I

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- .. .

,

NATIONALADVISORYCOMMITTEEFORAERONAUTICS

TECHNICALNOTE2184

INVESTIGATIONOF FREQUENCY-RESPONSE CHARACTERISTICSOF

ENGINE SPEED FOR A TYPICAL TURBINE-PROPELLER ENGINE

By BurtL.Taylor,III,andFrankL.Oppenheimer

LewisFlightPropulsionLaboratoryCleveland,Ohio

.

0

Washington

‘,-..= _.kl,

, ,.,

..,4

. . . . --- -------- ..—. . . . . .. . . . . .. .

I

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14I

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.. . . .. . --. .,-- . . . .. ----- ..

TECH‘ilBRARYKAFB,NM

Innllmlllllllllllml!Ior?ALADVISORYCxmmTEEFORAERONAUTICS

.0tlb5L40

IIVV339TIGATION

OF

TEOHNICALNO!EE2184

~ ~_-IIESP(lIISE~l@

ENGIX5Sti FORATKPICJL

TURBINE-EROPEKERENGI&’

ByBurtL.Taylor,III,and

Experimentalfrequenoy-response

l%anlcL.Oppenheimer

characteristicsofenginespeedfora typicalturbine-propellerenginearepresented.Thesedatawereobtti bysubjettingtheenginetosinusoidalvaria-tionsoffuelflowandpropeller-blade-angleinputs.CorrelationIsmadebetweentheseexperimentaldataandanalyticalfrequency-responsecharacteristicsobtainedfroma~ineardifferentialeqza-tionderivedfromsteady-statetorque-speedrelations.

Theresultsofthisinvestigationindicatethatenginespeed.isa linearf’umtionoffuelflowandpropeller-liladeangleforlimitedvariationsoftheseparameters.Algebraic expressionsapproximate-describingthefrequency-responsecharacteristicsoftheenginetestedareprimarilyfirstotier.Apprmimate frequency-responsecharacteristicslimitedtofirst-ordereffectsq becal.culatedfromsolutionsofa lineardifferentialequationande@.-libriumtorquqcharacteristicsoftheengineandthe~peller.

INTRODUCTION

Inofiertodesigncontrolsthatwillprovidedesirabletransientoperation,thedynamicsofthevariouscomponentsofthecontrolledsystemshouldbeknown.Themostgeneraldescriptionofthedynamicsofanysystemisthedifferentialequationsofmotion,butuseofthedetailedandpreciseaerodynamicandthermo-dynamicequationsforanalysisof@namicbehaviorofa gas-turbineengineandcontrolsystempresentsinsurmountableproblems.Onwe otherhad, anydescriptionwherebylinearbehaviorofthesystemcomponentsisassumedcanbeplacedina fom thatisresd-ilyhardled”andproductive of results. m advantages of me 1~-earassumptionaresofar-reachingthatitsuseisjustified,even

.

.. ..-——-.. — -. ._. _ _—__ .— --— . . ... .—. - -—— - ._. ___ . .— . -__ —-. —

2 . l’iACA5’J21,84

wherethesystemisofknownnonline-tybutwherehim pwcision ‘ofresultsisnotrquired,suohasintheinwstigathnofmn-trohforgas-turbineengines.

Inthelinsarfield,severaldesorlptiveformsmn beused.Themostgenemlof these is thefrequencyresponse,wherebytheattenuationandthephaselagofsteadysinusoidalinputsaregivenforthemntinuouefrquenoyspeotrum.Whenthesedataareavail- Eableforeaohmmponentofa dosed-loopcontrolsystem,thetime 3responsed thesystemoanbecalculated.Byemployingtechniquesofanalysisandsynthesisdescribedinreferenoe1,systembehaviormn bealteredtoobtaindesiredfrequenoy-respmseoharaoterletlosandtherelatedtimeresponse. .

A generalinvestigationisinprogressattheI’LICALewislab-oratorytodeterminethe nature ofthedynmiosofgas-turbinepowerplantsandtodeviseandtoevaluatemethodsfordescribingthisbehaviorina formapplicabletocontrolsynthesisandanaly-sis.Specialemphasishasbeenplacedonthedeterminationofthebehaviord rotativespeedbeoauseofitsimportameinthistypofengineforsafetyandoptimumoperatbn.Theresultsofaninvesti@ionofthefrequenoy-responseoharacteristiosofaturbimqropellerenginearepresentmiherein.J!kperimentalfreqmnoy-respmeedatawereenalyz~toobtainthegeneralformandnatmre& thefrequenoy-responseanddynamicohsraoterietiosofsuchengines,toverifythelinearassumption,andtoobtainabasisuponwhlohotherdescriptiveformsofthedynamicbehaviorq bemmpared.Onesuchcomparisonis madebetweenthelineardifferentialequationobtainedfrontequilibrium-torquemrvesandexperimentalfrequency-responseOharaoterietlos.o

Forpurposesofthisinvestigation,thedynamicresponseofengine”speedtod.isturbanoesinfuelflowandpropeller-bladeanglefora typtmlturbine-propellerengineisused.

with

AKAL3!WS

DevelopmentofLinearDifferentialEquations

thehypothesisthatquasi-staticmnditioneexletinanengineduringtramientrenditionsequationq bederivedto~ssa-functionoffuelflow(refezwnoethetorqueoutputoftheengineisables,enginespedandfuelflow,

ofoperation,a differentialthebehaviorofengineSPf3edU2). Anassumptionismadethat .{somefunotionofonlytwoverl-fora givenoonditional?

