politecnico di torino · possibility to reach sonic speeds on the blade, generating normal shocks...

POLITECNICODITORINO

MasterDegreeinAerospaceEngineering

Master’sDegreeThesis

AerodynamicperformanceofdifferentairfoilsforafutureMarsHelicopter

Supervisors: CandidateProf.AkiraOyama (JAXA) LorenzoBruniProf.GaetanoIuso(PolitecnicodiTorino)

July2020

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INDEX

ABSTRACT................................................................................................................................................3

1.INTRODUCTION..................................................................................................................................31.1.MOTIVATIONFORHELICOPTERFLIGHTONMARS.......................................................................................41.2.MARSHELICOPTERMISSIONS..........................................................................................................................41.3.JAXA’SHELICOPTER.........................................................................................................................................6

2.PURPOSEANDEXPERIMENTALSETUP.....................................................................................72.1.MATHEMATICS...................................................................................................................................................72.2.LIFTMEASURINGDEVICE.................................................................................................................................82.3.ELECTRONICS...................................................................................................................................................102.4.VACUUMCHAMBER.........................................................................................................................................112.5.GRAPHICALUSERINTERFACE.......................................................................................................................122.6.BLADEPROFILES.............................................................................................................................................132.6.1.DF102............................................................................................................................................................132.6.2.E178...............................................................................................................................................................142.6.3.E10.................................................................................................................................................................152.6.4.TriangularProfile....................................................................................................................................162.6.5.Nominal.......................................................................................................................................................19

2.7.DIFFICULTIES...................................................................................................................................................233.POSTPROCESSING..........................................................................................................................243.1.NEWROTORDESIGN.......................................................................................................................................33

4.CONCLUSIONS..................................................................................................................................344.1.POSSIBLEFUTUREWORK..............................................................................................................................34

5.APPENDIX..........................................................................................................................................355.1.HOVERINGHELICOPTERTHRUST................................................................................................................355.2.EXPERIMENTALAIRFOILFORULTRALOWREYNOLDSNUMBER............................................................375.3.WEIGHTREDUCTION......................................................................................................................................39

BIBLIOGRAPHY....................................................................................................................................40

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Abstract AerodynamicevaluationofmultiplebladeprofilesforafutureMarshelicopterispresentedusing vacuum chamber experiments.A small-scale Mars helicopter prototype, that waschosen to explore deep craters present on the red planet surface, was fastened to aspecifically designed test structure and the generated thrust was then measured fordifferent airfoils, rotor speeds and angles of attack. Afterwards, thrust coefficients wereextrapolated from these measurements and compared with one another leading tointeresting results such as performance improvement with decreasing thickness andincreasing camber, good performance of triangular airfoils and E178 airfoil improvingperformancewithdecreasingReynolds.

1.Introduction Inrecentyears,spaceexplorationagencieshaveincreasedtheirinteresttowardsMars,withtheintentionofstudyingitsevolution,searchingforlifeandfosteringhumanmissions.Asof2020onlywheeledrovershavebeenusedtoscouttheredplanet’ssurface.Thesevehicles,although being very slow, have been very useful for unprecedented exploration of theplanet.However, engineershavestarted to thinkaboutusing flyingobjects, suchas smallhelicopters or airplanes, to scout the surface or achieve operations that normal roverswould not be able to perform. There are many mission concepts regarding the use of acompact self-flyinghelicopter; however, all of themhave in common theneed foruniquedesign methods and flight configurations. This is due to Mars’ environment beingunfavorabletoflight,ascharacterizedbyanatmosphericpressurenearthesurfaceofabout1% of that of Earth and an atmospheric density that is as low as 0.0121!"

!!. Furthercompounding these issues isMars’ cold carbon dioxide atmospherewhich, given a lowerMachnumbercomparedtoEarth’s,inducescompressibilityeffectstoappearsoonerontherotor blades. Compressibility effects introduce a high increase in wave drag and thepossibilitytoreachsonicspeedsontheblade,generatingnormalshocksthatfurtherbreakup the airflow and could possibly damage the blade itself. Fortunately, not everything isagainst the success of Mars flight as gravity is nearly 3 times lower on Mars, wheregravitationalaccelerationis3.71!

!!comparedtoEarth’s9.81!

!!.

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1.1.MotivationforhelicopterflightonMarsThereareafewreasonofwhydeployingahelicopteronmarswouldbebeneficialforfuturemissions.Asastart,anaircraftwouldbeabletomovemuchfaster,coveringmoreground,whileflyingseveralmetersabovethegroundachievingbetterviewsofwhatithasaroundwithrespecttoarover.Inaddition,apeculiarfeatureoftheMarssurfaceisthepresenceoflavatubes.Alavatubeisanaturalconduitformedbyflowinglava,whichmovesbeneaththehardenedsurfaceofitsownflow.Tubescanbeextinct,meaningthelavaflowhasceased,andtherockhascooleddownleavingacavebehind.Somesectionsofthesetubes,theonesclosertothesurface,cancollapsecreatingaskylight.

Figure1PitcrateronMarssurface.[28]

Theseskylightsareattractingmanyscientistsbecauseitisconsideredthattheobservationof their inner wall could reveal the history of Mars’ volcanic activity. In addition, thepossibility of finding traces of life is discussedbecause of the lack of ultraviolet rays andother radiations from the Sun. For the same reason, engineers are considering thepossibility of using lava tubes as bases for future manned missions. Due to their nearlyverticalwalls, however, ordinary rovers cannot be used to explore these craters. In theirplace these small sized, lightweight helicopters are intended to be used,mainly for theirabilitytoexecuteVerticalTake-OffandLanding(VTOL).

1.2.MarshelicopterMissionsAlthoughtheirmissionconceptsandobjectivesarequitedifferent,twomaincompaniesarecurrentlyworkingonaMarshelicopter:JAXAandNASA.NASAisplanningtouseitsalreadybuiltandtestedMarshelicopterduringtheMars2020mission,whichislaunchinginJuly2020[29].The1.8Kghelicopter,infigure2,consistsoncounter-rotatingcoaxial rotorsofabout1.1m indiameter (tapered, twistedand10cm inchordlength)spinningbetween1900and2800rpm[20,30].Ahigh-resolutiondownward-looking camera for navigation, landing, and science surveying of the terrainmakesup itspayload, this isadded toa communicationsystem that relaysdata to the rover.Themainpurpose of this helicopter is to assist the main rover “Perseverance”. After two and halfmonthsfromthelanding,theaircraftwillflyuptofivetimesduringthe30-daytestperiod.Each flightwill lastupto3minutes,ataltitudesranging from3mto10maboveground,whilethedistancecoveredisthoughttoreachasfaras300mperflight.ItwillmakeuseofautonomouscontrolandwillcommunicatewiththePerseveranceroverdirectlyaftereachlandingwhilerechargingthroughitsownsolarpanel.Thehelicopterwillbeabletoprovideoverheadimageswithapproximatelytentimestheresolutionoforbitalimages,displayingfeaturesthataremostlikelyoccludedfromtherovercameras.Thisscoutingmethodcould

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enableimprovedroadmappingthusallowingfutureroverstosafelydriveuptothreetimesasfarpersol.Themissionisthusonlyasupportoperationfortherover,whichisthemainscopeoftheexpedition.

