Rolling bearing fault diagnosis based on fine-grained multi-scale Kolmogorov entropy and WOA-MSVM

Abstract

In allusion to solve the issue of fault diagnosis for bearing and other rotatory machinery, a technique based on fined-grained multi-scale Kolmogorov entropy and whale optimized multi-class support vector machine (abbreviated as FGMKE-WOA-MSVM) is proposed. Firstly, vibration signals are decomposed by fine-grained multi-scale decomposition, and the Kolmogorov entropy of the sub-signals at different analysis scales is calculated as the multi-dimension feature vector, which quantitatively characterize the complexity of the signal at multi-scales. Aiming at the problem of sensitive parameters selection for multi-class support vector machine model (abbreviated as MSVM), the whale optimization algorithm (abbreviated as WOA) is introduced to optimize the penalty factor and kernel function parameter, and constructing optimal WOA-MSVM model. Finally, an instance analysis is carried out with Jiangnan University bearing datasets to verify the effectiveness and superiority of this technique. The results show that compared with different feature vectors and models such as K nearest neighbors (abbreviated as KNN) and Decision Tree (abbreviated as RF), the proposed technique is superior with fast computation speed and high diagnostic efficiency.

1. I. Introduction

Rolling bearing is a commonly used component in rotating machinery. Due to the high intensity working condition and harsh working environment, it is prone to lose efficacy and then affect the operation of mechanical equipment, even leading to serious economic losses [1]. Online monitoring and diagnosis of bearing faults is of great significance to enhance the reliability of the whole machine operation and avoid important economic property losses.

Feature extraction is the basic step for fault diagnosis and the key lies in calculating the parameters that can characterize different fault modes [2]. In practice, as mechanical equipment inevitably operates under much friction, vibration and impact, the vibration signals often show nonlinear and non-stationary character. Traditional time-domain and frequency-domain analytical methods can only deal with linear and stationary signals, so the feature extraction of nonlinear and non-stationary signals has become an important hotspot in the field of mechanical fault diagnosis. In recent years, many scholars have proposed some analytical methods based on time-frequency analysis and information entropy [3]. As mentioned above, entropy theory is widely used in the field of fault diagnosis as a sensitive quantitative feature [4], some effective features have been studied and proposed including sample entropy (abbreviated as SE) [5], approximate entropy (abbreviated as ApEn) [6], permutation entropy (abbreviated as PE) [7] and so on. Among the entropy-based features, Kolmogorov entropy (abbreviated as K entropy) is an important index in delineating the chaotic system quantitatively [8]. The higher the K entropy, the higher the rate of information loss as well as the degree of chaos for the system. Besides, considering the complexity of the real system, it is difficult to fully reflect the fault information with a single entropy-based feature mentioned above. For this reason, multi-scale decomposition of time series is proposed to obtain features at multi-scales effectively [[9], [10], [11], [12], [13]]. Among different types of multi-scale decomposition, fine-grained multi-scale analysis (abbreviated as FGMA) has a constant sampling points at each scale to avoid information loss, which is more suitable and stable for signal analysis than coarse-grained multi-scale analysis (abbreviated as CGMA) and composite multi-scale analysis (abbreviated as CMA). Therefore, FGMA and Kolmogorov entropy is combined for feature extraction to calculate comprehensive features at multiple scales.

