Trchalík, J. and Gillies, E.A. and Thomson, D.G. (2008) Development of an aeroelastic stability boundary for a rotor in autorotation. In: AHS Specialist's Conference on Aeromechanics, 23-25 January 2008, San Francisco, USA. http://eprints.gla.ac.uk/4967/ Deposited on: 01 April 2009

Enlighten – Research publications by members of the University of Glasgow http://eprints.gla.ac.uk

Development of an Aeroelastic Stability Boundaryfor a Rotor in Autorotation

J Trchalík⋆, Dr E A Gillies†, Dr D G Thomson‡

⋆ PhD Research Studentemail: jtrchali@aero.gla.ac.uk

† Senior Lecturer,email: ericg@aero.gla.ac.uk

‡ Senior Lecturer,email: d.thomson@aero.gla.ac.uk

Tel.: +44 141 330 3575Fax: +44 141 330 5560

Department of Aerospace EngineeringUniversity of GlasgowGlasgow, G12 8QQ

Scotland

Keywords - autogyro, aeroelasticity, rotor, rotor blade, rotorcraft, stability, axial flight, forward

flight, FEM, MATLAB

January 9, 2008

Abstract

For the present study, a mathematical modelAMRA was created to simulate the aeroelasticbehaviour of a rotor during autorotation.Our model: Aeroelastic Model of a Rotor inAutorotation (AMRA) captures transversebending and teeter, torsional twist and lag-wisemotion of the rotor blade and hence it is usedto investigate couplings between blade flapping,torsion and rotor speed. Lagrange’s methodwas used for the modelling of blade flappingand chord-wise bending. Torsional twist of therotor blade was modelled with the aid of finiteelement method (FEM), and blade transversebending could also be modelled in FEM. Themodel can switch between using a full FEMmodel for bending and torsion, or a FEM modelfor torsion and simple blade teeter, dependingon the complexity that the user requires.

Presented at the AHS Specialist’s Conference on

Aeromechanics, San Francisco, CA, Jan. 23-25, 2008.

Copyright 2008 by the American Helicopter Society In-

ternational, Inc. All rights reserved.

The AMRA model was verified against exper-imental data obtained during a CAA sponsoredflight test programme of the G-UNIV autogyro.Published results of modal analysis of helicopterrotor blades and other data published in openliterature were used to validate the FEM modelof the rotor blade. The first torsional naturalfrequency of the ’McCutcheon’ rotor blades wasmeasured with the aid of high-speed cameraand used for validation of the FEM model ofblade torsional twist. As a further verificationof the modelling method, Aérospatiale Pumahelicopter rotor blade data were compared ona Southwell plot showing comparison betweenexperimental results and AMRA estimation.

The aeromechanical behaviour of the rotorduring both axial flight and forward flight in au-torotation was investigated. A significant partof the research was focused on investigation ofthe effect of different values of torsional and flex-ural stiffness, and the relative positions of bladeshear centre/elastic axis and centre of mass ofthe blade on stability during the autorotation.

1

The results obtained with the aid of the modeldemonstrate the interesting, and unique, charac-teristics of the autorotative regime - with insta-bilities possible in bending and torsion, but alsoin rotorspeed. Coupled rotor speed/flap/twistoscillations (flutter and divergence) occur if thetorsional stiffness of the blade is lower than acritical value, or if the blade centre of mass issignificantly aft of the blade twisting axis, as isthe case in helicopter pitch-flap flutter. The in-stability shown here, however, is specific to theautogyro, or autorotating rotor, as it is cou-pled with rotorspeed, and so differs from bothhelicopter rotor flutter and fixed-wing flutter.The coupling with rotorspeed allows a combinedflutter and divergence instability, where the ro-tor begins to flutter in rotorspeed, teeter angleand torsional twist and, once the rotorspeed haddropped below a critical value, then moves intodivergence in flap and rotorspeed. It was foundthat the aeroelastic behaviour of a rotor in au-torotation is significantly affected by the strongcoupling of blade bending stiffness and teeter an-gle with rotorspeed, and the strong coupling be-tween blade aeroelastic twist and rotor torque.

Nomenclature

[Λ] Dynamic inflow static gain matrix [-]

[τ ] Time constant matrix [-]

αD Angle of attack of the rotor disk [rad]

χ Wake skew angle, χ ≈ arctan µx

λ[rad]

γ Angle of climb of the vehicle [rad]

α Parameter of mass matrix of blade bend-ing FEM [1]

µ Blade weight per length [kg/m]

w Test (weighting) function [1]

ι Rotor disk longitudinal tilt, i.e. anglebetween rotor disk plane and horizontalplane [rad]

µx Advance ratio defined parallel to rotor

disk plane, µx =

√Vx+Vy

ΩR[1]

µz Advance ratio defined perpendicular torotor disk plane, µz = Vz

ΩR= λ− λi [1]

AB Sum of additional flexural forcing terms[N]

AT Sum of additional torsional forcing terms[N]

C0 Apparent mass factor, C0 = 1 for Pitt-Peters dynamic inflow model [1]

H Blade shape function [1]

li Length of i-th rotor blade element, li =ri+1 − ri [m]

