Active Aeroelastic Control of 2-D Wing-Flap Systems in anIncompressible Flowfield

Liviu Librescu∗1, Sungsoo Na†2, Piergiovanni Marzocca∗3, Chanhoon Chung†4and Moon K. Kwak‡5

∗Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061-0219, USA†Korea University, 136-701 Seoul, KOREA

‡Dongkuk University, KOREA

AbstractThis paper concerns the active aeroelastic control

of 2-D wing-flap systems exposed to an incompressibleflowfield. The goal is to implement an active flap con-trol capability as to suppress the flutter instability andenhance the subcritical aeroelastic response to gust orblast loads. To this end, a combined control law isimplemented and its performances toward suppressingflutter and reducing the vibrational level in the sub-critical flight speed range is demonstrated.

NomenclatureA System matrix for first order differential equationAi, BiAerodynamic lag state variablesAij Submatrix of Aα Elastic pitch angleα1,α2Coefficients of the two-term approximation ofthe Wagner function

b Semi-chordB Control input matrixβ Flap angleβ1,β2Coefficients of the two-term approximation ofthe Wagner function

c Dimensionless distance to flap hinge line from theelastic axis

C State matrixC (k)Theodorsen’s functiond Dimensionless distance to elastic axis from leadingedge

D (t) Duhamel’s integralΦ¡V tb

¢Wagner’s function

G Disturbance-input matrixg .h, g .αVelocity control gains in plunge and pitch

h Plunge displacementi√−1

Iα, IβInertia in pitch and of the aileronK Stiffness matrixKC Gain matrixKα,Kβ,Kh Stiffness of pitch, flap and plunge springL,M Lift and aerodynamic momentM Mass matrix

m Mass of airfoilP (t) Duhamel’s integralρ Air densityS Static momentSα, SβStatic moment of pitch and flap anglesσ Dummy time variableT Unsteady torque moment of flap springt Timeu Control vectorV Flight speedVf Flutter speedx, y Horizontal and vertical coordinatesxEA Elastic axis position from the mid-chord, positiverearward

Y Column vector of plunge, pitch, and flapdisplacements

wG Vertical gust velocityw External disturbance(·)0 ; (·)00 = d (·) /dτ ; = d2 (·) /dτ2.

(·),..

(·) = d (·) /dt; = d2 (·) /dt2[·]T Transpose of a matrix

1. IntroductionThe next generation of combat aircraft is likely to

operate in more severe environmental conditions thanin the past. This implies that such an aircraft will beexposed to blasts, fuel explosions, sonic-booms, etc.1,2

Under such circumstances, even in the condition offlight below the flutter speed, the wing structure willbe subjected to large oscillations that can result in itsfailure by fatigue. Moreover, in some special eventsoccurring during its operational life, such as escapemaneuvers, significant decays of the flutter speed canoccur, with dramatic implications for its structural in-tegrity. Passive methods which have been used to ad-dress this problem include added structural stiffness,mass balancing, and speed restrictions. However, allthese attempts to enlarge the operational flight enve-lope and to enhance the aeroelastic response result insignificant weight penalties, or in unavoidable reduc-tion of nominal performances. It is still correct to say

1Professor, Department of Engineering Science and Mechanics2Assistant Professor, Department of Mechanical Engineering3Visiting Assistant Professor, Department of Engineering Science and Mechanics4Graduate Student, Department of Mechanical Engineering5Professor, Department of Mechanical Engineering

1

44th AIAA/ASME/ASCE/AHS Structures, Structural Dynamics, and Materials Confere7-10 April 2003, Norfolk, Virginia

AIAA 2003-1414

Copyright © 2003 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

there is a need and a considerable interest in alter-native methods of increasing the flutter speed beyondthe original unaided value. All these facts fully un-derline the necessity of the implementation of an ac-tive control capability enabling one to fulfil two basicobjectives: a) to enhance the subcritical aeroelastic re-sponse, in the sense of suppressing the wing oscillationsin the shortest possible time, and b) to extend the flightenvelope by suppressing flutter instability and so, ofcontributing to a significant increase of the allowableflight speed. With this in mind, in this paper the activeaeroelastic control of a 2-D wing-flap system exposedto an incompressible flowfield will be investigated. Thismodel is able to capture most of the dynamics of a 3-D wing and for this reason is still well used in linearand nonlinear analysis.3,4 On the other hand, a clearunderstanding of control mechanism is of vital impor-tance for complex wing configurations. Moreover, thisstudy can constitute a preliminary work toward themorphing wing technology.In the last two decades, the advances of the active

control technology have rendered the applications ofactive flutter suppression and active vibrations controlsystems feasible.3−5

