EEMD-Based cICA method for single-channel signal separation and fault feature extraction of gearbox

Junfa Leng1 , Shuangxi Jing2 , Chenxu Luo3 , Zhiyang Wang4

1, 2, 3, 4School of Mechanical and Power Engineering, Henan Polytechnic University, Jiaozuo 454000, Henan, China

2Corresponding author

Journal of Vibroengineering, Vol. 19, Issue 8, 2017, p. 5858-5873. https://doi.org/10.21595/jve.2017.18115
Received 19 December 2016; received in revised form 4 June 2017; accepted 14 June 2017; published 31 December 2017

Copyright © 2017 JVE International Ltd. 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.
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Abstract.

This paper proposes a novel fault feature extraction method with the aim of extracting the fault feature submerged in the single-channel observation signal. The proposed method integrates the strengths of the constrained independent component analysis (cICA) extracting only the signals of interest (SOIs) with the advantage of ensemble empirical mode decomposition (EEMD) alleviating the mode mixing. The method, which is named EEMD-based cICA, not only enables gear fault feature extraction but also offers a new independent component analysis (ICA) mixing model with source noise and measured noise for the single-channel observation signal. The efficiency of the proposed method is tested on simulated as well as real-world vibration signals acquired from a multi-stage gearbox with a missing tooth and a chipped tooth, respectively.

Keywords: single-channel observation signal, gearbox, EEMD, cICA, fault feature extraction.

1. Introduction

In general, the goal of independent component analysis (ICA) [1-4] is to recover all the source signals from mixed signals at a time. ICA is one of the outstanding techniques for solving the signal blind source separation (BSS) problem, which has been widely applied to the source signals separation and feature extraction [3, 4] in the applications of biomedical engineering, telecommunications, mechanical engineering and audio. However, there are many problems to be solved for ICA applications: (1) classical ICA algorithm has some ambiguities, such as unknown number of source signals, undetermined the variance (energies) and the order of the independent components (ICs); (2) ICA model does not consider the source noise and measured noise simultaneously [3]; (3) It is desired to extract only the signals of interest (SOIs). (4) The difficulty of the single-channel observation signal signature extraction based on ICA, it belongs to the extreme case of the underdetermined BBS problem [4]. Therefore, it would be important to develop approaches to extract only the desired signal with given signature instead of all source signals from the single-channel observation signal.

ICA algorithm as the most important blind signal extraction (BSE) method has been used to extract the ICs, whose number is the same as the measured signals, but the SOIs are unknown. Hiroshi et al. [5] proposed time-frequency based ICA method to extract SOIs, but it needs some source signals to have dominant powers. W. Lu and J.C. Rajapakse [6, 7] proposed the constrained ICA (cICA) or ICA with reference (ICA-R) algorithms by incorporating a prior information into the conventional ICA algorithm, which means that only a single statistical IC will be extracted from the mixed signals, but it does not specifically discuss how to generate a reference signal. Zhi-Lin Zhang [8] developed a morphological cICA algorithm to extract weak temporally correlated signals from a pregnant woman ECG data, this method used second-order statistics based approach to design the suitable reference signal. Zhan-Li Sun et al. [9] proposed an improved cICA by using the reference based unmixing matrix initialization, which overcame the unstable problem encountered in cICA algorithm. Changli Li et al. [10] proposed an improved ICA-R algorithm for the non-invasive extraction of the fetal ECG (FECG), which alternately maximizes the negentropy contrast function for FastICA and the closeness measure function in ICA-R. Xiang Wang et al. [11] extended the conventional cICA framework to the case of complex-valued mixing model and presented different prior information, the method is named as ICA with cyclostationary constraint (ICA-CC) and ICA with spatial constraint (ICA-SC). Zhiyang Wang et al. [12, 13] introduced cICA into the machine fault diagnosis, and attained some successful applications.

