nationaladvisorycommittee foraeronautics
TRANSCRIPT
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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,
, ,.,
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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.
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..- ..-— ...--— .- —— . .. .—. — ——— --- .- —.. — —— .-— --- — -.—-——— -- -—--
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a IUACATN2184
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.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
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-- .—— -. -- —- ——-—— -—-—--- ——- —-- -.— -. -..
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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
—-. . --—— — -- —-- ——-——— -— -.
.—— .— ___ -- ._ ..__ .._. _ . ._ - _
.
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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.
[email protected]‘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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,
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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@)
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Fuel
“ax
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ltotor-3rlven orarikfor 81nu80Mal tnput
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fuel pump
manifold
3Lerge-clotmanifold
Figure1. - Systemfor provldlng slnusohlal fuel flow.
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(a) Typioal oircwit for (b) Circuit for transient-spe~dmeasurement of positlwh measurements
Figure 2. - Sc!hematiodiagrams of transient response instrumentation.
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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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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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(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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Rreqmnoy,w, radlud-
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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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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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(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.
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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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PercentageofmaximumratedfuelflOWSWf/hf(H)
(a)Conetantengine-epeedoharaoterlatioa.
Figure 8. - Relationbetweenengine-torqueparemeter,percentageofmaxlnumand roentageofaaxlmumratedfuelflowae obtained fromturb~e-pzwpellerengine. ~tal claw-f% g:h
o
‘\-.—— ..— _ _. .._ _. —-— -. ——-— .._-_ .__. ____,. —— —. —. —____ _____ .. . _ .
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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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