Using the orthogonal array with grey relational analysis to optimize the laser hybrid welding of a 6

984HRKimandKYLee

convertedintotherangebetweenzeroandoneby

usingalinearnormalizationprocedureandthispro-

cessiscalledgreyrelationalgenerating.Inthispro-

cessthedataaredescribedusingoneofthree

possibledescriptions:‘thelargerthebetter’,‘the

smallerthebetter’,and‘thenominalthebetter’.In

thisstudy,theultimatetensilestrengthisrelatedto

theform‘the-larger-the-better’followingitscharac-

teristic.Intheseexperimentsthenormalizedexperi-

mentalresultsoftheultimatetensilestrengthwere

expressedas

ÃxiðkÞ¼xðkÞÀminxðkÞ

maxxiið1Þ

wherexiðkÞisthekthexperimentalresultintheithÃexperiment,xiðkÞisthenormalizedresultxiðkÞafterthedatapre-processingstep,maxxiðkÞisthelargestvalueofxiðkÞ,andminxiðkÞimpliesthesmallestvalueofxiðkÞ.Alargevalueofthenormalizedresultimpliesagoodperformance.

Then,thegreyrelationalgrade[8–13]isdefinedas

followsÀÃÃÁDminþDmaxgx0;xiwi¼Gi¼ð2Þ0imax

n1XÃDðkÞð3ÞD0i¼nk¼10iÀÃÃÁð4Þ;xiwi610<gx0

whereD0iðkÞisthedeviationofthereferencevalueÃx0ðkÞ.ThecomparabilityvaluexiÃwiðkÞandweightingfactorwiforeachtensilestrengthareassignedtobe

equalwithvaluesof0.34,0.33,and0.33underthe

conditionthatthesumoftheweightingfactorsisone.

ThegreyrelationalgradeGirepresentsthelevelof

correlationbetweenthereferencevalueandthe

Table6specificvaluewithwhichitiscompared.Theevalu-atedgradeisbetweenzeroandoneandisequaltooneifthetwosequencesareexactlyidentical.BecausethegradeGiisestimatedthroughintegratedconsiderationoftherequirement,thehighestgradeamongall18groupsexactlyrepresentstheoptimizedcombinationforlaser-archybridwelding.Ahighergradealsoimpliesagreaterdominanceofthespecificvalueoverothervalues.Therefore,greyrelationalanalysisinvolvesaseriesofcalculationsthatincludemeasurementofthedifferencebetweenoriginaldatasequences.Table6showstheexperimentalresult,thenormalizeddata,andacomputedgradeforthe18groups.Experiment15hasthebestperfor-mancecharacteristicamongthe18experimentsbecauseithasthehighestgreyrelationalgrade.Thus,theperformancecharacteristicisstatisticallyconvertedintoasinglegreyrelationalgrade.Sincetheorthogonalarrayisusedfortheexperi-mentaldesign,itispossibletoseparateouttheeffectofeachweldingparameteronthegreyrelationalgradeatdifferentlevels.Forexample,themeanofthegreyrelationalgradefortheweldingwiretypeatlevels1and2canbecomputedbyaveragingthegreyrelationalgradeforexperiments1to9and10to18,respectively(Table5).ThemeanofthegreyrelationalgradeforeachleveloftheotherweldingoperationparameterscanbecalculatedinasimilarwayanditissummarizedinTable7.Figure3showsthegreyrelationalgradegraphandthedashedlineisthevalueofthetotalmeanofthegreyrelationalgrade.Basically,thelargerthegreyrelationalgrade,thebettertheperformancequality.Asaresult,thelevelofthelaserfocusshowsthelargestdifferencewhiletheeffectoftheshieldinggastypeshowslittledifference.Theexperimentalresult,itsnormalizeddataandgradefor18groups

Greyrelationalcoefficient

1

0.6532

0.0000

0.7784

0.2896

0.4936

0.7232

0.7094

0.7212

0.2877

0.9094

0.8680

0.3547

0.4257

0.9557

1.0000

0.8198

0.8493

0.634620.63930.00000.83020.30190.53430.81850.87830.71280.34510.85320.85480.37250.31460.71061.00000.81760.60870.593030.63110.00000.81450.27680.48390.87230.83800.73740.25870.88620.85210.39030.31340.78951.00000.82650.74610.6317Grade0.73980.50500.84120.58940.67270.83870.84130.78690.59200.89730.87810.61900.61150.85001.00000.85090.79430.7285Ultimatetensilestrength(MPa)Group1234567891011121314151617181216.81141.68231.20174.99198.45224.85223.27224.63174.77246.27241.51182.47190.64251.59256.69235.96239.36214.662217.62143.72239.69178.62205.48238.34245.25226.12183.61242.35242.53186.78180.09225.87259.32238.23214.09212.273216.52151.06235.54179.77201.25241.54237.98227.54177.89242.98239.44191.54183.57232.95254.78236.78228.45216.58

Proc.IMechEVol.222PartB:J.EngineeringManufactureJEM1070ÓIMechE2008