In practice, for most of the ICA-based methods, it should not be applied to the underdetermined BSS cases, in which the number of sensors is less than the source signals [4]. Especially in the extreme underdetermined BBS case, that is to say, single-channel observation signal separation, the number of sensor is only one. This is a very undesirable requirement for real-world applications because the number of active source signals is unknown in advance in most practical situations. In this case, single-channel observation signal mixing matrix is not invertible, and the traditional ICA or cICA methods fail to recover all sources, which also leads to the result that the desired signal cannot be extracted directly from the single-channel observation signal. Therefore, single-channel observation signal needs to be separated into several statistically independent components by using some approaches. Among these approaches, wavelet transform (WT) [14, 15] and empirical mode decomposition (EMD) [16, 17] are most usually employed to play the role of decomposing signal into various time scales. D.S Lee et al. [18] presented WT and PCA-based monitoring methods and illustrated its great potential in monitoring multiscale and multivariate processes. Wu, et al. [19, 20] combined continue WT with ICA to accomplish the early fault diagnosis of bearing. But WT requires choosing wavelet basis and decomposing layers, which makes it a non-self-adaptive signal processing method in nature. Empirical mode decomposition (EMD) algorithm [16, 17] can self-adaptively decompose any complex signal into a set of intrinsic mode functions (IMFs) according to the analyzed signal itself characteristic, and each IMF denotes a simple oscillatory mode in nature with different frequency component imbedded in the original signal. B. Mijovic et al. [21, 22] proposed a new method of sources separation from single-channel signal based on EMD and ICA. Q. Miao et al. [23] used EMD-based ICA method to extract the bearing fault feature. But EMD still has some disadvantages, such as end effects and modes mixing. Wu and Huang [24] developed and improved the EMD algorithm substantially, and proposed the ensemble empirical mode decomposition (EEMD) algorithm, which effectively alleviates the mode mixing of EMD algorithm. M. Žvokelj et al. [25] developed a method of multivariate and multiscale monitoring of bearings using EEMD and PCA, and then proposed an approach of non-linear multivariate and multiscale monitoring and signal denoising strategy using EEMD and KPCA [26]. Wang et al. [27] integrated EEMD and ICA to diagnosis wind turbine gearbox. After several years, Žvokelj et al. [28] again developed an EEMD-based multiscale ICA method to diagnosis the slewing bearing fault.

So far, the method of cICA combined with EEMD is seldom used to mechanical signals processing. Therefore, a so-called EEMD-based cICA method is proposed and applied to the BSE of single-channel observation signal. The validity and practicability of this proposed method are verified through simulation and experiments of gear fault characteristics extraction with a missing tooth and a chipped tooth, respectively.

This paper is organized as follows: Section 2 introduces the ICA model and the mixing model of the single-channel observation signal with source noise and measured noise. The single-channel signal separation and fault feature extraction method of EEMD-based cICA are elucidated in Section 3. Then, simulation and experiments are demonstrated in Section 4 and Section 5, respectively. Finally, Section 6 provides a conclusion.

2. Mixing model of single-channel measured signal based on ICA

2.1. Independent component analysis

In essence, ICA algorithm [1-4] assumes a set of m observable measured signals xt=x1t,x2t,,xmtT to be a linear combination of n unknown and statistically independent sources st=s1t,s2t,,sntT nm. The time series usually have unit variance and uncorrelation by using a linear “whitening” transform. Then ICA mixing model can be expressed as:

(1)
x t = A s t ,

where Am×n is the mixing matrix, usually m=n.

ICA algorithm must find a separating or de-mixing matrix W such that:

(2)
y t = W x t ,

where yt=y1t,y2t,,yntT is an approximate estimation of source signals st.

2.2. Mixing model of single-channel measured signal

In Eq. (1), if the row number m of the mixing matrix A is equal to 1, i.e m= 1, then the classical ICA mixing model is rewritten as:

(3)
x t = A s t ,

where A1×n is an unknown non-singular linear mixing vector, A=a1,a2,,an.

Consider the additional source noise and measured noise, and rewrite the Eq. (3) as:

(4)
x t = A s t + e s t + e m t ,

where est and emt represent the source noise and measured noise, respectively.