Nelem Number of blade span-wise elements [1]

q Torsional aerodynamic loading perlength [N]

t Blade thrust per length [N/m]

um Mass flow parameter [1]

vt Total velocity at the rotor disk centre[m/s]

vic Longitudinal component of induced ve-locity [m/s]

vis Lateral component of induced velocity[m/s]

xi Span-wise dimensionless coordinate of i-th rotor blade node, ri = Rxi [m]

1. Introduction

The autogyro represents the first successfulrotorcraft design and it paved the way forthe development of the helicopter during the1940s. Further development of the autogyrowas ceased during the following decades ashelicopters became more successful. Interestin the autogyro as a recreational vehicle wasresurrected in recent years thanks to simplicityof its design and low operational costs. Itswider use in military and civil applications isalso currently under investigation.

Autogyros use two-bladed teetering rotorsfor generation of lift. Unlike in helicopters,the rotor is not powered by an engine butrotor torque is generated by the aerodynamicforces and the rotor has to be pre-rotatedbefore take-off. Autogyros do not need a tailrotor as there is no torque acting on theirfuselage. Longitudinal and lateral tilt of therotor disk is used for longitudinal and lateralcontrol of the vehicle. Most of autogyros usethe combination of two-stroke or four-strokeengine and a propeller in pusher configuration,for thrust.

Unfortunately, autogyros in the UK have beeninvolved in number of fatal accidents during thelast two decades.1 Very little data on autogyroflight mechanics and handling qualities wereavailable in the literature at the time, which ledthe UK Civil Aviation Authority (CAA) and theDepartment of Aerospace Engineering, Univer-sity of Glasgow, to investigate the aerodynam-ics and flight mechanics of an autogyro.1–4 The

2

cause of some of these accidents still remains un-clear and rotor aeroelasticity has not yet beeninvestigated as a contributing factor. The aeroe-lastic behaviour of a rotor in autorotation is arelatively unexplored problem, and the aim ofthis investigation is to identify flight conditionsor configurations of the rotor that might causeinstabilities, and explore the effects of aeroelas-ticity on flight mechanics models of autogyros.

2. Aeroelastic Model of a

Rotor in Autorotation(AMRA)

There are substantial differences between theaerodynamics of a helicopter rotor and of a ro-tor in autorotation. During autorotation, bothtorque and thrust are generated exclusively byflow through the rotor disc, thus the rotor hasone extra degree of freedom. Thrust and torqueare functions of rotor speed and distributionof local angles of attack along the blade span.Further, angles of attack are dependent uponblade twist, rotor speed, speed of descent andinduced velocity. It can be easily shown thatboth speed of descent and rotor angular velocityare strongly dependent upon rotor torque androtor thrust. This makes modelling of rotoraeromechanics during autorotation relativelychallenging.

During steady autorotation, the overalltorque generated by flow through the rotordisc balances the profile drag and rotor thrustis equal to the weight of the vehicle.5,6 Thereare several design parameters of the rotorthat determine whether steady autorotationis possible. The most important are bladeincidence angle (i.e. angle of attack of theblade relative to the rotor disc plane) and bladetorsional stiffness. Torque equilibrium can notbe achieved for high incidence angles due to thehigh value of blade drag. If torsional rigidityis too low, extensive blade twist has the sameeffect. The extra degree of freedom in rotorspeed has significant implication for autogyrorotor stability. Decrement of the rotor speeddecreases centrifugal stiffness of the rotor andthe resulting higher deflections in flap and twistgenerate more drag and may cause further dropin rotor speed.

The AMRA model was developed with the aidof MATLAB programming language. A bladeelement method combined with quasi-steady orunsteady aerodynamics (Theodorsen’s theory)is used for calculation of aerodynamic forcesand moments generated by the rotor blade.

Two rotor blades and arbitrary number of bladespan-wise elements can be used. The aerody-namic characteristics of the aerofoil for the fullrange of angles of attack are approximated withthe aid of wind tunnel data.7 A NACA 0012aerofoil was chosen for the first version of theAMRA model since aerodynamic characteristicsfor the full range of angles of attack of theaerofoil are available.8 A semi-empirical methodof induced velocity calculation was used in thefirst versions of the AMRA model. The originalcalculation9 was improved in order to captureblade stall and compressibility of the airflow.A simplified version of Peters - HaQuanginflow model modified by Houston and Brown2

replaced semi-empirical approach in the laterversions of the AMRA model in order toimprove fidelity of forward flight simulations.

Lagrangian equations of motion were used todescribe dynamics of the rotor blade. Chord-wise locations of elastic axis (EA), centre ofgravity (CG) and aerodynamic centre (AC) canbe set in each span-wise station. Values of flex-ural and torsional rigidity of the blade can beset to investigate the behaviour of the rotor fordifferent physical properties of the blades. TheAMRA model also allows placement of singleconcentrated mass at any span-wise station ofthe blade.

