Application of EEMD Sample Entropy and Grey Relation Degree in Gearbox Fault Identification

In this paper, a new gearbox fault identification method was proposed based on mathematical morphological filter, ensemble empirical mode decomposition (EEMD), sample entropy and grey relation degree. First, the sampled data was de-noised by mathematical morphological filter. Second, the de-noised signal was decomposed into a finite number of stationary intrinsic mode functions (IMFs) by EEMD method. Third, some IMFs containing the most dominant fault information were calculated by the sample entropy for four gearbox conditions. Finally, due to the grey relation degree has good classify capacity for small sample pattern identification; the grey relation degree between the symptom set and standard fault set was calculated as the identification evidence for fault diagnosis. The practical results show that this method is quite effective in gearbox fault diagnosis. It’s suitable for on-line monitoring and diagnosis of gearbox.


INTRODUCTION
Gear is the common used part in mechanical transformation, their carrying capacity and reliability being prominent for the overall machine performance.Therefore, the fault identification of gear has been the subject of extensive research.Enveloping analysis and wavelet packages decomposition are commonly used fault feature extraction methods for gear signal [1].But the enveloping analysis needs to confirm the center frequency and frequency band of band-pass filter; it will impact the analyzing results [2].While the wavelet decomposition has finite length of basic function, energy will leak in the signal processing.Because the wavelet decomposition is based on the linear decomposition, so the good effectiveness will not be obtained in processing gear fault data due to the non-linear and non-stationary behaviors.
Mathematical Morphology is a subject concerning with the shape of an object based on set theory and integral geometry [3].In recent years, more and more studies have been done on the morphological filter.It is a non-linear filter with the advantage of better performance on rejection of white noise and pulse noise [3].The operations of the filter are mainly plus, minus and logic.So the implementation of the filter by software or hardware is very easy.Its filtering idea is based on the geometrical structure of the filtered signals and realized through moving predefined structure element to match and adjust the singular parts of the signals [4].It has been used in signal de-noising and purification of rotor axis [5,6].
Gear fault signal is the typical nonstational and nonlinear signal.How to extract feature parameter of different *Address correspondence to this author at the College of Engineering, Honghe University, Jinhua Road, Mengzi, Postcard: 661100, China; Tel: 15025218982; E-mail: 190322507@qq.comfault pattern is the key for gear fault diagnosis.Sample entropy is a good tool to evaluate complexity of non-linear time series, compared with other existing non-linear dynamic methods; it has many good characteristic, such as good residence of noise interference, closer agreement with theory for data sets with known probabilistic content.Moreover, sample entropy displays the property of relative consistency in situations where approximate entropy does not [7].These performances are suitable for fault extraction in practice.
In order to extract fault feature of gearbox, in this paper, a novel approach proposed based on mathematical morphological filter, EEMD and grey relation degree.The proposed method could extract gear fault feature by EEMD and sample entropy.Then we identify different gearbox fault pattern by calculating the grey relation degree between the fault sample and standard fault pattern.

