Transcription
gleyMarchFourierto HelicopterL. Santa MariaResearch Center,1999Hampton,VirginiaTransformFlyoverNoise
The NASASTI ProgramOfficeCONFERENCESince its founding,NASA has been dedicatedto the advancementof aeronauticsand spacescience. The NASA Scientific and TechnicalSPECIALtechnical,The NASA STI Program Office is operated byLangley Research Center, the lead center forNASA's scientific and technical information.analysis. Includes compilationsofsignificant scientific and technical dataand informationdeemed to be ofcontinuingreference value. NASAcounterpartof peer-reviewedformalprofessionalpapers, but having lessstringent limitationson manuscriptlength and extent of c and technical findings that arepreliminaryor of specializedinterest,e.g., quick release reports, workingpapers, and bibliographiesthat containminimal annotation.Does not ictechnical findings by NASA-sponsoredcontractorsand grantees.andPUBLICATION.Scientific,or historical informationfromNASA programs,projects, and missions,often concerned with subjects havingsubstantialpublic interest.The NASA STI Program Office providesaccess to the NASA STI Database, the largestcollection of aeronauticaland space scienceSTI in the world. The Program Office is alsoNASA's rtsof completedresearch or a majorsignificant phase of research thatpresent the results of NASA programsand include extensive data or theoreticalPUBLICATION.Collected papers from scientific andtechnical conferences,symposia,seminars, or other meetings sponsoredor co-sponsoredby NASA.Information(STI) Program Office plays a keypart in helping NASA maintain this importantrole.disseminatingthe results of its research anddevelopmentactivities. These results arepublishedby NASA in the NASA STI ReportSeries, which includes the following reporttypes:. in ProfileTECHNICALTRANSLATION.language translationsof foreignscientific and technical materialpertinentto NASA'sEnglish-mission.Specializedservices that complementtheSTI Program Office's diverse offeringsinclude creating custom thesauri, buildingcustomizeddatabases, organizingandpublishingresearch results . evenprovidingvideos.For more informationProgramabout the NASASTIOffice, see the following: Access the NASA STI ProgramPage at http://www.stLnasa.govHome E-mail your questionhelp@sti.nasa.gov Fax your question to the NASAHelp Desk at (301) 621-0134 Phone the NASA621-0390via the InternettoSTISTI Help Desk at (301)Write to:NASA STI Help DeskNASA Center for AeroSpace7121 Standard DriveHanover,MD 21076-1320Information
NASA/TM-1999-209114Two-DimensionalAppliedOdilynL. Santato irginiaTransformFlyoverNoise
The use of trademarksor names of manufacturersofficial endorsement,either expressed or implied,and Space Administration.Availablefrom:NASA Center for AeroSpace7121 Standard DriveHanover,in the report is for accurate reporting and does not constitute anof such products or manufacturersby the National AeronauticsMD 21076-1320(301) ion5285 Port Royal RoadSpringfield,VA 22161-2171(703) 605-6000Service(NTIS)
iiiABSTRACTA methodto ngledependsthesumthe other.ofis neitherFourieron the lengthsbe adequatelymainrotortwobecauseA two-dimensionalnorblocksbutpossiblychosenmethodin flightharmonizable.is usedfavorto showThebins whosefrequencieswouldisfrequencies,into frequencyIncommensurateblockanalysisa ary,any dataFouriernoisesignalsputs the signalof the datarepresentedperiodicperiodictransformand tail rotorsizewouldnotone frequencyorhelicopternoiseasharmonizable.The two-dimensionala singletone,two tonesa seriesof incommensurateDatafromare a Boeingblocksof lengthused in the analysis.be separatedalongtheindividualflightExplorertest is analyzedThe hein twomultiplesthat the mainonly informationrelatedof tonesThis initialrotorS-76C composedanalysisThetest(4-bladedrotorand tail rotorspectrum.signals:ofgivesand spectra.dimensions.of the maintransformfrequenciesto simulatedautocorrelationsand a Sikorskyto integerappliedand a seriesand their harmonics.(no tail rotor)equivalentis firsttones,of the two-dimensionalin the sa helicopterMD902analysisof periodicallyan idea of the characteristicsDataspectralTheof interest.to one particularbladesignalsaircrafttail rotor).passagecan re
