A novel fault diagnosis method for gearbox based on RVMD and TELM with composite chaotic grey wolf optimizer

Abstract

Fault diagnosis for gearbox by robust variational mode decomposition (RVMD) and twin extreme learning machine (TELM) with composite chaotic grey wolf optimizer (CCGWO) is proposed in this study. Robust variational mode decomposition is an advanced signal processing technique designed to decompose complex signals into intrinsic mode functions (IMFs) while maintaining robustness against noise and outliers,which addresses the limitations of variational mode decomposition (VMD), particularly its sensitivity to noise and its tendency to produce suboptimal results in the presence of outliers. The proposed twin extreme learning machine with composite chaotic grey wolf optimizer (CCGTELM) model can extract higher-level features and has higher classification accuracy than traditional ELM. A novel grey wolf optimization algorithm, named composite chaotic grey wolf optimizer (CCGWO), is used to optimize the kernel parameter of TELM. Thus, TELM with CCGWO (DGTELM) is used to fault diagnosis for gearbox.The experimental results demonstrates that fault diagnosis accuracy of RVMD–CCGTELM is higher than VMD-TELM, VMD–DNN, VMD–CNN, VMD–LSTM, EMD–ELM and WT–ANN, and RVMD–CCGTELM is suitable for fault diagnosis of gearbox.

Introduction

The gearbox is one of the most critical components of a station wagon. Any defects in the gearbox can cause the station wagon to malfunction, making the study of fault diagnosis methods for gearboxe highly important. In recent years, numerous fault diagnosis techniques have been developed^1,2, including artificial neural networks (ANNs)^{3, 4, 5, 6–7} and extreme learning machines (ELMs)^8,9. For instance, Wu et al.¹⁰ introduced a method combining wavelet transform and artificial neural networks (WT–ANN) for identifying faults in rotating machinery. To enhance the accuracy of fault diagnosis using artificial neural networks, deep neural networks have been proposed. For example, Li et al.¹¹ utilized deep neural networks for diagnosing faults in planetary gears, while Li et al.¹² proposed a method integrating variational mode decomposition and deep neural networks (VMD–DNN) for the same purpose. These studies demonstrate that deep learning methods can significantly improve model performance. Additionally, the network structure of extreme learning machines is simple, and unlike other methods, the connection weights and thresholds between layers do not require adjustment, which significantly enhances the learning and training efficiency of ELMs^{13, 14–15}. As a result, ELMs have been widely applied in fault diagnosis. For example, Silva et al.¹⁶ developed an intelligent embedded system for decision support in pulsed eddy current corrosion diagnosis using ELMs. Similarly, Silva et al.¹⁷ applied ELMs to study time-of-flight diffraction ultrasound for flaw diagnosis in welded steel plates. The proposed twin extreme learning machine (TELM) generates a pair of non-parallel hyperplanes in a random feature space, utilizing only the least squared loss to minimize empirical error in its optimization problem. This approach enables the extraction of higher-level features and achieves higher classification and recognition accuracy compared to traditional ELM.

Empirical mode decomposition is prone to mode mixing, where signals with different feature time scales may be mixed in the same intrinsic mode function (IMF) component, and endpoint effects are significant during the decomposition process. The decomposition results near the boundaries are unstable, affecting the accuracy of the results. Wavelet transform may generate false fluctuations at signal abrupt changes, resulting in distortion of the constructed time–frequency images. Robust Variational Mode Decomposition (RVMD) is an advanced signal processing technique designed to decompose complex signals into intrinsic mode functions (IMFs) while maintaining robustness against noise and outliers. It is an enhancement of the traditional Variational Mode Decomposition (VMD) algorithm, which is widely used for signal decomposition in various fields such as biomedical engineering, mechanical fault diagnosis, and financial data analysis. RVMD addresses the limitations of VMD, particularly its sensitivity to noise and its tendency to produce suboptimal results in the presence of outliers. By incorporating robust statistical methods and adaptive mechanisms, RVMD ensures more reliable and accurate signal decomposition. RVMD is designed to overcome the limitations of VMD by incorporating robust statistical methods and adaptive mechanisms. RVMD replaces the traditional squared loss function with robust loss functions, such as the Huber loss or the Tukey biweight loss, which are less sensitive to outliers and noise. RVMD introduces adaptive mechanisms for tuning the penalty parameter and the number of IMFs, ensuring optimal performance across different types of signals. RVMD employs nonlinear convergence factors to balance global and local search capabilities, preventing premature convergence and improving the accuracy of the decomposition.