-— -- .-. ... ..—— —-. ..— — —--- .. —- —— -- - —-- -.——

NACATN2184 3.

ualtitudeailrampressureratio. ThisfunotdonIUWbeexedandlinearizedsothatthefollowingequationapproximatesthebehaviorofenginespeedforopmationina SIUSJJ.region

(1)

wherea isthepartial derivativect engine torquewithreqeottofuelflowand b istheabsolutevalueofthepartialderiva-tiveofenginetorquewithrespeottoenginespeed.A deltapre-fixbeforea variablesigwlfiesdtiationframtheinitialopera-tingmint. (Allsymbolsusedinthisreportaretifsmaintheappendix.)

Ina similarmanner,thetorqueabsorbedbya propellermaybeexpressedassomefunotionofrotationalspeedandbladeangle

q = f’z(P,Np)AQP = oA$ +dANp.AQp= oA~+~#I?e}

(2)

whereo isthe,partialderivativeofmopellertorquewithrespmttobladeangleandwherea isthepartialderivativeofPrOpellmtorWewithrespeottopropellerspeed.

Anyunbalanoebetweentorquedevalopedbytheengineandtorqueabsorbedby the ~pellerresultsinanameleration.IXtheeffeotivemomentofinertiaisassumedtobemmprisedofthepropellerandtherotatingpartsoftheengine,theunbalamedtorquemaybeeqssd intermsofenginespeedandthemomentsofinertiabythefollowingequation:

H equation(2)isdividdbythegem?ratioR andsub-tractedfromequation(1),thefollmingequati~results:

A22Af&)-n-AWf .( )-:413- b+~ Al?eHz

(3)

(4)

.

:_. ~-.— — ---——- -—.-—- ——— — — —

4

Equationsferentialangle:

,

iWCATM2184 “

(5)- (4).= bemibinsd,yieldingthefollowingdif- “equationrelatingenginespeedtofuelflowandblade

(5)

Ifequation(5)isditiaea

-, ~g foil- a~~itiess equation expressing speed%?(max)allafuelfluwisObtainea:

t,

.

,,

.-. . . .-. —. —— ---.—. -

I{

.

tie‘e(max) +

ra~j]-[1Wf

a—% (mt3x)

AWf—-%’(Ilax)

Al/e

7

ala]‘[d

(6)

(n

Theformofequation(6)isthatofa first-orderlag,which &canlewrittenasfollows:

AIie ANe AWf

T‘e(m)—=Kl — -J@V . (7)— + ‘e(-) Wf(mm)

wherethedynsmiccharacteristicsareentirelydefinedbythetimeoonstantT and the equilibriumconditionisdefinedby K1 and zK2. Thus, 3

\1

T=bl+dl1’

al‘1= bl+dl

c1K2=—

bl+al

(8)

JfrequencyResponse

Frequenoy-responseohamoteristicsmaybemlculatedbysolv-ingtheMfferential.equationdescribinga systemsubJettedtosinusoidalforc~ functionsofvariousfrequemies.Thesameresults~ beobtainedbyapplyingtheoperationalcalculustothedifferentialeq~tion.Thus,theresponseofenginespeedtofaelflowatconstantbladeanglecanbe_ssed intermsoftheoperator~ asfollows:

AWe. 5&(iu)= ‘1- (9)

Wf(m&XX)

whiohisobtainedfmm equation(7]byreplaoingthefirstderiva-9tivewithrespeottotimeby (iu)andtreatingtheresultsalge-braically. The”complexoperatorrepresentsboththeratiooftheamplitudesoftheengine-speedandfuel-flowsinewswesandtherekrtivephaseangles.Theemplituderatioistheabsolutevalueoftheccunplexfunotion,asfollows:

. . . . . .— ...— .—. .-. ——- —-——---- ---- -.-— --

NACATN2184 7

IIANeM?!!dIIAWf% (msx)‘A2

(lo)

Theangle bywhiohtheengine-speedsinewavelagsthefuel-flowsinewaveisasfollows:

/

(u)

Thefrequenoyresponseofenginespeedtobladeangleatmn-stantfuelflowmaybeobtainedbythesameprocedure,

(1.2)

(13)

(14)

AmmATuSml),I!R(lcEmRE

!Q@??2”- Theprtnoipal componentsoftheengineusedfortheinvestigatimareanaxial-flowmmpressor,reve?.we-flowoonibustioncihsnikrs,anda single-staged~ct -coupledturbine.A two-stageplanetarygearsystemprovidesa speedreduotionbetweenthetur-bineandthepropeller.

