# CEEMDAN-Based Permutation Entropy: A Suitable Feature for the Fault Identification of Spiral-Bevel Gears

Authors: Lingli Jiang; Hongchuang Tan; Xuejun Li; Liman Chen; Dalian Yang
Journal: Shock and Vibration
 (2019)

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## Abstract

A spiral-bevel gear is a basic transmission component and is widely used in mechanical equipment; thus, it is important to monitor and diagnose its running state to ensure safe operation of the entire equipment setup. The vibration signals of spiral-bevel gears are typically quite complicated, as they present both nonlinear and nonstationary characteristics and are interfered with by strong noise. The complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN) method has been proven to be an effective method for analyzing this kind of signal. However, the fault feature information after CEEMDAN is not obvious and needs to be quantified. Permutation entropy can be used to quantify the randomness, complexity, and mutation of vibration time-series signals. This paper proposes to take the CEEMDAN-based permutation entropy as the sensitive feature for spiral-bevel gear fault identification. First, the raw vibration signal is decomposed by the CEEMDAN method to obtain a series of intrinsic modal functions (IMFs). The IMFs which included greater amounts fault information are selected as the optimal IMFs based on the correlation coefficient. Next, the permutation entropy values of the optimal IMFs are calculated. In order to obtain accurate permutation entropy values, the two key parameters, namely, embedding dimension and delay time, are optimized by using the overlapping parameter method. In order to assess the sensibility of the permutation entropy features, the support vector machine (SVM) is used as the classifier for fault mode identification, and the diagnostic accuracy can verify its sensibility. The permutation entropy of CEEMDAN-based/EEMD-based/EMD-based features, combined with SVM, is applied to identify three different fault modes of spiral-bevel gears. Their respective diagnostic accuracies are 100%, 88.33%, and 83.33%, which indicate that the CEEMDAN-based permutation entropy is the most sensitive feature for the fault identification of spiral-bevel gears.


