Vibration-basedfaultdiagnosisofspurbevelgearboxusingfuzzytechniqueN.Saravanan*,S.Cholairajan,K.I.RamachandranDepartmentofMechanicalEngineering,AmritaVishwaVidyapeetham,Coimbatore,TamilNadu641105,IndiaAbstractTo determine the condition of an inaccessible gear in an operating machine the vibration signal of the machine can be continuouslymonitoredbyplacingasensorclosetothesourceofthevibrations.Thesesignalscanbefurtherprocessedtoextractthefeaturesandidentifythestatusof themachine. Thevibrationsignal acquiredfromtheoperatingmachinehasbeenusedtoeectivelydiagnosethe condition of inaccessible moving components inside the machine. Suitable sensors are kept at various locations to pick up the signalsproducedbymachineryandthesesignalsareverymeaningfulinconditiondiagnosissurveillance.Todeterminetheimportantcharac-teristicsandtounravelthesignicanceofthesesignals,furtheranalysisorprocessingisrequired.This paper presents the use of decision tree for selecting best statistical features that will discriminate the fault conditions of the gearbox from the signals extracted. These features are extracted from vibration signals. A rule set is formed from the extracted features andfedtoafuzzyclassier.Therulesetnecessaryforbuildingthefuzzyclassierisobtainedlargelybyintuitionanddomainknowledge.This paper also presents the usage of decision tree to generate the rules automatically from the feature set. The vibration signal from apiezo-electrictransduceriscapturedforthefollowingconditionsgoodbevelgear,bevelgearwithtoothbreakage(GTB),bevelgearwith crack at root of the tooth (GTC), and bevel gear with face wear of the teeth (TFW) for various loading and lubrication conditions.The statistical features were extracted and good features that discriminate the dierent fault conditions of the gearbox were selected usingdecision tree. The rule set for fuzzy classier is obtained by once using the decision tree again. A fuzzy classier is built and tested withrepresentativedata.Theresultsarefoundtobeencouraging.2008ElsevierLtd.Allrightsreserved.Keywords: Featureselection;Statisticalfeatures;Decisiontree;Gearbox;Fuzzy;Faultdetection1.IntroductionAfaultygearsystemcouldresult inseriousdamageifdefectsoccurtooneofthegearsduringoperationcondi-tion.Earlydetectionofthedefects,therefore,iscrucialtoprevent the system from malfunction that could cause dam-age or entire systemhalt. Diagnosingagear systembyexamining the vibration signals is the most commonly usedmethod for detecting gear failures. The conventional meth-ods for processingmeasureddatacontainthe frequencydomain technique, time domain technique, and timefrequency domain technique. These methods have beenwidelyemployedtodetectgearfailures.Theuseofvibra-tionanalysisforgearfault diagnosisandmonitoringhasbeenwidelyinvestigatedanditsapplicationinindustryiswell established (Cameron &Stuckey, 1994; Gadd &Mitchell, 1984; Leblanc, Dube, &Devereux, 1990). Thisisparticularlyreectedintheaviationindustrywherethehelicopterengine,drivetrainsandrotorsystemsarettedwithvibrationsensorsforcomponent healthmonitoring.Therawvibrationsignalinanymodefromasinglepointon a machine is not a good indicator of the health or con-dition of a machine. Vibration is a vectorial parameter withthreedimensions andrequires tobemeasuredat severalcarefullyselectedpoints.0957-4174/$-seefrontmatter 2008ElsevierLtd.Allrightsreserved.doi:10.1016/j.eswa.2008.01.010*Correspondingauthor.Tel.:+914222656422;fax:+914222656274.E-mail addresses: n_saravanan@ettimadai.amrita.edu, nsaro_2000@yahoo.com(N.Saravanan).www.elsevier.com/locate/eswaAvailable online at www.sciencedirect.comExpertSystemswithApplications36(2009)31193135Expert Systemswith ApplicationsVibration analysis can be carried out using