issi

ma

Signal processing

is nplanin tcalssi

diagnose different conditions of the gear box. The wavelet transform is used to represent all possible

gear box in different conditions is used to demonstrate the application of various wavelets in feature

in theinclud

al com

modern days, there are powerful tools for signal processing pur-poses. The DWT is a knowledge based method characterized forits fast application and big capacity to concentrate information;those signicant marks make it optimal for on line process mon-itoring faults diagnosis in mechanical rotary parts, requiresincreasing the current efciency and performance. A realtime online monitoring can increase the early detection and fault diagno-

tied successfully peaks and phases of transitory signals. New-lands work made the WT popular in engineering applications,especially for vibration analysis. Liu and Ling (1997) proposed amethod based on Wavelet packets, for the diagnosis of failuresin ball bearings. The coefcients of the Wavelet packets wereused as typical features, and the results show that they possessa high sensibility to failures. Loparo et al., used a fuzzy classierto diagnose faults in bearings, based on the WT, achieving to gen-erate typical vectors, with the standard deviation of the DWTcoefcients. Finally, Peng (2004) carried out a bibliographical re-view about the WT application in the monitoring and the fault

* Corresponding author. Tel.: +968 2672 0101; fax: + 968 2672 0102.E-mail addresses: nsaro_2000@yahoo.com, snatarajan@soharuni.edu.om (N.

Expert Systems with Applications 37 (2010) 41684181

Contents lists availab

w

.eSaravanan).evaluate its internal state. From the beginning of the 20th cen-tury, different technologies have been used in order to processsignals of dynamical systems. From classic Fouriers analysis(Loparo, 2004) and its variations (Mallat, 1998), this gave a greatstep in this eld, up today, when Wavelet analysis (Polikar, 1999)plays the main role of a new time frequency evaluation methodfor fault diagnosis. The recent history of the Wavelet analysishas the origin in a narrow collaboration among several scientistsfrom different areas in the middle of last century, and it consistsof the next logical step of the processing signal evolution. In these

istics obtained in the previous step. The application of the Wave-let Transform (WT), for faults diagnosis in machines, has beenwidely developed in the last 15 years. In 1993, Wang and McFad-den (1993) were the pioneers in the WT application to the vibra-tion signal analysis in gears, results showed that WT is able todetect both mechanical incipient failures and different types offaults simultaneously. In 1999, Newland (Newland, 1999) intro-duced the WT in the engineering eld, with several calculationmethods and examples of his application in the vibratory signalanalysis; also, the author proposed a Harmonic Wavelet and iden-1. Introduction

All motor-driven machinery usedvelop faults. The maintenance plansnal relevant information of critic0957-4174/$ - see front matter 2009 Elsevier Ltd. Adoi:10.1016/j.eswa.2009.11.006extraction. In this paper fault diagnostics of spur bevel gear box is treated as a pattern classication prob-lem. The major steps in pattern classication are feature extraction, and classication. This paper inves-tigates the use of discrete wavelets for feature extraction and articial neural network for classication.

2009 Elsevier Ltd. All rights reserved.

modern world can de-e analyzing the exter-ponents, in order to

sis automation, but a more reliable and faster mathematicalmethod is required. There are two important phases to imple-ment in the fault diagnosis process: the rst is the signal process-ing for feature extraction and noise diminishing, and the secondphase consists of the signal classication based on the character-types of transients in vibration signals generated by faults in a gear box. It is shown that the transformprovides a powerful tool for condition monitoring and fault diagnosis. The vibration signal of a spur bevelIncipient gear box fault diagnosis using dfor feature extraction and classication u

N. Saravanan *, K.I. RamachandranFaculty of Engineering Mechanical Engineering, Sohar University, P.O. Box 44, Sohar, O

a r t i c l e i n f o

Keywords:Fault diagnosisDiscrete wavelet transformWavelet featuresArticial neural network

a b s t r a c t

An efcient predictive planand improve the economypercentage of breakdownstion from early stages, alsoin a neural network for cla

Expert Systems

journal homepage: wwwll rights reserved.crete wavelet transform (DWT)ng articial neural network (ANN)

n

eeded for any industry because it can optimize the resources managementt, by reducing unnecessary costs and increasing the level of safety. A greathe productive processes are caused for gear box, they began its deteriora-led incipient level. The extracted features from the DWT are used as inputscation purposes. The results show that the developed method can reliably

le at ScienceDirect

ith Applications

lsevier .com/locate /eswa

2. Data acquisition experiments

2.1. Experimental setup

Fig. 1 shows the fault simulator with sensor and Fig. 2 showsthe inner details of bevel gear box, respectively. A variable speedDC motor (0.5 hp) with speed up to 3000 rpm is used as the basicdrive. A short shaft of 30 mm diameter is attached to the shaft ofthe motor through a exible coupling; this minimizes effects ofmisalignment and transmission of vibration from motor.

The shaft is supported at its ends through two roller bearings.From this shaft the motion is transmitted to the bevel gear boxby means of a belt drive. The gear box is of dimension150 mm 170 mm 120 mm and the full lubrication level is110 mm and half lubrication level is 60 mm. SAE 40 oil was usedas a lubricant. An electromagnetic spring loaded disc brake wasused to load the gear wheel. A torque level of 8 N m was appliedat the full load condition. The various defects are created in thepinion wheels and the mating gear wheel is not disturbed. Gearbox vibration signals are obtained by mounting the accelerometeron the top of the gearbox.