BASIC CONCEPTS OF MATHEMATICAL MORPHOLOGICAL FILTER
A mathematical morphological filter is constructed by different morphological transforms.First, several important morphological transforms are introduced.
Dilation and erosion are two basic morphological transforms.While dilation is the transform used to expand the targeted object and shrink the hole, erosion is the transform used to shrink the targeted object and expand the hole.Let f(x) and g(x) denote 1-D input signal and structure element, where F= {0, 1, …, N-1} and G= {0, 1, …, M-1} denote sets on which signal f and g are defined, N ≥ M. Dilation and erosion of f and g are thus defined as follows: Usually, dilation and erosion are not mutual inverse.They can be combined through cascade connection to form new transforms.If dilation is next to erosion, such cascade transform is an opening transform.The contrary is a closing transform.The transforms can be computed using the following formulate respectively The opening and closing results of the signal f by the elliptical structure element g are shown in Fig. (1) [8].In order to reject both the positive noise and negative noise together, with the same structure element, the openclosing filter and close-opening filter can be realized through cascade connection of the opening and closing transform in different order.Two filers are defined as follows: Due to the expansibility of the opening transform and the inverse expansibility of the closing transform, the problem of the statistics deviation exists in the open-closing filter and close-opening filter.The output of the open-closing filter is small, while that of the close-opening filter is large.Under most circumstances, the best processing performance can't be achieved by using a single filter.In order to lower the output deviation, two filters can be cascaded to form a new combined filter, whose output is defined as follows Actually, the structure element acts as a filtering window, in which the data are smoothed to have a similar morphological structure as the structure element.The effectiveness and accuracy of the morphological filter depend on not only the combination mode of different transforms, but also on the shape and width of the structure element.Usually the shape of the structure element should be similar with the signal.The common used structure element has simple geometrical shape, such as line, round, triangle and other polygon etc.In general, the more complex the structure element, the better effectiveness will be obtained to reject the noises, but it will cost much time.
For structure element with the ensured shape, it's necessary to select proper height and length, especially the length is more important to the effectiveness of signal processing.In vibration signal processing, the selection of the height is based on the experience.For the triangular structure element, selecting 1 to 5 percents of original signal's height is appropriate [9].The length is mainly determined by the period and sampling frequency of signal's main wave.Meanwhile, only if the width of the structure element is longer than that of the widest pulse in the series, can all the pulse interferences be removed.

BASIC CONCEPT OF EEMD
The concept of the EEMD is the following: the added white noise constitutes components of different scale and would be uniform to inhabit the whole time-frequency space.When a signal is added to the uniformly distributed white noise background, the different scale components of the signal are automatically projected onto proper scales of reference established by the white noise in the background.Because each of noise-added decompositions contains the signal and the added white noise, each individual trial is certain to get very noisy results.As the noise in each trial is different in separate trials, the noise can be almost completely removed by the ensemble mean of entire trials.The ensemble mean is treated as the true answer because only the signal is persevered finally as more and more trials are added in the ensemble.The crucial principle advanced here is based on the following observations [10,11]: (1) A collection of white noise cancels each other out in a time-frequency ensemble mean; therefore, only the signal can continue to exist and remain in the final noise-added ensemble mean.
(2) White noise of finite amplitude necessarily compels the ensemble to discover all possible solutions.The white noise makes the different scale signals reside in the corresponding IMFs, controlled by dyadic filter banks, and renders the results of ensemble mean more meaningful. ( The decomposition result with truly physical meaning of the EMD is not the one without noise; it is assigned to be the ensemble mean of a large number of trials comprising the noise-added signal. Based on the aforementioned observations, the EEMD algorithm can be stated as follows [10,12]: (1) Initialize the ensemble number M and the amplitude of the added white noise, let M=1.
(2) Execute the mth trial for the signal added white noise.
(a) Add the white noise series with the given amplitude to the investigated signal, i.e.
x m (t)=x(t)+n m (t) (8) where n m (t) represents the mth added white noise, and x m (t) indicates the noise-added signal of the mth trial.
Where c i,m indicates the ith IMF of the mth trial; l is the number of IMFs and M means the number of the ensemble.
(c) If m<M, then let m=m+1 and repeat the step (a) and (b) again and again until m=M, but with different white noise each time.
(3) Compute the ensemble mean ci of the M trials for each IMF, and we obtain (4) Report the mean ci (i=1,2,…, l) of each of l IMFs as the final ith IMF.

DEFINITION OF SAMPLE ENTROPY
Let [x(n)]=x(1), x(2), …, x(N) denotes N-elements time series representing rotor vibration signal.Then, the estimation algorithm of sample entropy is consisted of the following steps [13]: Creating of m vectors defined as: (ii) Calculation of distance between two vectors in the following way: (iii) Calculation of number of similar segments in two vectors: where, r is a tolerance parameter.
(iv) Calculation of similarity measures of these segments: Calculation of mean measures of the similar signal segments:

GREY RELATION DEGREE
According to the grey theory, the relation degree evolves from the relation coefficient.The relation coefficient of the two series X i and X j , is represented by ζ ij (k), where k represents the sampling points [14,15].
ζ ij (k) is defined as: where, ρ is a constant with the range from 0 to 1.The value of ρ determines the classification capacity and is usually recommended to be 0.5.The relation degree of the two series X i and X j is as following: The relation degree represented by ζ ij shows the comparability of the X i and X j series.It is often applied to grey cluster in practice [16].Obviously, the bigger ζ ij is, the greater the inference of X i to X j would be.