2.1. Modelling of Rotor

Aerodynamics in Autorotation

The blade element method represents a widelyused tool for description of the flow through a ro-tor disc. The theory has to be modified in orderto capture the aerodynamics of a rotor in autoro-tation. During autorotation, the flow throughthe rotor has the opposite direction that in thecase of powered flight of a helicopter. Hence theblade aerodynamic angle of attack can has to beexpressed as9–12

α = φ+ θ (1)

Since autogyros use longitudinal and lateralrotor disk tilt for control, equations describinginflow velocity components have to be modifiedtoo. Inflow velocity is a function of angle of at-tack of the rotor disc that is given by sum ofincidence angle of the rotor disc (i.e. angle be-tween rotor disc plane and the horizontal plane)and pitch angle of the vehicle (see equation 2).Rotor disc angle of attack is 90deg during axialflight.

αD = ι+ γ

γ = atan

(

Vd

Vh

)

(2)

3

An assumption of a linear lift curve andparabolic drag curve is often used in order tosimplify the model and speed up computations.However, this approach does not allow theeffects of blade stall, drag divergence and mostimportantly air compressibility to be captured.Since the aerodynamic characteristics of rotorblade elements depend upon local values ofangle of attack and Mach number, it is con-venient to express aerodynamic characteristicsof the blade airfoil as functions of these twovariables. This can be achieved by expressingaerodynamic characteristics of the blade interms of polynomial functions of angle of attackand Mach number.5

It was shown by Prouty7 that it is possibleto obtain full-range angle of attack aerodynamicdata of an airfoil with the aid of polynomial fit.Prouty uses the example of the NACA 0012 air-foil in his book7. Prouty’s empirical equationswere derived from data published in Carpen-ter13. Since full-range AOA aerodynamic datafor the same airfoil are available from numer-ous sources8,14, Prouty’s approach was amendedand incorporated into the AMRA model.

2.2. Modelling of Dynamic Inflow

During Autorotation

Dynamic inflow models developed by Pittand Peters, Gaonkar and Peters, Peters andHaQuang and Peters and He represent one ofthe most popular dynamic inflow models. Mod-ern three-state dynamic inflow models can bedefined in the following form15,16

[τ ]

vi0

vis

vic

+

vi0

vis

vic

= [Λ]

TLR

MR

(3)

Peters - HaQuang inflow model was enhancedby Houston and Brown17 in order to capture in-flow of a rotor in autorotation. The time matrixand dynamic inflow static gain matrix can bewritten for a rotor in autorotative flight regimein the following form

[τ ]=

2

6

6

6

4

4R3πvtC0

0−R tan

χ2

12um

0 64R45um(1+cos χ)

05R tan

χ2

8vt0 64R cos χ

45um(1+cos χ)

3

7

7

7

5

(4)

and

[Λ]= 1ρπR3

2

6

6

6

4

R2vt

015π tan

χ2

64um

0 −4um(1+cos χ)

015π tan

χ2

64vt0 −4 cos χ

um(1+cos χ)

3

7

7

7

5

(5)

Total induced velocity at azimuth angle ψ andradial station x then is17

vi = vi0 + vicx cosψ + visx sinψ (6)

Although a full dynamic model with threedegrees of freedom was incorporated into theAMRA, its simplified version was used in orderto reduce computational time. From the systemof equations (3), only first equation is used in thesimulation as the remaining two components ofinduced velocity can be neglected.6,18 This mod-ification decreases computing time and reducescomplexity of the model significantly. The equa-tion below shows solution for the rate of changeof vertical component of induced velocity.

vi0=−

3C0

(

2πρR2vi0

s

V 2x +V 2

y +V 2z +

r

T

πρR2

„

1√

2−

√

2Vz

«

−T

)

8ρR3

(7)

Matrix equation 3 then becomes

vi =

∫

vi0dt (8)

2.3. Rotor Blade Physical

Properties

Since the majority of autogyro rotor bladesare manufactured by small private companies,it is difficult to get any information on theirstructural properties. A pair of McCutcheonblades were subjected to a series of experimentsin order to assess their physical propertiesand mass distribution. Data gathered duringthe experiments were used as input values ofthe simulations and also for validation of themodel of rotor blade dynamics. The majorityof experimental measurements and their resultsare described by the authors elsewhere19

The data obtained during the experimentalmeasurements are summarized as follows:elastic axis position is located between 35.5%(inboard) and 27.24% (outboard) of bladechord; position of the blade inertial axis isaround 30% of blade chord (inboard) and itgrows to 36% towards the tip of the blade.Mass per length of the blade is 4.25kg/min the root part of the blade and averages2.7kg/m outboards. Torsional stiffness of theblade changes linearly along the blade spanfrom 1534Nm2/rad to 1409Nm2/rad. Bladeflexural stiffness was estimated with the aid ofmeasurement of first bending natural frequencyof the blade. The value of transverse stiffnessis 1166Nm2 and it was assumed it is constantalong blade span.

4

The first torsional frequency of the rotor bladewas estimated as f1T = 34.8Hz.

