Fault diagnosis method of rolling bearing based on SSA-VMD and RCMDE

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Scientific Reports volume 14, Article number: 30637 (2024 ) Cite this article friction bearing material

As an important part of rotating machinery equipment, the performance of rolling bearings will directly affect the operating status and work efficiency of the entire unit1. With the development of rotating machinery and equipment in the direction of high speed, high precision and high reliability, the working environment of rolling bearings is becoming more and more harsh, resulting in a significant increase in the incidence of failure. According to statistics, about 50% of the failures of rotating machinery equipment are caused by rolling bearings2 . When a rolling bearing fails, it not only leads to the downtime of rotating machinery and equipment, but also can cause serious safety accidents. Therefore, it is of great economic and social significance to the research on the fault diagnosis of rolling bearings.

However, the vibration signals of rolling bearings collected by sensors usually contain various components such as fault shock, background noise, and environmental noise, which are nonlinear, non-stationary, and highly fluctuating, resulting in more challenging signal processing and fault identification3,4.

The purpose of signal processing is to accurately identify and enhance fault information from the mixed signals, and at the same time filter out the mixed noise to obtain clear fault signatures5. In order to achieve this purpose, there are two main types of signal processing methods commonly used: one is the traditional signal analysis method with Fourier transform (FT) as the core6; The other is a recursive signal decomposition method based on Empirical Mode Decomposition (EMD)7.

Zhang Q et al8 combined short-time Fourier Transform (STFT) and convolutional neural networks (CNNs) to effectively diagnose rolling bearing faults. Liang P et al.9 proposed a semi-supervised fault diagnosis method based on wavelet transform rolling bearings, which solved the unknown fault diagnosis problem under different working conditions. Cheng Z10 proposed a rolling bearing fault feature extraction and diagnosis model based on improved wavelet transform-support vector machine, and the experimental results show that the proposed method has good feature extraction performance and diagnostic accuracy. However, STFT does not handle stationary signals well, and the wavelet transform cannot meet the requirements for some signals due to its fixed window area11.

Meng D et al12 used EMD and kurtosis criterion filtering to extract fault features from wind power rolling bearings. Liu Y et al13 proposed a bearing fault diagnosis method based on the combination of EMD and genetic algorithm optimization BP neural network. However, due to the shortcomings of its own algorithms, EMD has problems such as endpoint effect and modal aliasing, which will directly affect the accuracy of its signal processing results, so researchers have proposed a variety of methods to solve the modal aliasing problem, among which the most widely used are ensemble empirical mode decomposition (EEMD) , and complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN)14,15. However, EEMD and CEEMDAN solve the problem of modal aliasing while generating the problem of residual noise.

Nevertheless, variational mode decomposition (VMD)16 has been favored by many researchers and has been applied in the field of rolling bearing fault diagnosis due to its solid theoretical foundation and effective solution to the problems caused by the above decomposition methods17. Yi K et al18 proposed that the rolling bearing fault feature extraction algorithm based on VMD and multi-point optimal minimum entropy deconvolution adjustment method can filter out interference such as background noise and accurately extract failure features. Jiang et al19 proposed a fault diagnosis method for rolling bearings based on VMD and gray wolf algorithm optimization extreme learning machine, which can effectively identify the abnormal state of rolling bearings.

Nonetheless, the decomposition effect of VMD is directly affected by the number of modal decompositions K and the secondary penalty factor \(\alpha\) . In the past, scholars have often determined these parameters manually and sequentilly, such as Wang et al20 used the energy loss coefficient to establish the number of modal decompositions K, and subsequently applied information entropy to determine the \(\alpha\) . In some cases, this method of manual parameter selection neglects the interaction between the two parameters, requiring significant computational resources and often resulting in inadequate outcomes. As a result, researchers have sought to employ intelligent optimisation algorithms to optimise both parameters of VMD simultaneously. For example, Li et al21 used the grey wolf optimization (GWO) algorithm to optimize the parameters of VMD, while Liu et al22 used the whale optimization algorithm (WOA) to obtain the optimal combination of the two parameters. The sparrow search algorithm (SSA), as a more advanced artificial intelligence algorithm, exhibits superior optimization ability, enhanced search speed, and improved resistance to local optima compared to both the WOA and GWO algorithms23.

How to obtain effective fault characteristics from signals is one of the difficulties in fault diagnosis. In recent years, in order to effectively extract fault features and quantify the complexity of rolling bearing vibration signals, nonlinear dynamic feature extraction methods such as permutation entropy (PE), sample entropy (SE), and dispersion entropy (DE) have been widely used in the study of rolling bearing fault diagnosis24,25,26. However, the above method only evaluates the complexity of the signal on a single time scale, and the obtained signal feature information is limited, and sometimes the ideal recognition effect cannot be achieved. To this end, researchers have proposed multi-scale permutation entropy (MSE)27 algorithms, such as multi-scale dispersion entropy (MDE)28, multi-scale permutation entropy (MPE)29, and multi-scale fuzzy entropy (multi-scale). fuzzy entropy, MFE)30, etc. However, when the data series is long, the above method may have problems such as slow operation speed, serious loss of fault information, and unclear influence of amplitude relationship between signals. To solve this problem, Azami et al31 proposed refined composite multi-scale dispersion entropy (RCMDE). This method refines the multi-scale analysis process and fully considers the relationship between coarse-grained sequences, which significantly improves the problem of information loss32. However, although RCMDE has a certain ability to express nonlinear fault characteristics, if the RCMDE is directly analyzed, the signals with different fault types will show similar distribution characteristics, which will reduce the accuracy of fault feature extraction, and the signals need to be processed to filter out the influence of interference components33.

