A NEW GEAR FAULT RECOGNITION METHOD USING MUWD SAMPLE ENTROPY AND GREY INCIDENCE

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1 A NEW GEAR FAULT RECOGNITION METHOD USING MUWD SAMPLE ENTROPY AND GREY INCIDENCE WENBIN ZHANG, 2 JIE MIN, 3 YASONG PU Assoc Prof, College of Engineering, Honghe Univ, Mengzi, Yunnan, 6600, China 2 Lecturer, College of Engineering, Honghe Univ, Mengzi, Yunnan, 6600, China 3 Lecturer, College of Engineering, Honghe Univ, Mengzi, Yunnan, 6600, China E-ail: georgezwb@sohuco, @qqco, @qqco ABSTRACT In this paper, we propose a new gear fault recognition ethod by using orphological undeciated wavelet decoposition (MUWD), saple entropy and grey incidence MUWD possesses both the characteristic of orphological filter in orphology and ulti-resolution in wavelet transfor Then we develop ulti-scale MUWD based on the characteristic of ipulse feature extraction in difference orphological filter At first, we use ulti-scale MUWD to process different gear fault signals in five levels, signal length is aintained invariable and inforation loss could be avoided in MUWD, siulation exaple tests the good effectiveness of its denoising capacity Second, we calculate the saple entropy of each level Different fault type corresponds with different saple entropy In the end, we serve saple entropy as the feature vectors and calculate the grey incidence of different gear vibration signals to identify the fault pattern and condition Practical exaple shows that the high efficiency of the proposed ethod It is suitable for condition onitoring and fault diagnosis of gear Keywords: Morphological Undeciated Wavelet Decoposition (MUWD), Grey Incidence, Saple Entropy (SE), Recognition, Fault Diagnosis, Gear INTRODUCTION Gear is the coon used part in echanical transforation, their carrying capacity and reliability being proinent for the overall achine perforance Therefore, the fault recognition of gear has been the subject of extensive research Enveloping analysis and wavelet packages decoposition are coon used fault feature extraction ethods for gear signal [] But the enveloping analysis needs to confir the center frequency and frequency band of band-pass filter; it will ipact the analyzing results [2] While the wavelet decoposition has finite length of basic function, energy will leak in the signal processing Due to the wavelet decoposition that is based on the linear decoposition, so the good effectiveness will not be obtained in processing gear fault data because of the non-linear and non-stationary behaviors Proposed by Goutisias and Heijans [3-4] in 2000, the orphological wavelet is a non-linear wavelet transforation based on atheatical orphology Morphological wavelet possesses both the characteristic of feature recognition and ulti-resolution in wavelet transfor; it also has good details keeping and noise resisting abilities Study of orphological wavelet in -D vibration signal is starting step by step [5] However, the traditional wavelet adopts a deciated wavelet decoposition ethod; inforation will be lost in decoposition procedures Literature [6] firstly proposed a orphological undeciated wavelet decoposition ethod based on undeciated wavelet schee Then a new orphological undeciated wavelet decoposition ethod is proposed and used in gear fault diagnosis [7-8] Gear fault signal is the typical non-stational and non-linear signal How to extract feature paraeter of different fault pattern is the key for gear fault diagnosis Saple entropy is a good tool to evaluate coplexity of non-linear tie series, copared with other existing non-linear dynaic ethods; it has any good characteristic, such as good residence of noise 86