3. Modelling of Rotor Blade

Dynamics

The AMRA model can use Lagrange’s equa-tions of motion or the finite element method(FEM) for modelling of rotor blade structuraldynamics in bending, teeter and torsion. Al-ternatively, combinations of both approachescan be used. Thus allowed comparison ofdifferences between results obtained with theaid of models with different levels of complexity.In Lagrange’s method, the blades are assumedto be perfectly rigid and blade stiffness ismodelled with springs located at the root ofeach blade. Bending and torsional deflectionsare assumed to be constant along the bladespan. This requires less computational timebut it does not provide a full picture of thedynamics of the system since it is significantlysimplified. Finite element analysis (FEA) ofcoupled bending-torsion of the blade on theother hand is highly complex and requiressignificantly higher computational time.

Lagrange’s method is an elegant way of ob-taining the equations of motion and it is usefulfor modelling of complex dynamics of rotorcraftrotor blades. In order to use of span-wisedistributions of blade physical properties andalso allow coupling of Lagrange’s equations ofmotion with FEA of blade dynamics, each rotorblade is discretized into a number of lumpedmasses. Lagrange’s equations of motion aregenerated for each lumped mass. Linearizedand simplified versions of equations of motionfor coupled torsion-flap of a rotor blade arepublished in the open literature20.

Galerkin’s method of solution is used in theAMRA model. The model can perform dy-namic FEA of blade torsion or blade bendingfor teetering rotors. Alternatively, FEA of cou-pled bending-torsion dynamics with inclusion ofteeter can be used. A 1D FEA was used in or-der to reduce complexity of the model. Eachblade element has two nodes and number of de-grees of freedom differs from one (torsion only)to three (torsion with flap-wise bending). Differ-ential equation of blade torsion can be writtenin following form21

∂w

∂rGJ

∂θ

∂r+ wixθ + wAT − wq = 0. (9)

The term AT represents sum of terms thatrepresent the effect of coupling of blade torsional

dynamics with degrees of freedom in flap androtation.

Although solution of the differential equationof blade torsion with the aid of FEM does not re-quire shape functions of higher order and linearshape functions can be used, cubic shape func-tion was chosen for modelling of blade torsionin AMRA. Cubic shape function is defined asfollows.21

H1 = 3 (xi+1 − x)2 − 2 (xi+1 − x)

3

H2 = 1 −H1 = 3 (x− xi)2 − 2 (x− xi)

3(10)

Corresponding mass and stiffness matricesand forcing vector are as follows21

[Ki] = GJ

[

−65li

65li

65li

−65li

]

(11)

[Mi] = ix,i

[

13li35

9li70

9li70

13li35

]

(12)

fi = fi

li2li2

(13)

Alternatively, diagonal (or diagonallylumped) mass matrix can be used. It speedsup the computations as inversion of this massmatrix is much easier than in case of consistentmass matrix21

[Mi] = ix,i

[

li2 0

0 li2

]

(14)

The finite element model of blade bending hastwo nodes per element but in contrary to theFEM model of blade torsion it requires two de-grees of freedom per node - vertical displacementand flap-wise rotation. Differential equation ofblade bending is shown below.21

∂2w

∂r2EI

∂2w

∂r2+ wµw + wAB − wt = 0 (15)

Again, the term AB,i in the above equationrepresents a sum of all coupling terms .

Unlike FEA of blade torsion, modelling ofblade bending with the aid of FEM requiresshape functions of higher order. Hence, two dif-ferent shape functions have to be used in orderto describe distribution of both vertical displace-ment of blade nodes and slope of blade elementsover a blade element. These shape functionsare called Hamiltonian shape functions and theyare based on the cubic shape function describedin equation (10).21 Application of Hamiltonian

5

shape functions results in the following formsof stiffness matrix, consistent mass matrix andforcing vector.21

[Ki] =EI

l3i

12 6li −12 6li6li 4l2i −6li 2l2i−12 −6li 12 −6li6li 2l2i −6li 4l2i

(16)

[Mi] = µi

li420

156 22li 54 −13li22li 4l2i 13li −3l2i54 13li 156 −22li

−13li −3l2i −22li 4l2i

(17)

fi =fi

12

6lil2i6li−l2i

(18)

A diagonal form of mass matrix is more con-venient for dynamic analysis. Parameter α hasto be a positive number smaller than 1

50 . Kwonet al 21 recommends α = 1

78 , which is used inAMRA.

[Mi] = µili

12 0 0 00 αl2i 0 00 0 1

2 00 0 0 αl2i

(19)

4. Verification of the AMRA

Model

Although the AMRA model is fully functionalstandalone model of rotor aeromechanics in au-torotation, its main purpose was to test modelof blade dynamics in autorotation so that it canbe incorporated into a rotorcraft flight mechan-ics model (i.e. RASCAL22, G-SIM 23 or other).Therefore, the main objective of the validationphase of the project was to make sure that bladedynamics are computed correctly by the AMRAmodel. FEM model of blade torsion representsthe key block of AMRA and extra care was takenduring its validation. Values of teetering an-gle predicted by the AMRA model were verifiedagainst G-UNIV autogyro flight data23. Esti-mations of flight performance during flight in au-torotation were also successfully compared withexperimental data published in open literature.5

4.1. Validation of Model of Rotor

Blade Teeter

Data gathered during CAA sponsored seriesof flight test of the University of Glasgow

Montgomerie-Parsons autogyro (G-UNIV) wereused for validation of model of blade teeter thatis included in the AMRA model.