In summary, this paper combines the superiority of VMD in signal processing with the effectiveness of RCMDE in fault feature extraction, and proposes a diagnostic method for rolling bearing fault signals based on SSA-VMD and RCMDE. Firstly, the SSA algorithm is used to automatically search for the two decomposition parameters of the VMD according to the signal characteristics, and the IMF components sensitive to the fault feature information are selected according to the intrinsic mode function (IMF) component selection method for reconstruction. Then, the RCMDE value of the reconstructed signal on the specified embedding dimension is calculated to form the state eigenvector. Finally, the KKNN classifier is used to diagnose the fault types of rolling bearings.

Step 1 Collect the vibration signals of rolling bearings under different states on the rolling fault simulation test bench. Step 2 The envelope entropy is used as the fitness function, and the key parameters of VMD are optimized by SSA. The fault signal was decomposed by using the optimized VMD to obtain multiple IMF components. Step 3 The discriminant algorithm of the sensitive IMF component of the comprehensive evaluation factor in the time-frequency domain is used to obtain the IMF component sensitive to the signal feature information for the reconstruction signal. Step 4 RCMDE is used to extract features from the reconstructed signal and form a state feature set. Step 5 The state feature set obtained in step 4 is input into the KKNN classifier for classification. Fig. 1 shows the troubleshooting flowchart.

The core idea of the VMD algorithm is to construct the signal into a variational problem model, and then search for the optimal solution of the model through iterative updates, that is, to continuously iterate the center frequency and bandwidth of each modal component to realize the adaptive decomposition of each frequency band of the signal, and to decompose the signal into IMF components with different center frequencies and bandwidths. A detailed theory of VMD can be found in refer16.

SSA algorithm is an intelligent optimization algorithm inspired by sparrow foraging and anti-predation behavior, which includes finders, predators and vigilantes in the population. The number of finders and predators varies dynamically, but their proportion of the population remains constant, and the two can be converted to each other according to the amount of energy reserves. The number of vigilantes accounts for 10%-20% of the sparrow population. During the foraging process, the location of the three is constantly updated to obtain the best resource location.

The finder has a high energy and mainly provides foraging directions and areas for the entire population, and its position update is expressed as.

where t is the number of iterations, \(X_{i,j}^{t}\) is the position of the i-th sparrow in the j-dimension after t iterations, \(\Phi \in (0,1)\) is the random number, \(R_2\in [0,1]\) is the early warning value, \(ST\in [0.5,1]\) is the safe value, Q is the random number that obeys the normal distribution of [0,1], L is the matrix of \(1\times d\) and each element in the matrix is 1.

The Predator follows the finder for food, and its location is updated as follows.

where \(X_{p}^{t+1}\) is the optimal position in the current discoverer, \(X_{w}^{t}\) is the global worst position, A is a matrix of 1\(\times\) d, and each element in it is 1 or -1, and \({A^ + } = {A^T}{(A{A^ + })^{ - 1}}\) .

Vigilantes act as guardians of the safety of the population, and if they become aware of potential danger, the population will quickly move to a safer area. Its location is updated as follows.

where \(X_b^t\) denotes the global optimal position, \(\omega\) is the step size control parameter, which is a random number obeying the normal distribution of [0,1], \(k\in [-1,1]\) is a uniform random number , \(\theta\) is a very small constant, and the avoided denominator is 0, \(f_i\) and \(f_g\) are the current global optimal and best fitness values, respectively.

The key to optimizing VMD parameters using AI search algorithms is to determine a suitable fitness function that can quantify the performance of the VMD algorithm under the current parameter configuration and update the parameters to optimize the overall performance based on this result34. As a fitness function, envelope entropy can provide a more comprehensive measure of information, with stronger global search performance and robustness. Compared with other fitness functions35,36, it has better ability to adapt to complex environments and changes and higher optimization efficiency37. In order to make the selection of VMD parameters reasonable, ensure the accuracy and stability of its decomposition results, and improve the robustness of the algorithm, the envelope entropy was selected as the fitness function. Steps to optimize VMDs with SSA. Step 1 Initialize the parameters of the SSA algorithm by initializing the parameter combination and setting its range. Step 2 The fitness values and corresponding positions of each sparrow were calculated. Step 3 Update the sparrow position according to fitness: the finder updates the location according to Eq. (1); The predator updates the position according to Eq. (2); The vigilant updates the location according to Eq. (3). Step 4 The fitness values of the current sparrows were newly calculated, excellent individuals were selected as candidates for the next-generation population, and the parameter combinations were updated. Step 5 Determine whether the stopping condition of the maximum number of iterations is satisfied. If so, the optimal parameter combination [K, \(\alpha\) ] is output; Otherwise, return to Step 2 and iterate until the stop condition is met.

Fig. 2 shows the flowchart of SSA optimizing the key parameters of VMD.

How to obtain the effective component from the IMF component decomposed by VMD is one of the decisive factors for the effectiveness of signal processing. The rejection and reconstruction of fake IMFs based on a single time-domain or frequency-domain criterion38,39 cannot take into account the time-frequency domain characteristics of the signal, resulting in the loss of key signal information. Therefore, this paper proposes a sensitive IMF component discrimination method based on time-frequency domain comprehensive evaluation factors. The kurtosis and frequency domain correlation number of each mode component after VMD decomposition is calculated, and the sensitive components that can effectively characterize the signal characteristics are selected after being weighted according to a certain coefficient. If the rolling bearing vibration signal x(t) is decomposed by SSA-VMD and obtains k IMF component \(u_i\) (i =1, 2,..., k), the specific calculation process of the sensitive IMF discriminant algorithm based on the comprehensive evaluation factor in the time-frequency domain is as follows.