Therefore,fault diagnosis for gearbox by robust variational mode decomposition and twin extreme learning machine with differential evolution-based grey wolf optimization (CCGWO) algorithm is proposed in this paper. As the features of the time–frequency image based on RVMD are clearer than those of VMD, RVMD is used to decompose the vibration signals of gearbox, and create the time–frequency images of the vibration signals of gearbox. Twin extreme learning machine with CCGWO (DGTELM) is used to fault diagnosis for gearbox. To address these issues, this paper proposes a composite chaotic grey wolf optimization algorithm, referred to as CCGWO. In the population initialization phase, the algorithm employs a composite chaotic sequence, Logistic-Cos, which combines the Logistic sequence and the Cos sequence, to initialize the population. This approach enriches population diversity and enhances the ability to find the optimal solution. Composite chaotic grey wolf optimization algorithm is used to optimize the kernel parameter of TELM. The experimental results demonstrates that RVMD–CCGTELM is suitable for fault diagnosis of gearbox of station wagon.

Robust variational mode decomposition

Robust Variational Mode Decomposition (RVMD) is an advanced signal processing technique designed to decompose complex signals into intrinsic mode functions (IMFs) while maintaining robustness against noise and outliers. It is an enhancement of the traditional Variational Mode Decomposition (VMD) algorithm, which is widely used for signal decomposition in various fields such as biomedical engineering, mechanical fault diagnosis, and financial data analysis. RVMD addresses the limitations of VMD, particularly its sensitivity to noise and its tendency to produce suboptimal results in the presence of outliers. By incorporating robust statistical methods and adaptive mechanisms, RVMD ensures more reliable and accurate signal decomposition.

Before delving into RVMD, it is essential to understand the foundational principles of VMD. VMD is a fully adaptive signal decomposition method that decomposes a signal into a finite number of band-limited IMFs. Each IMF is characterized by a specific center frequency and bandwidth. The key idea behind VMD is to formulate the decomposition process as a variational optimization problem^18,19, where the goal is to minimize the sum of the bandwidths of all IMFs while ensuring that their sum reconstructs the original signal.

The VMD algorithm solves the following constrained optimization problem:

1

where  denotes the Dirac distribution, denotes the convolution operator.

The VMD algorithm iteratively updates the IMFs and their center frequencies until convergence is achieved. While VMD is effective in many applications, it has limitations, particularly in noisy environments, where its performance can degrade significantly.The traditional VMD algorithm has several limitations that RVMD aims to address: (1) VMD is highly sensitive to noise, which can distort the extracted IMFs and lead to inaccurate signal decomposition. (2)The presence of outliers in the signal can adversely affect the performance of VMD, causing it to produce unreliable results. (3) VMD relies on predefined parameters, such as the number of IMFs and the penalty parameter, which may not be optimal for all types of signals. (4)VMD may converge to local optima, especially in complex or high-dimensional signals, leading to suboptimal decomposition.

RVMD is designed to overcome the limitations of VMD by incorporating robust statistical methods and adaptive mechanisms. RVMD replaces the traditional squared loss function with robust loss functions, such as the Huber loss or the Tukey biweight loss, which are less sensitive to outliers and noise. RVMD introduces adaptive mechanisms for tuning the penalty parameter and the number of IMFs, ensuring optimal performance across different types of signals. RVMD employs nonlinear convergence factors to balance global and local search capabilities, preventing premature convergence and improving the accuracy of the decomposition. RVMD incorporates regularization techniques to enhance the stability and robustness of the decomposition process.

The RVMD algorithm extends the traditional VMD formulation by introducing robust loss functions and regularization term. The optimization problem in RVMD is formulated as:

2

where denotes the robust loss function,and denotes a regularization term that enhances the stability of the decomposition.

The robust loss function is chosen to be less sensitive to outliers and noise. For example, the Huber loss function is defined as:

3

where  is a threshold parameter that determines the transition point between the quadratic and linear regions of the loss function.

In the RVMD algorithm, IMFs are updated as follows:

4

Update the center frequencies using the updated IMFs:

5

where  is the Fourier transform of .

