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0Sensors 2010,10,4602-4621; doi:10 3390/SL00504602
O PEN ACCESS
sensors
ISSN 1424-8220 www m dpicom / jumaVsengois
Communication
IntelligentGeaibox DiagnosisM ethods Based on SVM , W avelet Lifting and RBR
Lixin Gao 1, Zhiqiang Ren 1, W enVang Tang 1, Huaqing W ang2'*and Peng Chen 3
1 Key Laboratory of Advanced M anufacturing Technology, Beijing University of Technology, Chao Yang District, Beijing,100124,Chiina; E-M ails: Vead0003@ 163 com (L X G .);
renzhiqiang1221@ 163com (Z Q R ); twIXy@ emailsbjuteducn W LT.)
2 ■ ■ i i i 11 i i ■
Schoolof M echanical& Electrical Engineering, Beijing University of Chem xalTechnology,
Chao Yang District, Beiiing,100029,Chiina 3 Graduate Schoolof Bio2esounces, M ie University, 1577 Kurmamachiya-cho, Tsu, M ie, 514-8507, Japan; E-M ail: chen@ biomi^-^ac:.jD
* Autthortowhom correspondence should be addressed; E-M ail: wanghq_buct@ hotmailcom ; Tel: +86-10-6444-3037; Fax: +86-10-6444-6043.
Received: 22 March 2010; in revised form : 19 April 2010 /Accepted: 27 April 2010 / Published: 4 M ay 2010
Abstract: G iventhe problem s inin^telligen^t gearboxdiagnosis m ethods, it is difficult to obtain the desired inform atton and a large enough sam ple size to study; therefore, w e propose the application of various m ethods for gearbox fault diagnosis, including wavelet liftng, a supportvectormachine SVM ) andrule-basedreasoning (RBR). m a complex field environm ant, it is less likely for m achines to have the sam e fault; m oreover, the fault features can also vaty. Therefore, a SVM could be used for the initial diagnosis. First, gearbox vibration signals weie processed with wavelet.packet. decomposition, and the signal energy coefficients of each frequency band were extracted and used as input. feature vectors in SV M for norm al and faulty pattern recognition. S econd, precision analysis using wavelet lifting could successfully filter out- the noisy signals while m aitaining the im pulse characteristics of the fault;; thus effectively extracting the fault frequency of the m achine. Lastly, the know ledge base w as builtbasedonthe feldrules sum m arzedbyexperts to identify the detailed fault type. Results have shown that SVM is a powerful tool to accomplish gearboxfaultpattemrecognitionwhen the sample size is sm all, whereas the wavelet lifting schem e can effectively extract. fault features, and rule-based reasoning can
be used to identify the detaied fault type. Therefore, a method that combines SVM , wavelet lifting and rule-based reasoning ensures effective gearbox faultdiagnosis.
Keywords: gearbox; support vector machines SVM ); wavelet lifting; rule-based reasoning (RBR); intelligent diagnosis
l.Introductbn
W iththe continuous developm ent of m odemindustrial large-scale m anufacturing andprogr:ess in the sciences and technology, machinery, as the major production tool, tends to be large, complex, speedy, continuous andautom atc to m axim aDyim prove productinefficiency andproduct quality. M achine production efficiency is increasing, and their mechanical structures are becom ing more com plicated. O nce a m achine breaks down, the whole production process m ust stop, which can lead to enormous econom ic losses and serious personnel injuries. Therefore, reliablle and safe equipment operation is required. It has been proved that constantly monitoring equipment conditions and effectively im plem enting fault diagnosis techniques are the m ajor preventive m easures that guarantee safe equipm entoperation by detecting faults atan early stage to avoid m ajorand fatal accidents.
An intelligent m achine ffaultdiagnosis system has been developedrapidlyinthe past decades by successfully applying new theories. M eanwhile, the large scale and com plexity of m odern m achines, together w iththe urgent needs of real-tim e andautom atic m achine fault diagnosis, have driventhe transform ation of fault (diagnosis technology from artificial diagnosis to intelligent-diagnosis.
