On the use of control surface excitation in flutter testing · Abstract: Flutter testing is aimed at demonstrating that the aircraft ¯ ight envelope is ¯ utter free. Response measurements

On the use of control surface excitation in ¯uttertesting

J R Wright*, J Wong, J E Cooper and G DimitriadisDynamics and Aeroelasticity Research Group, Manchester School of Engineering, University of Manchester, UK

Abstract: Flutter testing is aimed at demonstrating that the aircraft ¯ight envelope is ¯utter free.Response measurements from deliberate excitation of the structure are used to identify and trackfrequency and damping values against velocity. In this paper, the common approach of using a ¯ightcontrol surface to provide the excitation is examined using a mathematical model of a wing andcontrol surface whose rotation is restrained by a simple actuator. In particular, it is shown that it isessential to use the demand signal to the actuator as a reference signal for data processing. Use of theactuator force (or strain) or control angle (or actuator displacement) as a reference signal is badpractice because these signals contain response information. It may also be dangerous in that theonset of ¯utter may not be seen in the test results.

Keywords: ¯utter testing, aircraft ¯ight envelope, excitation, ¯ight control

NOTATION

a two-dimensional lift curve slopeac lift per control angleam pitching moment/control angleA inertia matrixAF area of pressure stabilization pistonAP piston areab1 hinge moment/incidenceb2 hinge moment/control angleB aerodynamic damping matrixB…s† Laplace transform of control rotationc chord/subscript for control surfaceC aerodynamic stiffness matrixd differential operatord1, 2, 3 actuator parametersD structural damping matrixe normalized position of ¯exural axisE structural stiffness matrixF forceFP force applied to actuatorh actuator/control lever armH hinge momentH…s† transfer functionI moment of inertia

i, j integersJ actuator dampingk stiffnessK actuator stiffnessKF stiffness of the stabilization springKV valve ¯ow constantK0 static stiffnessK? oil bounce stiffnessL liftm mass/subscript for the main surfaceM pitching momentM _bb control damping derivativeM _yy torsional damping derivativeN bulk modulus of oilP…s† polynomial of system polesPJ pressure difference across the pistonPS, PR supply and return pressureP1, P2 pressures in chambers 1 and 2q generalized coordinateQ generalized forceQ1, Q2 valve ¯ows into chambers 1 and 2R matrixs span/Laplace transform operatorS matrixt timeT kinetic energy/matrixTD…s† displacement transmissibilityTF…s† force transmissibilityTb torque provided by actuatorU potential energyU matrix

The MS was received on 5 August 2003 and was accepted after revisionfor publication on 5 November 2003.* Corresponding author: Dynamics and Aeroelasticity Research Group,Manchester School of Engineering, University of Manchester, OxfordRoad, Manchester M13 9PL, UK.

317

G03303 # IMechE 2003 Proc. Instn Mech. Engrs Vol. 217 Part G: J. Aerospace Engineering

V velocityVo oil volumeW work done/matrixx chordwise coordinatexf distance to ¯exural axisxh distance to control hinge lineXi demanded displacement of actuatorXV valve displacementX0 actuator body displacementy spanwise coordinateY matrixY…s† Laplace transform of responseZ…s† polynomial of system zerosZA…s† actuator impedance

b control rotation anglebi effective demanded control angleg angle of ¯apd incrementy angle of twistl eigenvaluem real part of the eigenvaluem mechanical gearing of the valven frequency parameterr air densityo frequency (rad/s)oD , oF displacement and force cut-off frequency

overdot d/dtunderscore vector[ ] matrix

1 INTRODUCTION

Aircraft ¯utter is an instability that involves two ormore modes of vibration coupling together via aero-dynamic forces such that energy is extracted from theairstream [1]. Once the ¯utter velocity is reached,the amplitude can grow without limit and destroy theaircraft. Where non-linearity occurs, a limit cycleoscillation (LCO) may be encountered. Flutter is clearlya dangerous phenomenon and must be treated verycarefully, both by calculation and test.

In a ¯utter test, the idea is to progress through the¯ight envelope, gradually increasing velocity and Machnumber in order to examine the change in aeroelasticstability. At each test point, the aircraft is excited insome way so that it vibrates, and the excitation andresponse signals are used to obtain estimates of thefrequency and damping of all the modes that are active.The frequency and damping values are then plottedagainst velocity (or Mach number) and the trend ofdamping examined; any downward trend in dampingcan indicate the likely approach of ¯utter (which occurs

at zero damping) and so increased care needs to be takenwith subsequent tests.

A number of test signals may be used to excite thestructure at each test point deliberately, typicallyimpulse (or sequence of impulses), random or chirp (asine wave with increasing frequency); the chirp isarguably the most popular excitation signal as it canyield a high signal-to-noise ratio. The signals may beimplemented by use of an aerodynamic vane, an inertialexciter or, most commonly, a control surface (via the¯ight control system and control surface actuator).When using a special device to excite the structure (e.g.vane or inertial exciter), it is normal to use somemeasure of the excitation as a reference signal forcomputing the frequency response function (FRF), priorto estimating the parameters by some system identi®ca-tion approach [2]. In the case when the control surface isbeing used via a hydraulic (or other) actuator, it is mostcommon to use the demand signal to the actuator as thereference signal for data processing.

