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NATIONALADVISORYCOMMITTEEFORAERONAUTICS

TECHNICALNOTE3936

EXPERIMENTAL INVESTIGATION OF TEMPE&4TURE FEEDBACK

CONTROL SYSTEMS APPLICABLE TO

TURBOJET-ENGINE CONTROL

By C. E. Hart, L. M. Wenzel, and R. T. Craig

Lewis Flight Propulsion LaboratoryCleveland, Ohio

LIBRARY00PYMAR191957

MN(REYAMOUAIJtiCALLABORATORYUBRA.W,NAGA

UWJGLEYrts. u. vIHUINIA

Washington

Marchi957MYc

IIlrlllll!llilllllll[lllllllllllll31176014340872—.

NATIONALADVISORYCOWTTEE FOR AERONAUTICS.

TECHNICALNOTE3936.. z

EXPERIMENTALINVESTIGATIONOF TEMPERATURE

.

FEEDBACKCONTROL

SYSTEMSAPPLICABLETO TURBOJT?I-ENGINECONTROL

By C. E. Hsrt,L. M. Wenzel,andR. T. Craig

SUMMARY

Twobasictemperate feedbackcontrolsystemswereinvestigatedasmeansof controllingtail.pipegastemperatureof a turbojetengineduringtransientoperationin thehigh-speedregion.

A proportional-plus-integralcontrolin a temperature- fuel-flowcontrolsystemprovidedsatisfactorytransientresponseto a desiredstepincreasein temperature.For a temperature- exhaust-nozzle-areacontrolsystem,itwasnecessaryto addnonlinearcomponentsto thebasic

q proportional-plus-integralcontrolto providesatisfactorytemperatureEl0. responseduringtransients.

Severalcriteriafor selectingcontrol-loopparametersforoptimumtransientresponsewereinvestigated.For thetemperature- fuel-flowcontrolsystem,minimizationof timeintegralsof eitherthe squareorabsolutevaluesof temperatureerrorseemedto be moreselectivethanothercriteriain determiningoptimumcontrol-loopparameters.For thetemperature-areacontrolsystem,none-ofthe criteriaprovedadequateinselectingoptimumloopgain,buttheydidindicatea choiceof controlintegraltimeconstant.

Enginedynamicsin thehigh-speedregionweredeterminedby synthe-sizingtransferfunctionsto matchexperimentalfrequency-responsedata.Overthe controloperatingregioninvestigated,enginetemperature- fuel-flowdynamicscouldbe satisfactorilyrepresentedbyanda deadtime. Similarly,enginetemperature-arearepresentedby”first-ordertermsanda deadtime.

INTRODUCTION

first-ordertermsdymmics couldbe

M&t controlsystemsforturbojetenginesarebasedon useof enginespeedas theprimarycontrolledvariable.Enginetemperatureshavebe-

. comeimportantmainlythroughuseas damage-preventionlimitsandin ac-celerationschedules.Emphasison speedcontrolis,inpsrt,dueto theeaseof measuringspeedandthereliabilityof speedmeasuringdevices.

●

2 NACATN 3936—.

In orderto providesatisfactorycontrolin thehigh-epeed.operatingregion,it hasusuallybeennecessaryto adda temperature-limitingdeviceto the speedcontrolsystemto prevent–damageto turbinecomponents.In-steadof usingtemperaturelimitingas an auxiliaryto a speedcontrolsystem,it is possibleto usetemperatureas theprimarycontrolledvar-iable. Thisreportis concernedwiththeexperimentalinvestigationoftwobasictemperaturefeedbackcontrolsystems,one–usingfuelflowtocontroltemperatureandtheotherusingexhaust-nozzleareato controltemperature.Althougha temperature-meacontrolwouldnotbe feasibleas theprimaryenginecontrolsystem,it wouldhaveapplicationas partof a two-loopsystemwhichincludesspeed- fuel-flowcontrol.

Theobjectof thisinvestigationwasto deterulinesomeof thechar-acteristicsandcapabilitiesof temperaturefeedbackcontrolsystemsforturbojetengines.The abilityof thecontrolsystemto controltempera-tureduringtransientoperationwas evaluatedby consideringtimeinte-gralsof functionsof temperatureerroras wellas gvershoot-ratiosand

—

responsetimes. —

APPARATUS

Engine -.

A turbojetenginewithan axial-flowcompressor,vaporizing-.type

r.

fuelinjectors,anda two-stageturbinewas ifistalledin a sea-levelstatictestfacilityforthisinvestigation.

FuelSystem

Theoriginalfuelsystemof theengine,withtheexceptionof theflowdividersandinjectors,was supplantedby a research-facilityfuelcontrolwhichwasdesignedforveryrapidfuel-flowchanges.Thiscontrol,describedin reference1, consistedof a reducing-typedifferential-pressureregulatorthatmaintaineda constantpressuredropacrossathrottlevalve. An electr~hydraulicservosystem;as usedto varythe””throttle-valvepositionlinearlywithinput–signal.Throttleareavariedlinearlywiththrottle-valvepositionand”thusfuelflowwasproportionalto inputsignal.

——.

Frequency-”responseof throttle-valvepositionto inputsignal,givenin reference1, showsthatthistypeof fuelcentralwouldnot contributesignificantdynamicsto thecontrol.systemwithinthefrequencyrangeinvestigated.

.

NACATN 3936 3

Vaxiable-AreaExhaust-NozzleSystem

Nt-lN

A sketchof thevariable-areaexhaust-nozzlesystemis shownin fig-ure1. It consistedof dualbutterflyvalvesmountedin thetailpipeabout9 feetdownstreamof theturbine.The shaftsof thetwovalveswereconnectedby a mechanicalcrossarm-typelinkage.Theuppershaftwasdrivenby thepowerstageof an electrohydraulicservosystem.Areavar-iationsfrom73 to 13.3percentof ratedareawereobtainablewiththissystem.Thisexhaust-nozzlesystemwas designedforfastresponsewith-outconsiderationof goodthrustcharacteristics.

+Figure2

positionto a10percentof

showsthefrequencyresponseof the lowerbutterfly-valvesinusoidalinputsignalfora totalareaexcursionof aboutratedsrea.

Instrumentalion

~ Enginespeed.- To measureenginespeed,a magneticpickupwas in-2 stalledin the compressorhousingoppositea rowof steelcompressorrotorl-l blades. Eachbladepassingthepickupproduceda voltagepulsewhichwasA fedto an electroniccircuitwhoseoutputwas a voltageproportionalto.0 thenumberof pulsesper unittime. Thisvoltageprovideda goodindica-

tionof instantaneousenginespeedduringtransients,sincethepickupandelectroniccircuithadno measurabledynamicsin thefrequencyrange.of interest.