. .

.

..- ..-— ...--— .- —— . .. .—. — ——— --- .- —.. — —— .-— --- — -.—-——— -- -—--

.

a IUACATN2184

.

.A12-foot-l-in*~imter,4four-bladedpropeller-installedontheengine.The~itch-changingmechanismisa self-containedhydraulicunitlocatedinthepropellerhub. A lever,orbetaarm,onthestationaryportionofthehuboperatesthehydrauliccon-trolvalvesintherotatingpart-sothata linearrelationexists%etweenthebetaarmandthebladeangle.Ea& p=itionoftiebetaarmcorrespondstoa uniquebladeangleduringsteady-stateoperation.

Harmonic-motiongenerator.-Anapproximationofsimplehar-monicmotionbythebetaarmwasobtainedbyconnecti~ittoarotatingcrti-drivenbya direct-currentmotorthroughaworm-gearr~uctionunit.A ~ ratioti100:1betweentheconnect-ingrodand.thecranklengthmadethemotionofthebetaarmvarylessthan1 percentfroma truesinewere.Bycontrollingthearma-turecurrentandmibstitutinsvariousspeedreducersintheharmonic-motiongenercdxm,a rangeoffrequenciescouldbe‘obtained.Thecenterofamplitudeofthesinewavecouldbevaried‘byshift-ingthepositionofthehamonio-motiongenerator,@ vario~amplitudescouldbeobtainedbychanginsthecranklength.

In ofiertovaryfuelflowsinusoidally,thefuelsystemoftheenginewasdisconnectedfrantiemamlfoldailreplaoedbyanexternalsystem,asshowninfigure1. A linewaso-ecteaInparallelwiththemainthrottlein@er toobtaina smallvaria-tionoffuelinputabouta centerofemplitudesetbythemainthrottle.Theharmonic-motiongeneratorusedintheblade-angleruns%asusedto-varytheflowintheparallellinebymeanscd’alinearvalve.Ohemgesin~essuredropacrossthevalveweretoosmalltohaveaneffectupontheline=variationofflowaurlng—transientoperation.

Instrumentation

Steady-statemeasurementsweretakenoffuelflaw,enginetorque,a e-e spe~otomeasureallofthesevariables~oe@ thesientconditionsofoperation.

Propeller-bltieProvisionswere

angle,made

torqueduringtran-

Blade-mglepositionwasmeasumdbytheuseofa potentiometerattaohedtothepropellerbladethatvariedtheflowofcurrentInaneleotriccircuitindirectproporlxtontothebladeposition..Forsteady-statemeesurementi,thecurrentwasmeasuredbyamil-limeter;ati~ transientcotiitimethecircuitwasswitchedto ,‘a recordingoscillographelement.A conventionalslip-ring

.

-- .—— -. -- —- ——-—— -—-—--- ——- —-- -.— -. -..

.I

NA.CATN 21.S4

u arrangementwasusedtocompletetheciroultbetweeneteronthe~opellerhubad therecording,device.tionwasreoofiedina similarmsnner.

9

thepotentiom-Beta-azmposi-

In@er tomeasurefuelflow,anA.S.M.E.orificewasinstdleaInthemainfuellineimmediatelyupstreamoftheenginefuelmani-fold.A bellows-type&ifferential-pressuregagemeasuredthepres-suredropaorosstheorificeandaotuateda smallpotentiometerthatvarieathecurrentina oirouitshilartothatusedforblade-anglemeasurement.Inammzchasflowvariationsweresmallincom-parisontomsanflow,thenonlinearrelationbetweenpressuredropanaaotualflowwasinsignificant.

Theringgearoftheplanetaryreduotionunitoftheengineisrestrainedbya self-balancinghydraulicsystem.Thetorqueoutputoftheenginewasinaioatedbythepressurerequiredtoaotonthehydraulicpistontomaintainbalanoe.Thispressurewasmeaau&edbya Bou?dongage. .

Steady-statemeasurementofenginespeedwasaoooqpllshedbymeans@ a three-phasetachometergeneratorandindicator.For

. reoordingtransientsinenginespeedontheoscillograph,a dire&-currenttachometergeheratorwasutilizedina current-measuringClrouit●

A 10-cycle-per-secondtimingsignalgenerateabyanaudio-oscillatorwasrec~ed ontheoscillographfilmtoshowthetimevariationofthemeasuredvariables.Thesignalwasfrequentlytom-perdtoa standaml60-cycle-per-secondarrentona cathode-rayoscilloscopeforcalibrationpurposes.

