A novel intelligent method for inter-shaft bearing-fault diagnosis based on hierarchical permutation entropy and LLE-RF

Abstract

Since the transmission path of inter-shaft bearing-fault signal is complex, a fault feature extraction method based on hierarchical permutation entropy (HPE) and locally linear embedding (LLE) algorithm is proposed in this paper. In this method, HPE is utilized to extract fault information of signals, and LLE is utilized to reduce and fuse high-dimensional fault features of multi-sensors to construct fault samples. Then, the random forest (RF) model is established to diagnose the faults of the inter-shaft bearings. The fault simulation test rig with the inter-shaft bearing is built to simulate the normal bearing, inner ring fault, outer ring fault, and rolling ball fault, and the data are collected to verify the HPE-LLE-RF fault diagnosis algorithm of inter-shaft bearings established in this paper. The experimental results show that the proposed algorithm can extract the fault features of inter-shaft bearings effectively with a fault diagnosis accuracy of 93.3% without overfit phenomenon.

Keywords

Inter-shaft bearing Feature extraction Hierarchical permutation entropy Locally linear embedding Random forest

1. Introduction

The inter-shaft bearing is a key component of the aero-engine load-bearing trans-mission system, and it is very important to detect its fault status (Long et al., 2020). Inter-shaft bearings are easy to fail due to their high mobility and heavy load in the process of aero-engine operation. During the early stage of inter-shaft bearing failure, timely and accurate fault identification can effectively prevent the occurrence of catastrophic accidents (Tian et al., 2019).

The fault signal of inter-shaft bearing is a typical nonlinear and non-stationary signal which has obvious impact characteristics (Gao et al., 2011; Ke et al., 2017; He et al., 2022). Extracting useful fault feature information from non-stationary and nonlinear signals is the focus and difficulty of inter-shaft bearing-fault diagnosis. The entropy method can identify nonlinear parameters, such as approximate entropy (ApEn), sample entropy (SampEn), fuzzy entropy (FE), and permutation entropy (PE) (Chen et al., 2009; Zhao et al., 2013; Wang and Du, 2020). Among them, PE proposed by Bandt et al. (Bandt and Pompe, 2002) does not need to consider the value of time series, but it compares and analyzes adjacent sample points to obtain corresponding characteristic information; meanwhile, it has the advantages of short time series, being sensitive to signal mutation, fast calculation speed, and strong anti-noise ability, so it is widely used to detect the randomness and dynamic mutation of time series (Xie et al., 2021). Liu et al. (2017) introduced PE into mechanical equipment vibration signal mutation detection, and the influence of parameters on PE was studied according to the characteristics of rolling bearings in different states. Based on the PE method, Aziz and Arif (2005) proposed multiscale permutation entropy (MPE) to measure the complexity and randomness of time series at different scales. Li et al. (2016) utilized MPE and ISVM-BT to establish a fault diagnosis model of rolling bearings, and the results verified the stability and robustness of MPE. Based on the traditional univariate analysis method, multivariate multi-scale permutation entropy (MMPE) is proposed to measure the complexity of time series in multi-channel data by combining the multi-dimensional embedding reconstruction theory (Yin and Shang, 2017). However, the deviation of the MMPE entropy value increases as the length of the coarse-grained sequence decreases, resulting in the loss of much potentially useful information. Therefore, the composite multiscale permutation entropy (CMPE) was proposed to overcome the shortcomings of the MMPE algorithm by adopting the composite coarse-grained construction method (Si et al., 2019; Li et al., 2021). CMPE has been recently used as the characteristic parameter of rotor system fault diagnosis by Cheng et al. (2020), and the results verified that CMPE has advantages in the stability and accuracy of feature extraction. The frequency components of bearing-fault information are relatively rich, and its high-frequency components contain a large amount of fault information, while the traditional CMPE algorithm can only extract the low-frequency information of fault signal (Landauskas et al., 2020; Zhu et al., 2014). In this paper, in order to more comprehensively analyze the characteristics of nonlinear and non-stationary signals in various frequency bands, a hierarchical permutation entropy (HPE) algorithm is proposed as a tool for fault feature extraction and applied to the fault diagnosis of inter-shaft bearings.

