1、TECHNICALA SURVEY OF METHODS FORNOTE2340, :-.- P:-:;.-:.,” ,FE3,N. k.=.i.-*.-.,.DETRMINING STABILITYPARAMETERS OF AN AIRPLANE FROM DYNAMICFLIGHT MEASUREMENTSBy Harry GreenbergAmes Aeronautical LaboratoryMoffett Field, CaUf.WashingtonApril 1951-.:-,.=.Provided by IHSNot for ResaleNo reproduction or n
2、etworking permitted without license from IHS-,-,-TECH LIBRARY tiiFB, Nid1.TABLEOF CONTENTS 111111111111ilLlL51q2SUMMARY*.*.* a71 * *”INTRODUCTION. . . . . . . . . a71 = o= G=*= D* *o* a71NOTATION. . . . . . . . . . . a71 a71 a71 a71 *O s a71 a71 a71 a71 a71 a71 c a71 0*MATHEMATICALAND AERODYIWMICPRX
3、LIMINARIES. . . . . . . . .BasicAssumptions. . . . . . . . . . . . . . . . . . . a71 a71EquationsofMotionand Statementof “InverseProblemofAirplaneDynamics. . . . . . . . . . . . a71 . a71 a71 . . a71 a71TransferFunctionsforControlDeflectionInput. . . . . .COMPUTATIOI?OFDYNAMICPARAMETERS(TRANSFERCOEF
4、FICIENTS)FROMFLIGHl!DATA. . . . . . . . . . . . a71 . . a71 . a71Principleof LeastSquares . . . . . . . . . . . .Determinationof theParametersandTheirRelativeby Linearization. . . . . . . . . . . . . . . a71Methodsof Obtaininga FirstApproximationto theParameters. . . . . . . . . . . . . . . . . . .
5、 .a71a71.Accuracya71 a15a11a11a15a15a11a11Sinusoidal(frequency)response. . . . . . . . . . .Transientresponse. . . . . . . . . . . . . . . . a71 *(1)(2)(3)(4)DISCUSSION.Inspectionof thetransient. . . . . . . . . .Fouriertransform. . . . . . . . . . . . . . .Derivativemethod. . . . . . . . . . . . o
6、Pronysmethod. . . . . . . . . . . . . a71 . a71 “ComparisonofFourierTransformMethod,DerivativeMethod,andPronysMethodof ObtainingaFirstApproximation . . . . . . . . . . . . . . . .Page11356678810111113131415182020Provided by IHSNot for ResaleNo reproduction or networking permitted without license fro
7、m IHS-,-,-TABLEOF CONTENTS- Continued NACATN 231+0DeterminationofLiftDerivatives. . . . . . . . . . . .BasicLimitationsinDeterminationofMomentDerivatives. . . . . . . . . . . . . . . . . . . . . .InformationObtainedFromTailLoadMeasurements. . . Use ofNonaerodynamicForcingFunctions. .RelationBetweenS
8、taticandDynsmicTests .AerodynamicLag. . . . . . . . . . . . . .SuggestionsforFutureWork . . . . . . . .CONCLUDINGREMARKS.APPENDI X.*lm?EKENms. . . . .a71a71a15.a71a15a15.I.a71.a71a15. . . .a71a15a71a15.a71a15.,a71a15a71a15a15a71.a71a71.a71a71.a71. . . . . . . . *.* . .*.* .*. . . . . . . .a71a15a71a
9、15NUMERICALEXAMFLES(TABIJ3S-Iv).FIGURES. . . . . . . . . . . . . . . . . . . . . . . . . . .a71a71gage,212124242527292931363745?.a71Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NATIONALADVISORYCOMMITTEEFOR AERONAUTICSTECHNICALNOTE 23A SUKWEYOF MET
10、HODSFORDETERMININGSTABILITYPARAMETERSOF AN AIRPIANEFROMDYNAMICFLIGHTMEASUREMENTSBy HarryGreenbergSUMMARYVariousmethodsof reducingto stabilityparameterformtheresponseto sinusoidaland transientdisturbancesarediscussed,usingthesimplifiedlongitudinalmotionof an idealizedairplaneas an illus-trativeexampl
11、e. It is shownthattherearebasiclimitationsin thedeterminationof someof the stabilityderivativesas comparedwiththe%ransfez%?unctioncoefficients,whichare certaincombinationsof sta-bilityderivativesdirectlyrelatedto theairplaneresponse. Hencejmostof thereportis concernedwithmethodsof determiningtransfe
12、r-a71 functioncoefficientsratherthanstabilityderivatives.It is shownhowthemethodof leastsquarescanbe appliedto giveathedesiredparametersand alsotheratioof theirerrorto thatof thebasicdata. The determinationof theseparametersand theircorrespond-ing errorratiosisa nonlinearproblemwhichit is showncanbe