Figure2NASA’sMarshelicopterconcept.[17]

Ontheotherhand,JAXA’smissionforeseesahelicopterpiggybackingonarovertothecrestofalavapit,whereitthenproceedstohoverinsideofthetubemappingtheconduit,takingpictures and sampling the walls. It then returns on top of the rover where it will berechargedandeitheranewpathisplannedforthesameductortheaircraftisrelocatedforthenextassignment.ComparedtoNASA’smission,JAXA’sismorefocusedonthehelicopterandtheroverisonlyasupportforthemaintask.Thehelicopteristhusrequiredtobeableto fly for as much time as possible. This research is focused on improving the JAXA’shelicopteraerodynamicpropertiessincethemoreefficientitsaerodynamicsthelesspoweritwillrequiretogeneratethesameamountofthrust,thusincreasingflighttime.

Figure3JAXA’smissionconcept.[14]

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1.3.JAXA’sHelicopter

Figure4JAXA’sMarshelicopter.

Asshowninfigure4,this465gramshelicopterhastwocoaxialcounterrotatingpropellers.According to reference [21], this was chosen in order to maximize thrust productionoperatingatthesamepower,andtoeliminatetheeffectofhavinganonzerototaltorqueonthe body. The propellers are mounted on a carbon-fiber chassis and driven by a highefficiencyelectricmotorthroughappropriatelydimensionedgearssuchthatat100%motorusage, thepropeller reaches2500 rpm.Twodifferent remotes can control thehelicopter.One remote controls solely its angularvelocity.Theother controls itspitch, roll, yawandcollectorbydirectlymoving thehelicopter’ swashplateor, in caseof yaw,overriding thefirst controller triggering the propellers to rotate in the same direction affecting angularmomentum and creating a strong enough torque that induces the helicopter to turn onitself.

5Marscontrollers.RPMmanager(left)allmovementremote(right)

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2.PurposeandExperimentalSetUp Thegoal of these experimentswas tounderstandhowrotational speed,blade chord, andbladeprofile affect thrustproductionof a small-scalehelicopterprototype in a simulatedMartianenvironmentcharacterizedbyanultralowReynoldsnumberflow.ThiswascarriedoutbypositioningtheJAXA’shelicopterinadevicethatconstraintseverydegreeoffreedomexcept forverticalmotion.AnS-shapedstrainsensorwas then linked to thehelicopter inordertoblocktheverticalmotionwhilemeasuringhowmuchthrustwasbeinggenerated.Thiswholedevicewas thenput insideofavacuumchamberandbrought from1kPa to5kPawith a 1-kPa step.Once at thedesiredpressure the rotorwas spedup from2000 to2500rpmwitha250-rpmstep.Ateachstepthebladepitchanglewasshiftedfrom0to18degrees with 3 degrees step. This pitch angle was measured from the fundamentalhorizontalusingadigitallevel.Becauseofswashplatedesignandpositioningonthemainshaft,18degreeswasthemaximummechanicallyreachablepitchangle.

2.1.Mathematics Giventheequationforliftproducedbyadoublerotatingdisk(calculatedintheappendix): 𝑇 = 2𝜌𝐶!𝐴𝛺!𝑅! (1)Where,CT=Thrustcoefficientρ=AirdensityA=Diskarea,proportional=𝜋R2Ω=RotorangularspeedR=RotorradiusThrustcoefficientcanthusbeextrapolatedas:𝐶! =

𝑇2𝜌𝐴𝛺2𝑅2

(2) From equation (1) the formula for radius can also be composed. This equation was used at the end of the research to compute the ideal radius length:

𝑅 = !!!"!!!!

!! (3)

After all of the experiments, the results will be shown at the same Reynolds number conditions so that the flow characteristics are equal for each. Therefore this parameter’s equation is shownbelow:𝑅𝑒 = !"#

!(4)

Where,ρ=Airdensity[!"

!!]V=Rotorspeedat¾ofthebladelength,wherethemajorpartofthethrustisgenerated[!

!]

L=Chordlength[m] 𝜇=Airdynamicviscosity[𝑃𝑎 ∗ 𝑠]

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Thegasdensitywaseasilycalculatedbyusingtheidealgaslaw:𝜌 = !ℳ

!"(5)

Where,P=Airpressure[Pa] ℳ=Molarmass,29[ !"!"#

]forair T=Airtemperature[K] R=Gasuniversalconstant,8314[ !

!∗!"#]

On the other hand, the Sutherland law, shown below, was implemented to calculatedynamicviscosity:

𝜇 = 𝜇! ∗ (!! ! !!!!

) ∗ !!!

!!(6)

Where,𝜇!=Referenceviscosity,forair1.716*105[𝑃𝑎 ∗ 𝑠]T0=ReferenceTemperature,273.15[K]T=TemperatureatwhichviscosityiscalculatedC=Sutherland’sconstant,110.4[K]forair

2.2.LiftMeasuringDevice TheLiftMeasuringDevice (LMD)wasdesigned to stopanymovement except for verticalmotion. This way the helicopter was able to fly safely without any stabilization control,whichgavethepossibilitytotakeafewshotsoftheflyinghelicoptertoprovethefeasibilityof the project. After taking these shots the helicopter was anchored to the base of thestructure throughanS-shapedstrainsensor.At rest, thesensorcompletelywithstood thehelicopterweight.Whenthehelicopterstartedproducing thrust theweighton thesensorreduced letting the sensor expand. This expansion is translated from electrical signals togramsthusmeasuringthrustproduced.Thewholedevice,asdepictedinfigure6,wasfirstdesignedonAutodesk’sInventorProfessional,partbypartandthenassembled.Duetotheircriticality,somepartswerefirst3DprintedinPLAusingaMakerBotReplicator2printerinordertocheckthecorrectnessoftheirdesign.Onceeverypartwascheckedandconfirmedtobecorrect,thedevicewasbuiltresultinginwhatisshowninfigure10.Fromequation(1)itcouldbeeasilyseenthatthrustwasdirectlyproportionaltothesquaredpowerofΩandtothefourthpowerofR.ForpreliminarycalculationCTwaschosenas0.0063,valuetakenfromthepreviousworkdonebyAokiinreference[22].Thusinanidealenvironmentalargeradiusalongwithslowangularvelocityispreferred.However,therewereafewissues.Tostart, as radius increased the helicopter size increased making it difficult to performexperiments inside of the vacuumchamber. In addition, themotorused in thehelicopterhad better efficiency at high speeds (of around 10000 rpm or above [11]). It was alsodecided that theMachnumberat the tip shouldhavebeen lower than0.85 toavoiddragdivergenceduetoshockwaves.

