Numerical simulation of pararotor dynamics: Effect of mass displacement from blade plane

Contents lists available atScienceDirect

Aerospace

Science

and

Technology

www.elsevier.com/locate/aescte

Numerical

simulation

of

pararotor

dynamics:

Effect

of

mass

displacement

from

blade

plane

Joaquín Piechocki

a

,

∗

,

1

,

Vicente Nadal Mora

a

,

1

,

Ángel Sanz-Andrés

b

,

2

a_{UniversidadNacionaldeLaPlata,(1900)LaPlata,BuenosAires,Argentina} b_{UniversidadPolitécnicadeMadrid,E-28040,Madrid,Spain}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received1November2013

Receivedinrevisedform24November2015 Accepted3April2016

Availableonline21April2016

Keywords:

Decelerator Numericalsimulation Rotarywing

The pararotor is a biology-inspired decelerator device based on the autorotation of a rotary wing, whosemainpurposeistoguidealoaddescentintoacertainplanetaryatmosphere.Thispaperfocuses on a practical approach to the general dynamic stability of a pararotor whose center of mass is displaced from the blade plane. The numerical simulation tool developed is based uponthe motion equationsofpararotorflight,utilizinganumberofsimplifyinghypothesesthatallowthemostinfluencing factors on flight behaviorto be determined. Severalsimulated casesare analyzed to studytheeffect of different parameters associated with the pararotor configuration on flight dynamics, particularly the center of mass displacement from the blade plane. It was confirmed that the ability to reach stability conditionsdepends mainlyonalimitednumber ofparametersassociated withthe pararotor configuration: the relationshipbetween principal momentsof inertia, the planformshape (associated with blade aerodynamic coefficients and blade area)and thevertical distance between thecenter of massandthebladeplane.Asaresultdifferenttypesofequilibriumsolutionsarefoundandtheeffectof eachparameterischaracterized.Abifurcationinthestabilityshapetoaprecessing conicalrotation,not previouslyfoundinthelinearstability analysis,ispredictedbythisnumericalmodel.

©2016ElsevierMassonSAS.All rights reserved.

1. Introduction

Thepararotorisabiomimeticrotarywingsdecelerator, unpow-ered, and potentially deployable whose main practical use is to aerodynamicallydeceleratealoaddescendingintoacertain plane-taryatmosphereortoperformmeasurementsduringdescent.Such aprobe offersseveraladvantages overother recoverytechniques: simplicity, controlled deceleration, maneuvering capabilities and potentiallandrecovery.Also,thisdeceleratortypeisofinterest,for instance,for the measurement of atmosphericconditions around airportsforaviationoperationssupport,ortheexplorationof plan-etaryatmospheres.

Worksconcerningthedecelerationandcontroloffallingbodies were published by Shpund andLevin [1–4],in thearea of rotat-ing parachutes. Karlsen, Borgström and Paulsson [5] worked on wingedbodiesforsubmunitionapplications.Theyreportedonthe

*

Correspondingauthor.

E-mailaddress:[email protected](J. Piechocki).

1 _{AssistantProfessor,GrupodeIngenieríaAplicadaalaIndustria,Departamento} deAeronáutica,FacultaddeIngeniería,Calle116entre47y48.

2 _{FullProfessor,InstitutoUniversitariodeMicrogravedad“IgnacioDaRiva,”} Es-cuelaTécnicaSuperiordeIngenierosAeronáuticos,PlazaCardenalCisneros,3.

advantages of the pararotorover theparachute: lower sensitivity tolateralwinds,parachutedeploymentproblems,lowerprecession movements,andhigherfallingvelocity.

The flight of samara wings has similarities with pararotors. Seter andRosen[6,7] studiednumerically theinfluenceof differ-ent parameters onsamaraflight stability.Crimi [8]hasstudied a rotatingbodywithonlyonewingforsubmunitionapplications.He searchedforabodythatperformedperiodicmovements.

Previous work has been carried out concerning modeling the stability of a pararotor, mainly by Nadal Mora, Sanz-Andres and Piechocki [9–17]. They conducted investigations concerning the stability behavior ofpararotors whoseblades wherealignedwith the centerof mass ofthe wholedevice. They developedan ana-lytical modelthat predicts the dynamic behavior under different deviceconfigurations.

AnanalyticallinearizedmodelwasdescribedbyPiechockietal.

[17]presentingfourdifferentcasesofanalysesthatarerevisitedin thecurrentwork.

SeterandRosen[18]presentedthemodelingofmultibody sys-temsforahelicopter.

Pararotordecelerator potentialhasbeeninvestigatedby differ-ent authors[19–22],who considerapplications that are centered

