Krzysztof Sibilski - riunet.upv.es

Institute of Aeronautics and applied Mechanics

Final Project

Mechanics and Machine Design Modelling and Computer Simulations in Mechanics

Modelling and Simulation of resonance flapping wing resonance propulsion

Aurelio García Lloris

Krzysztof Sibilski

Warsaw, 2017

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ABSTRACT

This Project describes an analysis and design of aMAVMicro AirVehicle.Theaimoftheprojectistofindsomestructurethatcanallowtheresonanceoftheflapping-wingmovement.

Thestudy is focus inthestructureoftheconceptswheredifferentspringmaterialsarestudiedtoknowwhichonefitsbetterwiththeflexiblestructure desired. The concepts studied are based in the insect thorax-wingssystemwhichisatunedresonantsystem.

Thefirststepistheselectionoftwoprototypes.Secondly,weselectoneofthemofwhichisdoneastudyofthemotionequationwithdifferentsimulationstostandoutthemaindifferencesbetweenthem.

Theprojectisdividedintofiveparts.Thefirstonepresentsbrieflytheobjective of the project. Then, insect flight is described. The next partdescribes flapping wing MAVs where our two concepts are presented.Sectionbelowcontainssimulationsandcalculations.Lastpart,comprisesconclusionsbasedonexaminedinstances.

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Contents

1 Introduction...................................................................................................-4-1.1 MOTIVATION....................................................................................................-4-1.2 MICROAIRVEHICLE..........................................................................................-5-1.3 BACKGROUND..................................................................................................-6-1.4 PROJECTSTRUCTURE........................................................................................-7-

2 Insects...........................................................................................................-8-2.1 INSECTSANATOMY...........................................................................................-8-2.2 WINGMOVEMENT..........................................................................................-10-2.3 AERODYNAMICS.............................................................................................-12-2.4 KEYPOINTS.....................................................................................................-14-

3 FlappingWingMAVs....................................................................................-16-3.1 BACKGROUND................................................................................................-16-3.2 WINGS............................................................................................................-18-3.3 ACTUATORS....................................................................................................-19-3.4 DESIGN...........................................................................................................-21-3.5 CONCEPTS.......................................................................................................-22-

4 Analysis.......................................................................................................-23-4.1 MATHEMATICALMODEL.................................................................................-23-4.2 NUMERICALDATA...........................................................................................-25-4.3 TRANSMISSIONRATIO....................................................................................-29-4.4 RESPONSEANALYSIS.......................................................................................-32-4.5 FREQUENCYRESPONSEANALYSIS...................................................................-43-4.6 TRANSFERFUNCTION......................................................................................-44-

5 Conclusion...................................................................................................-48-

6 Bibliography................................................................................................-49-

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1 Introduction

1.1 MOTIVATION

Throughout history of evolution, the nature has optimized itsresources in order to maximize the performance of existing biologicalstructuresonourplanet.This facthasmotivated thesearchof conceptsbasedinanimals.Forexample,LeonardoDaVincidesignedaflyingmachinefromthestudyonthebirdanatomy,theOrnithopter.

Last years, there have been great advances in different areas ofscience related to the design of artificial devices from the study andobservation of organisms, systems and processes existing in nature. Inmanycases,acompletebiological imitationofaninsect,bird,orfishhasbeen attempted, resulting in disappointing results. It is important todistinguishbetweenwhatismeantbybiologicalimitation,biomimetic,andinspirationinbiology.Thebiomimeticorcopyofanorganismisverydifficult

Figure1:LeonardoDaVinci’sOrnithopter.

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to achieve with current technology, on the other hand, inspiration inbiology,isagoodstartingpointtotakeafirststeptowardsthedesignandmanufactureofrobotsthathaveaperformanceequalorgreaterthanthebiologicalentitiesexistinginnature.

Nowadays, scientific community is focus in the development ofMAVs,MicroAirVehicle,inspiredintheflightoftheinsectsandlittlebirds,likethehummingbird,thatareabletoperformcomplexmovements.

1.2 MICROAIRVEHICLE

The United States Defence Advanced Research Projects Agency,DARPA,hasdefinedtheMAVasanyflyingvehiclewhichislimitedto150mmorsmallerinanylineardimension.

WecandifferbetweenthreetypesofMAVs,wefindthefixed-wingsMAVs,therotary-wingaircraftandtheflappingwingflyers.

The fixed-wing is themost efficient kind ofMAV; it is able to runthroughlongerdistancesbutithaslessmobilitythantherest.Therotary-wingMAV,standsoutbecauseitseaseofmovements.Theflappingwingflyers, are inspiredby thenature. Theadvantagesof this typeof insect-inspiredvehiclesaregreaterversatility toperformcomplexmanoeuvres,greaterflightstabilityandlowerenergyconsumption.Itmeanslongerflighttimes.Amongthemanoeuvres, it isabletoperformlow-speedflights,aswellastake-offsandlandingsovershortdistances.Also,itcanproducethesameliftwithsmallerwingbytheincreasingofflappingfrequency.

Inthisproject,wearedesigningaflappingwingMAV.WetrytofulfiltherequirementsfromtheAtalantaProject.OurMAVshouldbeabletoflyautonomously,hoverandflyslowly,communicatewithothersandabasestation, manage power, provide payload capacity and be adaptable formanysituations.

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1.3 BACKGROUND

Theimplementationofflappingflightatthesmallerscales ismuchmore difficult than fixed-wing and rotary-wing implementations.Nevertheless,itisinterestingtostriveforinsectflapping-wingMAVsduetosomereasons.

Thefirstreason,itistheflyingspeed.Duetoflappingwingsystem,westayairborneinlowvelocitiesproducingliftwiththewingsmovement.Secondly,insectsandbirdscombinecontrolandpropulsionintheirflappingwing systems while rotary-wings and fixed-wingsMAVs need additionalactuatorsforcontrol,additionalrotorsinrotary-wingsandcontrolsurfacesinfixed-wings.

Becauseoftheirabilitytohover,wecanlookbirds,batsandinsectstogetsomeideas.ButthesizerangeofflappingwingMAVsmakesuspayclose attention in insects. Firstly, we notice the system that drives thewings. In insects the whole thorax structure functions as a spring andcombinedwith thewings essentially forms a tunedmass-spring-dampersystem. The wings in insects are predominantly passive structures. Theabsence of active elements in thewingsmake insectsmore suitable forinspirationofflapping-wingMAVs.

Along history, attempts have been made to imitate the insectsflapping-wing flight. In spite of the advances in this kind of MAVs lastdecade, the biological entities stay ahead of manmade prototypes. Themostcommonproblemsthatwecanfindaresizeproblemsandautonomyoftheflapping-wingMAV.