However, AMRA uses a NACA 0012 airfoilthat has different aerodynamics characteristicsin comparison with NACA 8-H-12 that are usedin McCutcheon rotor blades. In order to reachsimilar flight conditions during simulations (i.e.rotor speed and speed of descent), rotor speedwas set to mean value of rotor speed measuredduring the flight tests. Two different regimesof steady level flight were chosen for the vali-dation. Predictions of the model were found tobe in a good agreement with teeter angles mea-sured during flight trials as it can be seen fromFig. 1 and 2. Table 1 summarizes the results ofvalidation of AMRA model of blade teeter.

4.2. Validation of FEM Model of

Blade Torsion

Deflections of beams of several lengths andwith different torsional stiffness that wereloaded statically by a torsional moment at thetip were computed by the AMRA FEM modelof blade torsion. The results were then com-pared with analytical estimations of beam tiptorsional deflections according to the St. Venanttheory and were found to be in very good agree-ment. Mean relative deviation was less than 2%.

The shape of the first torsional mode pre-dicted by the FEM model was also compared tocorresponding torsional mode shapes that werepublished in open literature.20 Predictions ofthe model are in good agreement with publisheddata. Comparison of the first torsional modeshape computed by the FEM model of bladetorsion with data published in open literatureis depicted in Figure 3.

Since the span-wise distribution of bladetorsion has a strong influence on blade aero-dynamics, it is absolutely crucial that it ismodelled correctly and the aerodynamic forcingof the blade is estimated realistically. Figures 4and 5 depict a qualitative comparison of distri-bution of torque generated by rotor blade overthe rotor disk as predicted by the AMRA modeland qualitative sketch of torque distributionreproduced in open literature.6

Comparisons of the first natural frequencies intorsion and bending of two different rotor bladeswith results of experimental measurements andpredictions of other models of blade dynamicsrepresent another phase of AMRA validation.Data obtained during experimental measure-

6

ments of physical properties of the McCutcheonblade along with comprehensive data on phys-ical properties of Aérospatiale SA330 Pumahelicopter rotor blade were used.24 A Southwellplot of the McCutcheon rotor blade is shownin Figure 6. Although first natural frequencyin torsion is slightly under-predicted by theAMRA, general agreement with experimentaldata is good. Simple dynamic model of bladebending using spring stiffness and rigid bladeswas used during testing of the FEM modelof blade torsion. Predictions of first naturalfrequency in bending of McCutcheon rotor bladeis consistent with both theory and publishedshake tests of similar rotor blades.25–28

As it can be seen from Figure 7 predictions offirst natural frequency in bending of Puma ro-tor blade is in reasonable agreement with resultsof METAR/R85. CAMRAD and RAE/WHLmodels predict lower values of the first naturaltorsional frequency than both METAR/R85 andAMRA. Bousman et al 24, however, describes es-timations of modal frequencies of METAR/R85,CAMRAD and RAE/WHL models as consis-tent.

4.3. Validation of FEM Model of

Blade Bending

A coupled FEM model of blade torsion andbending was validated in similar manner as theFEM model of blade torsion. Predictions ofboth static and dynamic loading of the bladewere compared with analytical predictions,experimental measurements and results of otherstructural analysis codes.

Predictions of static bending as obtainedfrom the AMRA model are in good agreementwith analytical predictions. Values of bladevertical displacement obtained with the modelare roughly by 5% lower than analytical predic-tions and relative deviation of blade gradientsis roughly 6% and these values do not changewith loading, blade flexural stiffness or bladelength.

Estimations of blade bending behaviour werevalidated against the same set of data that wasused for validation of the FEM model of bladetorsion.24 The conclusion can be made that bothstatic and dynamic behaviour of AMRA struc-tural dynamics block was validated and that themodel is likely to give realistic estimations ofboth rotor blade torsion and bending. Figures 8and 9 depict distribution of torsional deflectionsand flexural vertical displacements over the ro-tor disk during forward flight.

5. Aeroelastic Stability of a

Rotor in Autorotation

Unlike helicopter rotors, rotors in autoro-tation can experience significant variations inrotor speed during manoeuvres. Decrement ofthe rotor speed decreases centrifugal stiffness ofthe rotor and the resulting higher deflectionsin flap and twist generate more drag and maycause further drop in rotor speed. It is clearthat thrust and torque of the rotor are functionsof rotor speed and distribution of local anglesof attack along the blade span. Further, anglesof attack are dependent upon blade twist, rotorspeed, descent rate and induced velocity. Itcan be easily shown that both descent rate androtor angular velocity are strongly dependentupon rotor torque and rotor thrust. The AMRAmodel has shown that the extra degree offreedom in rotor speed has significant effect onthe aeromechanics and aeroelastic stability ofan autorotating rotor. A series of parametricstudies that was carried out with the aid of themodel19 shows that blade induced twist (tor-sion), fixed angle of incidence of the blade andblade geometric twist have by far the strongestinfluence on the aeromechanical behaviour of arotor in autorotation.