The kurtosis values of each IMF component are calculated.

where \(\mu\) and \(\alpha\) are the mean and variance of the IMF component \(u_i\) , respectively; E(t) is the expected value of the variable t.

The frequency domain correlation number \(\rho _f\) between each IMF component and the original signal is calculated.

where \(G_u\) ,\({G_{{x_i}}}\) are the power spectrum of the signal \(u_i\) and x(t); \({{\bar{G}}_u}\) ,\({{\bar{G}}_x}\) is the mean value of the corresponding power spectrum; \(f_a\) is the frequency of analysis.

The time-frequency domain weighted value \(\gamma _i\) of the \(u_i\) is calculated using the following equation.

where \(\beta\) and \(\gamma\) are the weighting coefficients, and \(\beta +\gamma =1\) .

The comprehensive evaluation factor \(\delta _i\) based on the weighted value in the time-frequency domain is calculated.

The time-frequency domain weighted value of each component is calculated and the descending sequence is composed: \({\{\delta _i}\}={\{\delta _1,\delta _2,...,\delta _k}\}\) .

The comprehensive evaluation factor difference of two adjacent components was calculated.

The index m of the maximum difference is found, and the first m components of the sequence \({\{\delta }\}\) after reordering according to the comprehensive evaluation factor are the sensitive modal components containing the main characteristic information of the signal.

The flow chart of the discriminant algorithm of sensitive IMF components based on the comprehensive evaluation factors in the time-frequency domain is shown in Fig 3, which comprehensively considers the time-domain characteristics of each IMF component and the frequency domain cross-correlation of the original signal, and is less affected by interference factors in the discrimination process, which can more effectively identify the false IMF components that cannot characterize the fault information, weaken the influence of the modal components that are not related to the fault information, and increase the accuracy of fault feature extraction.

Discriminant algorithm of sensitive IMF components based on comprehensive evaluation factors in the time-frequency domain.

RCMDE is a further developed method based on MDE. Based on multi-scale coarse-graining, RCMDE further refines the time series, retains more effective information about the time series, solves the problem of data loss caused by MSE in coarse-graining, reduces the calculation error, has higher accuracy, and has a strong feature extraction ability for nonlinear nonstationary signals. A detailed theory of RCMDE can be found in refer31.

The k-means clustering algorithm is an unsupervised machine learning algorithm that can automatically group samples according to their similarity to each other to discover the internal structure and patterns of data40.

The basic principle of the KNN algorithm, proposed by Conver and Hart, is that each sample point in the data can be represented by k sample points closest to it41. When the k value is set to a large size, it is easier to rule out outliers, but with more samples, the model will be simpler and easier to form underfitting. When the k-value is set small, the underfitting problem can be solved, but the model will become complex, making it difficult to rule out outliers and causing overfitting. Therefore, when using the KNN algorithm, it is necessary to select an appropriate k value.

In order to fully verify the effectiveness of the proposed method, this paper first uses the proposed method to verify the experimental verification of different types of single faults and compound faults of rolling bearings. Then, in order to further explore its generalization performance and robustness, the single fault noise signal of the rolling element with different damage degrees was further used for experimentation. In order to highlight the superiority of VMD nonlinear signal processing and the effectiveness of the selection of key parameters based on SSA, a comparison method based on EMD-RCMDE, EEMD-RCMDE, CEEMDAN-RCMDE42 and RCMDE43 was established. In order to highlight the effectiveness of RCMDE feature extraction, VMD-MDE44, VMD-MFE45, and VMD-MPE46 comparison methods were established. In order to highlight the good performance of the KKNN classifier, the decision tree (DT), random forest (RF) and support vector machines (SVM) methods were selected for the comparison of classification accuracy.

The computer used in this study was equipped with an Intel Core i5-12500H processor, 16GB of RAM, and running Windows 11 operating system. In terms of software, MATLAB R2021a is mainly used for signal processing and feature extraction, and Python 2023.1 is used for fault classification.

Single-fault data acquisition test bench.

Fig. 5 shows the test bench used for composite fault diagnosis data acquisition. The platform consists of an AC motor, a motor speed controller, a digital force display, a hydraulic loading system, and a test bearing. The signal is acquired horizontally using a PCB352C33 accelerometer at a sampling frequency of 25.6 kHz. Rolling bearings of type LDKUER204 were used in the experiment. Select normal, inner and outer ring compound fault and inner ring, outer ring, rolling element, cage composite fault 3 state data, each state select 60 samples, each sample length is 6400, bearing state description is shown in Table 2.

Composite fault data acquisition test bench.

Because the signal acquisition system is not provided with a noise filtering device, the signal of the collected rolling bearing is interfered by factors such as background signal and noise and cannot effectively extract the signal characteristics, and it is difficult to accurately diagnose the working state and fault type of the rolling bearing if this is the basis, so the noise reduction process needs to be carried out on the collected vibration signal to highlight the signal characteristics.

Data labeled 0, 1, 2, and 3 were used for the test. Taking the fault data of the inner ring with label 1 as an example, a random signal is selected for analysis, and its time-domain waveform is shown in Fig. 6.

Time-domain waveform of the inner ring fault signal.

In Ref47,48,49, in the range of K\(\in [2,7]\) , \(\alpha \in [300,2000]\) , SSA was used to optimize the two parameters of VMD using entropy as the fitness function. In order to prove the good search performance of SSA, GWO and WOA were selected for comparison. In the experiment, the number of populations of each optimization algorithm is 50, and the maximum number of iterations is 10. The VMD parameters of the inner ring fault signal shown in Fig. 6 are optimized by using the above methods, and the fitness iteration curve and optimization results are shown in Fig. 7 and Table 3.