However, the authors are aware from anecdotal andpublished evidence that sometimes a different referencesignal is employed, namely the actuator force (or strain)or the measured control angle (or actuator displace-ment). Also, published references often do not makeclear what signal is actually being used as a reference forthe excitation. A limited number of examples of whereother signals appear to have been used will now begiven. In the AGARD Flight Instrumentation Series [3]measurements are made of the àctual input force’associated with use of a servo actuator. In reference [4],the term `control de¯ection signals’ is used for ¯uttertesting of a drone with the active ¯utter suppressionsystem off. In reference [5], the àileron rotationalposition’ was used for transfer function analysis, butonly in a qualitative sense to con®rm results from sinedwell tests, as it was recognized that this signal was not a`true force reference’. In reference [6], àileron motionsensors’ are employed for computing the frequencyresponse functions (FRFs).

In this paper, it is shown via a mathematical model ofa wing/control surface with a simple actuator that usingeither control angle or control force as a reference isincorrect and constitutes bad practice. It can even leadto a dangerous situation where the onset of ¯utter is notseen in the test results. It is therefore con-sideredessential to use the demand signal as a reference andpublications on ¯utter testing should always be explicitabout what reference signal was employed.

2 BASIC WING/CONTROL MODEL

2.1 Structural terms

The simple model chosen for this paper is a 3 degree-of-freedom system consisting of a rigid wing ¯apping with

J R WRIGHT, J WONG, J E COOPER AND G DIMITRIADIS318

Proc. Instn Mech. Engrs Vol. 217 Part G: J. Aerospace Engineering G03303 # IMechE 2003

an angle g about a hinge at the root, twisting with anangle y and with a rigid control surface rotating throughan angle b, as shown in Figs 1 and 2. The model is basedon that used in reference [7], albeit augmented to includea control surface. In inertial terms, the wing and controlsurface are considered to be of uniform thickness anddensity but aerodynamically the model would have aconventional aerofoil section. The rotational motionsare constrained by discrete springs of stiffness kg, ky andkb respectively, such that there are no stiffness couplingterms. The total potential energy of the system, U, canthen be expressed as

U ˆ 12kgg2 ‡ 12 kyy

2 ‡ 12kbb2 …1†

The total kinetic energy of the system, T, can beexpressed as

T ˆ 12$s0$xh0

dm _ggy ‡ x ¡ xf… † _yyh i2

‡ 12$s0$cxh

dm

6 _ggy ‡ x ¡ xf… † _yyh i

‡ …x ¡ xh† _bbn o2

…2†

where the `overdot’ refers to d/dt, dm refers to anelemental mass, s is the span, c is the chord, xf de®nesthe position of the ¯exural axis, xh de®nes the hinge lineposition and the two integral terms refer to the mainwing surface and control surface respectively.

Applying Lagrange’s equations to the energy terms inequations (1) and (2) leads to the expression

Ig Igy Igb

Iy Iyb

symmetry Ib

2

64

3

75 gg yy bb

8><

>:

9>=

>;‡

kg 0 0

0 ky 0

0 0 kb

2

64

3

75g

y

b

8><

>:

9>=

>;

ˆQg

Qy

Qb

8><

>:

9>=

>;…3†

Here, Qg, Qy and Qb are generalized forces due toaerodynamic effects. Also, Ig, Iy and Ib are moments ofinertia and Igy, Igb and Iyb are product moments ofinertia, all taken about the relevant axes of rotation andde®ned as

Ig ˆ $$w‡cy2 dm

Iy ˆ $$w‡c…x ¡ xf †2 dm

Ib ˆ $$c…x ¡ xh†2 dm

Igy ˆ $$w‡cy…x ¡ xf† dmIgb ˆ $$cy…x ¡ xh† dmIyb ˆ $$c…x ¡ xf†…x ¡ xh† dm

…4†

where integrals are taken over the wing plus controlsurface …w ‡ c† or the control surface alone …c†.

2.2 Aerodynamic terms

The generalized force is de®ned by Qi ˆ q…dW†=qqi,where qi is a generalized coordinate (i.e. q1, 2,3 correspondto g, y and b respectively) and dW is the incrementalwork done when an incremental change of generalizedcoordinates occurs. This equation can now be used tocalculate the aerodynamic forces acting on a strip ofwidth dy, as shown in Fig. 1. The generalized forces Qg,Qy and Qb correspond to unsteady aerodynamic forcesand are a function of the frequency parameter

n ˆ ocV

…5†

where c is the chord length, o is the frequency and V isthe relative velocity. However, for simplicity, the quasi-steady form of the aerodynamics will be used to allow theforces and moments to be expressed in terms of anglesand angular velocities*. However, the quasi-steady

Fig. 1 Schematic layout of wing/control surface deformations

Fig. 2 Schematic layout of the wing/control surface withspring

* Quasi-steady aerodynamics neglect the attenuation and phase lagassociatedwith an increase in the frequencyof oscillation and are de®nedas the limit of unsteady oscillatory aerodynamics as the frequencyparameter tends to zero. Here, the quasi-steady assumption is that theaerodynamic characteristics of an aerofoil undergoing variable motionsare equal, at any instant of time, to the characteristics of the sameaerofoil moving with constant velocities equal to the instantaneousvalues.