Soheunatiodiagramsofthetransientmeasurementcircuitsareshownin figure2. TableI indicatesthesteady-stateandtrans-ientcharacteristicsoftheinstpnts uses.Theoscillographwasusedtorecofionlyincrementsofdeviationfromtheinitialoperatingpoint.Conwntionalsteady+tateinstrumentswereusedtomeasurethelevelatwhichthetransientoocurredandtoevalu-atetheincrements.Theoscillographwaacalibratedfrequentlyduringtherunsforknownvaluesofdeviationas~8suerlbythesteady-stateinstruments.

Prooeduzw.

Frequenoy-responseruns.-Inordertodeterminethefrqzenoyresponseofenginespeed,sinusoidalvariationsweremadein

.

.

— .— -.—. .-. —. ...— —.-— .—— —..-—. .- —.— ..— — . —.. .- —.- --- .-,,

10 XACATN2184..

propeller-bladeangleatconstantfuelflowandinfuelflowat .constantbladeangle.,Thefrequencyrangetobeinvestigatedwaedeterminedbythefollowingconsiderations:Itisshowninrefer-ence1 thatfora first-ordersystemthebreakfrequencyisinverselyproportionaltothesystemtimeconstantandthatanade-quatefrequencyrangetodescribetheunitIsO.1to10timesthisValus. Therefore,byasmimingthetimeconstanttobe5 secondsandc0n8ik%g theengineprtmarilyoffirstofier,therange ofdescriptivefrequencieswasddina tobe0.02to2 radians persecond.-.

-C b*tior ab,outtwoeqtilibfimrum.rhgconditions oftheenginswasinvestigated.Thefirstconditionwas65percentofmaximum.ratedfuelflowanda bladeangleof180(97.5percentofmaxhumenginqspeed).Theotherconditionwas81percentofmaximumratdfuelflow@ a bladeangleof28°(92.5percentofmaximumenginespeed).Theamplitudeofthefoz%ingfunctionwasthesameateachoperatingcondition;namely,a bladeangleof2°and3.9percentofm&imumratedfuelflow.Foreachrun,oscil-lographicrecordsweremadeofthetimevariationofpertinentvariablesduringsteady-statesinusoidaloperation.Typicalrecordsofthesedataarereproducedinfigure3. Thehighfrequencyvari-ationofthefundamentalwavesinthesefiguresresultedfromelec-tricalendmechanicaldisturbancestothesensingelements.

u

.

Equilibriumruns.-lR@libriumdatarequiredtoevaluatetheconstantsofequation(6)wereobtainedbythefollowlngp?ocedure:Fora setvalueof$uelflow,propeller-bladeanglewasadjustedtoobtainnineoperatingpointsbetween85and100percentofmaxi-mumenginespeed.Thesespeedpointsweieusedforfuslflows of

50.0,57.6,65.4,69.1,77.0,and84.5percentofmsximumfuelflowexceptwhentorqueorturbine-outlet-temperaturelimitationspro- ‘hibitedoperationthroughoutthecompleterenge.Enginetorque,enginespeed,fuelflow,andblede-anglemeasurementsweretakenateachoper&kingpointj

RT!SULTS

Experimentally. Response

,

ANDDIMXJSSIOIJ

Detemined3Wequency-Oharacteristics.

Resultsofsinusoidalinputs.-Resultsofthesinusoidalblade-angleandfuel-flowinpu%sasobtainedfrcmtheoscillogra~ic.tracesareshowninfigures4 and5. Amplituderatioofoutputto

. . .— ....— —.-L. .—. -— — — - ————— .. —- .-— ..- . . .,-

. NACATN21.84 11

L inputispresentedonlog-logoootiinatesasa functionoffre-quency.Pha8e-anglelag& theoutputrelatlvetoinputispre-sentedonsemilogcoordinatesasa functionoffrequency.

Theformofthesefrequengy-responsecurvesindioatesthenatureofthedynamiccharacteristicsofthispowerplant.The

~ regularityofdatapointsindicatesthatthesystafO11OWBthesepredictedcharacteristicsbetterthanmightbeanticipated.Fur-thermore,insupportoftheassumptionsmadefortheMfferential-equationanalysis,thesystemappearstobeprimarilya first-ozderlagsystem.Despitethecomplexityofthephysicalsystemandtheprocessesperfomedintheturbinecycle,a fairlysim@emathe-maticalformapproximatestheactualbehavioroftheengipe.

Algebraicformofdata.-ASshowninfigures4 and5,althoughthesystemisprimarilyfirstofier,higherder effectsarepresent.Increasedattenuationoftheoutputamplitudeatphaseshiftsapproaching1800indioatesthata second+fier approxima-

“ tion-maymoreexactlydescribethedata.Anequattonofthefol-lowingformhasthereforebeenfittedtotheexperimentaldata:.

Theeqmtionwasmatchedtotheexperimentaldataatthesteady-statepointandatphaseshiftsof450end900,therebydetetiningthethreevariablesoftheequation,K, T1, and T2.Resultsofthesecomputationsarepresentedincolumn3 O*tableII.