ivTABLEOF CONTENTSviLIST OF FIGURESVIIILIST OF SYMBOLSixACKNOWLEDGEMENTSCHAPTERSources1 - INTRODUCTIONof helicopter1noise1Noise of the main rotor3Noise of the tail rotor3Main Rotor-TailPeriodicityof helicopterCHAPTERRandomRotor InteractionNoise4noise2 - TWO ityCHAPTER113 - RESULTSWITH SIMULATEDPure ToneCorrelatedIncommensurateSignalFrequencies4 - FLIGHT TEST DESCRIPTIONTest 4182222LayoutFlight ParametersData Acquisition2324and Analysis24
CHAPTEROne5 - 6BoeingMD 902 Explorer-LevelFlight28SikorskyS-76C -LevelFlight28SikorskyS-76C - Takeoff31SikorskyTwoS-76C Dimensional- ApproachSpectra3131BoeingMD 902 Explorer-LevelFlight35SikorskyS-76C -LevelFlight39SikorskyS-76C - TakeoffSikorskyS-76C - Approach44CHAPTER6 - CONCLUSIONS4749REFERENCESAPPENDIXAAPPENDIXB5152
viLIST OF FIGURESHelicopterNoiseVenn diagramStationary6of class relationsand periodicallySum of two sinusoidsSupport2Sourcescorrelatedwith frequenciesof 2-D powerspectral8processes10of 22 and 121 Hz13densityPure tone at 20 HzTimehistory15of periodicallyOne f orrelatedhistory17signalof sum of periodicsignals18with incommensuratefrequencies11One dimensionalpowerwith incommensurate12Sum of periodic13Centerand "tail"incommensuratespectrumof sum of periodicsignals19frequenciessignalswith incommensuratediagonal20frequenciesfrom two-dimensionalspectrumof sum of21frequencies14Boeing15Sikorsky16Microphonearray layout17Spectrogramof Boeingcenterlinedensity23MD 902 Explorer23S-76C at CrowsMD 902Landing,ExplorerCA, duringlevel flyoverflight24testrecordedby25500 ft27microphone18Boeingaltitude19SikorskyS-76C level flyover20SikorskyS-76C takeoffMD 902 Explorerspectrafor level flyoverat 136 knots,at 74 knotsat 115 knots,492 ft altitude2930
vii21SikorskyS-76C 6-degapproachat 74 knots3222BoeingMD 902Explorerspectrafor level flyover at 115knots,500ftaltitude3323BoeingMD 902level flyover, centerandmainrotor BPFdiagonalsfrom two-dimensionalspectra3424SikorskyS-76C level flyover at 136knots,492ft altitude3625Centerandmainrotor diagonalsfrom two-dimensionalspectra,SikorskyS-76C level flyover at 136knots3726Centerandtail rotor diagonalsfrom two-dimensionalspectra,SikorskyS-76C level flyover at 136knots3827SikorskyS-76C takeoffat 74knots4028Centerandmainrotor diagonalsfrom two-dimensionalspectra,SikorskyS-76C takeoffat 74knots4129Centerandtail rotor diagonalsfrom two-dimensionalspectra,SikorskyS-76C takeoffat 74knots4230SikorskyS-76C 6-degapproachat 74 kts4331Centerandmainrotor diagonalsfrom two-dimensionalspectra,SikorskyS-76C 6-degapproachat 74 kts4532Centerandtail rotor diagonalsfrom two-dimensionalspectra,SikorskyS-76C 6-degapproachat 74 kts46
viiiLIST OF SYMBOLSajConstantcoefficientAAmplitudecomponentof autocorrelation,R (t)BAmplitudecomponentof autocorrelation,R (t)El,]Expectedtransformof egressiveprocessvalueFourierDiracfor correlationof characteristicequationof randomtransformvalueof randomprocessprocessXXdelta functionFrequencyvariableFrequencyin in two-dimensionalFouriertransform
ksthedescribedU.S.Agreementin this wishesstaffof thefacultyandPhilipMorris,Langleyto expressGraduateand the membersResearchCenter,sincereProgramof the Fluidespeciallygratitudeforin AcousticsMechanicsDr. Jay C. Hardinsupportandat PennState,and Acoustics(ret.)guidanceto theespeciallyDivisionand Dr. Feri Farassat.Dr.at NASA