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## Body

## 1. Introduction

Spiral-bevel gear transmission is a transmission method commonly used in mechanical equipment, which has the advantages of large overlap coefficient, strong carrying capacity, high transmission ratio, smooth transmission, low noise, etc., and is widely used in aviation, automobile, mining, and other fields. In the transmission process, as spiral-bevel gears are affected by manufacturing errors and installation errors, poor lubrication, excessive speed, and overload working conditions, they are prone to damage and failure. This results in abnormal vibration, which will affect the normal operation of the entire transmission system and even the entire mechanical equipment setup, thus leading to safety accidents and causing significant economic losses. Therefore, the monitoring and diagnosis of the running state of spiral-bevel gears can ensure that the entire transmission system operates safely, efficiently, and reposefully, which has important practical significance. The vibration signal of spiral-bevel gears is extremely complex, as it is affected by the constant change of meshing logarithm, meshing point position, and instantaneous transmission ratio. In particular, as failure occurs, the signal exhibits strong nonlinear and nonstationary characteristics and is interfered with by strong noise. Fault characteristic information is submerged in strong noise, which is difficult to identify [1, 2]. At present, some researchers are interested in the typical fault diagnosis of spiral-bevel gears [3–5], which mainly focuses on the feature extraction based on wavelet decomposition, such as the discrete wavelet-based method for the fault diagnosis of arc tooth wimble gears [4] and adaptive multiwavelet-based method for the fault diagnosis of spiral-bevel gears [5]. Generally speaking, compared with the parallel shaft gear system and planetary gear system, research on the fault diagnosis of spiral-bevel gear systems is insufficient [6, 7].Signal processing methods have been developed rapidly in recent years. The traditional signal processing method is based on Fourier transform, but this method cannot obtain the time-domain- and frequency-domain-analyzed results simultaneously. For nonlinear and nonstationary vibration signals, important feature information is embedded in the time-frequency domain, so the time-domain- and frequency-domain-analyzed results need to be displayed at the same time. To effectively extract the fault features from the nonlinear and nonstationary vibration signals, many time-frequency analysis methods have been applied to fault diagnoses for vibration signal decomposition, such as short-time Fourier transform (STFT), Wigner–Ville distribution (WVD), wavelet decomposition (WD), empirical mode decomposition (EMD), and so on [8]. Empirical mode decomposition (EMD) is one of the most powerful signal processing techniques that can be used and have been extensively studied and widely applied in fault diagnosis [9]. EMD is based on the local characteristic timescales of a signal and can decompose the signal into a set of complete and almost orthogonal components known as intrinsic mode function (IMF). The IMFs indicate the natural oscillatory mode imbedded in the signal and serve as the basis functions, which are determined by the signal itself, rather than by predetermined kernels. Therefore, it is a self-adaptive signal processing technique that is suitable for nonlinear and nonstationary processes and is suitable for fault feature extraction of spiral-bevel gears [9, 10]. However, EMD has a major drawback, namely, the frequent appearance of mode mixing, which signifies that a single IMF consists of either signals of widely disparate scales, or a signal of a similar scale residing in different IMF components. In order to reduce this drawback of EMD, Wu and Huang proposed the ensemble empirical mode decomposition (EEMD) method based on EMD, which defines the true IMF components as the mean of an ensemble of trials, each consisting of the signal plus a white noise of finite amplitude [11]. The added white noise would populate the whole time-frequency space uniformly with the constituting components of different scales. When the signal is added to this uniformly distributed white background, the bits of signal of different scales are automatically projected onto proper scales of reference established by the white noise in the background. Undoubtedly, EEMD is a breakthrough in the development of EMD algorithm and works well to enhance the stability of EMD algorithm remarkably. Practicality, however, limits the number that one could employ in the ensemble; therefore, the resulting IMFs derived from EEMD would inevitably be contaminated by the added noise, especially when the number of ensembles is relatively low. In light of this problem, Yeh and Huang proposed a novel noise enhanced algorithm to improve the efficiency of the original noise-assisted algorithm of EEMD, by using each noise in pairs with plus and minus signs. Contrary to the requirements of EEMD, which call for independent and identically distributed noise, the present paired noises are perfectly anticorrelated. The advantage of this approach, however, is to have an exact cancellation of the residual noise in the reconstruction of the signal, which is known as complementary ensemble empirical mode decomposition (CEEMD) [12]. Furthermore, Torres proposed a noise-enhanced data analysis method known as complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN), in which the residue of added white noises can be extracted from the mixtures of data and white noise via pairs of complementary ensemble IMFs with positive and negative added white noise [13]. CEEMDAN is the development of the EMD, EEMD, and CEEMD, which is more effective in terms of decomposition ability and can effectively reduce the residual noise in the reconstructed signal. Until now, scholars have studied CEEMDAN and achieved good results [14, 15]. Therefore, this paper chooses CEEMDAN to decompose the vibration signals of the spiral-bevel gears in different fault states, and the raw signal is decomposed into a series of IMFs. In order to reduce the amount of calculation, the IMFs including relatively large amounts of fault information are selected as the optimal IMFs based on the correlation coefficient.The fault information in the optimal IMFs is not obvious and must be quantified. Using the quantified value of energy, entropy and energy entropy as a typical fault sensitive feature is very popular in the fault diagnosis domain [16–18]. Along with the development of entropy theory, permutation entropy (PE) has been proposed [19]. Compared with other entropy theories, permutation entropy is characterized by less computation and strong robustness, which can be used to detect the randomness, complexity, and mutation time-series signal [20, 21]. It can be used to quantify the fault features of mechanical vibration signals and has been preliminarily applied in mechanical fault diagnosis. For example, Ding et al. [22] studied the difference of permutation entropy among three states of a planetary gearbox and proposed the fault diagnosis method of the planetary gearbox based on the permutation entropy of local mean decomposition. Zheng et al. [23] proposed the multiscale permutation entropy to quantify the fault features of the bearing signal. When the spiral-bevel gear is running under fault conditions, its vibration signal is complex and changeable, and the permutation entropy can reflect the rule embedded in the vibration signal. Therefore, in the present paper, CEEMDAN-based permutation entropy is used as the fault sensitive feature for spiral-bevel gear fault state recognition. In the calculation process of permutation entropy, the different parameter settings have a great impact on the calculation results. The optimized parameters of embedding dimension and time delay are also studied in this paper.After the raw vibration signal is decomposed by CEEMDAN and a series of intrinsic modal functions (IMF) is obtained, the permutation entropy of the optimal IMFs is quantified. With this, the fault feature of spiral-bevel gears is extracted. In order to assess the sensibility of the permutation entropy features, the support vector machine (SVM) is used as the classifier for fault mode identification, and the diagnostic accuracy can verify its sensibility. The permutation entropy of CEEMDAN-based features is applied to identify three different fault modes of spiral-bevel gears, and these are then compared with the permutation entropy of EMD-based and EEMD-based features. This paper is structured as follows. Section 2 states the feature extraction and its sensibility assessment procedure. Section 3 presents experiments for the vibration signal of the acquisition of spiral-bevel gears. Section 4 presents the experiment analysis. Section 5 proposes the conclusion of this paper.


## 2. Theoretical Frameworks

### 2.1. CEEMDAN and Selection of IMFs

#### 2.1.1. CEEMDAN Theory

To better describe the CEEMDAN algorithm, we define an operatorEk⋅, the function of which is to solve the kth modal component IMFk of the EMD decomposition. wi is the white noise satisfying the distribution of N0,1, and εk is the amplitude coefficient of white noise added for the Kth time. The decomposition process for CEEMDAN is as follows:(1)
The white noiseXt+ε0ωit is added to the original signal, and Ith EMD decomposition is performed. Then, the average operation is performed on the result to obtain IMF1:(1)IMF1=1I∑i=1IE1Xt+ε0wit.(2)
The first-stage residual component can then be calculated:(2)r1t=Xt−IMF1.The white noiser1t+ε1E1ωit, i=1,2,…,I is added to the first-stage residual component, and the EMD is performed. Then, IMF2 can be calculated using the mean value of the first IMF:(3)IMF2=1I∑i=1IE1rt1+ε1E1wit.Fork=1,2,…,K, the Kth residual component can be calculated as follows:(4)rkt=rk−1t−IMFk.(3)
The white noiser1t+ε1E1ωit, i=1,2,…,I, is added to the kth residual component and EMD is performed. Then, IMFk+1 can be calculated with the mean value of the first IMF:(5)IMFk+1=1I∑i=1IE1rtk+εkEkwit.(4)
Repeat equation (4) and (5) until the value of residual component is less than two extremes, and then the decomposition is stopped. Eventually, the residual variable is obtained:(6)rt=Xt−∑k=1KIMFk,whereK is the total number of modes in the decomposition process, and the reconstructed signal can be expressed as follows:(7)Xt=rt+∑k=1KIMFk.