A m ong all kinds of intelligent diagnosis m etthods, pattern recognition based on an Artificial Neural Network (ANN) has been widely used because of itspowerin self- organizing, unsupervisad-leaming, and nonlinear pattern classification [1]. However, in practice, it is difficult to obtain the large quantity of typical fault sampies that is required by an ANN . Because m achinery m alfunctions, especially large-scale m achinery and equipm ent m alfunctions, can lead to huge econom ic losses, few fault samples are available. Thus, these fault diagnosis methods, although excellent in theory, do not perform well in practice [2]. The newly developed supportvectormachine (SVM ) opens a new path to resolve problem s in fault diagnosis. The SVM was originally proposed by Vapnik in 1979. It is a new tool that supports m achine learning with an optim al approach and has show n unique advantages and a prom ising future in resolving sm all sam pie size issues in pattern recognition. SVM s specifically target the issue of lim id sam ples andaim to obtainthe optim al solutionw iththe available inform ation, lather than the optim al value w ith a sam ple size close to infinity; its algorithm is converted to a quadratic optim izaton problem and, theoretically, produces a globally optim al solution, which solves the inevitable local extrema problem in neural network methods. The algorithm transforms the problem to a high-dim ensional eigenspace through a kemel-basad nonlinear transform ation., and a linear discrim inant function is subsequently constructed in high-dim ensional space to achieve the nonlinear discrim inant function in the original space. Thus, the dim ension problem is solved cleverly and the com plexity of the algorithm is independent of the sam pie dim ension [3]. Although the support vector data are significantly low er than the num ber of training sam pies, there are still som e problem s.
For example, the support vector data grow linearly with the number of training samples, whichmay lead. to excessive fitting and is tim e-consum ing in carnation; probability prediction can not be obtained with a SV M ; users of SV M m ust give an error param eter, which significantly influences the results. Unfortunately, the value of the given param eter is highly subjective, and all its possible values have to be guessed in orderto find the best result. M oreover, the kernel function of SVM must fulfill Mercer'scondition [4].
Itis wellknownthat the bottleneck of iaultdiagnosis is a lackof iaultsampies, which provides SVM a bright application future in m achine fault diagnosis. Jack has used SV M to detect the rolling bearing condition [5]; he also optim ized the SVM param eter with a genetic algorithm and achieved a good generalization [6]. Thukaram etal. compared the differences between neural networks and SVM s in recognizing faults, and dem onstrated the advantages of SVM in situations with am all sam pie size. N onetheless, m ost studies are still lim ited in laboratory tests; there are not m any applications of SV M in intelligent fault diagnosis system s in practice. M ore research and field tests are required for application of SVM in practice. W e investigate furtherin this field.
The wavelet transform is a breakthroughinsignal processing technologyin the past two decades [7]. Currently, the w avelet lifting analysis algorithm has been successfullyapplied inm any fields, even though it was only recently proposed. Calderbank, Daubechies and Sweldens et al. have applied wavelet lifting analysis to the im age compressing field and have achieved better compression com pared to first generation w avelet analysis [8,9] .H ow let. and N guyen have applied the w avelet lifting transform to audio signal analysis to reduce the signal Shannon entropy by approximatly 6% [10]. M IT investigators Sudarshan et al. combined the wavelet lifting transform with finite element analysis and proposed a novel multiresolution finite elementmethod. In fault diagnosis, application of the wavelet. lifting transform has just begun. Based on Claypoole's self-adaptive wavelet transformation, Samuel and Pines at the University of M aryland developed a new m ethod using the wavelet lifting com bined with m atching pursuit gear fault features, which has led to satisfactory results in helicoptertransm issbn fault diagnostics [11]. Zhengjia He and Chendong Duan etal in Xi'an Jiaotong University have also done considerable research in this field. They have deduced several construction m ethods of w avelet lifting and obtained excellent analysis results in signal processing, tim e-frequency analysis and feature extraction when com bining wavelet lifting with otherm ethods [8,12].
