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Expert Systems with Applications 36 (2009) 4862–4875

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Expert Systems with Applications

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A novel technique for selecting mother wavelet function using an intelligentfault diagnosis system

J. Rafiee a,*, P.W. Tse b, A. Harifi c, M.H. Sadeghi d

a Department of Mechanical, Aerospace and Nuclear Engineering, Jonsson Engineering Center, Rm. 2049, 110 8th Street, Rensselaer Polytechnic Institute, Troy, NY 12180-3590, USAb Smart Engineering Asset Management Laboratory, City University of Hong Kong, Hong Kongc Faculty of Electrical and Computer Engineering, University of Tabriz, Irand Faculty of Mechanical Engineering, University of Tabriz, Iran

a r t i c l e i n f o

Keywords:
Condition monitoringSignal processingVibration signalFault diagnosisNeural networkGenetic algorithmMother waveletGearPiecewise cubic hermite interpolationDaubechies
0957-4174/$ - see front matter � 2008 Published bydoi:10.1016/j.eswa.2008.05.052

* Corresponding author. Tel.: +1 (518) 276 6351; faE-mail address: [email protected] (J. Rafiee).

a b s t r a c t

This paper presents an optimized gear fault identification system using genetic algorithm (GA) to inves-tigate the type of gear failures of a complex gearbox system using artificial neural networks (ANNs) witha well-designed structure suited for practical implementations due to its short training duration and highaccuracy. For this purpose, slight-worn, medium-worn, and broken-tooth of a spur gear of the gearboxsystem were selected as the faults. In fault simulating, two very similar models of worn gear have beenconsidered with partial difference for evaluating the preciseness of the proposed algorithm. Moreover,the processing of vibration signals has become much more difficult because a full-of-oil complex gearboxsystem has been considered to record raw vibration signals. Raw vibration signals were segmented intothe signals recorded during one complete revolution of the input shaft using tachometer information andthen synchronized using piecewise cubic hermite interpolation to construct the sample signals with thesame length. Next, standard deviation of wavelet packet coefficients of the vibration signals considered asthe feature vector for training purposes of the ANN. To ameliorate the algorithm, GA was exploited tooptimize the algorithm so as to determine the best values for ‘‘mother wavelet function”, ‘‘decompositionlevel of the signals by means of wavelet analysis”, and ‘‘number of neurons in hidden layer” resulted in ahigh-speed, meticulous two-layer ANN with a small-sized structure. This technique has been eliminatedthe drawbacks of the type of mother function for fault classification purpose not only in machine condi-tion monitoring, but also in other related areas. The small-sized proposed network has improved the sta-bility and reliability of the system for practical purposes.

� 2008 Published by Elsevier Ltd.

1. Introduction

The application of a condition-based maintenance strategy usedfor detection and diagnosis of incipient faults can help us to mini-mize avoidable costs and impediments caused by the need to per-form ad hoc repairs. Gearboxes consisting of three majorcomponents (gears, bearings, and shafts) are considered to beone of the essential systems on account of its influential applica-tions in a wide domain of industries (e.g. machine tools and vehi-cles). As more than half of the failures in gearboxes are due to geardefects (Yesilyurt, 2004), the gear faults have been considered as acase study for the proposed technique.

In general, acoustic emission (Wu, Chiang, Chang, & Shiao,2008) and vibration analysis (Tse, Gontarz, & Wang, 2007) aretwo main centers of fault detection and diagnosis systems inmachine condition monitoring. In this paper, vibration signals

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established as the way of recording required dataset because ofthe ease of measurement and the rich contents of information.To analyze raw vibration signals, a simultaneous time–frequencyanalysis method which maps one-dimensional signals into a two-dimensional space of time and frequency was applied because ofnon-stationarity of vibration signals. Wavelet analysis, the mostpopular one for analyzing the non-stationary signals (Peng et al.,2005), overcomes the drawbacks of other techniques by means ofanalytical functions that are local in both time and frequency (Tse,Yang, & Tam, 2004). Therefore, in pre-processing and feature extrac-tion phase of the research, wavelet transform (WT) (Tse, Peng, &Yam, 2001) has been used to take out proper features. To bolster thisrecommendation, in early 1990, Leducq applied wavelet transformto analyze the hydraulic noise of the centrifugal pump which isprobably the first paper in application of wavelet transform inmachine diagnostics (Peng & Chu, 2004). Momoh and Dias (Momoh& Dias, 1996), moreover, used both fast Fourier transform (FFT)and wavelet transform (WT) to obtain processed vibrationsignals to identify and pinpoint the location of the faults in power

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DB 9DB 8DB 7

DB 4 DB 5 DB 6

DB 9DB 8DB 7

DB 4 DB 5 DB 6

Fig. 1. Low-order Daubechies.