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ThatMachnumbervaluewaschosenforthefollowingreasons:

• LocalMachnumbershouldhavealwaysbeenlowerthan1toremainsubsonic.

• Oncedeployedthehelicopterwillmoveforwardorbackwardcreatingrelativevelocities,whichwillincreaseanddecreasetheblades’Machnumber.

• Duetoitslowtemperaturecarbondioxidebasedatmosphere(γ=1.28,M=44.01 !

!"#),Mars’speedofsoundislowerthantheoneonEarthwhichmeans

thatalowerflightspeedisrequiredtoreachthecriticalMachnumber.

Amaximumradiusof0.5mwithanangularvelocityofmaximum5000rpm(Mach=0.77)wasconsidered,causingtheidealgeneratedthrusttobearound8.21N.ForlongerradiitheusableΩwastoolow,andmoreimportantly,thehelicopterwouldnothavefitinsideofthevacuumchamber.However,theweightofthe0.5mbladeswastoolargetobehandledbythesmallhelicopter:thecentrifugalforcewouldhavebrokentheaircraftapart.Thusitwasdecided to use small rotorswith different airfoils, explained in details in chapter 2.6. Anadditional factor was taken into account: Mars has a lower gravity pull than Earth. Theactualweightofthehelicopterdropssubstantiallyfrom4.52Nto1.71N.Inordertostudythe feasibility of the flight, the helicopter should have been able to lift only its Martianweight.Thus,theneedwasraisedtoalleviatethehelicopterfromallofthestructureweightaswell as part of its own. Thiswas carried out securing a total of 1.84 Kg (mass of thestructureconnectedtothehelicopteraddedtotheweightthatthehelicopterwouldloseonMars.EquationfoundintheAppendix)evenlydividedbetweenthetwoarmssupportingthehelicopter.Thedirectionof the force inducedby theseweightswasdirectedupwardwithrespecttothearmsusingpulleysatthetopofthestructureinsuchawaythatthemasseswouldnothitthegroundbeforethehelicopterreacheditsimposedheightlimit.

Figure6LiftMeasuringDeviceprojectasseenfromInventor3D.

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2.3.Electronics

Figure7ArduinoUNOboard(left)andHX711amplifier(right)

Thewhole case study’s objective is based on liftmeasuring. This brought up the need tochoose the right sensors, interfaces andmicrocontrollers. Arduino UNOwas chosen as amicrocontrollerduetothelowstrainofthetaskandtheboard’slowpriceandhighqualityperformance. As explained before, the maximum lift expected was at first 8N. Since thehelicopterwill not always be able to lift itself up, a sensor able tomeasure both its owncompression and tension was chosen. A sensor of this kind translates, through aWheatstonebridge,thechangeinresistance,duetoitsowndeformationaftercompressionor tension, in electrical signals. The sensor selected was a 10N Unipulse USM fullWheatstone bridge sensor with a Rated Output (R.O.) of 0.4159 mV/V. Although theArduinosensitivityisquitehigh,duetosuchlowR.O.,thissensorcouldn’tbedirectlylinkedwith themicrocontroller. Thus theHX711boardwas used to amplify the readings of theUSMsensor.This10Nsensorwasonlyusedtomeasurethrustatstandardpressure.Insideof thevacuumtestchamber,becauseof the forcedchoiceofradiusandspeeds,a2NUSMsensor,showninfigure8,wasoperated. Inordertominimizeelectricalnoise,thesensorcircuitwascompletelyembeddedintothevacuumchamber.SinceArduinopermitsconnectionthroughUSB-AtoUSB-Bportsasinglecable was cut and extended in such a way that the thrust readings were done a fewcentimeters from the sensorandstored into theArduino,which then relayed it to thePCthrougha10mcable.Thiswaytheanalogvoltageshift,whichiseasilycorruptiblebynoise,wasconvertedatonceintogramsandpromptlysentthroughadigitalsignaltotheoutsidecomputer.This,inadditiontothefringelayoutexplainedinthefollowingsection,ensuredelectricalnoisetoaminimum.Eachexperimentwascarriedoutat leasttwice,andthe2Nsensor’shigherprecisionwasusedtoconfirmthevaluesmeasuredbythe10Nsensor.Asanexample, if the 10N sensor detected a thrust of 10 g, the 2N sensor would perceiveapproximatelythesamevaluewitharangeof±0.8g.

Figure8USMUnipulse2Ncompression/tensionsensor

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2.4.Vacuumchamber

Figure9Vacuumchambersetupschematics.

Thechamberusedwasa cylindrical steel structure1.3mwideand7m longpoweredbytwo rotary pumps.Alongside these twomachines, two additional boost pumpswere alsopresent in order to lower the pressure below 2 kPa. The set up used two independentfringesof16and10cables inorderto interfacethehelicopterwiththeoutsidecomputerand controllers. This interface tried to separate power lines from data lines in order tolowerasmuchaspossibleelectricalnoisecreatedbyflowinghighcurrent.The12and24Voltcables,themotordriverscablesandtheservocableswereall inonefringe,whiletheRadio frequency cable, the rpm control cables, and the Arduino USB cables were in theother. As explained before, the sensor circuit was completely inside the chamberminimizingcablelengthandthusnoise.

Figure10LMDfullyconstructedandinsideofthevacuumchamber

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2.5.GraphicalUserInterfaceThisexperiment’sGUIwasfairlysimpleastheonlycheckedparameterswerepressureandthrust.“LabView”softwarewasusedtomonitorpressureinstantvaluesinordertocheckifitremainedconstantovertime,asonecanseefromfigure11.Thisoutputwasalsousedtosetthepressureatthedesiredvalueforeachstepoftheexperiment.

Figure11LabViewwindowmonitoringinstantandwaveformvalueforpressure.

Asfarastheforcesensorisconcerned,theArduinoIDEwasused.Arduinopermitsmachineserial communicationwhose output is shown in figure 12A,where an unloaded sensor’soutputisbeingdisplayed.Theslightbutconstantdecreaseonmeasuredthrustisduetoaninfinitesimalexpansionofthesensorduetoatemperatureincrease,whichiscausedbytheelectriccurrentrunningthroughtheWheatstoneresistors.Itisworthnotingthatthissmallchange in thrust isbarely sensedwhenmeasuringanactual loadas canbe seen in figure12B.