A maximumerrorinthevalueofthesecondtimecon&antT2of0.1secondcouldbeattributedto “instrumentlags;thenumericalvaluesforthistimeconstantT2 arethereforepresentedonlytoindicatetheorderofmagnitudeofthissecoti-ofiereffect.Becausethefuel-measurementdevicewasslmwerthantheblade-anglemeasurementcircuit,instrumentationerrorwouldreducethe ‘valued T2 inthefunctionsofenginespeedtofuelflowinput.Theoppositetrendofexperimentalresultsindicatesthatthesecond-ofiereffectisanenginephenomenon.Thefactthataratiointheorderof10:1extstsbetweenthetwotimecon-stants71 and T2 makesthesecond-ofiereffeetnegligiblefor- problm ofcontrolanalysis.

Inordertocomparetheformofthealgebratcapproximations. withtheactualdata,crossplotsoffigures4 and5 arepresentedonfigures6 and7,respectively,togetherwithcurvesofthealge-braicexpressions.Thecalculatedfirst-ordercurvesoffigures6. and7willbesubsequentlydiscussed.

—— .--— .- . —— .— —- — ..— — -—..—

12 lJACATN2184

Comparisonoffrequency-responsefunctions.-Variationofthefirst-ordertimeconstantwithoperatingconditionsmaybeofmoreimportancethanthesecond-ofiereffect..A comparisonoffirst-omiertimeconstantsintableIIshowsthattheseconstantsnotonlyvarywiththeengine-~owerTointebutalsovarywiththeforcingfunctionattheseinepowerpoint.Foranexactdescriptionofdynsmiobehatiorofenginespeed,extensivedeta~ berequiredtodetezminethetrendaftransientcharacteristics.

Linearityof@ amicbehavior.-Forpurposesofanalysis,anassumptionwasmalethatengines~ed.wasa linearfunctionoffuelflowendbladeangleforsmalltransients.Inordertoevaluatethisassumptionfromtheresultsofexperimentaldata,theformoftheoutputsreoordedonoscillographicfilmwerecomparedtoan_icti sinewave.Itwesfoumlthatforinputsresultinginengine-speedchangesofasmuchas6 percentofmaxlrnumspeedtheformofthespeedsinewme variednegligiblyframanexactsinewave.&eaterinputamplitudesresultinginlargerspeedoscil-lationscausedappreciabledistortion,theeffectbeinggreaterat .low-powerpoints.Anassumptionoflinesrityistherefore@ti-fiedonlyforrestricteddeviationsontheorderof3 or4 percentfrcmanequilibriumpoint.

An&l@icallyDeterminedFrequency-ResponseChemcteristics . .

Resultsofequilibriumruns.-Therelationbetweensteady-statevaluesofenginetorqueandfuelflowisplottedinfig-ure8(a)forseveralvalues-ofconstantspeed.Crossplotsforlinesofconstantfuelflowareshowninfigure8(b).I’i~ 9(a)isa plotofpropellertorqueagainstbl~e angleforvariousval-uesofconstantspeedandcrossplotsforlinesofconstantbladeangleareshowninfigure9(b).Inofiertoshowtheequilibrium~ cations aboutwhichsinusoidal-dataweretaken,twolines&fconstantbladeangle,18°andfigure8(b).

Lossestotheaccessories@ theneglectedincalculationsofpropeller

28°,aresuperimposedon

gear-reductionunitweretorqlle.

R- e ofllnearity.-Inthederivationofequations(1)and.(2),anassumptionwasmadethattherelationspresentedinfigures-8 and9~ beconsideredasstraightlinesforsmalldevi-ationsfroma givensteady-state point.Theactualdataindicatethatthisassumptionisjustifiedfortheenginechaxmteristics

.,

overratherlargeregions,butthepropellercharacteristicsshould

—-. . --—— — -- —-- ——-——— -— -.

.—— .— ___ -- ._ ..__ .._. _ . ._ - _

.

.

.

.

NACATN2184

beconsideredline~onlyin.verysmallregions.theseenginedataasobtainedundersteady-state

Furthermore,conditionsof

13

operationmaynotbeaccurateundertransientconditionsiftheturbineisrequiredtooperateata @nt farfromthedesigncon-dition(reference2).

Analyticalsinusoidalresponse.-l&eqwncy-respcnsecharac-teristicsoftheenginewerecalculatedforoperatingpointsatwhichexperimentalsinusoidaldataweretaken.SValuesofthecon-stantsal, bl, cl, and dl foranoperatingconditionwereobtainedfromtheslopesofthecurvesoffigures8(a),8(b),9(a),and9(b),respectively.Withvaluessodetermined,equations(10),(11),(13),and(14)wereevaluatedfora rangeoffrequenciescoincidingwiththatusedintheexperimentalruns.Results.&thesecalculationsareincltiedinfigures6 and7.