In a classical sense, the active flutter and vibrationsuppression control is a method employing as a pri-mary control, a control surface. Its deflection is com-manded by a suitable control law, i.e. by a relationshipbetween the motion of the 2-D wing section and thecontrol surface deflection. In the present paper, sev-eral control strategies, i.e. Plunging/Pitching Veloc-ity Feedback Control and their combination, Linear-Quadratic Regulator (LQR),6−8 Modified Bang-Bang(MBB)9 and Fuzzy Logic Control (FLC)10 are imple-mented, and some of their performances are put intoevidence. From a physical point of view, the activecontrol is achieved by deflecting the control surface ina manner that alters the overall nature of the aerody-namic forces on the wing, as to change in a beneficialway the dynamics of the wing structure.In the present paper, the aerodynamic forces for

time-dependent arbitrary motions of a 3-DOF airfoilfeaturing plunging-pitching-aileron deflection are de-rived from the Theodorsen’s equations using Wagner’sfunction. The equations of motion are presented instate-space form, suitable for control purposes.

2. Configuration of the 2-D Wing-Flap Struc-tural ModelFigure 1 shows the typical wing-flap section that

is considered in the present analysis. This model hasbeen well established for 2-D aeroelastic analyses, seee.g. Refs. 11,12. The three degrees of freedom associ-ated with the airfoil appear clearly from Fig. 1. Thepitching and plunging displacement are restrained by a

pair of springs attached to the elastic axis of the airfoil(EA) with spring constants Kα and Kh, respectively.The aerodynamically unbalanced control flap is lo-

cated at the trailing edge. A torsional flap spring is alsoattached at the hinge axis of the control surface whosespring constant is Kβ; h denotes the plunge displace-ment (positive downward), α the pitch angle (measuredfrom the horizontal at the elastic axis of the airfoil, pos-itive nose-up) and β is the aileron deflection (measuredfrom the axis created by the airfoil at the control flaphinge, positive flap-down).

3. Governing Equations of the AeroelasticModelThe governing equations pertinent to the three de-

grees of freedom aeroelastic systems can be found inthe classical aeroelasticity monographs. In matrix formthe equations governing the aeroelastic motion of a 2-Dwing-aileron system write as:12,13

M..

Y (t) +KY (t) = − £ LT (t) MT (t) TT (t)¤T(1)

In this equation the column vector of plung-ing/pitching/flapping displacements is defined as

Y (t) =

·h (t)

bα (t) β (t)

¸T(2)

while

M =

bm Sα SβbSα Iα Iβ + bcSβbSβ Iβ + bcSβ Iβ

(3)

K =

bKh 0 00 Kα 00 0 Kβ

(4)

denote the mass, and stiffness matrices, respectively.The second order aeroelastic governing equation

can be cast in a first order state-space form as:.X (t) = AX (t) +Bu (t) +Gw (t) (5)

Here A is the aerodynamic matrix, see Appendix A.The state vector is given by

X (t) =h .h (t) /b

.α (t)

.

β (t) h (t) /b

α (t) β (t) B1 (t) B2 (t) A1 (t) A2 (t)]T (6)

B1 (t) , B2 (t) , A1 (t) and A2 (t), are the aerodynamic-lag states; u(t) is control input, w (t) is an externaldisturbance represented in the present case by a time-dependent external excitation, such as a blast, sonic-boom or step pressure pulse;G is the disturbance-input2

matrix, whileB is the control input matrix that is givenby

B =1

Iβ

·³M−1 [0 0 1]T

´T0 0 0 0 0 0 0]T (7)