Rule-based reasoning (RBR) is a traditional intelligent diagnosis method. Experience and knowledge willbe represented inthe form of rules w hich willbe savedinknow ledge base, andthe reasoning m echanism s w illbe used to get the diagnosis conclusions w iththe rules;. Considering the engine wear process, Peilin Zhang, Bing LiandShubao Liang etal. have established a fuzzy rule based on typical wear faults for certain engines. They have introduced a symmetric fuzzy cross-entropy m ethod for fault reasoning and established a m odel of engine fault diagnosis based on a com bined m ethod of sym m etric fuzzy cross-entropy and rule-based reasoning [13]. In orderto perform a flexible, rapid and precise case adaptation in a case-based reasoning design system , Xin Song, W ei Guo and Zhiyong W ang have proposed a case adaptation mechanism that is based on regression analysis and rule-based reasoning [14].
This study presents a m ethodthat com bines w avelet lifting, an SVM and rule-basedreasoning to diagnose gearbox faults. Gearbox vibration signals sre initially processed by wavelet packet
decom position. Then, the energy coefficients of each frequency band are calculated and used as input vectors to the SVM to recognize norm al and faulty gearbox patterns. Precise analysis from the wavelet lifting schem e w as then utilized to obtainthe m achine fault feature frequency. Finally, basedonthe fault feature frequency, the existing diagnostic know ledge and rules were used for logical reasoning to establish a know ledge base to identify fault types. The diagnosis schem e based on an SVM , wavelet lifting and rute-based reasoning m ethods is shown in Figure 1.
Figure 1. The principle of intelligent fault diagnosis based on SVM , wavelet lifting and RBR.
2.Applicattion of SVM in M achine Fault Diagnosis 21.P rinciple of SVM
A support vector machine is based on m inim izing structural risks. Its algorithm was initially designedfor two-class classification. Inthe feldof machine faults, an SVM can simplydeterm ine whether there is a fault
The SVM m etthod is developed by determ ining the optim al separating hyperplane in for linear separability. The optim al separating hyperplane is not onlyaible to classfyall training samples, but also m axim izes the distance betw een the separating hyperplane and points in training sam pies that are closest to the separating plane.
The fault training sample set is txi,yi),i= 1,...,n,xe Rd ,ye {+1,-1} , where n is the number of
training sam ples, anddis the dim ensionof fault feature vectors. A ssum ing the follow ing equation is satiidied.:
yi((wx1) + b]> 1,i— 1,...,n 1)
the m inimum value of f (w) is:
<p(w) = — ||w |2 = - (ww) 2)
The solution of this optm al problem is the saddle point of the Lagrange function, and the optm al discrim inant function is obtained as:
f(x) = sgn ((w*x) + b*] - sgn^«Cy [(xix) + b*] (3)
Nonlinear problems can be converted to high-dimensional linear problems with a nonlinear transform ation. mhigh-dim ensional space, onlythe inner-product com putation is needed, which can be obtained by using functions in the original low dim ensions. A ccording to the relative principles of functional analysis, if one kernel function K XiX) fulfills Mercer's conditions, it corresponds to the inner product of one dimension. Such functions are called kernel functions, and the optimal dia^rim inant function in this situation is changed to:
f(x) = sgn(^a*yK (xi,x)+b*) (4)
The kernel functions com m only used are the RB F kernel, M LP kernel and M ultinom ial kernel.
|xi — x 2
However, the RBF described as K (xi,x)-exp{--} is used most widely. It will be described
in detail in reference [16]. B ecause the RBF kernel perform s better in recognition than the M LP kernel or M ultinom ial kernel, and the SVM algorithm has higher recognition accuracyand is m ore suitable thana BP neuralnetworkto deal with a smallsample data set 15,16], this studyemploys the RBF kernel in the LS-SV M toolbox to diagnose the fault.