J. Rafiee et al. / Expert Systems with Applications 36 (2009) 4862–4875 4863

distribution systems. The prior research demonstrates the wavelettransform as the most reliable technique for gear condition moni-toring and a capable technique to recognize the incipient failuresand to identify different types of faults simultaneously.

Fig. 2. (A) Experimental set-up, (B) slight-worn gear, (C) broken-tooth gear, (D) accelermechanism.

The most indispensable challenge in wavelet analysis is theselection of the mother wavelet function as well as the decompo-sition level of signal. As a result of the use of dyadic discrete wave-let transform (DWT), orthogonal wavelets have been applied in thisresearch; among them Daubechies (DB) wavelets have been widelyimplemented as it matches the transient components in vibrationsignals. Another main issue in wavelet analysis is the order of themother wavelet function, which was previously determined bytrial-and-error methods based on intrinsic characteristics of thedata in several papers (Wang & McFadden, 1995; Samanta & Al-Balushi, 2003; Tse et al., 2004; Liu, 2005; Kar & Mohanty, 2006;Saravanan et al., 2007; Rafiee, Arvani, Harifi, & Sadeghi, 2007). Tomore explanations, the range of DB2 and Db20 has been exten-sively used in machine condition monitoring. Wang and McFadden(1995) used DB4 orthogonal wavelets to disclose abnormal tran-sients generated by early gear damage from gearbox vibration sig-nal. Sung et al. and also Gaborson used DB20 and DB4, as themother wavelet, in their works, respectively (Kar & Mohanty,2006). Samanta and Al-Balushi (2003) and Rafiee et al. (2007) pro-cessed vibration signals through discrete wavelet transform (DWT)

ometer location, (E) slight-worn model, (F) schematic of the gearbox and (G) load

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045-100

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Fig. 3. Raw vibration signals of four gearbox conditions recorded during one revolution of input shaft.

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Fig. 4. Synchronized vibration signals of four gearbox conditions recorded during one revolution of input shaft.

4864 J. Rafiee et al. / Expert Systems with Applications 36 (2009) 4862–4875

DA2AA2

D1A1

Signal

Level 1

Level 2

Level n

DD2AD2

Fig. 5. Approximations and details in wavelet packet analysis.


using DB4 to obtain the wavelet coefficients. The most importantissue as it has been shown in Fig. 1, is that it can not be functionalif the adjacent wavelet functions (e.g. DB6 and DB7) are comparedas they do not differ from each other tremendously.

In general, the current fault detection and diagnosis techniquesare predominantly based on intelligent systems, modeling usingclassical techniques in time or frequency domain, and statisticalanalysis. Fuzzy-based (Wang & Hu, 2006), ANN-based (Wuxing,Tse, Guicai, & Tielin, 2004), GA-based (He, Guo, & Chu, 2001) meth-ods as well as intelligent hybrid systems such as fuzzy-GA-based(Lei et al., MSSP 2007), (Lo, Chan, Wong, Rad, & Cheung, 2007),Nero-fuzzy-based (Lei et al., ESWA 2007), Nero-GA-based (Saman-ta, 2004) systems can be categorized as the intelligent fault diagno-sis systems. In this category, ANN-based systems consisting ofsignal preprocessing, feature extraction and recognition processhave been introduced as one of the most noteworthy methods inseveral papers. To name a few, Li, Chow, Tipsuwan, and Hung(2000) presented an approach for motor rolling bearing fault diag-nosis using both simulation and experiment to acquire vibrationsignals which was used to obtain a feature vector to adjust ANNweights. Samanta and Al-Balushi (2003) introduced a new tech-nique for fault detection of rolling bearing elements through ANNsin which feature vector was obtained from statistical properties oftime-domain vibration signals of the rotating machinery with nor-mal and defective bearings. In 2003, Samanta, Al-Balushi, and Al-Araimi (2003) also enhanced the proposed procedureusing GA tometiculously optimize the mentioned feature vector for gear faultdetection using the experimental vibration data of a gearbox sys-tem. Inspired by Jack and Nandi (2002) innovations, the paper pre-sents an optimized ANN-based system for identifying the gearfaults of a gearbox system. It also presents a novel solution to findthe best mother wavelet function for fault classification purpose aswell as the best level of decomposing the vibration signals bywavelet analysis in machine condition monitoring and relatedareas. In addition, the small structure optimized network has im-proved the stability and reliability of the system in practicalimplementations.