#### 2.1.2. Selection of IMFs

The raw signal is transformed to a series of IMFs via CEEMDAN, but not all IMFs are related to fault. Some IMFs contain few fault information; thus, these IMFs have no contribution to the following fault identification but do impact the analysis speed. We must therefore develop an IMF selection methodology for selecting the better sensitive IMF related to fault.The parameter correlation coefficient with time can show the related degree between two signals. According to the Schwarz inequality, we can see the following:ρxyτ>0 represents positive correlation, ρxyτ<0 represents negative correlation, ρxyτ=0 represents no correlation, ρxyτ=1 represents perfectly positive correlation, and ρxyτ=−1 represents perfectly negative correlation, where ρxyτ is the correlation coefficient. The greater the absolute value of ρxyτ is, the higher the related degree will be.This paper proposes the selection of effective IMFs according to the correlation coefficient of the raw signal and each IMF. The IMFs with a higher correlation coefficient, whereρxyτ≥0.6 represents greater amounts fault information included [24], are selected as the optimal IMFs for the following analysis.


### 2.2. Permutation Entropy and Its Parameter Selection

#### 2.2.1. The Theory of Permutation Entropy

Entropy can be used to describe the uncertainty of data information. Permutation entropy is a random time sequence detection method and can reflect the one-dimensional time-series complexity, which has the advantages of simple design, strong antinoise ability, and high robustness. The following are the specific principles of the permutation entropy algorithm.Assuming a time seriesai,i=1,2,…,N with length N, the phase space reconstruction is carried out as follows:(8)H=H1H2⋮Hj⋮HQ=a1a1+t⋯a1+m−1ta2a2+t⋯a2+m−1t⋯⋯⋯ajaj+t⋯aj+m−1t⋯⋯⋯aQaQ+t⋯aQ+m−1t,where j=1,2,3,…,Q, m is the embedded dimension, t is the delay time, and Q+m−1t=n. Next, rearrange the jth component of refactoring matrix in ascending order, so that(9)Hj=aj+i1−1t≤aj+i2−1t≤⋯≤aj+im−1t,where i1,i2,…,im is the column position of Hj. If aj+ie−1t=aj+ic−1t exists, they are arranged by the values of e and c, while if e>c, aj+ie−1t≥aj+ic−1t and vice versa. Therefore, any component Hj of the reconstructed matrix H can be used to obtain the corresponding position sequence:(10)Sj=i1,i2,…,im,j=1,2,3,…,q,where q≤m!; due to m different codes i1,i2,…,im, there are m! different permutations, namely, m! different symbol sequences. Sj is just one of the m! code sequences.Next, calculate the probability of each sequenceP1,P2,…,Pq,∑j=1qPj=1. The permutation entropy of the signal time series ai,i=1,2,…,N can be defined in the Shannon entropy form as follows:(11)LPEm=−∑j=1qPjlnPj.WhenPj=1/m , LPEm will maximize lnm!. Then, use lnm! to standardize the permutation entropy LPEm:(12)LPE=LPEmlnm!,where 0≤LPE≤1. The value of LPE represents the degree of randomness of the one-dimensional time series. When LPE is larger, the randomness of the time series is stronger. On the contrary, this indicates that the regularity of the time series is stronger.


#### 2.2.2. Calculation of Parameters Selection

In the permutation entropy calculation process, there are two important parameters, namely, the embedded dimensionm and the delay time t, where the different settings of these two parameters will have an impact on the calculation results. Selecting the proper embedding dimension m and time delay t is the key to extracting the permutation entropy feature of spiral-bevel gears.By comparing the original vibration signals of spiral-bevel gears with different fault states, and combining with an analysis of existing literature, it is observed that the embedded dimensionm = 4, 5, 6 and time delay t = 1, 2, 3 are more suitable to the slight mutation of vibration signals, and then the optimal parameters are selected from these values. Due to the fact that embedding dimension and delay together impacts the calculation results, it is not reasonable to consider the change of a single parameter. In this paper, the overlapping grouping method is adapted to optimally select the embedding dimension and time delay, meaning that a section of original vibration signal of the spiral-bevel gear is divided into a series of subsequences w1,w2,…,wn according to the time series, where w1 moves back one data point to obtain w2, and so on. Then, the permutation entropy is calculated by the combination of embedded dimension and time delay. Since larger w cannot accurately reflect the signal changes and smaller w has low efficiency and no statistical significance, w = 128 is selected.Figure1 shows the combination of the original vibration signals in the normal state and the 2/3 broken tooth fault state of the spiral-bevel gear with a sampling frequency of 16384 Hz, where 0-1 s is the normal gear state, and 1-2 s is the two-third broken tooth state.Figure 1
Normal state and 2/3 broken tooth state vibration signal combination.Figure2(a) shows the calculation values of permutation entropy when t = 1 and m = 4, 5, and 6. Figure 2(b) shows the calculation values of permutation entropy when t = 2 and m = 4, 5, and 6. Figure 2(c) shows the calculation values of permutation entropy when t = 3 and m = 4, 5, and 6.Figure 2
The calculation values of permutation entropy with differentm and t. (a) t = 1 and m = 4, 5, and 6. (b) t = 2 and m = 4, 5, and 6. (c) t = 3 and m = 4, 5, and 6.
(a)
(b)
(c)It can be seen from Figure2 that the permutation entropy under the normal state is larger than that under the 2/3 broken tooth fault states. This is because the vibration signal of the spiral-bevel gear under normal states is more random, and when the tooth appears to be broken, it will result in regular shock signals, and thus the vibration under 2/3 broken tooth fault states is more regular. Comparing Figures 2(a), 2(b), and 2(c), it can be found that when t = 1 and 2 and m = 4 and 5, all permutation entropies can amplify weak information mutation and distinguish normal states and 2/3 broken states, and when t = 1 and m = 4, the permutation entropy mutation is more obvious when the states undergo mutation. Therefore, the time delay t = 1 and embedded dimension m = 4 are selected for the following permutation entropy calculation in this paper.