For steady-statespeedmeasur~ent,thepulsesfromthemagneticpickupwerefedto a digitalcounterwhichdeterminedanddisplayedactu-al enginerpmwhenthepropertimeintervalwas selectedforthe countingcycle.

Tailpipetemperature.- Fouxrakes,eachconsistingof fourchromel-alumelthermocouples,wereusedto measuretailpipegastemperatureduringtransients.Adjacentthermocoupleson eachrskewereconnectedin seriesto formeightpafrs. Fromoneto eightpairswereparalleledto providethe inputsignalto thetemperaturepreamplifier.

As a compromisebetweenfastresponseanddurability,22-gaugewirewas usedforthesethermocouples.Withthissizewire,thetimeconstantof thethermocoupleswas approximately0.3secondin theregionof opera-tionselectedforthe controlstudy.

Thermocoupleson twoadditionalrakes,similarto the onesjustde-scribed,wereconnectedto a self-balancingpotentiometerfor steady-statetemperaturemeasurement.

4 .L - NACATN 3936.

Fuelflow.- T& positiono~the throttlevalvein thefuelcontrolwasmeasuredby meansof a linearvariabledifferentialtransformer.Theresultingpositionsignalwasthebestobtainabl%lndicationof fuelflowduringtransients.Actualfuelflowintotheengineis affectedby thedynamicsof thefuelmanifoldandflowdividersbetweenthefuelvalveandtheengine.

A rotameterwa~ usedforsteady-statefuel-flowmeasurement.

Exhaust-=nozzlearea.- A rotaryvariabledifferentialtransformerwas usedto measurethepositionof the lowerbutterflyvalve. The sig-nalobtainedwasusedas an indicationof exhaust-nozzleareaduringtransients,althoughactualareadidnotvaryexactlylinearlywithposi-tionof thebutterflyvalves.

i%PM

For steady-stateareameasurementthepositionof the lowerbutter-flyvalvewas indicatedby a precisiondirect-currentmeterwhichmeas-uredthevoltageon the armof a linesrwire-woundpotentiometercon-nectedto the lowershaft. Accwrate&ureadeterminationcouldbe madebyusingthemeterreadingandan areacalibrationcurve.

Recording’equipment.- Transientdatawererecordedon a direct-writingsix-channeloscillographrecorder.Therecorderpens,whenusedwithequalizinga~lifiers,hada frequencyresponseessentiallyflatto100cyclesper second.A recorderchsr%peed of 25 millimetersper sec-ondwas used.

For sinusoidaldataa galvanometriclight-beamoscillographrecorderwas used. Flatfrequencyresponsebeyond100cyclesper second,highsensitivities,andhighchartspeedswereobtainablewiththisequipment.

Analogcomputing-eqgi~ent.- CircuitsforcalculatingtimeIntegralsof thesquareandabsolutevalueof-temperatureeirorwerasetup usinglinearandnonlinearelementsof anelectronicanalogcomputer.These.circuitsareshownin figure3. A two-stageR-Cfilterwas usedto re-ducenoisein thetemperature-errorsignal.An electronicmultiplierper-formedthe squaringoperation.The integralvaluesobtainedwereusedinevaluatingtransientresponseof controlledtemperature,as willbe dis-cussedin the sectionControlledEngineData.

CONTROLSYSTEMS

A blockdiagramrepresentativeof thebasictemperaturefeedbackcon-trolsystemsusedforthisinvestigationis shownin figure4. In these .systemstheindicatedtemperaturesignalwas comparedwiththedesiredtemperaturereferencelevel,andthedifferenceor ~ror signalwas appliedto theinputof thecontrol.The controlmodifiedthe errorsignalac- *cordingto thecontroltransferfunction.Theresultingcontrolsignal

NACATN 3936 5

was appliedto the inputof thefuelor sreaservosystem,whichchangedfuelflowor areauntilthetemperature-errorsignalbecamezero. Pro-visionfor stepchangein temperaturereferencelevelwas includedfortransientoperation.

Temperature- Fuel-FlowControl

Thetemperaturesummingnetworkandcontrolcomponentsforthetemperature- fuel-flowcontrolsystemwereset--upon an electronicana-logcomputeras shownin the circuitdiagramof figure5(a). Controlgainandintegraltimeconstantcouldbe variedseparately.Alsoshownin figure5(a)is a rate–limitingcircuitthatwas insertedat pointBin the controlloopto limittherateof changeof fuelflow. The ef-fectsof slowerfuel-valveactionon control-loop-parameterselection ,willhe discussedin the section“Rate-limitedfuelcontrol.”

Temperature-AreaControl

FigureS(b)showsthe computercircuitdiagramof thetemperaturesummingnetworkandcontrolcomponentsusedforthetemperature-areacontrolsystem.The controlcircuitryforthissystemwasmorecom-plexthanthatof thetemperature- fuel-flowcontrolsystem.Becauseof resultsencounteredin a previousspeed-areacontrol-systemstudy(ref.2),it was foundnecesssrytoaddnonlinesrcomponentsto the con-trol. Thesecomponentswerea diodelimiteron the inputto the areaservocontrol,anda relayandrelaytriggeringcircuit.

Duringthepreviousspeed-areacontrol-systemstudy(ref.2),it wasfoundthat,whenthe areabutterflyvalvesopenedto a positioncorre-spondingto approximately113percentof ratedarea,furtherincreaseininputsignalto theareaservosystemdidnot increasearea. Thiswasdueto the limitationon maximumareaobtainablewiththebutterflyvalvesfullyopen. Thus,whenoperatingwitha closed-loopsystemandaproportional-plus-integralcontrol,thecontroloutputsignalat timeswouldexceedthe areaservo-systeminputlevelcorrespondingto the satu-rationlimit(113percentratedarea). Whenthishappened,an excessivechargewouldbuildup on the integralcomponentof the control,whichdelayedthe closingof theareavalvewhenthe errorsignaldecreased,.Thisproducedpoorcontrolperformance,thatis, largeovershoots.

Forthetemperature-areacontrol-systemstudy,thebiasvoltagesofthe limiterdiodeswereadjustedto limit---theinputsignalto the areaservosystemto valuescorrespondingto approximately85 and 113percentof ratedarea. Themaximumareawas determinedby theparticulararea-valvedesignandtailpipethatwereused. Theminimumareawas arbitrar-ilychosen.