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1.4 PROJECTSTRUCTURE

TheProjectisdividedinfivemainparts.

• TheIntroductiondescribesbrieflygeneralpointsthatwearelookingintherestofthework.

• Secondly,wegetinformationabouttheinsects,itsanatomyanditsflight.

• ThethirdpartisastudyoftheMAVs,inparticulartheflappingwingMAVs.Basedintheinsects,wechoosetheprototypesthatwewillstudy.

• Consecutivelyweput theprototypes to some test to carryout ananalysis.

• Finally,wegetsomeconclusionsaboutthestudyperformedinthefourthpart.

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2 Insects

2.1 INSECTSANATOMY

When you are designing and constructing a MAV, an importantaspect is to understand how insects and small birds flap their wings toproduceenoughliftandthrusttoflyandtodrivearoundtheair.

There are a lot of experimental works about the study of insectaerodynamics,whichprovideagooddescriptionofthenon-conventionalaerodynamicmechanismsusedbythesecreaturestostayinflight.Ontheother hand, a precise quantitative description of such phenomena isdifficult to achieve due to the errors in the techniques used to performmeasurementsofthemagnitudesofinterest.

Whenlookingat insectsas inspiration,twoaspectsareinteresting.First,theinsectthorax,itproducesthepoweranddrivesthewings.Second,thewingswhichareanintegralpartoftheflightmechanism.

Figure2:Bodyofinsects.

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The insect consists of three body parts, the head, thorax andabdomen. The head carries the brain and perceive vision, olfactory andhearing. The abdomen contains the reproductive organs, breathingapparatusandmostofthedigestivesystem.Aswementionedbefore,thethorax is one of themost interesting aspects in the flight of the insectsbecause it produces the power and drives the wings. It has the wingactuationmechanism.

From an engineering viewpoint the wing-thorax structure is acompliant amplification mechanism. Two different thorax designs existwhichdifferinthewaythewingsaredriven:directandindirect.

Inthedirect-drivemechanism,themusclesdrivethewingsdirectly.Thedifferentwings,itispossiblewiththismechanismtofindtwoorfourwings,canhavedifferentflappingfrequenciesandamplitudes,allowingaverylargerangeofforceproductionandlargeaerialagility.

The indirect-drive mechanism produces higher wing-beatfrequenciesthandirect-drivemechanism.Thedeformationofthethoraxisgovernedbytwomusclegroups.Theelasticpropertiesofthethoraxandmusclescombinedwiththeinertialpropertiesofthewingsandthethoraxboxformatunedmass-spring-dampersystem.

Forapplicationinflapping-wingMAVs,theindirect-drivemechanismismostinterestingbecausetheresonantproperties.

Thewingsare,as seen fromanengineeringviewpoint,a structurewhich consistsof loadbearingmembers, longitudinal veins, crossbeams,cross-veins, and spanning between the veins and cross-veins is a sheet,membrane.

Beingpassivestructures,thewingsaredrivenatthebasewheretheyconnect to the thorax through the wing joint. The wings constitute themajorpartofeffectiveinertiaofthemovingthorax-wingsystem.

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Theresonantpropertieshelptoreducetheinertialcostofthewingmovement.Aswementioned,theinsect-thoraxwingsystemcanthenbeseenasatunedmass-spring-dampersystem.Themassisdominatedbytheinertia of thewings. The spring function is a combination of the elasticthorax shell and themuscleproperties. Theaerodynamic loadingon thewingscanbeseenasthedampingforce.

2.2 WINGMOVEMENT

The insectswingmovementcanbeseparated in twoparts:a rigidbodymotionandanelasticdeformation.Therigidbodymotionconsistsofthreerotationsfacilitatedbythewing-rootjoint.Theelasticdeformationconsistsmostlyoftorsionaldeformationandbending.Themagnitudeoftheelasticdeformationissmallcomparedtotherigidbodymovement.

Thethreerotationsthatcompoundtherigidbodymotionsarethemain sweeping angle, the wing pitching motion and the out of planemotion.

Themainflappingorsweepingangle,whichisthemostcharacteristicmotionofthewing,makesthewingtraceaplanewhichisapproximatelyhorizontalformosthoveringinsects.Themainflappingistheresultoftheresonantmotionandassuchisdescribedbyaharmonicmotion.

Thesecondmotion,thewingrotationorwingpitching,isdefinedasthe rotationof thewingabout its radialaxis.Thismotion, togetherwithdirectioninducedflow,definestheangleofattackofthewing.Thewingpitching is generally a motion close a pure sinus. The frequency of thepitchingmotion is always the same as themain flappingmotionwhit aphaseshiftbetweenboth.Theoriginofwingpitchinghasbeenthesubjectofargumentamongbiologists.Analysisoftheinsectthoraxsuggestthatthecontrolmuscles are too small to generate the required power for wingpitching.However,pitchingcouldbetheresultofasmartwing-roothinge

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configuration.Thesecondoptionispassivepitchingcontrolledbyinertialand aerodynamics loads. The general consensus among biologists andengineers is that during steady-state flying or hovering wing pitching ispassive. However, when the insects perform extreme manoeuvresindicationsarethatthereconfigurationofthewingjointissuchthatwingpitchingisatleastpartiallyactivelycontrolled.

Thelastmotionistheoutofplanemotionorheavingmotionwithdescribesthedeviationsfromthestrokeplane.

Atypicalwingstrokecyclevanberoughlydividedinfourphases.Thefirst are the up and downstroke. In these phases the wing moves withrespect to the body with approximately constant wing rotation. Theremainingtwophasesarethetwostrokereversals,calledsupinationafterthedownstrokeandpronationaftertheupstroke.

Forthemajorityofapplications,therigidbodydescriptionsuffices.However,theelasticdeformationcarriedoutbybendingandtorsionplayaroleintheinsectwingkinematic.

Theapplicationofloadsonthewings,bothinertialandaerodynamic,resultsinwingdeformationduringtheflappingmotion.

While bending is present in insect-wings, the aspect of torsion ismore important. Torsionandmore specifically the timingof torsionanddeformation during thewing stroke can directly influence the timing oftorsiondeformationduringthewingstrokecandirectlyinfluencethelocalwingpitchingangleand,consequently,thelocalangleofattack.

Whenwetalkabouttheelasticdeformationofwingsitisnotedthatthesizeisanimportantfactinasmuchasinthesmallinsectstheeffectsofbendingandtorsionaremorereducedthaninlargerinsects.

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2.3 AERODYNAMICS

FlappingflightbyinsectsischaracterizedbyunsteadyincompressibleflowatintermediateReynoldsnumber.ForlargeinsectstheReynoldsliesbetween 5000 and 10000. The smallest flying insects have Reynoldsnumberswhich cangoas lowas10. The range is largedue to thewidevariationinthebodymassoftheflyinginsects.