Via the use of the feedback control, the aeroelastic sys-tem can be stabilized and the flutter prevented. Inthis context, different types of control methodologieshave been applied, such as plunging/pitching and com-bination of velocity feedback controls, LQR, MBB andFLC.For a stability purpose, an infinite-time LQR that

leads to a full state-feedback controller of the formu(t) = −KCX, has been used. Herein, KC is thegain matrix, while X is the full state vector to be de-fined next. In particular, we use the Riccati equationto solve for an optimal control uopt.The aerodynamic load vector appearing in Eq. (1)

is expressed in terms of its components as

LT (t) = L (t) + LG (t) (8)

MT (t) =M (t) +MyG (t) (9)

TT (t) = T (t) + TyG (t) (10)

where L,M and T denote, respectively, the aerody-namic lift (positive in the upward direction), the pitch-ing moment about the one-quarter chord of the airfoil(positive nose-down), and the flap torque applied tothe flap hinge.The second terms in the expressions (8)-(10) are

due to the gust. In this respect, for the gust loadingwe have14

LG (t) =

Z t

0

ILG (t− σ)wGVdσ (11)

MyG (t) =

Z t

0

IMG (t− σ)wGVdσ (12)

TyG (t) =

Z t

0

IfG (t− σ)wGVdσ (13)

where wG is the gust vertical velocity, while ILG, IMG

and IfG are the related indicial impulse functions. Forthe incompressible flow, we have:14

ILG = 4π.

ψ (14)

IMG = ILG (1/2 + xEA/b) (15)

IfG = 0 (16)

The Küssner’s function ψ is approximated by:14

ψ (t) = 1− 0.5e−0.13t − 0.5e−t (17)

In the time domain, the aerodynamic loads have theform

L (t) = πρb2·..

h (t)− bxEA..α (t) + b

2πΦ4

..

β (t)

+V.α (t) +

V

πΦ3

.

β (t)

¸+ 2πρV bD (t) (18)

M (t) = πρb3·−xEA

..

h (t) + b

µ1

8+ x2EA

¶..α (t)

+b

4πΦ7

..

β (t) +

µ1

2− xEA

¶V.α+

V

2πΦ6

.

β (t)

+V 2

πbΦ5β (t)

¸− 2πρb2

µ1

2+ xEA

¶V D (t) (19)

T (t) = πρb2·µ

b

2πΦ4

¶..

h (t) +

µb2

4πΦ7

¶..α (t)

+

µb2

2π2Φ12

¶..

β (t) +

µbV

2πΦ9

¶.α

+

µbV

2π2Φ11

¶.

β (t) +

µV 2

π2Φ10

¶β (t)

¸+πρV bP (t) (20)

where Φi (φ) are Theodorsen’s constants,12 while φ =arccos (−xflap/b) . These transient aerodynamic loadsare obtained from a Fourier transform of the two-dimensional incompressible oscillatory coefficients, seee.g. Ref. 15,16. The functions D(t) and P (t) areDuhamel integrals given by

D (t) =

Z t

0

Φ

·(t− σ)V

b

¸Q01 (σ) dσ (21)

P (t) =

Z t

0

Φ

·(t− σ)V

b

¸Q02 (σ) dσ (22)

where Φ [V (t− σ) /b] is the Wagner’s function. In ad-dition, for a system that is initially at rest, Q01 (τ) andQ02 (τ) are expressed as13,16

Q01 (τ)µ≡ dQ1 (τ)

dτ

¶= h00 (τ) +

µ1

2− xEA

¶bα00 (τ)

+b

2πΦ2β

00 (τ) + V α0 (τ) +V

πΦ1β

0 (τ) (23)3

Q02 (τ)µ≡ dQ2 (τ)

dτ

¶=b

πΦ8h

00 (τ) +b2

π

µ1

2− xEA

¶Φ8α

00 (τ) +b2

2π2Φ2Φ8β

00 (τ) +V b

πΦ8α

0 (τ)

+V b

π2Φ1Φ8β

0 (τ) =b

πΦ8Q

01 (τ) (24)

where the dimensionless time τ is expressed as τ = tVband (·)0 ; (·)00 = d (·) /dτ ; = d2 (·) /dτ2. Q1 (τ) andQ2 (τ) are measures of the circulation about the airfoilwith flap. The standard two-term Jones’ exponentialapproximation of the Wagner’s function (incompress-ible flow field) is adopted here and is given by12