AMRA simulations were performed for threedifferent levels of model complexity. The sim-plest configuration of the model used perfectlyrigid rotor blades in both bending and torsionand blade flexibility was modelled with theaid of spring stiffness located at blade root.The second configuration of AMRA employedthe FEM model of blade torsion while onlyspring stiffness was used for modelling of bladebending. The most complex variant of themodel utilized coupled FEM models of bothtorsion and bending. Comparison of resultsobtained for these three model configurationsallowed assessment of the effect of complexityof the structural dynamics model on perfor-mance and fidelity of AMRA simulations. Inorder to investigate rotor stability boundary intorsion, AMRA simulations for various valuesof torsional stiffness and chord-wise positions ofcentre of gravity (CG) were carried out.

The results of the simulations have revealedthat low torsional stiffness of the blade leads toan aeroelastic instability (flutter) that manifestsas coupled rotor speed / pitch / flap oscilla-tions. These oscillations result in catastrophicdecrease of rotor speed. This is a demonstra-tion of strong rotor speed / pitch / flap couplingthat exists only during autorotation. Decrementof the rotor speed decreases centrifugal stiffness

7

of the rotor and the resulting higher deflectionsin flap and twist generate more drag and causefurther drop in rotor speed. This type of flutteris unique for rotors in autorotation since it dif-fers from both helicopter rotor flutter and flutterof a fixed wing.

5.1. Aeroelastic Stability of a Rotor

in Axial Autorotative Flight

Since blade aerodynamic forcing duringsteady axial autorotative flight is not dependentupon blade azimuth, it becomes constant afterthe rotor reaches equilibrium state. Duringsteady vertical autorotation, overall torque gen-erated by flow through the rotor disc is zero androtor thrust is equal to the weight of the vehicle.The rotorspeed converges towards its steadyvalue during torsional equilibrium. A character-istic span-wise distribution of blade torque fora rotor in the autorotative regime is observed.The inboard part of the blade generates positivetorque and the outboard part of the bladegenerates negative torque.5,6 The AMRA modeldescribes all major features of aerodynamics ofa rotor in autorotative axial descent very well.19

Results of the AMRA simulations obtainedfor different levels of complexity of the modelshow that torsion is the most important param-eter to compute accurately; torsion thereforerequires a FEM.

The aeroelastic behaviour of the rotor for twodifferent levels of model complexity is shown inFig.10 and 11. As it can be seen in the figures,reduction of rotor speed from a steady value tozero takes only few seconds. Speed of descentincreases to unacceptable value during this timedue to dramatic decrease of rotor thrust.

Parametric studies carried out with an ear-lier generation of the model (AMRA) and pub-lished in 32nd ERF proceedings19 had shownthat chord-wise position of CG seems to havemuch stronger influence on the stability of au-torotation than chord-wise position of EA.

5.2. Aeroelastic Stability of a Rotor

in Forward Autorotative Flight

Since autogyros operate mostly in the forwardflight regime, modelling of forward autorotativeflight represents the key task in investigation ofaeroelastic behaviour of a autogyro rotor blade.Since both direction and value of the inflowvelocity are functions of azimuth if horizontalspeed is not zero there is no torque equilibriumduring steady forward flight and the value of

torque oscillates around the zero value. Theamount of vibration induced by the rotor bladeduring steady forward flight is therefore signifi-cantly higher than in axial descent. In addition,free-stream velocity at the advancing side of therotor disc is higher than at the retreating side,and thus the values of the forcing moments arehigher and also asymmetrical.

Since aerodynamic forcing of the blade duringsteady forward flight has harmonic character,blade motion in both bending and torsionhas harmonic components too. As in caseof axial flight, simplification of the model ofblade torsion seems to significantly degradepredictions of the model. Only small changesin behaviour are seen when bending is modelledby an equivalent spring stiffness rather than aFEM.

Similarly as in case of axial flight in autorota-tion, simulations for various torsional stiffnessand chord-wise positions of centre of gravitywere performed. Computations carried out withthe aid of the AMRA model have shown that therotor suffers of aeroelastic instability if CG liesaft EA. Aeroelastic behaviour of the rotor forall three different levels of model complexity hasvery similar character to the aeroelastic insta-bility predicted in autorotative vertical descentand it is shown in Fig.12 and 13. The resultingaeroelastic stability boundary can be found inFig.14. It can be seen from the figure that po-sition of CG aft EA is destabilizing, similar tofixed wing classical flutter and helicopter pitchflap flutter.

6. Conclusions

An aeromechanical model of a autogyrorotor AMRA was developed in the MATLABprogramming language and used in predictingthe aeroelastic behaviour of a rotor. Tworegimes were investigated - autorotative axialflight (vertical descent) and forward flight inautorotation. Simulations have shown thatautorotation is a complex aeromechanicalprocess with auto-stabilizing characteristics.In order to obtain input parameters for thestructural model of the blade, a series ofexperimental measurements were carried outon a typical autogyro blade. Blade massdistribution, position of elastic axis, span-wisedistribution of CG and torsional and flexuralstiffness was determined during the experiments.