NACA’IN3936 *

Thebiasvoltageof’the relaytriggeringcircuitwasadjustedsothattherelayclosedwhentheinputsignalto theareaservomotorcontrol -decreasedbelowthevaluecorrespondingto 85 percentof ratedsrea.Closingtherelay,in effect,stoppedtheintegralactionof thecontrolby addingan equalbut opposite-signerrorsignalto the inputof thein-tegrator.A similartriggeringcircuitcouldhavebeenaddedto oneratetherelayat themaximumareastudybecausethe initialandtowardthe closedposition.

The experimentalprogram

limit-butwasnotdeemednecessaryf& thismaximumexcursionof theareavalveswas

PROCEDURE

consistedof (1]a determinationdynamicsin thehigh-speedregionof%perationand (2}a study

of engineof con-

trolledenginetransientresponseandcontrolstabilitylimitforthetwobasictemperaturefeedbackcontrolsystems.

Enginedynamicswereobtainedfromfrequency-responsedata. Sinu-soidalinputsignalswereappliedto thefuelcontrolandareacontrol,andresponsesof enginevariableswererecorded.Frequency-responseplotsweremadeandanalyticaltransferfunctions,madeup of engineandinstrumentationdynamics,werefittedto theseplots. .

The enginewas operatedon controlusingthesystemsshownin fig-ures4 and5. Stepinputswereappliedto increasethetemperatureref- .

erencelevelan amountequalto approximately7 percentof ratedtempera-ture. Controlgainandintegral_timeconstantwerevariedQvera rangeof values,andresponsesof enginevariableswere--recorded.Timeint%-gralsof the squareandabsolutevalueof.temperatureerrorwerecalcu-latedandrecorded.A rate-limitingcircuitwas addedto thetemperature- fuel-flowcontrolsystem,-andthe sae procedurefollowed.The stabilitylimitforbothcontrolsystemswasinvestigatedby increas-ingcontrolgainuntilthe systembecsmeandremainedoscillatory.

RESULTSANDDISCUSSION

EngineDynamics

Responseof temperatureto fuelflow.- Experimentaldatarepresent-ingthe frequencyresponseof indicatedtailpipegastemperatureto indi-catedfuelflowat constantareaareshownin figu& 6. Thesedataweretakenin the ssmeregionof engineoperationas selectedforthecontrQlstudy. Thedatashownin figure6(a)andallsubsequentamplitude-ratio .datahavebeennortilized(thatis,theproductof allconstantgaintermshavebeensetequalto unity). .

NACATN 3936

Thenormalizedtemperature- fuel-flowtransf’eringto thedataof figure6 canbe representedby

7

functioncorrespond-

1where 1 + ‘c~s representsthethermocoupledynamics(symbolsarede-

‘td,fs:Q finedin appendixA),G(s)e representsthefuel-distribution-system* andcombustiondynsmics,andtheremainingtermsme includedin the actu-

al temperature- fuel-flowenginedynsmics.Combustiondynamics,usuallyconsideredpsrtof enginedynamics,couldnot easilybe separatedfromfuel-distribution-systemdynamicsbecauseof the difficultyof measuring

1 + Tlsactualfuelflowintothe engineduringtransients.Theterms ~ + =9E

&

wereincludedto representa longtime-constantphenomenonbelieveddueto a heating-expansioneffectin theturbinesection.Thisphenomenonhasusuallybeenneglectedinpreviousdynamicstudies.

.

.

.

In figure6 areshowncalculatedfrequency-responsecurvesof equa-tion(1). Numericalvaluesforthevsrioustermswereobtainedby graph-icalmethodsdescribedin appendixB.

Responseof temperatureto area.- Figure7 showsdtiarepresentingthefrequencyresponseof indicatedtailpipegastemperatureto exhaust-nozzleareaat constantfuelflow. Theseexperimentaldatawerealsotakeninthe sameregionof engineoperationas selectedforthe controlstudy.

Thenormalizedtemperature-areatransferfunctioncorrespondingtothe dataof figure7 canbe representedby

(2)

Lwhere 1 -(-Tts representsthethermocoupledynamicsandthe remaining

termsrepresenttheactualtemperature-meaenginedynamics.Alsoin-cludedin thistransferfunctionarethetwo longtime-constanttermspreviouslymentioned.

Calculatedfrequency-responsecurvesof equation(2)arealsoshownin figure7. Numericalvaluesusedforthevsrioustermswerealsoob-tainedby graphicalmethodsdescribedin appendixB.

8 NACATN 3936.

Stability-limitcheck.- Fromthefrequency-responseplots,predic-tionsof stabilitylimitsforthebasictemperaturefeedbackcontrolsys- -ternscanbe made. For thetemperature- fuel-flowcontrolsystem,thedatapresentedin figure6(b)indicatethata 170°to 180°phaseshiftoc-cursin the-frequencyrangeof 2.0to 2.2cycles~er second.Consideringnegligible.phaseshiftin thefuelsystemandlesg.than10°phaseshiftin the control(for ‘rC~ 0.5 see),a totalopen-loopphaseshiftof 180°wouldoccurin thissamefrequencyrange,2.0to 2.2cyclesper second.Fromtheamplitude-ratiodatapresentedin figure6(a),the loopgainatinstabilityshouldbe in therangeof 5.7to 6.9. For thisreport,loopgainis definedas theproductof allthefrequencyinvariantgainsaroundthe loop.

,-

itPN

Recordingsof enginespeedandtailpipegastemperaturetakenduring .experimentaldeterminationof stabilitylimitfor.atemperature- fuel-flowcontrolareshownin figure8. The loopgainnecessaryforsustainedoscillationwas 6.3andthe oscillationfrequency,determinedfromthetemperaturetrace,was approximately2.2cyclesper second.Thusthestability-limitpredictionsfromopen-loopfrequency-responsedataaresubstantiatedverywellby experimental”measuremetis.

Similarpredictionscanbe madefromthetemperature-areafrequency-responsedat=jbnwever,thearea-systemdynamicsmustalsobe considered. &For thiscontrolsystem,loopgainat instabilityshouldbe 40 andos-cillationfrequencyshouldbe 7 to 10 cyclesper second.Experimentaldeterminationof stabilitylimitforthissystemwas unobtainableduetonoiselimitationsof thetemperaturesignalat thehighloopgainsrequired.

ControlledEngineData

Temperature- fuel-flowcontrol.- Goodcontrolperformanceduringtransientsdependson properselectionof control-loopparameters(that1s,in thiscase,loopgainandcontrol.integraltimeconstant).Twooften-usedcriteriafor evaluatingtransientresponseareovershootratioandti~-to first-zero.Figure9 showsplotsof thesecriteria(definedon fig.10(a))obtainedfromtransient-responserecordingswhenoperatingwitha temperature- fuel-flowcontrol.It is oftendifficultto selectthebestcombinationof control-loopparametersusingthesecriteria.Thisisborneoutby examinationof thetransient-responserecordingspresentedin figure10.