Theproductionofliftandthrustfollowsfromthecomplexinteractionbetween the compliant wings, wing kinematics and the aerodynamics.Aerodynamictheoryiswelldevelopedforlargerscalesandcanbeappliedatinsectscalewiththeincorporationofeffectsthatarepresentatthesesmaller scales and, consequently, lower velocities. The flapping winginducesunsteadyflowwithacyclicnature.

The production of lift and drag in insects is based on threemechanisms, the leading edge vortex, the rotational circulation and thewakecapture.

Figure3:InsectAerodynamic.

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Theleadingisthemostimportantaerodynamiceffectininsectflight.DuetothisLEVtheflappinginsectwingproducesfarmoreliftthancanbeexplainedusingnormalairfoiltheory.Experimentsshowthatwhenawingmovesthroughtheair,alargevortexstartsattheleadingedgeofthewingandformsonthetopofthewing.Therotationinthisvortexisveryhigh.Theassociatedhighvelocitycreatesapatchof lowpressureonthewingsurfacefacilitatingliftproduction.Thisvortexgrowsinsizeduringwingflapandaftersomechordlengthsreleasesfromthewingandremovingthelowpressurefieldandthuslift.

Therotationalcirculationeffectisonlypresentduringphaseswheretherotationalorpitchingvelocitiesarelarge,thestrokereversal.Byvaryingthe wing rotation timing and rotation speed, causing advancements ordelays in thewingkinematicpattern, thewing forcescanbevaried.Theforcewillincreasewhentherotationisadvancedanddecreasedwhentherotationisdelayed.Therotationalspeedofthewingcontributedtoextralift.

Thewakecaptureistheeffectwhereflappingwingsinteractwiththevortexthatwasshedattheendofthepreviousstroke.Thiscreateslargeloadpeaksatthebeginningofthenewstroke.Thisincreaseinliftiscausedbytheincreaseinfluidflowbythecaptureofthepreviousvortex.Thewakecaptureeffect ismostpronouncedduringstrokereversalandnotduringtheperiodofrelativelylowpitchingvelocitiesduringmidstroke.

Two other effects are present in insects but are not directlyresponsibleforliftproduction.Thefirstistheeffectofaddedmassorvirtualmass.Thevirtualmasscanbeseenasaregionofinfluencewheretheairaround the wing moves with the wing and contributes to the effectiveinertiaofthewing.Theothereffectisknownasclapandflight.Theeffectconsistsofexpandingthewingsweepinganglesofarthewingstouchattheendoftheupstrokeabovethebodyoftheinsect.WhenthedownstrokestartsthewingsarerolledopeninordertojumpstarttheLEV.Sincethe

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formationoftheLEVis independentofwingsize, jumpstartingmightbebeneficial for very small insects which have a very small wing stroke inabsolutesense.

2.4 KEYPOINTS

As we mentioned, direct copy of insects is not possible withnowadays science. However, to produce a great flapping-wingMAVweneedtolookindetailtheinsectsbecausetheyaresuchsuccessfulflyers.

ThemostimportantpointsarethebodyorthethoraxoftheflappingwingMAV,thewingtransmissionsystemandthewings.

Thegoalaboutthethoraxistheapplicationoftheresonantprincipleswhicharepresentintheinsects.Therewith,wereducetheenergyimpactofwingmovementandweobtainalargewingsweepingstroke.Besides,weneedtotakecarebecausesamedrawbacksareintroduced.

Thewingsweepingfrequencybecomesboundedbyhighsensitivetochangesindrivefrequency.Thismechanicalequivalentcouldbeanelasticelementtostorethekineticenergy.Besides that,weneedamechanicalequivalent for the thorax structure in insects that participates in theresonant thorax-wing system, it should strive for a very high degree ofintegration.

Thethoraxisdrivingthewingssothethoraxdeformationhastobetransformedandamplifiedintoalargesweepingmotioninthewingbase.Ininsectsthispropertyisintegratedintothethorax-wing-rootcombination.Thethoraxandwingroottogetherformafullycompliantsystemcapableofgeneratingcomplexkinematics.

Wing pitching also finds its origin in the wing root. MAVs thoraxwouldbeamendabletoaccomplishthesamefeatures.

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Thewingistheplacewhereliftisproduced.Itwillposeasignificantchallengetodesignamechanicalequivalentiftheinsectwing.Asweseebefore, thewingpitchingmaybeofpassiveorigin. In the idealcase, thetiming and amplitude of wing pitching will have to be obtained by acombinationofahighlycompliantwingrootwithawingwhichisabletosupportabeneficialtorsionalwavetoaugmentliftproduction.Thesystemhastobedesignedinahighlyintegratedfashion.

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3 Flapping Wing MAVs

3.1 BACKGROUND

The design of flapping wing MAVs is still a very active area ofresearch. While the model plane world has known small rubber bandpoweredornithopterssincethe1870s,thefirstelectricpoweredflappingwingMAV,namedtheMicroBat,onlyflewin1998.Thedesignofflappingwing MAVs until now mainly progressed by means of trial-and-error.Automatic optimization is still very unreliable due to a lack of accuratetheoreticalmodels.Especiallythedesigndecisionsconcerningtheshape,tension,andmaterialsofthewingscannotbemadepurelyonthebasisofsimulationduetoalackinknowledgeontheaerodynamicsaroundflexibleairfoil.

Historically,flapping-wingMAVshavebeenbasedonbird-likeflightfocussingonforwardflight.Nowadays,thesedesignsareinspiredbysmallbirds and insects because hovering and slow moving is interesting forpossibleindoorusage.

Toanalysethepastdesignsisagoodoptiontostart.Wecanseewhatisbeingdoneinthefieldandwecanextractsomeideasandgoodpracticesimilartotheanalysisofinsects.

An aircraft design is highly dependent on the intended use of theplatform.Thedrivingforceinthedesigncanbetooptimizeformaximumenduranceononehand,oronthecontrarytooptimizeforminimumsize.Typically,thegoalalsoincludesotheraspectssuchasstabilityorpayloadcapability. Different combinations of goals can lead to very differentdesigns. Common to almost all of these goals is that they are harder toattainatsmallerscales.Therearecoarselytwoapproachestofinallyarriveatfullyfunctioningfly-sizedflappingwingMAVs:bottom-upandtop-down.

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Thebottom-upapproachfocusesonconstructingandtestingthetinypartsnecessaryfordirectlyconstructingafly-sizedrobot.Researchstudiesadoptingthisapproachoftentackleextremelydifficultsub-taskssuchastheconstructionoftheinsectthorax,orthegenerationofsufficientthrust.

