Shock and Vibration

Volume 2019, Article ID 7806015, 13 pages

https://doi.org/10.1155/2019/7806015

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

Correspondence should be addressed to Xuejun Li; moc.361@jxlxdjknh

Received 26 July 2019; Revised 26 September 2019; Accepted 22 October 2019; Published 2 December 2019

Guest Editor: Franco Concli

Copyright © 2019 Lingli Jiang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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.

#### 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 operator , the function of which is to solve the *k*th modal component IMF_{k} of the EMD decomposition. is the white noise satisfying the distribution of , and *ε*_{k} is the amplitude coefficient of white noise added for the *K*th time. The decomposition process for CEEMDAN is as follows:(1)The white noise is added to the original signal, and *I*th EMD decomposition is performed. Then, the average operation is performed on the result to obtain IMF1:(2)The first-stage residual component can then be calculated: The white noise , is added to the first-stage residual component, and the EMD is performed. Then, IMF_{2} can be calculated using the mean value of the first IMF: For , the *K*th residual component can be calculated as follows:(3)The white noise , , is added to the *k*th residual component and EMD is performed. Then, IMF_{k+1} can be calculated with the mean value of the first IMF:(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: where *K* is the total number of modes in the decomposition process, and the reconstructed signal can be expressed as follows:

###### 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: represents positive correlation, represents negative correlation, represents no correlation, represents perfectly positive correlation, and represents perfectly negative correlation, where is the correlation coefficient. The greater the absolute value of 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, where 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 series with length *N*, the phase space reconstruction is carried out as follows:where , *m* is the embedded dimension, *t* is the delay time, and . Next, rearrange the *j*th component of refactoring matrix in ascending order, so thatwhere is the column position of . If exists, they are arranged by the values of **e** and **c**, while if , and vice versa. Therefore, any component of the reconstructed matrix *H* can be used to obtain the corresponding position sequence:where ; due to *m* different codes , there are different permutations, namely, different symbol sequences. is just one of the code sequences.

Next, calculate the probability of each sequence . The permutation entropy of the signal time series can be defined in the *Shannon* entropy form as follows:

When , will maximize . Then, use to standardize the permutation entropy :where . The value of represents the degree of randomness of the one-dimensional time series. When 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 dimension *m* 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 dimension *m* *=* 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 according to the time series, where moves back one data point to obtain , and so on. Then, the permutation entropy is calculated by the combination of embedded dimension and time delay. Since larger cannot accurately reflect the signal changes and smaller has low efficiency and no statistical significance, = 128 is selected.

Figure 1 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.