As the fault features obtained using HPE often have high dimensions in multi-point sensor fault information fusion processing, they may contain a large amount of redundant information (Gisbrecht et al., 2015; Ingram and Munzner, 2015). The recognition effect will be affected by the fault features being directly input into the classifier for training and testing, so a dimensionality reduction method is needed to promote the classification, visualization, and compression of high-dimensional data (Wang and Chen, 2009). Traditional dimensionality reduction methods have poor dimensionality reduction effect on nonlinear data, which cannot reveal the correlation between data and the characteristics of internal structure (Xi et al., 2018). As a classical nonlinear dimension reduction method of manifold learning, LLE can achieve dimension reduction and data visualization; by using these, the internal relationship of nonlinear data can be revealed (Liu et al., 2016). Li et al. (2018a) applied LLE to estimate high-resolution images, and the experiment verified the superiority of LLE. Thus, LLE is applied to the dimension reduction process of HPE fault features in this paper. To realize intelligent fault diagnosis, it is necessary to select an appropriate classifier. As a typical combined classifier model, RF has the advantages of fast calculation speed, high classification accuracy, and strong generalization ability (Breiman, 2001; Wan et al., 2021). Sexton and Laake (2009) compared the estimation results of bagging and RF, and their research indicated that the effect of RF estimation based on bootstrap was better. In this study, LLE is introduced to reduce the dimension of features after obtaining the fault features using HPE. In addition, RF is applied to the automatic recognition of failure mode. On the basis of feature extraction, the multi-source fault features are effectively fused based on LLE, the redundant information is removed, and the multi-source information is complementary. RF is used for decision fusion to realize the accurate diagnosis of inter-shaft bearing fault. Overall, a novel fault diagnosis method based on HPE, LLE, and RF is proposed to classify different fault types of inter-shaft bearings in this paper. The proposed method is applied for the analysis of test data, and the results show that it can diagnose different bearing faults effectively with a high recognition rate.

The rest of this paper is organized as follows: Section 2 describes the basis of HPE and validates the superiority of HPE using simulation signals, and then discusses the effect of each parameter of HPE. Section 3 presents the dimension reduction method based on LLE. The HPE and LLE-based fault diagnosis method for inter-shaft bearings is proposed in Section 4. Section 5 applies the proposed method to classify the fault types of bearings. Finally, conclusions are drawn in the final section.

2. Hierarchical permutation entropy

2.1. Basic principles of hierarchical permutation entropy

To take advantage of hierarchical segmentation, Jiang et al. (2011) proposed the algorithm of HE and successfully verified the effectiveness of the HE method by analyzing biological signals. Based on the advantages of hierarchical segmentation and the definition of PE, the HPE algorithm is proposed in this paper. The details of HPE can be summarized as follows:

(1) For the time series $x = {x (1), \dots, x (i), \dots x (N)}$ with length N, length $N = 2^{n}$ and an averaging operator $Q_{0}$ is defined by

Q_{0} (x) = \frac{x (2 j) + x (2 j + 1)}{2} j = 0,1, \dots, 2^{n - 1}

(1)

Similarly, a difference operator $Q_{1}$ for time series $x$ is defined by

Q_{1} (x) = \frac{x (2 j) - x (2 j + 1)}{2} j = 0,1, \dots, 2^{n - 1}

(2)

where

Q_{1} (x)

represents the high-frequency components after the decomposition of the original sequence. The matrix representation of the averaging operator

Q_{j} (x)

is as follows:

Q_{j} (x) = ({\begin{array}{c} \frac{1}{2} & {(\frac{- 1}{2})}^{j} & 0 & 0 & \dots & 0 & 0 \\ 0 & 0 & \frac{1}{2} & {(\frac{- 1}{2})}^{j} & \dots & 0 & 0 \\ 0 & 0 & 0 & 0 & \dots & \frac{1}{2} & 0 \end{array}}_{2^{n - 1} \times 2^{n}})