The essence of fault diagnosis is multi-classification problem for fault patterns. Some machine learning models can effectively solve such problems, and the emergence of these models such as convolutional-vector fusion network [14], Non-parallel bounded support matrix machine [15], multi-class fuzzy support matrix machine [16] also provide better solutions for improving efficiency in this field. In typical machine learning models, support vector machine (abbreviated as SVM) is suitable for binary classification problems and performs better in solving small samples when dealing with nonlinear and high-dimensional pattern recognition issues [17]. The SVM model can be effectively generalized to multi-classification problems with Multi-class SVM model (MSVM for abbreviation). However, the selection of penalty factor and kernel function parameter are important for MSVM model to improve the diagnosis accuracy, and for this reason, some swarm intelligence optimization algorithms have been introduced and applied to optimize the parameters [18,19]. Compared with the above methods, Whale Optimization Algorithm (WOA for abbreviation) was proposed in 2016 by Mirjalili [20]. it is simple and easy to implement theoretically, what is more, it requires few adjustment parameters with the advantage of fast convergence. WOA has been partially applied to optimizing parameters so far [21,22]. Considering its excellent performance, WOA is introduced into parameter optimization of MSVM, and the optimized WOA-MSVM model is selected as fault diagnosis model.

Above all, aiming at the issue of fault diagnosis of rolling bearings, the paper achieved the following two contributions.

• (1)Fault feature based on fined-grained multi-scale K entropy is proposed.
• (2)Fault diagnosis method based on whale optimization multi classification support vector machine (abbreviated as WOA-MSVM) is constructed. Datasets from Jiangnan University is used to analyze the influence of parameters and verify the model's effectiveness.

The paper is organized as follows: Section II proposed the feature extraction method based on FGMKE. In Section III, WOA is introduced into the parameter optimization of MSVM. The whole fault diagnosis procedure is discussed in Section IV and the technique is verified with bearing datasets in Section V. Finally, the conclusion of this paper is given in Section VI.

The abbreviation table of this paper is shown in Table 1.

Table 1.

The samples division.

Abbreviation	Full name	Abbreviation	Full name
MSVM	multi-class support vector machine model	DT	Decision Tree
WOA	whale optimization algorithm	KNN	K nearest neighbors
SE	sample entropy	FGMA	fine-grained multi-scale analysis
ApEn	approximate entropy	CGMA	coarse-grained multi-scale analysis
PE	permutation entropy	CMA	composite multi-scale analysis
K entropy	Kolmogorov entropy	MSE	multi-scale sample entropy
CGMKE	coarse-grained multi-scale K entropy	FGMKE	fine-grained multi-scale K entropy
CMKE	composite multi-scale K entropy

2. II. Feature extraction based on fine-grained multi-scale Kolmogorov entropy

2.1. Theory of fine-grained multi-scale analysis

Multi-scale analysis originates from multi-scale sample entropy (abbreviated as MSE) proposed by Costa in 2005 for analyzing the complexity of time series. The method considers the character at different scales for time series and constructs sub sequence at different scales, thus obtaining a richer quantitative representation combined with information theory [23]. According to different sub sequence constructing methods, multi-scale analysis can be further classified into coarse-grained multi-scale analysis (abbreviated as CGMA), composite multi-scale analysis (abbreviated as CMA) and fine-grained multi-scale analysis (abbreviated as FGMA), typical principles of multi-scale analysis are as follows [24,25].

(1) Assuming X = {x₁,x₂, …,x_N} is time series with length N, coarse-grained sequences at scale factor can be constructed as the following formula:

$$y_j^{\left(\tau \right)}=\sum _{i=\left(j-1\right)s+1}^{js}{x}_i,j=1,2,...,N/scale$$ (1)

The parameter scale is a positive integer which means the maximum scale, and the original time series is partitioned into sub sequence with length N/scale. The schematic diagram is shown in Fig. 1. As the scale factor increasing, the length of the coarse-grained sub sequence becomes smaller, as a result, the accuracy of some entropy-based features will be reduced, and this is the obvious disadvantage for coarse-grained multi-scale analysis.

Fig. 1.

The sketch map of coarse-grained multi-scale analysis.