2. Summary of the developed procedure

Since gears are the most challenging components in conditionmonitoring, a complex gearbox system has been considered asthe case study. Of paramount importance is to synchronize the

raw vibration signals which usually don’t have the same lengthin a complete revolution of the shaft. To overtake the impediment,piecewise cubic hermite interpolation (Rafiee et al., 2007) was sug-gested to synchronize the signals. Afterwards, wavelet packet anal-ysis (Liu, 2005) has been considered to preprocess thesynchronized signals for presenting a distinguished feature vector.Then, selecting of the mother wavelet function and decompositionlevel has been reformed by means of genetic algorithm as an obsta-cle in this regard. To present a more accurate fault classificationsystem, number of neurons in hidden layer has been also chosento be optimized by GA.

2.1. Experimental set-up

A proper data acquisition process can extensively improve thesensibility for the detection of failures. Therefore, selecting anappropriate method is important for acquiring vibration signaldata. The experimental set-up (Rafiee et al., 2007) to collect datasetin this research consists of a four-speed motorcycle gearbox con-taining the oil during acquiring dataset, an electrical motor witha constant nominal rotation speed of 1420 RPM, a load mechanism,multi-channel pulse analyzer system, a triaxial accelerometer,tachometer and four shock absorbers under the bases of test-bed.The vibration signals were recorded by mounting the accelerome-ter on the outer surface of the gearbox’s case near input shaft of thegearbox. Ten different configurations were synthesized and tested.For each configuration, three different fault conditions were testedthat were slight-worn, medium-worn, broken-tooth of gear. Forevaluating the meticulousness of the technique, two very similarmodels of worn gear have been taken into account with partial dif-ference. The rotational speed of the system was measured bytachometer used as a measure to compensate the fluctuations ow-ing to uncertainties of the load mechanism which was a constantstatic load. Moreover, the signals were also sampled at 16384 Hzlasting eight seconds. Fig. 2 clearly shows all parts of fault simula-tor and related components in detail.

2.2. Preprocessing of raw vibration signals

It is important to pre-process any raw data before use since itcontains redundant information. The pre-processing of vibrationsignals was involved in synchronization of the signals and compu-tation of standard deviation of wavelet packet coefficients per-

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Fig. 6a. Wavelet packet coefficients of a sample synchronized broken-tooth signal in four levels.


formed in the following two steps to extract the feature vector.Accurate pre-processing of the data to feed neural network canmake ANN training more efficient because of a considerable reduc-tion of the dimensionality of the input data and therefore improv-ing the network performance.

2.2.1. Synchronization using piecewise cubic hermite interpolationIn the recorded signals, as a consequence of non-synchronous

attribute of vibration signals, the number of data-points per eachrevolution may not be exactly equal from one revolution to an-other one, due to shaft speed fluctuations which can influencethe time-domain average adversely. To overcome the drawback,piecewise cubic hermite interpolation (P.C.H.I.) (Fritsch & Carlson,1980) was exploited to resample the obtained data in each revolu-tion as it outperforms the other methods (Rafiee et al., 2007). Amenace of interpolation is that the fitting function may tend to

demonstrate unsolicited fluctuations, essentially following randomerrors in the data and oscillations of the data points. This problemis particularly persisting if the fitting function is a polynomial de-fined over the entire region of interest, and becomes more criticalas the number of data points increases. Consequently, P.C.H.I. wasimplemented as the interpolating scheme. In this method, thesame sample spans in each revolution could be guaranteed. In thispaper, the average length of the 150 segmented sample signals wasapplied to synchronize the whole signal set. Given that hk is thelength of the kth subinterval,

hk ¼ xkþ1 � xk ð1Þ

Then the first difference, dk is calculated by the following equation,

dk ¼ ðykþ1 � ykÞ=hk ð2Þ

Assume that dk designate the slope of the interpolant at xk

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Fig. 6b. Wavelet packet coefficients of a sample synchronized broken-tooth signal in four levels.