FouriertransformtechniqueslikeFourierseriesexpansion(FSE),Fourier integral transform(FIT) and discrete Fouriertransform (DFT)(Collacott, xxxx). After the developmentoflarge-scaleintegration(LSI) andtheassociatedmicro-processortechnology, fast Fouriertransform(FFT) ana-lyzersbecamecost eectivefor general applications. Theraw signatures acquired through a vibration sensor neededfurther processingandclassicationof the datafor anymeaningful surveillance of the condition of the systembeingmonitored.Articial neural network (ANN), support vectormachine (SVM) andFuzzyclassier are widelyusedasclassication tool and reported in literature (Burgess,1998; Jack&Nandi, 2000a; Nandi, 2000; Samanta&Al-Baulshi, 2003; Samanta, Al-Baulshi, &Al-Araimi, 2003;Shi et al., 1988). Amongthem, ANNhas limitations ongeneralizationof the results inmodels that canover tthe data (Samanta et al., 2003). SVM has high classicationaccuracy and good generalization capabilities for crisp data(Burgess, 1998;Jack&Nandi,2000a;Shi etal., 1988).Intheproblemathand,thenatureofthefaultitselfisfuzzyin nature. Fuzzy classier models the physical problemunder study more closely. The ow chart of the fault diag-nosticsystemisshowninFig.1.1.1.DierentphasesofpresentworkThesignalsobtainedareprocessedfurtherformachineconditiondiagnosisasexplainedintheowchartFig.1.2.ExperimentalstudiesThefaultsimulatorwithsensorisshowninFig. 2andthe pinionandgear showninFig. 3. Avariable speedDCmotor(0.5 hp)withspeedupto3000 rpmisthebasicdrive. Ashortshaftof30 mmdiameterisattachedtotheshaft of themotorthroughaexiblecoupling; thisistominimize eects of misalignment and transmission of vibra-tionfromthe motor. The shaft is supportedat its endsthroughtworollerbearings. Fromthisshaft thedriveistransmitted to the bevel gear box by means of a belt drive.The gear box is of dimension 150 mm 170 mm 120 mmand the full lubrication level is 110 mm and half lubricationlevelis60 mm.SAE40 oilwas used as a lubricant. An electromagneticspring-loaded disc brake was used to load the gear wheel. Atorquelevel of 8 N-mwasappliedat thefull-loadcondi-tion. Thevariousdefectsarecreatedinthepinionwheelsand the mating gear wheel is not disturbed. With the sensormountedontopof the gear box vibrations signals areobtainedforvariousconditions. Theselectedareaonthetopof thegearboxfor mountingthesensor is madeatand smooth to ensure eective coupling between the sensorand the gearbox. The sensor used is a piezoelectric acceler-ometer (Dytran model) which is mounted on the atVibration Signals Feature Selection Using J 48 Algorithm Rule Generation Test data set Modeling Fuzzy systemFuzzy inference engine Fuzzy output Machine Condition Diagnosis Fig.1. Flowchartforbevelgearboxhealthdiagnosis.BevelGear Box SensorFig.2. Faultsimulatorsetup.PinionWheelGearWheelElectromagneticspring loaded disc brakeFig.3. Innerviewofthebevelgearbox.3120 N.Saravananetal. / ExpertSystemswithApplications36(2009)31193135surface using direct adhesive mounting technique. Theaccelerometerisconnectedtothesignal-conditioningunit(DACTRAN FFT analyzer), where the signal goes throughthechargeamplierandananalogue-to-digital converter(ADC). Thevibrationsignal indigital formisfedtothecomputer through a USB port. The software RT Pro-seriesthat accompaniesthesignal-conditioningunit isusedforrecordingthesignalsdirectlyinthecomputerssecondarymemory. The signal is then read from the memory and pro-cessedtoextractdierentfeatures.2.1.ExperimentalprocedureInthepresent study, fourpinionwheelswhosedetailsareas mentionedinTable1wereused. Onewas anewwheel and was assumed to be free from defects. In the otherthree pinion wheels, defects were created using electron dis-chargemachine(EDM) inorder tokeepthesize of thedefect under control. The details of the various