13、 solvedby linearizationusinga firstapproximationto theurilmownparameters.A numberof methodsof obtaininga goodfirstapproximation,whichalsoinvolvea leastsquaresprocedure,are explainedand illustratedin thenumericalexamples.Althoughthe examplesare confinedto a simplifiedcaseof longi-tudinalmotion,themet
14、hodspresentedareapplicablein generalto othermore complicatedtypesofmotion.INTRODUCTIONThe scarcityof reliabledataon the stabilitycharacteristicsofaircraftat transonicand supersonicspeedsandthe difficultiesofobtainingthis informationfromwind-tmel tests(particularlywithregardto dynamicparameters,sucha
15、s therotarydampingderivatives),haveacceleratedinterestinmethodsof obtainingsuchdatafromflighttests. Alsothe extensiveuse of automaticstabilizationand controlequipmentandtheuncertainand generallypoorerdymamic-stabilitycharacteristicsarisingfromthe use of unconventionalconfigurationsnecessitatecompreh
16、ensiveandrefinedmeasurementsof the stabilityderivatives.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-2 NACATN 23bIn thepast,limitedinformationhasbeenderivedfrommeasurementsof airplanecharacteristicsin steadywtraightand steady-turningflight. Recent
17、ly,frequency-responsemeasurementsto sinusoidalcontroldeflectionshavebeenemployedtoevaluateadditionaldynamic-stabilityparameters.Thebrieftestingperiodavailableduringflightsofmissilesandhig similartheoreticaland flight+testinvestigationsforthe lateralmotionshavealsobeenreported. Themethodof analysisof
18、reference1 is onlyapplicableto simpledynamicalsystems- thatis,systemswhichmathematicallyare similarto onewitha singledegreeoffreedom,and is incapableof reducingthestabilityparametersto thebasicstabilityderivativeform.Theobviousadvantage,fromthe standpointof testsplicityandtime,of usingtheresonseto a
19、 stepelevatorinputinsteadof thefrequency+esponsetestswas soonrealized,andtheworkreportedinreference2 showshowthe step-responsedatacanbe convertedintothefrequency-responseform. Subsequentwork (reference3) extendedthismethodto theresponseto a pulse elevator input. Anothermethodofanalysisof responseto
20、an arbitraryelevatormotionwas suggestedbyLoringandJonahof Chance-VoutAircraftCompany. Thismethoddoesnot requiretransferencefromthe Ttimedmainito the “frequencydomain”as is the casewithreferences1 through3, and isreferredtolaterin thisreportas the“derivativemethod.” In thisderivativemethodthereappear
21、sforthefirsttimetheapplicationof themethodof leastsquaresto obtainthemostreasonablevaluesof aLlaneparametersfromredundantmeasurements.Examinationof theavailableliteratureindicatesa lackof infopmationconcerningthegeneralmethodsof emalysisapplicabletomorecomplicatedsystems,forexample,systemswithmorede
22、greesof freedomorwithhighe=rder derivatives.The purposeof thepresentreportisto establishmoregeneralandrigorousmethodsfordeterminingaerdynamicparametersfromdynamicflightmeasurements.The followingprincipalandbasicproblemsare studied: therelationbetweenthenumberandtypeof appliedforcingfunctionsendmeasu
23、redresponsesandthe correspondingnumberand typeof determinableaerodynamicparameters;variousmethodsof convertingflightdatato a formsuitablefor9.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACATN 23h0 3determinationof aerodynamicparameters;andthecor
24、rectapplicationofmthemethodof leastsquaresto computetheaerodynamicparameters. Althoughthemethodspresentedareapplicabletomorecomplicatedsystems,theexamplesin thepresentreportare confinedto the simpli-fiedlongitudinalmotionsof an idealizedairplanehorder to facilit-ate computations.NOTATIONGeneralaatA,