Normal Gearbox

Output layer

Broken-tooth

Medium-worn

Slight-worn

Hidden layer

Input layer

StandardDeviation of Wavelet Packet Coefficients

Fig. 7. Structure of the MLP network for fault classification.

0 1 1 0 1 1 10 1 1 0 1 0 1

Number of Neurons

DBorder

Decomposition Level

Fig. 8. Proposed chromosome for GA.


dk ¼ P0ðxkÞ ð3Þ

for the piecewise linear interpolant, dk = dk�1 or dk

Suppose the following function on the interval xk 6 x 6 xk+1,expressed in term of local variables s = x- xk and h = hk,

PðxÞ ¼ 3hs2 � 2s3

h3 ykþ1 þh3 � 3hs2 þ 2s3

h3 yk þs2ðs� hÞ

h2 dkþ1

þ sðs� hÞ2

h2 dk ð4Þ

which is a cubic polynomial in s, and, hence in x, that fulfilled fourinterpolation conditions, two of those on function values and therest two on the possibly unknown derivative values. The raw andsynchronized signals have been depicted in Figs. 3 and 4, respec-tively. As shown, the signal lengths which are not equal in raw oneshave been precisely synchronized after implementing of P.C.H.I.

2.2.2. Wavelet packet analysisFeature extraction, aimed to take out certain characteristics

from original measured signals, plays the vital role in neural net-works. In simple words, badly implemented feature extraction orimproper features will be led to poor classification results evenusing the potential classifiers. Accordingly, special attention hasbeen paid to the selection of the feature vector or input layer inANNs.

Wavelet analysis, which is mainly categorized into continuousand discrete types, includes transforming a signal from the time

Yes

No

Selection of parameters by GA

Gearbox

Accelerometer Tachometer

Synchronized Signals by P.C.H.I

Raw vibration signals

Stopping

Criterion

StD of WPC

Training ANN Testing ANN

MotherWavelet

Decomposition Level

Number of Neurons

Training Time ANN Performance

FitnessFunction

Intelligent Fault Identification System with Optimized Parameters

Fig. 9. Intelligent fault identification system with optimized parameters.

Table 1GA parameters

Number of generation 200

Population 30Chromosome length 12Selection operator RouletteFitness normalization RankElitism 1Crossover Pc = 0.8, two-point, uniformMutation Pm = 0.1, Gaussian, mean = 0.0, std = 1.0

Table 2GA variables

Variable name Range Optimized value

Daubechies order DB2–DB20 DB11Decomposition level 3–6 4Number of the hidden-layer neurons 10–35 14


domain into time–frequency domain. Wavelet transform (WT) of-fers a satisfactory time and poor frequency resolution at high fre-quencies and a satisfactory frequency and poor time resolution atlow frequencies (Peng et al., 2005). It is used to detect transient

Synchronized Signal

4,1 4,2 4,3 4,4 4,5 4,74,64,0

3,1 3,2 3,33,0

2,12,0

1,0Level 1

Level 2

Level 3

Level 4

Fig. 10. Decomposition tree of sy

components because it is able to simultaneously impart time andfrequency structures.

Continuous wavelet transform (CWT), a function of two param-eters entitled ‘‘translation and scale”, bears a remarkable dataredundancy for the signal or function to be analyzed. By skippingseveral steps rather than continuously adjusting two parameters,in other words, by reducing the volume of calculations, discretewavelet transform (DWT) has been defined to analyze the signalswith a smaller set of scales and specific number of translations ateach scale (Soman & Ramachandran, 2004), as efficient as CWTwith identical accuracy. In DWT, the signals are decomposed intoa hierarchical structure of detail and approximations at limited lev-els as follows:

f ðtÞ ¼Xi¼j

i¼1

DiðtÞ þ AjðtÞ ð5Þ

where Di(t) denotes the wavelet detail and Aj(t) stands for the wave-let approximation at the jth level.