CHAPTERIINTRODUCTIONThesounddistinctiveof a helicopternoisebe consideredreducingis a causea highimpact where what condition how the noise(flightTheoperations,conveysthe noisegraphicchallengedthe most relevantThean illustrationwouldthenin orderto createthisillustrationin the analysisof helicopterflyovermeaninga tailare definedthat visualizesto identifythe noisethetheareasfor the listener.or graphicthatefficientlyof usinga two-dimensionalnoise.noise1, from a NASAwithofinformation.transforma helicopterprocessidentifying:or graphicto reducethe advantagesFigurecouldis generated;be a tool usedis to identifyof helicopterstep in theItsgenerated;of this paperSourcescan identify.and thereforeThis involvesThe purposeFourierfirstwhen the noiseto createtips, etc) to modifyis thuspeopleto the ground.can be usedbladesource.is beingis flyingpropagatesThenoisemoston the ground,is its characterization.the aircraftof a helicopter.researchernoiseis one thatto the listenercommunityon the aircraftThis informationoverheadof annoyancefar field llustratessourcesin [1] and are brieflythe sourcesof helicopterdescribedof ctic NoisePeriodicnoise,noisetail rotorrotatingdisplacedor harmonicnoisefrom eraction,noiseas the rotor.by the rotatingblade,anda helicoptertail rotormechanismsThicknesspropagatescomesnoise,in variousand the noiseare a functionnoiseis causedpredominantlyforms:fromthicknessmain rotor-of the rotationalby thein thespeedmovementrotorplane.ofof airThe
2extremecase of thicknesstransonic,is call high speedBlade-vortextip-trailingvorticesusuallywith a muchtail rotorshedtravelnoisecan be periodicwhennoise(BVI)noise,atis causedhighernoiseROTOR-TAIL(HSI).of the advancingHSI generallyalso calledbladetipby the lift and dragCompressorbladeimpactsand are intersectedfrequencyinteractionINTERACTIONa rotordownwardin nature.the tip ctingfromthe helicopterthan the mainrotornoise.furtherROTOR.: -. .whenblades.theTrailingflight.on the rotatingengineTail rotorbladeis alsonoiseandperiodic,and mainrotor-below.HIGHNOISE7"-// ,//toin level flight.rotorin descendingforcesare closeis generatedothernoiseare SEBLADE VORTEX,'INTER- ,CTIONNOISEQ ,rTAI L-ROTOR ,)TpO&RT n.-' 1'" ' -.COMPRESSORJAND JET NOISFbroadbandThe enceINTERACTIONNOISENOISE/1. esNoiseSomerandomly-HIGH/FigureBroadband/ BLADE-WAKEingestedcut throughnoiseis causedis broadbandin naturethe rotorblades.as doesthefromby turbulencebecauseTurbulenceflowoverthe atmosphereit. Blade-wakeinteractioninteractingwiththe generatingmechanism,in the boundarythesharpalso producesis es.turbulence,of the rotorof thenoiseof broadbandrotorwhenbladeblade.the rotornoise,as the
3turbulentwakefroma precedingfrom the helicopterThisenginesthesisfrequenciesTo explorewill yin ades,Rotortail rotoritselfrotor,in forwardhub,Somethosetheone witha tailflight test in 1996, will beFFT,Fourierthis thesisas wellas withtransformthe two-will be usedas anand tail rotor noise.Thesenoise.However,flightis quiteturbulence,and fuselage.to ys:interactionit is alsonotinteractionencounterfromonly1) highnoise,andthatencounteringdue to its orientation,thusthese4)whenthey encounterand generatewill be discussedon theaircraft,of the mainwakestheany noisesamein moretypesofits ifferenttip vorticesgeneratenoisephenomenonrotorandthe tail rotorto its orientationdifferentand measurethe wakestip vorticesThis latteris a smallerin several3) vegeneratesnoise,studiesbut theyTailatmosphericontransform,helicopters:an acousticnamelyrotorconductedinteractionas the main rotor.fieldbeenblades.rotormain5) loadingand the highlyof the1, thenoise,interactionNoisefocusingFouriertwo differentduringmethods,2) thicknesshaveof main rotorby ainJet noiseand tail rotor rotation.The two-dimensionalto distinguishshownofof theanalysisblades.aft of the aircraft.noisedata, collectedtransform.of the MainAsTheseconventionaldimensionalNoiseperiodicthe two-dimensionalusingalternatethethe followingmostlywith the main rotorand one without.shownimpactsand is heardaddressesassociatedbladeIt is not onlyvorticesgeneratedfromthein the tail