### 2.3. Sensibility Assessment for CEEMDAN-Based Permutation Entropy

At present, fault diagnosis can be realized by two procedures, namely, feature extraction and mode recognition. Both the sensitivity of features and effectiveness of the classifiers determine the diagnostic accuracy. By selecting the same classification and different feature vectors, the final diagnostic accuracy can reflect the sensitivity of the features. Based on this ideology, the SVM is used as the classifier for mode recognition, and the final diagnostic accuracy is used to assess the sensibility of CEEMDAN-based permutation entropy in this paper.At present, the analysis procedure of this paper is carried out as follows:(1)
Original vibration signal acquisition: the vibration signals of spiral-bevel gears under different typical fault states are collected as the raw time-series signals for subsequent analysis.(2)
CEEMDAN: a series of IMFs are obtained by CEEMDAN of the raw vibration signals under different failure states.(3)
Effective IMFs selected: the correlation coefficient of raw signal and IMF is calculated and is combined with the signal-to-noise ratio value. The IMFs’ larger correlation coefficient and high signal-to-noise ratio value are selected for subsequent analysis.(4)
Permutation entropy calculation: the permutation entropy of the optimal IMFs is calculated using the time delayt = 1 and embedded dimension m = 4, and then the multidimensional eigenvector LPE1,LPE2,…,LPEh is constructed.(5)
SVM classifier for mode recognition: the characteristic vector obtained in Step (4) is used to train and test the SVM and obtain diagnostic accuracy.(6)
Sensibility assessment: the accuracy of CEEMDAN-based permutation entropy is verified based on the diagnostic accuracy.


## 2.1. CEEMDAN and Selection of IMFs

### 2.1.1. CEEMDAN Theory

To better describe the CEEMDAN algorithm, we define an operatorEk⋅, the function of which is to solve the kth modal component IMFk of the EMD decomposition. wi is the white noise satisfying the distribution of N0,1, and εk is the amplitude coefficient of white noise added for the Kth time. The decomposition process for CEEMDAN is as follows:(1)
The white noiseXt+ε0ωit is added to the original signal, and Ith EMD decomposition is performed. Then, the average operation is performed on the result to obtain IMF1:(1)IMF1=1I∑i=1IE1Xt+ε0wit.(2)
The first-stage residual component can then be calculated:(2)r1t=Xt−IMF1.The white noiser1t+ε1E1ωit, i=1,2,…,I is added to the first-stage residual component, and the EMD is performed. Then, IMF2 can be calculated using the mean value of the first IMF:(3)IMF2=1I∑i=1IE1rt1+ε1E1wit.Fork=1,2,…,K, the Kth residual component can be calculated as follows:(4)rkt=rk−1t−IMFk.(3)
The white noiser1t+ε1E1ωit, i=1,2,…,I, is added to the kth residual component and EMD is performed. Then, IMFk+1 can be calculated with the mean value of the first IMF:(5)IMFk+1=1I∑i=1IE1rtk+εkEkwit.(4)
Repeat equation (4) and (5) until the value of residual component is less than two extremes, and then the decomposition is stopped. Eventually, the residual variable is obtained:(6)rt=Xt−∑k=1KIMFk,whereK is the total number of modes in the decomposition process, and the reconstructed signal can be expressed as follows:(7)Xt=rt+∑k=1KIMFk.


### 2.1.2. Selection of IMFs

The raw signal is transformed to a series of IMFs via CEEMDAN, but not all IMFs are related to fault. Some IMFs contain few fault information; thus, these IMFs have no contribution to the following fault identification but do impact the analysis speed. We must therefore develop an IMF selection methodology for selecting the better sensitive IMF related to fault.The parameter correlation coefficient with time can show the related degree between two signals. According to the Schwarz inequality, we can see the following:ρxyτ>0 represents positive correlation, ρxyτ<0 represents negative correlation, ρxyτ=0 represents no correlation, ρxyτ=1 represents perfectly positive correlation, and ρxyτ=−1 represents perfectly negative correlation, where ρxyτ is the correlation coefficient. The greater the absolute value of ρxyτ is, the higher the related degree will be.This paper proposes the selection of effective IMFs according to the correlation coefficient of the raw signal and each IMF. The IMFs with a higher correlation coefficient, whereρxyτ≥0.6 represents greater amounts fault information included [24], are selected as the optimal IMFs for the following analysis.