http://dx.doi.org/10.1016/j.ast.2016.04.004

Nomenclature

CD dragcoefficientoftheblade

CLα slopeofthecurveliftvs.angleofattackoftheblade CD M modeldragcoefficient

CM centerofmass

CP centerofpressure

Fgc weightforcevector . . . N

h kineticmomentum . . . kg m/s2

Ii principalmomentsofinertia,i

=

1

,

2

,

3 . . . kg m2

k verticalvelocitytoblade-tip-speedratio

ki j ratio of coordinates of the center of pressure of the

blade jinaxisi direction,ri j

/

r11

ke dimensionlessparameterusedforthestabilityanalysis

ni directionperpendiculartobladei surface,i

=

1

,

2

S areaofoneblade . . . m2

TM transformation matrix to convert vectors from the

bodyfixedframetotheinertialframe

Vri relativevelocitytotheiblade . . . m/s

Vt pararotordescentvelocity . . . m/s

vh axialhoveringvelocity . . . m/s

vi inducedvelocity . . . m/s

α

angleofattack . . . rad

β

0

, β

1

, β

2 meanpitchangle,pitchangleofblade1,2,

respectively . . . rad

θ

nutationangle . . . rad

ϕ

spinangle . . . rad

ψ

precessionangle . . . rad

ω

angularvelocity . . . rad/s

ω

12 angularvelocityprojectedinthe1–2plane,or

transversalangularvelocity . . . rad/s

in mission concepts requiring controlled descent, low-velocity landing,andatmosphericresearchcapabilityonplanetexploration. Themainobjectiveofthecurrentworkistodevelopand vali-dateanumericalsimulationtooltodescribepararotorflightmodes todeterminethestabilitybehavior ofa pararotorconsidering dif-ferentconfigurationparameters,particularlythe effectofthe dis-tancebetweenthebladeplaneandthecenterofmass(illustrated inFig. 1) andto study dynamic effects of parameters associated withthedeviceconfiguration.

Itisworthmentioningthatthepresentpaperextendsthe stud-iespreviously developed[17] consideringnonlinear effectsinthe pararotors dynamics and their results on pararotor flight behav-ior. Those nonlinear terms are the resultof using Newton–Euler equationsforarigidbodyexcitedwithaerodynamicloadscoupled withinertialphenomena.Intheanalyticalmodel[17]secondorder termswereconsistentlyneglected byconsideringthat the follow-ingparametersareofsmallmagnitude:bladepitchangle, relation-shipbetweendescentratetothevelocityinducedinthebladesby rotation,distanceofbladecenterofpressurein

e

2 directionfrom

centerofmass,bladedrag aerodynamiccoefficient. Thenatureof Newton–Eulerequationsofarigidbody undertheeffectof aero-dynamic forces (that are blade relative velocity dependent)is in generalnonlinear. Thevalidity ofthelinearanalyticalmodel[17]

isrestrictedtothevicinityofanequilibriumpoint.Thisnumerical approachisausefulpotentialtool thatallows thestability limits, theeffects of configurational parameters, andthe flight mode of futurepracticaldevicesto bemore accuratelydefined. This mod-elization includes neither simplificative assumptions (magnitude ordersneglectingcriteria)norlinearizationofthesystemnearthe equilibriumsolutions,asdevelopedin[17].Thesignificanceofthis work isthat the nonlinear factors effects onflight dynamics can beconsidered practically,not onlyintermsofresponsetime, but alsoinstabilitylimitsandflightmode,particularlywithpararotors flyingneartheirstabilitylimitsdefinedin[17].

The current research shows solutions that cannot be reached by previous modelization, andso, a deeper understanding ofthe phenomena canbe reached, with itspositive practical impact on pararotordesign.

2. Mathematical and numerical models

The approach chosen in this work to studythe pararotor dy-namic behavior was to build an analytical model of a pararotor basedonthe developmentof thecompletemotionequations, in-cludingaerodynamic forces andtorques generatedby the blades andto compute numerically the evolution ofthe systemfrom a

particularinitialflightattitude,differentfromthefinalequilibrium state. So, the analyses performedresponds to the observation of patterns over specific cases.The effect of k31, the dimensionless

distance of the center of mass to the blade plane in the falling direction, isanalyzedtogether withanumberof geometricaland aerodynamicalparameters.

Induced velocity, vi, was not directly included in the model,

followingsimilar assumptions offormerstudies [16]. Other stud-ies [21] indicated that real low aspect ratio rotary wing shows an induced power correction factor (that can be defined as the factorthatmodifiesrotor classicmomentumtheorywhen consid-ering nonideal blade physical effects),

κ

, of about 2 or greater. This factindicates that the relationship ofdescent rateandaxial hoveringvelocity, vh, isgreater than2, thusin a windmillbrake

stateaxialdescent[22].Thecurrentmodelfocusesonthe assump-tion ofsmallinduced velocity, in thewindmill brakestate. Large induced power correction factors associated with low aspect ra-tiowingare consistentwiththishypothesis. Verticaltunneltests performed[20]usingdifferentsmallaspectratiowingspararotors showed induced velocities of 15% of the descent velocity on av-erage, evidencing that the simplificative hypothesis is associated witharepresentativeflightrange.

Flow model foraxial descent inwindmill brakestate [22] in-dicatesthat thesensibility ofinducedvelocity, vi,todescentrate,

decreaseswiththedescentvelocitytothehoveringvelocityratio,

Vt

/

vh.Thismeansthatverticalvelocity disturbancesinfluenceon

inducedvelocitydependsonthevi

/

vh magnitude,andsoon

κ

.

However,theeffectofinducedvelocityisincludedinthevalues oflift,liftslopeanddrag coefficients, CL,CLα andCD,whichare

determined from experiments [16]. This experimental work was performedin operationalconditionsof apararotor model,sothe effect of vi will resultin an incrementof themagnitudes of lift

anddrag iffreestream velocityover theblades isconsidered.As a consequence,themodel willbe developedassociated withthis experimental data,thatwill definean operatingrangeofvalidity, overwhichtheliftslopecanbeapproximatedaslinear.Thisrange isdefinedintheincompressibleregime.

The system analyzed is a pararotor flying in an autorotation regime, modeled as an inertial cylindricalbody withtwo identi-callow-aspect-ratioflatbladeswhichrotatesatangularvelocity

ω

andfallsverticallyatspeed Vt.ThegeometryisdefinedinFigs. 1 and2.Thebody-fixedreferenceframe,1,2,3,hasitsoriginatthe centerof massanddirections

e

1,

e

2,

e

3.The axes1, 2,3are the

Fig. 1.Schemeofthestudiedpararotor.Position ofthe centerofmass, CM,and centerofpressure,CP1,inbodyaxes.