Fig. 1. Fault simulator setup.

ystems with Applications 37 (2010) 41684181 4169diagnosis in machines. In summary, many kinds of fault featurescan be obtained, principally with the Wavelet coefcients or theWavelet energy. Since the Wavelet coefcients will highlightthe changes in signals, which often predicate the occurrence ofthe fault, the Wavelet coefcients-based features are suitablefor fault detection.

The early work in wavelets was in the 1980s by Morlet, Gross-mann, Meyer, Mallat and others, but it was the paper by Daube-chies (1988) in 1988 that caught the attention of the largerapplied mathematicians in signal processing, statistics and numer-ical analysis. Much of the early work took place in France (Combes,Grossmann, & Tchamitchian, 1989; Meyer et al., 1992) and the USA[Dau88a, RBC*92, Dau92, RV91]. As in many new disciplines, therst work was closely tied to a particular application or traditionaltheoretical framework. Now we are seeing the theory abstractedfrom application and developed on its own and seeing it relatedto other parallel ideas.

Wavelet based analysis is an exciting new problem solving toolfor the mathematician, scientist and engineer. It ts naturally withthe digital computer with its bias functions dened by summationsnot integrals or derivatives. Unlike most traditional expansion sys-tems, the basis functions of the wavelet analysis are not solutionsof differential equations. In some areas, it is the rst truly new toolwe have had in many years. Indeed, use of wavelet transforms re-quires a new point of view and a new method of interpreting rep-resentations that we are still learning how to exploit.

In the early studies, Fourier analysis has been the dominatingsignal analysis tool for gear box fault detection. But, there are somecrucial restrictions of the Fourier transform (Peng & Chu, 2004) thesignal to be analyzed must be strictly periodic or stationary; other-wise, the resulting Fourier spectrumwill make little physical sense.Unfortunately, the gear box vibration signals are often non-station-ary and represent non-linear processes, and their frequency com-ponents will change with time. Therefore, the Fourier transformoften cannot fulll the gear box fault diagnosis task pretty well.On the other hand, the timefrequency analysis methods can gen-erate both time and frequency information of a signal simulta-neously through mapping the one-dimensional signal to a two-dimensional timefrequency plane. Among all available timefre-quency analysis methods, the wavelet transforms may be the bestone and have been widely used for gear box fault detection (Wang,1996).

For gear box fault detection, the frequency ranges of the vibra-tion signals that are to be analyzed are often rather wide; andaccording to the Shannon sampling theorem, a high samplingspeed is needed, and sequentially, large size samples are neededfor the gear box fault detection. Therefore, it is expected that thedesired method should have good computing efciency. Unfortu-nately, the computing of continuous wavelet transform (CWT) issomewhat time consuming and is not suitable for large size dataanalysis and on-line fault diagnosis. The discrete wavelet trans-form (DWT), which is based on sub-band coding, is found to yielda fast computation of Wavelet Transforms. It is easy to implementand reduces the computation time and resources required. Hence,it is taken up for study.

In present work, a discrete wavelet, daubechies wavelets (db1to db15) is used for feature extraction and their relative effec-tiveness in feature extraction is compared. For each Daubechieswavelet, wavelet features (in this study 13 features) are extractedat different levels. Naturally, not all features contribute to the clas-sication; there are many techniques available for feature selec-tion. The commonly used techniques for selection of features areDecision Tree (Samanta & Al-Baulshi, 2003) and Principal Compo-

N. Saravanan, K.I. Ramachandran / Expert Snent Analysis (PCA) (Salido & Murakami, 2004; Gunn et al.,1998). In the present study ID3 Decision Tree is used for featureselection and articial neural network is used for classication. Fig. 2. Inner view of bevel gear box.

2.2. Experimental procedure

In the present study, four pinion wheels whose details are asmentioned in Table 2 were used. One wheel was new one andwas assumed to be free from defects. In the other three pinionwheels, defects were created using EDM in order to keep the sizeof the defect under control. The details of the various defects aredepicted in Table 1 and its views are shown in Fig. 3.

The size of the defects is a little bigger than one can encounterin the practical situation; however, it is in-line with work reportedin literature. The vibration signal from the piezoelectric pickupmounted on the test bearing was taken; after allowing initial run-ning of the gear box for some time. The sampling frequency was12,000 Hz and sample length was 8192 for all speeds and all con-ditions. The sample length was chosen arbitrarily; however, thefollowing points were considered. Statistical measures are moremeaningful, when the number of samples is more. On the otherhand, as the number of samples increases the computational timeincreases. To strike a balance, sample length of around 10,000 waschosen. In some feature extraction techniques, which will be usedwith the same data, the number of samples is to be 2n. The nearest2n to 10,000 is 8192 and hence, it was taken as sample length.Many trials were taken at the set speed and vibration signal wasstored in the data.

The raw vibration signals acquired from the gear box when it isloaded with various pinion wheels discussed above using FFT areshown in Fig. 4ad. The vibration signal from a piezoelectric trans-ducer is captured for the following conditions: Good Bevel Gear,Bevel Gear with tooth breakage (GTB), Bevel Gear with crack atroot of the tooth (GTC), and Bevel Gear with face wear of the teeth(TFW) for various loading and lubrication conditions.

3. Feature extraction

The process of computing some measures which will representthe signal is called feature extraction.

3.1. Theoretical background of wavelet transform

The wavelet transform (WT) is a timefrequency decompositionof a signal into a set of wavelet basis functions. In this section, we

a a

Yoshioka, & Ueno, 1996; Mallat, 1989). Under suitable conditions

4170 N. Saravanan, K.I. Ramachandran / Expert Systems with Applications 37 (2010) 41684181Table 1Details of faults under investigation.