(A) (B)

Figure12ArduinoIDEvisualoutputforzeroload(A)andappliedthrust(B).

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2.6.BladeProfiles

Becauseofthelimitedusableradiusitwasdecidedtousea3Dprintertoalsobuildvariousbladeprofiles inorder tounderstandtheirperformance inanultra lowReynoldsnumberenvironment.DF102,E178andE10profileswerechosen,afterabroadsearchofmostlowReynolds number airfoils (designed for Re of about 70000) currently being used, due totheir better performance [23, 27]. On the other hand, the triangular airfoil, specificallydesignedforReynoldsnumber3000,waschosentocontinuesomepromisingpreviousandongoing researches [6, 9, 24, 25, 26]. TheMakerBotReplicator2, showcased in figure 18,was again used. Although initially the blades should have been0.23m long tomatch thenominalbladelength,duetosome3Dprintinglimitationstheyhadtobeshortenedto0.17m. Normal 1.75 mm PLA was used. Before being printed all the airfoils were carefullydesignedwithAutodesk Inventor3D. This software supports importing point coordinatesthatweredownloaded from “Airfoiltools”website [23].Afterbeingprinted,eachbatchofblades was meticulously sandpapered to eliminate as much surface imperfection aspossible.Eachbladeisnowdescribedandtheirexperimentalvalues,alreadytranslatedintoCT,documentedbelow.

2.6.1.DF102TheDF102isalowReynoldsnumberairfoilwithan11%maximumthicknesspositionedat29.1%chordandamaximumcamberof2.4%at43.5%chord.

Figure13DF102profile.

ValuesofCTforachamberpressureof4kPaandtwospeedsarereportedbelow.

RPM PitchAngle[°] CT

2250

0 0.000003 0.000396 0.000789 0.0023412 0.0054615 0.0078018 0.01170

2500

0 0.000003 0.001266 0.002539 0.0031612 0.0056815 0.0082118 0.01011

Table1DF102CTvaluesat4kPa,2250and2500rpm,andallAOA.

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2250

0 0.000003 0.000626 0.001259 0.0025012 0.0056115 0.0081118 0.01123

2500

0 0.000003 0.000516 0.001019 0.0025312 0.0050515 0.0080818 0.01061

Table2DF102CTvaluesat5kPa,2250and2500rpm,andallAOA.

2.6.2.E178TheE178 isa lowReynoldsnumberairfoilwithan8%maximumthicknesspositionedat29.8%chordandamaximumcamberof2.8%at38.8%chord

Figure14E178profile.



2250

0 0.000003 0.001566 0.002349 0.0039012 0.0070215 0.0109218 0.01326

2500

0 0.000003 0.001266 0.001899 0.0037912 0.0069515 0.0094718 0.01200

Table3E178CTvaluesat4kPa,2250and2500rpm,andallAOA.

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2250

0 0.000003 0.001876 0.002509 0.0037412 0.0062415 0.0093618 0.01248

2500

0 0.000003 0.001016 0.002029 0.0030312 0.0060615 0.0091018 0.01162


2.6.3.E10The E10 is a lowReynolds number airfoilwith amaximum 10% thickness positioned at31.9%chordandamaximumcamberof0.3%at31.9%chord.

Figure15E10profile.



2250

0 0.000003 0.000396 0.000789 0.0015612 0.0039015 0.0062418 0.00936

2500

0 0.000003 0.000326 0.000639 0.0018912 0.0037915 0.0063218 0.00821


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2250

0 0.000003 0.000626 0.001259 0.0018712 0.0034315 0.0068618 0.00936

2500

0 0.000003 0.000516 0.001269 0.0022712 0.0040415 0.0065718 0.00859


2.6.4.TriangularProfileThetriangularairfoilchosenhasa10%maximumthicknesspositionedat25%ofthechord.

Figure16Triangularprofile.

ValuesofCTforachamberpressureof1kPaandthreespeedsarereportedbelow.


2000

0 0.000003 0.000006 0.000399 0.0007912 0.0015815 0.0043418 0.00513

2250

0 0.000313 0.001256 0.001569 0.0025012 0.0049915 0.0068618 0.00936

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2500

0 0.000513 0.001776 0.002539 0.0053112 0.0063215 0.0080818 0.00935

Table7TriangularairfoilCTvaluesat1kPa,allspeeds,andallAOA.



2000

0 0.000003 0.000006 0.001189 0.0015812 0.0041415 0.0067118 0.01066

2250

0 0.000473 0.001406 0.002969 0.0039012 0.0079515 0.0102918 0.01310

2500

0 0.000763 0.001896 0.002659 0.0034112 0.0061915 0.0093518 0.01200




2000

0 0.000133 0.000536 0.001849 0.0032912 0.0065815 0.0090818 0.01118

18

2250

0 0.000313 0.000836 0.002609 0.0047812 0.0078015 0.0116418 0.01393

2500

0 0.000343 0.001016 0.002449 0.0040412 0.0070715 0.0099418 0.01263




2000

0 0.000303 0.000896 0.001889 0.0034512 0.0064115 0.0094718 0.01174

2250

0 0.000783 0.002036 0.003909 0.0049112 0.0085015 0.0120918 0.01497

2500

0 0.000633 0.001336 0.002659 0.0041112 0.0070115 0.0101718 0.01288


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2000

0 0.000393 0.000956 0.002299 0.0036312 0.0067915 0.0099518 0.01216

2250

0 0.000563 0.001256 0.003189 0.0047412 0.0086715 0.0122318 0.01566

2500

0 0.000513 0.001166 0.002589 0.0040412 0.0072815 0.0104618 0.01319


2.6.5.NominalNominalbladeisaNACA0015airfoil.Maximumthicknessis15%positionedat30%ofthechord.

Figure17NACA0015profile.



2000

0 0.000003 0.000226 0.000449 0.0015412 0.0019815 0.0026418 0.00352

20

2250

0 0.000003 0.000436 0.000709 0.0017412 0.0024315 0.0035618 0.00426

2500

0 0.000003 0.000426 0.000919 0.0012712 0.0024615 0.0031718 0.00359

Table12NACA0015CTvaluesat1kPa,allspeeds,andallAOA.