defectsaredepictedinTable2anditsviewsareshowninFig.4.Thesizeofthedefectsisin-linewithworkreportedinliterature (Gadd&Mitchell, 1984). The vibrationsignalfromthepiezoelectricpickupmountedonthetestbearingwas taken, after allowing initial running of the bearing forsometime. The sampling frequency was 12,000 Hz andsamplelengthwas8192forall speedsandall conditions.The sample length was chosen arbitrarily, however, the fol-lowing points were considered. Statistical measures aremoremeaningful, whenthenumber of samples is more.Onthe other hand, as the number of samples increasesthe computation time increases. To strike a balance, samplelength of around 10000 was chosen. In some feature extrac-tion techniques, which will be used with the same data, thenumber of samples is tobe2n. Thenearest 2n10,000is8192 and hence, it was taken as sample length. Many trialswere taken at the set speed and vibration signal was storedTable1DetailsoffaultsunderinvestigationGears Faultdescription Dimension(mm)G1 Good G2 Geartoothbreakage(GTB) 8G3 Gearwithcrackatroot(GTC) 0.8 0.5G4 Gearwithfacewear 0.5Table2GearwheelandpiniondetailsParameters Gearwheel PinionwheelNo.ofteeth 35 25Module 2.5 2.5Normalpressureangle() 20 20Shaftangle() 90 90Topclearance(mm) 0.5 0.5Addendum(mm) 2.5 2.5Wholedepth(mm) 5.5 5.5Chordaltooththickness(mm) 3.930.1503.920.110Chordaltoothheight(mm) 2.53 2.55Material EN8 EN8a bcFig.4. (a)Viewofgoodpinionwheel.(b)Viewofpinionwheelwithfacewear(GFW).(c)Viewofpinionwheelwithtoothbreakage(GTB).N.Saravananetal. / ExpertSystemswithApplications36(2009)31193135 3121in the data le. The raw vibration signals acquired for var-iousexperimentalconditionsformthegearboxusingFFTareshowninFig.5.3.FeatureextractionStatistical analysis of vibrationsignals yields dierentprimary and secondary Parameters. Research workreported (James Li & Wu, 1989) use these in combinationstoelicitinformationregardingbearingfaults.Suchproce-dures use alliedlogic oftenbasedonphysical Consider-ations.Afairlywidesetoftheseparametersisselectedasabasisforourstudy,asdetailedbelow.a.Meanb.Standarderrorc.Mediand.Standarddevi-atione. Samplevariance f.Kurtosisg. Skewnessh. Rangei.Minimumj.Maximumk.Sum.All the above mentioned statistical features wereextractedfor the vibration signals obtained for variousconditions and fed as an input to J 48 algorithm for select-ing the best features which classify the dierent faultconditions.4.DescriptivestatisticsThestatisticalfeaturesareexplainedbelow.4.1.StandarddeviationThisisameasureoftheeectiveenergyorpowercon-tentofthevibrationsignalandclearlyindicatesdeteriora-tioninthebearingcondition. ThefollowingformulawasusedforcomputationofstandarddeviationStandarddeviation nPx2 Px 2nn 1s:0 2000 4000 6000 8000-0.4-0.200.2Sample No.AmplitudeGood-Dry-Unload0 2000 4000 6000 8000-0.4-0.200.2Good-Dry-FullLoadSample No.Amplitude0 2000 4000 6000 8000-0.4-0.200.2Good-HalfLub-UnloadSample No.Amplitude0 2000 4000 6000 8000-0.200.2Good-HalfLub-FullLoadSample No.Amplitude0 2000 4000 6000 8000-0.200.2Good-FullLub-UnloadSample No.Amplitude0 2000 4000 6000 8000-0.200.2Good-Full-FullLoadSample No.Amplitude0 2000 4000 6000 8000-0.4-0.200.2Sample No.AmplitudeGTB-Dry-Unload0 2000 4000 6000 8000-0.4-0.200.2GTB-Dry-FullLoadSample No.Amplitude0 2000 4000 6000 8000-0.4-0.200.2GTB-HalfLub-UnloadSample No.Amplitude0 2000 4000 6000 8000-0.4-0.200.2GTB-HalfLub-FullLoadSample No.Amplitude0 2000 4000 6000 8000-0.4-0.200.2GTB-FullLub-UnloadSample No.Amplitude0 2000 4000 6000 8000-0.4-0.200.2GTB-FullLub-FullLoadSample No.Amplitude0 2000 4000 6000 8000-0.100.1Sample No.AmplitudeGTC-Dry-Unload0 2000 4000 6000 8000-0.100.1GTC-Dry-fullLoadSample No.Amplitude0 2000 4000 6000 8000-0.100.1GTC-HalfLub-UnloadSample No.Amplitude0 2000 4000 6000 8000-0.100.1GTC-HalfLub-FullLoadSample