ON THE USE OF CONTROL SURFACE EXCITATION IN FLUTTER TESTING 319


aerodynamics are modi®ed to include the non-dimen-sional aerodynamic torsional damping derivative, M _yy, assuggested in reference [7], to yield a more realistic ¯utterphenomenon for the quasi-steady case; a control aero-dynamic damping derivative, M _bb, is also included. Thequasi-steady assumption, while not yielding accurate¯utter predictions, does not affect the arguments madeherein about the different approaches to ¯utter testingand is simple to derive for this basic model.

Applying two-dimensional `strip theory’ results forthe wing/control surface leads to the lift and pitchingmoment (positive leading edge up) about the ¯exuralaxis for a strip dy as

dL ˆ1

2rV 2c dy a y ‡

_ggyV

³ ´‡ acb

µ ¶…6†

dM ˆ 12rV2c2 dy ae y ‡ _ggy

V

³ ´‡ amb ‡ M _yy

_yycV

" #…7†

where _ggy is the effective `heave’ velocity of the strip,ac ˆ qCL=qb, am ˆ qCM=qb, a is the two-dimensionallift curve slope, ec is the eccentricity (distance of the¯exural axis aft of the aerodynamic centre) and M _yy < 0.The aerodynamic hinge moment on the strip (based onthe wing chord and positive trailing edge down) can beshown to be

dH ˆ 12rV 2c2 dy b1 y ‡

_ggyV

³ ´‡ b2b ‡ M _bb

_bbcV

" #…8†

where b1 ˆ qCH=qa, b2 ˆ qCH=qb and M _bb < 0. Theincremental work done on the wing/control, dWaero, byaerodynamic forces dL, dM and dH is

dWaero ˆ ¡ $s

0dL…y dg† ‡ $s

0dM…dy† ‡ $s

0dH…db† …9†

where dg, dy and db are the incremental changes inangles associated with dL, dM and dH respectively. Theaerodynamic matrices will be derived in section 2.4.

2.3 External excitation

However, in addition, a way to input an excitation forceto the wing model is needed for the ¯utter test to beperformed. Therefore, using the geometry of the wing,the incremental work done by a direct force acting onthe wing main surface or control surface can be obtainedas

dWdirect force ˆ Fm‰ ym dg ‡ …xm ¡ xf†dyŠ‡ Fc‰ yc dg ‡ …xc ¡ xf†dy ‡ …xc ¡ xh†dbŠ

…10†

where the subscripts m and c refer to the main andcontrol surfaces respectively.

2.4 Aeroelastic equations

The full incremental work term dWtotal is therefore

dWtotal ˆ dWaero ‡ dWdirect force …11†

The generalized forces can be calculated by differentiat-ing the incremental work terms de®ned in equations (9)and (10) by the corresponding incremental rotations.Combining these results with that obtained fromequation (3) yields the full aeroelastic equations

Ig Igy Igb

Iy Iyb

symmetric Ib

2

64

3

75 gg yy bb

8><

>:

9>=

>;

‡1

2rVs

as2

30 0

¡aesc

2c2M _yy 0

¡ b1sc2

0 ¡ c2M _bb

2

666664

3

777775

_gg_yy_bb

8><

>:

9>=

>;

‡1

2rV2s

0as

2

acs

20 ¡ aec ¡ amc0 ¡ b1c ¡ b2c

2

664

3

775

g

y

b

8><

>:

9>=

>;

‡kg 0 0

0 ky 0

0 0 kb

2

64

3

75g

y

b

8><

>:

9>=

>;

ˆ Fmym

xm ¡ xf0

8><

>:

9>=

>;‡ Fc

yc

xc ¡ xfxc ¡ xh

8><

>:

9>=

>;…12†

This matrix equation can be written in the classical formas follows:

A‰ Š qq ‡ rV B‰ Š ‡ D‰ Š… † _qq ‡ rV2 C‰ Š ‡ E‰ Š¡ ¢

q ˆ F …13†

where ‰AŠ is the inertia matrix, ‰BŠ is the aerodynamicdamping matrix, ‰DŠ is the structural damping matrix,‰CŠ is the aerodynamic stiffness matrix, ‰EŠ is thestructural stiffness matrix, F is the external forcingvector and q ˆ fg, y, bgT is the vector of generalizedcoordinates. The structural damping has been ignoredin equation (12) in order to gain conservative answers,but the aerodynamic damping terms usually tend todominate anyway.

2.5 Flutter solution

In order to determine the exact ¯utter characteristics ofthe model to use for evaluation of a simulated ¯uttertest, a classical eigenvalue approach needs to beadopted. Firstly, the excitation force F is set to zerofor free vibration and then a solution for the generalized



response is taken as

q ˆ q0 elt …14†

where q0 is the response amplitude, t is time and l is theexponent that dictates the stability of the system asvelocity V increases. Substituting this response intoequation (13) yields a complex eigenvalue problem

l2‰AŠ ‡ l…rV ‰BŠ ‡ ‰DŠ† ‡ …rV2‰CŠ ‡ ‰EŠ†£ ¤

q0 ˆ 0…15†

but this is normally solved by transforming to ®rst-orderform. The resulting eigenvalues are then of the formlj ˆ mj + ioj … j ˆ 1, 2, 3† and may be used to determinethe frequency and damping of the three roots of thissystem. Parameters for the model (see Appendix 1) werechosen to cause a `control surface ¯utter’, i.e. oneinvolving control surface rotation; this introduces thecomplexity of simultaneously using the control surfacefor excitation while it is active in the ¯utter mechanism.A plot of the frequency and damping values againstvelocity is shown in Fig. 3a, where it may be seen thatthe ¯utter mechanism involves bending and controlrotation and that the approximate ¯utter velocity is40 m/s. Note that the modes tend to couple as the ¯uttervelocity is approached so descriptions of bending, etc.,become less meaningful.

