1、 copy 223RM A55J28RESEARCH MEMORANDUMAPPLICATION OF TCHEBICHEFFORM OF HARMONICANALYSISTO THE CALCULATION OF ZERO-LIFT WAVE DRAGOF WING-BODY-TAIL COMBINATIONSBy George H. HoldawayandWilliam A. Mersma.nAmes AeronauticalLaboratoryMoffett Field, Calif.I%ianmtmimlcOntalnsI.Obrmatkmatfectlngb NWOnalDnfmae
2、ofb Udt8dStateswlthlnb mefdnKoftinMMIEWS,W 18,TLLI.C.,SSCS. s.nd?W,tbtrnnsmtdonor rwelatbuofwklchlmuy toanCmnnuulizedmn MprOMMtedbylaw.NATIONALADVISORY COMMITTEEFOR AERONAUTICSWASHINGTONFebruary 13, 1956b?i.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS
3、-,-,-111111111111G NACARMA55J28 01J4335BNATIONALADVISORYCOMMITTEEFORAERONAUTICSRESEARCHMEMORANDUMAPPLICATIONOFTCHESICEEFFORMOFHARMONICANALYSISTOTHECALCULATIONOFZERO-LIFTWAVEDRAGOFWING-BODY-TAILCOMBINATIONSByGeorgeH.HoldawayandWilliamA.MersmansuMMARYThetechniquesofthecamputingprocedureofNACARMA53HL7h
4、avebeensignificantlyimprovedbya newprocedureofharmonicanalysisusing. Tchebichefpolynomials.Thisimprovedmethodisdescribedindetail.withillustrationsofitstwomainadvantages;theseare,simplificationofthecomputingprocedures,andtheprovisionfora comprehensivecheck0solutionwhichticludesa directcheckofhowwellt
5、henumberofharmonicsusedrepresentthearea-distributioncurve.Forthepresent,nospecificrecommendationcanbemadeastothenumberofharmonicswhichshouldbeusedforallconfigurationsining wave-dragcomputations;however,certainguidesaregivenintheconcludingremarksofthereport.Thenewprocedureisalsoevaluatedbycomparisons
6、withanal2calsolutions,resultsfromthepreviousmethod,andexperimentalresults.INTRODUCTIONThecomputingmethodofreference1hasbeeneffectivelyusedtoestimatethezero-lj.ftdrag-risecoefficientsofvsriousrelativelysmoothwing-body-tailcombinations(refs.2,3, and4). Howeverthisoriginalmethodinvolvesseveraloperation
7、s,andthecheckingprocedures,suchaswereusedinreference3,sretime-consumingsadcheckbacktoonlythslopesofthearea-distributioncurvesandnottothearea-distributioncurvesthemselves.Itisthepurposeofthispapertopresentandtoanalyzeanother. methodforrepresentingtheslopeofarea-distributioncurves,whichwillallowforamo
8、rerapidcomputationofwavedragandwillpermittheuseofanimprovedmethodofcheckingthecomputations.Thebasicmethodfor computingthewavedragisfundamentallythesameasinreference1 andisbasedonthetheoryofreferace5. The mati differenceisthat9Provided by IHSNot for ResaleNo reproduction or networking permitted witho
9、ut license from IHS-,-,-2Tchebichefpolyaomia.ls(refs.torepresenttheFouriersinedistributioncurves.6 and,alsospelledChebyshev)areus4-seriesdefiningtheslopesofthearea-Thenew.methodisevaluatedbycomparisonofresultswithanalyticalsolutions,resultsframthepreviousmethodMachnumbersupto1.8. ,andexperimentalres
10、ultsforTheconfigurationss-elected,forwhichexperi-mentaldatawereavailable,includedmodelsofa triangular-winginterceptor-typeairplane,a swept-winginterceptor-typeairplane,andabody-tailconfigurationwitha scoop-inletduct.SYMBOLS AnCDOCDOrACDOcM.A.C.do1M.mNnqscoefficientsdefim.ingthemagnitude.oftheharmoni
11、csofaFouriersineserieszero-liftdragcoefficient,dragatzerolift!dicul.arto x axisb. .-.*FProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACARMA55J28 3.s areaformedbycuttingconfigurationswithplanesperpendicularorobliquetothe x axis.s?(x) slopeof S curv
12、esasa flmctionof x,Udx%? totalwingareax distancealongthe x axismeasuredframthemid-lengbhposition.XYz Cartesiancoordinatesasconventionalbodyaxese anglebetweenthe z axisandtheintersectionofthecuttingplanesX withthe yz plane(Seeref.1 fordescriptivesketchesanddetafleddefinitions.)E distancealongthe x ax