2000

0 0.000003 0.000166 0.000499 0.0012112 0.0023115 0.0029718 0.00357

2250

0 0.000003 0.000266 0.001009 0.0015612 0.0028715 0.0040018 0.00487

2500

0 0.000003 0.000216 0.000539 0.0014412 0.0022515 0.0031718 0.00391


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2000

0 0.000003 0.000156 0.000709 0.0015012 0.0023115 0.0033718 0.00414

2250

0 0.000003 0.000176 0.000729 0.0018012 0.0027215 0.0035318 0.00492

2500

0 0.000003 0.000166 0.000709 0.0012912 0.0023215 0.0033118 0.00411




2000

0 0.000003 0.000166 0.000699 0.0012112 0.0023115 0.0034618 0.00421

2250

0 0.000003 0.000356 0.000899 0.0015612 0.0028015 0.0040418 0.00517

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2500

0 0.000003 0.000196 0.000699 0.0012812 0.0023915 0.0035718 0.00442




2000

0 0.000003 0.000266 0.000779 0.0012812 0.0023115 0.0033418 0.00433

2250

0 0.000003 0.000216 0.000909 0.0014812 0.0028015 0.0041718 0.00528

2500

0 0.000003 0.000236 0.000779 0.0013912 0.0024615 0.0036918 0.00469


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2.7.Difficulties

The blades could not be printed horizontally because of two main reasons. First, whenprinted horizontally, the necessary support material would become stuck to the surfacemaking it very rough and uneven. At that point the time needed to sand off theseimperfections would have been too great and the result would not have been as clean.Secondlythebladewouldsometimesdetachfromthebedwhilebeingmade,creatinganarcthatwarpedthedesignoftheprofileitself.InadditionfortheDF102,E10andE178profilesonlyexperimentsat4and5kPaandspeedsof2250and2500couldbecarriedout.Thiswas due to the increase of vibrations that damaged the LMD base at the end of theexperiments and the lack of a sensitive enough strain sensor at the beginning of theexperiments.Futureresearchshouldseektoresolvetheseissuessoastofurther improveontheexperimentaldesign.

Figure18MakerBotReplicator2printingtwoairfoils.

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3.PostProcessing

Figure19CloseupofthehelicopterfixedtotheLMD.

Aspreviouslyexplained, inorder tobeable to compare thevariousCT, all the resultsareshown at the same Reynolds number, meaning that for the nominal blade, which has achordofonly2cm(halfofthatoftheotherprofileswhichhavea4cmchord),thespeedsandpressureswerequitedifferent.Itisalsoimportanttonotethatifforthenominalbladetheradius lengthwas0.234m,foralloftheotherbladesthatnumberdroppedto0.17m.Forexample, inorder toachieveRe=2400, the triangularairfoilwasspinning ina3kPaenvironment at a speed of 2250 rpm, while the nominal airfoil was spinning in a 4 kPaenvironment at a speed of 2500 rpm. Although Mach number should also be taken intoconsideration when comparing results, during these experiments this number waspracticallyconstant:minimumMachnumberwas0.10andmaximumwas0.18.Asmaybeseenfromthegraphedresultsinfigure20to25,thetriangularairfoilisthebestperformingairfoil of all, as its values of CT are always greater than the other profiles. Right after thetriangular,theE178profile,whichhasthesmallestthicknessofall,isalsoperformingquitewell.Thedifference inCTbetweenthenominalairfoiland theothers isalwaysmore thandouble,meaning that chord length plays a very important role in thrust production. It isworth noting that although CT for the nominal airfoil was always much lower than theothers, due to its longer radius, the thrust producedwas consistentlymuch greaterwithrespecttotheshorterblades.Values of CT shown in tables 1 through 16 are now graphed in the following figuresprovidingvisualmeanstograspthebehavioroftheairfoils.

25

Figure20CTcomparisonbetweenvariousairfoilsatRe4500.

Figure21CTcomparisonbetweenvariousairfoilsatRe4000.

0 2 4 6 8 10 12 14 16 18AOA

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

CT

Re = 4500

E10(10%)DF102(11%)E178(8%)T10

0 2 4 6 8 10 12 14 16 18AOA

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

CT

Re = 4000

E10(10%)DF102(11%)E178(8%)T10

26

Figure22CTcomparisonbetweenallairfoilsatRe3000.

Figure23CTcomparisonsbetweenTriangular(T10)andNACA0015(nominal)airfoilsatRe3000.

0 2 4 6 8 10 12 14 16 18AOA

0

0.005

0.01

0.015

CT

Re = 3000

T10Nominal

27

Figure24CTcomparisonbetweenTriangular(T10)andNACA0015(nominal)airfoilsatRe2400.

Figure25CTcomparisonbetweenTriangular(T10)andNACA0015(nominal)airfoilsatRe1600.

0 2 4 6 8 10 12 14 16 18AOA

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

CT

Re = 2400

T10Nominal

0 2 4 6 8 10 12 14 16 18AOA

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

CT

Re = 1600

T10Nominal

28

Bylookingatfigures26and27,itisnoticeablehowthenominalbladehasindeedalowCTbut it is rather constantwith changingReynoldsnumber: the total displacement is about0.001,which corresponds to a 25% decrease. On the other hand, the triangular airfoil isbehaving quite well until Reynolds 800 where it has a huge drop in performance: totaldisplacement isnow0.006whichcorresponds toa40%decrease.ThatReynoldsnumberconditioncorrespondedtoaspeedof2000rpmatapressureof1kPa,whichisnotlikelytohappenasthatrotationspeedisrathersmall.Therefore,ifweignorethatlastcondition,themaximumdropforthetriangularairfoilbecomes0.002ora13%decrease.Ontopofthis,the triangular airfoil has another advantage on the nominal blade: thrust production isessentiallylinearatlowReynoldsnumbersascanbeobservedcomparingthecurvesatRe800forthetriangularandRe500forthenominalblade.This linearitykeepstherequiredcontrolrulefairlysimple,easingthecalculationsforoptimizationpurposes.

Figure26CTcomparisonatvariousReynoldsforTriangularairfoil.

0 2 4 6 8 10 12 14 16 18AOA

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

CT

T10

Re = 800Re = 1600Re = 2400Re = 3000Re = 4000

29

Figure27CTcomparisonatvariousReynoldsforNACA0015airfoil.

Anotherpeculiarfeatureofthetriangularairfoilbecomesevidentafterfurtherobservationoffigure26.Itcanbenoticedhowtheslopesofeverycurvehaveasharpincreaseoncetheangle of attack reaches a value around 9°. This rise in slope converts into higher thrustproduction.IthasbeenconfirmedbyCFDanalysis(reference[24,25,26])thatthisisduetoa separation bubble that is created between the leading edge and the point ofmaximumthicknessorbetweenthelatterandthetrailingedge.Asportrayedinfigure28,aftersuchanangle,theairflowdetachesassoonasitreachesthebodyandthenreattachesgeneratingthe separation bubble. On the other hand, in figure 29 the second possibility is reportedaccordingtoCFDcalculationcarriedoutinreference[25].Thesebubblesarecharacterizedbyaratherlargepressuredrop,whichinturn,coupledwiththealmostunchangedpressureundertheairfoilcreatesaconsiderableamountofthrust.