## 2.1.1. CEEMDAN Theory

To better describe the CEEMDAN algorithm, we define an operatorEk⋅, the function of which is to solve the kth modal component IMFk of the EMD decomposition. wi is the white noise satisfying the distribution of N0,1, and εk is the amplitude coefficient of white noise added for the Kth time. The decomposition process for CEEMDAN is as follows:(1)
The white noiseXt+ε0ωit is added to the original signal, and Ith EMD decomposition is performed. Then, the average operation is performed on the result to obtain IMF1:(1)IMF1=1I∑i=1IE1Xt+ε0wit.(2)
The first-stage residual component can then be calculated:(2)r1t=Xt−IMF1.The white noiser1t+ε1E1ωit, i=1,2,…,I is added to the first-stage residual component, and the EMD is performed. Then, IMF2 can be calculated using the mean value of the first IMF:(3)IMF2=1I∑i=1IE1rt1+ε1E1wit.Fork=1,2,…,K, the Kth residual component can be calculated as follows:(4)rkt=rk−1t−IMFk.(3)
The white noiser1t+ε1E1ωit, i=1,2,…,I, is added to the kth residual component and EMD is performed. Then, IMFk+1 can be calculated with the mean value of the first IMF:(5)IMFk+1=1I∑i=1IE1rtk+εkEkwit.(4)
Repeat equation (4) and (5) until the value of residual component is less than two extremes, and then the decomposition is stopped. Eventually, the residual variable is obtained:(6)rt=Xt−∑k=1KIMFk,whereK is the total number of modes in the decomposition process, and the reconstructed signal can be expressed as follows:(7)Xt=rt+∑k=1KIMFk.


## 2.1.2. Selection of IMFs

The raw signal is transformed to a series of IMFs via CEEMDAN, but not all IMFs are related to fault. Some IMFs contain few fault information; thus, these IMFs have no contribution to the following fault identification but do impact the analysis speed. We must therefore develop an IMF selection methodology for selecting the better sensitive IMF related to fault.The parameter correlation coefficient with time can show the related degree between two signals. According to the Schwarz inequality, we can see the following:ρxyτ>0 represents positive correlation, ρxyτ<0 represents negative correlation, ρxyτ=0 represents no correlation, ρxyτ=1 represents perfectly positive correlation, and ρxyτ=−1 represents perfectly negative correlation, where ρxyτ is the correlation coefficient. The greater the absolute value of ρxyτ is, the higher the related degree will be.This paper proposes the selection of effective IMFs according to the correlation coefficient of the raw signal and each IMF. The IMFs with a higher correlation coefficient, whereρxyτ≥0.6 represents greater amounts fault information included [24], are selected as the optimal IMFs for the following analysis.


## 2.2. Permutation Entropy and Its Parameter Selection

### 2.2.1. The Theory of Permutation Entropy

Entropy can be used to describe the uncertainty of data information. Permutation entropy is a random time sequence detection method and can reflect the one-dimensional time-series complexity, which has the advantages of simple design, strong antinoise ability, and high robustness. The following are the specific principles of the permutation entropy algorithm.Assuming a time seriesai,i=1,2,…,N with length N, the phase space reconstruction is carried out as follows:(8)H=H1H2⋮Hj⋮HQ=a1a1+t⋯a1+m−1ta2a2+t⋯a2+m−1t⋯⋯⋯ajaj+t⋯aj+m−1t⋯⋯⋯aQaQ+t⋯aQ+m−1t,where j=1,2,3,…,Q, m is the embedded dimension, t is the delay time, and Q+m−1t=n. Next, rearrange the jth component of refactoring matrix in ascending order, so that(9)Hj=aj+i1−1t≤aj+i2−1t≤⋯≤aj+im−1t,where i1,i2,…,im is the column position of Hj. If aj+ie−1t=aj+ic−1t exists, they are arranged by the values of e and c, while if e>c, aj+ie−1t≥aj+ic−1t and vice versa. Therefore, any component Hj of the reconstructed matrix H can be used to obtain the corresponding position sequence:(10)Sj=i1,i2,…,im,j=1,2,3,…,q,where q≤m!; due to m different codes i1,i2,…,im, there are m! different permutations, namely, m! different symbol sequences. Sj is just one of the m! code sequences.Next, calculate the probability of each sequenceP1,P2,…,Pq,∑j=1qPj=1. The permutation entropy of the signal time series ai,i=1,2,…,N can be defined in the Shannon entropy form as follows:(11)LPEm=−∑j=1qPjlnPj.WhenPj=1/m , LPEm will maximize lnm!. Then, use lnm! to standardize the permutation entropy LPEm:(12)LPE=LPEmlnm!,where 0≤LPE≤1. The value of LPE represents the degree of randomness of the one-dimensional time series. When LPE is larger, the randomness of the time series is stronger. On the contrary, this indicates that the regularity of the time series is stronger.