2 interference, closer agreeent with theory for data sets with known probabilistic content Moreover, saple entropy displays the property of relative consistency in situations where approxiate entropy does not [9] These perforances are suitable for fault extraction in practice The arrangeent of the paper is as follows Introduction elaborates in section- Section-2 describes elaborately the concepts of orphological filter and coupled wavelet, section-3 gives the definition of saple entropy, section-4 introduces the definition of grey incidence, the proposed ethodology is explained in section 5, section 6 and section 7 give the siulation and practical application of the proposed ethod and finally, the conclusion is given in section 8 2 BASIC CONCEPTS OF MORPHOLOGICAL FILTER AND COUPLED WAVELET Morphological Filter A orphological filter is constructed by different orphological transfors Dilation and erosion are two basic orphological transfors While dilation is the transfor used to expand the target object and shrink the hole, erosion is the transfor used to shrink the target object and expand the hole Let f(x) and g(x) denote -D input signal and structure eleent, where F= {0,,, N-} and G= {0,,, M-} denote sets on which signal f and g are defined, N M Dilation and erosion of f and g are thus defined as follows (f Å g)(n) = ax = 0,,, M- {f(n - )+g()} () (f Q g)(n) = in = 0,,, M- {f(n + ) - g()} (2) (n=0,,,n-m) (n=0,,,n+m-2) Where Å denotes dilation transfor and Q denotes erosion transfor Usually, dilation and erosion are not utual inverse They can be cobined through cascade connection to for new transfors If dilation is next to erosion, such cascade transfor is an opening transfor The contrary is a closing transfor The transfors can be coputed using the following forulas respectively (f g)(n)=[fqgå g](n) (3) (f g)(n) = [f Å g Q g](n) (4) Where denotes opening transfor and denotes closing transfor The opening and closing results of the signal f by the elliptical structure eleent g are shown in figure [0] (a) opening and its result (b) closing and its result Fig Opening And Closing Results Of The Signal F By The Elliptical Structure Eleent G Fro figure (a), when g oves under f closely, the parts of f that do not contact with g will fall into the upper edge of g So the opening transfor can be used to reove the peaks in the signal Fro Figure (b), when g oves over f closely, the parts of f that do not contact with g will roll into the lower edge of g So the closing transfor can be used to fill the valleys in the signal Both transfors can be cobined to for a orphological filter because they have the capacity of low-pass filtering Principle of difference orphological filter In orphological operations, open and close operations have different processing perforances Open operation could restrain positive ipulse and keep negative ipulse, while close operation will have inverse functions So in the practical application, we should select relative orphological operation corresponding with processing ai But soeties it is difficult to get transcendental knowledge of practical positive and negative ipulses; the universal situation is that the positive and negative ipulses are contained in practical data at the sae tie [] Therefore, it is necessary to construct orphological filtering algorith with cobination of open and close operations The definition of difference orphological filter is (5) DIF(f)=f g - f g 862

3 Fro equation (5) we can see that the difference orphological filter ay be used to extract the ipulsive coponents of the signal The reason is shown in equation (6) (6) f g - f g=(f g - f)+(f - f g) Here, f g- f and f- f g operations are two fors of orphology Top-Hat transfor The forer operation is called black Top-Hat transforation, which is used to extract negative ipulse; while the latter is called white Top-Hat transforation, which is used to extract positive ipulse So the different orphological filter could extract the positive and negative ipulses at the sae tie Principle of coupled wavelet decoposition The synthesis and analysis operators ust satisfy the pyraid condition which plays an iportant role in constructing the operators of the decoposition schee Consider a faily V j of signal spaces where j is a finite or an infinite index set Here the signals consist of two failies of operators, a faily y - j of analysis operators apping V j into V j+ and a faily Y j of synthesis operators apping V j+ back into V j The analysis and synthesis operators y - j, Y j are said to satisfy the pyraid condition if y - jy j = id on V j+ where id is an identity operator The pyraid condition guarantees that no inforation is lost in two consecutive steps: synthesis and analysis It is the fundaental principle used to construct pyraid and wavelet operators The coupled wavelet is constructed according to the pyraid condition A coupled wavelet coprises of two analysis operators and one synthesis operator The analysis operators include a signal analysis operator and a detail analysis operator Figure 2 shows the coupled wavelet decoposition schee In Figure 2, V j and W j are two sets: V j is the signal space at level j and W j is the detail space at level j: y - j :V j V j+ is the signal analysis operator, w - j : V j W j+ is the detail analysis operator and Y j is the synthesis operator apping the inforation back to the lower level In order to guarantee that no inforation is lost and the decoposition is non-redundant, the analysis operators y - j, w - j and synthesis operator Y j of the coupled wavelet ust satisfy the pyraid condition below y - j( Y j( xy, )) = x, if xî Vj+, yî Wj+ (7) w - j( Y j( xy, )) = y, if xî Vj+, yî Wj+ (8) In order to yield a coplete signal representation, the aps ( y - j, w - j ): Vj Vj+ Wj+ and Y j : Vj+ Wj+ Vj are the inverses of each other This leads to the following perfect reconstruction condition Y j( y - j( x), w - j( x)) = x, if xî Vj (9) Fig2 Coupled Wavelet Decoposition Schee The coupled wavelet provides a standard structure for the ulti-resolution signal decoposition schees Based on this structure, the MUWD schee is presented by regenerating the analysis operators and synthesis operators [8] 3 DEFINITION OF SAMPLE ENTROPY Let [x(n)]=x(), x(2),, x(n) denotes N-eleents tie series representing gear vibration signal Then, the estiation algorith of saple entropy consists of the following steps [2-3]: (i) creating of vectors defined as: (i)=[x(i), x(i+),, x(i+-)] (0) (i=,2,,n-+) (ii) calculation of a distance between two vectors in the following way: d[ (i), (j)]= () ax k= 0,, x(i+k)-x(j+k) (iii) calculation of nuber of siilar segents in two vectors: n=# d[ (i), (j)] r,while i j; n+=# d[ + (i), + (j)] r,while i j where r is a tolerance paraeter 863