RVMD offers several advantages over traditional VMD: (1) By using robust loss functions, RVMD is less sensitive to noise and outliers, ensuring more accurate signal decomposition. (2) RVMD adaptively tunes its parameters, making it suitable for a wide range of signals and applications. (3) The use of nonlinear convergence factors and regularization techniques prevents premature convergence and enhances the accuracy of the decomposition. (4) RVMD can be applied to various types of signals, including biomedical signals, mechanical vibrations, and financial time series.

Figure 1 compares the time–frequency images generated by VMD and RVMD for the no-damage condition, Fig. 2 provides the comparison for the abrasion condition, Fig. 3 shows the comparison for the tooth pitting condition, and Fig. 4 illustrates the comparison for the snaggletooth condition. It is evident that the time–frequency images produced by RVMD exhibit clearer features compared with those generated by VMD. As a result, RVMD is employed in this paper to create the time–frequency images of the gearbox vibration signals.

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Fig. 1

Comparison of time–frequency image between VMD and RVMD in the situation of no damage.

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Fig. 2

Comparison of time–frequency image between VMD and RVMD in the situation of abrasion.

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Fig. 3

Comparison of time–frequency image between VMD and RVMD in the situation of tooth pitting.

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Fig. 4

Comparison of time–frequency image between VMD and RVMD in the situation of snaggletooth.

The CCGTELM model

The classification model of ELM,, where is the mapping function matrix, is the weight vector connecting the output layer and the hidden layer, , is the Hidden layer output matrix, is the penalty factor, and I is the identity matrix.The output of extreme learning machine is expressed as .

The core of TELM algorithm is to solve the following two quadratic programming:

6

where denotes the output, denotes the output of hidden layer,, , denotes the weight, denotes the bias, denotes the activation function, denote error vector, denote the penalty parameters.

Introducing regularization term , can be obtained as follows:

7

where denotes the Lagrange multipliers.

Finally, TELM can be obtained as follows:

8

As the penalty parameters, of kernel function needs to be determined, a differential evolution strategy-based GWO is employed to determine the penalty parameters. Traditional GWO algorithm limits the multi class of the grey wolf population,which will make the algorithm easily converge and fall into local optima.

Although the Grey Wolf Optimization (GWO) algorithm is characterized by its simplicity and efficiency, its performance is highly sensitive to the initial population settings when dealing with complex problems such as high-dimensional and multi-peak optimization. The linear variation of the convergence factor in the search mechanism may lead to premature convergence, making the algorithm prone to falling into local optima and suffering from low search accuracy.

To address these issues, this paper proposes a composite chaotic grey wolf optimization algorithm, referred to as CCGWO. In the population initialization phase, the algorithm employs a composite chaotic sequence, Logistic-Cos, which combines the Logistic sequence and the Cos sequence, to initialize the population. This approach enriches population diversity and enhances the ability to find the optimal solution. During the search phase, a nonlinear convergence factor is introduced to replace the traditional linear convergence factor, balancing global and local search capabilities. This improves the algorithm’s search efficiency, enhances its comprehensiveness and accuracy, and avoids getting trapped in local optima. Furthermore, Cauchy mutation is introduced, leveraging the perturbation capability of the Cauchy operator to provide a broader search space and increase the probability of finding the global optimal solution.

In the Grey Wolf Optimization (GWO) algorithm, the initial distribution of the population significantly impacts its performance^20,21. However, the random strategy used in the initialization phase of the traditional GWO algorithm may result in an initial grey wolf population that is either overly concentrated or unevenly distributed. Both scenarios are detrimental to the algorithm’s ability to find the optimal solution. Chaotic systems are highly sensitive to initial conditions, meaning even minor changes can lead to significant differences in the system’s final state. This characteristic enables chaotic initialization to generate an initial population with high randomness and diversity, ensuring uniform distribution and coverage of the entire solution space. This approach helps prevent premature convergence of the optimization algorithm, avoids getting trapped in local optima, and enhances the algorithm’s global search capability, thereby improving the likelihood of finding the global optimal solution.

Here, the objective function can be given as follows:

9

where is the total number of the validation samples, and is the total number of the validation samples with correct validation.

The fitness is defined as follows:

10

The Logistic chaotic mapping is widely used in the initialization process of swarm optimization algorithms due to its simplicity, ease of implementation, and single control parameter, which makes it convenient to adjust and embed. However, the Logistic mapping has a limited number of folding iterations, and the generated random numbers tend to cluster at the extremes, which somewhat restricts its effectiveness in improving the initial population distribution. To address this limitation, the CCGWO algorithm adopts a composite chaotic mapping model, Logistic-Cos, which combines the Logistic mapping and the Cos mapping, for population initialization during the initialization phase.