2 2.F eature vector analysis of w aveletpackets energy
Using m ultiesolutioh analysis and the wavelet packet technique, signals can be decom posed into different frequency bands. Analyzing signals in these frequency bands is called frequency bandw idth analysis. Usually, based on the frequency range where signals of interest are located, users can decom pose signals to a certain scale and obtain inform ation from the corresponding frequency bands. A dditionally, signals in different frequency bands can be further subject to statistical analyses to obtain feature vectors that represent signal characteristics. Analyzing the signal energy in different frequency bands is called frequency band energy analysis. It is characterized by wie-frequency-range responses when processing nonstationary, transient signals w ith higher frequency resolution at low frequency and higher time resolution at high frequency. Compared to the FFT, it contains a great deal of non-stationary and nonlineardiagnostic inform ation.
The theoretical basis for w avelet frequency bandw idth analysis is Paisevalfe theorem . The tm e domain energy of f(x) is || f |f = J |f(x) 2 dx ; the wavelet transform of f(x) is
d(j,k) = W (2j,2jk) = 22 |R^(2~jx-k)f(x)dx ; and thesetwoarelinkedbyParseval'sequation:
J |f(x) 2 dx = ^|djt2 (5)
Thus, vibration signals are decom posed into independent frequency bands of different levels by using a conjugate quadrature filter.Not only are these decomposed signals in quadrature to each other inagream ent w iththe law of cbn£ervatbnof energy, but theyalso containa large quanttyof non-stationary and nonlinear diagnostic inform atton com pared to an FFT. Therefore, the signal energy in every frequency band can be used as a feature vector to represent the operation condition of the m achine and is useful form achine fault diagnosis.
The procedure for feature vectorextaction using wavelets is the follow ing:
Step 1: process vibration signals for waveletpacket decom position ;
Step 2: reconstruct each wavelet packet coefficient and extract signals in different frequency ranges;
Step 3: acquire Ej, the signal energy at different frequency bands and the total energy E.
e. ¿XjJ
and E= ^E. , whala k = 1, 2, 3...n is defined as the discrete points at
k-1 j=1
frequency band j;jis the num ber of frequency bands; and x. represents the am plitude of the discrete points.
Step 4: use the percentile ratio of the signal energy Ej ateach decom posed frequency band and the total energy E as elem ents to construct feature vectors..
3.W avelet Lifting Scheme
The wavelet lifting transform includes two stages: decom position and reconstruction. Decomposition consists of splitting, predicting and updating. As shown in Figure 2 (a), given data seriesS = {s(k),ke Z}, the decom position stage of the w avelet lifting transform based on the lifting
schem e is shown below :
(1) Split: the data series (s(k),ke Z}is split into an odd sample series so (k) and even sample series Se(k)
so(k) = s(2k+1) k£ Z (6)
ss(k)= s(2k) ke Z 7)
(2) Predict: suppose P (• )is the predictor; then use se (k)to predict so (k), and define the predictive deviation as the detail signal d (k) :
d(k)= sok)- PSe(k)] k e z (8)
Then, the detail signal series is D = |d(k),ke Z}
(3) Update: assum e U (•) is the updater, then se (k) is updated basedonthe detail signal d (k).ls result is defined as the approxim ation sdgnalc (k) :
c(k)= se (k)+U[dk)] ke Z (9)
Then, the approxim aton signal series is C = (c(k),ke Z}
R econ^struction of the wavelet lifting is the revere process of decom position., and is com posed of recovery prediction, recovery updating and m erging:
se (k) = ck)-U[d(k)] ke Z (10)
so (k) = dk)+ P s k)] ke Z
The recon^structbnsignal s is obtainedbym erging the oddandeven sample series, as shownin Figure 2 b).
Figure 2. Decomposition and reconstruction process of the wavelet lifting. () decom position of the wavelet lifting; b) reconstruction of the wavelet lifting.
4. Fundam ental Ideas of RBR Diagnostic Strategy 41.Fuzzy reasoning m echaniOT of typical faults in RBR
B ecau^^ of the differences in m achine working conditions, vibration signals can provide significant qualitative inform ation. However, there is no one-to-one correspondence between fault features and conclusions because of the com plexity of the m achines. Therefore, in the diagnosis ^stem used in this study, the fuzzy reasoning strategy was used to perfect rule-based diagnosis m ethods. The know ledge base is represented by the production rule. The fundam ental ideas of fuzzy reasoning are as follow s:
Suppose G is a ^t of a fuzzy proposition, fuzzy characteristics and a fuzzy relation. For sim plicity, the fuzzy proposition, fuzzy characteristics and fuzzy relation are together called the fuzzy assertion.. Then, a piece of factual inform ation can be presented by a binary group (P,jff).