### 2.2.2. Calculation of Parameters Selection

In the permutation entropy calculation process, there are two important parameters, namely, the embedded dimensionm and the delay time t, where the different settings of these two parameters will have an impact on the calculation results. Selecting the proper embedding dimension m and time delay t is the key to extracting the permutation entropy feature of spiral-bevel gears.By comparing the original vibration signals of spiral-bevel gears with different fault states, and combining with an analysis of existing literature, it is observed that the embedded dimensionm = 4, 5, 6 and time delay t = 1, 2, 3 are more suitable to the slight mutation of vibration signals, and then the optimal parameters are selected from these values. Due to the fact that embedding dimension and delay together impacts the calculation results, it is not reasonable to consider the change of a single parameter. In this paper, the overlapping grouping method is adapted to optimally select the embedding dimension and time delay, meaning that a section of original vibration signal of the spiral-bevel gear is divided into a series of subsequences w1,w2,…,wn according to the time series, where w1 moves back one data point to obtain w2, and so on. Then, the permutation entropy is calculated by the combination of embedded dimension and time delay. Since larger w cannot accurately reflect the signal changes and smaller w has low efficiency and no statistical significance, w = 128 is selected.Figure1 shows the combination of the original vibration signals in the normal state and the 2/3 broken tooth fault state of the spiral-bevel gear with a sampling frequency of 16384 Hz, where 0-1 s is the normal gear state, and 1-2 s is the two-third broken tooth state.Figure 1
Normal state and 2/3 broken tooth state vibration signal combination.Figure2(a) shows the calculation values of permutation entropy when t = 1 and m = 4, 5, and 6. Figure 2(b) shows the calculation values of permutation entropy when t = 2 and m = 4, 5, and 6. Figure 2(c) shows the calculation values of permutation entropy when t = 3 and m = 4, 5, and 6.Figure 2
The calculation values of permutation entropy with differentm and t. (a) t = 1 and m = 4, 5, and 6. (b) t = 2 and m = 4, 5, and 6. (c) t = 3 and m = 4, 5, and 6.
(a)
(b)
(c)It can be seen from Figure2 that the permutation entropy under the normal state is larger than that under the 2/3 broken tooth fault states. This is because the vibration signal of the spiral-bevel gear under normal states is more random, and when the tooth appears to be broken, it will result in regular shock signals, and thus the vibration under 2/3 broken tooth fault states is more regular. Comparing Figures 2(a), 2(b), and 2(c), it can be found that when t = 1 and 2 and m = 4 and 5, all permutation entropies can amplify weak information mutation and distinguish normal states and 2/3 broken states, and when t = 1 and m = 4, the permutation entropy mutation is more obvious when the states undergo mutation. Therefore, the time delay t = 1 and embedded dimension m = 4 are selected for the following permutation entropy calculation in this paper.


## 2.2.1. The Theory of Permutation Entropy

Entropy can be used to describe the uncertainty of data information. Permutation entropy is a random time sequence detection method and can reflect the one-dimensional time-series complexity, which has the advantages of simple design, strong antinoise ability, and high robustness. The following are the specific principles of the permutation entropy algorithm.Assuming a time seriesai,i=1,2,…,N with length N, the phase space reconstruction is carried out as follows:(8)H=H1H2⋮Hj⋮HQ=a1a1+t⋯a1+m−1ta2a2+t⋯a2+m−1t⋯⋯⋯ajaj+t⋯aj+m−1t⋯⋯⋯aQaQ+t⋯aQ+m−1t,where j=1,2,3,…,Q, m is the embedded dimension, t is the delay time, and Q+m−1t=n. Next, rearrange the jth component of refactoring matrix in ascending order, so that(9)Hj=aj+i1−1t≤aj+i2−1t≤⋯≤aj+im−1t,where i1,i2,…,im is the column position of Hj. If aj+ie−1t=aj+ic−1t exists, they are arranged by the values of e and c, while if e>c, aj+ie−1t≥aj+ic−1t and vice versa. Therefore, any component Hj of the reconstructed matrix H can be used to obtain the corresponding position sequence:(10)Sj=i1,i2,…,im,j=1,2,3,…,q,where q≤m!; due to m different codes i1,i2,…,im, there are m! different permutations, namely, m! different symbol sequences. Sj is just one of the m! code sequences.Next, calculate the probability of each sequenceP1,P2,…,Pq,∑j=1qPj=1. The permutation entropy of the signal time series ai,i=1,2,…,N can be defined in the Shannon entropy form as follows:(11)LPEm=−∑j=1qPjlnPj.WhenPj=1/m , LPEm will maximize lnm!. Then, use lnm! to standardize the permutation entropy LPEm:(12)LPE=LPEmlnm!,where 0≤LPE≤1. The value of LPE represents the degree of randomness of the one-dimensional time series. When LPE is larger, the randomness of the time series is stronger. On the contrary, this indicates that the regularity of the time series is stronger.


## 2.2.2. Calculation of Parameters Selection

In the permutation entropy calculation process, there are two important parameters, namely, the embedded dimensionm and the delay time t, where the different settings of these two parameters will have an impact on the calculation results. Selecting the proper embedding dimension m and time delay t is the key to extracting the permutation entropy feature of spiral-bevel gears.By comparing the original vibration signals of spiral-bevel gears with different fault states, and combining with an analysis of existing literature, it is observed that the embedded dimensionm = 4, 5, 6 and time delay t = 1, 2, 3 are more suitable to the slight mutation of vibration signals, and then the optimal parameters are selected from these values. Due to the fact that embedding dimension and delay together impacts the calculation results, it is not reasonable to consider the change of a single parameter. In this paper, the overlapping grouping method is adapted to optimally select the embedding dimension and time delay, meaning that a section of original vibration signal of the spiral-bevel gear is divided into a series of subsequences w1,w2,…,wn according to the time series, where w1 moves back one data point to obtain w2, and so on. Then, the permutation entropy is calculated by the combination of embedded dimension and time delay. Since larger w cannot accurately reflect the signal changes and smaller w has low efficiency and no statistical significance, w = 128 is selected.Figure1 shows the combination of the original vibration signals in the normal state and the 2/3 broken tooth fault state of the spiral-bevel gear with a sampling frequency of 16384 Hz, where 0-1 s is the normal gear state, and 1-2 s is the two-third broken tooth state.Figure 1
Normal state and 2/3 broken tooth state vibration signal combination.Figure2(a) shows the calculation values of permutation entropy when t = 1 and m = 4, 5, and 6. Figure 2(b) shows the calculation values of permutation entropy when t = 2 and m = 4, 5, and 6. Figure 2(c) shows the calculation values of permutation entropy when t = 3 and m = 4, 5, and 6.Figure 2
The calculation values of permutation entropy with differentm and t. (a) t = 1 and m = 4, 5, and 6. (b) t = 2 and m = 4, 5, and 6. (c) t = 3 and m = 4, 5, and 6.
(a)
(b)
(c)It can be seen from Figure2 that the permutation entropy under the normal state is larger than that under the 2/3 broken tooth fault states. This is because the vibration signal of the spiral-bevel gear under normal states is more random, and when the tooth appears to be broken, it will result in regular shock signals, and thus the vibration under 2/3 broken tooth fault states is more regular. Comparing Figures 2(a), 2(b), and 2(c), it can be found that when t = 1 and 2 and m = 4 and 5, all permutation entropies can amplify weak information mutation and distinguish normal states and 2/3 broken states, and when t = 1 and m = 4, the permutation entropy mutation is more obvious when the states undergo mutation. Therefore, the time delay t = 1 and embedded dimension m = 4 are selected for the following permutation entropy calculation in this paper.