(2) Composite multi-scale analysis is proposed according to the above deficiency. Composite multi-scale sequences series at scale factor can be constructed as follows:

$$y_{k,j}^{\left(\tau \right)}=\frac{1}{\tau }\sum _{i=\left(j-1\right)\tau +k}^{j\tau +k-1}{x}_i,j=1,2,...,N/scale,k=1,2,...scale$$ (2)

The parameter k is the sub sequence number at the scale, the principle of composite multi-scale analysis is shown in Fig. 2. The composite multi-scale analysis is better than coarse-grained analysis in terms of accuracy, however, the time consumption is significant because of the complex calculation, and time series at different scales still cannot avoid the problem of length reduction essentially.

Fig. 2.

The sketch map of composite multi-scale analysis.

(3) Fined-grained multi-scale analysis is proposed to avoid the length reduction problem. The fined-grained multi-scale time series can be constructed as follows:

$$y_j^{\left(\tau \right)}=\sum _{i=j}^{j+\tau -1}{x}_i,j=1,2,...,N-scale+1$$ (3)

The sub sequence at each scale is constructed with smoothing, ensuring the sub sequence's length is constant, thus improving the accuracy and stability of the entropy features. The principle is shown in Fig. 3.

Fig. 3.

The sketch map of fined-grained multi-scale analysis.

2.2. Kolmogorov entropy

Kolmogorov entropy (abbreviated as K entropy) is an important feature to characterize chaotic system. The value of K entropy can be used to distinguish regular, chaotic, and random motion. The higher the K entropy, the higher the rate of information loss and the higher the degree of chaos of the system. There are two main methods for calculating K entropy, the first one is the correlation integral algorithm and the second is maximum likelihood algorithm [26,27]. In this paper, we adopt the first method and some parameters need to be set before calculation including phase space reconstruction dimension m, delay factor τ and threshold factor r. considering the calculation amount and stability, we usually set m = 4, τ = 1 and r = 0.2˟std(x) where x represent the signal to be analyze and std means standard deviation value.

2.3. Presentation of FGMKE

In order to effectively characterize different modes of mechanical components such as rolling bearings, a feature extraction method based on FGMKE is proposed by combining the FGMA and K entropy theory so that the parameter accurately reflects the complexity of the signals at different scales. Firstly, the signal is decomposed by fine-grained multi-scale decomposition, and then K entropy of each sub sequence is calculated separately to obtain FGMKE. Meanwhile, in combination with other multi-scale analysis methods, coarse-grained multi-scale K entropy (abbreviated as CGMKE) and composite multi-scale K entropy (abbreviated as CMKE) are proposed for comparative analysis. The basic flow of feature extraction is shown in Fig. 4 below.

Fig. 4.

The flowchart of fine-grained multi-scale K entropy.

In the process of feature parameter calculation, the scale parameter is generally set as scale = 5 considering the amount of calculation.

• III.Multiclass SVM optimized with whale optimization algorithm

2.4. Theory of whale optimization algorithm

Whale Optimization Algorithm (abbreviated as WOA) was proposed by Mirjalili in 2016, which mainly stems from the blister net feeding mechanism of humpback whales, and this algorithm has some advantages such as high optimization seeking accuracy, few parameters, and simple structure. WOA can be divided into three stages including enclosure hunting, bubble net predation and spiral updating, and searching for prey [28].

The basic flowchart of WOA is shown in Fig. 5.

Fig. 5.

Basic flowchart of WOA.

2.5. MSVM parameters optimization with WOA

Support Vector Machine (abbreviated as SVM) is a widely used machine learning model. The basic model is to define the linear classifier with the maximum interval on the feature space, so as to map the feature vectors to the high-dimension space, and utilize the linear discriminant function to achieve the classification in the low-dimensional space [29]. In the process of model training and testing, parameter selection of the penalty factor C and the kernel function parameter g directly affects the results of model classification. Therefore, the optimization of MSVM model parameters has been a research hot spot in this field. Combined with the above analysis, this paper introduces WOA with parameters optimization of MSVM model, and the process is shown in Fig. 6 which mainly includes the WOA optimization and classification with MSVM model.

• IV.Fault diagnosis based on FGMKE-WOA-MSVM

Fig. 6.