2.6 Simulated ¯utter test

In order to demonstrate that a ¯utter test carried outusing a conventional excitation force applied to the wingor control surface is able to predict the ¯utter velocity, asimulated test was carried out on the model in equation(12). A SIMULINK model was developed and a chirpexcitation was applied to the wing surface, moving at avelocity of 30 m/s, so yielding the typical response at theexcitation point shown in Fig. 4. The FRF between theresponse and excitation at the excitation point wascalculated and the rational fraction polynomial identi-®cation method [8] was used to estimate the frequencyand damping values for the system at this velocity.Table 1 shows good agreement between the results, socon®rming accurate simulations.

The process was repeated at a range of velocities andthe results for the simulated test are shown in Fig. 3b tofollow the exact results and so be able to predict the¯utter velocity adequately. A similar result occurred foran excitation force applied on the control surface. Thusa measured excitation applied to the wing or controlsurface (e.g. inertia exciter, bonker, vane, etc.) issatisfactory for ¯utter testing provided the applied forceis measured or some characteristics of the force areassumed (e.g. a ¯at spectrum). For these excitationtypes, the excitation signal is independent of theresponse and so does not interfere with the ¯uttercharacteristics.

3 WING/CONTROL MODEL WITH ACTUATOR

In order to examine the use of an actuator (e.g.hydraulic) to drive a control surface by providing theexcitation stimulus for a ¯utter test, a modi®ed modelneeds to be developed where the rotational spring kb isreplaced by an actuator as shown in Fig. 5. The actuatorchosen is a hydraulic device with pressure feedbackstabilization, as shown in Fig. 6. The ¯ow equations willbe linearized in order to allow the exact ¯uttercharacteristics of the linear model to be determined.The type of actuator employed is considered to beimmaterial to the results of this paper, provided that theequations may be linearized. The impact of using a non-linear actuator in ¯utter testing is not addressed in thispaper, but it is believed that using the actuator as anexcitation device may well affect the ¯utter behaviour.

3.1 Actuator model

The piston is assumed to be at the centre of its stroke,and the external load and input demand are assumed tobe small and oscillatory. Thus the pressure differencebetween the chambers is

PJ ˆ P1 ¡ P2 …16†

where P1, 2 are the pressures in each chamber and anydisplacements of the valve or piston are small. Also, theinertia force associated with the mass of the actuatorbody is assumed to be small for the frequency range ofinterest.

The valve opening XV will be given by

XV ˆ m…Xi ¡ X0† ¡ PJAFKF

…17†

where m is the mechanical gearing, Xi is the demandeddisplacement, X0 is the displacement of the actuatorbody and AF and KF are the area and stiffness of thepressure feedback stabilization components. The ¯ows

Table 1 Comparison of theoretical and estimated values at30 m/s

Theoreticalfrequencies(Hz)

Estimatedfrequencies(Hz)

Theoreticaldampings(%)

Estimateddampings(%)

1.8793 1.8836 0.3625 0.36892.4570 2.4551 0.6254 0.63119.1130 9.1147 0.2411 0.2397



Q1,2 into and out of the chambers through the valves are

Q1 ˆ KVPS ¡ P1

pXV, Q2 ˆ KV

P2 ¡ PR

pXV

for XV50

Q1 ˆ KVP1 ¡ PR

pXV, Q2 ˆ KV

PS ¡ P2

pXV

for XV < 0

…18†

where KV is the valve ¯ow constant and PR and PS arethe return and supply pressures respectively. Theexpressions are clearly non-linear in the pressures. For

small oscillatory loads and displacements, and with thereturn pressure assumed to be negligible …PR ˆ 0†, therewill be a small oscillation of the pressures about PS=2.Thus, equations (18) can be linearized as

Q1 ˆ Q2 ˆ KV

PS2

rXV …19†

without losing the basic ¯ow mechanism that iscontrolled by the valve motion. The ¯ow through thevalve must balance the rate of change of the oil volumeVo (i.e. Vo=2 per chamber) due to compression of the oil

Fig. 3 Variation of frequency and damping against velocity for the basic wing/control surface model:(a) solid line, exact values from the ¯utter solution; (b) points, estimated values from the simulated¯utter test



and due to piston movement, so that

Q1 ˆVo2N

_PP1 ‡ AP _XX0

Q2 ˆ ¡Vo2N

_PP2 ‡ AP _XX0

…20†

where leakage across the piston and ¯ow into the

pressure stabilization sensing chambers are neglected.Adding these equations together and using equation (19)gives

2KV

PS2

rXV ˆ

Vo2N

… _PP1 ¡ _PP2† ‡ 2AP _XX0 …21†

which, after substituting for PJ from equation (16) andXV from equation (17), gives a ®rst-order differentialequation for the pressure difference PJ:

Vo4N

_PPJ ‡KVAF

KF

PS2

rPJ

ˆ ¡ AP _XX0 ¡ mKV

PS2

rX0 ‡ mKV

PS2

rXi …22†

Finally, the relationship between the force FP applied tothe actuator and the differential pressure is

FP ˆ ¡ APPJ …23†

Fig. 5 Schematic layout of the wing/control surface withactuator

Fig. 4 Sample response to a chirp at 30 m/s velocity for the wing/control model

Fig. 6 Schematic layout of the actuator



In Appendix 2, the characteristics of the actuator areexamined further.