It can be seen from Fig. 7 that GWO has the slowest convergence speed in the early stage of algorithm operation, while WOA has a slightly faster convergence speed than GWO. However, in the middle and late stages, the convergence rate of GWO begins to surpass that of WOA, and its fitness value is relatively low. In contrast, the SSA had the best performance throughout the optimization process, having already converged at the 5th iteration, with a significantly faster convergence speed and smaller fitness values. The reason why SSA has such an advantage is mainly due to the clear division of labor and cooperation mechanism within its population. Under this mechanism, multiple roles of finder, predator, and vigilante work together, taking into account various factors, so that the population can efficiently move towards the global optimal solution, so that the population can move towards the global optimal solution efficiently, thus achieving rapid convergence.As can be seen from Table 3, SSA-VMD has the best fitness values. Therefore, SSA-VMD was used to decompose the inner circle fault signal shown in Fig. 6, and the time-domain waveform diagram and spectrum of IMF component were shown in Fig. 8.

Time-domain waveform and spectrum of the inner ring fault signal after SSA-VMD decomposition.

It can be seen from Fig. 8 that the VMD decomposition is reasonable, and the IMF components are mainly concentrated near the center frequency, and there is no obvious frequency divergence, which indicates that the proposed parameter selection method can effectively suppress the modal aliasing problem in the decomposition process, reduce the information leakage between the modal components, and provide a guarantee for the subsequent signal purification.

SSA-VMD parameters were optimized for labels 0, 1, 2, and 3 to obtain the best parameter combinations, as shown in Table 4.

When the frequency component of a signal changes rapidly, time-domain analysis may not accurately capture the characteristics of the signal, while frequency-domain analysis can reveal more fully the frequency components contained in the signal. The spurious modal components usually have unique frequency-domain characteristics, such as sharp peaks or abnormal frequency components. Therefore, the frequency domain information is more critical than the time domain information in the discrimination of false IMF components, according to which 0.4 is \(\beta\) taken and 0.6 is taken \(\gamma\) in Eq. 6. The time-frequency domain weighted values of each IMF component shown in Fig. 8 are calculated and the descending sequence \({\{\delta }\}\) is composed, as shown in Fig. 9.

A comprehensive evaluation factor \(\delta _i\) for each modal component based on SSA-VMD.

It can be seen from Fig. 9 that the maximum difference is between IMF1 and IMF3, with IMF3 and IMF4 being the sensitive IMF components containing the inner ring fault features shown in Fig. 6, and IMF1 and IMF2 being the interfering components containing noise and background signals. Therefore, IMF3 and IMF4 were selected as sensitive IMF components for reconstruction, and the time-domain waveform of the reconstructed signal is shown in Fig 10.

Time-domain waveform of the inner ring fault reconstruction signal.

Comparing Fig. 6 and Fig. 10, it can be seen that the proposed method effectively filters out most of the interference components such as noise, enhances the characteristics of the fault signal, and highlights the impact component of the signal, which provides support and guarantee for the subsequent fault feature extraction.

The RCMDE algorithm is used to extract the state features of the reconstructed signal shown in Fig. 10. In the process of calculating the RCMDE value of the reconstructed signal, the main parameters that affect the solution accuracy are embedding dimension m, category c, delay d and maximum scale factor \(\tau _{max}\) . According to refer31 and50, the embedding dimension m is used to detect dynamic changes in the signal, and if the value of m is too small, it will be difficult to detect the dynamic changes in the signal, and if the value of m is too large, the small changes in the signal will be ignored. Therefore, m can be 2, 3, or 4. Category c is used to determine the dispersion type of the signal, if the value of c is not appropriate, it may cause two sequences with large amplitudes to be classified into the same class or sequences with similar amplitudes to be classified as heterogeneous, so the value range of c is an integer between [4, 8]. The delay d can usually be selected as [1,3]. When the delay d > 1, the integrity of the signal frequency information acquisition may be affected, resulting in the loss of part of the frequency information, so the value of d is generally 1. The maximum scale factor \(\tau _{max}\) needs to be selected according to the specific analysis needs. In some cases, this parameter may need to be adjusted for better results.

In summary, this article selects m=2, 3 and 4; Analyze the performance of RCMDE with c=4-8; d=1; \(\tau _{max}\) =20.

Different parameters were selected to calculate the RCMDE value of the reconstructed signal shown in Fig. 10, and the results are shown in Fig. 11.

The RCMDE value of the inner ring fault reconstruction signal when different parameters are selected.

It can be seen from Fig. 11 that the overall trend of the RCMDE value of the reconstructed signal shown in Fig. 10 is basically the same under the condition of selecting different parameters. In other words, when category c is constant, the RCMDE value decreases with the increase of the scale factor \(\tau\) and gradually tends to be stable. When m=2,3,4; c=4,6,8, the stability of the curve is better.

Fig. 12 shows the calculation time for the RCMDE value when different parameters are set. It can be seen from Fig. 12 that the computation time t increases with the increase of category c and the embedding dimension m.

Time taken to calculate the RCMDE value when different parameters are used.

Combined with Figs. 11 and 12, it can be seen that when m=2, 3, 4 and c=4, the stability of the RCMDE value distribution curve is the best (it tends to be stable after the scale factor \(\tau\) \(\ge\) 5), and when m=2; c=4, the calculation time is shorter and the calculation efficiency is high. Considering the calculation results and time of RCMDE, m=2, c=4, d=1, \(\tau _{max}\) =20 are selected in this paper.