This processing of improving the GWO algorithm by using composite chaotic mapping model is shown as follows:

11

To avoid poor convergence and falling into local optima, the CCGWO algorithm introduces Cauchy mutation, leveraging the perturbation capability of the Cauchy operator to increase the probability of finding the global optimal solution.

As shown in Fig. 5,the processing of determining the parameters of TELM by using differential evolution strategy-based grey wolf optimization algorithm,which is described as follows:

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Fig. 5

The processing flow of determining the parameters of TELM by using differential evolution strategy-based grey wolf optimization algorithm.

Step 1 Initialize the range of parameters and configure the parameters of CCGWO. Set the grey wolf population size to 20 and the maximum number of iterations to 100.

Step 2 Define the fitness of each grey wolf individual according to Eq. (10).

Step 3 Calculate the fitness values of each grey wolf individual, determine their hierarchical roles (α, β, and δ), and record their respective positions. Based on the fitness values, classify the grey wolf population into α, β, and δ wolves. During each iteration of the CCGWO search process, the α, β, and δ wolves guide the remaining grey wolf individuals in the hunting process.

Step 4 Update the position of each individual and select the α, β, and δ wolves based on their fitness values. Use composite chaotic mapping to generate a new chaotic population for the α wolf, which has the highest fitness value in the current population. This population is inversely mapped back to the original solution space to assess its fitness. The optimal solution from this population then replaces the corresponding solution in the original population space.

Step 5 The remaining grey wolves locate their target by following the guidance of the α, β, and δ wolves.

Step 6 The process terminates when the maximum number of iterations is reached. Otherwise, loop to Step 3.

Step 7 The optimal parameters of TELM are obtained.

The comparison of the optimization process among CCGWO, GWO, and PSO is given in Fig. 6. It is obvious that the convergence of CCGWO is better than that of GWO,and PSO.

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Fig. 6

Comparison of the optimization process among CCGWO, GWO, and PSO.

Experimental analysis

The gearbox fault diagnosis test setup is illustrated in Fig. 5. Vibration signals from the gearbox were captured using a vibration signal acquisition device. As depicted in Fig. 7, the fault diagnosis system for the gearbox comprises a motor, gearbox, coupling, brake, acceleration sensor, corresponding acquisition card, and a computer. The brake can be used to generate the load. The sampling frequency is 5 kHz, and the vibration signals are collected using an acceleration sensor paired with its acquisition card. The datasets include the vibration data from the following two conditions: 1200 rpm with no load and 1800 rpm with a medium load. The gearbox fault diagnosis model is implemented in MATLAB. Three primary gearbox faults—tooth pitting, snaggletooth, and abrasion—are investigated in this experiment.The vibration signals of the state of no damage and the faults of gearbox including tooth pitting, snaggletooth, abrasion are collected,respectively.

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Fig. 7

Fault diagnosis test rig of gearbox.

In this experiment, a total of 800 samples are used for training, consisting of 200 samples each for no damage, tooth pitting, snaggletooth, and abrasion. Additionally, 200 samples are used for testing, with 50 samples each for no damage, tooth pitting, snaggletooth, and abrasion. Firstly,as the features of the time–frequency image based on RVMD are clearer than those of VMD, RVMD is used to decompose the vibration signals of gearbox,and create the time–frequency images of the vibration signals of gearbox.Secondly,analysis the correlation of the texture features, the texture features with weak correlation of the time–frequency images are extracted as the input vector of the diagnosis model,and the corresponding state of gearbox is used as the output of the diagnosis model.Thirdly,set the grey wolf population size to 20 and the maximum number of iterations to 100,the ranges of the penalty parameters are respectively set to [1 10000],and the optimal parameters of TELM are obtained by CCGWO.Finally,the optimal TELM can be obtained and used to fault diagnosis for gearbox.For different gearbox’s faults,the bigger the discriminability of the feature is,the more important the feature is. As shown in Figs. 8 and 9, the discriminability ability for gearbox’s faults by RVMD is better than by VMD.Obviously,the correlation between these features is relatively weak, and there are more features with discriminability ability for gearbox’s faults by RVMD than by VMD. In order to highlight the superiority of RVMD–CCGTELM, VMD–TELM, VMD–DNN, VMD–CNN, VMD–LSTM, EMD–ELM and WT–ANN are respectively compared with RVMD–CCGTELM. The parameters of RVMD–CCGTELM are selected by CCGWO, and the parameters of other models are selected by GWO.