3.2 Coupling the actuator into the wing/control equations

Having found equations for the actuator, the two sets ofequations must now be coupled using the kinematicrelationship between the control angle b and theactuator displacement X0, namely

X0 ˆ ¡ bh …24†

where h is the lever arm for the actuator/control surfaceconnection seen in Fig. 5. Thus

_XX0 ˆ ¡ _bbh …25†

and

Xi ˆ ¡ bih …26†

where bi is the effective demanded control angle and Xiis the demanded actuator displacement.

Because the control spring has been removed, the re-storing torque from the spring …kbb† needs to be replacedby a torque Tb, now provided by the actuator, namely

Tb ˆ ¡ FPh ˆ hAPPJ …27†

where FP is the force applied to the actuator body, asshown in Fig. 6. Combining equations (12) and (22) andusing the coupling equations (24) to (26) and (27) can beshown to yield the aeroelastic equations incorporating theactuator, namely

Ig Igy Igb 0

Iy Iyb 0

Ib 0

symmetric 0

2

6664

3

7775

gg yy bb PPJ

8>>><

>>>:

9>>>=

>>>;‡ 1

2rVs

as2

30 0 0

¡ aesc2

c2M _yy 0 0

¡ b1sc2

0 ¡ c2M _bb 0

0 0 0 0

2

666666664

3

777777775

_gg_yy_bb_PPJ

8>>><

>>>:

9>>>=

>>>;

‡

0 0 0 0

0 0 0 0

0 0 0 0

0 0 ¡ hAPVo4N

2

66664

3

77775

_gg_yy_bb_PPJ

8>>><

>>>:

9>>>=

>>>;‡ 1

2rV 2s

0as

2

acs

20

0 ¡ aec ¡ amc 00 ¡ b1c ¡ b2c 00 0 0 0

2

66664

3

77775

g

y

b

PJ

8>>><

>>>:

9>>>=

>>>;

‡

kg 0 0 0

0 ky 0 0

0 0 0 hAP

0 0 ¡ hmKVPS2

rKVAF

KF

PS2

r

2

666664

3

777775

g

y

b

PJ

8>>><

>>>:

9>>>=

>>>;ˆ bi

0

0

0

¡ hmKVPS2

r

8>>>>><

>>>>>:

9>>>>>=

>>>>>;

…28†

Note that the excitation force vector has now beenremoved because the actuator will provide the ¯uttertest excitation via the control surface rotation.The coupling terms between the actuator and the

wing/control structure may be seen more clearly ifequation (28) is rewritten compactly as

‰AŠ 00 0

µ ¶ qq

PPJ

» ¼‡

‰rVBŠ 0S R

µ ¶_qq

_PPJ

» ¼

‡‰rV2C ‡ EŠ T

U W

µ ¶q

PJ

» ¼ˆ

0

Y

» ¼bi …29†

where the matrices ‰AŠ, ‰BŠ and ‰CŠ are the same as shownearlier in equation (13), ‰EŠ is de®ned differently as

‰EŠ ˆkg 0 00 ky 00 0 0

2

4

3

5

and other actuator terms are introduced, namely

T ˆ0

0

hAP

8><

>:

9>=

>;, S ˆ 0 0 ¡ hAP‰ Š, R ˆ

Vo4N

U ˆ 0 0 ¡ hmKVPS2

rµ ¶, W ˆ KVAF

KF

PS2

r

Y ˆ ¡ hmKV

PS2

r

3.3 Flutter solution

The exact ¯utter solution for the wing/control withactuator may be found by a similar approach to that insection 2.4, where a ®rst-order eigenvalue problem is setup to determine the roots of the system. In this case,

because the additional equation is only of ®rst order,there will be three complex conjugate pairs as before,together with a real root that occurs because of the lagin the actuator frequency characteristics. The same



wing/control parameters were chosen as before but theactuator cannot exactly replace the control springstiffness, kb, because it is a far more complex systemwith frequency-dependent stiffness and damping. Forthis reason, the actuator parameters were chosen asdiscussed in Appendix 3. Once again, the aim was toachieve a `control surface ¯utter’. A plot of thefrequency and damping values against velocity for thewing/control surface with actuator model is shown inFig. 7a, where it may be seen that the ¯utter mechanisminvolved bending and control rotation and that theapproximate ¯utter velocity is 41 m/s, a value close tothe earlier case. The ¯utter is `softer’ than before, i.e. therate of change of damping with velocity as ¯utter isapproached is more gradual. This is probably because of

the additional actuator damping present in the controlrotation. It should be noted that use of an actuator willin practice lead to a higher control frequency, but thismodel is simply being used to illustrate a point that isindependent of the relative natural frequencies of themodes.