Results from the AMRA model were verifiedand found to be in reasonable agreement withexperimental measurements. Several paramet-

8

ric studies were performed so as to gain moreknowledge on the effect of blade geometry andstructural properties on performance of therotor during autorotation.

It was found that blade twist / bending /rotor speed coupling has major effect on thestability of autorotation when the rotor is ina stable configuration. Computations wereperformed for three different levels of complex-ity of the model of blade structural dynamics.Results of the AMRA simulations have shownthat detailed modelling of rotor blade torsion(i.e. blade induced twist) has major influenceon aeromechanics of a rotor in autorotation. Itwas shown in the paper that it is sufficient tomodel only first bending mode, i.e. to assumeconstant span-wise distribution of flap angle.The conclusion can be made that any aeroelasticsimulation of a rotor in autorotation shouldcontain model that gives realistic predictions ofblade dynacmics in torsion.

Occurrence of a type of flutter that is uniquefor autorotating rotors was predicted by themodel both during axial descent in autorotationand autorotative forward flight. This aeroelasticinstability is driven by blade pitch / bending /rotor speed coupling and differs from both flut-ter of a helicopter rotor and flutter of a fixedwing. The instability results in catastrophic de-crease of the rotor speed and significant increaseof speed of descent.

Acknowledgments

The authors would like to acknowledge thecontinued support for autogyro research pro-vided by the UK Civil Aviation Authority. Thiswork is funded through a CAA ARB Fellow-ship. The support and advise from Steve Grif-fin, Jonathan Howes, Alistair Maxwell, AndrewGoudie and Joji Waites is highly appreciated.Many thanks go also to Dr Richard Green andthe departmental team of technicians from AcreRoad laboratories for help with the experimentalmeasurements.

References

1. Thomson, D.G., Houston, S.S., Spathopou-los, V. M., Experiments in Autogiro Air-worthiness for Improved Handling Qualities,Journal of American Helicopter Society, Pg.295, No. 4, Vol. 50, October 2005.

2. Houston, S. S., Longitudinal Stability of Au-togyros, The Aeronautical Journal, Vol. 100(991), 1996, pp. 1-6.

3. Houston, S. S., Identification of AutogiroLongitudinal Stability and Control Character-istics, Journal of Guidance, Control and Dy-namics, Vol. 21, No. 3, 1998, pp. 391-399.

4. Coton, F., Smrcek, L., Patek, Z., Aerody-namic Characteristics of a autogyro Config-uration, Journal of Aircraft, Vol. 35, No. 2,1998, p. 274 - 279.

5. Leishman, J.G., The Development of the Au-togiro: A Technical Perspective, Journal ofAircraft, Vol. 41, (2), 2004, pp. 765-781.

6. Leishman, J.G., Principles of HelicopterAerodynamics, Cambridge University Press,2nd Edition, 2006, ISBN 0-521-85860-7.

7. Prouty, R. W., Helicopter Performance, Sta-bility and Control, Robert E. Krieger Pub-lishing Co., Malabar, FA, USA, 1990.

8. Sheldahl, R. E., Klimas, P. C., Aerody-namic Characteristics of Seven Aerofoil Sec-tions Through 180 Degrees Angle of Attackfor Use in Aerodynamic Analysis of Ver-tical Axis Wind Turbines, SAND80-2114,Sandia National Laboratories, Albuquerque,New Mexico, USA, 1981.

9. Nikolsky, A.A., Seckel, E., An Analyt-ical Study of the Steady Vertical Descentin Autorotation of Single-Rotor Helicopters,NACA TN 1906, Washington, 1949.

10. Wheatley, B., Bioletti, C., Wind-TunnelTests of a 10-foot-diameter autogyro Rotor,NACA TR 536.

11. Wheatley, J.B., The Aerodynamic Anal-ysis of the autogyro Rotating-Wing System,NACA TN 492, Langley Memorial Aeronau-tical Laboratory, 1934.

12. Wheatley, J.B., An Aerodynamic Analysisof the Autogiro Rotor with a Comparison be-tween Calculated and Experimental Results,NACA TR 487, 1934.

13. Carpenter, P.J., Lift and Profile-drag Char-acteristics of an NACA 0012 Airfoil Sectionas Derived from Measured Helicopter-rotorHovering Performance, NACA TN 4357,Langley Memorial Aeronautical Laboratory,Langley Field, VA, USA, 1958.

14. www.cyberiad.net/foildata.htm, visited inthe fall 2005

15. Houston, S. S., Modelling and Analysis ofHelicopter Flight Mechanics in Autorotation,Journal of Aircraft, Vol. 40, No. 4, 2003.

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16. Chen, R. T. N., A Survey of NonuniformInflow Models for Rotorcraft Flight Dynam-ics and Control Applications, NASA TM102219, Ames Research Centre, California,USA, 1989.

17. Houston, S. S., Brown, R. E., Rotor WakeModelling for Simulation of of HelicoptersFlight Mechanics in Autorotation, Journal ofAircraft, Vol. 40, No. 5, 2003.