JC@braioformofanalgticalresults.-‘l!healgebraicformofthefirst-ofiercurves~esentedinfigures6 and7 isshowninequations(9)ahd(12).Theseequationswereevaluatedfortheselectedoperatingpointsandtheresultsarepresentedincol-umn4 oftableII.

Fortheassumptionsmadeintheanalysis,theen$inetk con-stantisindependentoftheforcingfunction,SEshowninequa-tion(7).Variationofthe constantsforthetwopowerpoints .resultsfromthenonlinearityoftorque-speedcharacteristicsof .theengineandthepropeller.Themagnitudeofthisvariationissufficienttolimittheutilityofananalysisinwhichlinearapproximatlonaareassumed.Forstudiesofbehaviorwithinrestrictedregions,theanalysisissatisfactory.

ComparisonofDifferential-EquationSolutionwith~erimmrklResults

Comparisonof form.-Calculatedfrequency-responeechal%oter-isticsaresimilartotheexperimentalresultsatlow-frequencyPalues,ssshownbyfigures6 and7. Becauseofhigher-ordereffects~ibitedbytheexperimentaldata,deviationbetweenthetwosetsofcurvesincreasesathigherfrequencies.!l?hiseffectisparticularlyevidentinthephaseshiftoccurringathighfre-quencies.R?esenceofphaseshifts&eaterthan90°intheenginesystem~ l~t theover-allloopgainforstabilityina combinedengineandcontrolsystem.Fortheusualcaseinwhichthecontrolcontributesat least 90°additionalphaseshiftto-tieenginepolardiagram,thegainatthe180°phaseshiftpointhasa definitelimltaocofilngtotheNy@st criterion(refemzmce1).

,

.

-.. . . . .. . . . .. .. . . . . ------ .. . .. . . ______ _____ _____ . . .—- ——.. .-- . . . ..—. . —.

14 NACATN21.84# ,

Comparisenofnumericalvaluea.-Ingeneral,thevalueoftheexperimentaltdmeconstantislowerthanthevalueoftheana-lyticaltimeconstantfora givenpoweroperatingpoint.Thesig-nificanceofthisvariationdependsonth6particularapplicationofthedata;forsaneproblemsofcontrolanalysisa rangeofvar-iationof2 to1 intheenginetimeconstantcauseslittlechangeintheccmibinedengine-controlsystem.Discrepanciesbetweensteady-stategainvalues,analyticalandexperimental,cau%0accountedforbythefactthattheanalyticalvaluesarecalculatedframslopesofequilibrium-torquecurves.

CONCLUSIONS

Froma sea-levelstaticinvestigationofthefrequency-responsecharacteristicsofa turbine-propellerengihe,thefol-lcwingconc?luaionsmaybedrawn:

1. Enginespeed3.sa linearfunct’ionoffuelflowand *pzmpeller-bladeangleduringtransientoperation,U deviationsfromtheequilibriumoperatingWintareontheorderof3 or4 percent.

2.Thefrequency-responsefunctionsofenginespeedtofuelflowandenginespeedtobladeanglebuthigher-ofiereffectsexist.

3.A differentialequationthatspeedasa functionoffuelflowandfromsteady-statedata.

ImisFlightPropulsionLaboratory,NationalAdvisoryCommitteefor

Cleveland,Ohio,March24,

areprimarilyfirstorder,

clGaelydescribesenginebladeanglecanbederived

Aeronautics,1950.

,

..— -— —.—. — ,—. — . -—. - — ——. ..- —- -...

NACATN2184. 15#

APPENDIX- SYM80LS

Thefollowingsymbolsareusedthroughoutthisreport:

psrtialderivativeofenginetorquewithrespecttofuela

al

b

bl

c

c1

partialderivativeofengine-torqueparsmeterwithrespect%

to

1

u

~ JqI R2‘e(max)le 1+1

npercentagemaximumfuelflow, L A

*f#

a—‘f(max)

1sec

absolutevalueofps@ialderivativeofenginetorquewithrespect

to

()a% lb-fttoenginespeed,~ rpm

partialderivativeofengine-torquepmmeterwithrespect .

Qea‘e(mix)le

percentagemaximumenginespee~ 11+%1 1Ne ‘=\

‘=partial derktiveofpropellertorquewithrespecttoblade

psxtialderivativeofpropeller-torqueparameterwithrespect

a %

‘[ 1,Np(max)% 1 + *

tobladeangle,P (secj(deg)

.

,

-. --— —...- . ..— — —.-— — ———— .— —-. ... ———. ---——---

.