P is the fuzzy assertion, P e G is the reliability of P, ft e 0,1] •
One fault symptom may correspond to multiple causes, while one fault cause may alto correspond to m ultple fault ^m ptom s. Therefore, the relationship betw een cause and sym ptom is com plicated.
Table 1. The rules of know ledge.
Fault cause Shaft imbalance Shaft m isaügnm ent Shaftbending Foundation deform atton GearTooth profile error Gearwear Gear tooth breakage Dam age of bearing
f 08 0.4 0.7 0.4 02 0 02 0 4
2fr 0 03 0 02 01 0 01 01
3f, 0 02 0 02 01 0 0 01
6fr 0 0 0 0 0 0 0 01
9fr 0 0 0 0 0 0 0 01
fa 0 0 0 0 02 0 4 02 0
2fn 0 0 0 0 01 02 01 0
3fn 0 0 0 0 01 02 01 0
Xq >3 0 0 0 0 01 01 02 02
Radial vibration direction 02 0 03 0 01 01 0 0
Axial vibration direction 0 01 0 02 0 0 0 0
f divided by peak value < 0.4 0 0 0 0 0 0 0 0
For proper diagnosis, the m em bership degree betw een fault causes and fault sym ptom s needs to be pree-determ ined. The value of this m em bership degree can be obtained based on expert experience or theoretical research. Based on years of axpelianca in our lab in field diagnosis, we summ arize the rules and establish the know ledge base.
The fuzzy rules of the know ledge base were used for fault cause reasoning to determ ine the reason for the faults. Then, according to the typical gearbox fault features, the rules of the know ledge base are constructed as shown in Table 1. In Table 1, f,fm and xq are rotation frequency, gear meshing frequency and kurtosis, respectively.
Forgearbox fault diagnosis, a fuzzy m atrix was established:
08 0 0 0 0 0 0 0 0 02 0 0
0.4 03 02 0 0 0 0 0 0 0 01 0
0.7 0 0 0 0 0 0 0 0 03 0 0
0.4 02 02 0 0 0 0 0 0 0 02 0
02 01 01 0 0 02 01 01 01 01 0 0
0 0 0 0 0 0 4 02 02 01 01 0 0
02 01 0 0 0 02 01 01 02 0 0 01
0.4 01 01 01 01 0 0 0 02 0 0 0
In the fuzzy m atrix for gearbox fault diagnosis, row s represent sets of fault causes, colum ns represent sets of faults sym ptom s, and the values in the m atrix represent the m em bershp degree betw een fault sym ptom s and causes.
W hen im plem enting fault diagnosis w ih the fuzzy reasoning approach, the fuzzy m atrix R is established first. G iven a fault sym ptom A, if the fault conclusion is B, then the fuzzy reasoning form ula can be show n as the follow ing :
The final diagnosis result includes the vectors with relatively large values upon conclusion of the diagnosis. If there are several relatively large values, the existing fault sym ptom should be considered for the final conclusion.
¥1 Ï2 •• rm
B = R * A = Î1 ** r2m Sale 2
_ r ** ^Tim _ _ am _
5.E xam ples of Diagnosis
In this study, SVM , wavelet lifting and diagnosis rules were used to analyze the vibration acceleration signal, according to a broken cog fault of the Z5 gear (tooth 31) in Shaft II of 22 gear-boxes of a highspeedw ire rolling m ill. The gearboxtransm issnon chainof a high-speed wire rolling m ill is shown in Figure 3.
Figure 3.The gearbox transm issbn chain chart.
Figure 4 (a) show s the broken cog fault signals of one rolling m ill. The fault happened in November 2006. The motor was running at 1,169 r/nin..The sampling ftequencywas 5,000 Hz, the sam pling num berwas 2,048, and the Z5/Z6 m eshing frequency was 1,197.676 Hz.