The flowchart of MSVM optimization based on WOA.

To improve the accuracy and effectiveness, A fault diagnosis technique based on FGMKE and WOA-MSVM (abbreviated as FGMKE-WOA-MSVM) is proposed for bearing and other rotatory machinery. The multi-domain feature constitutes with FGMKE and time-domain statistical parameters, and MSVM models optimized with WOA is construct for fault diagnosis. The basic flowchart is shown in Fig. 7.

Fig. 7.

The fault diagnosis flowchart.

The flowchart of diagnosis mainly comprise four basis steps below.

• (1)Vibration signal acquisition. Monitoring and collecting vibration signals of rolling bearings on different conditions as a fault diagnosis datasets and we mark it as Dataset = {X₁, X₂, …. ,X_n}.
• (2)Multi-domain fault feature extraction. Performing fine-grained multi-scale analysis and time-domain analysis for each samples X_i, extracting FGMKE features Ke_fea_i = [KE₁,KE₂, …,KE_s], which quantitatively characterizes complexity of the signal at different scale. Calculating two-dimension time-domain statistical feature Time_fea_i = [rms_i, var_i], which quantitatively characterizes statistical character on the time domain. Finally, multi-domain features are constructed as multi_domain_fea_i = [Ke_fea_i, Time_fea_i]to comprehensively characterize different conditions of rolling bearings.
• (3)WOA-MSVM model optimization. Dividing the samples into training sets and testing sets and optimizing the parameters C and g of MSVM model based on WOA to build the optimal WOA-MSVM model.
• (4)Sample testing and comparative analysis. Testing the model accuracy and making a comparison with different features models to verify the superiority of the technique proposed.
• V.Instance analysis on JN bearing datasets

2.6. Bearing datasets introduction

The experimental data were obtained from the Jiangnan University and abbreviated as JN datasets [30]. The data sampling frequency is 50 kHz and the rotational speed is 600r/min, 800r/min, 1000r/min respectively. The dataset is divided by 5000 sampling points as the window width, and four groups of samples with different fault conditions are obtained including normal (300 groups), inner ring faults (100 groups), outer ring faults (100 groups), and rolling element faults (100 groups). The time-domain wave forms of the four types of samples are shown in Fig. 8(a) ∼ Fig. 8(d) below. The signals show different time-domain amplitude, impact frequency, and distribution characters.

Fig. 8.

Time domain wave forms for different conditions of rolling bearings.

Taking the signal in Fig. 8(a) as an example for multi-scale analysis. The maximum analysis scale is set as scale = 4. The time domain wave forms of fine-grained, coarse-grained and composite multi-scale are show in Fig. 9 respectively. It is clear that the length of sub sequence at different scale will keep constant in Fig. 9(a), however, the sub sequence sampling points of the coarse-grained and composite multi-scale analysis will decrease as the scale increasing, which brings about information loss and reduces the accuracy of the K entropy calculation.

Fig. 9.

Different type of multi-scale analysis on signal.

The maximum analysis scale is an important parameter for multi-scale analysis. The complexity information specific to sub sequence is embedded in each scale. Taking the above four signals as an example, the maximum analysis scale is set as scale = 16, the phase space reconstruction dimension is set as m = 4, and the tolerance factor is set as r = 0.2. After the fine-grained multi-scale decomposition, the K entropy values of the sub sequence at different scales are shown in Fig. 10 below. It can be seen that the larger the analysis scale, the smaller the value of K entropy obtained, indicating that the complexity of the sub sequence gradually decreases. And from the perspective of value, the K entropy of normal condition and inner ring fault are similar so as to K entropy of outer ring fault and rolling element fault, which indicates that it may be difficult to accurately distinguish different fault modes by analyzing the K entropy at a single scale alone, and it is necessary to synthesize the K entropy of multiple scales as the feature vector.

Fig. 10.

Multi-scale K entropy at different scale parameters.