(3)

(2) The averaging operators $Q_{0} (x)$ and $Q_{1} (x)$ are repeatedly used to decompose the time series $x$ hierarchically to construct a vector $[h_{1}, h_{2}, \dots, h_{n}]$ with length $n$ . Then, there is a non-negative integer $e$ , expressed as follows:

e = \sum_{j = 1}^{k} h_{j} 2^{k - 1}

(4)

(3) Based on vector $[h_{1}, h_{2}, \dots, h_{n}]$ , the hierarchical components of a time series $x$ is defined by

x_{n, e} = Q_{h_{n}} • Q_{h_{n - 1}} • \dots • Q_{h_{1}} (x)

(5)

$w h e r e x_{n, 0}$ is the low-frequency component of time series $x$ in scale $n + 1$ ; $x_{n, 1}$ is the high frequency component of time series $x$ in scale $n + 1$ ; and $x_{0,0}$ is the original time series $x$ .

(4) By calculating the PE of each obtained hierarchical component, the PE value of $2^{n}$ hierarchical components will be obtained, which is called HPE analysis. The HPE can be expressed as follows:

HPE (x, n, e, m, r) = PE (x_{n, e}, m, r)

(6)

MPE only analyzes the pattern information of the low-frequency part of the time series, ignoring that of the high-frequency part, while HPE analyzes the pattern information of the high-frequency part as well as the low-frequency part of the time series. The calculation process of HPE is shown in Figure 1.

2.2. Simulation analysis

In this section, white noise and pink noise are used to verify the effectiveness and superiority of HPE. In order to further study the statistical behaviors of HPE and MPE, the white noise and pink noise are analyzed for comparison. MPE and HPE are used to extract features from white noise and pink noise, respectively, with the following parameters: N=4096, m=5, τ=10, and K=3. The results of feature extraction using MPE and HPE for white noise and pink noise, respectively, are shown in Figure 2.

From Figure 2, it can be found that when dealing with white noise, the MPE decreases with the increase of scales, while the HPE is relatively stable. The energy distribution of white noise in each frequency band is similar, so the complexity of its time domain sequence should be similar. Therefore, MPE cannot reflect the characteristics of the signal accurately, while the values of HPE proposed in this paper are basically consistent in each frequency band, which conforms to the characteristics of white noise. According to the processing results of pink noise, it is also proved that HPE is superior to MPE. Therefore, the HPE method can extract more accurate signal features than the MPE method.

2.3. Parameter analysis of HPE

The accuracy of HPE depends on four parameters: embedding dimension $m$ , time scale $τ$ , data length $N$ , and the decomposition layer $k$ . To study the influence of embedding dimension $m$ , time scale $τ$ , data length $N$ , and the decomposition layer $k$ on HPE, the white Gaussian noise signal is applied for analysis (Zheng et al., 2019; Li et al., 2018b; Zheng and Pan).

(1) Embedding dimension

To study the effect of embedding dimension $m$ on HPE, the white Gaussian noise signal with length $N = 4096$ is utilized for tests. The decomposition layer $k = 3$ and the time scale $τ = 20$ are set, and the embedding dimension of each sample is 2, 3, 4, 5, and 6. The results are shown in Figure 3.

As the embedding dimension $m$ increases gradually, the difference between reconstructed signal and original signal is prominent. The calculated HPE cannot reflect the signal characteristics of the original signal accurately, so it is not that the bigger the m the better. From Figure 3, it can be found that HPE with embedding dimension $m = 5$ is relatively stable, and the calculation time of HPE is within an acceptable range.Therefore, the embedding dimension $m = 5$ is set in this paper.

(2) Data length

The Gaussian noise with lengths of 512, 1024, 2048, 4096, 8192, and 16834 are constructed as examples to investigate the influence of data length on HPE. The decomposition layer $k = 3$ , the embedding dimension $m = 5$ , and the time scale $τ = 20$ are set. The results are shown in Figure 4.