ALGORITHM OF FAULT IDENTIFICATION OF GEARBOX
The detailed algorithm of fault diagnosis can be seen as below: Step 1: The sample data is obtained from the experimental testing of gearbox.The four conditions were tested that were normal, slight-worn, medium-worn, brokenteeth.
Step 2: Using mathematical morphological filter to denoise the white noise and other interferences in the original signal.
Step 3: Using EEMD to process the de-noised signals.Select some IMFs which contains the most dominant fault information as research objects.
Step 5: Build the feature vector by equation ( 16): Here, i refer to the number of selected IMFs.
Step 6: The grey relation degree between the symptom set and standard fault set is calculated as the identification evidence.

PRACTICAL APPLICATION
To verify good effectiveness in gearbox fault identification, all vibration signals were collected from the experimental testing of gearbox using the accelerometer which was mounted on the outer surface of the bearing case of input shaft of the gearbox.The speed of the motor is 1420 RPM and the sample frequency is 16384 Hz.The four conditions were tested that were normal, slight-worn, medium-worn, broken-teeth.Now we get five sampled data sets of each condition.First, we use mathematical morphological filter to process the original signal.      1 as the standard fault set, we recognize different gear fault pattern by calculating the grey relation degree between the fault sample and standard fault pattern.Table 3 gives the final identification results.We can see that each fault pattern has been identified by the proposed method.

CONCLUSIONS
In this paper, a novel gearbox fault identification way is proposed by using mathematical morphological filter, EEMD, sample entropy and grey relation degree.First, mathematical morphological filter is used to eliminate the noise interferences in original gearbox vibration signal.
Second, EEMD is used to decompose the processed signal adaptively into a finite number of stationary intrinsic mode functions.Third, the sample entropy of the first five IMFs containing the most dominant fault information is calculated and served as the fault feature.Finally, the grey relation degree between the fault sample and standard fault pattern is obtained as the evidence of fault identification.Practical examples verify that the proposed method is very useful in gearbox fault type diagnosis.It has great application value in fault diagnosis.

CONFLICT OF INTEREST
The authors confirm that this article content has no conflict of interest.

Fig. ( 1 ).
Fig. (1).Opening and closing results of the signal f by the elliptical structure element g.From Fig. (1a), when g moves under f closely, the parts of f that do not contact with g will fall into the upper edge of g.So the opening transform can be used to remove the peaks in the signal.From Fig. (1b), when g moves over f closely, the parts of f that do not contact with g will roll into the lower edge of g.So the closing transform can be used to fill the valleys in the signal.Both transforms can be combined to form a morphological filter because they have the capacity of low-pass filtering.
Fig. (2a) shows the waveform of the original broken-teeth signal in time and frequency domain.Fig. (2b) shows the processed signal.
Fig. (2).Waveform of original signal and de-noised results comparison.Comparing the above two figures, we can see that the high frequency noises are eliminated and the fault feature is obtained.It is very useful in the next procedure.Next, we use EEMD to decompose the same signal.Fig. (3) gives the processed results.From the above figure, we can see that denoised vibration signals are decomposed into a finite number of stationary intrinsic mode functions (IMFs); and IMF 1 to IMF 5 contain obvious shocking components.So we calculate the sample entropy of these IMFs.Table 1 gives the mean calculated values of ten data sets in four fault conditions.From Table 1, we can see that different fault pattern has different sample entropy.

Table 1
gives the mean calculated values of ten data sets in four fault conditions.From

Table 1 ,
we can see that different fault pattern has different sample entropy.

Table 2
gives five sample data of each data set selected randomly.Then we set the values of Table