Wavelet packet analysis (Avci, Turkoglu, & Poyraz, 2005), ageneralization of wavelet decomposition offering a richer rangeof possibilities for signal analysis, is based upon DWT. Every wave-let detail component is further decomposed to get their own

0,0

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Fig. 11. Wavelet packet coefficients in one sample broken-tooth signal in level 4 (decomposition tree and standard deviation of the coefficients in title).


approximation and detail components as shown in Fig. 5. Waveletpacket includes a set of linearly combined usual wavelet functions.Therefore, wavelet packet inherits the attributes of their corre-sponding wavelet functions such as orthonormality and time–fre-quency localization. In this research, after synchronizing thevibration signals, wavelet packet has been used to pre-processthe signals. A wavelet packet is a function with three indices ofintegers i, j and k which are the modulation, scale and translationparameters, respectively,

wij;kðtÞ ¼ 2j=2wjð2jt � kÞ; i ¼ 1;2;3; . . . ð6Þ

The wavelet functions wj are determined from the following recur-sive equations:

w2jðtÞ ¼ffiffiffi2p X1

�1hðkÞwið2t � kÞ ð7Þ

w2jþ1ðtÞ ¼ffiffiffi2p X1

�1gðkÞwið2t � kÞ ð8Þ

The original signal f(t) after j level of decomposition are defined asfollows:

f ðtÞ ¼X2j

i¼1

f ij ðtÞ ð9Þ

while the wavelet packet component signal f ij ðtÞ are stated by a lin-

ear combination of wavelet packet functions wij;kðtÞ as follows:

f ij ðtÞ ¼

X1k¼�1

cij;kðtÞw

ij;kðtÞ ð10Þ

where the wavelet packet coefficients cij;kðtÞ are calculated by:

cij;kðtÞ ¼

Z 1

�1f ðtÞwi

j;kðtÞdt ð11Þ

providing that the wavelet packet functions satisfy theorthogonality:

wmj;kðtÞw

nj;kðtÞ ¼ 0 if m–n ð12Þ

In this paper, standard deviation of wavelet packet coefficients(Rafiee et al., 2007) has been proposed to identify the faults as thefeature vector for ANN training.

The wavelet-based techniques are absolutely dependent uponthe mother wavelet function. Ingrid Daubechies invented what iscalled ‘compactly supported orthonormal wavelets’– thus makingdiscrete wavelet analysis practical. In machine condition monitor-ing, Daubechies family functions are often selected for signal anal-ysis and synthesis arbitrarily by trial and error (Wang andMcFadden, 1995; Samanta and Al-Balushi, 2003; Tse et al., 2004;Liu, 2005; Peng et al., 2005; Kar and Mohanty, 2006; Saravananet al., 2007; Rafiee et al., 2007. In other words, there is no compu-tational logic behind the selection of Daubechies order which istherefore considered to be optimized for fault classification in thisresearch. It is of paramount significance to note that calculating of

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Fig. 12. Wavelet packet coefficients in one sample broken-tooth signal in level 4 (decomposition tree and standard deviation of the coefficients in title).


the wavelet coefficients is directly dependent on the shape of themother wavelet such that the correlation between mother waveletand signal is calculated as wavelet coefficients. Consequently, thestandard deviation of wavelet coefficients (Rafiee et al., 2007) isamong ideally suited features for fault detection and classification.As demonstrated in Fig. 6, the time and frequency contents of thesignals have not been lost with down-sampling of the lengthy sig-nals into 16 sub-signals.

In Daubechies family functions, the difference between twowavelet functions with adjacent order (e.g. DB5 and DB6) is slightand they are similar to each other as depicted in Fig. 1. Therefore,the research has been focused more on a novel technique not onlyto eliminate the mentioned shortcoming but also to acquire themore efficient ANN-based system by proposing an accurate featurevector.

2.3. Artificial neural networks

Artificial neural networks (Haykin, 1998) or parallel-distrib-uted-processing systems are made of simple processing units,called neurons which are a rough imitation of architecture of bio-logical nervous system. ANNs have been shown to be suited tomodel highly complex and nonlinear phenomena. ANNs are alsouseful in machine condition monitoring since they can learn thesystem’s normal operating conditions and determine if incomingsignals are significantly different as it is handy when there is lots

of training data available and can manage the imprecision of con-dition monitoring process thanks to adaptability dynamism. Unlikethe classic digital-processing techniques, ANNs possess the abilityto perform parallel processing of data, cope with noisy data andadapt to different circumstances.

The most popular neural network is the multi-layer perceptron,which is a feed-forward network and frequently exploited in faultdetection and diagnosis systems, which has found an immensepopularity in machine condition monitoring applications as it con-stitutes more than 90% (Bartelmus, Zimroz, & Batra, 2003) of thecurrent ANNs in the field, and therefore, in this study, has beenused as the base of failure classifier system. The feed-forward net-work first learns through already existing data by iterativelychanging its weights using the ‘back propagation’ of error algo-rithm which is performed by iteratively seeking to minimize theroot-mean-square-error (RMSE) using the gradient search tech-nique (Kwak & Ha, 2004).