According to the method of selecting parameters, the RCMED values of the reconstructed signals labelled 0, 1, 2 and 3 were calculated respectively, and the calculation results are shown in Fig. 13.

RCMDE value of the fault type reconstruction signal.

It can be seen from Fig. 13 that the normal state rolling bearing signal has strong randomness, so the entropy value is the largest, and the overall change trend of the curve is positively correlated with the change of the scale factor \(\tau\) value, which is very different from the change trend of other fault signal curves. For inner ring faults, rolling element faults and outer ring faults, although the RCMDE entropy curves fluctuate to a certain extent, the trend is about the same as the whole, the RCMDE value decreases with the increase of scale factor, and the speed gradually slows down, and the RCMDE value curves of various states within the inner ring show obvious differentiation, which is conducive to the final classification.

By observing Fig. 13, it is found that when the value of \(\tau\) is less than 5, there is a crossover of the curves, which increases the difficulty of effectively distinguishing different bearing faults51. In addition, if the RCMDE value of the reconstructed signal at more scales is selected as the fault feature vector, although the purpose of state classification can be realized, it will cause redundancy of feature information and reduce the efficiency of fault diagnosis. If the value of the fault feature vector is small, the state characteristics of the signal may not be fully reflected, and the accuracy of fault diagnosis will be reduced. Considering comprehensively, when \(\tau\) =6-20, the RCMDE value of the rolling bearing reconstruction signal is selected as the signal feature to form the state feature vector and compose the state feature set.

(1) Comparison of different classification methods based on VMD-RCMDE. The state feature sets extracted from different types of fault data by VMD-RCMDE are input into the KKNN, DT, RF and SVM classifiers respectively. In the KKNN classifier, Euclidean distance is used as a measure of distance between samples. In order to determine the optimal k value, a grid search strategy was used to iterate through different k values in the set {3,5,7,9,11,13}, and combined with 3-fold cross-validation to evaluate the model performance corresponding to each k value. The DT classifier selects entropy as the splitting criterion. In the RF classifier, the number of trees and the maximum depth of the trees are optimized in the range of {120, 200, 300} and {5, 8, 15}, respectively, and the performance of each combination of parameters is evaluated by using 3-fold cross-validation to select the best parameters. The SVM classifier uses a linear kernel function.

A variety of evaluation indexes (EI) were established, including accuracy (Acc), precision (Pre), recall (Rec), and F1-Score (F1)52,53, in which the accuracy reflected the overall classification performance of the model. Each result was averaged 10 times, and the standard deviation (SD) was calculated as a measure of classifier stability. In order to evaluate the performance of different classifiers, the corresponding confusion matrices, four important evaluation indexes and their SDs were constructed under different test numbers (TN), as shown in Fig. 14 and Table 5, respectively. In Fig. 14, a-e is the confusion matrix of the KKNN classifier when the number of test sets decreases. Similarly, f-j, k-o, and p-t are the confusion matrices of DT, RF, and SVM classifiers when the number of test sets decreases, respectively.

Confusion matrices of different classifiers for different fault types.

It can be seen from Fig. 14 that when the number of test sets is 50, KKNN predicts 1 label 0 as label 1, while DT, RF and SVM predict 2 labels 0 to label 1, respectively. When the number of test sets is 40 and 30, respectively, KKNN, RF, and SVM all predict correctly, while DT predicts 1 label 3 as label 2. From the above, KKNN, RF and SVM are close in performance and better than DT.

From Table 5, it can be seen that the SD of the three classifiers, KKNN, RF and SVM, is 0 in different test sets. When the number of test sets is 10, 20, 30 and 40, the classification results are consistent. When the number of test sets is 50, Acc: KKNN (99.5\(\%\) )> RF (99\(\%\) ) \(=\) SVM (99\(\%\) ), Pre: KKNN (99.5\(\%\) )> RF (99.07\(\%\) ) \(=\) SVM (99.07\(\%\) ), Rec: KKNN (99.5\(\%\) )> RF (98.98\(\%\) ) \(=\) SVM (98.98\(\%\) ), F1: KKNN (99.5\(\%\) )> RF (99.01\(\%\) ) \(=\) SVM (99.01\(\%\) ).

Based on the above analysis, it can be concluded that in the classification tasks of different types of faults, the KKNN classifier has good performance in both classification accuracy and classification stability, which is better than the other three methods, and can stably achieve accurate classification of different faults. (2) Comparison of different decomposition methods In order to evaluate the effectiveness of the VMD method, the EMD, EEMD and CEEMDAN methods were used to decompose different types of fault signals, and the sensitive IMF components were selected for signal reconstruction by the time-frequency domain comprehensive evaluation factor algorithm, and the RCMDE values were calculated and compared with the RCMDE method. The state feature sets extracted by the above methods are input into the KKNN classifier for different types of fault data, and the results of the four evaluation indicators obtained are shown in Table 6.

From Table 6, it can be seen that the classification results of the signal decomposition by VMD are significantly better than those of other comparison decomposition methods. EMD-RCMDE, EEMD-RCMDE, CEEMDAN-RCMDE, RCMDE comparison methods compared with VMD-RCMDE method, the largest differences in the classification results on Acc were 35.62\(\%\) , 35\(\%\) , 7.5\(\%\) and 6.87\(\%\) , respectively. The largest gaps in the classification results on Pre were 32.9\(\%\) , 41.67\(\%\) , 7.05\(\%\) and 5.74\(\%\) , respectively. The largest gaps in the classification results on Rec were 34.59\(\%\) , 36.11\(\%\) , 6.67\(\%\) and 7.59\(\%\) , respectively. The largest gaps in the classification results on F1 were 35.61\(\%\) , 39.58\(\%\) , 7.4\(\%\) and 7.32\(\%\) , respectively.