0 2 4 6 8 10 12 14 16 18AOA

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

CT

10-3 Nominal

Re = 500Re = 1000Re = 1200Re = 2500Re= 3000

30

It becomes clear that even though the triangular airfoil presents sharp edges, the fluidarounditmanagestocreateacushionlikeflowthatgivesadifferentshapetotheairfoilasshown by the streamlines in figure 29. This cushion tends to take the shape of a moreefficientairfoilthusincreasingCTproduction.Atfirstglance,itwouldseemthatwithlowerReynoldsnumberthisbehaviorbecomesweakerandweaker,withabigsetbackwhentheReynoldsnumberdrops to800,which is reasonable since thedesignReynoldsnumber is3000forthisairfoil.

Figure28Triangularairfoilseparationbubbleformationwhilehorizontal(A)AOA<9°(B)andAOA>9°(C)

(A) (B)

Figure29TriangularairfoilCFDfor12°(A)and14°(B)accordingtoYang’sresearch[25]

31

However,acounter-intuitiveresponseisobservedfortheE178airfoil,showninfigure30.ThebehaviorofthisprofileactuallyimprovesasReynoldsnumberdecreases.ThisincreaseisclosetothetriangularreductionastheE178’scoefficientofthrustundergoesamaximumgrowthof0.0016or11%.ThesefindingsshouldbeconfirmedwithfurtherexperimentingconsideringthattheReynoldsnumberwasnotaslowasthetriangularornominalairfoils.

Figure30CTcomparisonatvariousReynoldsforE178airfoil.

In addition, after observing figure 31 and 32, it becomes clear how a bigger camberprovideslargerthrustproduction.E10andDF102profilewerechosenassamplesbecausetheir thickness isalmost identical (10%and11%respectively).Comparing the twoat thesameReynoldsnumberclearlyshowstheeffectoftheirdifferentcamber:DF102’scamberis2.4%positionedat43.5%ofthechordwhileE10’scamberisaslowas0.3%positionedat31.9% of the chord. At a Reynolds number of 3200 and an angle of attack of 12°, themaximumdifference in CT production due to camber is 40%: E10’s CT is 0.003899whileDF102’sis0.005458.WhenincreasingReynoldsnumberto4000,CTenhancementjumpsto63%:inthiscasetheCTforE10is0.003431whileforDF102thisvalueraisesto0.005614,while angle of attack remains 12°. These experiments draw the conclusion that camberpositiveeffectsweakenswithlowerReynoldsnumbers.

0 2 4 6 8 10 12 14 16 18AOA

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

CT

E178 (8%)

Re = 3200Re = 4000Re = 4500

32

Figure31EffectofCamberonCTatRe3200.

Figure32EffectofCamberonCTforRe4000

33

3.1.NewrotordesignThemainconcernofthisprojectiswhetherthehelicopterwouldbeabletoflyornotinaMartianenvironment.Atthecurrentstate,therotoristooshorttogenerateenoughthrustto lift up the aircraft in a low-pressure environment; however, with all the informationunveiledaboveitispossibletocomeupwiththenecessaryrotorradius.Tohelpvisualizethestepstakentoreachtheoptimalvaluethefollowingflow-chartisshownandexplained.

Tocalculatethisnewparameter,equation(3)wasusedwithinitialvaluesof:

• T=300g=3N• Ω=2500rpm• ρ=1kPa• CT=0.009347(fromtable7)

ExceptforCTalloftheaboveparameterswillremainconstantduringeveryiteration.Usingequation (4), the Reynolds number linked to this new radius is calculated. Knowing theReynoldsnumber,theactualCTisfoundinthetablesandsubstitutedtothepreviousone.Atthispoint thewholecalculation is carriedoutagainandagainuntil theReynoldsnumberremains basically constant. Eventually this leads to a radius length of 0.462m correlatedwithRe=2400aCTof0.01393,Mach=0.35andanactualthrustgeneratedof3.3N.Whentranslated onMars’ conditions (previously cited gas constants and temperature of 5 °C),thesevalueschange toRe=3000,CT=0.01497,Mach=0.47andgenerated thrust3.4N.Thereforeconfirmingthat,ashypothesizedinitially,aradiusof0.5misenoughtoallowthehelicoptertohover intheharshMarsenvironment. It is importanttopointout thatbladeextensioncomeswithaweightincreasethatshouldpartiallyberemovedusingthecounterweights.

CalculateRadiuseq.3

CalculateReynoldseq.4

CheckdatafornewCT

IsReconstant?

Calculatefinalthrust

eq.1

Yes

No

34

4.ConclusionsThis research has experimentally confirmed several aerodynamic properties of adevelopmentMarshelicopteranddeterminednewairfoilscharacteristicsthatareworthyofa further investigation.For starters, thrust isdirectlyproportional to the fourthpowerofradius. As an example, at Re = 2400 and an angle of attack of 18° even if the triangularairfoil(0.17mradius)hadaCTof0.01393andthenominalairfoil(0.234mradius)hadonly0.00442 the actual thrust produced was respectively 13.4g and 25.1g. Anotherdemonstrated property is how thinner airfoils show to have better performance whenReynolds number diminishes. This is indicated by higher CT values of the triangular andE178airfoils(10%and8%thickness)withrespecttoalloftheotherprofiles(whichrangefrom11%to15%thickness). Samebehaviorisobservedforincreasingcambershownbythe better performance of the DF102 airfoil (thickness 11%, camber 2.4%) over the E10(thickness10%,camber0.3%).Also,duetothepossibleformationonthetriangularairfoilof a reattached separation bubble, which rounds off the profile sharp edges, betterperformance is achieved by this particular profile with respect to the other airfoils.Reynoldsnumberreduction isshowntobedetrimental forallairfoilsexcept fortheE178though this should be confirmed through further experiments in future studies. Anotherpoint that needs to be checked is how chord length affects thrust production. It can bededuced from thisdata that longer chordsmeanhigher thrustalthough theextentof thisphenomenonisnotyetclear.

4.1.PossibleFutureWorkSeveral possibilities for future investigations arise as a result of this research project.AdditionaltrialsandmeasurementsshouldbeperformedtoperfectlygraspthebehaviorofairfoilsatsuchlowReynoldsnumber.Someadditionalexperimentsideasthataroseduringthisworkare:

• Increaserotorradiustoallowthehelicoptertohoverinsideofthevacuumchamber.Astrengtheningoftheswashplatewillprobablybenecessary,ascentrifugal forcewillbemuchhigher.

• Effect of airfoil chord length on thrust and needed power, and understanding atwhatpointthislengthactuallybecomesdetrimental.

• A specific ultra low Reynolds number airfoil, shown in appendix, should bemanufactured and tested to check if that design is compatiblewith the helicopterneeds.ItcouldbethatCTishighbutmaterialtensileorshearstrengthsaretoolowduetotheverysmallthickness.