It can be seen from Figure 4 that when data length is 4096, the HPE values of white noise signals are basically the same in each scale, which is consistent with the energy distribution of white noise. The HPE values have met the requirements when data length is 4096.

(3) Time scale

To investigate the stability of HPE with different time scale, time scale $τ$ on HPE, the white Gaussian signal with length $N = 4096$ is taken for example, the decomposition layer $k = 3$ , and the embedding dimension $m = 5$ are set. Figure 5 shows that the HPE values of white noise signals are calculated under time scales $τ = 10,20,20,40$ , and 50.

As can be seen from Figure 5, when the time scale increases gradually, the HPE values of white noise signals begin to separate, and the curve separation of HPE becomes more and more obvious. When $τ = 10$ and 20, the HPE values of white noise signals in different scales are close to each other, and the variation trend of HPE in the whole scale interval is basically the same. Therefore, to reflect the difference of the reconstructed space, the accuracy and stability of the calculated entropy value, $τ = 20$ , is set in this paper.

(4) Decomposition layer

The white Gaussian noise signal with length $N = 4096$ is used as example, and the embedding dimension $m = 5$ and the time scale $τ = 20$ are set. The decomposition layer of each sample is set to 1,2, 3, 4, and 5. The results are shown in Figure 6.

It can be seen from Figure 6 that the stability of HPE decreases and the fluctuation of HPE increases with the increase of decomposition layer. This is because the increase of the decomposition layer makes the scale factor increase greatly, and the frequency band contained in each scale factor becomes narrower, which leads to less data in each scale factor. The reduction of the data volume leads to the deterioration of data statistical performance, resulting in certain differences in the entropy of different scale factors. Considering that the increase of decomposition layer will reduce the calculation accuracy of HPE, $k = 3$ is set in this paper.

3. Locally linear embedding

3.1. Principles of locally linear embedding algorithm

The basic idea of the LLE algorithm is that the nonlinear data in high-dimensional space maintain global nonlinearity and local linear relationship at the same time. The LLE algorithm breaks through the limitation of principal component analysis in processing nonlinear data and can process and analyze nonlinear signals. The algorithm can reduce the amount of calculation and improve the accuracy of fault diagnosis.

Usually, LLE involves the following three steps:

• Select k nearest neighbors of each sample point. For the sample $X = [x_{1}, x_{2}, \dots x_{n}]$ , $X$ is a D-dimensional column vector. The distance formula of sample points is used to calculate the distance of the nearest neighbors as follows:

d_{i j} = {[\sum_{k = 1}^{D} {| x_{i k} - x_{j k} |}^{p}]}^{\frac{1}{p}}

(7)

When $p = 1$ , the distance between samples is the City lock distance. When $p = 2$ , the distance between samples is the Euclidean distance. When $p = \infty$ , the obtained distance $d_{i j}$ is the Dominion distance. In this paper, the Euclidean distance is used to calculate the nearest neighbor distance, which can effectively improve the calculation efficiency, that is, $p = 2$ .

• Compute the weight, $W$ , that best linearly reconstructs $X$ , from its neighbors, solving the constrained least-squares problem in equation (7).

Φ (W) = {\sum_{i} ‖ x_{i} - \sum_{j = 1}^{k} w_{i j} x_{i j} ‖}^{2}

(8)

where

x_{i j}

is the jth nearest neighbor of the sample point

x_{i}

, and

w_{i j}

is the weight coefficient between the sample point

x_{i}

and its neighbor point

x_{i j}

• Compute the low-dimensional embedding vectors $Y$ best reconstructed by $W$ , minimizing equation (8) by finding the smallest eigenmodes of the sparse symmetric matrix in equation (9).

Φ (Y) = {\sum_{i} ‖ y_{i} - \sum_{j = 1}^{k} w_{i j} y_{i j} ‖}^{2}

(9)

where

Φ (Y)

is the loss function,

y_{i}

is the output of

x_{i}

, and

y_{i j}

is the

K

nearest neighbors of

y_{i}

in low dimensions.