The top-downapproach startswith relatively larger scalebut fullyfunctioning flapping wing MAVs. The idea behind the approach is thatstudyingsuchMAVscanleadtoinsightsfortheconstructionofafollowing,smallerorsmarterversion.Oneadvantageofthisapproachisthatitallowsinterplaybetweentheoryandpractice.Foraerodynamicsresearch,havinga flyingsystemensuresthat theresearch isdirectedtoaspects thatalsohave a practical relevance. For artificial intelligence research, having aphysicalandfullyfunctioningMAVisofgreatvalue:real-worldtestsforcetheexperimenterstotakeintoaccountallaspectsoftheroboticsystem.Inaddition,theyrevealphysicalpropertiesofthesystemthatcanbeexploitedbythealgorithms.

The bottom-up and top-down approach have complementaryadvantagesandrisks.Forexample,ariskofthetop-downapproachisthattheresearchwillfocustoomuchonincrementalmodificationsoftheMAV,ignoringpossibledisruptiveimprovements.Ontheotherhand,ariskofthebottom-up approach is that research may focus too much on detailedaspectsthatmightturnoutirrelevantforafullyflyingsystem.Thebottom-upapproachcanleadtofundamentalnewunderstandingandtechniques,whilethepractical ‘surprises’of thetop-downapproachgive insight intopressing problems of lacking scientific knowledge or technology. Webelievethatprogressinflappingwingresearchrequiresbothapproachestoexist.

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3.2 WINGS

Variousmethodsexisttomanufacturewingsforflapping-wingMAVs.Themost commonmethodof constructing is themethodof spanning amembrane by stiff rod type structures. Other methods for constructingwings usually rely on using a membrane and stiffening structure whichrepresentsaveinlikestructure.

Passive rotation characteristics have a strong dependence on thedetailedshapeandmassdistributionofawing.Naturalandartificialwingsdisplay great variation in size, platform, material composition, and veinstructure.Developmentofaconcisewingparameterizationisrequired.

Among the existing and functional MAVs, three types of wingconfigurations,monoplanes,biplanesandfoldingwingsaredistinguished.The monoplanes consist of a pair of wings that oscillate in phase. Theadvantagetheypresentisalowerfinalweightandasadisadvantageitismoreinstableandithasmoreamplitudeofrolling.

Biplanes consist of two pairs of wings thatmove in phase. Thesewingscanbeonebehindtheotheroroneovertheother.Theadvantageofthis configuration is a lower rolling range and greater stability. Thedisadvantageisanincreaseintheweightofthevehicle.

Figure4:TheMicroBat,exampleofmonoplane.

Figure5:TheDelfly,exampleofbiplane.

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Finally,thefoldingwingsarefoldedtosimulatetheflightofbirds.Thewingsextendwhenthemovement isdownwardand fold to thebodyastheyrise,thusreducingtheareawhichminimizesthenegativeliftingeffect.ThedisadvantageofthisconfigurationistheexcessiveweightofactuatorsmakingitdifficulttoimplementinMAV.

Thewingdesignisveryimportantfortheaerodynamicperformance,and hence the lift that can be generated by the flapping wingMAV. Itinvolvesa choiceof thematerials for the structuringelementsandwingmembrane,andthewaytocombinethesetoformthewing’sshapeandstructure.AtraditionalchoiceforthewingmaterialsconsistsofPET-foilandcarbonfibberreinforcedpolymer(CFRP)rods.Thesematerialshaveproventheirworth,arewidelyavailable,anddonotrequirespecificinfrastructureforconstruction.Thedownsideisthattheytypicallyrequiresomemanualwork, which can limit repeatability. Moreover, the design options withthesematerialsarerelativelylimited.MostdesignswithPET-foilandrodsarelimitedtogeometricshapeswithastiffleadingedgeandafewstiffenersaddedtothewing.Inordertoallowformoreintricate,andyetrepeatabledesigns,othermaterialsandfabricationmethodshavebeeninvestigated.

Thecurrentlyavailablewingtechnologiesaresuitableforthecurrentinsect sized flapping-wings MAVs, however, improvements can beexpected.

3.3 ACTUATORS

Thefunctionoftheactuatorsistoactivatesomeprocessinordertoachieve thecontrolof theprocess.Therearevarious typesofactuators,mechanical,hydraulic….ThemostcommoninthedesignofMAVareservosmotorsandmagneticactuatorsduetolowweight.Theyareabletodeliverasmoothandproportionalcontroltothesignal.

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Inordertochooseoneactuatortechnologyoranother,theactuatorshavetobechecked if they liveupto theboundaryconditionswhichareimposedbyaflapping-wingMAV.Thedesignoftheflapping-wingMAVishighlyinfluencedbythechoiceoftheactuator.

Magneticactuatorsarepulsewidthmodulation(PWM)drivenwithadutycycleproportionaltothetransmittercontrolstick,whichresultsinaproportionalcurrent.Thiscurrentproducesamomentinthemagnetwhichagain translates toaproportional force thatgoes to thecontrol surface.However,theforceissmallandwiththeairpressureonthecontrolsurfacesbeingproportionaltotheairvelocitysquared,thecontrolthrowgetsmuchlower at higher airspeed. This can cause inability to pull up from fastdescendingflight.

Analternativeconsistsofconventionalservosthatuseasmallelectricmotor,gearingandapotentiometerormagnetic-hall sensor forpositionfeedback. This type of actuator has more force and a higher accuracycomparedtothemagneticactuator.

Due to the trend of miniaturization and the availability ofmanufacturing techniques, piezoelectric actuators have been used formicro devices. Application in flapping-wing MAVs is complex due tomediumtohighvoltagesneededforactuation,andthecorrespondingneedfor high voltage electronics. Piezoelectric actuators allow a high velocitymovementofthewingswithasmallamplitude.

Not all mentioned actuator technologies are equally feasible.Implementationeffortfortheapplicationofaspecificactuatortechnologydependsonthesizeoftheintendedflapping-wingMAV.

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3.4 DESIGN

Aswehaveseen,thesourceofinspirationistheinsects.However,adirect translationfromthe insect thoraxtoengineeringequivalent isnotpossibleordesired,becausetechnicalsolutionsfollowdifferentguidelinesandusedifferentmaterialsratherthanbiologicalstructures.

The requirements for the current developments of the wingactuationmechanismfor the flapping-wingMAVaregearedtowards theimplementation of resonant principles while maintaining a very simpledesign.

Theapplicationof resonance in flapping-wingMAV isbased in thestorageofpotentialenergyinadeformablestructure.Thisisfocusinfourproperties:Thematerialused,thevolumeofthematerial,thedeformationmodeandamountofdeformation.