No.Amplitude0 2000 4000 6000 8000-0.100.1GTC-FullLub-UnloadSample No.Amplitude0 2000 4000 6000 8000-0.100.1GTC-FullLub-FullLoadSample No.Amplitude0 2000 4000 6000 8000-0.200.2SampleAmplitudeTFW-Dry-Unload0 2000 4000 6000 8000-0.200.2TFW-Dry-FullLoadSampleAmplitude0 2000 4000 6000 8000-0.200.2TFW-HalfLub-UnloadSampleAmplitude0 2000 4000 6000 8000-0.200.2TFW-HalfLub-FullLoadSampleAmplitude0 2000 4000 6000 8000-0.200.2TFW-FullLub-UnloadSampleAmplitude0 2000 4000 6000 8000-0.200.2TFW-FullLub-FullLoadSampleAmplitudeabcdFig.5. (a)Vibrationsignalforgoodpinionwheelunderdierentlubricationandloadingconditions. (b)Vibrationsignalforpinionwheelwithteethbreakageunderdierent lubricationandloadingconditions. (c)Vibrationsignal forpinionwheel withcrackatrootunderdierent lubricationandloadingconditions.(d)Vibrationsignalsforpinionwheelwithteethfacewearunderdierentlubricationandloadingconditions.3122 N.Saravananetal. / ExpertSystemswithApplications36(2009)311931354.2.SkewnessSkewness characterizes the degree of asymmetryof adistribution around its mean. The below shown expressionwas used to calculate the skewness, where n is the samplesizeandsisthesamplestandarddeviationSkewness nn 1n 2Xxi xs 3:4.3.KurtosisKurtosis indicates the atness or the spikiness of the sig-nal.ItsvalueisverylowforgoodbevelgearboxandhighforfaultygearboxduetothespikynatureofthesignalKurtosis nn 1n 1n 2n 3Xxi xs 4( )3n 12n 2n 3 :wheresisthesamplestandarddeviation.4.4.StandarderrorStandarderrorisameasureoftheamountoferrorinthepredictionof yforanindividual xintheregression,wherexandyarethesamplemeansandnisthesamplesize.Standarderrorofthepredictedy 1n 2Xy y2Px xy y 2Px x2" #vuut:4.5.SamplevarianceIt is variance of the signal points and the following for-mulawasusedforcomputationofstandardvariance.Samplevariance nPx2 Px 2nn 1:4.6.RangeIt refers to the dierence in maximum and minimum sig-nalpointvaluesforagivensignal.4.7.MinimumvalueItreferstotheminimumsignal point valueinagivensignal. Asthegearparts(crack, breakage, facewear)getdegraded,thevibrationlevelsseemtogohigh.Therefore,itcanbeusedtodetectfaultygears.4.8.MaximumvalueItreferstothemaximumsignal pointvalueinagivensignal.4.9.SumItisthesumofallsignalpointvaluesinagivensignal.5.UsingJ48algorithminthepresentworkAstandardtreeinducedwithc5.0(orpossiblyID3orc4.5) consists of a number of branches, one root, a numberof nodes and a number of leaves. One branch is a chain ofnodes from root to a leaf; and each node involves one attri-bute.Theoccurrenceofanattributeinatreeprovidestheinformationabouttheimportanceoftheassociatedattri-bute as explained by Peng, Flach, Brazdil, and Soares(2002). A Decision Tree is a tree based knowledge represen-tationmethodologyusedtorepresent classicationrules.J48algorithm(AWEKAimplementationof c4.5 Algo-rithm) is awidelyusedone toconstruct DecisionTreesas explainedby Sugumaran, Muralidharan, andRama-chandran(2006).The DecisionTree algorithmhas beenappliedtotheproblemunderdiscussion.Inputtothealgorithmissetofstatistical featuresof vibrationsignatures. It isclearthatthetopnodeisthebestnodeforclassication. Theotherfeatures in the nodes of Decision Tree appear in descendingorder of importance. It is to be stressed here that only fea-tures that contribute to the classication appear intheDecision Tree and others do not. Features, which have lessdiscriminatingcapability,canbeconsciouslydiscardedbydecidingonthethreshold. This concept is madeuseforselectinggoodfeatures. Thealgorithmidentiesthegoodfeatures for the purpose of classicationfromthe giventrainingdataset,andthusreducesthedomainknowledgerequiredtoselect goodfeatures for patternclassicationproblem. ThedecisiontreesshowninFigs. 