Figure10 showstransientresponsesof fuel-valveposition,tempera-tureerror,andenginespeedto stepchangesin desiredtemperat~e. The .transientof figure10(a)wasobtainedusinga controlintegraltimecon-stant Zc of 0.25secondanda loopgain KL of 1.14= Thiscombinationof control-loopparametersgavea temperatureovershootratioof 1.12and ,.

NAM TN 3936 9

.

.

a timeto firstzeroof 0.46second.Thetimeto firstzerocouldbereducedby increasingthe loopgain;however,thiswas accomplishedonlyat theexpenseof greaterovershootratio,as shownin figure10(b).Forthistransient,Zc waskeptat 0.25secondand KL was increasedto3.03. Thetimeto firstzerowasreducedto 0.23second,but theover-shootratiowas increasedto 1.58. Thisovershootis consideredtoogreatandtheresponsetoooscillatoryforgoodcontrolperformance.

Figure9 indicatesthatat a constantloopgain(greaterthan2.5)the overshootratiocanbe decreasedwithoutgreatlytifectingthetimeto firstzeroby increasingZc. Figure10[c)showsthetransientre-sponsesfor Zc of 4.0 secondsand KL of 3.03. Theovershootratiowas decreasedto 1.10andthetimeto firstzerowas increasedslightlyto 0.26second.Accordingto theplotsof figure9, thesecriteriaval-ueswouldindicategoodcontrolperformance.However,a closerlookatthetemperatureresponsein figure1O(C)showsa longdrift-inor set-tlingtime. For manycontrolsystemsthiswouldbe consideredundesir-able. Thus,thisexampleshowsa shortcomingof usingthe overshootratioandthetimeto firstzeroas criteriaforselectingcontrol-loopparametersforgoodcontrolperformance.

A discussionandevaluationof criteriaforoptimizingtransientresponseof controlsystemsaregivenin reference3. Twoof the cri-teriadiscussedin thisreferencewereusedin thistemperaturecontrol-systeminvestigation.Thesewerethetimeintegralsof thesquareandabsolutevalueof temperatureerror. Circuitsforautomaticallycalculat-ingtheseintegralsduringtransientsweresetup on an electronicanalogcomputer,as shownin the diagramin figure3.

Shownin figure11 areplotsof theintegralcriteriavaluesobtainedduringtransientoperationwitha temperature- fuel-flowcontrolsystem.The curveswhichgivethe leastvaluesforthe integralsindicatethe ap-proximatevaluesof controlintegraltimeconstantandloopgainto use

foroptimumtransientresponse.The criterionJ

‘T2 dt indicatesaOe

rangeof ‘rCfrom0.75to 1.0secondandof KL from2.5to 3.5as giv-

ingoptimumtransientresponse.TheothercriterionJ

‘[Teldt indi-0

catesa rangeof Zc from0.5to 0.75secondandof KL from2.0to 3.0.

Althoughthesetwo criteriaindicateslightlydifferentrangesofcontrol-loopparametersforoptimumtransientresponse,bothchoicesofoptimumresponsearecharacterizedby a satisfactorycompromisebetweenovershootratioandtimeto firstzero.

10

Transient

etersselected

shootratiois

NACATN 3936

responsesfor ‘rC of 0.75secondand KL of 2.65~param- .

byJ

*2Te dt criterion)areshownIn figure12(a). Over-0

1.21andtimeto firstzerois 0.27”second.Figure12(b)showstransientresponsesfor ‘CC of 0.5secondand KL Of’2.27(ptiSm-

metersselectedby J’1 ITe dt criterion).Hereovershootratiois 1.15

0andtimeto firstzerois 0.30second.Settlingtimeforbothof thesetransientsismuchshorterthanforthetransientof figure1O(C).

An analysisof optimumcontrolsforlinearfeedbackcontrolsys-temswasmadeandis present%din reference. One-”ofthecontrolsys-temsconsideredin thisanalysiswas a speed- fuel-flowcontrolforaturbojetengine.Linearizedenginedynamicswererepresentedby a fi.rst-orderlag term, andthefuelsystemwas consideredas havingonlydeadtime. The controlactionof theoptimumcontrolforthissystemwasshownt-obe essentiallyproportional-plus-integral.Optimlzatlonofcontrol-loopparameterswasbasedon minimizationof theintegralofspeederrorsquared.

Thetemperature- fuel-flowcontrolsystemcanbe consideredin thelightof thisanalysisiflcertainsimplifyingassumptionsaremadere-

--

gardingthe systemdynamics.An approximationof thefreqhency-responsedatapresentedin figure6 canbe madewitha first-orderlagtermhavinga timeconstant-o~approximately0.4secondanda deadti”meof approxi-mately0.115second.“Byusingthese”simplified&namLcs,thi&control””systemis comparablewiththe speed- fuel-flowcontrolsystemof ref-erence4. .-

Accordingly,theoptimumloopgainshouldbe tbesystemtimecon-stant,0.4second,dividedby thedeadtime,0.1L5second,or

‘L,opt= 0“4 = 3.5. Also,theoptimumcontrolintegraltimeconstant0.115shouldequalthe systemtimeconstant,or ~c,opt= 0.4second.Compar-ingthesetheoreticaloptimumparameterswithther@ngeof parameters(’cCfrom0.75to 1.0secondand KL from2.5to 325)obtainedfromthe

experimentalintegralcurvesof figuren(a) showsgoodagreementin spiteof thesimplifiedapproximationof thesystemdyaamlcs.

Rate-limitedfuelcontrol.- Sincetheresearch-facilityfuelcontrolwas considerablyfasterthananypracticalfuelcontrolnowin useon tur-bojetengines,a controlstudysimilarto thatdisc~sedin previoussec-tionswasmadewitha rate-limitedfuelcontrol.A rate-lhd.tingcircuitwas insertedbetweenthe“controlandthefuelservosyst-em.Thiscircuitlimitedthemaximumrateof changeof fuelflowto approximately140

9s’m

NACATN 3936 11.

.percentof ratedfuelflowin 0.1second.Althoughthisratemaybegreaterthanthatof a practicalfuelcontrol,it was consideredsuffi-cientlylessthanthe originalrateto showtheeffectson controlper-fotianceof a slowerfuelcontrol.