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Fig. 8

Comparison of features between tooth pitting and abrasion by RVMD.

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Fig. 9

Comparison of features between tooth pitting and abrasion by VMD.

The fault diagnosis results for the gearbox using RVMD–CCGTELM, VMD–TELM, VMD–DNN, VMD–CNN, VMD–LSTM, EMD–ELM and WT–ANN are presented in Fig. 10. In Fig. 10a, the results from RVMD–CCGTELM show that 2 samples are misdiagnosed. Figure 10b displays the results from VMD–TELM, where 13 samples are misdiagnosed. Figure 10c illustrates the results from VMD–DNN, with 16 samples incorrectly diagnosed. Similarly, Fig. 10d shows the results from VMD–DNN, also with 13 misdiagnosed samples, Fig. 10e presents the results from VMD–LSTM, where 14 samples are incorrectly diagnosed, Fig. 10f presents the results from EMD-ELM, where 18 samples are incorrectly diagnosed, and Fig. 10g presents the results from WT–ANN, where 24 samples are incorrectly diagnosed.

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Fig. 10

Comparison of the diagnosis results among RVMD–CCGTELM, VMD–TELM, VMD–DNN, VMD–CNN, VMD–LSTM, EMD–ELM and WT–ANN.

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As summarized in Table 1, fault diagnosis accuracy of RVMD–CCGTELM is 99%, fault diagnosis accuracy of VMD–TELM is 93.5%, fault diagnosis accuracy of VMD–DNN is 92%, fault diagnosis accuracy of VMD–CNN is 93.5%, fault diagnosis accuracy of VMD–LSTM is 93%,fault diagnosis accuracy of EMD–ELM is 91%,fault diagnosis accuracy of WT–ANN is 88%.

Table 1. Comparison of the diagnosis accuracies among RVMD–CCGTELM, VMD–TELM, VMD–DNN, VMD–CNN, VMD–LSTM, EMD–ELM and WT–ANN.

Method	The number of the samples with correct diagnosis	Accuracy (%)
RVMD–CCGTELM	198	99
VMD–TELM	187	93.5
VMD–DNN12	184	92
VMD–CNN	187	93.5
VMD–LSTM	186	93
EMD–ELM	182	91
WT–ANN10	176	88

Fault diagnosis accuracy of RVMD–CCGTELM increases by 5.5% compared to VMD–TELM, by 7% compared to VMD–DNN, by 5.5% compared to VMD–CNN, by 6% compared to VMD–LSTM, by 8% compared to EMD–ELM, and by 11% compared to WT–ANN.

These results indicate that RVMD–CCGTELM achieves higher accuracy in fault diagnosis compared with VMD–TELM, VMD–DNN, VMD–CNN, VMD–LSTM, EMD–ELM and WT–ANN. The experimental findings confirm that RVMD–CCGTELM is well-suited for fault diagnosis of gearbox.

To evaluate the anti-interference capability of RVMD–CCGTELM, comparisons were made with VMD–TELM, VMD–ELM, VMD–DNN, and WT–ANN under white noise interference,impulsive noise, respectively.

The fault diagnosis results for the gearbox using RVMD–CCGTELM, VMD–TELM, VMD–DNN, VMD–CNN, VMD–LSTM, EMD–ELM and WT–ANN under white noise interference are illustrated. As summarized in Table 2, fault diagnosis accuracy of RVMD–CCGTELM is 98.5%, fault diagnosis accuracy of VMD–TELM is 93%, fault diagnosis accuracy of VMD–DNN is 91%, fault diagnosis accuracy of VMD-CNN is 93%, fault diagnosis accuracy of VMD–LSTM is 92%, fault diagnosis accuracy of EMD–ELM is 90%,fault diagnosis accuracy of WT–ANN is 86.5%.

Table 2. Comparison of the diagnosis accuracies among RVMD–CCGTELM, VMD–TELM, VMD–DNN, VMD–CNN, VMD–LSTM, EMD–ELM and WT–ANN in the case of white noise interference.