7, 13, 18, 22,27and31areforvariouslubricationandloadingcondi-tionsofdierentfaultscomparedwithgoodconditionsofthepiniongearwheel.Based on the output of J 48 algorithm, the decision treevariousstatistical parametersareselectedforthevariousconditionsof thegearbox. Thevaluesappearingbetweenvariousnodesinthedecisiontreeareusedforgeneratingthe fuzzy rules toclassifythe various conditions of thegearboxunderstudy.5.1.ApplicationofdecisiontreeforfeatureselectionThealgorithmhasbeenappliedtotheproblemunderdiscussionforfeatureselection. Input tothealgorithmisthe set of statistical features described in Section 3extractedfromrawvibrationsignatures, theoutputistheDecisionTree. It iscleartherefromthatthetopnodeisthebestnodeforclassication.TheotherfeaturesappearN.Saravananetal. / ExpertSystemswithApplications36(2009)31193135 3123in the nodes in Decision Tree in descending order of impor-tance.Itistobestressedherethatonlyfeaturesthatcon-tribute tothe classicationappear inthe DecisionTreeandothersdonot. Thelevel of contributionisnot sameandall statistical featuresarenotequallyimportant. Thelevel of contribution by individual feature is given by a sta-tistical measure within the parenthesis in the Decision Tree.The rst number in the parenthesis indicates the number ofdata points that can be classied using that feature set. Thesecondnumber indicates the number of samples againstthisaction. Iftherstnumberisverysmall comparedtothetotal numberofsamples, thenthecorrespondingfea-tures can be considered as outliers and hence ignored. Fea-tures that have less discriminating capability can beconsciouslydiscardedbydecidingonthethreshold. Thisconcept is made use of in selecting good features. The algo-rithm identies the good features for the purpose of classi-cationfromthegiventrainingdatasetandthusreducesthe domain knowledge required to select good featuresforpatternclassicationproblem.6.MethodologyadoptedforfuzzyclassicationSeeFig.6.7.Fuzzylogic(classier)Fuzzy Logic provides a precise approachfor dealingwithuncertainty. Fuzzyinferenceisamethodthat inter-prets thevalues intheinput vector and, basedonsomeset of rules, assigns values to the output vector. Thepoint of fuzzylogicistomapaninput spacetoanout-putspace, andtheprimarymechanismfordoingthisisalist of if-then statements called rules. Rules are theinputsforbuildingafuzzyinferenceengine.Themethod-ologyadoptedforfuzzyclassicationisshowninFig. 6.All rules are evaluatedinparallel, andthe order of therules is unimportant. The real worlddata donot havesharply dened boundaries where information is oftenincomplete or sometimes unreliable. In quest for preci-sion, scientists have generally attempted to manipulatethe real world into articial mathematical models thatmake noprovisionfor gradation. Because Fuzzy Logicprovides the tools to classify information into broad,coarsecategorizations or groupings, it has innitepossi-bilities for application which have proven to be muchcheaper, simpler and more eective than other systemsinhandlingcomplexinformation(Cox, 1994).Fortheproblemathand,theconditionofthegearbox,good or faulty is basically fuzzy in nature. All the faults donotoccurinthegearboxinstantly.Itcomesgradually. Inthat case, there is nothresholdvalue (crispdata) basedonwhichthe decisiononthe conditionof the gearboxcanbetaken. (Whether gearboxis nowgoodor faulty).The problems of this kind can be modeled using fuzzy logicmore closely (Huang, 1997; Rao, 1996; Zeng &Wang,1991).8.MembershipfunctionA membership function (MF) is a curve that denes howeachpointintheinputspaceismappedtoamembershipvalue (or degree of membership) between 0 and 1. Observ-ingthevaluesofthefeature,basedonwhichthebranchesof the Decision Tree are created for dierent conditions ofthe gearbox, the membership functions for the correspond-ingfeaturesaredened.Therearefourpossibleoutcomesfrom a fuzzy