The deformation mode used to store potential energy has largeinfluence of the thorax structure. The deformationmodeswhich can beusedarecompression,bending,torsionandshearofamaterialwhichcanstoreenergyefficiently.Combinationsofdeformationmodesarepossible.

In order to put boundaries on the designs, four main aspectsinfluencethedesignandprovidedesigndirections.

• Simplicity:Basedonthefactthatsimplerdesignsseemtobemore successful. Low complexity designs are usually of lowmassandthereforeprovidebetterchancesoflift-off.

• Resonance: The design has to fully exploit resonance formaximumreductionofthepowerexpenditureinacceleratinganddeceleratingthewings.

• Passivewingpitching:Thisisatrade-offbetweenaerodynamicefficiencyandalowmassstructure.Althoughactivelypitched

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wings are able to produce lift, the increase in mechanismcomplexityleadstoanoverallincreaseinmass.

3.5 CONCEPTS

Followingthemainconceptsthatwehaveseenalongtheproject,wehave the design of two concepts. These two concepts, are two wingeddesign.

Figure6:Concept1

Figure7:Concept2

Faero Faero

Factuator

Factuator

Inertia

Faero Faero

Factuator

Factuator

Inertia

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4 Analysis

4.1 MATHEMATICALMODEL

Theanalysiswillbedoneonlyfortheconcept1duetothecomplexityofthestudyforthesecondconcept.

Inthismodelstudy,weassumethatthebodyweightishigherthanthewingweight.Thesystemisreviewedasbeingsingledegreeoffreedom

representedbythewingsweepingangleφ.

Theone-dimensionalequationofmotionis:

𝐼𝜙 + 𝑘𝜙 = 𝑀'()(𝜙) − 𝑀-(𝜙)

Where:

• Inertia 𝐼 represents the equivalent inertia of the systemincludingbothwings.

• Krepresentsthespringconstant.

• Thetorqueoftheaerodynamicforcesarerepresentedby𝑀-.

• Theactuation𝑀'()isappliedasatorqueatthewingroot.

We suppose aharmonic forcedmotionbasedon simpleharmonicwingsweepingmotionso:

𝜙 = 𝐴 sin 𝑤𝑡

𝜙 = 𝐴 cos 𝑤𝑡 ∗ 𝑤

𝜙 = −𝐴 sin 𝑤𝑡 ∗ 𝑤8

MaximumamplitudeAisrestrictedbythewingsetupoftheflapping-wingMAV.

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TheinertiaIofabar,wings,is:

𝐼 = 𝑚: ∗ 𝐿8

• Themassofthewingisrepresentedby𝑚:.

• L represents the length between the inertia centre andattachmentpointoftheaerodynamicsforces.

Usingsimplifications,wehave:

𝑀'()(𝜙) = 𝐼𝜙 + 𝑘𝜙 +𝑀-(𝜙)

𝑀'()(𝜙) = 𝑚: ∗ 𝐿8 ∗ (−𝐴 sin 𝑤𝑡 ∗ 𝑤8) + 𝑘 ∗ 𝐴 cos 𝑤𝑡 ∗ 𝑤 + 𝑀-(𝜙)

𝑀'()(𝜙) = 𝑚: ∗ 𝐿8 ∗ (−𝐴 sin 𝑤𝑡 ∗ 𝑤8) + 𝑘 ∗ 𝐴 sin 𝑤𝑡 + 𝑀-(𝜙)

𝑀'()(𝜙) = (𝑘 − 𝑚: ∗ 𝐿8𝑤8) ∗ 𝐴 sin 𝑤𝑡 + 𝑀-(𝜙)

K can be replaced by natural frequency equation to highlight theimportanceoftheresonance.

𝑘 = 𝑤<8*𝑚: ∗ 𝐿8

𝑀'()(𝜙) = 𝑚: ∗ 𝐿8(𝑤<8 −𝑤8) ∗ 𝐴 sin 𝑤𝑡 + 𝑀-(𝜙)

First part of equation,𝑚: ∗ 𝐿8 𝑤<8 −𝑤8 ∗ 𝐴 sin 𝑤𝑡 , representtheovercomewinginertia.

Theaerodynamicforces𝑀-(𝜙)canbeapproximatedby:

𝑀- = 𝐶-,( ∗ 𝐿? ∗ 𝜙8 ∗𝜙

𝜙8

Aftersimplifications,weobtaintheovercomedrag:

𝑀- = 𝐶-,( ∗ 𝐿? ∗ 𝐴8 ∗ 𝑤8 ∗12 +

12 ∗ cos 2 ∗ 𝑤𝑡

• 𝐶-,( isdefinedasB8𝐶C ∗ 𝑆 ∗ 𝜌

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𝑀'()(𝜙) = 𝑚: ∗ 𝐿8(𝑤<8 −𝑤8) ∗ 𝐴 sin 𝑤𝑡 + 𝐶-,( ∗ 𝐿? ∗ 𝐴8 ∗ 𝑤8

∗12 +

12 ∗ cos 2 ∗ 𝑤𝑡

We need to add the lift force to the model to keep the systemhovering.Theminimumlifttosustainthesystemhoveringisgivenby:

𝐹G = 𝑚: +𝑚H ∗ 𝑔

Finalexpressionofthetorqueatthewingrootis:

𝑀'()(𝜙) = 𝑚: ∗ 𝐿8(𝑤<8 −𝑤8) ∗ 𝐴 sin 𝑤𝑡 + 2 ∗𝐶-,(𝐶G,(

∗ 𝐿 ∗ 𝑚: +𝑚H

∗ 𝑔 ∗12 +

12 ∗ cos 2 ∗ 𝑤𝑡

4.2 NUMERICALDATA

We identify the known variables and we think about consistentmeasures.

The first step is the sizing of the concept.We consider the valuesappliedbyBolsmaninhisPhDandtheDelFlyproject.Ourlongitudesdatawillbe:

Figure8:Concept1longitudes.

Faero Faero

Factuator

Inertia

L1

L3 L2

-26-

Firstsizedconcept

L1 40mm

L2 10mm

L3 40mm

Wewillexperimentwithtwodifferentsizedconcepts.Theothersizesare:

Secondsizedconcept

L1 50mm

L2 15mm

L3 60mm

When we have sized the concept, we estimate the mass of thesystem.

Theoverallmassofthesystemisthesumofmassofthewingandmassofthebody.

Looking the numerical of Bolsmanwe assume a totalmass of thesecondsizedof10’5grams.

ThewingsmaterialsassumedisthePET.Thismaterialhasadensity

of ρ=1’34 J(KL. The width and the thickness of the wings are based on

Bolsmannumericaldata.

Withthisdatawecancalculatethevolumeofthewingsandwehavethedensitysowecancalculatethemassofthewings.