## 2.3. Sensibility Assessment for CEEMDAN-Based Permutation Entropy

At present, fault diagnosis can be realized by two procedures, namely, feature extraction and mode recognition. Both the sensitivity of features and effectiveness of the classifiers determine the diagnostic accuracy. By selecting the same classification and different feature vectors, the final diagnostic accuracy can reflect the sensitivity of the features. Based on this ideology, the SVM is used as the classifier for mode recognition, and the final diagnostic accuracy is used to assess the sensibility of CEEMDAN-based permutation entropy in this paper.At present, the analysis procedure of this paper is carried out as follows:(1)
Original vibration signal acquisition: the vibration signals of spiral-bevel gears under different typical fault states are collected as the raw time-series signals for subsequent analysis.(2)
CEEMDAN: a series of IMFs are obtained by CEEMDAN of the raw vibration signals under different failure states.(3)
Effective IMFs selected: the correlation coefficient of raw signal and IMF is calculated and is combined with the signal-to-noise ratio value. The IMFs’ larger correlation coefficient and high signal-to-noise ratio value are selected for subsequent analysis.(4)
Permutation entropy calculation: the permutation entropy of the optimal IMFs is calculated using the time delayt = 1 and embedded dimension m = 4, and then the multidimensional eigenvector LPE1,LPE2,…,LPEh is constructed.(5)
SVM classifier for mode recognition: the characteristic vector obtained in Step (4) is used to train and test the SVM and obtain diagnostic accuracy.(6)
Sensibility assessment: the accuracy of CEEMDAN-based permutation entropy is verified based on the diagnostic accuracy.


## 3. Experimental Setup and Data Acquisition

The experimental data are collected to evaluate the effectiveness of the method presented in this paper. The spiral-bevel gearbox test equipment used for vibration signal acquisition under typical fault states is shown in Figure3. The test equipment is composed of a governor, a motor, a coupling, a pair of spiral-bevel gears, and a load. The respective numbers of driving gears and driven gears are 10 and 30. The driving gear is used as the test gear to simulate three typical fault gears, normal tooth, 1/3 broken tooth, and 2/3 broken tooth, respectively, as shown in Figure 4. The accelerometers were placed at three positions: one at the input shaft support bearing, one at the output shaft support bearing, and one at the top of the gearbox. The vibration signals are collected by using B&K’s PULSE data acquisition system. During the test, the speed of the motor is kept constant by adjusting the governor to 1200 r/min, and the vibration signals with three different degrees of broken teeth and the same load are collected three times at each mode. The sampling frequency is 16384 Hz.Figure 3
Spiral-bevel gearbox test equipment.Figure 4
Gears with different degrees of broken teeth.
(a)
(b)
(c)The time-domain vibration signals under three states of spiral-bevel gears are shown in Figure5. It can be seen from Figure 5 that the vibration signal under the normal state shown in Figure 5(a) is relatively scattered. The 1/3 broken tooth state and 2/3 broken tooth state shown in Figures 5(b) and 5(c) show that the amplitude peak value increases significantly compared with the normal state, and as the degree of broken tooth increases, the vibration impact of tooth engagement becomes greater.Figure 5
The time-domain vibration signals under three states of spiral-bevel gears. (a) Normal tooth. (b) 1/3 broken tooth. (c) 2/3 broken tooth.
(a)
(b)
(c)The axial vibration signals from the input shaft support bearing are divided into segments and then processed according to the method described in Section2.