Figure 4. Comparison between waveform and spectrum with fault or fault-free, (a) time-domain waveform when the fault occurr^; (b) spectrum when the fault occurr^; (c) time domain waveform undernormal condition; (d) spectrum undernormal condition.
..£ <i.a u.J 0 500 1CCO 13DC
Frequency Hi
A s seen inFigure 4 (b), the spectrum onthe dayof the fault show s single ftequencyanddouble frequency, and the amplitude of the single frequency is very high, which indicates a severe gear problem at that tm e. Figure 4 (c) show s the tm e dom ain waveform under norm al condition and Figure 4 (d) show s spectrum under norm al condition.
51.SVM estimation
As m entoned in Section 2 2, the ' 'db10" waveletwas used to decom pose the signal into three layers and the energy of each of the eight decomposed frequency bands Ej and the total energy E were acquired. The feature vector can be established using the ratio betw een Ej and E . The horizontal axis represents energy of each of the eight frequencies afterthe signals have been decom posed. The vertical axis represents the ratio.
Figure 5.W aveletenergy. a) normalwaveletenergy profile; (b) faulty wavelet energy profile.
Percent Percent
12 3 4 5
Signal Energy Bands
1 2 3 4 5 6 7 £ Signal Energy Bands
As shown in Figure 5, wavelet energy was concentrated in frequency bands 1 and 2 under norm al conditions, and tended to m ove to higher frequency bands when a faultoccured.
Figure 6. SVM TestResult
+ clarr 1 —i- i - ■ + ■ + +
□ clan 2 ++ +
+ *> +
■ □ -
□ -"■a □
□ i i
The wavelet energyfrom the hourlydata obtainedin earlyJune is set to class 1, which indicates norm al conditions; the wavelet energy ftom data obtained when the trolling m ills experienced a fault is œtto class 2. Fifteen sets of data were used as SVM input ibr tuaining.The test data includeddata ftom June and Septem ber, and each had 15 œts of datai. B ecause gears have different crack patterns, data from Septem berwere identified as fault class 2 by the SVM and wets significantly different from data in the norm al condition. Figure 6 show s the testresults.
5 2.W avelet lifting analysis
Unorder to verifythe effectiveness of w avelet lifting ondata analysis, the field data obtained74 days before the m achine m alfunction were analyzed. Through wavelet lifting, the original signals with the spectrum ranging from 0 to 2,500 Hz were decomposed at two levels, asshown in Figure 7. Two different- bands can be obtained aftter carrying out decom position at level 1, am ong which the gpectrum range of c0 is 0~1,250 Hz, and that of d0 is 1,250~2,500 Hz. Four different bands can be obtained aftter carrying outdecom position atlevel2, am ong which the spectrum range of c1 is 0~625 Hz, that of d1 is 625-1,250 Hz, that of C2 is 1,250-1,875 Hz, and that of d2 is 1,875-2,500 Hz. The approximation coefficient of the w avelet lifting decom position at level 2 c1, the approxim ation coefficient of the wavelet decom position at level 2 c2, and the detail coefficient of the wavelet decom position at level 2 d are shown in Figure 8. The approxim aton coefficient of wavelet lifting decomposition contains the low frequency inform ation of the signals, and the detail coefficient contains the high frequency inform ation of the signals.
A ccording to the rotational speed of m otor, the frequency of all parts in a rolling m ill can be calculated, am ong which f, the m eshing frequencyof highspeedaxis gear pairZ5/Z 6 is 1,140 H z, and the double frequency is 2,280.4 Hz. Both of the frequencies are included in the reconstructed gpectrum d1 (625-1,250 Hz) andd2 1,875-2,500 Hz) respectively, after decom positionat level one and level two. Thus the wavelet lifting only at level one and two are decom posed without any other m ore decom positions in this paper:.
Figure 7. D ecom positionof the original signals (the spectrum range 0-2500H z) through wavetetlifthg.