4 (iv) calculation of siilarity easures of these segents: B i (r)= n N + N + n A i (r)= + while i=,,n- (v) calculation of ean easures of the siilar signal segents: B N i= B i () r N A = N i= A i () r N (vi) calculation of saple entropy estiation A B () r () r Sap-En(,r)= In (2) 4 GREY INCIDENCE According to the grey theory, the relation degree evolves fro the relation coefficient The relation coefficient of the two series i and j, is represented by ζ ij (k), where k represents the sapling points [4-5] ij (k)= j (k)- i (k) k {,2,,N} (3) in= in in j k ij (k) ax= ax ax j k ij (k) ζ ij (k) is defined as: ξ (4) + ρ in ax ij = k {,2,,N} ij ( k) + ax ρ Where ρ is a constant with the range fro 0 to The value of ρ deterines the classification capacity and is usually recoended to be 05 The relation degree of the two series i and j is as following: (5) N N ij = [ ij ( k ) + ij ( k )] N 2 k= k= 2 ξ ξ ξ The relation degree represented by ζ ij shows the coparability of the i and j series It is often applied to grey cluster in practice [6] Obviously, the bigger ζ ij is, the greater the inference of i to j would be 5 GEAR FAULT RECOGNITION METHOD In order to extract gear fault feature, let φ represent orphological closing operator and γ represent opening operator Based on the MUWD schee proposed by literature [6], we construct a new MUWD ethod for fault feature extraction It is shown that x = - j y j( x ) = j f ( xj, ( j + ) g0) - + g( xj, ( j + ) g0) ( 6) y = - j wj( xj)=id -[ f ( xj, ( j + ) g0) - g( xj, ( j + + ) g0)] ( 7) Y j( y j( xj), wj( xj)) = y j( xj) + wj( xj) = xj (8) Equation (0) represents processing signal x j with structure eleent (j+)g 0 by selecting ulti-scale difference orphological filtering In which, (j+)g 0 denotes the dilation operation for j ties with structure eleent g 0 The approxiate signal fro equation (0) represents the ipulsive feature by difference orphological filter with different scale structure eleents [8] Using the grey relation degree to identify fault, the first step is to build standard fault feature atrix Assue the nuber of typical faults is k, and each typical fault contains one feature vector which is constructed by soe feature paraeters Then a standard fault feature atrix could be constructed by these feature vectors The second step is to assue there are soe groups of pending series, and a pending series feature atrix could be constructed The third step is to calculate the grey siilar relation degree between the pending series feature atrix and each typical fault by using equation (5) Set the standard fault feature atrix is [ ri ]: 864

5 ri = r rn = r rn () () () rn (2) (2) (2) ( 9) Here, r refers to the standard fault feature vector, n refers to the nuber of the standard fault feature vector, and k refers to the diension of each standard fault feature vector Let the jth pending fault feature vector is [ tj ]=[ tj (), tj (2),, tj (k)] (20) r rn x(t)=7x (t)+07x 2 (t)+i(t) (22) Here, x (t) is a typical series of exponentially decaying ipulses with the ipulse function of f(t)=e -25tsin(25πt)) and the ipulse frequency of 6Hz; x 2 (t) is the su of two haronic waves: ( k) x 2 (t)=cos(2π 20t)+cos(2π 40t); i(t) is the Gauss ( k) white noise with the standard deviation of Figure 3 shows the wavefor in tie doain and spectrus of the siulated signal and the ipulse coponents ( k) (A) Wavefor In Tie Doain Of Ipulse Signal Then the grey relation degree rank will be [ξ tjri ]=[ξ tjr,ξ tjr2,,ξ tjrn ] (2) According to the biggest principle of relation degree, set ξ tjr (=,2,,n) is the biggest one, the potential fault type could correspond with [4] The detailed steps of this algorith as follows: (i) The original signal is decoposed by orphological undeciated wavelet decoposition into different levels (ii) We calculate the saple entropy of each level with equation (2) and then we get a set of fault feature atrix (iii) We build the standard fault feature atrix fro the calculated data (iv) The grey relation between the sypto set and standard fault set is calculated as the identification evidence 6 SIMULATION To verify the effectiveness of the new MUWD schee on noise suppression and ipulsive feature extraction, the siulated signal is forulated as follows [8]: (B) Spectru Of The Ipulse Signal (C) Wavefor In Tie Doain Of Siulated Signal (D) Spectru Of Siulated Signal Fig3 Wavefors In Tie Doain And Spectrus Of Ipulse Signal And Siulated Signal Figure 3(d) is the partially enlarged spectru of the signal, fro which the haronic waves of 25 and 50 Hz are obvious, while the ipulse coponents can not be seen clearly because it ixes with the haronic waves and Gauss white noise In order to stand out the ipulsive feature, the haronic waves and noises ust be suppressed Insert the flat structure eleent g 0 ={0,0,0} into equation (6)-(8), the new MUWD 865