Figure13 showsplotscouparingintegrals,overshootratios,andtimesto firstzerofortheoriginalandrate-limitedfuelcontrol.Inte-gralcriteriavalueswerehigherfortherate-limitedfuelcontrol,butminimumsindicatedaboutthe samechoiceof controlintegraltimecon-stant’andloopgainforoptimumtransientresponse.Therate-limitedfuelcontrolgaveslightlygreatertimesto firstzerothanthe originalcontrol.At smallvaluesof %C therate-limitedcontrolgavemoretem-peratureovershootthanthe originalcontrol,andat largevaluesof Zctherate-limitedcontrolgavelesstemperatureovershootthanthe originalcontrol.The optimumtransientresponsefortherate-limitedfuelcontrolstillindicateda satisfactorycompromisebetweenovershootratioandtimeto firstzero.

Temperature-areacontrol.- Fourcriteriaforevaluatingtransienttemperatureresponseforthetemperature-areacontrolsystemareshowninfigure14. Threeof thesecriteriaarethe sameas usedforthetemperature- fuel-flowcontrolsystem.Sincetherewasno temperatureovershootformanyof thetransientstakenwiththetemperature-sreacon-trolsystem,anothercriterion,timefor Te to reach10percentof itsinitialvalue,wasusedinsteadof timeto firstzero.

The integralcriteriacurvesof figures14(a)and (b)showno defi-niteminimums,in contrastwiththe integralcriteriacurvesforthetemperature- fuel-flowcontrolsystem(fig.11). Thesecurvesindicatethata small Zc shouldbe usedbutarenotveryselectiveas to loopgainforoptimumtransientresponse.

Theovershoot-ratiocurvesof figure14(c)showa reversetrendtothatusuallyfoundin controlsystems.Overshootratiodecreaseswithincreasingloopgain. Thesecurvesarenotveryhelpfulin selectingcontrol-loopparametersforoptimumtransientresponseotherthanindi-catingthatloopgainshouldprobablybe >3 and %C? 0.25second.

Thefourthcriterion,presentedin figure14(d),alsoindicatesthata small ‘c shouldbe used,at leastfor loopgainslessthan6. Athigherloopgainsthesecurvessrenotveryselectiveas to Zc. Thesecurvesshowthattheminimum10-percentresponsetimeobtainablewiththetemperature-areacontrolsystemwas aboutthreetimestheminimumtime tofirst zero obtainable with the temperature- fuel-flow control system(fig.9).

.

12 -. NACATN 3936

In figure15 areshowntransientresponsesof engine-speed,exhaust-valveposition,andtemperatureerrorforthetemperature-areacontrolsystem.To obtainthedesiredincreasein temperature,thecontrolclosedtheareavalves;butin thiscasethespeeddecreasedinsteadofincreasingas i&did withthetemperature- fuel-flowcontrolsystem(fig.12).

Figure15(a)showsthetransientresponsesfo~zc of 0.25secondand KL of 3.74. Thiscombinationof control-loopparametersgaveveryslighttemperatureandspeedovershoots.Thetimefor Te tu reach10perce~tof.itsinitialvalu~.was approximately0.66second.Theexhaust-valve-positiontraceshows“thatthesreaclosedquicklyandremainedatthe limitingposition(85percentof ratedarea)forapproximately0.4secondandthenopenedslowlyto the steady-statepositioncorrespondingto thefinaldesiredtemperature.

Transientresponsesfor ‘CC of 0.25secondand KL of 9.96areshownin figure15(b). Therewasno temperatureovershootandveryslightspeedovershoot.Thetimefor Te to reach10percentof itsinitialvaluewas apjioximately0.63second.Duri@ thistransient,theexhaustvalvesremainedat thelimitingpositionforapproximately0.58secondandthenvery“quicklyopenedto thefinalsteady-stateposition.

The10-percentresponsetimesobtainedfromthetransientdataoffigure15 do notclearlyindicatea choice‘ofiloopgain;bu~consideringthe-sreatraces,it wouldbe preferableto usethelowervalueof loopgain. At thehigherloopgaintheareavalveismoresensitiveto thenoisein the–temperaturesignal.

CONCLUDINGREMARK6

In thehigh-speedoperatingregion,tailpipegastemperaturecouldbe satisfactorilycontrolledduringsmalltransientsby usingtemperaturefeedbackcontrolsystemswhichvariedeitherfuelflowor exhaust-nozzlearea. A proportional-plus-integralcontrolin a temperature- fuel-flowcontrolsystemprovidedsatisfactorytransientresponseto a desiredstepincreasein temperature.For a temperature-areacontrolsystem,it wasnecessaryto addnonlinearcomponentsto thebasicproportional-plus-integralcontrolto providesatisfactorytemperatureresponseduringtransients.

For theessentially

errorfunctionintegrals

is thetemperatureerror

lineartemperature- fuel-flowcontrolsystem,

J’ Jll

w

suchas = T: dt and Te dt (whereTeo 0

and t is time)appearedto be bettercriteria

NACATN 3936 13

thanovershootratioandtimeto firstzerofor selectingcontrol-loopparametersforoptimumtransienttemperatureresponse.Control-loopparametersselectedon thebasisof theseintegralcriteriaagreedverywellwith‘optimumpredictionsof a previousanalysis..

Limitingtherateof changeof fuelflowdidnotappreciablyalterthe selectionof control-loopparametersforoptimumtransienttempera-tureresponse.

Forthetemperature-areacontrolsystem,whichis nonlinearbecauseof areasaturationlimits,theintegralcriteria,theovershootratio,andthetimefor Te to reach10percentof itsinitialvalueweresome-whatinadequatefor selectingloopgainforoptimumtransienttemperatureresponse.However,thesecriteriadid indicate that a small control inte-gral time constant shouldbe used.

Thetransferfunctionsgivenby equations(1)and (2)providedasatisfactorydescriptionof engineandinstrumentationdynamicsin thehigh-speedoperatingregion.

LewisFlightPropulsionLaboratoryNationalAdvisoryCommitteeforAeronautics

Cleveland,Ohio,January18,1957

14 — IW3ATN3936

APPENDIXA

SYMBOLS

A

a

b

E

G(s)

KL

‘L,opt

N

s

Te

Ti

t

‘d,a

‘d,f

Wf

‘c

‘C,opt

‘e

Tt

T1

‘C2

exhaust-nozzlearea,sq In.

initialriseratioof temperatureto stepchangein fuelflowat constantarea

initialriseratioof temperatureto stepchangein areaatconstantfuelflow

voltage .

transferfunction

loopgain,productof-frequencyinvariantgainsaround10UP

optimumloopgain

enginespeed,rpm

complexvariableusedin Laplacetransformationmethods

errorbetween.indicatedandreferencet~mperaturesl‘F

indicatedtailpipegastemperature,W

time,sec

deadtimeof temperatureor speedto area,sec

deadtimeof temperatureor speedto fuelflow,sec

indicatedfuelflow(afuel-valveposition),lb/hr

controlintegraltimeconstant>sec

optimumcontrolintegraltimeconstant,sec

enginetimeconstant,sec

thermocotipletimeconstant,sec

timeconstantof a leadtermof enginedynsmics,sec

timeconstantofa lagtermof enginedynamics,sec

NMATN3936 15

APP~IX B

MRTHODOF OBTAININGENGINE

FREQUENCY-RESPONSE

DYNAMICSFROM

DATA

Graphicalmethodscanbe appliedto the speedandtemperaturefrequency-responsedatato evaluatethevarioustimeconstants,deadtimes,andinitialriseratiosof equations(1)and (.2).Thesemethods,whichwillbe describedin thisappendix,arebaseduponthemanipulationof varioustransferfunctions.Derivationof transferfunctionsdescrib-ingenginedynamicbehaviorcanbe foundin reference5.