The schematic diagram of the LLE algorithm is shown in Figure 7.

3.2. Verification of local linear embedding algorithm

To verify the dimension reduction ability of the LLE algorithm for nonlinear data and to determine whether it can be used for fault classification, a typical simulation signal data set is used to reduce the three-dimensional data to two-dimensional data in this paper. The signal is Swiss roll signal. The original model of the signal is shown in Figure 8 (A), and the color coding reveals how the data is embedded in two dimensions.

In this paper, 3000 data points are randomly collected for signal, and the data collection result is shown in Figure 8 (B). The LLE algorithm is used to reduce the dimensions of Swiss roll signal, and the processing result is shown in Figure 8 (C). It can be seen from the result of LLE that the uniformity and continuity of data samples are effectively maintained in the two-dimensional space after embedding, and the color transition in the two-dimensional space is consistent with the original signal, which also indicates that the manifold structure does not change.

4. The fault diagnosis method based on HPE and LLE

According to the non-stationary and nonlinear characteristics of fault vibration signals, a fault diagnosis method of inter-shaft bearing based on HPE, LLE, and RF is established, and the flowchart of the proposed method is shown in Figure 9. In this paper, the HPE method is used to extract the complex fault features of the vibration signal at different states. At the same time, due to the high feature dimensions which may cover up the effective feature information, the LLE algorithm is utilized to reduce the dimension of the high-dimensional feature vector matrix and mine the low-dimensional manifold features with inherent laws. Finally, to realize the intelligent diagnosis, the random forest algorithm with fast calculation speed, high classification accuracy, and strong generalization ability is used to identify the fault feature vectors of the low-dimensional manifolds after dimension reduction.

5. Application

5.1. Fault simulation test of inter-shaft bearing

The experiment data of inter-shaft bearing are collected on the bearing test rig in Shenyang Aerospace University. As shown in Figure 10, the experimental rig consists of driving motor, rotor system, supporting system, and data acquisition system.

Considering that the inter-shaft bearing does not bear axial force, the NU202EM cylindrical roller bearings are used in the test. The structural parameters of the inter-shaft bearing are shown in Table 1.

Table 1.

NU202 bearing parameters.

Parameter name	Numerical value
Inner ring diameter (mm)	15
Outer ring diameter (mm)	35
Pitch diameter (mm)	25
Roller diameter (mm)	5
Bearing thickness (mm)	11
Roller number (mm)	11
Contact angle (°)	0

To simulate the typical fault of inter-shaft bearing, rectangular surface defects are artificially implanted on the inter-shaft bearings by using wire-cutting technology. The fault types include the local rolling ball fault (BF), outer race fault (ORF), and inner race fault (IRF), with widths of 0.1 mm, 0.5 mm, and 0.5 mm and depths of 0.1 mm, 0.5 mm, and 0.5 mm.

The vibration acceleration sensor is fixed above the bearing seat of the supporting shaft of the driving motor, and the vibration signals are collected under four different fault modes. The vibration acceleration signals are collected under the following conditions: outer ring speed is 300 r/min, inner ring speed is 1500 r/min, sampling frequency is 6400 Hz, and sampling time is 20 s.

5.2. Vibration signal analysis and fault feature extraction

The vibration data used in this test is composed of vibration signals in three fault states and normal states. The time domain waveforms of vibration signals under four fault modes are plotted in Figure 11.

As can be seen from Figure 11, due to the background noise interference, the difference of time domain waveform curves of vibration signals is not obvious, and the fault symptoms may be hidden in the noise. Therefore, HPE is used to process the vibration data under different states, and the results are shown in Figure 12.

It can be seen from Figure 12 that when the inner and outer rings of the inter-shaft bearing rotate in reverse, the HPE curves extracted from the four typical faults of the inter-shaft bearing are separated. Although the curve crossing phenomenon appears in some scales, there are obvious differences in the HPE of the four faults. When the scale is 5–8, the four kinds of faults are completely separated. This is because the bearing fault information is mainly concentrated in the high-frequency band, and it is easy to separate when the fault information is abundant.