Eq. (4) shows the noisy ICA mixing model of the single-channel observation signal xt. It belongs to the extreme case of the underdetermined BBS problem, and cannot be solved directly. For this reason, we developed an EEMD-based cICA method to separate fault signal from the single-channel observation signal xt.

3. Single-channel signal separation and fault feature extraction

3.1. Ensemble empirical mode decomposition

3.1.1. Empirical mode decomposition

Empirical mode decomposition (EMD) was pioneered by Huang et al. [16] in 1998. EMD has the ability of nonlinear multi-resolution self-adaptive signal processing, and is very applicable to processing the nonstationary data. A complicated signal xt can be decomposed into the sum of n IMF components cjt,j=1, 2,,n and a residue rnt by EMD method:

(5)
x ( t ) = j = 1 n c j t + r n t .

3.1.2. EEMD algorithm

EMD method has been successfully applied to mechanical signal processing [17]. Nevertheless, EMD cannot extract mechanical fault feature accurately because of the mode mixing phenomenon, which can make physical meanings unclear. To alleviate this drawback, Wu and Huang [24] developed and improved the EMD algorithm substantially, and proposed the ensemble empirical mode decomposition (EEMD) algorithm. Y. H. Wang et al. discussed the computational complexity of EMD/EEMD algorithms [29]. The decomposition procedures of EEMD are expressed briefly as follows:

1. Add a differently generated white noise eit with a different magnitude σei to the original signal xt each time to generate a new signal:

(6)
x i t = x t + σ e i e i t .

2. Decompose the newly generated signal xit into IMFs using the EMD method:

(7)
x i t = j = 1 n i c i , j t + r i , n t ,

where ci,jt, ri,nt and ni represent the jth IMF, the residue and the IMFs’ number during the ith trial, respectively.

3. Calculate the ensemble means of the corresponding IMFs of N1 times decompositions, and take it as the final result:

(8)
c - j t = 1 N 1 i = 1 N 1 c i , j t , r - n t = 1 N 1 i = 1 N 1 r i , n t .

4. Finally, the original signal xt is formed as follows:

(9)
x t = j = 1 n c - j t + r - n t .

3.1.3. Criterions of IMF selection

EEMD method can effectively alleviate the mode aliasing, but it will produce false components during its decomposition procedures. Therefore, we propose the following criterions of IMFs selection in order to eliminate the influence of false IMFs.

3.1.3.1. Correlation coefficient-based

The correlation coefficient ρ between IMF c-jtand original signal xt is as follows:

(10)
ρ c - j , x = c o v c - j t , x t σ c - j σ x = k = 1 N c - j k x k σ c - j σ x .

When IMF includes some fault characteristics, the correlation coefficient between the IMF and the original signal is relatively larger, on the contrary, it is much smaller.

3.1.3.2. Kurtosis-based

However, when the signal-to-noise ratio (SNR) of the observation signal xt is extremely low, that is to say, the concealed fault information is very weak. In this case, even if the IMF includes effective fault information, the correlation coefficient between the corresponding IMF and the original signal could be also very small. Therefore, we must introduce another criterion of IMF section, i.e. kurtosis-based combined with the correlation coefficient-based criterion. The kurtosis of the IMF is expressed as:

(11)
K c - j = 1 N k = 1 N c - j k σ c - j 4 .

In Eqs. (10-11), c-jk and xk are zero-mean, i.e. μx=μc-j= 0, σ denotes the standard deviation, and N is the data length.

Usually, the larger the kurtosis value of the IMF, the more prominent the effective fault information of the corresponding IMF.