The phase space reconstruction dimension m and the tolerance factor r are two key parameters in K entropy calculation. Taking the inner ring fault signal shown in Fig. 8(b) as an example, setting the parameters scale = 5 and r = 0.2, the FGMKE curves when m is taken in section [3,9] are shown in Fig. 11(a). As the value of m is larger than 7, the K entropy will have Null value and K entropy relations at different scales will be overlapped at the same time. Keeping the maximum analyzing scale as 5, the embedding dimension m = 4, and the tolerance factor r takes values in section [0.1, 0.9]. The FGMKE curves are shown in Fig. 11(b). The larger the tolerance parameter, the smaller the value of FGMKE, and the relationship of the K entropy at different scales remains constant.

Fig. 11.

The influence of different parameters.

In summary, the three parameters of maximum analysis scale, phase space embedding dimension and tolerance factor will affect the value of FGMKE, but the value are relatively stable within a certain range. Therefore, taking into account the stability, computational volume and other factors, this paper generally selects the maximum analysis scale as scale = 5, the embedding dimension m = 4, and the tolerance factor r = 0.2.

2.7. Multi-domain feature extraction

According to the process of feature extraction, five-dimension complexity features (FMGKE) and two-dimension time-domain statistical features (RMS & VAR) are extracted from each sample respectively. Ten groups of samples are taken from each bearing condition, and the distribution of the two types of features is shown in Fig. 12 below. The comparison shows that both type of features can numerically distinguish different fault conditions. For complexity features, the complexity of the healthy condition is the highest, the feature of the outer ring fault has the lowest value, and the value of the inner ring fault and outer ring fault are similar overall. For time domain statistical features, signal under the outer ring fault has the highest root mean square and variance, while the statistical value under normal condition is the lowest, the values of the inner ring fault and rolling element fault are similar overall. Therefore, the extraction of multi-domain features can combine the advantages of the two types of features to distinguish different fault modes.

Fig. 12.

The distribution of multi-domain features.

The distributions of CGMKE and CMKE are comparatively analyzed in Fig. 13. Compared with the FGMKE distribution shown in Fig. 12(a), both types of features are also able to numerically distinguish different fault conditions. Relatively speaking, the relative size distinction of FGMKE features at different scales is more obvious, the larger the scale, the smaller the value, and this consistent rule will enhance the effectiveness of the feature vectors and improve the accuracy of diagnosis. By comparison, CGMKE and CMKE curves have obvious value mixing at different scales, so the extracted multi-dimension features will be conflicted on different dimensions, thus reducing the accuracy of fault diagnosis to a certain extent.

Fig. 13.

The distribution of different multi-scale K entropy.

2.8. Diagnosis based on WOA-MSVM model

Through the multi-scale analysis and calculation of the signals, a feature matrix with a dimension of 600 × 7 is obtained, where 600 means the samples number and 7 means the dimension number. The training and testing samples are divided according to the ratio of 7:3 and the labeling matrix is created accordingly as shown in Table 2 below.

Table 2.

The samples division.

	Normal condition	Inner race fault	Outer race fault	Rolling element fault
Training samples	210	70	70	70
Testing samples	90	30	30	30
The labels	1	2	3	4

The MSVM model with Gaussian kernel function is built and the training samples are used for model training. The diagnosis results of the model on the testing samples are shown in Fig. 14, in which a total of four samples are misjudged and the accuracy reaches 97.8%.

Fig. 14.

The diagnosis result for MSVM model.

The WOA is used to optimize the penalty factor C and the kernel function parameter g of the MSVM model. The parameters are set as follows: the population size parameter group = 10, the iteration generation is set as 100, the parameter optimization range are set within section [0.001,100] and the 10-CV loss rate of the training samples is used as the objective function. The optimization curve of the WOA is shown in Fig. 15 and the WOA-MSVM model is built according to the optimization results. The diagnosis results for the testing samples are shown in Fig. 16. After the parameter optimization of WOA, only one sample is misjudged and the accuracy reaches 99.4%.