Method	The number of the samples with correct diagnosis	Accuracy (%)
RVMD–CCGTELM	197	98.5
VMD–TELM	186	93
VMD–DNN12	182	91
VMD–CNN	186	93
VMD–LSTM	184	92
EMD–ELM	180	90
WT–ANN10	173	86.5

Fault diagnosis accuracy of RVMD–CCGTELM increases by 5.5% compared to VMD–TELM, by 7.5% compared to VMD–DNN, by 5.5% compared to VMD–CNN,by 6.5% compared to VMD–LSTM, by 8.5% compared to EMD–ELM,and by 12% compared to WT–ANN.

These results indicate that RVMD–CCGTELM achieves higher accuracy in fault diagnosis compared with VMD–TELM, VMD–DNN, VMD–CNN, VMD–LSTM, EMD–ELM and WT–ANN. The experimental findings confirm that RVMD–CCGTELM is well-suited for fault diagnosis of gearbox.The experimental findings confirm that RVMD–CCGTELM is well-suited for fault diagnosis of gearbox, even in the presence of white noise interference.

The fault diagnosis results for the gearbox using RVMD–CCGTELM, VMD–TELM, VMD–DNN, VMD–CNN, VMD–LSTM, EMD–ELM and WT–ANN under impulsive noise interference are illustrated. As summarized in Table 3, fault diagnosis accuracy of RVMD–CCGTELM is 98%, fault diagnosis accuracy of VMD–TELM is 92%, fault diagnosis accuracy of VMD–DNN is 90%, fault diagnosis accuracy of VMD–CNN is 91%, fault diagnosis accuracy of VMD–LSTM is 91.5%, fault diagnosis accuracy of EMD–ELM is 88%, fault diagnosis accuracy of WT–ANN is 85%.

Table 3. Comparison of the diagnosis accuracies among RVMD–CCGTELM, VMD–TELM, VMD–DNN, VMD–CNN, VMD–LSTM, EMD–ELM and WT–ANN in the case of impulsive noise interference.

Method	The number of the samples with correct diagnosis	Accuracy (%)
RVMD–CCGTELM	196	98
VMD–TELM	184	92
VMD–DNN12	180	90
VMD–CNN	182	91
VMD–LSTM	183	91.5
EMD–ELM	176	88
WT–ANN10	170	85

Fault diagnosis accuracy of RVMD–CCGTELM increases by 6% compared to VMD–TELM, by 8% compared to VMD–DNN,by 7% compared to VMD–CNN,by 6.5% compared to VMD–LSTM,by 10% compared to EMD–ELM,and by 13% compared to WT–ANN.

These results indicate that RVMD–CCGTELM achieves higher accuracy in fault diagnosis compared with VMD–TELM, VMD–DNN, VMD–CNN, VMD–LSTM, EMD–ELM and WT–ANN. The experimental findings confirm that RVMD–CCGTELM is well-suited for fault diagnosis of gearbox.The experimental findings confirm that RVMD–CCGTELM is well-suited for fault diagnosis of gearbox, even in the presence of impulsive noise interference.

Conclusions

This paper proposes a novel fault diagnosis method for the gearbox utilizing robust variational mode decomposition (RVMD) and twin extreme learning machine (TELM) optimized by chaotic composite grey wolf optimization (CCGWO). The key contributions of this paper are as follows: (1) Since the time–frequency image features derived from robust variational mode decomposition are more distinct than those derived from traditional variational mode decomposition, RVMD is employed to decompose the vibration signals of the gearbox and generate corresponding time–frequency images. (2) Twin ELM is constructed by stacking multiple layers of extreme learning auto-encoders, enabling it to extract higher-level features and achieve superior classification accuracy than extreme learning machines. Additionally, CCGWO is utilized to optimize the kernel parameters of TELM and enhance its performance. Consequently, the proposed twin extreme learning machine with CCGWO (CCGTELM) is applied to fault diagnosis for gearbox. The experimental results demonstrate that RVMD–CCGTELM is highly effective and suitable for fault diagnosis for gearbox of station wagon.

Acknowledgements

This work was supported by 1.Natural Science Foundation of Hainan province (No. 722RC719); 2. Key Laboratory of Island Tourism Resource Data Mining and Monitoring,Ministry of Culture and Tourism (KLITRDMM 2024-5); 3. Scientific Research Foundation of Hainan Tropical Ocean University (No. RHDRCSK202411).

Author contributions

All authors wrote and reviewed the manuscript.

Data availability

The data that support the findings of this study are available from the corresponding author upon request.

Declarations

Competing interests

The authors declare no competing interests.

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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