Gears Fault description Dimension (mm)

G1 Good G2 Gear tooth breakage (GTB) 8G3 Gear with crack at root (GTC) 0.8 0.5 20G4 Gear with face wear 0.5

Table 2Gear wheel and pinion details.

Parameters Gear wheel Pinion wheel

No. of teeth 35 25Module 2.5 2.5Normal pressure angle 20 20Shaft angle 90 90Top clearance 0.5 mm 0.5 mmAddendum 2.5 mm 2.5 mmWhole depth 5.5 mm 5.5 mmChordal tooth thickness 3:930:150 mm 3:920:110 mmChordal tooth height 2.53 mm 2.55 mmMaterial EN8 EN8Fig. 3. (a) View of good pinion wheel, (b) pinion wheel with faceEq. (3.1.3) is an orthonormal basis of L2R, and the original time func-tion can be expressed as:is a member of the wavelet basis, derived from the basic analysiswavelet wt through translation and dilation. As seen in Eq.(3.1.2), the transformed signal Ta; b is dened on the ab plane,where a and b are used to adjust the frequency and the time loca-tion of the wavelet in Eq. (3.1.2). A small a produces a high-fre-quency (contracted) wavelet when high-frequency resolution isneeded. The WTs superior time- localization properties stem formthe nite support of the analysis wavelet: as b increases, the analy-sis wavelet transverses the length of the input signal, and a in-creases or decreases in response to changes in the signals localtime and frequency content. Finite support implies that the effectof each term in the wavelet representation is purely localized. Thissets the WT apart from the Fourier Transform, where the effects ofadding higher frequency sine waves are spread throughout the fre-quency axis.

3.1.2. Discrete wavelet transform (DWT)Discrete methods are required for computerized implementa-

tion of the WT. The DWT is derived from the CWT through discret-ization of the wavelet wa;bt. The most common discretization ofthe wavelet is the dyadic discretization, given by:

wj;kt 12j

p w t 2jk2j

!3:1:3

where a has been replaced by 2j and b by 2jk (Mori, Kasashima,review the continuous wavelet transform (CWT) and the discretewavelet transform (DWT).

3.1.1. Continuous wavelet transform (CWT)The continuous wavelet transform of a time function f(t) is gi-

ven by the equation:

Ta; b Z 11

f twa;btdt 3:1:1

where * denotes complex conjugation and

wa;bt 1p w t b a; b 2 R; a 0 3:1:2wear (GFW), and (c) pinion wheel with tooth breakage (GTB).

ystemN. Saravanan, K.I. Ramachandran / Expert Sf t X1j1

X1k1

Cj;kwj;kt 3:1:4

Cj;k Z 11

f twj;ktdt 3:1:5

where Cj;k are referred to as wavelet coefcients. A second set of ba-sis function /t called scaling function is then obtained by applyingmulti-resolution approximations to obtain the orthonormal basis ofwt:

Fig. 4. (a) Vibration signals for good pinion wheel (Good) under different lubrication andunder different lubrication and loading conditions, (c) vibration signals for pinion wheelvibration signals for pinion wheel with teeth face wear (TFW) under different lubricatios with Applications 37 (2010) 41684181 4171/j;kt 12j

p / t 2jk2j

!3:1:6

The original time function can now be written as:

dj;k Z 11

f twj;ktdt 3:1:7

Here, the dj;k, which are called the scaling coefcients, d0;k is thesampled version of f(t), represent a jth order resolution discretiza-

loading conditions, (b) vibration signals for pinion wheel with teeth breakage (GTB)with crack at root (GTC) under different lubrication and loading conditions, and (d)n and loading conditions.

the same for feature selection is explained as follows.

most useful feature for classication using appropriate estima-tion criteria. The criterion used to identify the best featureinvokes the concepts of entropy reduction and information gainas explained by Sugumaran, Sabareesh, and Ramachandran(2006).

The Decision Tree algorithm has been applied to the problemunder discussion. Input to the algorithm is set of statistical featuresof wavelet coefcients. The output is the Decision Tree, a sampledecision tree obtained in classifying the statistical features of db5wavelet Coefcients of different faults in half lubrication and no-load condition is show in Fig. 7a.

stemtion of f(t). The scaling coefcients and the wavelet coefcients forresolutions of order greater than j can be obtained iteratively by

dj1;k X1i1

hi 2kdj;k 3:1:8

cj1;k X1i1

gi 2kdj;k 3:1:9

The sequences h and g are low-pass and high-pass lters derivedfrom the original analyzing wavelet wt. The scaling coefcientsdj;k represent the lower frequency approximations of the originalsignal, and the wavelet coefcients Cj;k represent the distributionof successively higher frequencies. The inverse DWT yields a differ-ence series representation for the input signal d0;k in terms of the l-ters h and g and the wavelet coefcients Cj;k:

dj;k X1i1

hk 2idj1;i X1i1

gk 2iCj1;i 3:1:10

The wavelet lters adopted determine the quality of the waveletanalysis. We use the Daubechies system of wavelets (Daubechies,1988), because it has the advantages of orthogonality, compact sup-port in the time domain and computational simplicity. The Daube-chies wavelets of length 2 chosen by the authors are:

h0 1=2

p; h1 1=

2

p; g0 h1; g1 h0

3:1:11Since the input signal f(t) is discretized into N samples, Eqs. (3.1.8)and (3.1.9) can be written in the form of matrix:

d1;1C1;1d1;2C1;1

d1;N=2C1;N=2

0BBBBBBBBBBBBB@

1CCCCCCCCCCCCCA TN

d0;1d0;2d0;3d0;4

d0;N1d0;N

0BBBBBBBBBBBBB@

1CCCCCCCCCCCCCA

where

TN

h0 h1 0g0 g1 00 h0 h1 00 g0 g1 0

00 h0 h10 g0 g1

h1 0 0 h0g1 0 0 g0

0BBBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCCA

3:1:12

The scaling coefcients dj1;kk 1 N=2j1 and the wavelet coef-cients Cj1;k of the j 1th order resolution can be obtained byapplying the N=2jXN=2j matrix TN=2j to the scaling coefcients ofthe jth order dj;kk 1 N=2j. When the number of data pointsis N 2i, all of the wavelet coefcients are obtained after i 1 iter-ations of Eq. (3.1.12). The inverse DWT is performed in a similarmanner by straight forward inversion of the orthogonal matrix TN .

The wavelet analysis has the advantage of better performancefor non-stationary signals, representing a time signal in terms ofa set of wavelets. They are constituted for a family of functionswhich are derived from a single generating function called motherwavelet, although dilation and translation processes. Dilation is re-

4172 N. Saravanan, K.I. Ramachandran / Expert Sylated with size, and it is also know as scale parameter while trans-lation is the position variation of the selected wavelet along thetime axis. This process is illustrated in Fig. 5.1. The set of features available at hand forms the input to the algo-rithm; the output is the Decision Tree.

2. The Decision Tree has leaf nodes, which represent class labels,and other nodes associated with the classes being classied.

3. The branches of the tree represent each possible value of thefeature node from which they originate.

4. The Decision Tree can be used to classify feature vectors bystarting at the root of the tree and moving through it until a leafnode, which provides a classication of the instance, isidentied.

5. At each decision node in the Decision Tree, one can select the3.2. Daubechies wavelet

The Daubechies wavelets are a family of orthogonal waveletsdening a discrete wavelet transform and characterized by a max-imal number of vanishing moments for some given support.

With each wavelet type of this class, there is a scaling functionwhich generates an orthogonal multi-resolution analysis. Daube-chies wavelet functions are shown in Fig. 6a are widely used in solv-ing a broad range of problems, e.g. self-similarity properties of asignal or fractal problems, signal discontinuities, etc. Fig. 6b showsthe scaling functions of Daubechies wavelets db1 to db15.

4. Classication using ID3 algorithm

A standard tree induced with c5.0 (or possibly ID3 or c4.5) con-sists of a number of branches, one root, a number of nodes and anumber of leaves. One branch is a chain of nodes from root to aleaf; and each node involves one attribute. The occurrence of anattribute in a tree provides the information about the importanceof the associated attribute as explained by Peng, Flach, Brazdil,and Soares (2002). A Decision Tree is a tree-based knowledge rep-resentation methodology used to represent classication rules. ID3algorithm (A WEKA implementation of c4.5 Algorithm) is a widelyused one to construct Decision Trees as explained by Quinlan(1996). The procedure of forming the Decision Tree and exploiting

Fig. 5. Wavelet transform execution.

s with Applications 37 (2010) 41684181It is clear there from that the top node is the best node for clas-sication. The other features in the nodes of Decision Tree appearin descending order of importance. It is to be stressed here that

ystemN. Saravanan, K.I. Ramachandran / Expert Sonly features that contribute to the classication appear in theDecision Tree and others do not. Features, which have less discrim-inating capability, can be consciously discarded by deciding on thethreshold. This concept is made use for selecting those featureswhich contribute more towards classication. The algorithm iden-

Fig. 6. (a) Daubechies wavelet functions of db1 to db15,s with Applications 37 (2010) 41684181 4173ties the good features for the purpose of classication from the gi-ven training data set, and thus reduces the domain knowledgerequired to select good features for pattern classication problem.The information gain and entropy associated with the decision treealgorithm has been discussed by Sugumaran et al. The confusion

and (b) Daubechies scaling functions of db1 to db15.

ANNs, which can be congured for a specic application, such aspattern recognition or data classication, through a learning pro-cess. An articial neuron is composed for some connections, whichreceive and transfer information, also there is a net function de-signed for collect all information (weights inputs + bias) andsend it to the transfer function, which process it and produces anoutput. The process is illustrated in Fig. 10.

There are two mean phases in the ANNs application: the learn-ing or training phase and the testing phase. The learning phase iscritical because it determines the type of future tasks able to solve.Once trained the network, the testing phase is followed, in whichthe representative features of the inputs are processed. After calcu-

and

stemmatrix given by the ID3 algorithm identies the number of cor-rectly classied and mis-classied instances. Fig. 7b shows thesample confusion matrix for the above said classication. The dif-ferent feature vectors corresponding to wavelet coefcients of thedifferent wavelets were analyzed one by one using the decisiontree ID3 algorithm and the relative efciency of the different wave-lets in feature extraction is tabulated in next section.

5. The process of optimized feature selection

In the current problem, a discrete wavelet, Daubechies waveletdb1 to db15 are chosen for feature selection and classication.In classication, we are given a set of example records, called thetraining data set, with each record consisting of several attributes.One of the categorical attributes, called the class label, indicatesthe class to which each record belongs. The objective of classica-tion is to use the training data set to build a model of the class labelsuch that it can be used to classify new data whose class labels areunknown. The abbreviations used in classication of differenttypes of bevel gears and for different conditions of lubricationand loading as said in section are given in Table 3.