13、ismeasuredfranthemid-lengthpositionditidedbyonehalfofthebodylength9 transformationofthelengthx toradians,arccos or. arccosx a series ofparallelcuttingplanestangenttotheMch cone. (AtM=1.0theseplanesareperpendiculartothe x axis.)$ angleinthe planeforwardbetweenthe Y axisinterceptofthecutt3ngplanesX on
14、the plane,arc tan(K cose)Tn()= COS I Tchebichefpolynomial,reference6Vn(g).= =-l() Tcheblchefpolynomial,reference6COMPUTINGMEZHODA summaryofthecomputingmethodispresentedhere,withthecampletedetailsgivenintheappendix.Asshowninreference1 (basedonthetheoryofref.5),thewave-dragequationmaybewrittenincoeffi
15、cientformasCDO=thenequation(2)maybewritten,.An=(E)Vn(E)d,n=l,2,3,. (4)-.“Becauseofthissimplificationthecomputationofthesecoeffi.clentsandNthesummationI llAn2canbeperformedbyonecontinuousoperationonn=la digitalcomputingmachine.Likewise,areversecheckcomputationcanbeperformedbyonemachineoperation.Thewa
16、ve-dragcoefficientsarecomputedfhm equation(1)bya simplelnteationaswasdoneinreference1.Therearedefiniteadvantageswhicharecharacteristicofthenewcamputingmethod.Iteliminatestheintermediatestepsofthepreviousmethod(ref.1)consistingofcamputingtheslopess(x),plottings(x)asa functionof (p,andthenreadingthiss
17、lopecurve.Thenewmethodworksdirectlyfromthearea-distributioncurvesofthemodel.Anaddi-tionalmachineoperationhasbeenprogramed(seetheappendix)whichpermitsa checkcomputationfromtheAn coefficientsbacktotheareacurve.Thisisanall-inclusivecheckwhichmayalsobeusedtoevaluate.Provided by IHSNot for ResaleNo repro
18、duction or networking permitted without license from IHS-,-,-NACAw A5w28 5.theadequacyoftheselectednumberofharmonicsusedinrepresentingaparticularcurve.Supervisionofthecomputationiscuttoaminimumby. thepreviouslymentionedimprovemembs.Themachinetimerequiredtomakecmputationsbythenewmethodisone-halfthato
19、fthepreviousmethod(ref.1). Thisccmprisondoesnotincludethetimelostusingthepreviousmethodduetodatahandlingbetweensteps.Thetimerequiredtodeterminetheareadistributionofthemodelsisnotconsideredinthisreportandwouldnotvarybetweenthewave-dragcomputationmethods.TocamputetheFouriersineseriessolutions(25harmon
20、ics)frompuncheddatacsrdsrepresentingoneareacurve,usingaMagneticDrumCalculatoryonly5minutesarerequired.Thechecksolutionrequiresabout10minutes.ThusontheassumptionthatfivesreacurvesarerequiredperWch numbercputition,thecetimerequiredfortheFouriersolutionusingtheimprovedmethodwouldbelessthan1/2hour;andth
21、echeckingtime,lessthan1 hour.A completederivationofthenewcomputingmethodandthecheckingprocedurearegivenintheappendixwhichcontains:1.2.3*4.5*6.Anparts:FourierTransformationtoTchebichefformforcomputingFouriercoeffi-cientsGeneralintegrationprocedureTchebichefintegrationcoefficients(a)Linearapproximatio
22、n(b)QuadraticapproximationCheckingprocedureConstructionandcheckingoftables(availableonpuncheddatacardsuponrequest)TheoryandPropertiesofTchebichefPolynomialsRESULTSANDDISCUSSIONevaluationofthenewcomputingmethodwillbediscussedinthreeknownanalyticalsolutionscamparedwithcomputedvaluesofthecoefficients;p
23、r-ous solutionsfromreference3 comparedwithIi. newcomputedvaluesofthedragparameterY fin2;andavailableexp=i-mentalvaluesofdrag-risecoefficientscomparedwithcomputedVsJ.uesofa wave-diagcoefficientsatzerolift.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-
24、,-6 NACARMA55J28.CheckofMethodbyAnalyticalSolutions.Thefirstlmownsolutionconsideredwill.bethatfora Sears-Haackbody.Theshapeofthisminimum-dragbodyforprescribedvolumeandlengthisdefinedinfigure2. Thefinenessratioof12.5andtheactualdimen-sionswerearbitrarilyselected.Thetheoreticalequationforthezero-iftve