After the above analysis, it can be concluded that compared with the recursive decomposition method, VMD can effectively reduce the influence of endpoint effect, modal aliasing and other problems, and the selection of sensitive IMF components through the time-frequency domain comprehensive evaluation factor algorithm can effectively filter out the influence of noise, which provides strong support and guarantee for the effective extraction of subsequent features. (3) Comparison of different feature extraction methods based on VMD In order to evaluate the effectiveness of the RCMDE method in feature extraction, a variety of commonly used MSE methods were used for comparison, including MDE, MFE and MPE. The state feature sets extracted by the above methods are input into the KKNN classifier for different types of fault data, and the results of the four evaluation indicators obtained are shown in Table 7. The \(\tau _{max}\) of the comparison method is taken as 20, and the other parameters are:MDE: m=2, c=4; MFE: m=3, r=0.1; MPE: m=4, t=1.

Through the observation of table 7, it can be seen that when the test set is 20 and 10, the four evaluation indicators of VMD-MDE can achieve 100% recognition rate like VMD-RCMDE. When the number of other data in the test set is obtained, VMD-RCMDE is the best. VMD-MDE, VMD-MFE, VMD-MPE comparison methods compared with VMD-RCMDE method, the largest gaps in the classification results on Acc were 0.83%, 2.5% and 17%, respectively. The largest gaps in the classification results on Pre were 0.74%, 2.5% and 16.1%, respectively. The largest gaps in the classification results on Rec were 0.76%, 2.78% and 17.21%, respectively. The largest gaps in F1 classification results were 0.76%, 2.79% and 16.92%, respectively.

After the above analysis, it can be seen that in the classification task of different types of faults, RCMDE can accurately extract the fault features in the reconstructed signal, and the classification results are significantly better than other comparative feature extraction methods, showing its superior performance.

The time-domain waveforms of labels 10, 11 and 12 are shown in Fig. 15 and Table 8 shows the key parameters of VMD optimization by using SSA. The time-domain waveform plot of the reconstructed signal is shown in Fig. 16. The RCMDE values for feature extraction of the reconstructed signal using RCMDE are shown in Fig. 17.

Time-domain waveform of composite faults.

Time-domain waveform of the composite fault reconstruction signal.

RCMDE value of a composite fault.

By observing Fig. 17, it can be seen that the RCMDE algorithm not only has a strong feature capture ability when extracting the composite obstacle features of rolling bearings, but also ensures the obvious discrimination between various fault states. It fully demonstrates its effectiveness in the extraction of composite fault features and provides support and guarantee for subsequent classification. When \(\tau\) is less than 5, the relative fluctuation of RCMDE value is large and there is a crossover phenomenon in some areas, which increases the difficulty of fault classification. In addition, in order to avoid the redundancy of feature information and ensure that the state features are fully reflected, the RCMDE value of \(\tau\) =6-20 is selected as the eigenvector. (1) Comparison of different classification methods based on VMD-RCMDE. The state feature sets extracted by VMD-RCMDE composite fault data are input into the KKNN, DT, RF and SVM classifiers respectively. In order to evaluate the performance of different classifiers, the corresponding confusion matrix and four important evaluation indexes and their SDs were constructed, as shown in Fig. 18 and Table 9, respectively.

Confusion matrices of different classifiers for composite faults.

By observing Fig. 18, KKNN predicts 5 labels 11 to label 10 when the number of test sets is 50, and all the rest are predicted correctly. The DT, RF, and SVM classifiers all have 5 prediction errors when the test set is 50. When the test set is 40, 4, 2, and 3 prediction errors, respectively. When the test set is 30, 4, 2, and 2 prediction errors, respectively; When the test set is 20, there is one prediction error.

Table 9 shows that the performance of KKNN is better than that of the other three classifiers. DT, RF, SVM compared with KKNN, the largest gaps in Acc were 4.77%, 2.22% and 2.5% respectively; The largest gaps in Pre were 5.33%, 1.85% and 2.36% respectively. The largest gaps in Rec were 5.29%, 3.03% and 3.03%, respectively. The biggest gaps in F1 were 5.31%, 2.54%, 2.54%.

Based on the above analysis, the KKNN classifier not only maintains an excellent high recognition rate in the field of composite fault identification, but also shows excellent recognition stability in the composite fault classification task. (2) Comparison of different decomposition methods In order to evaluate the effectiveness of the VMD method, the EMD, EEMD and CEEMDAN methods were used to decompose different types of fault signals, and the sensitive IMF components were selected for signal reconstruction by the time-frequency domain comprehensive evaluation factor algorithm, and the RCMDE values were calculated and compared with the RCMDE method. The state feature sets extracted by the composite fault data by the above methods are input into the KKNN classifier respectively, and the results of the four evaluation indicators obtained are shown in Table 10.

From Table 10, it can be seen that the classification results of the signal decomposition by VMD are significantly better than those of other comparison decomposition methods. the EMD-RCMDE, EEMD-RCMDE, CEEMDAN-RCMDE, RCMDE comparison method compared with VMD-RCMDE method, the largest differences in the classification results on Acc were 40%, 19.23%, 38.33% and 36.67%, respectively. The largest gaps in the classification results on Pre were 39.93%, 16.1%, 33.74% and 23.55%, respectively. The largest gaps in the classification results on Rec were 39.99%, 20.75%, 35.74% and 23.8%, respectively. The largest gaps in the classification results on F1 were 39.8%, 21.86%, 39.88% and 24%, respectively.