3.4 Simulated ¯utter test

In order to investigate how suitable the use of thecontrol surface is for providing excitation in a ¯utter testwith different reference signals used for estimating theFRF, a SIMULINK model was set up as before.Simulated ¯utter tests were performed at a series of

Fig. 7 Variation of frequency and damping against velocity for the wing/control surface with actuatormodel: (a) solid line, exact values from the ¯utter solution; (b) points, estimated values from thesimulated ¯utter test. (- - -) angle of ¯ap, g, (. . . . .) control rotation angle, b, (ÐÐ) angle of twist, y



velocities and with a chirp demand signal provided tothe actuator input. FRFs were calculated between theresponse on the wing or control surface and one ofseveral reference signals, as discussed below. Therelevant FRFs were used together with the rationalfraction polynomial method to identify the frequencyand damping values at each test point.

Each of the following three reference signals was usedin the simulated ¯utter test:

(a) the actuator demand signal itself,(b) the force in the actuator/control link and(c) the control rotation angle.

Sample time histories for the displacement response andreference signals at 20 m/s are presented in Figs 8 to 10.It may be noted immediately that the actuator force andcontrol rotation angle reference signals actually re ectthe dynamics of the system whereas obviously thedemand signal does not, being generated using aconstant amplitude chirp. FRFs corresponding to a

selection of the response/reference combinations arepresented in F igs 11 to 13. F igure 11, based upon thedemand angle reference signal, shows peaks at thefrequencies expected from the eigenvalue solution.However, it may be seen clearly in F igs 12 and 13,where the actuator force or control angle are used asreference signals, that the peak positions are notconsistent with the true frequencies; a similar storyapplies for the FRFs based on a response on the controlsurface itself. Indeed, the peaks actually occur at the‘anti-resonances’ of some of the other FRFs, as will beexplained later. F inally, the FRF for the actuator forcewith respect to the demand angle shown in Fig. 14demonstrates clearly that the actuator force actuallycontains the dynamics of the system; a similar resultoccurs for the control angle.

To complete the demonstration of the use of differentreference signals, the results from the curve ®t at a rangeof velocities is compared to the true ¯utter behaviourin Fig. 7b for the wing response/control demand angle.

Fig. 8 Sample wing response to a chirp at 20 m/s velocity for the wing/control with actuator model

Fig. 9 Sample actuator force reference for a chirp excitation at 20 m/s velocity for the wing/control withactuator model



The results compare very closely, so con®rmingaccuracy of the simulations, and demonstrate that asatisfactory ¯utter test could be carried out for thisreference signal. This result is repeated for all of theother FRFs using the control demand angle, even wherethe actuator force or control rotation are used asresponses rather than references. This con®rms that theactuator force and control rotation contain the essentialresponse dynamics of the ¯utter system and should notbe used as reference signals. It is also apparent that it ispossible to use an actuator to provide excitation for a¯utter test, even when it simultaneously acts as animpedance in the ¯utter model; it is not obvious that thisis also the case when non-linearities are taken intoaccount, which would require a separate study.

In order to con®rm this conclusion, the FRFs in Figs12 and 13, based upon using the actuator force orcontrol rotation as reference signals, were curve-®tted,noting that in most cases there were only two clear

peaks. The àpparent’ frequency values differed notice-ably from the true frequencies, though sometimes thevalues were actually quite close. Also, the àpparent’damping values did not indicate a ¯utter at all in thevelocity range up to 40 m/s; nor was there any sign fromthe damping trend that ¯utter was imminent. In mostcases the àpparent’ damping values increased withvelocity or remained essentially constant, as shown inFigs 15 and 16, which use actuator force and controlangle respectively as reference signals. In essence, pseudo`modes’ have been identi®ed; even though the frequencieslook quite close to the true values in this example, theyare incorrect and so are the damping values.

Thus the use of the actuator force (or strain) orcontrol rotation angle (or actuator displacement) as areference signal in a ¯utter test is totally inappropriateand could be dangerous, misleading the analyst tobelieve that there was no ¯utter problem when in fact a¯utter could be imminent. The only suitable reference

Fig. 10 Sample control angle reference for a chirp excitation at 20 m/s velocity for the wing/control withactuator model

Fig. 11 Sample FRF between wing response and demand angle at 20 m/s velocity for the wing/control withactuator model



signal, when employing an actuator-type device to movethe control surface, is the actual demand signal.

4 EXPLANATION IN TRANSFER FUNCTIONTERMS

In order to attempt to explain further why the actuatorforce and control rotation angle are entirely inappropri-ate as reference signals for ¯utter testing, consider anargument based on transfer functions. It is well knownthat a transfer function comprises poles in the denomi-nator and zeros in the numerator. The true poles of asystem are unique whereas the zeros depend upon theactual input and response positions.

Now consider the response Y…s† on the wing orcontrol surface, where s is the Laplace operator,expressed in terms of the input (or reference) signaldemanded of the actuator B…s†. As this is the source ofthe excitation to the system, the transfer function

between Y…s† and Bi…s† will contain the true poles ofthe system P…s†, so

HYBi …s† ˆY…s†Bi…s†

ˆ ZYBi …s†P…s† …30†

where ZYBi …s† contains the relevant zeros for thisresponse/reference combination. Any test using thistransfer function will inevitably yield the true stabilitycharacteristics of the ¯utter system, embedded in P…s†.