25、-g coefficient(CDjfig.2)isseentobeindependentofWch number(forslenderbodieorforMachnumbersnear1.0).Alsonotethatthederivativeoftheareacurveisexactlyequalto A=sin2q wherethereforeequation(1)simplifiestoZ A22CDO; %“ (5) _where,inthiscase,thereferencearea() isreplacedbythebodymaxi-nmmcross-sectionalareaf
26、ora body-alonetest;thatis,2A=2cDot= .g.+(*)2do=and(6) .,x Forthedimensionsselected, =53.9198inches.Forcomparisonwiththistheoreticalsolution,theareacurveinfure2wasanalyzedbythenewmethodusingTchebichefpolynomials.Theareaswerecamputedfor200equalincrementsof x,datacardswerepunched,andthe Anvalueswerecmp
27、utedfor25harmonicswiththeresultsasshownintable1. Evenforthelin”karapproximation(seeappendix)betweendatapointsof S(g),thevalueof2A22equalto206.502isinexcellentagreementwiththetheoreticalvalueof206.521,andthe25total summation1 nAn2 equalto206.731showsanerroronlyonthea“dern=xof1/10of1 percent.Forallhar
28、monicsotherthanthesecondthecoef-ficientsshouldbeZero.Theareasframthechecksolutionagreedwellwiththeoriginalvaluesforthiscase,witherrorslessthanthepossibleerrorsintheoriginaldata(0.05percentof G(g)maximum).Fora bodythatisclosedatbothendsthefirstcoefficientoftheFouriersineseriesisalwysequaltozero,asint
29、hesolutionpresentedintableI. .Iora bodywhichdoesnotcloseattheends,thefirsttermIA12representsthefunctionofwavedragfora vonK)definedby(ref.16): IOifr#jL(m,j,k;ak+ = =J(ink)Anexplicitrepresentationis-(ak)L(%iiJ% E)=;i=oak+-(ak+i#J-llm-jH (E- )for f()inequation(AIO)gives,byvirtueofeqpation(A14)theexplic
30、itformulaforthe oxnotthenlmast.Moref()fak+za(m,j,k)=L(m,j,k;3)g(E)dE (AL6)akgenerally,ifitisdesiredtoevaluateantitegralinvolvingbutanyoneofitsderivatives,formulas(A3.0)bereplacedbythrough(6) remainval.id,exceptthatin(u6)LthecorrespondingderivativedrL(m,j,k;)drwhilef isreplacedbythederivativeintheint
31、egrandsbutnotinthesums.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-18 NACARMA55J28Clearlytheeffectivenessofthisentireprocedurehingesuponthedifficultyofcalculatingthe a thesecondformofequation(Al)gives,for m =1,2,respectively:(A17)I%isa straightfo
32、rward,thoughtedious,tasktoextendsucha tabletoashigha degree,m,asisdesired.Thecalculationoftheintegralinequation(AL6) thendependsonthenatureofthefunctiong(.EJ).Thiscalculationcanbeperformedanalytica.llywheneverg()isa polynomial,trigonometric,hyperbolic,exponential,orlogarithmicfunction.Formorecomplic
33、atedfunctions,g(),itmaybenecessarytoresorttonumeri-calintegrationatthisstage,whichalmostdefeatsthepurposeoftheentireprocedure?However,theseintegrations(eq.(JiL6) are perfomvedonlyonce,andthenequations(A12)and(A13)areapplicabletoarbitraryfunctionsf().Inthenextsection,then,theseresultswillbeappliedtot
34、heintegralsofequation(A7).Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACARMA528 amE)=dSubstitutioninequation(A23)a(l,O,k)a(l,l,k)-1 dL(l,l,k;E) .Lz;d Agives,for j =0,1,respectively,JEk+l-1=A k1fk+l=-A k(A24)Provided by IHSNot for ResaleNo reprod
35、uction or networking permitted without license from IHS-,-,-20 NAcARMA55J28 (ITnNow, byequation(A),Vn=-n dandthereforeequation(A24)canbeintegrateddirectlytogivea(ll)k)n+ Tn(k+l)-Tn(k)1Substitutingtheseinequation(A13)andchangingthenotationtoagreewithequation4lhlr=tinA(A22) givesTn(0)-Tn(A1-Tn(Er-L)+2
36、Tn(E.r)-Tn(Er+=),r=I.,2) .,N-I (A25)L J IInthe linear case, then,tions()= 2-_(4k+_3)Ad 2A=dL(2,1,k;)=d A2dL(2,2,k;). 2j- (kk+l)Ade 2A2Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-N.AcA A55f28substitutioninequation(=3)givesfor J =0ly2yrespectivdy21