25、B9 D. 8eE%E1geIyLL.La.Lbangleof attack,radiansangleof attackof tail,radiansi and out-of-phasecomponentsof oscillationdifferentialoperator()aEelevatordownwashresidualresidualresidualdeflection,radiansangle,radianserrorin an equationerrorin a realequationerrorin an imaginaryequationaccelerationdueto g
26、ravityangleof pitch,radianspitchingmomentof inertia,slug-feetsquaredrootsof characteristicstabilityequationliftforce,poundsProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-4LtLvmMMaDaMb%MvPn$1R,qJstvV.wWqxu)N/WATN 2340taillift,poundsmassof airplane,sl
27、ugspitchingmoment,foot-pounds?IM.daaM35/ 2mrelativedensityparsmeter =( pS#a712 Xma.ss )densityXwing areaX wingchord) .normalacceleration,g unitsangularpitchingvelocity,radiansper secondamplitudeandphaseof complexnumbersumofweightedsquaresoftime,secondsvelocityof airplane,feettrimvelocityof airplane,
28、residualsper secondfeetper secondweightof airplaae,poundsweightingfactorindicatingaccuracyof q measurement,etc.longitudinaldistancebetweencenterof gravityandneutralpointof airplane,feetangularfrequency,radiansper secondProvided by IHSNot for ResaleNo reproduction or networking permitted without lice
29、nse from IHS-,-,-NACATN23m 5Subscriptsnc.mcalculatedmeasuredTransfer-TunctionCoefficientsdampingparameter(+q-MDct+ mv)VoCoq(Clq - Coa)+k stiffnessparameter(% %._=mVo Iy)MATHEMATICALANDAERODYNAMICPRELIMINARIESIn orderto illustratethe typesof dynamicparametersinvolvedandthe conditionsunderwhichtheymay
30、 be measured,the caseof longi-tudinalmotionof an airplanewillbe considered.In settingup theuequationsofmotion,the followingassumptionsaremade:.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-BasicAssumptions1. Linearequationswithconstantcoefficients2
31、 Measurementof forcingfunctionnot subjectto error3. Responsemeasurementssubjectonlyto randomerrorsk. Constantairspeedandlevelflight5. Aerodynamicliftequals + 8L6. Aerodynamicmamentequals cd theothersaremadeto simplifythenumericalexemplesgivenlaterandbecausetheydo approximatelydescribetheairplanemot
32、ionsin themaneuverswhicharecopsideredinthisreport.EquationsofMotionandStatementof tInverse”ProblemofAirplaneDynsmicsBasedon theaboveassumptions,thelongitudinalequationsofmotionmaybe written-anamely,giventheairplaneresponsein a, q,or n to a disturbance,to evaluatethe stabilityderivativesof theairplan
33、e.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-N/WATN 23koItwillbe provedthatthedetermination7laterin thisreport(seesection“Discussion”)of momentderivativesfromflightdatais subJectto certainbasictheoreticallimitations.Thereare ertaincombina-tionso
34、fmomentderivatives,however,whichdeterminethebehavioroftheairplanesndwhichcanbe computedfromtheflightdata. Thesecombinationsof derivativesare called“transfercoefficients”and aredefinedbelow. Mostof thisreportis devotedto thedeterminationofthesetramsfercoefficients.TransferFunctionsforControlDeflectio
35、nInputFor thiscase,inwhich L=5Ltionof equations(1)and (2) ist-xand M=5Mj theoperationalsol-% %!i+mVo Iv-.-8 j32+D % MIh+it( ) C andCoq suchthat.-Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACATN 2340.9, = /T(qfic) titLJo. is a minimum.Ifmorethan
36、oneairple responseismeasured,for example,thetransientresponsesin 9, q, andDq from t=O to T, then the valuesof theparameters b, k, Clq udcoq are tobe determinedsuchthatTf JTJTs = we (%r%) dt + Wq (%IHC)2 at + w (Dqm-Dqc)2dto 0 0is a minimum. Theweight w of a measurementis a numberindicatingtheaccurac