The enginetimeconstantcanbe obtainedfromthe speed-areafrequency-r=sponsefunction-ofe&inecanbe represented

datashownin figure16. Thenormali&dtransferspeedto exhaust-nozzleareaat constantfuelflowby

~ (s)e-td,as

a Wf = l+zes(Bl)

By fittinga first-orderresponsecurveto the experimentaldatain fig-ure16(a),an enginetimeconstantof 0.83secondwas determined.Also,by consideringthedifferencebetweenthe e~erimentaldataandthefirst-orderphase-shiftcurvein figure16(b),a speed-areadeadtimeof 0.021secondwas determined.

‘Thenormalizedtransferfunctionof enginespeedto indicatedfuelflowat constantareacanbe representedby

I-t~,&

~— (s] = G(s)eAwf A 1 + Tes

Experimentaldatarepresentingthe speed- fuel-flowfrequencyresponseareshownin figure17. Alsoshownin thisfigurearecalculatedfirst-

.Lorderresponsecurvesof 1 + ‘res with Ze = 0.83second.The differ-

encebetweentheseresponsecurvesandthe experimentaldatarepresents-t s

G(s)e “f , thefuel-distribution-systemandcombustiondynamics.Plotsof thesedynamicsareshownin figureU3.

.

ture“

Thenormalizedtransferfunctionof indicatedtailpipegastempera-to indicatedfuelflowat constantareacanbe representedby

16

Dividingequation:

. NAUA ‘XN3YJb -—.

-%,+(1 + a~es)(l + Tls)G(s)e-~ (s) ~= (1 + Tes)(l 1-72s )(1+Tts}

(1) -

equation(1)by equation(B2)resultsin thefollowing

(1 + a~es)(1 + ‘rls)~ (s) =

A (1 + Tts)(l +“T2S)

Datacorrespondingto equation(B3)canbe calculatedfromthe experi-mentaldataof figures6 and17. Plotsof thesecalculateddataareshownin figure19. Alsoshownin figure19 are_@culatedfrequency-responsecurvesrepresentingequation(B3). Numericalvaluesfortheriseratioandtimeconstantswerechosento make‘theresponsecurvesfitthe datawithreasonableaccuracy.Thusvaluesforinitialriseratio a of 1.3andforthermocoupletimeconstaiit~t of 0.3secondweredetermined.Timeconstantsof 9 andl.Gsecotidawereusedfortheotherleadandlagtermsof theenginedynamics.

Thenormalizedtransferfunctionof indicatedtailpipe gastempera-tureto exhaust-nozzleareaat constantfuelflowcanbe representidby .

(2) -

The deadtimeandallthetimeconstantsof thistransferfunctionhavebeendeterminedby themethodsdescribedin thep&cedingparagraphs.Thus,by extractingtheproducto~all thetermsexcept 1 + b~es fromthe datacorrespondingto thistransferfunction(fig.7),plotsof1 + b~es wereobtained.Theseplot=areshownin figure20. By fittinga first-orderresponsecurveto thesecalculateddata,a valuefortheinitialriseratio b of approximately0.4was determined.

Thenumericalvalues of thetimeconstants, deadtime, and initialrise ratios determinedby the methodsdescribedin this appendixarevalid in the range of operationwherethe frequewy-~esponsedata wereobtained. Since these values vsry with operatingrange, a morecompletedescription of enginedynamicscould be obtainedfromfrequency-responsedata taken at several operatingpoints.

.NACATN 3936 17

1.

2.

3.

4.

5.

REFERENCES

Otto,EdwardW., Gold,Harold,and.Hiller,Hrby W.: Designformanceof Throttle-TypeFuelControlsforEngineDynamicNACATN 3445,1955.

Wenzel,L. M.,Hart,C. E., andCraig, R. T.: comparison of’

Fuel-FlowandSpeed-&eaControlson a TurbojetEngineforStepDisturbances.NACATN 3926,1957.

andPer-Studies.

Speed-Small

Graham,FrankD.,andLatbrop,RichsrdC.: Synthesisof “Optimum”TransientResponse- CriteriaandStandardForms. WAIXTech.Rep.No. 53-66,WrightAirDev.Center,~r Res.andDev.Co_nd,Wright-PattersonAirForceBase,Aug.1953. @lONO. 206-11.)

Boksenbom,AaronS.,Novik,David,andHeppler,Herbert:OptimumControllersforLinearClosed-LoopSystems.NACATN 2939,1953.

Delio,GeneJ.: Raluationof ThreeMethodsforDeterminingDynamicCharacteristicsof a TurbojetEngine.NACATN 2634,1952.

.

.

-. . .— .

Cro6sarmllnksge

HydraulicpmerUpperbutterflyvalva

~’it’o”’en’orFi@re 1. - Schematicof variable-areaexhaust-nozzlesystem.

, .2-R* ‘ ‘

$31’-3baok 4212 , ,

FigureZ. - FrequencyCentOf rSted =eaj

Frequency,Cpa

(a)Amplituderatio.

respmse of exhaust-valmpositionto 6inusoldSliUpUtEi@. Mea ~iation, A Per-operatingpoint,103percentof ratedarea.

.- .—

.

g

~-

:

0

i-

.1 .2 .4 .6 .8 1 2 4 6 s 10 20 40 60Requency,CPM

(b)Ea.%@.shift.

F@ra 2.- Cmmluded . Frequmcyrmpmne ofexkmt-mlvepaitkm b sinui30iMinplt0iE8sl.Areamrti-,tiotl,& plwmt ofr@ellm; upmmingpint,1C4)peraentofIwt&larea.

, m!%’ t *,!

,4212 , ,

r —— —— ——— —0.01 0.01

II5 (!!5

-“1 1

Te j 0,5 Electronic 0“5

r“’”-multiplier

inputL__ –-zIEeL––_- .,J 1

Figure3. - Diagramofcomputercapacitanceinmlcrofarads.