18. Bramwell, A. R. S. et al, Bramwell’sHelicopter Dynamics, Second edition,Butterworth-Heinemann, 2001. ISBN 0-7506-5075-3

19. Trchalík, J., Gillies, E.A., Thomson, D.G.,Aeroelastic Behaviour of a gyroplane Ro-tor in Axial Descent and Forward Flight,32nd European Rotorcraft Forum, Maas-tricht, Netherlands, October 2006.

20. Bielawa, R. L., Rotary Wing Structural Dy-namics and Aeroelasticity, Second edition,AIAA Education series, 2006. ISBN 1-56347-698-3

21. Kwon, Y. W., Bang, H., The Finite Ele-ment Method Using MATLAB, Second Edi-tion, CRC Press, 2000, ISBN 0-8493-0096-7.

22. Houston, S.S., Validation of a RotorcraftMathematical Model for Autogyro Simulation,AIAA Journal of Aircraft, Vol. 37, No. 3, pp.203-209, 2000.

23. Bagiev, M., Autogyro Handling QualitiesAssessment Using Flight Testing and Sim-ulation Techniques, PhD Thesis, Dept. ofAerospace Engineering, University of Glas-gow, 2005.

24. Bousman, W.G., Young, C., Toulmany, F.,Gilbert, N.E., Strawn, R.C., Miller, J.V.,Maier, T.H., Costes, M., A Comparison ofLifting-Line and CFD Methods with FlightTest Data from a Research Puma Helicopter,NASA TM 110421, Ames Research Center,Moffett Field, CA, USA, 1996.

25. Friedman, P. P., Rotary-Wing Aeroelastic-ity: Current Status and Future Trends, AIAAJournal, Vol. 42, No. 10, October 2004.

26. Friedman, P. P., Hodges, D.H., RotaryWing Aeroelasticity - A Historical Perspec-tive, Journal of Aircraft, Vol. 40, No. 6,November-December 2003.

27. Wilkie, W.K., Mirick P.H., Langston,Ch.W., Rotating Shake Test and ModalAnalysis of a Model Helicopter Rotor Blade,NASA TM 4760, ARL TR 1389, Langley Re-search Center, Hampton, VA, USA, 1997.

28. Maier, T.H., Sharpe, D.L., Abrego, A.I.,Aeroelastic Stability for Straight and Swept-Tip Rotor Blades in Hover and ForwardFlight, AHS 55th Annual Forum, Montreal,Quebec, Canada, 1999.

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Figure 1: Comparison of predictions of rotorblade teeter and G-UNIV experimental data

Figure 2: Comparison of predictions of rotorblade teeter and G-UNIV experimental data

Table 1: Comparison of predictions of rotorblade teeter and G-UNIV experimental data

CASE VH Ω β exp β AMRA[m/s] [rad/s] [rad] [rad]

A 14 38 0.031 0.026B 27 41 0.058 0.056

Figure 3: Comparison of the first torsional modeshape computed by the AMRA and publisheddata

Figure 4: Distribution of blade torque over rotordisk during steady forward flight in autorotationestimated by AMRA

Figure 5: Distribution of blade torque over rotordisk during steady forward flight in autorotationpublished by Leishman6

Figure 6: Southwell plot of McCutcheon rotorblade showing correct qualitative behaviour.

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Figure 7: Southwell plot Aérospatiale SA330Puma helicopter rotor blade showing correctqualitative behaviour. Data in the plot are non-dimensionalised with Ω0 = 28rad/s.

Figure 8: Distribution of blade torsional deflec-tion obtained with the aid of AMRA.

Figure 9: Distribution of blade vertical dis-placement in bending obtained with the aid ofAMRA.

Figure 10: An example of an aeroelastic insta-bility during axial flight in autorotation as pre-dicted by the AMRA. The data shown in the plotwere acquired with the aid of simplified model ofblade dynamics that is using equivalent springstiffness for modelling both torsion and bend-ing. Calculated for the value of blade torsionalstiffness of 100N.m2/rad, elastic axis at 32% ofblade chord and blade centre of gravity at 40%of blade chord.

Figure 11: An example of an aeroelastic in-stability during axial flight in autorotation aspredicted by the AMRA. Coupled FEM modelof blade torsion and bending was used. Calcu-lated for the value of blade torsional stiffness of400N.m2/rad, elastic axis at 32% of blade chordand blade centre of gravity at 40% of blade chord.

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Figure 12: An example of an aeroelastic instabil-ity during forward flight in autorotation as pre-dicted by the AMRA. The data shown in the plotwere acquired with the aid of simplified model ofblade dynamics that is using equivalent springstiffness for modelling both torsion and bend-ing. Calculated for the value of blade torsionalstiffness of 300N.m2/rad, elastic axis at 32% ofblade chord and blade centre of gravity at 40%of blade chord.

Figure 13: An example of an aeroelastic insta-bility during forward flight in autorotation aspredicted by the AMRA. Coupled FEM modelof blade torsion and bending was used. Calcu-lated for the value of blade torsional stiffness of600N.m2/rad, elastic axis at 32% of blade chordand blade centre of gravity at 40% of blade chord.

Figure 14: Stability boundary for autorotativeflight

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