Table1:Concept1firstsizing.

Table2:Concept1secondsizing.

-27-

𝑉𝑜𝑙𝑢𝑚𝑒 = 𝑇ℎ𝑖𝑐𝑘𝑛𝑒𝑠𝑠 ∗ 𝑊𝑖𝑑𝑡ℎ ∗ 𝐿𝑒𝑛𝑔ℎ𝑡

𝑚: = 𝑉𝑜𝑙𝑢𝑚𝑒 ∗ρ

Thickness 12mm

Width 0’1mm

Length1 40mm

Length2 60mm

Volume1 0’048𝒄𝒎𝟑

Volume2 0’072𝒄𝒎𝟑

Mw1 0’064g

Mw1 0’096g

Table3:WingsMass.

-28-

Abouttherelationshipbetweenliftanddrag,weassumebyEllington

thatthetypicalratiobetweenthemis0^1 ≤ `a,b`c,b

≤ 0^25.

Duringtheprocess,wehadseriousproblemstogettheestimationofthestiffnessofthespring,K.

Atfirst,wechoosetwotypicalmaterials,springsteelAISI302andtheBerylliumCooper,forthespringsandwefindbothvaluesoftheYoungisModulus.

Material E

BerylliumCooper 124’105GPa

StainlessSteelAISI302 180GPa

Togetthespringisstiffness,wewillusethisequation:

𝐾 =𝐴𝑟𝑒𝑎𝐿𝑒𝑛𝑔𝑡ℎ

𝑁

𝑚

Due to that, we need to differentiate between the two differentconcept sizing. After an exhaustive search, we do not find informationaboutthesizingofthespring,soweassumeit.

Concept1Firstsizing

SpringLength 4cm

SpringArea 8𝑐𝑚2

BerylliumCooperK 2’48*10hiK

StainlessSteelAISI302K 3’6*10hiK

Table4:YoungisModulus

Table5:Firstsizingstiffness

-29-

Concept1Secondsizing

SpringLength 5cm

SpringArea 15𝑐𝑚2

BerylliumCooperK 3’72*10hiK

StainlessSteelAISI302K 5’4*10hiK

After our calculations, we compare the results with the springpropertiesusedbyBolsmanandwenotifythatourresultsarenotcoherent.As we have not many information about the springs, we will use thestiffnessusedbyBolsman.

K1 381’1iK

K2 403’4iK

4.3 TRANSMISSIONRATIO

First simulation in the project is the analysis of transmissionmechanism.Theequationis:

𝑇 =𝜙𝑢

𝜙istheoutput,itistheangleofdeflectionofthewing.𝑢,theinput,isobtainedfromthedeflectionofthering.𝜙willbeexpressedinrad,𝑢inmeters.

Table6:Secondsizingstiffness

Table7:Stiffnessfinalvalues

-30-

Inthissection,weplottherelationforadeflectionof90ºand120ºandaninputof20mmand30mm.Weassumethetotaldeflectionofthespringof20and30mmaccordingtopaststudiesabouttheflightoftheinsects.

-1,57

0

1,57

-0,01 0 0,01

⏀[rad]

u[m]

Input- Output

Figure9:Relationship20mm–90º

-2,09

0

2,09

-0,01 0 0,01

⏀[rad]

u[m]

Input- Output


-31-

Inthefigures9,10,11and12,wecanseethelinearapproximationoftheinput-outputrelationship.

Below,wewillstudyaboutthenonlinearmovementbecauseintherealcaseitbecomesnonlinearinthedegreesofdeflection.

-1,57

0

1,57

-0,015 0 0,015

⏀[rad]

u[m]

Input- Output


-2,09

0

2,09

-0,015 0 0,015

⏀[rad]

u[m]

Input- Output


-32-

4.4 RESPONSEANALYSIS

Aswehavecommentedalongtheproject,oursystemhasonedegreeoflibertyanditisnotdumped.Wecanrepresentoursystemas:

Themovementequationofthissystemis:

𝑚 ∗ 𝑢 𝑡 + 𝑘 ∗ 𝑢 𝑡 = 𝑓

Weknowthatoursystemresponseisaharmonicsimpleresponse:

𝑢 𝑡 = 𝐴 ∗ sin(𝑤 ∗ 𝑡)

Sothederivatetransactionsare:

𝑢 𝑡 = 𝐴 ∗ sin(𝑤 ∗ 𝑡)

𝑢 𝑡 = 𝐴 ∗ cos 𝑤 ∗ 𝑡 ∗ 𝑤

𝑢 𝑡 = −𝐴 ∗ sin 𝑤 ∗ 𝑡 ∗ 𝑤8 = −𝑤8 ∗ 𝑢 𝑡

As an external force we consider the aerodynamics force of ourmathematicaldevelopedatpoint4.1.

𝑓 = 2 ∗𝐶-,(𝐶G,(

∗ 𝐿 ∗ 𝑚: +𝑚H ∗ 𝑔 ∗12 +

12 ∗ cos 2 ∗ 𝑤𝑡

Thenextstep,isthecalculationoftheamplitude“A”oftheharmonicsimpleresponse.Forthismeasurement,weconsiderthetime“t”equalto

m

F(t)

u(t)

Figure13:Representationofasystemofonedegreeoflibertywithoutdumper.

-33-

zeroandwith thehelpof the softwareWolframAlphaMathematicawesolvetheequationtogettheamplitude“A”.

𝐴 =2 ∗

𝐶-,(𝐶G,(

∗ 𝐿 ∗ 𝑚)k)'G ∗ 𝑔

𝑘 − 𝑚)k)'G ∗ 𝑤8

Withthesystemdeveloped,wecanprogrammeacodetodosomeinterestingsimulationsaboutourconcepts.ForthissimulationsweuseaswecommentedbeforeMatlabsoftware.

The first comparison is between the application of the naturalfrequencyandotherfrequency.Therestofthevalueswillbecommonforbothsimulationsandwecanlooktheminthefollowingtable:

𝐦𝐭𝐨𝐭𝐚𝐥 10’5g

k 381’1iK

𝐂𝐝,𝐜𝐂𝐥,𝐜

0’1

L 0’1m

g 9’8Ktu

Thenaturalfrequencyis:

𝑤v =𝑘𝑚

𝑤v = 190^51𝑟𝑎𝑑𝑠

Table8:Commonvaluesforfrequencyrepresentations.