An important network design issue is the selection and imple-mentation of the network configuration. In the proposed system,the number of nodes in hidden layer, usually determined bytrial-and-error methods in several papers, has been optimizedusing GA to provide the maximum throughput. Hyperbolic tangentsigmoid transfer function was applied to all net layers with Resil-ient Backpropagation training algorithm (Riedmiller & Braun,1993). A matrix of a data set with 800 processed elements, com-prising 5 � 4 � 40 sampled data for every four gear conditions

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(4,13) (4,14) (4,15)(4,12)

NG

NG

NG NGSW

SWSWSW

MW

MW

MW

MW

BT BT BT BT

Fig. 13. Standard deviation of wavelet packet coefficients for 90 segmented sample signals in each condition (decomposition level: 4, gearbox conditions: normal gearbox,slight-worn, medium-worn, broken-tooth).


and five gearbox configurations, was applied to the network as thefeature vector (Rafiee et al., 2007).

The initial weights were, in addition, obtained randomly in arange of (�10, 10). Error function was chosen to be least meansquare. In many applications, training the network to perform faultdetection system is performed off-line (Chow & Sharpe, 1993). Theobtained signals for 10 configurations were divided into twogroups; each used for training and testing of the proposed ANN.Fig. 7 depicts schematically the utilized ANN structure.

2.4. Genetic algorithm

Genetic algorithms have become popular following Holland’swork (Holland, 1975). GAs consisting of continuous and binaryforms are designed to efficiently search huge, non-linear, discreteand poorly understood search spaces, where expert knowledge isscarce or difficult to model and traditional optimization tech-niques has failed (He et al., 2001). Typically, a simple GA consistsof three operations: (1) parent selection, (2) crossover, and (3)mutation.

In this research, Roulette wheel selection scheme has beenapplied among the selection operators (Wong & Nandi, 2004).Two-point crossover was used for each chromosome of thechromosome-pair having a 50% chance of selection, the twoparents selected for crossover exchange information lying be-tween two randomly generated points within the binarystring.

In addition that most of the successful applications of GAs(Tang, Man, Kwong, & He, 1996) is greatly dependent on findinga suitable method for encoding the chromosome, the creation ofa fitness function to rank the performance of a specific chromo-some is also of paramount importance for the success of the train-ing process. The genetic algorithm rates its own performancearound that of the fitness function; consequently, if the fitnessfunction does not adequately take account of the desired perfor-mance features, the genetic algorithm is unable to meet the de-served requirements of the user.

The proposed chromosome includes five genes for Daubechiesmother wavelet function, two genes for decomposition level of sig-nals, and five genes pertain to the number of neurons of hidden

Standard Deviation of Wavelet Packet Coefficients

30 s

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2 4 6 8 10 12 14 16

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Fig. 14. Standard deviation of WPC of 30 sample signals in each gearbox condition with different Daubechies order (gearbox conditions: normal gearbox, slight-worn,medium-worn, broken-tooth).


layer which is depicted in Fig. 8. Furthermore, the following linearequation was used as the fitness function F.

F ¼ a � pþ b � t ð13Þ

where a and b are the weights, p and t denote network perfor-mance and training time, respectively. In general, the networkperformance which is directly related to the accuracy of the net-work outweighs the training time constraint due to the signifi-cance of the system accuracy and off-line attribute of thenetwork in the fault detection and diagnosis systems. Therefore,to optimize the system, network performance has more impacton fitness function than training time; the weights were appliedwhere a was assumed to be 0.7 vs. �0.3 which was assigned to b.As the more training time has the more negative impact on thewhole fault identification system negative sign has been consid-ered for b in fitness function. In simple words, the network per-formance and training time have inverse proportion on the fitnessfunction. Depending on different case studies, these values forconstant coefficients a and b may deliberately be selected. Forexample, perhaps the training time may not be very importantfor a fault identification system having offline training process.Accordingly, the weight of training time can verge to be zero inthat circumstance.

To sum up, GA was used to optimally search Daubechies order,decomposition level of the signals, and the number of neurons in

hidden layer. Fig. 9 depicts the whole scheme of the fault identifi-cation system. Tables 1 and 2 explore, respectively, the parametersand variables of the GA exploited in the research.