Fig. 2.Systemgeometry.(X,Y,Z),inertialreferenceframe;(1,2,3),body-fixed ref-erenceframe; ψ, precessionangle; θ,nutationangle;ϕ, spinangle;ω,angular velocity;β1,β2,bladeP1andP2pitchangles,respectively;ζ,auxiliaryaxis;B, in-tersectionofbladeplanewithaxis3.

To develop the mathematical model used for the numerical simulationsitwasnecessaryto:statearepresentationsystemthat allows migration from a body fixed coordinate frame to inertial coordinateframeandviceversa,setthecompletedynamics equa-tionsforthepararotor,settheexpressionsforforcesandmoments actingonthepararotor,includingaerodynamicandgravityforces, considering the distance from the center of mass to the blade plane,andstatethenumericalcomputationorderanddata-flow.

The tool adopted to develop the numerical simulation is an opensourcesoftwarecalledScicos.Tostudythepararotordynamic responseby numericalsimulationa time intervalandinitial con-ditionswereestablished.

2.1. Pararotororientation

Averyimportantissueforthedevelopmentofapararotorflight dynamics simulator is to track the spatial position and attitude of the pararotor. This fact implies having the capability to ex-pressvectorsfromthebodyfixedcoordinatesystemintheinertial system,andtheotherwayaround,ina univocalway,avoiding sin-gularitiesalongthenumericalprocess.

Anormalized quaternion isused toexpress thepararotor ori-entation. Therefore, the relative attitude of the pararotor in the

inertial coordinate systemis expressed interms of four parame-ters(oravectorplusascalar).

The use of quaternions to express pararotor attitude implies settingarotationmatrixthatallowscoordinatesandvectorstobe transformedfromonereferencesystemtotheother,inagreement totheexpressionssummarizedbyDiebel[23].

Tonumericallycompute theorientation overa periodoftime, the derivative oftheorientation quaternion withrespectto time,

t,isintegratedfromtheinitial conditions(given byaninitial ori-entation quaternion, equivalent to the orientation described by initial Euler angles)in differential steps. As a result, an orienta-tion quaternion corresponding to each integrationstep over time is obtained, which will be the initial orientation quaternion for the next time step. The arithmetic operations over the quater-nion introduce a cumulative distortion on the orthonormality of the element. The algebraic method introduced by Ai Chun Fang and Zimmerman[25] was usedtoreduce thiseffect.Themethod introduces adimensionlessconstant Kε callednormalizationgain.

It isdemonstratedthat themethodisstableifhKε

<

1 s(bythe

directmethodofLyapunov),wherehisthetimestep.Sothe val-uesadoptedaftertestingtheperformanceoverdifferentnumerical examplesare Kε

=

1,andh

<

1

/

10 s.

2.2. Initialconditions

The initial orientation conditionsforthe numericalsimulation aredefinedbyasetofEulerangles

E

ul0

=

(

ϕ

0

,

θ

0

,

ψ

0

)

,which

cor-respondstoaninitialorientationquaternion.Aftereachtimestep, thesimulationtooloutputsanorientationquaternion, correspond-ingtoasetofEulerangles.Tofacilitatethespatialrepresentation, theseEuleranglesareexpressedintheso-called3-1-3order[24]: rotationisdescribed byarotationalong 3-axis,asequential rota-tionalong1-axis,andafinalrotationalong3-axis.

2.3. Dynamicsmodel

The complete pararotor dynamics system has six degrees of freedom, andincludescoupled rotationsandtranslations.The ex-pressions,intheprincipalinertiaaxesbodyreferenceframe,are:

M

=

I

·

dω

/

dt

+

ω

×

(I

·

ω

),

(1a)

F

=

m

(

dVb

/

dt

+

ω

×

Vb

),

(1b)

where

ω

= [

ω

1

,

ω

2

,

ω

3

]

is the rotational speed vector, Vb

=

[

Vb1

,

Vb2

,

Vb3

]

is the center of gravity velocity vector, M

=

[

M1

,

M2

,

M3

]

is the momentum vector, F

= [

F1

,

F2

,

F3

]

is the

thrustvector,

I

istheprincipalmomentsofinertiatensorandmis themassofthepararotor.

The numerical simulation is based on the integration of the dynamics equations, starting from initial values for angular ac-celeration and translational acceleration. The integration process is sequenced so that it results in an angular velocity, a transla-tional velocity anda position at every time step. The simulation parametersconsideredare:integrationtime,numericalintegration method,timestep,initialconditions(orientation,position,speed).

The relative velocity of the flow with regard to the blades 1 and 2,onaquietatmosphere,isgivenby:

Vr1

= −

ω

×

r1

−

Vb

,

Vr2

= −

ω

×

r2

−

Vb (2)

where r1

= [

r11

,

r12

,

r13

]

T and r2

= [

r21

,

r22

,

r23

]

T are the vector

positionsofthecenterofpressureoftheblades1and 2.

Normalizedvectorshavingthelift(eL)anddrag(eD)directions

Fig. 3.RepresentationofnormalizedvectorseD1,eL1inthedirectionsoftheliftand dragoftheP1blade.e1,e2,e3,areparalleltotheaxes1,2,3;β1,bladepitchangle forblade1;Vr1flowvelocityrelativetoblade1.

