method based on multiscale permutation entropy and improved support vector machine based binary tree. Measurement 2016, 77, 80\u0026ndash;94.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eZhang X, Han P, Xu L, et al. Research on bearing fault diagnosis of wind turbine gearbox based on 1DCNN-PSO-SVM[J]. IEEE Access, 2020, 8: 192248\u0026ndash;192258.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eYu X, Li Z, He Q, et al. Gearbox fault diagnosis based on bearing dynamic force identification[J]. Journal of Sound and Vibration, 2021, 511: 116360.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eZhou L, Duan F, Corsar M, et al. A study on helicopter main gearbox planetary bearing fault diagnosis[J]. 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Signal Process. 2004, 18, 1077\u0026ndash;1095.\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Fault diagnosis, variational mode decomposition, convolutional neural network, vibration signal, bearing","lastPublishedDoi":"10.21203/rs.3.rs-4137819/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-4137819/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eGearbox bearings are crucial components in numerous mechanical systems. These gearboxes typically operate in environments characterized by significant noise, causing their fault signals to be obscured by background interference, vibrations, and signals from other mechanical parts. This interference complicates the accurate extraction and diagnosis of fault characteristics from complex data. To address this challenge, we propose a novel bearing fault diagnosis model that integrates Variational Mode Decomposition (VMD), Convolutional Neural Network (CNN), and advanced optimization algorithms. Initially, the Squirrel Search Algorithm (SSA) is employed to automatically optimize VMD parameters, enabling efficient extraction of denoised signal features. VMD decomposes vibration signals into multiple Intrinsic Mode Functions (IMFs), which are then analyzed and reconstructed using kurtosis and cross-correlation criteria. Subsequently, these processed signals serve as input feature vectors for the CNN model, facilitating both training and testing phases. The model is designed to construct a singular value vector matrix that reflects the current fault state based on the position of each submatrix. Simulation verification of our model demonstrates an accuracy exceeding 95% in bearing fault diagnosis, a substantial improvement over traditional methods. This advancement offers a new perspective for the health monitoring and maintenance of critical mechanical equipment, such as gearboxes. It holds significant potential for application in intelligent manufacturing and automated monitoring systems in the future.\u003c/p\u003e","manuscriptTitle":"Research on gearbox bearing fault diagnosis based on SSA-VMD-CNN algorithms","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2024-04-15 04:04:42","doi":"10.21203/rs.3.rs-4137819/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"e6e4efd0-71e4-4517-b7ac-b16efdd92556","owner":[],"postedDate":"April 15th, 2024","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[{"id":30632461,"name":"Physical sciences/Engineering"},{"id":30632462,"name":"Physical sciences/Engineering/Mechanical engineering"}],"tags":[],"updatedAt":"2024-07-22T10:26:27+00:00","versionOfRecord":[],"versionCreatedAt":"2024-04-15 04:04:42","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-4137819","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-4137819","identity":"rs-4137819","version":["v1"]},"buildId":"qtupq5eGEP_6zYnWcrvyt","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}