16 NACATN 2184 .,

d

K

partial derivativeofpropellertorquewithrespecttopro-—

()%lb-f%pellerspeed, ~ —P-

partialderivativeofpropeller-t&queparemeterwithrespecttopercentagemaximumpropellerspeed,

%a-”

[- 118

‘p(-).% 1 + ~@2 ~.—Ifp Jsec

b~p(mu)

fUIICtiOIlof Wf>Ne

functionof 13,1?p

polar momentofine$tiaofrotating-s ofengine,(lb)(ft)(see)(rad)(min)/revolution

polarmomentofinertiaofpropeller,(lb)(ft)(see)(red)(min)/revolution

Srbitraryconstantal

K1 constantdefiningequilibriumcondition, ‘1 = bl +al>

K2 c1 ~constantdefiningequilibriumcondition,K2= —bl+al’deg

Iie engine speed., rpn

‘P propellerspeed,rpm

~ engine tor~e,lb-ft

R gearratioofengines~edtopropellerspeed,

—.. —_....—_.__——_ .- —— ——— . . ..— —--—.-—_ .. ..

NACATN21842

17

Wf fuel flow,lb/hr

P propeller-bladeangle,deg

$~ betaarm,usedtodenoteleverinputtomechanismforchang-ingpropeller-bl~eangleonengineusedtorundata

A deviationfrominitialoperatingpoint

T enginetimeconstant,sec

(Ll frequency,radians/see

Subscript:

max maxilmn

REFERENm

1. Brown,GofionS.,andCempbell,DonaldServomechanisms.JohnWiley& Sons,

P.: PrinciplesofInc., 1948.

2.Otto,EdwardW.,arilTeglor,jetEngineConsideredasa1950.

BurtL.,~: -CS ofa Turbo-Quasi-StaticSystem.NAOATN2091,

#

.

.

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18

,

NACATN2184

Ieasuredvariable Steady-etate!hSIIBiSlltchmacterletice,accuracy

Bladeengle,B “$0 MmitedbyOsciuographelement(40cps).

Betaam> Pa 1 percentof LlmltedbyosclUographfulltravel element(40cps)

Rlelflow,Wf d percentfor 1O-I.5Cpsfullflow

T~us,Q - percentfor -------------fullscale

2t@nespeedJNe * of1 percent Limitedbyfilterdrcuitforfullscale (1.59Cps)

.L

.

TABLEII- I?REQmcY-mmm FumTIoN8

1 2 3 4

Frequency- Operatingpoint Algebraloformoffre- Algebraicformofreeponee qumcy-reeponsefuuc- frequency-responeefunct@n tionfrom~erimemtalfunctionfromcal-

aata culatedequation

K %

(l+iT@(l+iT~@ (l+iwo)

‘efie(uax)(iu) wf=65percent -1.06 -1.06M . (l+i3.05@)(l+10.283@) (1;13.756$

of madmumrated(percent/tieg)

Wfdxlpercelt - -1.73 -1.76(l+i3.35@(l+io.&@ofIcaxtmumX73ted (l+i2.86@

K1General . (1+i@71+i~2@ (l+iT@

m-w) g4.8° (1+13.32@~~+iO:375@)(l+i:75@)(percent)

~=28°(l+i2.73@~+i0.332@)(l+if86@)

\N A-

m

— --- .-—----- . . . ——. -.. ----— .—— ...— .

,I

Fuel

“ax

a

Es!

ltotor-3rlven orarikfor 81nu80Mal tnput

!i

fuel pump

manifold

3Lerge-clotmanifold

Figure1. - Systemfor provldlng slnusohlal fuel flow.

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Set zeroingvoltage

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Ckdvanotneter

(a) Typioal oircwit for (b) Circuit for transient-spe~dmeasurement of positlwh measurements

Figure 2. - Sc!hematiodiagrams of transient response instrumentation.

*

NACATN2184 21

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Betaarm,pa

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(a)

Figure3.-

“

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Typioaldatashowingsteady-statesinusoidalresponseofenginespeedtobladeangle.Fuelflow,81peroentofmaximumratedjmeanenginespeed,92.5peroentofmaximum;meanbladeangle,280;frequenay,0.986radianperseoond4

Typicalsteady-statesinusoidaldataforturbine-propellerengine.

.. . . .— . ...... —-—— .-—-”—y ———— ——— — r-————— —— —.—--,..

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22

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(b)

Figure

Typioal datashowingsteady-statesinusoidalresponseof enginespeedto fuelflow. Bladeangle,18°;meanenginespeed,97.5percentofmaxhum;meanfuelflow,65percentof msxhmm rated;frequency,0.873radianper second.

3* - Concluded.l’ypiealsteady-stateturbine-propellerengine.

sinusoidal datafor

..—. _ -....- .- ------ —-— --—— .- .= __ — —

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maxiammratea;mean

angleamplltude,*.ms%lmmjmeanbladeengle,MO; blade-

Flgwe4.- Fropell=-bladeangle- engine-speedrelaticmsatmnetanttielflowobtained l’rmnexP@-tsl EhQSOidEl respensedata for typicaltvrbim-propellereoglne.