The vibration signal associated with various conditions of gearbox when the gear box is loaded with different gears explainedin Section 2.2, have been decomposed using db1 to db15 wave-lets individually. The approximated and detailed coefcients whichdecomposition to 8 levels by db5 for various conditions of gearbox with various fault gears in run-up conditions are shown inFig. 8. From Fig. 8ad, the signal s represents the actual vibrationsignal where as a8 represents the approximation at level 8 of db5wavelet and d1 to d8 represents the coefcients details at level 18, respectively.

The wavelet tree representation of the vibration signals gives aclear idea about how the original signal is reconstructed using theapproximations and details at various levels. The wavelet tree rep-

Fig. 7. (a) Decision tree

4174 N. Saravanan, K.I. Ramachandran / Expert Syresentation of the good pinion wheel under dry condition and no-load is shown in Fig. 8e.

The coefcients obtained using this wavelet transforms are fur-ther subjected to statistical analysis and the statistical features areextracted for all the signals and for all the wavelet coefcients fromdb1 to db15. These features are used for classication and faultdiagnosis. Here the training was done with 80 data set attributesand the cross validation is done using 20 data sets.

The classication was done in such a way that the lubricationcondition and loading were xed and then different defects havebeen analyzed accordingly. The average efciency of classicationobtained using ID3 algorithm and using these wavelets for all con-ditions is shown in Fig. 9. It was found that DB5 wavelet has thehighest potential to bring out the hidden pattern in the vibrationsignal extracted from the gear box under investigation. The bestfeatures suggested by ID3 algorithm is fed as an input for the arti-cial neural network to develop the machine learning process.6. Articial neural network

The pattern classication theory has been a key factor in faultdiagnosis methods development. Some classication methods forprocess monitoring use the relationship between a set of patternsand fault types without modeling the internal processes or struc-ture of an explicit way. Nowadays, the ANNs constitute the mostpopular method. An articial neural network (ANN) is an informa-tion processing paradigm inspired by biological nervous systems.The human learning process may be partially automated with

(b) confusion matrix.

Table 3Abbreviations and their description.

Abbreviation Gears considered Lubrication Load

d_noload Good, GTB, TCW, TFW Dry Noloadd_fulload Good, GTB, TCW, TFW Dry Fulloadh_noload Good, GTB, TCW, TFW Half Noloadh_fulload Good, GTB, TCW, TFW Half Fulloadf_noload Good, GTB, TCW, TFW Full Noloadf_fulload Good, GTB, TCW, TFW Full Fulloads with Applications 37 (2010) 41684181lated the weights of the network, the values of the last layer neu-rons are compared with the wished output to verify the suitabilityof the design.

6.1. The back propagation algorithm of ANN

The back propagation of an ANN assumes that there is a super-vision of learning of the network. The method of adjusting weightsis designed to minimize the sum of the squared errors for a giventraining data set.

j identies a receiving node;i denotes the node that feeds a second node;I denotes input to a neuron;O denotes output of a neuron;

Wij denotes the weights associated with the nodes.

00.05a

Decomposition at level 8 : s = a8a

ystem0.1-0.058-0.4-0.2

0sN. Saravanan, K.I. Ramachandran / Expert SEach non input node has an output level Oj where

Oj 1=1 eIjIj RWijOj

6:1:1

where Oi is each of the signals to node j (i.e., the output of node of i)The derivation of the back propagation formula involves the use

of the chain rule of partial derivatives and equals:

1000 2000 3000-0.1

00.1

d1

-0.10

0.1d2

-0.10

0.1d3

-0.20

0.2d4

-0.050

0.05d5

-0.050

0.05d6

-0.10

0.1d7

-0.10d8

1000 2000 3000 4-0.1

00.1

d1

-0.10

0.1d2

-0.10

0.1d3

-0.10

0.1d4

-0.050

0.05d5

-0.050

0.05d6

-0.050

0.05d7

-0.10

0.1d8

-0.10

0.1a8

-0.20

0.2

s

Decomposition at level 8 : s = a8 +b

Fig. 8. (a) Decomposed signal of good pinion wheel under dry, noload condition using dbdb5, (c) decomposed signal of TCW under full lubrication and noload condition using db5db5, and (e) wavelet tree. + d8 + d7 + d6 + d5 + d4 + d3 + d2 + d1 .

s with Applications 37 (2010) 41684181 4175dij @SSE@Wij

@SSE@Oj

@Oj@Ij

@Ij

@Wij

6:1:2

where by convention the left hand side is denoted by dij, the changein the sum of squared errors (SSE) attributed to Wij. Now error is gi-ven by

ei Dj OjSSE RDj Oj2

6:1:3

4000 5000 6000 7000 8000

000 5000 6000 7000 8000

d8 + d7 + d6 + d5 + d4 + d3 + d2 + d1 .