• Increase collector range to reach stall point and observe how Reynolds numberaffects it. Throughout this new experiment,maximum CTwill be reached and theairfoil’sfullpotentialwillbeexploited.

• Groundeffectandrotor-on-rotorinfluenceextent.• Perform new experiments with the E178 airfoil to further verify the previous

results.• Performtorquemeasurements.

35

5.Appendix

5.1.HoveringHelicopterThrust

Figure33Helicopterinhoveringflight[]

Taking as reference figure 33 we can distinguish: section 0–0 which is the plane farupstreamoftherotor(whereairvelocityisnull);sections1–1,and2–2thataretheplanesjustaboveandbelowtherotordisk,andthefarwakesection∞.Betweensection1-1and2-2- the flux moves at the induced velocity vi and in the far wake the flux velocity is w.AccordingtotheReynoldsTransportTheorem,foracontrolvolumesurroundingtherotoranditswake(reddashedlinesinfigure33)anddefining𝑑𝑆 = 𝑛 ⋅ 𝑑𝑆,wheretheunitnormalarea vector𝑛is oriented outward, for any extensive parameter K,whereK = k ⋅ mass (m),thefollowingequationisvalid:

!"!" !"!#$%

= !!"

𝜌 𝑘𝑑𝑉 + 𝜌𝑘𝑉 ∙ 𝑑𝑆 (A1)

Where,𝑉=localflowvelocity.m=massofthefluid.𝜌=densityofthefluid.

36

Ifweconsiderasteadyflowthepreviousequationsimplifiesto:

!"!" !"!#$%

= 𝜌𝑘𝑉 ∙ 𝑑𝑆(A2)

Substitutingtok,inequationA2,thevalues1,𝑉,and!

!𝑉!weobtainrespectivelythe

conservationofmass,momentumandenergy.Conservationofmass(k=1) !"

!" !"!#$%= 𝜌𝑉 ∙ 𝑑𝑆(A3)

Fromthisequationwecanderivethemassflowrate,whichissetat0sinceweconsideredasteady-flow.0 = −𝜌𝑣!𝑑𝑆

!!!! + 𝜌𝑤𝑑𝑆!

! (A4)Consequently,

𝜚𝑣!𝐴 = 𝜚𝑤𝐴! (A5)

Conservationofmomentum(k=𝑉)

!(!!)!"

!"!#$%= 𝜌𝑉𝑉 ∙ 𝑑𝑆(A6)

TheleftendsideofequationA6isthesummationofalloftheforcesappliedonthecontrolvolume, in this case thrust (T)while the righthand side introduces themass flow rate𝑚anddevelopsasfollows:

𝑇 = 𝑤 𝜌𝑤𝑑𝑆!! = 𝑤𝑚(A7)

Conservationofenergy(k=𝟏

𝟐𝑽𝟐)

!"

!" !"!#$%= 𝜌 !

!𝑉!𝑉 ∙ 𝑑𝑆(A8)

!"!"

isthepowerabsorbedbytherotoranditisequaltoTvi,thus,

𝑇𝑣! = 𝜌 !!𝑉!𝑉 ∙ 𝑑𝑆 = !

!𝑤!𝑚(A9)

Wheretherightendsideisnoneotherthantheworkdoneontherotor.SubstitutingequationA7intoA10wederive:𝑚𝑤𝑣! =

!!𝑤!𝑚 → 𝑣! =

!!𝑤(A10)

37

FinallycombiningequationA1andA7weget: 𝑇 = 𝑚𝑤 = 𝑚2𝑣! = 𝜌𝐴𝑣!2𝑣! = 2𝜌𝐴𝑣!! (A11)Theinducedvelocityisthennormalizedusingrotor’sradiusandangularvelocity(𝑅Ω)alsointroducingCTasTnormalizedbydynamicpressureobtaining:𝐶! = !

!"!!!! (A12)

Notethatsinceintheexperimenttherearetworotordisks,thrustmeasuredwashalvedinordertoworkwithsinglerotorreadings.

5.2.ExperimentalAirfoilforultralowReynoldsnumber

Figure34ExperimentalUltralowReynoldsnumberairfoil

Thisairfoil isespeciallymade forultra lowReynoldsnumberbutcouldn’tbeproduced intime. Therefore it would be interesting to see how it behaves and if its mechanicalcharacteristics are good enough for the helicopter’s needs. Coordinates for thisexperimental airfoil are shown below. If the produced CT is much higher then the otherairfoil,thisprofilecoulddrasticallyreducerotorlength,thusweightofthesingleblade.Dueto the lower bladeweight a higher rotor speed can be reachedwithout reaching criticalcentrifugalforces.

XY1.000000-0.0020000.995000-0.0010750.990000-0.0001500.9850000.0007750.9800000.0017000.9750000.0026250.9700000.0035500.9650000.0044750.9600000.0054000.9550000.0063250.9500000.0072500.9450000.0081750.9400000.0091000.9350000.0100250.9300000.0109500.9250000.0118750.9200000.0128000.9150000.0137250.9100000.0146500.9050000.0155750.9000000.0165000.8950000.0174250.8900000.0183500.8850000.0192750.8800000.0202000.8750000.0211250.8700000.0220500.8650000.0229750.8600000.0239000.8550000.0248250.8500000.0257500.8450000.0266750.8400000.0276000.8350000.0285250.8300000.0294500.8250000.0303750.8200000.0313000.8150000.0322250.8100000.033150

XY0.8050000.0340750.8000000.0350000.7950000.0349960.7900000.0349920.7850000.0349880.7800000.0349830.7750000.0349790.7700000.0349750.7650000.0349710.7600000.0349670.7550000.0349630.7500000.0349580.7450000.0349540.7400000.0349500.7350000.0349460.7300000.0349420.7250000.0349380.7200000.0349330.7150000.0349290.7100000.0349250.7050000.0349210.7000000.0349170.6950000.0349130.6900000.0349080.6850000.0349040.6800000.0349000.6750000.0348960.6700000.0348920.6650000.0348880.6600000.0348830.6550000.0348790.6500000.0348750.6450000.0348710.6400000.0348670.6350000.0348630.6300000.0348580.6250000.0348540.6200000.0348500.6150000.034846

XY0.6100000.0348420.6050000.0348380.6000000.0348330.5950000.0348290.5900000.0348250.5850000.0348210.5800000.0348170.5750000.0348130.5700000.0348080.5650000.0348040.5600000.0348000.5550000.0347960.5500000.0347920.5450000.0347880.5400000.0347830.5350000.0347790.5300000.0347750.5250000.0347710.5200000.0347670.5150000.0347630.5100000.0347580.5050000.0347540.5000000.0347500.4950000.0347460.4900000.0347420.4850000.0347380.4800000.0347330.4750000.0347290.4700000.0347250.4650000.0347210.4600000.0347170.4550000.0347130.4500000.0347080.4450000.0347040.4400000.0347000.4350000.0346960.4300000.0346920.4250000.0346880.4200000.034683