classier, namely: good bevel gear, bevel gearwithtoothbreakage(GTB),bevelgearwithcrackatrootofthetooth(GTC), andbevel gearwithfacewearoftheteeth(TFW) for various loading andlubricationcondi-tions. Hence, fourmembershipfunctionsaredenedwithequalrangefortheoutput.9.RulegenerationfromdecisiontreeArticial neural networkandsupport vector machineare used to generate rule for classication problems(Andrews et al., 1995; Haydemar, Cecilio, &Andreu,xxxx). Inthis study, DecisionTreeis usedfor that pur-pose. DecisionTree shows the relationbetweenfeaturesandtheconditionofthegearbox. Tracingabranchfromtherootnodeleadstoaconditionof thegearbox(ReferFigs. 7, 13, 18, 22, 27and31)anddecodingtheinforma-tionavailableinabranchintheformof if-then state-ment gives the rules for classication using fuzzy forDry/Half/FullNo LoadFull LoadGOOD/GTC/GTB/TFWFig.6. MethodologyofclassicationusingFuzzy. Fig. 7. Decision tree fromJ 48Algorithmfor dry-lubno-load condition.3124 N.Saravananetal. / ExpertSystemswithApplications36(2009)31193135various conditions of the gearbox. Hence the usefulness ofthedecisiontreeinformingtherulesforfuzzyclassica-tionisestablished.10.GenerationofrulesforvariousgearboxconditionsanddiscussionsThe precedingsectiondescribes howthe classicationhasbeencarriedoutusingfuzzytechnique.10.1.Dry-lubricationandno-loadconditionFrom Fig. 7 we can see that standard error, kurtosis andvariance play a decisive role in classifying the various gear-boxfaults under drylubricationandno-loadcondition.This output of the decision tree is used to design the mem-bership function for fuzzyclassierasshownin Fig.810.Amembershipfunction(MF)isacurvethatdeneshoweachpointintheinputspaceismappedtoamembershipFig.8. Membershipfunctionforstandarderror.Fig.9. Membershipfunctionforkurtosis.Fig.10. Membershipfunctionforvariance.N.Saravananetal. / ExpertSystemswithApplications36(2009)31193135 3125value(ordegreeofmembership)between0and1. Inthepresent study, trapezoidal membershipfunctionis used.The selection of this membership function is to some extentarbitrary. However, thefollowingpointswereconsideredwhile selectingmembershipfunction. The DecisionTreefor the selected three features is shown in Fig. 7. Observingthevaluesofthefeature, basedonwhichthebranchesofthe Decision Tree is created, the membershipfunctionsfor all three features are dened for standard error, kurto-sisandvariance,respectively.10.1.1.Rulesdesignedforthedry-lubricationandno-loadconditionFigs.1416.1. If(stderrisnotstderr)then(Output1isGTC).2. If (stderr is stderr) and (kurtosis is Kur) then (Output1 isGOOD).3. If(stderrisstderr)and(kurtosisisnotKur)and(vari-anceisVar)then(Output1isGTB).4. If(stderrisstderr)and(kurtosisisnotKur)and(vari-anceisnotVar)then(Output1isTFW).Themembershipvalueof theconditionbeingGTCiswhenthe standarderror value is less thanor equal to0.000175 (from Fig. 7) which is the threshold value. Hence,uptothisthresholdvaluethemembershipfunctiongener-ates the value 0 and afterwards it increases linearly(assumption). Thetrapezoidal membershipfunctionsuitsthisphenomenonandhenceit wasselectedtomapeachpoint in the input space to a membership value. To review,the threshold values are given by decision tree and the slopeis denedbythe user throughheuristics. The thresholdvalue (0.000175) is dened based on the representativetraining dataset. If standarderror value is less thanorequal to 0.000175, a membership function which is denedon a 01 scale gives a value of 0 which means that it is not astandard error. If threshold value is greater than 0.000175,themembershipfunctiongeneratesavalueof1.Similarlymembership functions for other features are designedaccordinglyandshowninFigs.9and10.There are four possible outcomes from a fuzzy classier,namely: Good, GTC, GTB