Thenormalized liftvector, eLi,isnormal tothe plane defined

bytherelativespeed,

V

r1,andtheaxisalongthespandirectionof

theblade,

e

1.The componentofthe aerodynamicloadalongaxis e1 is not considered because aerodynamic forces on the blades

in this direction are neglected as a hypothesis. The relative air speedtoeachblademayhaveacomponentalong

e

1 whena

lat-eralpararotordisplacementsappear.Thiscomponentis,ingeneral, considerablysmallerthantheothers,andappearswithacyclic be-havioraroundanullaveragemagnitude.Theimpactofneglecting aerodynamicloadalong

e

1 onthe modelwas tested,initially

ap-pearingtovalidatethehypothesisatfirstlookasitseffectsdidn’t show a remarkablecontribution to dynamics. The impact ofthis second order effectmay be subjectof further research. The nor-malizeddrag vector,

e

D1,hasthedirectionoftherelativevelocity

ofeachblade.

Theangles ofattack,

α

1,

α

2, aregivenby the following

equa-tions:

sin

α

1

=

n1Vr1

/

norm

(V

r1

),

sin

α

2

=

n2Vr2

/

norm

(V

r2

),

(3)

where n1

= [

0

,

sin

β

1

,

cos

β

1

]

, n2

= [

0

,

−

sin

β

2

,

cos

β

2

]

, are unit

vectorsnormaltoblade1,2respectively.

Theaerodynamic forcesoflift,Li,anddrag, Di,andmoments,

Mi,generatedbytheblades,aswellasthegravityforcegenerated

atthecenterofmass,aretheinputelementsthatactovertherigid body dynamics.Theaerodynamic forces are easilydescribed in a bodycenteredreferenceframe,andthegravityforceinaninertial referenceframe. The used globalaerodynamic coefficientsof the blades include the effectsof the induced velocity. These aerody-namiccoefficientscouldbeadjustedtomatchexperimentalresults withthetheoreticalmodel.Asareferenceintheaerodynamic co-efficientdetermination,previouswork[15]hasbeenconsidered.

The aerodynamic coefficient is determined experimentally. These coefficients are linearized around a pararotor operating a steadystate.

Thisfactimpliesdefiningan operatingrangeofvalidityofthe aerodynamiccoefficientsoverwhichtheliftslopecanbe approxi-matedaslinear.

Thesemomentsandforcesaregivenby:

Li

=

1

2

ρ

SCLα

α

i

|

Vri

|

2_e

Li

,

(4)

Di

=

1

2

ρ

SCD

|

Vri

|

2_e

Di

,

(5)

Fgc

=

TM

[

0

,

0

,

−

mg

],

(6)

M

= [

r1

×

L1

+

r1

×

D1

+

r2

×

L2

+

r2

×

D2

]

.

(7)

Table 1

Physicalcharacteristicsofthemodel[19].

Parameter Value

Mass, kg 0.31

Blade chord, m 0.74 Blade span, m 0.15

I1, kg m2 _2.229·10−4

I2, kg m2 _9.930·10−3

I3, kg m2 _1.010·10−3

I21;I12, kg m2 _−5.860·10−5

I13;I31, kg m2 _5.690·10−7

I23;I32, kg m2 _−1.170·10−5

Table 2

Meanmeasuredperformanceofsamaratests.Comparisonwithnumericalresults.

Performance parameter Free flight[18] Numerical simulation Error

ω3, rad/s 17.1 17.1 0.1%

Vt, m/s 2.8 2.8 0.1%

θ, rad 0.52 0.48 7.6%

2.4. Integrationmethod

Anintegrationmethod isusedtointegrate thecompletely de-veloped equation system (1) and (4)–(7) with respect to flight time.Differentmethodswerecomparedinpracticalsimulationsto determinethesolutionerrormagnitude.Inparticular,theRunge– Kutta non-adaptive iterative methods, studied by Butcher [26], were analyzedandtested. Finally aDormand–Princemethod(5th order)waschosen,togetherwithatimeintegrationstepof0.005 s, as a compromise between error magnitude and computational cost.

To validate thenumerical simulation ofthe pararotor dynam-ics with a known solution, the free motion of a rigidbody was chosen. A rigidbody moving without external forcesor moment action,whosemainparametersaredefinedinTable 1,was numeri-callysimulatedandcomparedwiththeanalyticalsolution,givenin

[27].Theexcellentagreementbetweentheangularvelocitiesgiven by the analytical solution andthe numerical simulation, showed the efficiency of the simulation tool to represent body dynam-ics. Anerror parameterto compareanalyticalresults and numer-icalsimulations ateach integrationstepwasdefinedasE123

(

t

)

=

sqrt

((

ω

1T

(

t

)

−

ω

1N

(

t

))

2

+

(

ω

2T

(

t

)

−

ω

2N

(

t

))

2

+

(

ω

3T

(

t

)

−

ω

3N

(

t

))

2

)

wheresubscripts TandNrepresentthe analytical andnumerical results,respectively. The result ofthe errorparameter evaluation over60sremainedboundedbyamagnitudeof10−4rad

/

s.

2.5. Comparisonbetweenthenumericalsimulationandexperimental results

Experimentalmeasurements oversamaraflights havebeen re-ported by different authors. These data can be used to compare thenumericalsimulationwithexperimentalmeasuredparameters, as a complementary validation approach that is not intended to be fullyconclusive. Thesamaratype,single bladedpararotor,free flight case taken by Kellas [19] is adopted, because it includes enoughinformationtoobtainthenecessarysimulationparameters. ThephysicalcharacteristicsofthemodelaregiveninTable 1. Theperformanceparametersusedinthecomparisonare:spin, vertical velocity and nutation angle. Results are summarized in

Table 2. They correspond to an equilibriumsolution so that the constant parametercould be measuredinan experiment. Agood agreementcan alsobe appreciatedinthiscase. Differencescould beduetotheaerodynamicmodelemployed.

tun-Table 3

Meanmeasuredperformanceofpararotortests(C05P02).Comparisonwithnumericalresults.