38

XY0.4150000.0346790.4100000.0346750.4050000.0346710.4000000.0346670.3950000.0346630.3900000.0346580.3850000.0346540.3800000.0346500.3750000.0346460.3700000.0346420.3650000.0346380.3600000.0346330.3550000.0346290.3500000.0346250.3450000.0346210.3400000.0346170.3350000.0346130.3300000.0346080.3250000.0346040.3200000.0346000.3150000.0345960.3100000.0345920.3050000.0345880.3000000.0345830.2950000.0345790.2900000.0345750.2850000.0345710.2800000.0345670.2750000.0345630.2700000.0345580.2650000.0345540.2600000.0345500.2550000.0345460.2500000.0345420.2450000.0345380.2400000.0345330.2350000.0345290.2300000.0345250.2250000.0345210.2200000.0345170.2150000.0345130.2100000.0345080.2050000.0345040.2000000.0345000.1950000.0336380.1900000.0327750.1850000.0319130.1800000.0310500.1750000.0301880.1700000.0293250.1650000.0284630.1600000.0276000.1550000.0267380.1500000.0258750.1450000.0250130.1400000.0241500.1350000.0232880.1300000.0224250.1250000.0215620.1200000.0207000.1150000.0198380.1100000.0189750.1050000.0181130.1000000.0172500.0950000.0163870.0900000.0155250.0850000.0146630.0800000.0138000.0750000.0129380.0700000.0120750.0650000.0112130.0600000.0103500.0550000.0094870.0500000.0086250.0450000.0077620.0400000.0069000.0350000.0060370.0300000.0051750.0250000.0043120.0200000.0034500.0150000.0025870.0100000.0017250.0050000.0008630.0000000.0000000.0050000.0005000.0100000.0010000.0150000.0015000.0200000.0020000.0250000.0025000.0300000.0030000.0350000.003500

XY0.0400000.0040000.0450000.0045000.0500000.0050000.0550000.0055000.0600000.0060000.0650000.0065000.0700000.0070000.0750000.0075000.0800000.0080000.0850000.0085000.0900000.0090000.0950000.0095000.1000000.0100000.1050000.0105000.1100000.0110000.1150000.0115000.1200000.0120000.1250000.0125000.1300000.0130000.1350000.0135000.1400000.0140000.1450000.0145000.1500000.0150000.1550000.0155000.1600000.0160000.1650000.0165000.1700000.0170000.1750000.0175000.1800000.0180000.1850000.0185000.1900000.0190000.1950000.0195000.2000000.0200000.2050000.0200450.2100000.0200910.2150000.0201360.2200000.0201820.2250000.0202270.2300000.0202730.2350000.0203180.2400000.0203640.2450000.0204090.2500000.0204550.2550000.0205000.2600000.0205450.2650000.0205910.2700000.0206360.2750000.0206820.2800000.0207270.2850000.0207730.2900000.0208180.2950000.0208640.3000000.0209090.3050000.0209550.3100000.0210000.3150000.0210450.3200000.0210910.3250000.0211360.3300000.0211820.3350000.0212270.3400000.0212730.3450000.0213180.3500000.0213640.3550000.0214090.3600000.0214550.3650000.0215000.3700000.0215450.3750000.0215910.3800000.0216360.3850000.0216820.3900000.0217270.3950000.0217730.4000000.0218180.4050000.0218640.4100000.0219090.4150000.0219550.4200000.0220000.4250000.0220450.4300000.0220910.4350000.0221360.4400000.0221820.4450000.0222270.4500000.0222730.4550000.0223180.4600000.0223640.4650000.0224090.4700000.0224550.4750000.0225000.4800000.0225450.4850000.0225910.4900000.022636

XY0.4950000.0226820.5000000.0227270.5050000.0227730.5100000.0228180.5150000.0228640.5200000.0229090.5250000.0229550.5300000.0230000.5350000.0230450.5400000.0230910.5450000.0231360.5500000.0231820.5550000.0232270.5600000.0232730.5650000.0233180.5700000.0233640.5750000.0234090.5800000.0234550.5850000.0235000.5900000.0235450.5950000.0235910.6000000.0236360.6050000.0236820.6100000.0237270.6150000.0237730.6200000.0238180.6250000.0238640.6300000.0239090.6350000.0239550.6400000.0240000.6450000.0240450.6500000.0240910.6550000.0241360.6600000.0241820.6650000.0242270.6700000.0242730.6750000.0243180.6800000.0243640.6850000.0244090.6900000.0244550.6950000.0245000.7000000.0245450.7050000.0245910.7100000.0246360.7150000.0246820.7200000.0247270.7250000.0247730.7300000.0248180.7350000.0248640.7400000.0249090.7450000.0249550.7500000.0250000.7550000.0255000.7600000.0260000.7650000.0265000.7700000.0270000.7750000.0275000.7800000.0280000.7850000.0285000.7900000.0290000.7950000.0295000.8000000.0300000.8050000.0292000.8100000.0284000.8150000.0276000.8200000.0268000.8250000.0260000.8300000.0252000.8350000.0244000.8400000.0236000.8450000.0228000.8500000.0220000.8550000.0212000.8600000.0204000.8650000.0196000.8700000.0188000.8750000.0180000.8800000.0172000.8850000.0164000.8900000.0156000.8950000.0148000.9000000.0140000.9050000.0132000.9100000.0124000.9150000.0116000.9200000.0108000.9250000.0100000.9300000.0092000.9350000.0084000.9400000.0076000.9450000.006800

39

XY0.9500000.0060000.9550000.0052000.9600000.0044000.9650000.003600

XY0.9700000.0028000.9750000.0020000.9850000.0004000.9800000.001200

XY0.990000-0.0004000.995000-0.001200

5.3.WeightreductionWhenMartiangravityisconsideredthehelicopterwillsurelyweightlessthenonEarth.Iffeasibility is concerned this is an important parameter that can determine whether amissionisdoableornot.TheweightreductionwascalculatedconsideringthedifferenceinweightbetweenthehelicopteronEarthandonMarsaddedtothestructureweightasso:

𝑊!"#$!! = 𝑚!!"# ∙ 𝑔!"#$! −𝑚!!"# ∙ 𝑔!"#$ +𝑊!"#$%" = 𝑚!!"# 𝑔!"#$! − 𝑔!"#$ + 𝑊!"#$%"ItwasthentransformedintotherequiredEarth’smassanddividedintwocounterweightsasexplainedinsection2.2.

40

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politecnico di torino · possibility to reach sonic speeds on the blade, generating normal shocks...

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