3.2. cICA principle

Constrained independent component analysis (cICA) [6, 7] method is derived from independent component analysis (ICA) algorithm. By incorporating an interesting priori information into the traditional ICA algorithm, cICA algorithm forms a constraint optimization problem, and ensures that the ICA model output is a necessarily desired independent component (IC), which is closest to a corresponding reference signal rt [12]. The reference signal rt with interesting fault feature denotes the inequality constrained condition but need not be a perfect match with the desired IC. We take εr,y as the closeness measure norm between the IC yt and the corresponding reference signal rt. Note that the desired IC, which is extracted from the new observation signal vector xt=x1t,x2t,,xmtT, is the one and only the one closest to the corresponding constructed reference signal rt, which is satisfied the following the inequality relationship:

(12)
ε r , w * T x < ε r , w 1 T x ε r , w l - 1 T x ,

where w* is the optimum de-mixing vector corresponding to the desired output IC, and wi, i=1, 2,,l-1 wiw0 are any other l-1 local optimal solutions corresponding to the undesired output ICs. Thus, an inequality constraint, only when the optimum equation y=y*=w*Tx is satisfied, is expressed as follows:

(13)
g y = ε r , y ζ 0 ,

where ζεr,w*Tx,εr,w1Tx is a threshold parameter, the closeness measure norm εr,y is usually expressed by εr,y=Er-y2.

The model of cICA framework [5, 6] as a constrained optimization problem is defined as:

(14)
max       J y ρ E f y - E f υ 2 ,
s . t .             g y 0 ,
                        h y = E y 2 - 1 = 0 ,         h r = E r 2 - 1 = 0 ,

where Jy denotes the negentropy function, f· is an any non-quadratic function, ρ is a positive constant, υ is a Gaussian variable with zero-mean and unit variance, gy is the closeness constraint described in Eq. (13), and the equality constraints h· ensure that the output yt and the reference signal rt have unit-variance.

The model of cICA algorithm is efficiently solved by the use of an augmented Lagrangian function [7]. At the same time, we use the signal-to-interference ratio (SIR) index [12] to evaluate the extraction quality of the cICA algorithm. The larger the SIR, the better the extraction effect of cICA algorithm. More details about the model of cICA framework are expressed as a constrained optimization problem in Refs. [6-13].

3.3. Constructing reference signal for cICA in gearbox diagnostics

The faulty signal in gear transmission system mostly appears as a periodical impact sequence. Hence, we may select a series of pulses or square wave as the suitable reference signal, such as Eq. (13) below:

(15)
r t = s q u a r e 2 π f m t + θ , w ,

where fm is the gear meshing frequency, θ is the initial phase angle or time-delay and w is the duty ratio or impulse-wide.

3.4. Procedures of the proposed approach

The proposed method is a good candidate for extracting the desired source signal from the single-channel measured signal with source noises and measured noise. Its procedures can be described as follows:

Step 1: Decompose the gearbox single-channel measured signal xt according to Eq. (6), and obtain n IMF components.

Step 2: Compute the kurtosis of each IMF and correlation coefficient between each IMF and the original signal xt, select the IMF components with greater kurtosis and correlation coefficient to compose a new observation vector with the original signal xt, then take the new vector as the cICA algorithm input, given the new vector is xt=x1t,x2t,,xmtT, mn.

Step 3: Construct the reference signal rt with the desired fault signature, then extract the fault signal y*t with cICA method.

Step 4: Analyze the extracted fault signal y*t with Hilbert envelope spectrum and obtain the desired fault feature.

4. Simulation analysis

The aim of the simulation is to extract the desired low-frequency weak fault signal from the mixed data set. According to Eq. (16), we generated three source signals, s1, s2 and s3, whose time domain waveforms are shown in Fig. 1:

(16)
s 1 t = X 1 1 + m 11 A c o s 2 π f C t + m 12 A c o s 2 π 2 f C t c o s 2 π f p m t + θ 1 , s 2 t = X 2 1 + m 21 A c o s 2 π f r t + m 22 A c o s 2 π 2 f r t c o s 2 π 2 f m t + θ 2 , s 3 t = X 31 c o s 2 π f 1 t + X 3 2 c o s 2 π 2 f 1 t ,

where signal s2 is desired to be extracted, but its energy is weak. The parameter values of three simulated source signals, s1, s2 and s3 in Eq. (16) are listed in Table 1.