Pitch angles Experimental results Numerical simulation results

β1, rad β2, rad ω3, rad/s Vt, m/s θeq, rad ω3, rad/s Vt,m/s θeq, rad

−2 −2 287 4.2 0 287 4.2 0

−4 −4 308 4.3 0 296 4.8 0

−6 −6 341 4.7 0 307 5.5 0

Table 4

Generalparametersofthepararotorcases.

Parameter Value

CLα 1.35

CD 0.15

ρ, kg/m3 _1.21

S, m2 _0.0254

r11,r21, m 0.037

r12,r22, m 0.016

β1,β2, rad 0.07

ϕ0, rad 0.00

θ0, rad 0.10

ψ0, rad 0.00

nelwherea pararotorwas testedinastableflightcondition. The experimental conditions are described by NadalMora et al.[28]. The testedmodels consisted ofa hollow cylinder withtwo rect-angular blades of bare aluminum alloy, named in [28] withthe code C05P02, with k31

=

0. The following parameters havebeen

measured:thespinvelocity,measuredbymeansofastroboscopic lamp(resolution50 rpm);thebladepitchangle,witha goniome-ter(resolution 1 deg); theflow velocity by a standardpitot-tube NPLtypeandamicromanometer(resolution0.5 Pa).Acomparison betweenexperimentsandnumericalsimulationresultsareshown inTable 3.

InTable 3agoodfitbetweenexperimentsandsimulationscan be observed, for the case where k31

=

0 and the Straight

Solu-tionispresented.Thenutationanglefortheequilibriumpoint,

θ

eq,

presentsnodifference betweenexperimentalresultsand numeri-cal simulation.The descent rate, Vt,displays differencesfrom0%

to11%.Thespinvelocity,

ω

3,exhibitsdifferencesfrom0%to15%.

Tohavemore conclusive results overthe simulationapproach topararotorflight,freeflighttests, orverticaltunnel tests,should be performed extensively,considering differentpararotors config-urations, which are beingcarried out. Transient response should alsobecontrastedwithexperimentalresults.

3. Results of the numerical model

In the present section four groups of cases are considered regarding the relationships between principal inertia moments: A

(

I3

>

I2

,

I1

)

,B

(

I3

<

I

,

I1

)

;C

(

I1

>

I3

>

I2

)

andD

(

I2

>

I3

>

I1

)

.

The groups of cases are the same as those presented in previ-ous studies [17]; they correspond tothe pararotorconfigurations thatcanbebuilt.OnlyCaseAresultsarepresented.Thegroupsof caseswere chosen inorderto explorethedynamical response of thepararotorpresentedby PiechockiandNadalMora[9,10].Each group includescasevariants toexplore the effectofthe distance betweenthecenterofmassandthebladeplane,andthepitch an-gles,onthepararotordynamics.

CasesAandBarepararotorconfigurationswheretheprincipal momentofinertia I3 isthehighestandthesmallest,respectively.

Cases C and D are configurations where I3 has an intermediate

valuebetweenI1 andI2.

The physical parameters of the baseline configuration consid-eredarepresentedinTable 4.Thisbaselineconfigurationissetto attainpossibletechnologicalinterestofpararotoruse.Theanalysis

Table 5

Principalmomentsofinertia.

Variable Case A Case B Case C Case D

I1, kg m2 _5.4·10−6 _6.0·10−6 _22.0·10−6 _21.0·10−6

I2, kg m2 _21.2·10−6 _22.0·10−6 _6.0·10−6 _23.0·10−6

I3, kg m2 _25.9·10−6 _16.0·10−6 _16.0·10−6 _15.0·10−6

ofthedynamicsisfocusedmainlyontheevolutionofthenutation angle.

Theinitialconditionsforthenumericalsimulationsweregiven by:

ω0

= [

ω

10

,

ω

20

,

ω

30

]

T

= [

0

.

00 rad

/

s

,

0

.

00 rad

/

s

,

ω

eq

]

T.

In all cases,the total duration of the numerical simulation is set insuch awaythat theequilibriumsolution iscompletely es-tablished.Themoments ofinertiavaluesaregiveninTable 5.

3.1. CaseA(I3

>

I2,I1)

ThepresentcaseconsiderstheconditionofI3beingthebiggest

momentofinertia.

Analytical results ofthe simplified analytical method are pre-sented together with the numerical simulation results. The ana-lytical results were found usingthe expressions ofthesimplified linear analytical model developed by Piechocki, Nadal Mora and Sanz Andrés [13]. The differences between numerical simulation andanalyticalresultsareconsideredtobeduetothehigherorder dynamics effects andnon-linear behavior. The numerical simula-tionconsidersanumberofeffectsneglectedonthesimplified ana-lyticalmodel,thatextendsthevalidityrangeandthetoolstrength. These neglected effects on the simplified model are originated fromtruncating termsintheaerodynamic forcesandmomentum equations, considering only the terms of highest magnitude for what was considered a general configuration [10]. Also, on the simplified analytical model,the linearizationstrategy followed to generatea coefficientmatrix inordertoevaluate thesystem sta-bilityimpliesneglectingnonlinear effects.

Thefirstsetoftests(Table 6)showthatforcaseswherek31

=

0

thereisonlyoneequilibriumsolutiontype,characterizedbya nu-tation angle,

θ

, which is 0or is very close to 0 radians (axis of rotation aligned or almost aligned with axis 3 in the body ref-erenceframe). Thisfamilyofequilibriumsolutions includessmall angular velocities

ω

eq1,

ω

eq2 intheorder of1rad/s. For

simplic-ity this solution is named Straight Solution, or Type 1 Solution. The results ofthe simulations andof the linearanalytical model arealsoshowninTable 6.Itcanbenotedthatbothwaysof com-puting the equilibrium solutions are coherent if the equilibrium solution reached is the Straight one. In this case relative errors betweenvariablesare small,increasing asthedifferencebetween

β

1 and

β

2 increases(evidencingtheinfluenceofnonlinearterms).