5, (b) decomposed signal of GTB under half lubrication and full load condition using, (d) decomposed signal of TFW under full lubrication and full load condition using

d6 +

stem0.02-0.05

00.05

d8

-0.050

0.05a8

-0.20

0.2

s

Decomposition at level 8 : s = a8 + d8 + d7 +c

4176 N. Saravanan, K.I. Ramachandran / Expert SyTherefore,

@SSE@Oj

2RDj Oj 6:1:4

From the output of the output node, we obtain,

@Oj@Ij

Oj1 Oj 6:1:5

The input to an input node is Ij RWijOiTherefore, the change in the input to the output node resulting

from the previous hidden node, i, is

1000 2000 3000 4000-0.05

00.05

d1

-0.050

0.05d2

-0.050

0.05d3

-0.050

0.05d4

-0.050

0.05d5

-0.050

0.05d6

-0.020d7

1000 2000 3000 4000-0.2

00.2

d1

-0.10

0.1d2

-0.10

0.1d3

-0.10

0.1d4

-0.10

0.1d5

-0.050

0.05d6

-0.050

0.05d7

-0.10

0.1d8

-0.10

0.1a8

-0.20

0.2

s

Decomposition at level 8 : s = a8 + d8 + d7 + d6 +d

Fig. 8 (cont d5 + d4 + d3 + d2 + d1 .

s with Applications 37 (2010) 41684181@Ij@Wij

Oi 6:1:6

Thus from above equations, the jth delta is

dij 2ejOj1 OjOi 6:1:7Now the old weight is updated by the following equation:

DWijnew gdijOj aDWijold 6:1:8For the hidden layers, the calculations are similar. The only changeis how the ANN output error is back propagated to the hidden layernodes. The output error at the ith hidden node depends on the

5000 6000 7000 8000

5000 6000 7000 8000

d5 + d4 + d3 + d2 + d1 .

e

inued)

date. A critical parameter is the speed of convergence, which isdetermined by the learning coefcient. In general, it is desirableto have fast learning, but not so fast as to cause instability of learn-ing iterations. Some previous works (Samanta & Al-Baulshi, 2003)have had a successful training with a learning rate of 0.05. The stop-ping criteria are based in the minimum error reached, in this case

Fig. 11. ANN architecture.

Avg. Classification Eff. of DB

98.4

98.45

98.5

98.55

98.6

98.65

98.7D

B1

DB2

DB3

DB4

DB5

DB6

DB7

DB8

DB9

DB1

0

DB1

1

DB1

2

DB1

3

DB1

4

DB1

5

% A

vg. E

ffici

ency

Fig. 9. Average classication efciency of db1 to db15.

N. Saravanan, K.I. Ramachandran / Expert Systems with Applications 37 (2010) 41684181 4177output errors of all nodes in the output layer. This relationship is gi-ven by

ei RWijej 6:1:9

Fig. 10. Articial neuron diagram.After calculating the output error for the hidden layer, the updaterules for the weights in that layer are the same as the previous up-

Table 4Network statistics of articial neural network for dry-noload condition.

No. of neurons RMS error Training epochs No. of data items

In hidden layer2 0.0212744 10,234 203 0.0379643 9634 204 0.0245643 127,643 205 0.045954 12,456 206 0.0345217 8767 207 0.0675401 107,057 208 0.035429 11,676 209 0.0325693 6674 2010 0.0756492 54,671 2011 0.0486432 14,589 2012 0.0876432 104,589 2013 0.0459321 14,987 2014 0.096543 6875 2015 0.0459321 5789 2016 0.0569342 14,789 2017 0.0965431 10,467 2018 0.085643 4567 2019 0.045632 29,453 2020 0.075432 53,212 2021 0.085436 48,965 2022 0.0435216 5634 2023 0.0564321 4234 2024 0.075432 6785 2025 0.0659432 10,876 2026 0.085621 11,567 2027 0.0432671 4378 2028 0.0956432 10,345 2029 0.075432 4765 2030 0.0865432 4879 205%. The network training is also limited to 25,000 epochs and thevalidation dataset may affect the training, with a maximum of1000 iteration failed. By changing the weights given to theses sig-nals, the network learns in a process that seems similar to thatfound in nature. i.e., neurons in ANN receive signals or informationfrom other neurons or external sources, perform transformations onthe signals, and then pass those signals on to other neurons. Theway information is processed and intelligence is stored dependson the architecture and algorithms of ANN. Fig. 11 shows the archi-tecture of ANN.

Number right Number wrong % Right % Wrong

18 2 90 1017 3 85 15

16 4 80 2018 2 90 1019 1 95 516 4. 80 2017 3 85 1519 1 95 518 10 90 1017 6 85 1516 4 80 2017 3 85 1519 1 95 519 8 95 518 4 90 1018 2 90 1019 1 95 518 4 90 1017 3 85 1517 6 85 1519 1 95 519 1 95 519 1 95 518 2 90 1018 2 90 1019 1 95 518 4 90 1019 1 95 519 1 95 5

Table 6Network statistics of articial neural network for half-noload condition.

No. of neurons RMS error Training epochs No. of data items Number right Number wrong % Right % Wrong

In hidden layer2 0.0212744 10,234 20 18 2 90 103 0.0379643 8734 20 16 4 80 204 0.0145643 139,643 20 18 2 90 105 0.545954 12,456 20 18 2 90 106 0.0245217 8647 20 19 1 95 57 0.0375401 107,057 20 17 3 85 158 0.235429 10,575 20 18 2 90 109 0.0125693 6274 20 19 1 95 510 0.0656492 52,071 20 18 2 90 1011 0.0486432 12,389 20 19 1 95 512 0.0676432 122,589 20 16 4 80 2013 0.0459321 10,987 20 18 2 90 1014 0.096543 6776 20 19 1 95 515 0.0259321 5489 20 19 1 95 516 0.0469342 17,789 20 18 2 90 1017 0.1665431 10,467 20 18 2 90 1018 0.045643 4267 20 16 4 80 2019 0.035632 28,453 20 18 2 90 1020 0.065432 50,212 20 17 3 85 1521 0.035436 41,965 20 17 3 85 1522 0.0235216 5034 20 18 2 90 1023 0.0464321 4234 20 20 0 100 024 0.045432 6085 20 19 1 95 525 0.069432 11,576 20 18 2 90 1026 0.085621 18,567 20 18 2 90 1027 0.0232671 4478 20 19 1 95 528 0.0956432 10,345 20 17 3 85 1529 0.045432 4065 20 19 1 95 530 0.0165432 4879 20 19 1 95 5

Table 5Network statistics of articial neural network for dry-fullload condition.