Φ (τ) = 1− α1e−β1τ − α2e

−β2τ ;α1 = 0.165;α2 = 0.335;β1 = 0.041;β2 = 0.32. (25)

D (t) and P (t) are evaluated following the procedurehighlighted in Refs. 15 and 16. Substitution of Eq.(25) in Eqs. (21) and (22) yields:

D (t) = Q1 (t)− α1B1 (t)− α2B2 (t) (26)

P (t) = Q2 (t)− α1A1 (t)− α2A2 (t) (27)

where Q1 (t) and Q2 (t) are obtained in time domainfrom Eqs. (23) and (24) via integration:

Q1 (t) =.

h (t) +.α (t)

µ1

2− xEA

¶b

+b

2πΦ2

.

β (t) + V α (t) +V

πΦ1β (t) (28)

Q2 (t) =b

πΦ8

.

h (t) +b2

π

µ1

2− xEA

¶Φ8

.α (t)

+b2

2π2Φ2Φ8

.

β (t) +V b

πΦ8α (t) +

V b

π2Φ1Φ8β (t) (29)

Herein overdots denote the derivatives with respect tothe time t. One should remark that, the aerodynamicsincorporated in the model is in the form of the indi-cial function. The motivation for its use stems fromthe fact that it enables to obtain linearized unsteadyaerodynamic loads in time domain via Duhamel’s con-volution.The indicial functions can be derived via var-ious approaches, such as rational approximation, com-putational fluid dynamics (CFD), or in an experimen-tal way. Based on the concept of indicial functions, aunified representation of linear unsteady aerodynamicloads in incompressible, compressible subsonic and su-personic flows can be developed, see Refs. 17-20.

4. Numerical SimulationThe geometrical and physical characteristics of the

2-D wing-flap system to be used in the present numeri-cal simulations are those considered in Ref. 12 and arepresented in Table I.

Table I: 2-D wing-flap section parameters, from Ref.12

b = 3ft; Kh = 502m;

xEA = −0.4; Kα = 1002Iα;

c = 1.0; Kβ = 3002Iβ;

m = 2.6883 slugsft ; ρ = 0.002378 slugsft3

Iβ = 0.151217slugs-ft2

ft ; Sβ = 0.10081slugs;

Sα = 1.61298slugs; Iα = 6.04868slugs-ft2

ft ;

Corresponding to these data the flutter speed isVF = 890 ft/sec. The critical value of the flutter speedis obtained herein via the solution of both the complexeigenvalue problem and from the subcritical aeroelasticresponse analysis. The flutter characteristics obtainedvia the present approach, are in excellent agreementwith Refs. 12,13.Within the present simulations, due to the fact that

the proportional and acceleration controls were provento be less efficient, only combined velocity feedbackcontrol laws were used. As a result, herein, plung-ing/pitching velocity feedback control and their com-bination are used. These relates the control input u (t) ,i.e. the required flap delection angle, to the decoupledplunging and pitching velocities of the main airfoil sur-face. Hence u (t) is represented according to the law:

u (t) = g .h

³ .h/b´+ g .α

.α (30)

where g .hand g .α are the corresponding control gains.

A more encompassing control law, that includesplunging/pitching deflections and plunging/pitchingaccelerations is presented in Ref. 21. As remarkedin Ref. 21, from the mathematical point of view, it canbe assumed that, instead of moving the flap with a re-quired deflection, an equivalent control hinge momentcan be incorporated into the open-loop aeroelastic gov-erning equation (5). This is analytically valid since thisexternal moment act on the flap-hinge and affect onlythe β DOF.There is no doubt that more encompassing control

laws can be implemented. In this context, two opti-mal feedback control methodologies, namely LQR, theMBB, and a FLC have been applied.In the presence of external time-dependent exci-

tations, the determination of the time history of thequantities (eh (≡ h/b) ,α,β), at any flight speed lowerthan the flutter speed, requires the solution of aboundary-value problem.6 In the absence of the controlinput, the open-loop aeroelastic response is obtained,whereas in the presence of the control, the closed-loopaeroelastic response is derived.4