Fig. 15.

The optimization curve of WOA.

Fig. 16.

The diagnosis result for WOA-MSVM model.

2.9. Contrastive analysis

• 1.Analysis on feature vector superiority

The multi-domain features adopted in this paper include the FGMKE and time-domain statistical parameters, which combined the advantages of the two types of features to improve the diagnosis accuracy. Keeping the WOA-MSVM diagnosis model unchanged, five-dimension CGMKE, five-dimension CMKE, and two-dimension time-domain features are used as feature vectors respectively. The confusion matrices before and after the optimization are shown in Fig. 17 below. Among the three contrastive schemes, The CMKE-WOA-MSVM model has the highest accuracy of 93.9% for the testing samples, which means the CMKE features has a better effect compared with the other two types.

Fig. 17.

Confusion matrix for MSVM model with different features.

Proceeding cross-validation with 10-CV method and average diagnosis accuracy comparison with different features are shown in Fig. 18. It can be seen that from the perspective of the type of features, the complexity features are better than the time-domain features, probably due to the fact that the multi-dimension complexity features contain more information about the fault modes. Among the complexity features, the fined-grained multi-scale K entropy is optimal due to its better distinction at different scales, which is consistent with the results analyzed in the previous section. On the whole, the multi-domain feature vectors proposed in this paper have the optimal diagnosis effect.

Fig. 18.

The contrast of accuracy with different features.

32. Analysis on diagnosis model superiority

Keeping the FGMKE feature unchanged, two typical supervised learning techniques are introduced for comparison including KNN [31] and DT [32]. The key parameters of the two models are also optimized by WOA, where the KNN optimizes the num_neighbors and distance_type parameters, and the DT model optimizes the min_leafsize parameter. The testing accuracy of the models as well as the average 10-CV fault diagnosis accuracy are shown in Fig. 19 and Fig. 20. It can be seen that all the three supervised learning techniques have good classification accuracy, and the accuracy is improved by the optimization of WOA. Comparatively speaking, the MSVM model has superior performance, with an optimized accuracy of 99.33%.

Fig. 19.

Confusion matrix for different models with multi-domain features.

Fig. 20.

The accuracy comparison with different diagnosis models.

3.1. VI conclusion

In allusion to the fault diagnosis of rolling bearings, a fault diagnosis model based on fine-grained multi-scale K entropy and WOA-MSVM is proposed, and JN bearing datasets is used to verify the validity and comparative analysis. The MSVM model with Gaussian kernel function is built and the training samples are used for model training. The diagnosis accuracy of the model on the testing reaches 97.8%.

Combined with the statistical time domain features, the multi-domain features can comprehensively reflect the character of different modes of the signal. Compared with different models such as KNN and DT, the MSVM model has superior performance, with an optimized accuracy of 99.33%.

Entropy based feature parameters can reflect the complexity of signals, thereby characterizing the essential characteristics of mechanical equipment. This article introduces K entropy as its characteristic parameter. Considering the universality of this type of parameter, it is necessary to comprehensively compare the characteristics and advantages of this type of feature in the next step.

Fundings

This research received specific grant from National High Technology Research Development Plan (2013AA041106), National Natural Science Foundation (62073213), China Postdoctoral Science Foundation(2014M561458), Shanghai Natural Science Foundation of China (23ZR1426700), Shanghai Engineering Technology Research Center Construction projects (20DZ2253300).

Data availability statement

The data that support the findings of this study are openly available in https://github.com/ClarkGableWang/JNU-Bearing-Dataset.

CRediT authorship contribution statement

Bing wang: Writing – review & editing, Writing – original draft, Software, Funding acquisition, Formal analysis. Huimin li: Data curation. Xiong Hu: Conceptualization. Cancan Wang: Resources. Dejian Sun: Validation.

Declaration of competing interest

The authors declare no conflict of interest in preparing this article