Table 1. Parameter values of the simulated signal

f p m
f C
f m
f r
X 1
X 2
θ 1 , θ2
m 11 A m 21 A
m 12 A , m22A
f 1
X 31
X 32
530 Hz
5.3 Hz
46.5 Hz
1.5 Hz
6
2
0
1
0.5
25 Hz
2
1

The source noise es1, es2 and es3 are respectively added to the three source signals s1, s2 and s3 with SNR of –5dB. Three noisy signals are randomly mixed by a mixing vector A and get a single-channel mixed signal. Then the mixed signal is added a Gaussian white noise emt with the amplitude standard deviation of 2. Finally, we obtain a single-channel simulated signal xt, whose time-domain waveform, FFT spectrum and envelope spectrum are shown in Fig. 2. Among the three source signals, signal s2t without source noise es2t is expected to be extracted from the mixed signal xt by using the proposed method.

From Fig. 2, the low-frequency modulation frequency fr (1.5 Hz) is invisible except for the frequency components 2fm (93 Hz), fpm (530 Hz) and the modulated frequency fc (5.3 Hz).

Fig. 1. Time domain waveforms of three simulated source signals without noise

 Time domain waveforms of three simulated source signals without noise

Fig. 2. Mixed signal xtand its spectrum and envelope spectrum with SNR of –5 dB

 Mixed signal xtand its spectrum and envelope spectrum with SNR of –5 dB

a) Mixed signal xt

 Mixed signal xtand its spectrum and envelope spectrum with SNR of –5 dB

b) FFT spectrum of mixed signal xt

 Mixed signal xtand its spectrum and envelope spectrum with SNR of –5 dB

c) Envelope spectrum of mixed signal xt

Fig. 3. EEMD decomposition results of the mixed signal xt with SNR of –5 dB

 EEMD decomposition results of the mixed signal xt with SNR of –5 dB

a)

 EEMD decomposition results of the mixed signal xt with SNR of –5 dB

b)

Fig. 3 depicts the decomposition results with EEMD method for the mixed signal xt. The kurtosis of each IMF and the correlation coefficients between each IMF and the signal xt are listed in Table 2. Among the IMFs, although the correlation coefficient value of c1 is very big, it is a high frequency noise and not to be considered. So, based on the criterions of kurtosis and correlation coefficient, we select the IMFs c2, c3, c4 and c5 (K> 3.0 and ρ> 0.2) combined with the original signal xt to construct a new observation vector. We generate a suitable reference signal rt (shown in Fig. 4(a)) with frequency fm (46.5 Hz) of signal s2, and then use the cICA method to successfully extract a desired source signal y*t (shown in Fig. 4(b)) as the closeness of the simulated signal s2t. The SIR value of the extracted signal y*t is 3.16 dB.

Table 2. Kurtosis and correlation coefficients of IMFs by EEMD method with SNR of –5 dB

IMFs
c 1
c 2
c 3
c 4
c 5
c 6
c 7
c 8
c 9
r 9
K
2.16
3.16
3.00
4.22
3.58
2.51
2.75
2.21
3.19
3.74
ρ
0.685
0.552
0.296
0.298
0.216
0.192
0.055
0.031
0.014
0.008

Fig. 4. Suitable reference signal rt and its extracted desired signal y*t using EEMD-based cICA

 Suitable reference signal rt and its extracted desired signal y*t using EEMD-based cICA

a) Constructed suitable reference signal rt

 Suitable reference signal rt and its extracted desired signal y*t using EEMD-based cICA

b) Extracted signal y*t with EEMD-based cICA method

The FFT spectrum and envelope spectrum of the extracted signal y*t are shown in Fig. 5. Apparently, the modulated sidebands of low-frequency fr (1.5 Hz) around the center frequency 2fm(93 Hz) is very evident. Of course, the original source signal s2 is not completely recovered for the strong source noise, but the low-frequency weak feature has been extracted from the mixed signal xt with other strong signals and noise influence.