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Amplitude(m/s )
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U> "f_i o^ U)
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со I N
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through monitoring spectrum . f means shaft-frequency. The calculated frequency is3700 Hz obtained through rotational speed of the m otor in the field, while the feature frequency is 36.751 Hz obtained through m onitoring spectrum .
in Figure 10, there are many 37 Hz side frequencies around f, which is 1,139277 Hz representing the gear m eshing frequency. These frequencies are definedas side frequency, mthis case, the side frequency (37 Hz) is very close to the shaft-frequency (36 751 Hz) of Shaft II.
The ratio can be obtained through the follow ing calculation. The closer this ratio is to 1, the m ore consistent is the feature frequencyof the m onitoring spectrum w iththe calculating frequencyof the faultpart andthe m ore possibilitythere is of a partw ithsom e fault. The process of calculatbnis shown as follow s: 1,139 277/L,140.00 = 0999, 36 751/37 00 = 0993.
By analyzing Septem ber data inTable 2, it is noted that the single and double frequencies of the Z5/Z6 meshing frequency, as well as the double shaft-frequency of Shaft II, are outstanding. The calculated kurtosis is greater than 6. Compared to fault symptom s w ithkurtosis greater than3, fault symptom s w itha kurtosis over 3 are verylkely.The Jm^f^f-andpeak values are 0 2337, 0 3037, 05685 and 09855 respectively. The ratio of the fm to the peak value is about 0237, which is smaller than 0.4. The sym ptom can be quantified by com bining the above values, and the phasor of this fault symptom A = [0 99, 099, 0, 0, 0, 099, 099, 0, 095, 0, 0, 09]. The calculation of the fault conclusion phasorB is show n below :
B = R =
0.8 0 0 0 0 0 0 0 0 02 0 0 " 0 "0 7792
04 03 02 0 0 0 0 0 0 0 02 0 0 0 693
07 0 0 0 0 0 0 0 0 043 0 0 0 0 693
04 02 02 0 0 0 0 0 0 0 02 0 * 0 99 0 594
02 01 01 0 0 02 01 01 01 01 0 0 0 99 0 689
0 0 0 0 0 04 02 02 01 01 0 0 0 0 689
02 01 0 0 0 02 01 01 02 0 0 02 0 95 0 874
0 4 01 01 01 01 0 0 0 02 0 0 0 0 0 09 0 685
A s calculated in the final result, the m axim um value that the fault conclusion corresponds to is 0 874; thus, the corresponding fault cause can be confirm ed to be a broken gear tooth. A broken cog was found in gearbox Z5 when the machine was disassembled in the field, which is consistentwith the diagnosis conclusion.
Together w ith the fuzzy reasoning approach in the above fault, w e have proved that, in fault diagnosis, the application of fuzzy logic can effectively present som e fuzzy inform ation and construct a fuzzy m atrix; furtherm ore, the fault type can be effectively diagnosed with fuzzy reasoning.
5 ¿.Analysis of the faultdiagnosiis ability
in order to describe the ability of intelligent diagnosis m etthod put forward in this paper, we carried out two cases and m ade com paratve analysis with traditionalm etthod of Fourier transform .
5 41.C ase 1: faultdiagnosiis for tooth collision of helical gear
At 14 00 on Nov. 30th, 2008, through Fourier Transform and wavelet lfting analysis of the original vibration signals!, both of these two methods show that the gear meshing frequency of Z5/Z6 in Shaft. III in the sixth rack of rolling m ill in som e factory w as 45 9 Hz. Figures 11 (a,b) show s the original signal andthe spectrum after Fourier Transform . Figures 11 (cd) show s the spectrum after autocorrelation analysis about the approxim atton signals which were obtained after wavelet transform reconstruction and decom position to the data at level three. It can be seen that the SN R of the wavelet transform is higher in the spectrum analysis. The device disintegrated four days later; the broken cog tooth is shown in Figure 12.