and TFW Hence, four member-shipfunctionsaredenedwithequal rangeandshowninFig.11.10.1.2.FuzzyinferenceengineAfter dening membership functions and generating theif-thenrules,thenextstepistobuildthefuzzyinferenceengine. ThefuzzytoolboxavailableinMATLAB7wasusedfor buildingfuzzyinference engine. Eachrule wastaken at a time and using membership functions and fuzzyoperatorstheruleswereentered. Theruleswereobtainedfroma training data set (150 trials ineach condition).For testing the built model a portion of the data (100 trialsin each condition) called testing data was kept aside. Usingthetestingdata, thefuzzyinferenceenginewasevaluatedanditsperformancewaspresentedasconfusionmatrixinTable 3. The diagonal elements inthe confusionmatrixtable(3) show the numberof correctly classiedinstances.In the rst row, the rst element shows the number of datapoints belonging togood class andclassiedby fuzzylogicasgood. Thesecondelementshowsthenumberofdata points belonging to GTC class and classied by fuzzylogicas GTC. Thethirdelement shows thenumber ofdata points belonging to GTB class and classied by fuzzylogicasGTB. Thefourthelementshowsthenumberofdata points belonging to TFW class and classied byfuzzylogicasTFW. Table3illustratesthepowerfulnessof the fuzzy rules designed with the aid of the decision treesbytheauthors.Fig.12illustratestheapplicationoftherulesdesigned.Here eachrowcorresponds toeachrule as discussedinFig.11. Membershipfunctionsforcondition.(output).Table3Condition GOOD GTC GTB TFWGOOD 100 0 0 0GTC 0 100 0 0GTB 0 0 100 0TFW 0 0 0 1003126 N.Saravananetal. / ExpertSystemswithApplications36(2009)31193135section6.4.2.Therstthreeblocksinrowsrepresentsthemembership function of standard error, kurtosis, variance,respectively. The fourth block corresponds to the member-shipfunctions for output as showninFig. 11. Withthehelp of sample inputs for standard error, kurtosis and var-iancetherulesaretestedasfollows,forasampleinputofstandarderror as 0.0005, kurtosis as 10andvarianceas0.005whichsatises the secondrule completelyandthecorresponding output condition is GOOD, which is shownintheoutput blockof thesecondrowintheruleviewershowninFig.12.10.1.3.ConfusionmatrixIn lieu with the above discussions the fuzzy rules, mem-bershipfunctions, confusionmatrixandrule viewer areshowninSections10.210.5and10.6.10.2.Dry-lubricationandfull-loadconditionSeeFig.13.10.2.1.Rulesdesignedforthedry-lubricationandfull-loadconditionSeeFigs.1416.1. If(stderrisstderr)then(output1isGTC).2. If (stderris notstderr)and (variance is var3)then(out-put1isGTB).3. If (kurtosis is kur) and (variance is var2) then (output1 isGOOD).4. If(kurtosisisnotkur)and(varianceisvar1)then(out-put1isTFW).Heretherearethreemembershipfunctionstorepresentthreethresholdvaluesofvarianceinthedecisiontree.Fig. 17istheruleviewerforthefollowingtestdata.Ifstderr = 0.0005, kurtosis = 10, Variance = 0.005, thentheoutputis6.25,i.e.,theconditionisGTB.10.2.2.ConfusionmatrixTable4.Fig.12. Ruleviewerforoneofthetestdata.Fig.13. DecisiontreefromJ48Algorithmfordry-lubfull-loadcondition.N.Saravananetal. / ExpertSystemswithApplications36(2009)31193135 312710.3.Half-lubricationandno-loadconditionSeeFig.18.10.3.1.Rulesdesignedforhalf-lubricationandno-loadconditionSeeFigs.1921.1. If(stderrisstderr1)then(output1isGTC).2. If(stderrisnotstderr2)then(output1isGOOD).3. If (stderr is stderr2) and (kurtosis is kur) then (output1 isGTB).4. If(stderrisstderr2)and(kurtosisisnotkur)then(out-put1isTFW).10.3.2.ConfusionmatrixTable5.10.4.Half-lubricationandfull-loadconditionSeeFig.22.10.4.1.Rulesdesignedforhalf-lubricationandfull-loadconditionFigs.2326.Fig.14. Membershipfunctionforstderr.Fig.15. Membershipfunctionforkurtosis.Fig.16. Membershipfunctionsforvariance.3128 