37、yof thatmeasurement(asregardsrandomerrors). Morespecifically,theweightis thereciprocalof themean squareerror(normaldistributionin the errorsof measurementassumed,seerefe cence 4) . If thefrequencyresponse(bothamplitudeandphase)ismeasuredovera rangeof frequenciesu, thentheparametersare tobe detemined
38、by theconditionthats = WR (%IAC)2 + I wp(%Pc)2is a minimumwherethe summationis takenoverthe frequenciesat whichtheresponseismeasured.The problemsabovearenonlinearin theunknowns b, k, Cl,andCo.The onlypracticalmethodof solutionis to linearizetheproblemanditeratefroma firstapproximation.For thecaseinw
39、hichonlyonequs.ntityis subjectto error(likethe firstcasementionedabove)themethodsof linearizationand iterationare explainedin reference4pages214and 84, respectively.Themethodof linearizationis discussedbrieflyinthe nextsectionand in theappendix. The subsequentsection(whichconstitutestheprincipalpart
40、of thisreport)dealswithmethods,mostof whichalsoinvolveleastsquaressolutions,of obtaininga goodfirstapproximation.The ideaof determiningtheparametersfroma trsnsientresponsebylinearizationand iterationfroma firstapproximationis dueto Shinbrotof AmesLaboratoryand is discussedand exploitedmore fullyin r
41、efez-ence5.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-10 NACATN23koDeterminationof theParametersandTheirRelativeAccuracyby LinearizationTo determinetheparametersby k Clqs=d COq from q and 5measurements,it is firstnecessaryto determineem initiala
42、pproxim somespecialtypesof inputfunctions(tipulse,step,and ramp)are shownin figure3. Severalmethodsofanalysis,of varyingdegreesof generalitywillbe explained.the(1)1.- If 5 becomesconstantaftera brieftransientperiod, b and k canbe determinedfrmnthedamping andperiodof theoscillations(assumingthatthe s
43、ystemislessthancriticallydamped). If T1J2 is thetimefor thefreeoscillationsto dampto halfamplitudeand P is theperiod,thenbIf, in addition,valueof 8 areIf thevaluesof_ 1.386 ati k_ 0.48 + 39.48%/2 T1/22 p2the steady-statevalue of q andthe steady-stateknown,thentheirratio / hencea slightlydifferentfre
44、quencyresponseisobtainedThe frequencyresponseobtainedas an intermediatestepinthecomputationof transfercoefficientsby thisprocedureis oftenitselfof interestinproblemsof automaticstabilization.The so-called“incompleteFouriertransforms” qe-titdt andf: e-titdt canalsobe usedto computetheparametersfroma
45、pulse.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACATN 23h0 15.:,whichhasnotreacheda steadystateandfornonzeroinitialcondi-tions. AssumingthetransferfunctionC1qD+Coqs=5 D2+bD+kthenit canbe shownthatT(im)=+bim+kJ qe-itdt + e-o(DqT+bqT+iT)- Dqo- b
46、qo- iQo= (ch.p+coq) J 5e-imt dt + Clq8Te-iut- Clq?50*fromwhicha realand an imaginaryequationcanbe setup. Thesecanbe usedto setup foursimultaneousequationsusingcomputationsattwofrequencies,to get approximatevaluesfor b k clqandCoq.Ifmorethantwofrequenciesare involved,thenthemethodof leastsquarescanbe
47、 usedas in thecaseof frequency-responsemeasurements.(3) Derivativemethod.- Anothermethodfor computingthe transfercoefficientsfromthe transientresponseconsistsinusingthemeasuredvaluesof a sufficientnumberof higherderivativesof inputandresponsein theassumedtransferfunction. For example,if theassumed-f
48、ormof the transferfunctionis_=y(Di(o)ik tZ(Di(5)i C=q _2(DiifdTc =2 (DG)i(D20)ii io q iZ (Di(ib + z (e)i2k - tiZ(6)i(5)iCl - X(8)ii f 5dT Co =-z (G)i(D26)ii i qi o qiAn exampleof theapplicationof thederivativemethodto thedetermitionof theparametersfromtheresponse 6, IX3 and D% to a stepinputin 8 is shownin figure5, and workedout intableIII. Fora stepresponsethedifferentialequationrelating (3 and 5 i.sD2e+b