Finally,itcanbe notedthattheanalyticalmodeltendsto system-aticallyoverestimatethemagnitudeofthevariablesincomparison with the numerical simulation,when the equilibrium solution is a Straight Solution.These resultsare verifiedfor simulationtests performed fork31

=

0

.

0.

In Table 7 the existence oftwo equilibrium solution types is shown. One of these solutions, for example for

β

2

=

0

.

07 (with

β

1

=

0

.

07), showsa nullor a verysmall nutationangle, andthe

Table 6

CaseA–equilibrium resultsfromnumericalsimulationandanalyticalmodelfork31=0.

Bladespitch Numerical simulation Analytical model

ωeq1, rad/s ωeq2, rad/s ωeq3, rad/s Vt, m/s θeq, rad ωeq1, rad/s ωeq2, rad/s ωeq3, rad/s Vt, m/s θeq, rad

β1=0.07;β2=0.07 0.0000 0.0000 292.8 4.57 0.000 0.0000 0.0000 292.8 4.57 0.000 β1=0.07;β2=0.09 0.0303 0.0581 294.0 4.66 2.229·10−4 _{0.0327 0.0618 294.0 4.66 2.377·10}−4 β1=0.07;β2=0.14 0.1072 0.2056 297.2 4.88 7.802·10−4 _{0.1165 0.2198 297.2 4.88 8.370·10}−4 β1=0.07;β2=0.19 0.1856 0.3561 300.2 5.11 1.338·10−3 _{0.2030 0.3830 300.2 5.11 1.444·10}−3 β1=0.07;β2=0.21 0.2174 0.4170 301.4 5.21 1.560·10−3 _{0.2384 0.4497 301.4 5.21 1.689·10}−3 β1=0.07;β2=0.25 0.2816 0.5403 303.7 5.41 2.006·10−3 _{0.3106 0.5859 303.7 5.41 2.184·10}−3

Table 7

CaseA–equilibrium resultsfromnumericalsimulationandanalyticalmodel−k31=0.7.

Bladespitch Numerical simulation Analytical model

ωeq1, rad/s ωeq2, rad/s ωeq3, rad/s Vt, m/s θeq, rad ωeq1, rad/s ωeq2, rad/s ωeq3, rad/s Vt, m/s θeq, rad

β1=0.07 0.0000 0.0000 292.8 4.57 0.000 0.0000 0.0000 292.8 4.57 0.000

β2=0.07

β1=0.07 0.0322 0.0184 294.0 4.66 1.250·10−4 _{0.0298 0.0170 294.0 4.66 1.167·10}−4 β2=0.09

β1=0.07; 0.0650 0.0355 295.3 4.75 2.490·104 _{0.0599 0.0330 295.3 4.75 2.316·10}−4 β2=0.11

β1=0.07 0.0750∗ 0.0375∗ 295.0∗ 4.77∗ 0.117∗∗ 0.0643 0.0349 295.0 4.77 2.481·10−4 β2=0.113 (3.025∗∗) (2.908∗∗) (0.007∗∗) (0.007∗∗) (0.005∗∗)

β1=0.07; 0.0827∗ 0.0400∗ 294.7∗ 4.81∗ 0.156∗ 0.0747 0.0396 294.7 4.81 2.870·10−4 β2=0.12 (4.017∗∗) (3.840∗∗) (0.010∗∗) (0.011∗∗) (0.007∗∗)

β1=0.07; 0.1000∗ 0.0534∗ 295.1∗ 4.87∗ 0.199∗ 0.0898 0.0463 295.1 4.87 3.423·10−4 β2=0.13 (5.120∗∗) (4.917∗∗) (0.019∗∗) (0.021∗∗) (0.009∗∗)

β1=0.07 0.1000∗ 0.0650∗ 295.2∗ 4.93∗ 0.236∗ 0.1048 0.0524 295.2 4.93 3.968·10−4 β2=0.14 (6.100∗∗) (5.825∗∗) (0.073∗∗) (0.0315∗∗) (0.010∗∗)

β1=0.07 0.2250∗ 0.1075∗ 295.0∗ 5.21∗ 0.385∗ 0.1793 0.0753 295.0 5.21 6.593·10−4 β2=0.19 (9.875∗∗) (9.463∗∗) (0.075∗∗) (0.081∗∗) (0.017∗∗)

β1=0.07 0.2500* 0.1200∗ 294.7∗ 5.33∗ 0.437∗ 0.2089 0.0809 294.7 5.33 7.601·10−4 β2=0.21 (11.150∗∗) (10.710∗∗) (0.095∗∗) (0.105∗∗) (0.019∗∗)

β1=0.07 0.3750∗ 0.1750∗ 293.9∗ 5.55∗ 0.539∗ 0.2677 0.0863 293.9 5.55 9.569·10−4 β2=0.25 (13.730∗∗) (13.125∗∗) (0.144∗∗) (0.156∗∗) (0.024∗∗)

smallerthan1 rad/s.Thissolutionisqualitativelyidenticaltothat foundfork31

=

0,theStraightSolution.Theothersolutionappears

inCase Afor k31

≥

0

.

7 or fork31

≤ −

1

.