Several results displaying the closed/open-loopaeroelastic response to selected types of loads are sup-plied next. Details on the blast, sonic-boom and gustloads are available in the Refs. 1,2,6,10,19.In Figs. 2a,b,c, the open/closed response time-

histories of the quantities³eh,α,β´ of the aeroelastic

system operating in the close proximity to the flutterboundary (Vf = 889 ft/sec) and subjected to a blastload, represented in the inset of the respective figures,are presented.The result reveal that in the absence of the control,

the amplitudes of the response quantities are on theverge of increasing in time, implying that the systemis in the close proximity of the flutter instability. How-ever, in the presence of the plunging velocity feedbackcontrol (identified by the gain g .

h), and especially, of the

combined pitching velocity feedback control (identifiedby the gain g .α, g .h), as time unfolds, the amplitudesdecay rapidly. In this analysis, a combined plunging-pitching velocity feedback control law will produce acombination of the effects, implying that this type ofcontrol law is more effective than that based either onplunging or pitching velocity feedback controls, and isable to suppress the flutter instability and simultane-ously, the oscillations of the wing. In addition, it can bealso shown that the pitching velocity feedback controlalone is more efficient than the plunging one.

The time-histories of quantities³eh,α,β´ of the

aeroelastic system subjected to a step pulse are de-picted in Fig. 3. It is clearly revealed that, for theopen-loop system, at the flight speed V > Vf , oscilla-tions with exponentially growing amplitudes occur.In Figs. 4a,b,c there are depicted the open-loop

plunging aeroelastic time-histories of the airfoil withflap exposed to a graded gust, for selected flight speedsbelow the flutter speed.In this connection it should be noticed that, for eh

and α, higher amplitudes correspond to lower flightspeeds, and, for the same quantities, the flutter ap-pears when the oscillation amplitudes are lower thanthose featured at lower flight speeds. This phenomenawas also highlighted in Refs. 1,18. However for β anopposite trend occurs. It is also evident that, for valuesof the flight speed larger than the flutter speed, oscil-lations with exponentially growing amplitudes occur.This is even more evident in Fig. 5 where the

open/closed-loop plunging aeroelastic time-histories ofthe airfoil with flap exposed to a step pulse, for selectedflight speeds are displayed. Whereas for flight speedsbelow the flutter speed, a very little influence of thecontrol is visible, in the sense of a marginal influenceon the time-history, at V > Vf the flutter response isconverted, by its action, into a stable response.In Figs. 6a,b,c the open-loop dimensionless plung-

ing, pitching and flapping time-histories of the aeroe-lastic system operating at three different flight speeds(V = 800 ft/sec; 880 ft/sec;V = Vf ) and exposed to ablast load are presented. With the increase of the flightspeed, the growth of the aeroelastic response deflectionis experienced.The counterparts of Figs. 6a,b for the open-loop

system exposed to a sonic-boom, r = 2, for the sameflight speed regime are depicted in Figs. 7a,b. Similarconclusion can be reached. . For a speed V < Vf , itbecomes apparent that the amplitude of the responsedecreases with the increase of the speed. However, forV = Vf the response becomes unbounded, implyingthat the occurrence of the flutter instability is impend-ing. It should be noticed that the response to sonic-boom pressure pulses involves two regimes: one thatcorresponds to the forced motion, and the other thatcorresponds to free motion. The jump is due to thediscontinuity in the load occurring at τp = 10. Forexplosive pressure pulses, where r = 1, the jump doesnot occur (Figs. 6a,b).The influence of a combined velocity feedback con-

trols on the aeroelastic response of a typical sec-tion with flap exposed to an asymmetric sonic boom(r = 1.5) is presented in Fig. 8. The results revealthat by increasing the feedback gains from g .

h= g .α = 3

lb/(ft sec) to g .h= g .α = 10 lb/(ft sec), the system is

stabilized in a shorter time.In Figs. 9 and 10 the influence on the dimensionless

displacements in pitching and flapping time-historiesto a graded gust load for control laws characterized by(g .h= 3 lb/(ft sec) ; g .

h= 10 lb/(ft sec)) and (g .α = 3

lb/(ft sec) ; g .α = 10 lb/(ft sec)) are presented. In bothcases, if the flight speed is larger than the flutter speed,the system remains unstable even in the presence of thecontrol.The optimal control methodology is likely to be

more effective from the point of view of the per-formance of the control. Figure 11 displays theopen/closed-loop pitching aeroelastic time histories ofthe airfoil impacted by a step pulse for selected valuesof the flight speed. From this plot the high efficiency ofthe application of the LQR toward stabilizing the air-foil at the flutter speed is appearing. Also this resulthas been validated, see Refs. 6,7. On the same plot,also the uncontrolled response is present for V = Vf .For the same type of external excitation, Fig.