4rotor tip path plane would propagateto the ground directly underneaththe tail.References[4], [5], [6], and[7] addresstail rotor noise.This type of noisewas observedto be strongestduringtakeofffor the SikorskyS-76in Ref. teractionby[8]. Figuremainthe tail ingto ontransformThefromverywouldit mayclearly.Chapter3on theinteractionof multiplesrotor-tailandbe his dependentonas a "burbling"sound,typeof aircraftmaneuver.noisewas identifiedof therotormaininteractionin detailmultiplefrequency,rotornoisethehighlyInon powerandtailappearsrotorin thein this thesis.cannotis neverIt is shownbe adequatelyproposedrotationaldesignedcan be characterizedand Miameefrequencyto be a wholeeach otheras thesum of twoby thenumber.and resonating.in [3] that this summedrepresentedinperiodicsignalsingleisFourierin [3] that a two-dimensionalFourierthe signal.discussesto distinguishpreliminarysignalsof tail rotorfrom reinforcingfrequencies.chapteron io,twoHardinbetternoiseto this noiseof the two rotorstheseTherefore,rotorthe maininteractiondependentof mainwith incommensurateexplainsperformedalsorotationalnot periodicbeenNoisethe harmonicsthe noisetransform.[8] referredrotorrotora mainfrom from6, it will not be addressedrotor-tailto the maintheoreticallyrotoror differencesof Helicoptermainsinusoidsrotor-tailevidencein ChapterThis preventsThusandas sumsshed fromhastip path plane.mainlistener,interactionis the wake[7] and [8], main rotor-tailspectraharmonics.track.rotorthe tip vortexNot shownalso may intersectboth referencesrotor-tail1 illustratesthe tail rotor tip path plane.Noiseto what"ideal"noiseFourierfromFouriertransformtail alysesshould
5looklike.analyzed.the measureddimensionalin ChapterChapter4 describesChapter5 showsflyoverFourierdata,helicopterthe resultsfollowedtransform6, and suggestionstheacousticsof two-dimensionalby conclusionsanalysisfor futureflighton helicopterstudytestFourierin Chapterflyoverare also presented.fromtransform6. Theacousticwhichanalysisadvantagesdatadataisonof two-are discussed
r CAR.is given[3] defineThea classCAR process,X(t),of randomprocesseswith autocorrelationcalledcorrelationfunctionRx(tl,t2) by the relationshipNRx(tx't2) E a;Rx(tx ;'t2 ;)(1)j lfor all tl and t2, whereCARaj and E are fixedSubsetsofprocesses-definedin [3] by the behaviordiagramfrom[3], representingreal numbersstationary,of thetheseperiodicallyautocorrelationclassesof randomand N is a fixed,correlated,in eq.positiveharmonizable1. Figure2 showsprocesses./Correlation \'-4\\/Figure 2. Velm diagram of class relations [3] integer.-area Venn
7A stationaryrandomprocessis definedin [3] as a randomNRx(tl't2) ZaJ Rx(tlcorrelatedwhereN t't2Rx(tl't2)Z t) j lA periodicallyprocessa1for all t.(2)Jfor all n.(3)j lprocessis one such thatNRx(tl,t2) Rx(tl np,t 2 np) Rx(tl,t2) ,aj lThis type of processessentiallyrepeatsitselfover a finiteperiodp. For both stationaryNand periodicallycorrelatedrandomprocessesto satisfyeq. 1,aj 1.j lAn exampleConsidershownof a stationary,a sinusoidalsignal,in Fig. 3(a). FigureperiodicallyX(t),of finite3(b) showscorrelatedlengthwiththe autocorrelationprocess,is thatfrequency20 Hz.of the signal,of a sinusoid.Thissignaland partis(c) shows1its autospectrum.The periodof this signalis T -- 0.05 secas can be readilyseeninfFig.3 (a).A Hanningautospectrum(c). Thethe end of a signalin analysisin orderIt commensurateperiodic.to bettershownAn infinitelyis discreteusedwindowin calculatingtapersmatchdowntheautocorrelationthe amplitudethesetermination3(b)is plottedin Fig.as a dampedfor the autocorrelationa gradualwas(b)andat the beginningpoints.It is usedandextensivelysignals.autocorrelationt t 2 -t .showsHanningof periodicTheFig. 3(a)windowsinusoidto thelong and continuousand a delta functionand finite,dueasfinitesinusoidfor the Fourierthe autocorrelationa functionlengthwouldtransform.is dampedandofof theproduceSincethetimesignalanda sinusoidthe signalthe Fourierintransformpeak near 20 Hz.noiseForwasclassifieda helicopterfrequencies.For example,considerin [3] as a harmonizable,withThis typea tail rotor,of signala signal X(t),therearecorrelationtwo periodiccan be consideredneitherthat is the sum of two t) Z(t).