-34-

Figure14:w=10rad/s

Figure15:w=80rad/s

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-8

-6

-4

-2

0

2

4

6

8

Des

plaz

amie

nto

u(t),

m

10-6 Respuesta a excitación armónica de un sistema de 1 gdl amortiguado

u(t)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2Tiempo, s

0

1

2

3

Fuer

za F

(t), N

10-5 Excitación

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-6

-4

-2

0

2

4

6

Des

plaz

amie

nto

u(t),

m

10-6 Respuesta a excitación armónica de un sistema de 1 gdl amortiguado

u(t)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2Tiempo, s

0

1

2

3

Fuer

za F

(t), N

10-5 Excitación

-35-

Looking the different simulations, we can verify the sinusoidalresponse.

Inasmuchasweapproachtothenaturalfrequency,thedisplacementbecomehigher.

Aswecansee,whenthesystemisexcitedatthenaturalfrequency,figure 16,weobtain the phenomenonof resonance. This is thatwe arelookingforbecausewearetryingtoimitatetheflyoftheinsectandaswesaw,theyusethisphenomenontomovetheirwings.Duetothat,forthenextsimulationsweusethenaturalfrequency.

Second comparison is for different values of K, this represent thedifferencebetweentwokindofspringmaterials.BecauseofdifferentK,wehave different values of natural frequency. As we saw before the firstsimulation,thenaturalfrequencydependsonthemassandthestiffness.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-4

-3

-2

-1

0

1

2

3

4

Des

plaz

amie

nto

u(t),

m

1010 Respuesta a excitación armónica de un sistema de 1 gdl amortiguado

u(t)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2Tiempo, s

0

1

2

3

Fuer

za F

(t), N

10-5 Excitación

Figure16:w=naturalfrequency.

-36-



0’1

L 0’1m

g 9’8Ktu

The values of K studied are 381,1 and 403,4 iK so the natural

frequenciesare:

K=381,1𝑵𝒎 𝑤v = 190^51

𝑟𝑎𝑑𝑠

K=403’4𝑵𝒎 𝑤v = 196′01

𝑟𝑎𝑑𝑠

Table9:Commonvaluesforstiffnessrepresentations.

Table10:Naturalfrequenciesvaluesforstiffnessrepresentations.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-4

-3

-2

-1

0

1

2

3

4

Des

plaz

amie

nto

u(t),

m


u(t)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2Tiempo, s

0

1

2

3

Fuer

za F

(t), N

10-5 Excitación

Figure17:k=381’1iK.

-37-

Betweenthis twostiffness,wecannotseethedifference,sobothmaterials for the spring can be a good idea for our concepts. To seedifferencewiththevaluesofK,weprovevaluesthatwehavenotconsidertoourconcept.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-4

-3

-2

-1

0

1

2

3

4

Des

plaz

amie

nto

u(t),

m


u(t)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2Tiempo, s

0

1

2

3

Fuer

za F

(t), N

10-5 Excitación

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-8

-6

-4

-2

0

2

4

6

8

Des

plaz

amie

nto

u(t),

m


u(t)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2Tiempo, s

0

1

2

3

Fuer

za F

(t), N

10-5 Excitación

Figure18:k=403’4iK.

Figure19:k=100iK.

-38-

Aswecanconclude fromthe figures, less stiffnessproducesmoredisplacement.Inspiteofthat,maybewecannotobtainaspringwiththisproperties,ormaybea springwith this stiffnesshas lessduration.Morestiffness produces less movement as far as reverse movement of thesystem.

Third simulations are for compare the different values of the

relationshipbetweenliftanddrag|},~|�,~

.Thetwovaluesthatwesimulateare

thebordersoftherangegivenbyEllington,0^1 ≤ `a,b`c,b

≤ 0^25.

The rest of the values are the same as we use in the rest ofsimulationswiththenaturalfrequency.


k 381’1iK

𝐰𝐧 190’51�'-K

L 0’1m

g 9’8Ktu

As we can see down below, the increment of the value of therelationshipbetweenliftanddragproducesagrowthofthevalueoftheexternal force. It produces in turn an increment in the value of thedisplacementbecausethevalueoftheamplitudehasgrown.

Table11:Commonvaluesfor|},~|�,~

representations.

-39-

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-4

-3

-2

-1

0

1

2

3

4

Des

plaz

amie

nto

u(t),

m


u(t)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2Tiempo, s

0

1

2

3

Fuer

za F

(t), N

10-5 Excitación

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Des

plaz

amie

nto

u(t),

m


u(t)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2Tiempo, s

0

2

4

6

Fuer

za F

(t), N

10-5 Excitación

Figure20:|},~|�,~

=0’1.

Figure21:|},~|�,~

=0’25.

-40-

Thenextcomparisonisforthemasscase.Weconsideratotalmassof7and10’5g.Therestofvaluesforthesimulationsarethesamethanfortheothersimulations,thisvaluesareenclosedinthethefollowingtable:


0’1

k 381’1iK

𝐰𝐧 190’51�'-K

L 0’1m

g 9’8Ktu

Table12:Commonvaluesformassrepresentations.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-4

-3

-2

-1

0

1

2

3

4

Des

plaz

amie

nto

u(t),

m


u(t)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2Tiempo, s

0

1

2

3

Fuer

za F

(t), N

10-5 Excitación

Figure22:m�� =10’5g.

-41-

Inconclusion,forthisstiffnesstheconceptwithamassof7gramshasnotmovement.Ifwewantaprototypelighter,witharound7grams,weneedtoincludeaspringwithlessstiffnesstoallowthemovementoftheprototypeandproducetheflight.TheLengthalsoaffectstheforce.

Last simulations are to compare thedifferent length ideas for ourprototypes.Thetwolengthare10cmand15cm.Therestofvaluesare:


0’1

k 381’1iK

𝐰𝐧 190’51�'-K


g 9’8Ktu

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Des

plaz

amie

nto

u(t),

m

Respuesta a excitación armónica de un sistema de 1 gdl amortiguado

u(t)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2Tiempo, s

0

0.5

1

1.5

Fuer

za F

(t), N

10-5 Excitación

Figure23:m�� =7g.

Table13:Commonvaluesformassrepresentations.

-42-

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-4

-3

-2

-1

0

1

2

3

4

Des

plaz

amie

nto

u(t),

m


u(t)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2Tiempo, s

0

1

2

3

Fuer

za F

(t), N

10-5 Excitación

Figure24:L=10cm.

Figure25:L=15cm.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-6

-4

-2

0

2

4

6

Des

plaz

amie

nto

u(t),

m


u(t)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2Tiempo, s

0

1

2

3

4

Fuer

za F

(t), N

10-5 Excitación

-43-

Aswellastherestofthevariablesthatarepresentintheforce,ariseofthevalueofthelengthproducesariseinthevalueofthedisplacementandoftheexternalforce.