By analyzing the HPE of typical inter-shaft bearing fault signals, it can be known that HPE can classify the vibration signals of normal (Normal) bearing and bearings with faults (ORF, IRF, BF). Although HPE can be utilized for the fault feature extraction of vibration signals of inter-shaft bearings, it is still very difficult to distinguish the fault types intelligently by observing the HPE curves.

5.3. Bearing-fault diagnosis and result analysis

In the process of fault feature extraction, if decomposition layer $k = 3$ is set, the fault feature vector extracted from the time domain signal of each vibration sensor is 8-dimensional. If the fault feature dimension is too large, fault data will be redundant and the calculation time of the classifier will be increased in the process of fault recognition, which will reduce the fault recognition accuracy of the classifier. Therefore, an LLE algorithm is proposed to fuse and reduce the dimension of fault features of multiple sensors. In order to realize the visualization of fusion data, the dimension of LLE $D = 3$ is set. In this paper, the fault features of six vibration sensors are fused for dimension reduction, and the dimension reduction result is shown in Figure 13.

According to the criterion of inter-class separability, the smaller the intra-class distance of similar samples, the larger the distance between the subsets of different samples, which indicates that the better the separability of data, the more suitable it is to be used as the fault feature vector. It can be seen from Figure 13 that after reducing the eight-dimensional HPE feature vector to a three-dimensional one, the four typical faults tend to have good clustering.

In order to verify the superiority of this method, the method proposed in this paper is compared with the K-means method. The fault feature extraction results of 4-channel vibration signals using the two algorithms are shown in Figure 14. It can be seen from the result that after K-means processing, some samples deviate from the clustering center in the fault characteristics of the four types of samples, and the aggregation of four types of samples is not clear. The proposed method can distinguish the four types of samples basically, and the aggregation of all kinds of samples is excellent. The above analysis verifies the effectiveness of the proposed method in terms of fault feature extraction and feature dimension reduction. In addition, by comparing the two-dimensional (2D) and three-dimensional (3D) data of the same sensor, it can be found that the 3D feature vectors are obviously separated in the figure, and the 2D data are basically separated, while some data are confused. In general, the clustering of 3D feature vectors is obviously better than 2D feature vectors, which is consistent with the fact that 3D data contain more information than 2D data.

Finally, the four feature extraction results are input into the random forest classifier model for training and recognition. LLE is used to reduce the 8-dimensional HPE of each sensor sample to a 3-dimensional one, and the 3-dimensional fault data of sensors No. 2, No. 4, and No. 6 form a 9-dimensional fault vector. The fault samples of some vibration signals are shown in Table 2.

Table 2.

Vibration signal fault samples.

Fault type	Sample number	Fault sample					Type no.
Normal	1	0.362	1.774	−0.360	−1.286	−2.029	1
	1	−2.755	0.401	−0.340	−0.260		1
	2	0.476	2.124	−0.341	−0.293	−1.394	1
	2	−1.558	0.585	−0.867	−0.742		1
	3	−0.218	1.947	0.148	0.324	−1.057	1
	3	−0.809	0.237	−0.510	0.283		1
Inner ring fault	1	0.160	−1.013	−0.957	−0.565	0.709	2
	1	0.038	0.614	−0.008	−0.596		2
	2	0.033	−0.864	−1.250	−0.357	0.856	2
	2	−0.123	0.351	−0.055	−0.600		2
	3	−1.180	−0.391	−1.422	−0.837	0.249	2
	3	0.555	0.846	0.930	−0.588		2
Outer ring fault	1	0.033	−0.932	0.333	−0.992	2.411	3
	1	−1.697	0.248	0.294	1.191		3
	2	0.948	−0.725	0.881	0.485	1.146	3
	2	−0.249	1.190	0.230	−0.655		3
	3	0.586	−1.082	0.607	1.267	1.565	3
	3	−0.674	−0.034	−0.436	−0.269		3
Roller fault	1	−1.806	0.166	1.141	1.050	−0.580	4
	1	1.161	0.344	1.693	0.238		4
	2	0.232	−0.308	1.279	0.701	0.110	4
	2	0.684	0.827	−0.131	0.097		4
	3	1.476	−0.565	0.238	1.880	−0.697	4
	3	1.139	1.139	−3.223	3.860		4