~..—— — +————.I

I

22 I 1

12

1

*

*

1

11

IAbsOlute-value

0.2 II

2

‘Cfrcutt1-

-200v——— ——— —— —— 1

circuitusedto obtainerrorIntegrals.Resistanceinmegohms,

.

IvN)

===’(

I

Temperature-error signal *

Indicatedtailpipe gas temperature Temperaturesensor

I I I

Fuel flow or

variablearea

LIhgim

Stepinput

Figure 4. - Block diagram of temperaturefeedback control By-stem.

. ,

, , ,4212 , ,

1Indicatedtemperaturesignal~ B

10 0-1o R1 s

fi~

Q

!Cemperature10 w

referencelevel ;

~0.5

Step input =

___— ——— ——— —— ——— — —— 1

Figure 5. - Computerin microfarads.

O-&

I

1

-’l -

(e) Temperature- fuel-flow controlsystem.

circuitdiagramof aummlngnetworkand control. ResistanceIn megohms, capacitance

A--- Indicatedtemperaturesigoal10

10 Temperature

Te referencelevel

El0.5 ~

.L

_d ‘

rRetiy A-’

R 11

R

Figure5. - Concluded. Computercapacitsmcein mlcrofarads.

.

0-1o

DA&-l Al

1 To area

servosystem

I +El -E2-.—— ——— ——— ——— ——

I IRelay triggeringcticuit

L——— ———— ——. . . -J

(b) Tam~ature-area control.syatem.

circuitdiagramof sting networkand control. Resistancein megohms,

I

W4

Q *# .+

.8 b n n xI==e=GW ~ — ‘ — –

\

.6Calculatedtransferfunction,

t1

i

.4

FH

g! (s)

f

(1+ aTeE](I.+~ls)@s)e-td)fB

A= (1+ Te6)(l+ ~2s)(l+~~d

●

a n 1,3~s E 0.63sec m,+ - 0.5sec

.2 - # = 9 sec / ‘Z=losec

~(s)e-td,faobtainedfromcurvesof fig.17/

.1

,09

,06

.04.01 .02 .C-4 .C6.ml.1 .2 .4 .6

4212 ‘ ‘

Frequency,CPE

(a)Ar@ituderatio.

Figure6. - Frequencyre6pon8eof indicatedtailpipegastemperatureto indicatedfuelflowat condantarea.Temperaturevariation,&.5 percentof ratedteraperaturejoperatingpoint,79 percentof ratedtemperature.

tom

.—

-3(

-6(

~ -9[

$■

~ -~,

-m

-MO

-210

1 t ZRV , .

, (h-4 baok 4212 I .

.01 .02 .W .0s.m .1 .2 .4 .6 .8 1 2 4 6 8 10Frequancy, CPB

(a)Amplltuderatio.

Fi@.re7.- fiequenc~rtqome ml indicatedtailpipsgastemperaturetiexhaust-nozzl.aareaat conetantfuelflnw,l@er&3UY V’Uiatinn,M.7 peroentof rated t~rU~j opsratingpnint,78,5percent,tirated~eratll~.

, , , I 1

0

-3

-E

-ls

?requency,Wdl

(b)Phasesbifr“

Figure7. - C0nclude3.Frequencyre.swseof indicatedtailpipegast~ture toexhaunt-nomleareaat consta@fuelflow.Teqwwkwe varifltion,H.7 percentofratedtempsratum;operatingpoint,78.5percent of mted temperature.

. r

, 1

.-

, ,

t--l see-l

Figure El.- Stability-1imitcheckfortemperature- f’uel-flovcontrol.Loopgain,6.3;con-trolIntegralttieconstaut,0.5seccmd.

NACATN 3936

2.0.

G3ntrolintegraltimeconstant,

1.8.,.5

~c)sec /1.0

.0.25 2.04.0

~‘P 1.6 /6!.!

/

+0

3 /m / /

i 1.4 / /

/

/

/

1.2

/

1.o-

11.0 8 1Central Integraltimeconstant,

sec

.8

I

.6

\

.4\

\

.2

0 1 2 3 4 5 6 i’Loopgain,KL

.

.

.

.

.Figure 9. - Criteriaforevaluatingtransienttemperatureresponsefortemperature-fuel-flowcontrol.

NACATN 3936 31

.

mlA2

(a)ControlIntegral.the wnstant,0.25second;loopgain,1.14.

I!lgure10.- !&armientresponsestostepIncreaaeinreferencetemperaturefortemperature- fuel-flowcontrolsystem.

32 NACATN 3936.

●

—

Fi&yre10.- Continued.Transientre~eee tostepinfreaseinreferencetemperaturefortemperature-fuel-flowcontrolsystem.

.

.

NACATN 3936 33

.

03d$

Figure 10.- Ccm?huied. !3Ycankientresponsesta stapincrease5nreferencetemperaturefortemperature-fuel-flowcontrolsystem.

. .

2R

I,

Controlintsgral 4.0

Z4- time constat, 6

Te,

Sec /1

: 0.25 2.0 /

cdHal 20 \ / f io .75

\o

1.0,5, /

\ //

16./\

/

12:0 1 2 3 4 5 6 7

LOOP gain,KL

(a)J

‘~ dt.

Figure11.- Criteriaforewluatingtraneienttemperatureresponsefortemperature-fuel-flowcontrolByatem.

Z-R? *,

CJl

$m

NACATN 3936 35

.

36

4.0!

32

28\

24

Controlinte~altimeconetant, t / ‘

~c)

/1.0

/ :/.?

/ ‘

I.25\ I \l \l I 1/1 I 7r/ /

20 sec I \l /1 II I I I 1

Figure11.- Concluded.Criteriaforevaluatingtranaienttemperatureresponsefortemperature- fuel-flowcontrolsystem.

36 NACATN 3936.

(a)Control-integral.timeconetant-,0.75second;loopgain,2.65.

FYgure12.- Transientresponeeetostepincreaseinreferencetemperat&efortemperature- fueLflowcontrolsystem.

.

.

.

NACATN 3936 37

Cul-lNd+

(b)Controlintegralthusconstant,0.5~tij loop gain, 2.27.

i?lgureE. - Concluded.mansientreeponeestostepincreaseinreferencetemperaturefortemperature- fuel-flowcontrolsystem.

38 NACATN 3936

;ak

II Puel valve ]m — Original‘—— Rate-limitedI

2.2

df

1.8/ 8

/

1.44‘

y /

1.00 2 4 6

.

Imp gain,KL

(a)ControlIntegraltimeconstant;-0.25second.

Figure13.-Cornparlsouoftransient-temperature-responsecriteriaforrate-llmitedand originalfuel-flowcontrol,eysteme.