4.5 FREQUENCYRESPONSEANALYSIS

The frequency response is a characteristic of a system that has ameasuredresponsethatistheresultofaknownappliedinput.Inthecaseofamechanicalstructure,thefrequencyresponseisthespectrumofthevibrationofthestructure,dividedbythespectrumoftheinputforceintothesystem.Tomeasurethefrequencyresponseofamechanicalsystem,wemustmeasure the spectra of the input force to the system and thevibrationresponse.ThisisdonemoreeasilywithaTRFanalyser.Frequencyresponse measurements are widely used in the modal analysis ofmechanicalsystems.

The frequency response function is a three-dimensional quantityconsistingofamplitudevs.phasevsfrequency.Thatiswhyatruegraphofitneedsthreedimensions,whichisdifficulttorepresentonpaper.Onewayto do this is the Bode diagram, which consists of two curves, one ofamplitudevsfrequency,andoneofphasevsfrequency.

H(w)isthefrequencyresponsefunction.Tocalculateit,weusethefollowingequation:

𝐻 𝑤 =1

(𝑘 − 𝑚 ∗ 𝑤8) = 1𝑘

1 − 𝑟8

Where:

𝑟 =𝑤𝑤v

-44-

Lookingatthephasegraphic,thelowerplot,wecanseeachangeofphaseof180ºat thenatural frequency.This isdue toour system isnotdumped.Ifoursystemweredumped,thechangeofphasewouldhavethesamemagnitudebutthechangewouldbesofter.

The H(w) representation shows us a maximum in the naturalfrequencyofthesystem.At lowerfrequencieswecanobservethestaticdeflexionduetoatthesefrequenciesthestiffnesshasmoreimpact.

4.6 TRANSFERFUNCTION

Tocheckourresults,wedoothersimulationswithMatlabsoftware.ThistimeweuseSimulink.Thebaseoftheequationstogetthetransferfunctionisthemathematicalmodeldevelopedinpoint4.2.

𝐼𝜙 + 𝑘𝜙 = 𝑀'() − 𝑀-𝜙

100 101 102 103 10410-5

10-4

10-3

10-2

10-1

100

Rece

ptan

cia

H(

)

Magnitud de la función de respuesta en frecuencia FRF

100 101 102 103 104

Frecuencia , rad/s

0

100

200

Fase

(g

rado

s)

Fase de la función de respuesta en frecuencia FRF

Figure26:Frequencyresponsefunctionanalysis.

-45-

Where:

𝐼 = 𝑚:�vJ ∗ 𝐿)k)'G

𝑀'() = 𝐿:�vJ�kk) ∗ 𝐹 ∗ sin(𝑤v ∗ 𝑡)

𝑀- = 2 ∗𝐶-,(𝐶G,(

∗ 𝐿:�vJ ∗ 𝑚)k)'G ∗ 𝑔

𝑀'()isatorqueappliedinthewingroot.

DoingLaplacetransformandsolvingtheequationwegetourtransferfunction.

𝐺 𝑠 =𝜙𝑢 =

𝜙𝐹 ∗ sin(𝑤v ∗ 𝑡)

=𝐿:�vJ�kk)

𝑠8 ∗ 𝐼 + 𝑀- + 𝑘

𝐺 𝑠 =𝐿:�vJ�kk)

𝑠8 ∗ 𝑚:�vJ ∗ 𝐿)k)'G + 2 ∗𝐶-,(𝐶G,(

∗ 𝐿:�vJ ∗ 𝑚)k)'G ∗ 𝑔 + 𝑘

Thenumericaldataare:

𝐋𝐰𝐢𝐧𝐠𝐫𝐨𝐨𝐭 1’5cm

𝐦𝐰𝐢𝐧𝐠 0’096g

𝐋𝐭𝐨𝐭𝐚𝐥 15cm


0’1

𝐋𝐰𝐢𝐧𝐠 6cm


g 9’8Ktu

k 381’1iK

Table14:Valuesfortransferfunction.

-46-

Soourtransferfunctionis:

𝐺 𝑠 =0′015

𝑠8 ∗ 0′0012492 + 381′1

WepreparetheSimulinkwithasinusoidalentryandweget:

Wesimulatefornaturalfrequencyandforfrequencyof50�'-t:

Figure27:Simulinkmodel.

Figure28:𝐺(s)withnaturalfrequencyinput.

-47-

We can observe how for the natural frequency we obtain fasterresonance and bigger displacements comparing with the response with

w=50�'-t.Thesameaswiththeotherprocess.

Also,wecanseetheplotofthesinusoidalinput:

Figure29:𝐺(s)withw=50rad/s.

Figure30:Inputwithnaturalfrequencyandamplitudeof1’5cm.

-48-

5 Conclusion

After the process ofmaking the project.We can conclude that ispossibletodoaprototypeswhichcanimitatetheflightoftheinsects.

Aswecanreadinfirstspoints,itistoodifficultnowadaystoproduceanexactcopyof the insects,but technology isdevelopingandmaybe insomeyearsweareabletomakeanexactimitationtotheinsects.

Aboutour idea for theprototypes.After theexperimentationwithMatlab,wecanaffirmsomeaspects:

• Bothvalues forKaregood.Thestiffness thatwehave thinkallowsufficientdisplacementandmovementoftheprototypesandarematerialsthatwecanfindinthemarket.

• Thesizeoftheconceptneedstobethesecondsizingmade.With the first sized concept, we obtain some problemsbecauseistoolightforthisstiffness.

• TheliftanddragratiointervalgivenbyEllingtoncanbeused.Ifwechoosetheuppervalue,theforceappliedismajorsothedisplacementismajor.

• Following those guidelines, our prototype can produce thenatural frequency and in that way imitate the flight of theinsects.

-49-

6 Bibliography

Flapping wing actuation using resonant compliant mechanism.Author:CasparTitusBolsman.

Función de transferencia y respuesta en frecuencia. Escuela SanFelipe.

Práctica2:ModelizadoySimulacióndeSistemasNoLineales.EscueladeIngenieríadeCádiz.

Onthequasi-steadyaerodynamicsofnormalhoveringflightpart I:the induced power factor. Author: Mostafa Nabawy and William J.Crowther.

Biologically inspired hovering flight stabilization for the flappingwingsmicroairvehicles.Author:PawełCHOROMAŃSKI,KrzysztofSIBILSKI.

Diseño y construcción de un robot volador bio-inspirado. Author:SebastiánIgnacioRoblesGebauer.

Aerodinámicainestacionariayno-linealdemicrovehículosaéreosdealasbatientesinspiradosenlabiología.Author:BrunoA.Roccia.

Design and Performance of Insect-Scale Flapping-Wing Vehicles.Author:Whitney,JohnPeter.

http://www.acxesspring.com/constante-de-un-resorte.html

http://www.engineeringtoolbox.com/young-modulus-d_417.html

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