Fig. 5. FFT spectrum and envelope spectrum of the extracted signal y*t

 FFT spectrum and envelope spectrum of the extracted signal y*t

a) FFT spectrum of the extracted signal y*t

 FFT spectrum and envelope spectrum of the extracted signal y*t

b) Envelope spectrum of the extracted signal y*t

Fig. 6 shows the decomposition results and its FFT spectra with EMD-based cICA method for the mixed signal xt. The SIR value of the extracted signal yt is 1.58 dB. Obviously, EMD-based cICA method can also expresses the feature frequency fr (1.5 Hz) of signal xt, but its effect is a bit worse than the EEMD-based cICA method. The simulated results show that the proposed method can effectively extract the low-frequency weak gear fault signals from the single-channel observation signal.

Fig. 6. Extracted signal y*t using EMD-based cICA and its FFT spectrum and envelope spectrum

 Extracted signal y*t using EMD-based cICA and its FFT spectrum and envelope spectrum

a) Extracted signal yt with EMD-based cICA method

 Extracted signal y*t using EMD-based cICA and its FFT spectrum and envelope spectrum

b) FFT spectrum of the extracted signal yt

 Extracted signal y*t using EMD-based cICA and its FFT spectrum and envelope spectrum

c) Envelope spectrum of the extracted signal yt

5. Experimental signals analysis

Next, we use the real-world signal from a multi-stage gearbox to verify the effectiveness of our approach, the single-channel vibration signals with a missing tooth and a chipped tooth localized on the gear Z3 (= 36) of the two-stage fixed-shaft gearbox in this experiment are studied, respectively. The schematic diagram of gearbox test rig is shown in Fig. 7.

Fig. 7. Schematic diagram of gearbox test rig

 Schematic diagram of gearbox test rig

Table 3. Characteristic frequencies of gearbox

Items
Single-stage planetary gearbox
Two-stage fixed-shaft gearbox
Tooth number Z
Z S = 28
Z P = 36
Z R = 100
Z 1 = 29
Z 2 = 100
Z 3 = 36
Z 4 = 90
Shaft rotating frequency f / Hz
f S = f r = 24
f C = 5.25
f r 1 = 5.25
f r 2 = 1.52
f r 3 = 0.6
Meshing frequency f/ Hz
f p m = 525.0
f m 1 = 152.3
f m 2 = 54.8

Experimental fault gear photos are shown in Fig. 8. The rotating frequency fr of motor is 24.0 Hz, sampling frequency is 5120 Hz and data length is 15 kB samples. Characteristic frequencies are listed in Table 3, where fr. fS and fC denote the rotating frequency of motor, sun gear, planet carrier, respectively, fm and fpm denote the meshing frequency of fixed-shaft gearbox and planetary gearbox, respectively. The gear fault characteristic frequency is fr2 (1.52 Hz), and its corresponding meshing frequency is fm1 (152.3 Hz).

Fig. 8. Photos of faulty gear Z3: a) missing a tooth; b) a chipped tooth

 Photos of faulty gear Z3: a) missing a tooth; b) a chipped tooth

a)

 Photos of faulty gear Z3: a) missing a tooth; b) a chipped tooth

b)

5.1. A missing tooth signal analysis

Fig. 9 illustrates the FFT spectrum and envelope spectrum of the gear vibration signal x1t with a missing tooth. The main frequency components are the meshing frequency fpm (525 Hz) of planetary gearbox and its harmonics, the modulated frequency is the planet carrier rotating frequency fc (5.25 Hz), which does not mean that the planetary gearbox has any fault according to the reference [30]. However, it is difficult to distinguish any obvious fault feature frequency fr2 (1.52 Hz) because the fault feature with a missing tooth is not apparent.

Fig. 9. Single-channel signal x1t with a missing tooth and its FFT spectrum & envelope spectrum

 Single-channel signal x1t with a missing tooth and its FFT spectrum & envelope spectrum

a) Single-channel observation signal x1t with a missing toth

 Single-channel signal x1t with a missing tooth and its FFT spectrum & envelope spectrum

b) FFT spectrum of x1t

 Single-channel signal x1t with a missing tooth and its FFT spectrum & envelope spectrum

c) Envelope spectrum of x1t

Table 4. Kurtosis and correlation coefficients of IMFs by EEMD with a missing tooth