NACATN 3936 39

28

24

C.-l-la)

<

200

16

.7

c1IJm

sF+v .5-N

g

$ .3’E

\

z ~

.10 2 4 6

1--t-iFuel valve

uriginal——— Rete-llmitea

20

i

16

4‘

II //‘12 /

\. /~t

8

2.2

/,1.8

/v

1.4

////

1.00 2 4 6LCIOpgain,KL

(b)Controlintegralthe constant,0.5.eecond.

Figure175.- Continued.Comparisonof transient-temperature-res~nsecriteriaforrate-limitedandoriginalfuel-flowcontrolsystems.

.

.

40 NACATN 3936.

24

Fuelvalve

— Original

\——— Rate-limited

20\

/1

\ / /16. /

12-

U .5-al <m \~N

\

\\

g “3’ \.

i

\

H.1.

c1 2 4 6

22

18 \ A

\

\\

14 /T

10

I-.8

/

1.4/ ‘/

/

1.00 2 4 f

.

ICOPgain,KL

(c) Controlintegraltimeconstan&l.Osecond.

Figure13.- Continued.Comparisonof transient-temperature-responsecriteriaforrate-limited andoriginalfuel-flowcontrolsystems.

t 1 423.2

26

Fueltin— 01’lginal‘—— Rute-llmlted

24

\ .‘I

\ t

\ /20

\ \ /

\. /

\ /

* \. /16

w ~ -— H

u

1.8

//

/

1.4 /

/

// /

/ L --1.0

2 4 6 8 10 12

IODP win, KL

(d)C0nt301i.te~ltl= mrwt.ant,2.0.ecende.

Figugumu. - Concluded. Cmparimn of tmnnlent-teqperatwm-mnpmse criteria for rate-limitd and original fuel-flow control

42 NACATN3936.

.—

i

!

I

Controlintegraltimeconstant,

sec

- .5

0 2. 4 6 8 10Loopgain, KL

(a)rl IT=dt.o

Figure14.- Criteriaforevaluatingtransienttemperatureresponeefortemperature-areacontrolsystem.Lower-areaexcursionlimit,85percentof-ratedarea.

.

.

.

NACATN 3936 43

.

.

12

10

8

6

4

1.4

1.2

1.(3

Controlintegraltimeconstant,

‘c

sec

‘0.25-2.0

\!

(b)J

‘T2 ato e“

Controlintegral0.25 timeconstant,

Tc>sec

.5

. \

\—.- 0 2 ~ 6 8 10

Loopgain,KL

(c)Overshootratio.

Figure14.- Continued.Criteriafor evaluatingtransienttem-peratureresponsefortemperature-areacontrolsystem.Lower-areaexcursionlimit,85 percentof ratedarea.

44 NACATN 3936

2.E

2.4

2.0

1.6

1.2

.8

~ontrol integra~timeconstant,

‘cc,sec

2.0

1.0\

.5 >25

\\

\— ~ ~

.4 0 2 4 6 8 10Loopgain,KL

(d) Timeto reach 10 percent of initial temperatureerror.

Figure 14. - Concluded. Criteria for evaluating transienttemperatureresponsefor temperature-areacontrol system.Lower-area excursion limit, 85 percent of rated area.

.

.

.

1—

4212, -,”* t ,

1--1 see-d

Figure I-5.- Emnsientrespo~esto stepinereaBeinreferencetemperaturefortemperature-areacontrolsystem.Controlinte~altimeconstant,0.25second.

----- — .—

1---1 see-+-l

(b) LOOP &in, 9.96.

Figure 15. - Concluded, Transient responses to etep increase In reference temperature fortemperature-area control system. Control integral time constant, 0,25 second.

,, , 4 1

z-m

* ,

1(

.0

.6

.4

.2

.1

.0%

.%

.04

.02

.01 .02 .04 .ffi .@ .1

t ,4C.LL 1

, #

.2 .4 .6 .8 1 2 4 61Frequency, cpr.

(a)Amplitudemtio.

Figure 16. - Frequency respmse o!? engine WSW3 to mhawt-nozzle area at constant fuel flow. Spee6 WriatiOn, M Per-centofratedspee6joperating~int, 95.5 percent of ratul aped.

Frequen~,cps

(b)Fhaseshift.

.6 .8 1 2 4 6 8:

,.,ZZz? ‘

,

1

.8

.6

,4

.2

.1

.

“-I=D=l= =?Ill

I 7== T=+

.02

.01 0Frequency, CP

(a) Amplituderatio.

Figure 17. . h3qUeII~ iY8pOIMe Of .31giD9 BpOd tn iII&iCat.d fid fbW .9t COIIBtiUt EUWR. Bpd ~TiatiOU, & wXC,Ut *w

of rated speea~ operating pint, 95.5 percent ur ratecl Bwa,

mo

g

iism

!

sFrequemy, Cpa c

(b) hm shift. . ~

Figura 17. - Cmcluded. RuqwrIcy reapoma of en@ne speei to icdicated fuel flov at conatmt UEU. _ **, ~+S WC* 03?TWtUd.Ep8dj mtiUE Wtik, 95.5 ~mt & mtd UPWd. u

m

.“ “ ZL3 .

, W-7 back 4212 . *

.

Frequency,CN

(a) Amplitude ratio.

Figure 18.-t@

- hsqueney response of fual-distribution-systemad ccmbuationdynamics G(s)e .

I

.— .

c ZTZV ● *

.

t 1

4

2

1

.8

.6.01 .02 .04 .06 .06.1 .2 .4 .6 .8 1 2 4

Frequency,cps

(a)Amplituderatio.

Figure19. - Frequencyresponseofindicatedtailpipegastemperalmretoenginespeedat constantarea.

(n(J!

-. — —. —.

—

—

—

/

—

—

—

—

—

—/

.04 .06 .06 .1 .2 .4 .6 .8 1 2 4Frequency,cps

(b)Phaseshift.

Figure19.- Concluded.Frequencyresponseof indica~edtailpipegastemperatureto enginespeedat constantarea.

● I

I * 1 ,

4Llii -1

“ ,

10

8

6E

r\

.02 .04 .00 .06 .1 .2 .4 .6 .E 1 z 4 6

Frequency,.CPS

(a)@utuae ratio.

Figure20. - Cal,culatdfrequencyrespmse of engine leadterm 1 + b’c~.

r.)1

$m

I

mm

i!!

.1 .2 .4Frequency, cpe

I

(b) Fhaae shifi.

—

—

Q

/r

—

—

—

F@re 20. - Concln&d. Calculated freqnency response of engine 1.4 term 1 + b’reB.

‘6

.

8 10

r, , . #