1.00.50.0-0.5-1.000.050.10.150.20.25Time (sec)(a) Sinusoidal12000 '8000-'0osignal,tAA::4ooo:vv t-8000--16000X(t)vv:,.-0.4-0.200.20.4Time (sec)(b) Autocorrelation,02040Frequency(c) AmplitudeFigure3. StationaryRx, of X(t).60801O0(Hz)of Auto spectrum,and periodicallySx(f) of X(t).correlatedprocess.
9Figure 4 (a) showsthe signal,X(t).signal,andFig. 4(c)calculationshowsits auto spectrum.of the autocorrelationto pletelyassumingis notthe signalTo showthatareperiodicsignal,theprocess.theFourierin Fig.theseof theusedthis signalsuch as X(t)as shownRather,has beenAlthoughsinceprocessplotswindowof X(t).fact,the autocorrelationin theappearstransformis a function4 (a) andplotshaveandof two(b) arebeennotproducedis stationary.that thesinusoidsInautoregressiveofshowsA Harmingperiodic.t2 [3], one-dimensionalrepresentative4(b)and auto spectrumof a correlationtl andFiguresignalperiodicX(t)within Fig. 4(a)periodp.is not periodic,Giventwoexaminefrequenciesa andsinat sin sina(t p) sin p)sin at sin t sin at cosapsin ap sin t cos/3p cosatthesumof two/3, thenfor a cos/3t sin/3pThis is true if and only ifcosapThus,the signal cos/6pis periodic 1andtypefrequencies,breaksequalwoulda tedthesetones,signal,notthesumbe ideallyin the time domainto the blockis chosenTheofFourierlength.an 0.twotreatssignalofany periodprocessrotorsignalssingleof sinusoidsthe signaltheperiod.representedas if it wereWhenchosenas the blockare derivedin Ref.interaction.withFourieris thus best representedin Fig. 4 is the sum of two "purei.e., main rotor-tailperiodicby theinto a seriesmultiplefrequencies,/3p 2nrc.representedA periodicintegerandoftransformfor a non-periodicsignal sin/6pif and only ifap rm,in the frequencyperiodicwhentwowith periodthe blocksinusoidswill result[9]. Additionally,it lackswhichlengthareofin an error.becauseany interactionthebetween
lO01t!!!!t!!!-200.050.10.150.20.25Time (sec)(a) )0.20.4(sec)Autocorrelation,Rx(t),of X(t).20dB/'''i50100150200Frequency (Hz)(c)Figure4. SumAmplitudeof twoof Autosinusoidsspectrum,withSx(f),frequenciesof X(t).of 22and121Hz
11Thenextcorrelated,sectionthe signaland thus be analyzedexplainsthatdespitebeingfromFig. 4 and rynormay be consideredperiodicallyharmonizable,on periodicity.HarmonizabilityReferenceFourier[3] definestransform,dimensionala randomSx(0.,)1,0.,)2)Fourierprocess,a harmonizabletransform,is definedRx(tl,t:)a processwoulditsas one in whichautocorrelation,also calledRx(tl,t2)the two-dimensionalthe two-dimensional, exists.powerThespectraltwo-densityofin [10] asS ( '/ 2) GivenofprocessI I Rx(tl't2)e-J(h'-x '2)dtflt2of two time variables,X(t),(4)its two-dimensionalautocorrelation,be(5)whereEl.] is the expectedTo understandvalue.the contentthe sum of two periodicsignalswith incommensurateX(t) If thetwo-dimensionalinvolvingof a two-dimensionalautocorrelationA2eir l('l '2)-iThe two-dimensionaltransformfrequencies,letfor the caseA e ' '1' A2 e' vthe two time variables,E[X(I1)X(I2)] FourierFourier(6)is obtained,and the two frequency- A2ei%('1 '2)-ij a2.a 1 e"i091(tl t2) a 22 e i092(tl t2) ala2theresultis a sumof fourvariables,- A1A2ei(r l'l %'2)-itransform,of- A1A2e i(%,1 r 1,2)/ , of eq. 7 can be expressed"'( J t J2t2) ala2i( J2t J t2)(7)as(8)terms