IMFs
c 1
c 2
c 3
c 4
c 5
c 6
c 7
c 8
c 9
c 10
K
3.29
4.15
3.79
10.09
3.32
2.95
1.91
2.96
2.40
2.64
ρ
0.693
0.795
0.530
0.155
0.139
0.078
0.048
0.017
0.007
0.0005

Fig. 10 depicts the decomposition results of the single-channel observation signal x1t in Fig. 9(a) by using EEMD method. The kurtosis of each IMF (c1-c10) and the correlation coefficient between each IMF (c1-c10) and the fault signal x1t with a missing tooth are listed in Table 4. Based on the criterions of kurtosis and correlation coefficient, we select the IMFs c1-c5 (K> 3.2 and ρ> 0.1) combined with the original signal x1t to construct a new observation vector. Through generating a proper reference signal rt (shown in Fig. 11(a)) with the meshing frequency of fm2 (54.8 Hz), we successfully extract the desired fault signal y1*t (shown in Fig. 11(b)) with cICA method. Obviously, the periodical impacts at T= 0.67 s (1/fr2= 1/1.52) in time domain are very evident. The corresponding FFT spectrum and envelope spectrum of the extracted signal y1*t are shown in Fig. 12(a) and (b), respectively. As shown in Fig. 12, it can be clearly distinguished that there are plentifully modulated sidebands around the right side of frequency 2fm2 (109.6 Hz). The obvious fault feature frequency is fr2 (1.52Hz), which is corresponding to the shaft 2 rotating frequency fr2of the fault gear Z3 (= 36) with a missing tooth on the fixed-shaft gearbox.

Fig. 10. EEMD decomposition results of the original signal x1t with a missing tooth

 EEMD decomposition results of the original signal x1t with a missing tooth

a)

 EEMD decomposition results of the original signal x1t with a missing tooth

b)

Fig. 11. Proper reference signal rt and its extracted desired fault signal y1*t using EEMD-based cICA method with a missing tooth

 Proper reference signal rt and its extracted desired fault signal y1*t  using EEMD-based cICA method with a missing tooth

a) Proper reference signal rt

 Proper reference signal rt and its extracted desired fault signal y1*t  using EEMD-based cICA method with a missing tooth

b) Extracted fault signal y1*t

Fig. 12. FFT spectrum and envelope spectrum of the extracted desired fault signal y1*

 FFT spectrum and envelope spectrum of the extracted desired fault signal y1*

FFT spectrum y1*t

 FFT spectrum and envelope spectrum of the extracted desired fault signal y1*

b) Envelope spectrum y1*t

To compare the effect, the extracted result of EMD-based cICA method is shown in Fig. 13, the result is not as good as the EEMD-based cICA method.

Fig. 13. Extracted results using EMD-based cICA method with a missing tooth

 Extracted results using EMD-based cICA method with a missing tooth

a) Extracted signal y1t

 Extracted results using EMD-based cICA method with a missing tooth

b) FFT spectrum of y1t

 Extracted results using EMD-based cICA method with a missing tooth

c) Envelope spectrum of y1t

5.2. A chipped tooth signal analysis

Fig. 14 demonstrates the FFT spectrum and envelope spectrum of the gear fault vibration signal x2t with a chipped tooth. The main frequency components are also the meshing frequency fpm (525 Hz) of planetary gearbox and its high order harmonics, and the fault modulated frequency is still the planet carrier rotating frequency fc (5.25 Hz), which is uninterested for us. However, it is much difficult to identify the fault feature frequency fr2 (1.52 Hz) because the fault signal with a chipped tooth is much fainter.