6 procedure was applied to the siulated signal using two levels of analysis operation Figure 4 shows the approxiate signal and its spectru in second level Fro it we can see the firstly three orders of the ipulse frequencies It is shown that the new MUWD ethod has good effectiveness in ipulsive feature extraction (A) Spectru Of Broken-Teeth Signal Fig4 Spectru Of Siulated Signal After New MUWD 7 PRACTICAL EAMPLE To verify good effectiveness in fault recognition, all vibration signals were collected fro the experiental testing of gearbox using the acceleroeter which was ounted on the outer surface of the bearing case of input shaft of the gearbox [7-9] The speed of the otor is 420 RPM and the saple frequency is 6384 Hz The four conditions were tested that were noral, slight-worn, ediu-worn, broken-teeth Now we get five sapled data sets of each condition First, we use MUWD ethod to process the original signal in five levels Figure 5(a) shows the spectru of the original broken-teeth signal Figure 5(b) shows the processed signal in the first level Table Saple Entropy Of Different Fault Pattern (B) Spectru Of Processed Signal Fig5 Spectru Coparison Of Broken-Teeth Signal By MUWD Method Coparing the above two figures, we can see that the high frequency noises are eliinated and the fault feature is obtained obviously It is very useful in the next procedure Next, we calculate the saple entropy of each level Table gives the ean calculated values of five data sets in four fault conditions Fro Table, we can see that different fault pattern has different saple entropy Gear condition SE SE 2 SE 3 SE 4 SE 5 Noral Slight-worn Mediu-worn Broken-teeth Table 2 gives the saple entropy of each data set in four fault conditions Then we set the values of Table as the standard fault set, we recognize different gear fault pattern by calculation the grey incidence between the fault saple and standard fault pattern Table 2 Saple Entropy Extracted By MUWD In Different Conditions Gear condition Saple SE SE 2 SE 3 SE 4 SE Noral Slight-worn

7 Mediu-worn Broken-teeth Table 3 gives the final recognition results We can see that each fault pattern has been identified by the proposed ethod Table 3 Grey Incidence Between Fault Saple And Standard Fault Pattern Saple Noral Slight-worn Mediu-worn Broken-teeth Recognition Result Noral Noral Noral Noral Noral Slight-worn Slight-worn Slight-worn Slight-worn Slight-worn Mediu-worn Mediu-worn Mediu-worn Mediu-worn Mediu-worn Broken-teeth Broken-teeth Broken-teeth Broken-teeth Broken-teeth 8 CONCLUSION In this paper, we propose a new fault recognition ethod by using orphological undeciated wavelet, saple entropy and grey incidence Its advantages are as follows: () The orphological undeciated wavelet is a non-linear wavelet transfor ethod based on orphology It possesses both the characteristic of orphological filter in orphology and ulti-resolution in wavelet transfor (2) Based on the characteristic of difference orphological filter, the proposed ethod has better effectiveness in fault feature extraction of gear and the fault feature is obtained obviously (3) Because of the siplicity of the MUWD algorith, the proposed ethod is suitable for the on-line onitoring and fault diagnosis of gear in rotating achines (4)Though the proposed ethod achieves good effectiveness in gear fault recognition, selection of the artificial intelligent recognition ethod is the core for future research ACKNOWLEDGMENT This research is financially supported by Innovation Experiental Plot Project Fund (No MS02) of Honghe University 867

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