0 (Table 9). Thisclass of

solutionis not predictedby the analytical model,andrepresents a bifurcation of the solution. This solution type is characterized by:1) nutationanglesconsiderablylarger thantheprevious case, 2)asinusoidalperiodicbehaviorof

ω

eq1,

ω

eq2 withan amplitude

ofconsiderablemagnitude;and3)asinusoidalfluctuationof

ω

eq3

showingsmallamplitude. Thissolutionishereafternamedasthe ConicalSolution, orType2Solution because the angularvelocity vector,

ω

, moves along a cone with elliptical cross-section with smallexcentricity.Simulationtestsfork31

>

0

.

7 showed astrong

tendencytoconicalsolutions.

As shown in Fig. 4, the equilibrium solution of the numeri-calsimulationshowsabifurcationfromtheStraightSolution ata givenlimitofthepitchangle,

β

2,inthecasek31

=

0

.

7.Ifthislimit

issurpassed, the Conical Solutionappears abruptly. The nutation angle,

θ

eq,forConicalSolution(

θ

eq

>

5◦)issignificantlylargerthan

forStraight Solution (

θ

eq

≈

1◦). These large angles could reduce

theprediction capacityofthe aerodynamicsmodel.To substanti-atethispointa campaignofexperimentationwouldberequired.

Forsimplicity,henceforthwewillcallCaseAtheBaseCase.

Table 8summarizesthetimeresponseandequilibriumnutation angle, with different k31 values. In these calculations, the

inte-gration time was increased until the nutation solution remained boundedbetweena

±

0

.

1 degreerange.

Fig. 4.Bifurcationdiagram.Variationofthenutationangleatequilibrium,θeq,with blade2pitchangleβ2,forβ1=0.07 andk31=0.7.

Thesimulationsperformedshowedthatwhenk31

= −

10,

nuta-tionanglesarehigherthan90 deg.Thismeansthatthepararotor turnsupsidedown, anda stablesolution(thatcanbe considered anomalousfromapracticalpointofview)isreachedwithnutation anglesofconsiderablemagnitudeinthatattitude,asintheConical Solutioncase.ThissolutionisdenotedType3Solution.

Table 8

Timeneededtoreachtheequilibriumpoint,tre,fordifferentpararotorcasesovertheCaseA.

Case β1=0.01,β2=0.01 β1=0.07,β2=0.07 β1=0.07,β2=0.14

tre, s θe, deg tre, s θe, deg tre, s θe, deg

k31=100 4.20 0.0T1 3.9 0.0T1 2.6 0.0T1

k31=10 12.0 45.8T2 12.1 45.8T2 16.2 40.7T2

k31=1.0 63.2 34.3T2 49.9 34.3T2 44.4 38.7T2

k31=0.5 30.1 0.0T1 32.2 0.0T1 37.5 0.0T1

k31=0.0 7.8 0.0T1 8.1 0.0T1 8.5 0.0T1

k31= −0.5 6.8 0.0T1 7.9 0.0T1 9.6 0.0T1

k31= −1.0 14.3 0.0T1 24.1 0.0T1 47.5 0.0T1

k31= −2.0 43.5 39.2T2 28.2 39.2T2 23.5 45.4T2

k31= −10 7.7 132.1T3 10.6 128.3T3 11.0 126.1T3

Table 9

Effectonthedynamicbehavior ofthepararotor(timetoreachequilibrium,tre)oftheparametersk31,k21,I1/I3,CLα/CD.

k31 k21 I1/I3 CLα/CD Equilibrium solution Results

0 Increase Type 1 treincreases.

0 Decrease Type 1 tredecreases.

0 Increase Type 1 treincreases.

0 Decrease Type 1 tredecreases.

0 Increase Type 1 treincreases.

0 Decrease Type 1 tredecreases.

>0 Type 1 or 2 treincreases. Nutation angle increases.

>0 Increase Type 1 tredecreases a second order magnitude.

>0 Increase Type 2 treincreases. Nutation angle decreases.

>0 Decrease Type 1 tredecreases.

>0 Decrease Type 2 treincreases. Nutation angle decreases.

>0 Increase Type 1 or 2 treincreases. Nutation angle increases to instability.

>0 Decrease Type 1 or 2 tredecreases.

>0 Increase Type 1 or 2 treincreases. Nutation angle increases to instability. >0 Decrease Type 1 or 2 tredecreases. Nutation angle decreases.

<0 Type 1, 2 or 3 treincreases.

<0 Increase Type 1, 2 or 3 tredecreases.

<0 Decrease Type 1, 2 or 3 treincreases to instability.

<0 Increase Type 1, 2 or 3 tredecreases.

<0 Decrease Type 1, 2 or 3 treincreases to instability. <0 Increase Type 1, 2 or 3 treincreases to instability.

<0 Decrease Type 1, 2 or 3 tredecreases.

Fig. 5.Variationofthenutationangle,θ,withtime,t,fordifferentvaluesofthe positionparameterk31.Initialnutationangleissetasθ0=0.1 rad.SuperscriptsT1, T2,T3denotetheequilibriumtype.

From theanalysesof casesshown,the modelbehaviorcan be summarizedby usinga certainnumber ofparameters,whose in-fluencepatternsareshowninTable 9.

Itcanbeconcludedthatfor I3

>

I2,I1 fourdifferentsolutions

canbe reached:Straight,Conical,Type3andDynamicinstability.

For k31

=

0 only two possibilities exist: Straight solution or

Dy-namicinstability.

The effect ofk31 (if different from0) depends on its

magni-tude andsign, geometrical configuration, and aerodynamics per-formance.Theeffectcanbedescribedintermsofthestabilization timeandofthesolutiontype.