## 4. Results and Discussion

The raw vibration signals of the spiral-bevel gear under three failure states are decomposed by the CEEMDAN method, in which the noise standard deviation of CEEMDAN is 0.19, and the overall average frequency is 100. Due to space limitations, only the CEEMDAN results of the 1/3 broken tooth are shown in Figure6. It can be seen from the figure that the original vibration signal is decomposed into 13 IMFs, of which the 13th component is the residual term. The mode-mixing phenomenon of the waveform gradually weakens from IMF1 to IMF12, and the frequency is also distributed from high to low.Figure 6
CEEMDAN results.The correlation coefficients and signal-to-noise ratio of each IMF shown in Figure6 are calculated, and the results are shown in Table 1. It can be seen from Table 1 that the first two orders of IMFs are highly correlated with the raw vibration signal, both being above 0.6; thus, the first two orders of IMFs are selected.Table 1
Correlation coefficient.
IMF component
Correlation coefficient
IMF1
0.731
IMF2
0.634
IMF3
0.506
IMF4
0.292
IMF5
0.176
IMF6
0.184
IMF7
0.079
IMF8
0.013
IMF9
0.0031
IMF10
0.0059
IMF11
0.0035
IMF12
−8.68e-04
IMF13
7.35e-04The permutation entropy of the first two IMFs under three failure states is calculated with time delayt = 1 and embedded dimension m = 4. The comparison diagram of three failure states is shown in Figure 7. From Figure 7, it can be seen that the permutation entropy of the optimally selected IMFs can clearly distinguish among three typical failure states of spiral-bevel gears.Figure 7
Comparison of CEEMDAN-based permutation entropy of three states. (a) Permutation entropy of IMF1. (b) Permutation entropy of IMF1.
(a)
(b)In order to further verify the sensitivity of CEEMDAN-based permutation entropy, the permutation entropies of IMF1 and IMF2 are used to construct a two-dimensional feature vector, then the SVM is used to classify the mode recognition, and the final diagnostic accuracy is used to assess the sensibility of CEEMDAN-based permutation entropy. With 40 samples of each fault state, a total of 120 samples are used as the training samples. The normal state label of spiral-bevel gears is set as “1,” the state of 1/3 broken tooth is set as “2,” and that of 2/3 broken tooth is set as “3.” The RBF Gaussian kernel radial basis function is used for the SVM kernel function, and the grid search method of cross validation is used to determine the optimal penalty factor and kernel function parameters. With 40 samples of each fault state, a total of 120 samples are used as the test samples and input into the trained SVM classifier for classification and recognition. The classification results are shown in Figure 8, with an accuracy rate of 100%, which indicates that the CEEMDAN-based permutation entropy has a high sensitivity for fault identification in spiral-bevel gears.Figure 8
The spiral-bevel gear fault diagnostic results by using CEEMDAN-based permutation entropy and SVM.In order to verify the superiority of CEEMDAN-based permutation entropy, we compare it with EEMD-based permutation entropy. The same vibration signals of the spiral-bevel gear in the aforementioned results are decomposed by the EEMD method, and the permutation entropy values of the first two IMFs are calculated with the same parameter set. The comparison diagram of EEMD-based permutation entropy of the three failure states is shown in Figure9. It can be seen that the permutation entropy of the optimal selected IMFs of EEMD can distinguish three typical failure states of the spiral-bevel gear, but the distinction ability is not as good as the CEEMDAN-based method. The reason for this is that the resulting IMFs derived from the EEMD method would inevitably be contaminated by the added noise, especially when the number of ensembles is relatively low.Figure 9
Comparison of EEMD-based permutation entropy of three states. (a) Permutation entropy of IMF1. (b) Permutation entropy of IMF2.
(a)
(b)The two-dimensional feature vectors constructed by EEMD-based permutation entropy are input into the SVM classifier for training and testing. The test results are shown in Figure10. From Figure 10, it can be seen that the overall classification result is only 88.33%. In addition, for label “1,” i.e., the normal state of the spiral-bevel gear, 7 of the 40 test samples are incorrectly identified, and for label “2,” i.e., one-third of the broken tooth, 7 of the 40 test samples are also incorrectly identified, while for label “3,” i.e., two-third of the broken tooth, the identification results are completely accurate.Figure 10
The spiral-bevel gear fault diagnostic results by using EEMD-based permutation entropy and SVM.As discussed earlier, the two-dimensional feature vectors constructed by EMD-based permutation entropy are input into the SVM classifier for training and testing. The test results are shown in Figure11, where it can be seen that the overall classification result is only 83.33%. For label “1,” i.e., the normal state of the spiral-bevel gear, 5 of the 40 test samples are incorrectly identified, and for label “2,” i.e., one-third of the broken tooth, 5 of the 40 test samples are also incorrectly identified, while for label “3,” i.e., two-third of the broken tooth, 10 of the 40 test samples are incorrectly identified. The reason for this is that EMD has the major drawback of mode mixing.Figure 11
The spiral-bevel gear fault diagnostic results by using EMD-based permutation entropy and SVM.As illustrated in Section2.3, which features the same classification tool but different feature vectors for the fault mode recognition of spiral-bevel gears, the diagnostic accuracy can reflect the sensitivity of the features. Therefore, compared with the diagnostic results by using CEEMDAN-based/EEMD-based/EMD-based permutation entropy and SVM, it is easy to reach a conclusion, and the CEEMDAN-based permutation entropy is the most sensitive feature for fault identification of spiral-bevel gears.


## 5. Conclusion

This paper proposes to take the CEEMDAN-based permutation entropy as the sensitive feature for fault identification of spiral-bevel gears. An experimental study has been carried out on a spiral-bevel gearbox to verify the effectiveness of the feature. From observations of the analytical results, the following conclusions are drawn:(1)
The CEEMDAN-based permutation entropy of the first two IMFs can be used to clearly distinguish three typical failure states of spiral-bevel gears.(2)
Using the CEEMDAN-based permutation entropy to construct the feature vector and the SVM as the classifier for fault mode recognition of the spiral-bevel gears, the accuracy of the high spiral-bevel gears fault diagnostic verifies the sensitivity of the feature.(3)
Compared with the diagnostic results based on CEEMDAN-based/EEMD-based/EMD-based permutation entropy and SVM, the diagnostic accuracies are 100%, 88.33%, and 83.33%, respectively, which indicates that the CEEMDAN-based permutation entropy is the most sensitive feature for fault identification of spiral-bevel gears.In this paper, the sensitive feature is used to analyze the fault mode of spiral-bevel gears. Our future research will focus on the analysis of the fault severity.



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*Source: 7806015-2019-12-02.xml*