3. Results and discussion

In this paper, by running GA, DB11, level 4 and 14 neuronshave been selected as the best values for Daubechies order,decomposition level, and the number of nodes in hidden layer,respectively. Decomposition tree of wavelet packet has been de-picted in Fig. 10 and related wavelet coefficients of two samplesignals for normal gearbox and broken-tooth gear have beenillustrated in Figs. 11 and 12. With close scrutiny to the ampli-tude of the wavelet packet coefficients, it has been shown thatthere is a considerable correlation between DB11 and the failurecondition (broken-tooth gear). The values of standard deviationof wavelet packet coefficients in 16 sub-signals are mostly high-er than those of the normal one. On the other hand, as shown inFig. 13, the fluctuations of the standard deviations in differentsample segmented signals are considerable and this is the mostappropriate factor for training the neural networks. Stability andconvergence of the network are directly dependent on the distin-guished feature vector with enough samples in ample ranges. Asa matter of fact, the range of the samples in feature vector ismore important than the number of samples in training that

Table 3Various combinations of the three optimized parameters and related results

Wavelet functionorder

Decompositionlevel

No. of hidden-layerneurons

Samples fortesting

Slight-worn

Medium-worn

Broken-tooth

Faultless Best netstructure

1 DB 8 3 13 5 � 4 � 75 94 90 98.06 96.66 8:10:42 DB 9 3 15 5 � 4 � 75 93.33 92.66 97.13 92.33 8:12:43 DB 10 3 13 5 � 4 � 75 95.66 93.33 97.33 95.66 8:10:44 DB 11 3 13 5 � 4 � 75 98.53 97.66 99.33 97.66 8:10:45 DB 12 3 15 5 � 4 � 75 97.33 96.06 98.2 96.73 8:12:46 DB 13 3 15 5 � 4 � 75 93.33 95.33 98.06 99 8:12:47 DB 8 4 17 5 � 4 � 75 94.33 100 99.66 96 16:14:48 DB 9 4 19 5 � 4 � 75 92 98.46 99.13 94.66 16:16:49 DB 10 4 17 5 � 4 � 75 100 98.66 100 93.33 16:14:4

10 DB 11 4 14 5 � 4 � 75 100 100 100 100 16:11:411 DB 12 4 15 5 � 4 � 75 100 100 100 100 16:13:412 DB 14 4 14 5 � 4 � 75 97.86 100 99 100 16:14:413 DB 8 5 24 5 � 4 � 75 97.33 96.86 100 98.66 32:21:414 DB 9 5 25 5 � 4 � 75 90.66 92 99.4 97.53 32:22:415 DB 10 5 25 5 � 4 � 75 96 100 100 98.66 32:22:416 DB 11 5 23 5 � 4 � 75 100 100 100 100 32:21:417 DB 12 5 23 5 � 4 � 75 100 96 100 100 32:23:418 DB 13 5 23 5 � 4 � 75 90.66 100 100 94.66 32:23:4

Table 4Various arrangements of the three optimized parameters with a constant number of neurons and the related results

Wavelet functionorder

Decompositionlevel

No. of hidden-layerneurons

Samples fortesting

Slight-worn

Medium-worn

Broken-tooth

Faultless Netstructure

1 DB 8 4 14 5 � 4 � 75 97 93.33 100 98.66 16:11:42 DB 9 4 14 5 � 4 � 75 96.33 97.33 100 100 16:11:43 DB 10 4 14 5 � 4 � 75 98.66 100 100 100 16:11:44 DB 11 4 14 5 � 4 � 75 100 100 100 100 16:11:45 DB 12 4 14 5 � 4 � 75 100 100 100 100 16:11:46 DB 13 4 14 5 � 4 � 75 96 100 100 100 16:11:47 DB 8 5 14 5 � 4 � 75 89.66 91 96.66 95 32:11:48 DB 9 5 14 5 � 4 � 75 94.33 89.66 95.33 96.66 32:11:49 DB 10 5 14 5 � 4 � 75 95.06 97 98.66 100 32:11:4