Now consider using the control rotation B…s† (oractuator displacement) as the reference signal. Becausethis is actually part of the response of the system b ( justas much as g or y), it may be related to Bi…s† as follows:

B…s† ˆ HBBi …s†Bi…s† ˆZBBi …s†

P…s† Bi…s† …31†

Thus, using equations (30) and (31), the transferfunction (or FRF) between the response Y…s† and

Fig. 12 Sample FRF between wing response and actuator force at 20 m/s velocity for the wing/control withacutator model

Fig. 13 Sample FRF between wing response and control angle at 20 m/s velocity for the wing/control withactuator model



control rotation angle B…s† may be written as

HYB…s† ˆY…s†B…s†

ˆ

ZYBi …s†P…s†

Bi…s†

ZBBi …s†P…s† Bi…s†

ˆ ZYBi …s†ZBBi …s†

…32†

From this equation it may be seen that the apparentpoles of this transfer function are actually the zeros ofthe control rotation/control demand transfer functionand will therefore not contain the essential stabilityinformation in P…s† for the ¯utter system. A similarargument applies to using the force in the actuator FP…s†

Fig. 14 Sample FRF between actuator force and demand angle at 20 m/s velocity for the wing/control withactuator model

Fig. 15 Simulated ¯utter test results over the velocity range using actuator force as the reference for thewing/control with actuator model



as a reference signal, but here the force is less directlylinked to the response. The force derives from thepressure difference across the actuator chambers PJ(which is actually one of the states of the system) andthus the displacement of the actuator body, which is inturn directly linked to the control rotation by kine-matics.

Thus the reason that the apparent test results wereincorrect is that the characteristics of some of the systemzeros were being followed instead of the poles. Becausesome of the zeros are close to the poles, some of the`apparent’ frequencies happened to be close to the trueones.

5 IMPACT ON THE FLUTTER TESTPHILOSOPHY

The lesson for ¯utter testing is very clear and simple. Ifan actuator is used to provide excitation via a controlsurface, then the actuator demand signal should be usedas the reference signal for FRF calculation. To use theactuator force (or strain) or control rotation angle (oractuator body displacement) as a reference could yieldincorrect and potentially misleading results and couldeven result in ¯utter occurring without warning.

In reference [5], it is recognized that the `aileron

Fig. 16 Simulated ¯utter test results over the velocity range using control angle as the reference for thewing/control with actuator model



rotational position’ is not a true force signal, butnevertheless the resulting transfer function was analysedand results used for qualitative purposes only, tosupport sine dwell trends. In this case, because theaileron rotation angle turned out to be fairly constantwith frequency, the results would have been satisfactory;however, had this not been the case, the availability ofpotentially incorrect data could have been misleadingand could have given different trends.

It is interesting that in reference [6] results fromanalysis of a resonant dwell based on the accelerationresponse were compared to those from curve-®tting anFRF generated using an aileron motion sensor and asigni®cant difference in frequency was found. Non-linearity was blamed, but it could well be that thedifference could also have been due to the referencesignal chosen.

One check as to whether the use of, say, the controlangle is likely to lead to incorrect results is whether thetime history (in the case of a constant amplitude chirp)displays any variation in amplitude. If the amplitude isessentially constant then the control and actuator arenot signi®cantly involved in the ¯utter mechanism andtest results will be satisfactory. For example, in reference[4], the time histories for the control de¯ections are fairly¯at over the lower frequency range of interest andtherefore the results shown are not likely to be incorrect.

6 CONCLUSIONS

In this paper, simulations of a 3 degree-of-freedom¯utter system have been performed with control rotationrestrained ®rstly by a spring and secondly by a simplehydraulic actuator. The actuator was used to provideexcitation for a simulated ¯utter test and it was shownthat the only appropriate reference signal for dataprocessing was the control angle demand input to theactuator. Using the actuator force or measured controlangle could lead to incorrect and potentially dangerousresults in the ¯utter test.

REFERENCES

1 Scanlan, H. and Rosenbaum, R. Introduction to the Study ofAircraft Vibration and Flutter, 1951 (Macmillan, London).

2 Cooper, J. E. Parameter estimation methods for ¯ight ¯uttertesting. AGARD-CP-566, Advanced Aeroservoelastic Test-

ing and Data Analysis, 1995.

3 van Nunen, J. W. G. and Piazzoli, G. Aeroelastic ¯ight testtechniques and instrumentation. AGARDograph 160,

Vol. 9, Flight Test Instrumentation Series, 1979.

4 Bennett, R. M. and Abel, I. Application of a ¯ight test anddata analysis technique to ¯utter of a drone aircraft. In

AIAA Dynamics Specialists Conference, Atlanta, Georgia

1981, AIAA paper 81-0652.

5 Dunn, S. A., Farrell, P. A., Budd, P. J., Arms, P. B., Hardie,

C. A. and Rendo, C. J. F/A-18A ¯ight ¯utter testingÐlimitcycle oscillation or ¯utter. In International Forum on

Aeroelasticity and Structural Dynamics, Madrid, Spain,

2001.6 Dickenson, M. CF-18 ¯ight ¯utter test (FFT) techniques.

AGARD-CP-566, Advanced Aeroservoelastic Testing and

Data Analysis, 1995.7 Hancock, G. J., Simpson, A. and Wright, J. R. On the

teaching of the principles of wing ¯exure/torsion ¯utter.

Aeronaut. J., 1985, 89(888), 285±305.8 Richardson, M. and Formenti, D. L. Parameter estimation

from frequency response measurements using rational

fraction polynomials. In Proceedings of the 1st IMAC, 1982.