No. of neurons RMS error Training epochs No. of data items Number right Number wrong % Right % Wrong

In hidden layer2 0.0112744 10,434 20 17 3 85 153 0.1379643 8634 20 19 1 95 54 0.2145643 129,643 20 17 3 85 155 0.045954 144,56 20 18 2 90 106 0.0145217 8667 20 19 1 95 57 0.0575401 117,057 20 17 3 85 158 0.535429 10,675 20 18 2 90 109 0.0325693 6574 20 19 1 95 510 0.0756492 52,671 20 18 2 90 1011 0.1486432 14,389 20 17 3 85 1512 0.0676432 102,589 20 16 4 80 2013 0.0359321 12,987 20 18 2 90 1014 0.076543 6675 20 19 1 95 515 0.0359321 5289 20 19 1 95 516 0.0569342 13,789 20 18 2 90 1017 0.1965431 11,467 20 18 2 90 1018 0.075643 4067 20 19 1 95 519 0.025632 26,453 20 18 2 90 1020 0.075432 51,212 20 17 3 85 1521 0.065436 46,965 20 17 3 85 1522 0.0235216 5234 20 19 1 95 523 0.0464321 4034 20 20 0 100 024 0.085432 6285 20 19 1 95 525 0.059432 10,576 20 18 2 90 1026 0.095621 10,567 20 18 2 90 1027 0.0332671 4078 20 19 1 95 528 0.0856432 12,345 20 17 3 85 1529 0.095432 4265 20 19 1 95 530 0.0765432 4279 20 19 1 95 5

4178 N. Saravanan, K.I. Ramachandran / Expert Systems with Applications 37 (2010) 41684181

Table 8Network statistics of articial neural network for full-noload condition.

No. of neurons RMS error Training epochs No. of data items Number right Number wrong % Right % Wrong

In hidden layer2 0.0212744 10,234 20 18 2 90 103 0.0379643 8734 20 16 4 80 204 0.0145643 139,643 20 18 2 90 105 0.545954 12,456 20 18 2 90 106 0.0245217 8647 20 19 1 95 57 0.0375401 107,057 20 17 3 85 158 0.235429 10,575 20 18 2 90 109 0.0125693 6274 20 19 1 95 510 0.0656492 52,071 20 18 2 90 1011 0.0486432 12,389 20 19 1 95 512 0.0676432 122,589 20 16 4 80 2013 0.0459321 10,987 20 18 2 90 1014 0.096543 6775 20 19 1 95 515 0.0259321 5489 20 19 1 95 S16 0.0469342 17,789 20 18 2 90 1017 0.1665431 10,467 20 18 2 90 1018 0.045643 4267 20 16 4 80 2019 0.035632 28,453 20 18 2 90 1020 0.065432 50,212 20 17 3 85 1621 0.035436 41,965 20 17 3 85 1522 0.0235216 5034 20 18 2 90 1023 0.0464321 4234 20 20 0 100 024 0.045432 6085 20 19 1 95 525 0.069432 11,576 20 18 2 90 1026 0.085621 18,567 20 18 2 90 1027 0.0232671 4478 20 19 1 95 528 0.0956432 10,346 20 17 3 85 1529 0.045432 4065 20 19 1 95 530 0.0165432 4879 20 19 1 95 5

Table 7Network statistics of articial neural network for half-fullload condition.

No. of neurons RMS error Training epochs No. of data items Number right Number wrong % Right % Wrong

In hidden layer2 0.0612744 10,134 20 18 2 90 103 0.0279643 8234 20 16 4 80 204 0.105643 119,643 20 18 2 90 105 0.025954 10,556 20 18 2 90 106 0.0245217 8047 20 19 1 95 57 0.1565401 127,057 20 17 3 85 158 0.435429 11,775 20 18 2 90 109 0.0125693 5074 20 19 1 95 510 0.1956492 64,071 20 18 2 90 1011 0.0486432 12,389 20 19 1 95 512 0.0476432 112,589 20 16 4 80 2013 0.1459321 11,487 20 18 2 90 1014 0.496543 2075 20 19 1 95 515 0.0169321 5989 20 19 1 95 516 0.0239342 10,789 20 18 2 90 1017 0.1665431 24,467 20 18 2 90 1018 0.015643 2067 20 20 0 100 019 0.025632 21,453 20 20 0 100 020 0.015432 50,212 20 17 3 85 1521 0.025436 42,965 20 17 3 85 1522 0.0735216 5024 20 18 2 90 1023 0.0164321 4904 20 20 0 100 024 0.045432 6085 20 19 1 95 525 0.069432 14,576 20 18 2 90 1026 0.045621 10,567 20 18 2 90 1027 0.0132671 4278 20 19 1 95 528 0.0356432 16,345 20 17 3 85 1529 0.045432 4765 20 19 1 95 530 0.0165432 4579 20 19 1 95 5

N. Saravanan, K.I. Ramachandran / Expert Systems with Applications 37 (2010) 41684181 4179

ems wron

19 116 4

stem 1Table 9Network statistics of articial neural network for full-fullload condition.