10 DB 11 5 14 5 � 4 � 75 95 98.66 96.66 98 32:11:411 DB 12 5 14 5 � 4 � 75 95.66 94 96.33 96.66 32:11:412 DB 13 5 14 5 � 4 � 75 93.66 91.66 96.66 94 32:11:4


the standard deviation of wavelet coefficients has fulfilled theneed. Fig. 14 clearly shows the difference between different or-ders of Daubechies to discriminate the failures. As it can be seen,the failures and normal condition can be clearly discriminate inall of them, and it is related to the appropriate feature vector.Nonetheless, the broken-tooth failure is not as clear as the oth-ers in DB2 even though there is enough severity in the fault. Be-sides, the uniformity among the sample signals in each conditionis not as satisfactory as DB11. In DB15, the drawbacks are thesame as DB2 but more appropriate results. DB20 is not a worthycase for neural network because in many level there is no differ-ences among the failures and normal condition. Most signifi-cantly, the preciseness of discriminating the faults can not becompared with DB11. In DB20, the slight- and medium-worngear, which is more important for us than the broken-toothone are not distinguished as easily as DB11 especially in incipi-ent stages of the fault. Consequently, Daubechies wavelet func-tion with the order of 11 was selected as the most properfunction by GA for fault classification. It means that for differentwavelet-based methods, the selection of mother wavelet func-tion is mandatory to increase the performance of the systemalthough as mentioned in this research it may be possible tohave enough system performance with a specific mother waveletfunction.

To validate the estimated values, the performance of the neu-ral network with three above-mentioned parameters has been

also calculated manually with several trials and errors. Relatedresults which are matched with the estimation of GA have beenshown in Table 3. Although, the network structure is compact indecomposition level of 3 (due to 8-element network input), obvi-ously no satisfactory results were obtained. In spite of the higherperformance and better results, the decomposition level of 5leads to a larger number of neurons of the hidden layer (32-ele-ment network input) which leads to the complexity of the sys-tem. Therefore, fourth decomposition level was determined tobe the most appropriate level for this case study because ofthe better outcomes than those of the others and the introduc-tion of a well-formed network structure. The data in Table 3has been calculated for each case without genetic algorithm. Inaddition, the optimal number of the neurons of the hidden layerwas determined using several trials and errors runs for each casein Table 3.

Hence, the end results obtained using genetic algorithm notonly does the high capabilities of this algorithm prove in orderto find the optimal solution but also verifies the soundness ofthe parameters for the adjustment of the algorithm presentedin Table 1.

Table 4 puts forward the results of a diagnostics system withoutusing GA and manually with 14 neurons in hidden layer with dif-ferent function orders and decomposition levels to better clarifythe aptness of the results of the previous case (DB11 with a decom-position level of 4).

10

1010110101

Synchronization by P.C.H.I.

Fitness Function

DecompositionLevel

DB order

No. ofNeurons

10

1010110101

. .F p tα β= +

Genetic Algorithm

Crossover Mutation

GenerationSelection

END

Yes

No StoppingCriteria Fulfilled

SegmentationRaw signal

Fig. 15. Schematic of the entire proposed algorithm.


4. Conclusion

An intelligent gear fault identification system was developedand implemented to pinpoint the gearbox faults; Three parameterswere recognized to be of paramount significance to be optimizedusing GA i.e. Daubechies wavelet function order, decomposition le-vel, and number of neurons in hidden layer of ANN, which play akey role in size and performance of the network and the soundnessof the feature vector. The Feature vector was also obtained fromoptimized standard deviation of wavelet packet coefficients ac-quired from DB11 at fourth level of decomposition following apiecewise cubic hermite interpolation for synchronizing. Eventu-ally, a MLP network with well-formed and optimized structure(16:14:4) and remarkable accuracy was presented providing thecapability to identify slight-worn, medium-worn and broken-toothof gears faults perfectly in 100% of the five tested configurations ofgearbox. By the proposed system which has been elaborately de-picted in Fig. 15, the drawback of implementing mother waveletfunction through trial-and-error based methods has been also im-proved for fault classification purposes.

Acknowledgements

This work described in this paper was partially supported byVibration and Modal Analysis Lab at University of Tabriz and par-tially supported by a grant from the Research Grants Council ofHong Kong Special Administrative Region, China (Project no. CityU120506). The authors would like to write in memoriam of two ded-icated mentors, Professor Vahhab Pirouzpanah and Professor Sa-mad Nami Notash who devoted more than four decades toenlighten a myriad of students. They would also like to thank the

anonymous reviewers for spending their valuable time to reviewthe current research.

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