After the above analysis, it can be seen that compared with other methods, VMD can effectively decompose components of different frequencies in the composite fault classification task, and also shows good applicability to composite faults, which provides strong support and guarantee for the subsequent discrimination and feature extraction of sensitive IMF components. (3) Comparison of different feature extraction methods based on VMD In order to evaluate the effectiveness of the RCMDE method in feature extraction, a variety of commonly used MSE methods were used for comparison, including MDE, MFE and MPE. The state feature sets extracted by the composite fault data by the above method are respectively input into the KKNN classifier, and the results of the four evaluation indicators obtained are shown in Table 11.

From Table 11, it can be seen that the fault features extracted by RCMDE are the most effective, and the classification results are the best in the four evaluation indexes. VMD-MDE, VMD-MFE and VMD-MPE signal decomposition methods compared with VMD-RCMDE, the largest gaps in the classification results on Acc were 13.33%, 10% and 15.56%, respectively. The largest gaps in the classification results on Pre were 17.78%, 13.33% and 14.97%, respectively. The largest gaps in the classification results on Rec were 17.04%, 14.81% and 14.26%, respectively. The largest gaps in the classification results on F1 were 17.52%, 14.35% and 16.66%, respectively.

After the above analysis, it can be seen that compared with other feature extraction methods, RCMDE can effectively extract the fault information in the composite fault signal in the composite fault classification task, and shows good applicability.

In order to further validate the generalization ability and robustness of the proposed method, noise with a signal-to-noise ratio of 10dB was added to the rolling element fault signals labeled with different damage degrees of 2, 5 and 8 in a single fault dataset. In addition, the same signal-to-noise ratio noise components was added to the inner ring fault and outer ring fault signals with different damage degrees, which has also achieved satisfactory results, but this paper will not introduce it in detail limited by its length. SSA was used to optimize the VMD parameters, and the optimization results were K=6 and \(\alpha\) =1340. Fig 19 shows the RCMDE value curve for feature extraction from the reconstructed signal using RCMDE.

RCMDE values of rolling element faults with different degrees of impairment with added noise.

(1) Comparison of different classification methods based on VMD-RCMDE. The rolling element fault data with different damage degrees with added noise were input into the KKNN, DT, RF and SVM classifiers respectively through the state feature sets extracted by VMD-RCMDE. In order to evaluate the performance of different classifiers, the corresponding confusion matrix and four important evaluation indexes and their SDs were constructed, as shown in Fig 20 and Table 12, respectively.

Confusion matrices of different classifiers for rolling element faults with different damage degrees of added noise.

As can be seen from Fig 20, all predictions made by the KKNN classifier are correct. Both DT and RF all have prediction errors at 10, 20, 30, 40, and 50 in the test set; However, SVM only predicts correctly when the test set takes 10, and the rest of the predictions are incorrect.

From Table 12, it can be seen that when the number of test sets is 50, on Acc: KKNN(100%)>RF(98%)=SVM(98%)>DT(97.07%); On Pre: KKNN (100%)> RF (98.15%) = SVM (98.15%)> DT (97.28%); On Rec: KKNN (100%)> RF (98%) = SVM (98%)> DT (97.07%); On F1: KKNN (100%)> RF (98.02%) = SVM (98.02%)> DT (97.1%). No matter how many test sets are taken, the recognition rate of KKNN is as high as 100% on the four evaluation indicators, which is significantly better than the other three classifiers.

Based on the above analysis, the KKNN classifier can take into account the high recognition rate and recognition stability in the rolling element fault classification task with different damage degrees of added noise, and shows good classification performance. (2) Comparison of different decomposition methods In order to evaluate the effectiveness of the VMD method, the EMD, EEMD and CEEMDAN methods were used to decompose the rolling element fault signals with different damage degrees of added noise, and the sensitive IMF components were selected for signal reconstruction based on the time-frequency domain comprehensive evaluation factor algorithm, and the RCMDE values were calculated and compared with the direct RCMDE method. The state feature sets extracted by the above method are input into the KKNN classifier by the rolling element fault data with different damage degrees of added noise, and the results of the four evaluation indexes obtained are shown in Table 13.

From Table 13, it can be seen that the recognition rate of VMD-RCMDE can be as high as 100% on the four evaluation indicators, and the highest recognition rate of other methods is 83.33%.

After the above analysis, it can be seen that compared with other decomposition methods, VMD shows excellent decomposition ability in the rolling element fault classification task with different damage degrees of added noise, and can accurately and effectively separate components of different frequencies from the signal, and will not affect the decomposition effect due to the addition of noise, thus providing a solid and reliable foundation for subsequent signal processing. (3) Comparison of different feature extraction methods based on VMD In order to evaluate the effectiveness of the RCMDE method in feature extraction, a variety of commonly used MSE methods were used for comparison, including MDE, MFE and MPE. The state feature sets extracted by the above method are input into the KKNN classifier by the rolling element fault data with different damage degrees of added noise, and the results of the four evaluation indexes obtained are shown in Table 14.

From Table 14, it can be seen that the effectiveness of VMD-RCMDE’s feature extraction is better than that of other comparison methods. Compared with the VMD-MDE, VMD-MFE, VMD-MPE and VMD-RCMDE feature extraction methods, the largest gaps in the classification results on Acc were 28.89%, 23.33% and 33.33%, respectively. The largest gaps in the classification results on Pre were 25.51%, 22.98% and 29.27%, respectively. The largest gaps in the classification results on Rec were 25.51%, 23.16% and 29.66%, respectively. The largest gaps in the classification results on F1 were 25.51%, 23.8% and 32.82%, respectively.