N.Saravananetal. / ExpertSystemswithApplications36(2009)311931351. If(stderrisstderr1)then(output1isGTC).2. If(stderrisnotstderr2)then(output1isGOOD).3. If(stderrisstderr2)and(kurtosisisnotkur)then(out-put1isTFW).4. If (stderr is stderr2) and (kurtosis is kur) and (minimumismin)then(output1isGTB).Fig.17. Ruleviewerforoneofthetestdata.Table4Condition GOOD GTC GTB TFWGOOD 100 0 0 0GTC 0 100 0 0GTB 0 0 100 0TFW 0 0 0 100Fig.18. DecisiontreefromJ48Algorithmforhalf-lubno-loadcondition.Fig.19. Membershipfunctionsforstderr.N.Saravananetal. / ExpertSystemswithApplications36(2009)31193135 31295. If (stderr is stderr2) and (kurtosis is kur) and (minimumisnotmin)then(output1isTFW).10.4.2.ConfusionmatrixTable6.10.5.Full-lubricationandno-loadconditionSeeFig.27.10.5.1.Rulesdesignedforfull-lubricationandno-loadconditionFigs.2830.1. If(stderrisstderr1)then(output1isGTC).2. If(stderrisnotstderr2)then(output1isGOOD).3. If(stderrisstderr2)and(varianceisvar)then(output1isGTB).4. If (stderris stderr2) and(varianceis notvar)then(out-put1isTFW).10.5.2.ConfusionmatrixTable7.10.6.Full-lubricationandfull-loadconditionSeeFig.31.10.6.1.Rulesdesignedforfull-lubricationandfull-loadconditionFigs.32and33.1. If(stderrisstderr1)then(output1isGTC).2. If(stderrisnotstderr3)then(output1isGOOD).3. If(stderrisstderr2)then(output1isGTB).4. If(stderrisstderr3)then(output1isTFW).10.6.2.ConfusionmatrixTable8.Fig.20. Membershipfunctionforkurtosis.Fig.21. RuleviewerforoneofthetestdataIfstderr = 0.0005,kurtosis = 10.Theoutputis6.25,hencetheconditionisGTB.Table5Condition GOOD GTC GTB TFWGOOD 100 0 0 0GTC 0 100 0 0GTB 0 0 100 0TFW 0 0 0 1003130 N.Saravananetal. / ExpertSystemswithApplications36(2009)31193135Fig.22. DecisiontreefromJ48Algorithmforhalf-lubfull-loadcondition.Fig.23. Membershipfunctionsforstderr.Fig.24. Membershipfunctionsforkurtosis.N.Saravananetal. / ExpertSystemswithApplications36(2009)31193135 313111.ConclusionFaultdiagnosisofGearboxisoneofthecoreresearchareas in the eld of condition monitoring of rotatingmachines. Theworkconducted, proposingthemethodofFig.25. Membershipfunctionsforminimum.Fig.26. Ruleviewerforoneofthetestdata.IfStderr = 0.0005,kurtosis = 10,minimum = 0,output1 = 6.25,i.e.,theconditionisGTB.Table6Condition GOOD GTC GTB TFWGOOD 100 0 0 0GTC 0 100 0 0GTB 0 0 100 0TFW 0 0 0 100Fig.27. DecisiontreefromJ48Algorithmforfull-lubno-loadcondition.3132 N.Saravananetal. / ExpertSystemswithApplications36(2009)31193135thegearboxfaultidenticationbasedonfuzzylogictech-nique, showsthegreat potentialityandthestrongabilitytoclassifyandidentifymachineryfaults. This workhasinvestigated the use of basic fuzzy logic principle as a faultdiagnostic technique for spur bevel gearbox. The work con-ductedhas demonstratedthe potential of fuzzylogic toFig.28. Membershipfunctionforstderr.Fig.29. Membershipfunctionforvariance.Fig.30. Ruleviewerforoneofthetestdata.Ifstderr = 0.0005,variance = 0.005,thenoutput1 = 6.25.Table7Condition GOOD GTC GTB TFWGOOD 96 0 0 4GTC 0 100 0 0GTB 10 0 90 0TFW 0 0 0 100N.Saravananetal. / ExpertSystemswithApplications36(2009)31193135 3133Fig.31. DecisiontreefromJ48Algorithmforfull-lubfull-loadcondition.Fig.32. Membershipfunctionforstderr.Fig.33. Ruleviewerforoneofthetestdata.Ifstderr = 0.0002thenoutput1 = 1.25,ie,theconditionisGood.3134 N.Saravananetal. / ExpertSystemswithApplications36(2009)31193135classifythelikelyfault conditionswhicharerepresent inrotatingmachinery. Fromthestudycarriedout andpre-sentedinthispaper,thediagnosistechniquebasedonthefuzzy logic principle is found to be practical for the condi-tionrecognitionofthegearbox.Alsothisworkbringsoutthepotential of decisiontreestogeneratetherulesauto-maticallyfromthefeaturesetwhichprovestobeagreatasset in generating fuzzy rules. 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