To improve the accuracy of fault diagnosis of inter-shaft bearings, a random forest algorithm is proposed to fuse the fault features of vibration signal and diagnose the fault types of inter-shaft bearings. The random forest model is established with 80 groups of fault samples, and then the random forest classifier is used to identify the 80 groups of fault samples and the remaining 120 groups of fault samples are used to verify the effectiveness of the model diagnosis. The confusion matrixes of the diagnosis results are shown in Figure 15 and Figure 16, respectively.

As can be seen from Figure 15, the accuracy of fault identification of the original 80 groups of fault samples using the established random forest classifier is 100%. This shows that the random forest algorithm has good fault classification performance and is not prone to the over-fitting phenomenon.

It can be seen from Figure 16 that the random forest algorithm proposed in this paper is completely correct for outer ring fault diagnosis, and only one of the normal bearings is diagnosed as outer ring fault by mistake. In the inner ring fault, 25 fault samples are diagnosed correctly (the accuracy rate is 83.3%). The accuracy rate of rolling ball fault is 90%. The overall diagnostic accuracy of fault samples is 93.3%. This proves that the multi-source heterogeneous information fusion algorithm can effectively diagnose the inter-shaft bearing faults.

6. Conclusions

This study proposes a new fault feature extraction method based on HPE. HPE is compared with MPE by analyzing simulation data, and the influence of parameters on HPE calculation is also studied. Then, a novel fault diagnosis method based on HPE, LLE, and RF to identify fault types of inter-shaft bearings is put forward by combining manifold learning and supervised learning. Finally, the proposed method is applied to analyze experimental data of inter-shaft bearing.

The major contributions of this paper are summarized as follows:

(1) In this paper, HPE is proposed to measure the complexity and dynamic behavior changes of time series. The simulation analysis results show that HPE can provide a more comprehensive estimation of signal complexity with more stable performance and higher distinguishability than MPE.

(2) By analyzing the noise signal, the selection and influence of HPE parameters are studied, and the selection criteria of HPE parameters are given as follows: $m = 5$ , $N = 4096$ , $τ = 20$ , and $k = 3$ .

(3) In this paper, the dimension reduction analysis of Swiss volume dataset simulation signal by the LLE algorithm is used to verify the effectiveness of LLE feature dimension reduction, and the optimal dimension is determined to be three according to the correlation dimension method.

(4) A feature fusion algorithm of HPE-LLE is proposed to extract the fault features of inter-shaft bearings. The experimental results show that using LLE to reduce the dimension of high-dimensional HPE can improve the separability of fault feature vectors as well as reduce the dimension of fault feature samples.

(5) A new approach based on HPE-LLE-RF for fault diagnosis of inter-shaft bearings is put forward in this study. The simulation analysis and experimental results show that the proposed method has obvious advantages in fault feature extraction, high-dimensional data visualization and dimension reduction, and pattern recognition accuracy. The overall diagnosis accuracy of fault samples reaches 93.3%.

Footnotes

Acknowledgements

The authors would like to thank the organizations that provided funding for this work.

Declaration of conflicts of interest

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: this work was supported by the National Natural Science Foundation of China (Grant no.12172231), the Natural Science Foundation of Liaoning Province of China (Grant no. 2020-BS-174), the Liaoning province Department of Education Fund (Grant no.JYT2020019), and the Research Start-up Funding of Shenyang Aerospace University (Grant no.19YB38).

ORCID iDs

Jing Tian

Yuwei Zhang

Fengling Zhang

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