After the above analysis, it can be concluded that in the rolling element fault classification task with different damage degrees of added noise, RCMDE can effectively extract the fault information in the rolling element fault signals with different damage degrees, so as to provide support and guarantee for the identification rate of fault classification.

Through the diagnosis of single faults and compound faults of rolling bearings with different fault types, the results show that the SSA optimization VMD method proposed in this paper can avoid the problems of endpoint effect and modal aliasing in EMD, EEMD and CEEMAN methods, and can reasonably and effectively decompose different frequency components in the signal according to the characteristics of the signal itself, which lays a foundation for the accurate elimination of interference components. Compared with MDE, MFE, MPE, RCMDE fully considers the relationship between coarse-grained sequences, significantly improves the problem of information loss in the process of entropy calculation, and can more effectively extract the fault features in the reconstructed signal, providing a more reliable guarantee for the accuracy of fault classification. Compared with SVM, DT, RF, KNNN classifier shows good recognition accuracy and generalization ability in multiple evaluation indicators of classification tasks, which is a more ideal fault classification model for small sample cases. In addition, noise interference components are added to the rolling element fault signals with different damage degrees to verify the robustness of the proposed method. The results show that the proposed method can still maintain an ideal fault identification accuracy for such faults, and show good generalization ability and anti-noise performance, which provides a theoretical reference for the accurate diagnosis of rolling bearing faults.

In order to solve the problems of weak information extraction of rolling bearing fault characteristics and poor generalization performance of diagnostic methods, a fault diagnosis method based on the combination of SSA-VMD and RCMDE was proposed. Through the tests of different types of single faults, compound faults and single fault noise signals of rolling elements with different damage degrees, and the comprehensive comparison of different methods through a variety of evaluation criteria, the following conclusions are obtained:

By comparing with a variety of single and combined methods, it is shown that the rolling bearing fault diagnosis method based on SSA-VMD and RCMDE proposed in this paper shows excellent performance in dealing with different types of single faults, compound faults and rolling element single fault noise signal classification tasks with different damage degrees, which not only has high classification accuracy, but also has strong generalization ability, can effectively deal with different types of rolling bearing faults, and provides strong technical support for the accurate diagnosis of rolling bearing faults.

The signal processing method combining SSA-VMD and the comprehensive evaluation factor algorithm based on the time-frequency domain effectively realizes the decomposition of fault signals and the selection of sensitive IMF components, and eliminates the interference signals such as background noise mixed in the fault-type signals. SSA-VMD avoids the cumbersomeness and complexity of manual parameter tuning. The comprehensive evaluation factor algorithm based on the time-frequency domain realizes the accurate discrimination of different IMF components. On the Acc index, the highest recognition rate: SSA-VMD (100%)> CEEMDAN (95.63%)> EEMD (91.11%)> EMD (85%).

The test results show that RCMDE can effectively extract the fault features in the fault signal in the classification task of different types of single faults, as well as in the classification task of single fault noise signal of rolling element with compound fault and different damage degrees, which realizes the extraction of the feature information of the multi-scale characterization fault signal, solves the limitation of single-scale entropy, and provides a guarantee for the next step of fault identification and classification. RCMDE can achieve a 100% recognition rate in any number of cases, which is significantly higher than the other three methods.

The KKNN classifier not only realizes the accurate classification of faults in different types of single fault tasks, but also maintains high-precision classification performance and strong classification stability in the face of more complex composite faults and single fault noise signals of rolling elements with different damage degrees, which fully demonstrates the strong robustness of the KKNN classifier. Under different test sets, most of the KKNN can achieve a recognition rate of 100% in the index Acc to evaluate the integrity of the model, and more than 90% for DT, RF, and SVM.

In this paper, the rolling bearing signal is studied, and the method described in this paper has a better processing effect than other methods. However, the method proposed in this paper has not been verified for the fault diagnosis of bearings under steady-state conditions or variable working conditions, and in the future, we will verify the applicability and robustness of the method in the above two situations. In addition, we may use this method to examine power system signals and compare them with many commonly used signal feature extraction methods, signal decomposition methods, and signal classification methods.

statement:Get access to the raw data for http://biaowang.tech/xjtu-sy-bearing-datasets and https://engineering.case.edu/bearingdatacenter/download-data-file.

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This research was funded by Jilin Science and Technology Development Plan Project (no. 20210203047SF), the Graduate Innovation Project of Beihua University ([2023]043),Natural Science Foundation of Jilin Province (YDZJ202401581ZYTS), Science and Technology Research Project of Jilin Provincial Department of Education (No. JJKH20230060KJ)

These authors contributed equally: X. Wang, Z.Xing, Z. Yang, L. Cao, X. Zhou.

College of Mechanical Engineering, Beihua University, Jilin City, Jilin, 132021, China

Xiangkun Wang, JiaHong Li, the Jing, Haoyu Li, Zhongyuan Xing, Zhilun Yang, Linlin Cao & Xiaolong Zhou

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X.W., Z.Y. and X.Z. conceived and designed this paper. Z.X., Z.Y. and L.C. offered advanced suggestions about the paper. J.H.L., Z.J. and H.L. analyzed the data. X.W., Z.X., Z.Y. and X.Z. wrote the draft of the paper.

the authors declare no conflict of interest.

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Wang, X., Li, J., Jing, Z. et al. Fault diagnosis method of rolling bearing based on SSA-VMD and RCMDE. Sci Rep 14, 30637 (2024). https://doi.org/10.1038/s41598-024-81262-9

DOI: https://doi.org/10.1038/s41598-024-81262-9

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