No. of neurons RMS error Training epochs No. of data it

In hidden layer2 0.0212744 20,134 203 0.0479643 4234 204 0.145643 109,643 205 0.015954 14,506 206 0.0745217 8007 207 0.2565401 117,057 208 0.035429 10,775 209 0.2125693 3074 2010 0.1956492 61,071 2011 0.0986432 16,389 2012 0.0176432 102,589 2013 0.1959321 10,487 20

4180 N. Saravanan, K.I. Ramachandran / Expert Sy6.2. Application of ANN for problem at hand and results

For each faults namely good Bevel Gear, Bevel Gear with toothbreakage (GTB), Bevel Gear with crack at root of the tooth (GTC),and Bevel Gear with face wear of the teeth (TFW) for various loadingand lubrication conditions, four feature vectors consisting of 100 fea-turevalue setswere collected fromtheexperiment. Twenty-ve sam-ples ineachclasswereused for trainingand5samplesare reserved fortesting ANN. Training was done by selecting three layers neural net-work, of that one is input layer, onehidden layer andoneoutput layer.The numbers of neurons in the hidden layer were varied and the val-uesofnumberofneurons,RMSerrorandnumberofepochsalongwithpercentage efciency of classication of various faults using ANN arecomputed. A total of six networks as mentioned above were createdfor classifying the faults namely dry lubrication-noload, dry lubrica-tion fulload, half lubrication noload, half lubrication fulload, fulllubrication noload and full lubrication fulload.

Fig. 12. RMS error vs. n

14 0.096543 2079 2015 0.0169321 1989 2016 0.0539342 10,189 2017 0.1065431 20,467 2018 0.515643 2067 2019 0.095632 26,453 2020 0.015432 50,512 2021 0.065436 46,965 2022 0.0735216 5064 2023 0.0264321 4904 2024 0.045432 6285 2025 0.069432 14576 2026 0.025621 10,567 2027 0.0132671 4078 2028 0.0256432 10,345 2029 0.035432 4065 2030 0.0065432 4779 2018 2 90 1019 1 95 516 4 80 2018 2 90 1019 119 17. Results of ANN

The architecture of the articial n

Network type

No of neurons in input layerNo of neurons in hidden layerNo of neurons in output layerTransfer function

Training ruleTraining toleranceLearning ruleMomentum learning step sizeMomentum learning rate

umber of neurons.

19 119 118 218 220 020 017 317 318 220 019 118 218 219 117 319 119 195 580 2095 595 619 118 295 590 1019 1 95 5

17 3 85 15g % Right % WrongNumber right Numbers with Applications 37 (2010) 4168418eural network is as follows

Forward neural networktrained with feed backpropagation4varied form 2 to 301Sigmoid transfer function inhidden and output layerBack propagation0.05Momentum training method0.750.75

95 595 590 1090 10100 0100 085 1585 1590 10100 095 590 1090 1095 585 1595 595 5

rma

N. Saravanan, K.I. Ramachandran / Expert Systems with Applications 37 (2010) 41684181 4181for training with minimum number of epochs for different classi-bearing by applying the discrete wavelet transform to vibration signals. Wear,195, 162168.

Newland, D. E. (1999). Ridge and phase identication in the frequency analysis ofThe efciency of classication of gear box faults using abovenetworks are reported in Tables 49 for classifying the faultsnamely dry lubrication noload, dry lubrication fulload, halflubrication noload, half lubrication fulload, full lubrication noload and full lubrication fulload, respectively. Fig. 12 depictsthe RMS error corresponding to different neurons in hidden layerfor all the six networks, respectively.

Fig. 12 shows the rms plot vs. the number of neurons. Fig. 13shows the performance rate of ANN. It was found that half noloadhalf full load and full noload was classied exactly ie, the learningrate is highly successful compared to other conditions. Initially thenumber neurons in hidden layer in each network were varied from2 to 30 and Fig. 13 gives the optimal number of neurons required

cations of faults. The overall average efciency of entire classica-tion using ANN was found to be 95%.

Fig. 13. Perfo8. Conclusions

This paper has outlined the denition of the discrete waveletstransform and then demonstrated how it can be applied to theanalysis of the vibration signals produced by a bevel gear box invarious conditions and faults. Wavelet features were extractedfor all the wavelet coefcients and for all the signals using theDaubechies wavelets db1 to db15. The features selection ofvarious discrete wavelets was carried out and the wavelet havingthe highest potential to describe the various conditions of gearbox was fed as input to neural network for classication of variousfaults of the gear box. The performance of the neural network bothin learning and classifying was found encouraging and can be con-cluded that neural networks has high potential in condition mon-itoring of the gear box with various faults.

References

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Incipient gear box fault diagnosis using discrete wavelet transform (DWT) for feature extraction and classification using artificial neural network (ANN)IntroductionData acquisition experimentsExperimental setupExperimental procedure

Feature extractionTheoretical background of wavelet transformContinuous wavelet transform (CWT)Discrete wavelet transform (DWT)

Daubechies wavelet

Classification using ID3 algorithmThe process of optimized feature selectionArtificial neural networkThe back propagation algorithm of ANNApplication of ANN for problem at hand and results

Results of ANNConclusionsReferences