12whereci( l(tl t2)zIICi( l(tl t2)c-i(ziti z2t2)dtldt2 ( ( )1-&)( ( )l-(9)2),ci(( ltl ( 2t2)---( (0 1--&)1(0 2-- 2), ' ' ) ( 0:--X ) ( OI-- :).fi is the DiracdeltaincommensurateHardin(o1 -and Miamee0-)2characteristicZ1 and Z2 are thefrequencyvariables,and0-)1 and(o2 arefrequencies.of a harmonizable,linesfunction, provedcorrelationf'k paralleltheoreticallyin [3] that the two-dimensionalautoregressiveto the line( 1process wouldwith the r/s0-)2,featurebeingspectrum"supportonlythe real rootsonof theequationN-ig),' ig(n) aje -l O.(lO)j lwheref ( 1These"supports"dimensionalprocess-0-)2.poweris zerodescribedabove.dependentare ereFromon (ol andlines that wouldin fig. 5, reproducedeq.wouldexceptonathewoulddiagonal,if (ol andbe thosebe exclusivelyto all harmonizablein fig. 5, the two-correlation( 1 0-)2, andthat theseparallellinesthosewithcan only bethattwothis conditionincommensuratefrequencies.For a finite, ,&) signal,the FourierZ.( I, Z I A,B,,5()transformnco2)5()mof eq. 7 is [from) B, ,5()eq. 9]nco2)5()asof eq. 8. That is, theof (ol or (o2. Notenot justlinesthe only parallelto the first two termsrepresentativeautoregressiveparallel(o2 are incommensurate,relatingprocesses,[3]. As shownharmonizable,8, it can be determined(o2. However,appearforfromrn(11)
ctranextdownfieldIt is simplydimensionaldiagonalstheof of veryof real helicoptersomesimplesignals.provideof thewillfor furthersignals,in Fig.amplitudescomponentsexamplesThesedensitythen be possiblecomponentsas a subjectgivesTheseIt woulddifferentspectralillustratedof Am, A,, Bin, and B,.to breakfromof theof 2-D powerbe for the case m n.ThisThe5. Support[3].5 wouldalongbe productsthe mainfrom thesecenteramplitudeof thenot be appliedto theofor centerdiagonalparalleldatain thisstudy.of two-dimensionalsomethus givingideaFourierof the characteristicsa simplisticpreviewtransformof twoof expected
14CHAPTERRESULTSTo get an ideaworkinitiallywithdimensionalsignal.of the spectrawindow,one-dimensionalanddimensionalarethe ,it is usefuldata. In the followinga puretone,a periodicallyspectralare madeto 20000usingHz samplingThe are generally8192 points.whileratecomplex,of 5000rate and HanningfrequencyThe lowerto be useda samplingsamplingretainingfrequencyof20for theautocorrelationsHz,rate allowscomputingHz and awindowsof the two-dimensionalfundamentalof the fundamentalin the two-dimensionalthetwo-dimensionalof thefrequencyMultiplesfromcalculationsas opposedspectrafromSIGNALSwill be shown.calculations.spectraresultsor computer-generatedSinceAll two-dimensionalrectangularSIMULATEDwill be presentedand a non-periodicthe amplitudeof the e-multiplesofandfrequencyat 20 Hz are used for betterresolutionvalues.Pure toneThegenerate(Eq.puretonethe two-dimensional5) describedspectrumshownin ChapterPeriodicallyusingto the singleCorrelatedA periodicallyamplitudeMATLABsoftwarespectrum.Using2 is usedto generatein Fig. 6. FromFig. 6 is equivalentdecreasinggeneratedMATLAB,Eq. 5, it is apparentFouriertransformin fig.3 hasthe powerspectralthe two-dimensionalthat the diagonalspectrumbeenusedestimationpowerdensityof the spectrumthat is showntoinin Fig. 3(c).Signalcorrelatedsignalhas also beenconsistinggenerated,of a pureusing MATLAB.toneandharmonicsFor this example,with19
15harmonicsin additionto thefundamentalat 20 Hzare generated,usingthe followingequation:20X(t) (1-.Oli)sin(2Tc2Oit)(12)iwheret 0 to 409.4msec in incrementsof 0.2 msec.Rx(t)(a) Two-dimensionalautocorrelation1::20.MAX;U"NiZ 402OMIN".":re#en t (i,-i{z)(b) Amplitudeof two dimensionalFigurepowerdensity6. Pure tone at 20 Hz.spectrum
16Figure0.05sec.tones7 showsFigure8 shows(fundamentalresemblesthe time historytheplusthe signalfrom19of the signalone-dimensionalharmonics)a helicopterup to 0.25spectrumcanwithoutbeof theobserved.a tail rotor,sec, whichsignal,Thishas a periodin whichparticularall 20examplesuch as the MD 902 Explorer.151050-5-10-1500.050.10.150.20.25Time (sec)Figure7. Time historyof uencyFigure8. One dimensionalpowerspectraldensity400500(Hz)of periodicallycorrelatedofsignal.