Fig. 14. Single-channel signal x2t with a chipped tooth and its FFT spectrum and envelope spectrum

 Single-channel signal x2t with a chipped tooth and its FFT spectrum and envelope spectrum

a) Single-channel observation signal x2t with a chipped tooth

 Single-channel signal x2t with a chipped tooth and its FFT spectrum and envelope spectrum

b) FFT spectrum x2t

 Single-channel signal x2t with a chipped tooth and its FFT spectrum and envelope spectrum

c) Envelope spectrum x2t

Fig. 15 shows the decomposition results of the single-channel observation signal x2t in Fig. 14(a) using EEMD method. The kurtosis of each IMF (c1-c10) and the correlation coefficient between each IMF (c1-c10) and the fault signal x1t with a chipped tooth are listed in Table 5. Based on the criterions of kurtosis and correlation coefficient, we select the IMF components c1-c5 (K> 3.2 and ρ> 0.1) combined with the original signal x2t to construct a new observation vector. The unchanged reference signal rt is shown in Fig. 11(a), then we utilize cICA method to successfully extract the desired fault signal y2*t, whose time domain waveform, FFT spectrum and envelop spectrum are shown in Fig. 16. From Fig. 16(a), the periodical impacts at T= 0.67 s (1/fr2= 1/1.52) in time domain is evident, but it is not as clear as that shown as in Fig. 11 (b). In Fig. 16 (b), we can clearly distinguish that there are some modulated sidebands around the right side of frequency 2fm2 (109.6 Hz). The fault feature frequency is 1.52 Hz from Fig. 16(c), which is also corresponding to the shaft 2 rotating frequency fr2 of the faulty gear Z3 (= 36) with a chipped tooth on the fixed-shaft gearbox.

Similarly, if we use the EMD-based cICA method to analysis the signal x2t, the effective low-frequency fault feature fr2 (1.52 Hz) will not be extracted, as shown in Fig. 17.

Table 5. Kurtosis and correlation coefficients of IMFs by EEMD with a localized chipped tooth

IMFs
c 1
c 2
c 3
c 4
c 5
c 6
c 7
c 8
c 9
c 10
K
4.14
3.44
4.22
3.63
3.29
2.71
2.59
2.74
2.53
2.48
ρ
0.650
0.696
0.168
0.161
0.107
0.085
0.065
0.005
0.003
0.001

Fig. 15. EEMD decomposition results of the original signal x2t with a chipped tooth

EEMD decomposition results of the original signal x2t with a chipped tooth

a)

EEMD decomposition results of the original signal x2t with a chipped tooth

b)

The experimental results indicate that the proposed method is effective and available for low-frequency fault feature extraction, especially for the weak fault feature extraction of the gearbox single-channel observation signal.

Fig. 16. Extracted results using EEMD-based cICA method with a chipped tooth

 Extracted results using EEMD-based cICA method with a chipped tooth

a) Extracted signal y2*t

 Extracted results using EEMD-based cICA method with a chipped tooth

b) FFT spectrum of y2*y

 Extracted results using EEMD-based cICA method with a chipped tooth

c) Envelope spectrum y2*t

Fig. 17. Extracted results using EMD-based cICA method with a chipped tooth

 Extracted results using EMD-based cICA method with a chipped tooth

a) Extracted signal y2t

 Extracted results using EMD-based cICA method with a chipped tooth

b) FFT spectrum of y2t

 Extracted results using EMD-based cICA method with a chipped tooth

c) Envelope spectrum of y2t

6. Conclusions

Aiming at the shortcomings of traditional ICA method and trying to solve the key problem of the extremely underdetermined single-channel blind source separation and fault feature extraction with source noise and measured noise, we proposed an approach combining the advantages of EEMD and cICA. Through simulation and experiments of gear low-frequency fault feature extraction for the single-channel observation signal, the results verify the effectiveness of this proposed method, which is suitable for the gearbox fault diagnosis, especially for the low-frequency and weak fault diagnosis of gearbox. Further study is yet required to introduce the additional denoising processes to enhance this proposed method performance in the low SNR case. Notably, this proposed method is also suitable for other signals feature extraction that show periodicity characteristics, such as the bearing fault signal, the internal combustion engine fault signal.

Acknowledgements

This work was supported by the Project of China National Coal Association (Grant No. MTKJ2015-261), Doctoral Fund of Henan Polytechnic University (Grant No. B2017-28) and Foundation of innovative research team of Henan Polytechnic University (Grant No. T2017-3). The authors would like to thank the reviewers for many valuable comments and suggestions.

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