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Flgwe4.- Continued.RweU~-b~e Q - e@M -meedrelationsatoonstaatfeelflowobtainedframexperimentalainuoldalresponeednt.efar@’@d81turbineqmowllaren@ie.

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(o)Amplitude-mtiorelaticui.Feel“ktow,61pereentofme%immrated;meena@ae epeed,96.5peroentefmsxlmum;meanbladeangle,260jblade-englea~lltude,*.

Figure.4.-Ceatiuued.Propeller-blademwle- ~e-SPe* M18tioMSteOIIStfLDtfuelfkmobtainedfrctne~lmaitalalaneoidelrewuieedatafor&Pl~l t=b~z-~= ewfi.

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ratodjmean

angleamplltude,~.mexlmm:mea bladeengle,W; blade-

F@ure 4.-Conoluded.Prepellem-bladeangle- mglae-speedrelatiensateenetentfuelflowobtalaed*●xperimentalaiaueoidalreepmae datafor~iunl turblxm+mqmller-e,

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Pated meen!_; bladeen@, 160jfuel-flowamp itude,,

3.9 paOm Or ~

F&nw 8. - Fnel-flow- mgine-speedrelhtlmeat!acmebantbladeangh obtained-cmexperimentalsinusoidalreencmeednbafortiionl turblae-pmpellerengine.

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ratedJmmllmnximmjbladea@e, @j fnel-flmamplitude,

5.9permit or ~

Figure5. . Oontinnad.Puel-flw- ●xine-speadrelationnat ocantantbladeangleobtained .fraue%perlnmnhlBimusoldalronpmmedatafor~ioal tnrblne~lla mighe.

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~WexmyS CdBradlana/aeo

Amplitude-ratiorelatlcm.Meanfuel flow,81percent ~fenginespeed,92.5peraentofmaximum;bladeangle,88”;3.9peroentof nmxlmum

maxlmwnrated;meanfuel-flowamplitude,

-Continued.~1-t’low-mglne-speedrelationeatoonatentbladeangleobtained.. -.frcauexperimentalsinusoidalreaponaedatafor typZoalImrblne-propellerengJne..

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(a) Fuel flow, 66 peroemt of maxlmmm rated; mean englae speed, 97.6 peroemtiofmaximumj mam blade engle, 160.

Figme 6. - Frequemey respcmae of emgine speed *O veriatlcmnlm propeller-bladeangle obtainedfrmo erperfmentaldate ad theoretdoaloaloulatloneror typicalturblne~opallem englna.

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(b)Fuel flow,81 p?meentof maxlmomrated;men enginespeed,92.5pel$omltor mximum; mean bladeamgle, 680.

Wure 6. - Coaeluded. Frequenoy responee of eobtained~ experimentaldataand thaoretio-.

&&l:Od to variationsin pyeller-blade angleatienefor ty’ploalturbn~~ellep !a

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% ys. o .4*Q

Experimentalt

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—--— Calculated, first order /“.

1’. 11 :

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(a)Mean fuel flow, 65peroantof maximumrafmdj mean engineepeed, .97.5 peroantof maxhmne;blade angle,18°.

Mgure 7* - Fraquenoyresponseof engine epeed to va+ations in fuel Slow obtainedfhmaxpwrimentaldata and &ear etioal oaleulatkne for tgpioalturbine-propellerengine.

E’la

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11-lo 0 10 80 M 40 so 4

1-00

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Figure 7. - Conolnded. Frequanoyreapcmeeof engineepeed to variatlme k fuel flowobtainedfrom experimentaldata and themetioal oalmlatione for typioalturbine-propellerengine,I

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t{ 14 } ‘ / 1/ / ‘1

.01 , I40 46 56 64 72 60 06 96 104

PercentageofmaximumratedfuelflOWSWf/hf(H)

(a)Conetantengine-epeedoharaoterlatioa.

Figure 8. - Relationbetweenengine-torqueparemeter,percentageofmaxlnumand roentageofaaxlmumratedfuelflowae obtained fromturb~e-pzwpellerengine. ~tal claw-f% g:h

o

‘\-.—— ..— _ _. .._ _. —-— -. ——-— .._-_ .__. ____,. —— —. —. —____ _____ .. . _ .

.-. —... .. . .. . .

I I I 1

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0 I 184 86 88 90 m 84 96 96 100

(b) oonatantfuel-flowcba=mtertetloatioludings,eleotedoperatln8lines.

Figure8.- Ocmolmded.Relationbetweenengine-torqueparameter,pwoentageofmaxlmom-e epe~,andPercentagemaximomratedfuelflowaaobtainedfromoxerlmantaldatafortypicalturblne~ller engine. Y(Oross-plottedfromfig.8(a● )

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(a) Ocmatonb mopeller-speed aharaoteristilos.

$’@me 9. - Relatlcuebehwm prowller-torqnePametmr, peroaitngenmxisumpropellerspeed,andbladeanglefor U3-foot-l-fnohdinmetqfour-bladedprqeller.

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