17Figurethe periodically9 showsthe ion9(a) showsTim:e l y(a) Two-dimensionaland powera matrixspectralof dots, whichdensityare equally(se }autocorrelation,Rx(t)dBii iii;iiiliiiiiiiiiiiiiiiiiiiiiiiiiiiii(b) TwoFiguredimensional9. f
18spacedby the period. The "support"lines describedin [3] may be drawn in Fig. 9(b) byfollowing the red "peaks"parallelto rfrequencieswithaare summedusing20X(t) (1-0.01i)sin(2zc20it)rotor-tail referredtones.msec in incrementsfrequenciesrotorratiodifferof thesinusoids.the n (2zc110jt)(13)simulatedtonessignalThe time historyof 0.2 msec.by a factorSikorskyto as the main rotorThetwojt 0 to 409.4The tworotor,2iwheretailof 5.5 (20S-76Cand theandhelicopter.110-Hzincludesharmonicsof this signalis shownTheharmonicswith110 Hz),20-Hzwhichharmonicswill be calleddecreasingis the mainwillthe tail rotoramplitudein Fig. 10. Thoughbeforit appearsbothperiodic150 -10-1500.050.10.150.20.25Time (see)Figurein nature,dimensional10. Time historyfrequencies.it is in fact non-periodic,spectrumof sum of periodicas explainedof this non-periodicsignal.signalsin ChapterNotewith incommensurate3. Figurethe higher11 showsamplitudethe one-at 220 Hz,
19wherethe11 th and 2 nd harmonics,respectively,of the two signalsare integermultiplesofeach other.01 l rotor)TwospectrumshowsrotorFig.diagonalwould13 showsof onthe centerof the tail rotordotsmarkedof thehigher(tailincludingtheto theone-dimensionalfrequencyand offsetby 110 Hz.diagonalsplottedsignal.haveThetail rotorat 110 Hz. The tailtogether.diagonal.at 110 and 330 Hz, respectively,peaksthetwo-dimensionalfrom the tail rotorin the tail rotortheofthein theand tail rotormarkonis equivalentthein red,all the generatedtailrotorand12(a)centerandof sum of periodicautocorrelationof 20 and 110 Hz.to the centerreducedspectrum500frequencies.as dots,all the400at multiplesTheonly containwereshowsinformationis parallelor 20 Hz tonesthird harmonics12(b).containsrotor )frequencypowerClosefrequency)in mFrequencywith incommensuratethenon-periodiclower30011. One dimensionalsignalsFigure200TheThe mainrotorfundamentalandretainedtheir
20. Rx(t)iiii iiiiii(a) Two-dimensionalautocorrelationdB.ii i! i! ii ! iiiiiiill . (b) Two-dimensionalFigure12. Sum of periodicsignalsspectrumwith incommensuratefrequencies.
21amplitudein the tail rotor diagonal.The secondandfourth harmonicsof the tail rotor at220 and440 Hz, respectively,arereducedby an averageof 4 dB. Becausethe tonesat220 and440Hz areharmonicsof both the mainrotor andthe tail rotor, the reductionsinthe tail rotor diagonaldenoteremovalof the contributionof the mainrotor to quencyFigure13. Centerand "tail"diagonalsum of incommensurateIn the nextchapter,a flightwithout,will be desc
the other. A two-dimensional Fourier analysis method is used to show helicopter noise as harmonizable. The two-dimensional spectral analysis method is first applied to simulated signals: a single tone, a series of periodically correlated tones, and a series of tones composed of two