37、JNow,framequation(A42),vn+(5)+vT_J-L(5.)theserelations are substitutedinequa-2EYn(5)1 dTn and,asbefore,Vn=.dtion(A26),theintegratiocanbeperfomnimmediatelytogivea(2,0,k)=2* 1w(n+l,k)+W(r -l,k)- (4k+ 3)R(rYk)Aa(2,1,k)=* (4k+2)W(n,k)A-W(n+l,k)-W(n-l,k)1a(2,2,k)=m 1)W(II+l,k)+W(n -l,k)- (4k+l)W(nk)A(A27
38、)whereTn( r =01,.,100)wascomputedforthegyadraticcasefromequation(A29jAsa prelry check,thetablesof Tn()andVn()werecheckedusingequations(A43):Tn+l(E) = ETn(E)- (- E2)vn(E)1n=o,l,. . .,98vn+l() = Wn() +Tn(&)Thetablesof I_Lmwerecheckedasfollows:Forthelinearcase,theintegrationformula(A22)shouldbeexactif
39、S(g)islinear.Thatis,eqyations(A20)and(A22)shouldgivethesameresultsfortheFouriercoefficients,An,wheneverS(g)islinear.Thischeckwasappliedforvariouslinearfunctionss(g). Forthequadraticcase,equations(A20)and(A22)shouldgivethesameresultfortheFouriercoefficients,An,wheneverS(g)islinearorqgadratic.Thischec
40、kwasappliedforvariouslinearandquadraticfunctionss(g).Thetablesofcheckcoefficients,b(),wereeedby coct-inganalyticfunctionsS(g)havingonlya finitenumberofnonvanishingProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-26 NACARMA55J28.l?ouriercoefficients,An
41、,W ofwhich-e calcatediY andthensubstitutedinthecheckeqyation(A35).l?hepurposeofthesechecksistoensurethatallthetablesarefreeofnumericalerrors.Therestillrem- thequestionoftwoothersourcesofmor: theapproximationof S(E)bya polynomial,andtheuseofonlya finitenumberoftermsAn. Thesetwoeffectsarediscussedinth
42、emainbodyofthepresentreport,particularlyinconnectionwiththeSears-HAackbody.Ingeneral,suchquestionscanonlybeexaminedempirically.Ifthecheckprocedureindicatgthepresenceoferrors,threethingscanbedone.MorecoefficientsAn canbetaken,a finerintervalA canbeused,orhigherdegreepolynomialapproximationcanbeused.A
43、nyofthesecoursesofactionrequiresex%endingthebasictables.JhmanypracticalcasesthedatafunctionS(g)isoriginallyobtainedbyfairinga curvethroughareaso-blenumberofpoints.Obvi-ously,theuseofa finerreadingintervalA isincapableofyielding”greateraccuracyinsuchcases.Useofhigherdegreepolynomialsissame-timesjusti
44、fied,butthisalso- giveillusoryimprovements.Forexample,a curvewithsharpchangesinslopeisbetterapproximatedbyshortstraight-linesegmentsthanbylongersrcsofhigherdegreepoly-ncmlials.FimaK1.y,theuseofmoretermsintheFourierseriescanbe .reducedtoanabsurdityoncethewavelegthsinvolvedbecomeshorterthanthereadingi
45、ntervalA. .TheoryandFtrop-iesofTchebichefPolynomialsTheTchebichefpolyncmdal.ssredefinedinreference6asTn(Ej) = cosnqUn()= sin(n+l)q/sinq1 (A37)where =cos(p.SinceUn-l()iscloselyrelatedtotheFouriercoef-ficient,An,itseemsconvenienttointroducethenotation.(A38)Sincetheprincipal.propertiesofthesepolymmials
46、,includingthefactthattheyarepolbmial.sin g,aretreatedextensivelyinreference6,thepresentsectionwilllistandproveonlythosethathavebeenusedinthisreport.First,thefollowingareobviousfrom,thedefinitions: .To()=1, U(5) = Vo(k)= 0, V=(EJ=1 (A39) .Provided by IHSNot for ResaleNo reproduction or networking per
47、mitted without license from IHS-,-,-MUXRMA55J28 27. N- considertheeffectofreplacing by -k,sothat q isreplacedby fi-ql,thatis,Cos(fi-q)= -C!osCp:-Tn(-) = COS II(YC - q) =Thusandcosnncosnq-cosnsrsinn-cosnsincpTn(-5i)=(-)nTn(E), v(-u = (-l)n%nm (AkO)forall.integralvaluesof n.Next,setting =1 gives(p=O,T
48、n=1,whileVn istheindeter-minateformsinncplim cp+ostiApplyingLtHospitalisrulegivesTn(l)=1, Vn(l)=n (A41)Recurrencerelationscanbeobtaineddirectlyfromthedefinitionsusingthetrigonometricadditionformulassin(ab)=sinacos bcosaslnbcos(ab)=cosacos bksinasinbByuseofthese,thefollowingareeasilyproved:Tn+z(6)=2Tn(3)oTn-l()n+l()=2Vn()-vn-1()1Tn+()= Tn()- (1- 2)Vn(6)(A42)VU+=(6)= Vn()+Tn()Numerousdifferential