APPENDIX 1

Wing/control model parameters

The wing/control structure was assumed to have auniform mass distribution of m ˆ 1320 kg=m2, span s ˆ3:5 m and chord c ˆ 0:7 m. Also, xf ˆ 0:4c ˆ 0:28 m andxh ˆ 0:75c ˆ 0:525 m were chosen. The moments ofinertia could then be calculated from ®rst principlesand had the following values: Ig ˆ 13 205, Iy ˆ 148,Ib ˆ 8:25, Igy ˆ 396, Igb ˆ 123:8 and Iyb ˆ 25:6 kg m2.

Stiffness values were chosen such that the uncouplednatural frequencies (i.e. ignoring product moments ofinertia coupling terms) for bending, torsion and controlrotation were 2.0, 6.0 and 2.2 Hz to ensure that the¯utter mechanism involved some control rotation. Thesenatural frequency choices led to stiffness values ofkg ˆ …2p62:0†2Ig ˆ 2:0856106, ky ˆ …2p66:0†2Iy ˆ2:106105, kb ˆ …2p62:2†2Ib ˆ 1:5766103 N m=rad.Clearly, the frequencies for the coupled system aresomewhat different to these assumed frequency values,namely 1.75, 2.61 and 9.11 Hz. The aerodynamicparameters were estimated using equations found inScanlan and Rosenbaum [1] and the followingvalues were determined: a ˆ 6:283, ac ˆ 3:826, am ˆ¡ 0:076, b1 ˆ ¡ 0:035, b2 ˆ ¡ 0:022, M _yy ˆ ¡ 0:220 andM _bb ˆ ¡ 0:035.

APPENDIX 2

Actuator characteristics

Equation (22) in the paper can be simpli®ed to

d1 _PPJ ‡ d3PJ ˆ ¡ AP _XX0 ¡ d2X0 ‡ d2Xi …33†

where the parameters d1, d2 and d3 are clear by directcomparison of equations (22) and (33). To examine thetransmissibility and impedance characteristics of thebasic actuator model, the differential pressure can beeliminated from equations (23) and (33). Using the



Laplace s domain,

FP ˆ APAPs ‡ d2d1s ‡ d3

³ ´X0 ¡

d2d1s ‡ d3

³ ´Xi …34†

Setting Xi ˆ 0 (i.e. no demand), then equation (34)yields the actuator impedance Z…s†:

Z…s† ˆ FPX0

ˆ APAPs ‡ d2d1s ‡ d3

³ ´…35†

Using s ˆ io and rationalizing the denominator of thecomplex equation yields the real and imaginary parts ofthe impedance:

Z…o† ˆ K ‡ ioJ

ˆ APd2d3 ‡ o2d1AP

d23 ‡ o2d21

³ ´‡ io APd3 ¡ d1d2

d23 ‡ o2d21

³ ´

…36†

where K and oJ are the stiffness and dampingcomponents respectively. The locus for Z with variationin o is the semicircle shown in Fig. 17. K0 and K? arethe static stiffness and oil bounce stiffness of theactuator respectively, namely

K0 ˆAPd2

d3ˆ mAPKF

AF, K? ˆ

A2Pd1

ˆ 4NA2P

d1…37†

In ¯utter calculations, the hydraulic actuator character-istics are not usually modelled in detail. Instead, valuesof K are used for a spring to study the effect of a rangeof actuator stiffness values, whereas the damping

contribution is often ignored, though it is by no meansinsigni®cant.

Putting FP ˆ 0 into the force expression yields theactuator displacement transmissibility TD for zero load,giving

TD…s† ˆX0Xi

ˆ d2APs ‡ d2

…38†

which is equivalent to a simple lag with cut-offfrequency

oD ˆd2AP

ˆ mKVAP

PS2

r…39†

In a similar way, putting X0 ˆ 0 yields the forcetransmissibility TF for the `blocked’ actuator body,with a cut-off frequency of

oF ˆd3d1

ˆ 4KVAFNKFV

PS2

r…40†

APPENDIX 3

Actuator model parameters

The actuator parameters were selected to yield astiffness value that gave a control rotation frequencysimilar to that for the basic spring, together withreasonable values of the cut-off frequencies. Also,because the transient solution of the equation in PJincludes the term exp…¡ d3=d1†, care had to be takenwith the decay rate compared to the time constant of thewing/control.

The actuator parameters were chosen as follows:AP ˆ 7:068610¡4 m2, V ˆ 5:301610¡5 m3, N ˆ 6:96108 N=m2, PS ˆ 200 000N=m2, KV ˆ 0:003, m ˆ 0:05,AF ˆ 7:854610¡7 m2, KF ˆ 34 000 N=m and h ˆ0:04 m. Thus K0 ˆ 1:536106 N=m, K? ˆ 2:66107 N=m, fD ˆ 10:7 Hz and fF ˆ 181 Hz. The naturalfrequencies for the coupled wing/control plus actuatorsystem at zero velocity are 1.87, 3.00 and 9.33 Hzcompared to the values of 1.75, 2.61 and 9.11 Hz for thebasic system with a simple spring. Note that the actualvalues of the parameters are not critical in demonstrat-ing the issues in this paper.

Fig. 17 Impedance of the actuator



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