Figure 11. Comparison on the spectrum of Fourier Transform and wavelet lifting analysis of the original signals. a) time domain waveform of the original signals; (b) spectrum of the original signals after Fourier Transform ; (c) tim e dom ain waveform of approxim ation signal reconstruction after decom position of w avelet lifting at level three; (d) spectrum analysis of waefetliftng.
-10 -15
J-0.5 <
Time(s)
Time(s)
0.25 0.2
£0.15
Ü. 0.1
0.25 0.2
]= 0.15
= 0.1 a.
0.05 0
14.65Hz
14.65Hz
34.18Hz
-45.9Hz
0 10 20 30 40 50 60 70 Frequency(Hz)
34.18Hz
45.9Hz
0 10 20 30 40 50 60 70 Frequency(Hz)
Figure 12.The real case of broken cog fault of Z5 (25-tooth) helical gear.
5.4 2.C ase 2: faultdiagnosis for broken cog
At 4 00 on Jan. 25tth, 2008, through Fourier Transform and wavelet lifting analysis of the original vibratonsignals at low frequency, it is found that the shaft-ftequencyof Shaft H ingear-boxinthe second rack of rolling m illwas 2.44 Hz. Figures 13 (a,b) show s the original signal andthe gpectrum after Fourier Transform . Figure 13 (cd) shows the spectrum after autocorrelation analysis of the approxim aton signals which were obtained affer wavelet transform reconstruction and decom position to the data at level three. It can be seen that the wavelet transform had m ore apparent features in gpectrum and of higherSNR . The device was opened and checked 18 days later.The broken cog tooth is shown in Figure 14.
Figure 13. Comparison on gciectrum of Fourier Transform and wavelet lifting analysisof the original signals, a) tim e dom ainw aveform of the original signal; (b) spectrum after Fourier transform of the original signal; (c) tim e dom ain w aveform of approxim ation signal reconstruction after decom position to w avelet lifting at level three; (d) spectrum analysis of waeltlifttng.
Figure 13. Cont
(c) (d)
Figure 14.The realcase of abroken cog faultofbevelgearZ2 (tooth 35).
Som e conclusions can be obtained through com parison of the above two cases, and are sum m arized in Table 3. It can be seen that the SNR of wavelet transform is higher, and the features extracted by wavelet transform is more apparent
Table 3. Improving SNR and extracting features ability compared between Fourier transform FT) and wavelet transform W T).
Faulttype
Position
Effect of noise reduction
Ability of picking-up features
Case 1
Broken cog fault of helical cylindrical gear
G ear-box of bloom ing mill
Non-reduction
M ean-square difference ff/SNR S 10)2/0 33
General
Broken cog faultof bevel gear
G ear-box of bloom ing mill
Non-reduction
M ean-square
difference
ff/SNR S 4 20/013
General
6.C onclisions
By using wavelet lifting, together w itth support vector m achines and rule-based reasoning fault diagnosis m ethods, a real fault exam ple of a broken cog ingearbox was analyzed and the follow ing conclusions were draw n:
SVM is suitable for pattern recognition of problems with small sample sizes. In this study, two-class pattern recognition of actual gearbox faults was accom pli^ed for diagnosis using SVM as the classifier. Based on the second generation wavelet packet feature extraction technology, by taking advantage of the fact that resonance occurs in the high frequency bands in the early stages of a fault, interference from noise signals from other frequency bands is effectively avoided through the decom position and reconstruction of signals at high frequency bands; thus, fault feature extraction was achieved. According to the features of gearbox faults, a fuzzy production approach was applied to reveal fault rules, and rule-based reasoning was achieved through the fuzzy m atrix. As dem onstrated with actual data, this approach effectively overcomes the difficulty that some rules are difficult to present precisely.
Integrating different diagnosis technologies has becom e popular ^intelligent diagnosis research.. Taking advantage of each m etthod in diagnosis inference arch that the m etthods com plem ant each other and create a hybrid diagnosis system is the goal for designing itelligentdiagnosis technology.
Acknow ledgm ent s
This workis supported byNatonal NaturalScience Foundatonof China (Grant No. 50705001), and parttysupported by the Fundamental Research Funds for the Central Universities (Grant No. JD 0904).
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