Conical solutionassumes a nutation angle significantly bigger than 0.Aconsiderableincrease ink31 tendsto diminishthe

sen-sibility of the pararotor of destabilizing factors. However, if the sign ofk31 is negative andits magnitudeis significant,the

solu-tiontendstobeType3.

Withregard to the effectofincreasing k21,fork31

=

0, it

im-pliesanincreaseinstabilizationtime.Ifk31

>

0,whenthesolution

isStraight,thereisavalueofk21thatmakesthestabilizationtime

maximum. When the solution is Conical, there is a value of k21

that makesthestabilizationtime minimum.Itcan beseenthatif

k31

<

0,increasingk21leadstoadecreaseinstabilizationtime.

Regarding I1

/

I3 and I2

/

I3 (in thecase I3

>

I2

>

I1) it can be

observed thatindependentlyofthevalue ofk31,theinfluenceon

the dynamic response is given by the relationship between the biggestandsmallestinertiamoments.

The Case A simulations indicate that an increase in I1

/

I3 for k31

=

0 impliesanincreaseintransienttime,uptoinstability.

Besides, theresultsshow thatan increase in I1

/

I3 fork31

>

0

implies an increase in transient time, up to instability, going through equilibrium solutions Straight and Conical. Also, if I1

/

I3

diminishes,fork31

>

0,thenasmaller transienttimeappearsthan

Finally,itwasalsoshownthatanincrementofI1

/

I3fork31

<

0

impliesanincrementof transienttime,uptoinstability.Also,that if I1

/

I3 diminishes, an absolute minimumtransient time can be

found,relatedwiththemomentsofinertia.

WithregardtotheparameterCLα

/

CD,itwasshownthat

inde-pendentlyofk31value,itsincrementimpliesagrowthontransient

time.

4. Conclusions

Thenumericalmodeldevelopedtoanalyzethestability ofthe motionhasbeenvalidatedby twocomparisons: 1) between ana-lyticalmodelresultsandnumericalsimulation,and2)between ex-perimentalresultsandnumericalsimulations.Takingintoaccount thisvalidationapproach,the numericalsimulation developedcan beconsidered tobe ausefulandreliable toolto describe pararo-torflight.Transientbehaviorshouldbevalidatedwithafreeflight testcampaign.

Fromtheobservationsmade,itcanbenotedthatfoursolution types are possible, denoted as Straight, Conical, Type 3 and Dy-namicinstability. From these solutions, when k31

=

0,numerical

simulationsshowedthatthepossibilities arereducedtosolutions Straight and Dynamic instability. Therefore, it can be said that therearenewphenomenaderivedfromtheexistence ofnon-zero valuesofk31that introduceConicalandType3equilibrium

solu-tions,alsodenotedasconicalsolutions.Itisworthnotingthatthe analyticalapproachisabletopredictaccuratelyenoughthe pararo-tordynamicresponse whenk31

=

0.Thedifferencesbetween

nu-mericalsimulationandanalyticalapproachcomefromthehigher orderdynamicseffectsandnon-linearbehaviorconsidered inthe simulationmodel.

Thishigherorderdynamicseffectsandnon-linearbehavior ap-pear whenthemodelflights nearthestability limitsdescribedin

[17].Thissimulationmodelallowsforconsiderationoftheeffectof notsmallanglesofattack,oftheexistence ofacomponentalong axis1oftherelativevelocity, ofexistence ofadistancefromthe aerodynamiccenteroftheblades tothecenterofmassinthe di-rectionperpendicularto thebladespan,andofangularvelocities alongaxis1and2,

ω

1 and

ω

2,notsmallcomparedwith

ω

3.

Thevariationofk31mainlyaffectstostability.Inthecasea

sta-bleconfigurationisgiven, k31 hasan impactonthetype of

equi-libriumsolutionandonstabilizationtime.Thestabilityisgivenby theratioofprincipalmomentsofinertiaasitwas determinedon theworkabovementioned[10].

While Straight Solution is characterized by nutation angles closeto zero,inConical solution a significant nutationangle ap-pears, configuring a conical motion of the rotation axis. Fork31

differentfromzero,thesystemshowsabifurcation fromStraight SolutiontoConical, dependingonthepararotorconfiguration,the aerodynamiccharacteristicsofthedevice,andk31magnitude.

Inanycase, it can be said that a significant incrementofk31

magnitudetendstoreducethesensitivityofthepararotorto desta-bilizingfactors.

The general conclusion is that the parameters k31, CLα

/

CD,

β

1 and

β

2 can be used to control the dynamic behavior of the

pararotor.Both,thestabilizationtimeofthesystemtoagiven per-turbationand the type of equilibrium solution,depend on these parameters.So,itispossibletocontrolthistypeofdeviceby ma-nipulatingtheseparameters.

Ingeneral,itcanbesaidthatthedistancefromtherotortothe centerofmass(k31indimensionlessform)isastabilizing

param-eterthatcanleadtothebifurcationtoconicalsolutions.

5. Discussion

Inthispaper,theeffectofdifferentconfigurationparameterson stability and stabilization time of pararotorshas been presented. Twokindsofstableresponsewerefound:straight andconical so-lutions. The numerical results obtained show a good agreement withthetestmeasurementsdonein previousworks[12,19].

Future investigations should be conducted to explore regions ofthe N_e,ke plane to define thestability regions limits.In-flight

testsshouldalsobedoneinordertocheckresults,tocomplement thewindtunneltests.Moreover,futurestudiesshouldinclude con-trolstrategiesandguidingdevices.Finally,anexperimental effort shouldbe devotedtocharacterizebladeaerodynamicsinorderto completetheaerodynamicloadingmodelofpararotorblades.

Conflict of interest statement

Nonedeclared.

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