12 shows that for the control input considered here,whereas FLC (sc1 = 9 ·107; sc2 = 1 ·107; sc3 = 3 ·10−7)is not able to stabilize the system, MBB (umax =3 · 10−7 lb/ft) proves to be efficient toward stabilizingthe aeroelastic system, .In Fig. 13a the performances of the application of

the LQR and MBB to stabilize the pitching aeroelasticresponse of the airfoil exposed to step pulse are pre-5

sented (umax = 5 · 10−6 lb/ft). Both control method-ologies appear to be very successful. However, as it isshown in Figs. 13b and 13c, for smaller control input(umax = 5 · 10−7 lb/ft in Fig. 13b and umax = 1 · 10−6lb/ft in Fig. 13c) the MBB control methodology ap-pears more efficient than the LQR that is not able tostabilize the aeroelastic system.It is worth remarking that the methodology pre-

sented here can be extended to the compressible flightspeed regimes. In this sense, proper indicial functionsfor the compressible subsonic, supersonic and hyper-sonic flight speed regimes have to be applied. In Refs.17-20 the concept of indicial function in subsonic com-pressible flow has been widely used in the case of a 2-Dairfoil with/without flaps, and in more complex wingconfigurations as well. In these works a derivation andvalidation of indicial functions for selected values of theMach number in the compressible speed range was ob-tained. Following the formulation in Refs. 17,18, a newset of indicial functions, for the plunging, pitching andflapping degrees of freedom can be adopted. However,the goal of this paper is to illustrate the methodologyand the importance of the implementation of the activecontrol on the lifting surfaces equipped with a flap.

ConclusionsResults related to the aeroelastic response and

control of 2-D wing-flap systems operating in an in-compressible flight speed and exposed to blast/gustloads are presented. Plunging/pitching velocity feed-back controls and combined controls, in the senseof plunging-pitching one, as well as optimal controlmethodologies such as LQR, MBB and FLC are consid-ered, and their relative performances are highlighted.It clearly appears that, in addition to its inherentvirtues of involving a constraint in the control input,MBB appears to be more efficient than LQR, FLC andthe velocity feedback control methodologies. Dealingwith the aeroelastic response, the necessity of address-ing it in the time-domain becomes evident. In this con-text, the aerodynamics based on the concept of indicialfunction proves to be of great efficiency. Moreover, itsuse can be extended to various flight speed regimes,i.e. compressible subsonic, transonic and supersonic,and as a result, an unified approach of this problembecomes feasible. Results addressing these additionalissues will be presented elsewhere.

AcknowledgmentL. Librescu and P. Marzocca would like to acknowl-

edge the partial support by the NASA Langley Re-search Center through Grant NAG-1-02-011. SungsooNa also acknowledges the partial support by the BasicResearch Program of the Korea Science & EngineeringFoundation, Grant No. R01-2002-000-00129-0.

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[11] Scanlan, R.H., and Rosenbaum, R., Introductionto the Study of Aircraft Vibration and Flutter, TheMacmillian Co., 1951.

[12] Edwards J.W., “Unsteady Aerodynamic Modelingand Active Aeroelastic Control,” SUDARR 504 (NASAGrant ngl-05-020-007), Stanford University, Feb. 1977.Also available as NASA CR-148019.6

[13] Olds, S.D., “Modeling and LQR Control of a Two-Dimensional Airfoil,” MS Thesis, Department of Math-ematics, Virginia Polytechnic Institute and State Uni-versity, Blacksburg, VA, April 1997.[14] Dowell, E.H. A Modern Course in Aeroelasticity,Sijthoff and Noordhoff, 1978.[15] Rodden, W.P., and Stahl, B., “A Strip Methodfor Prediction of Damping in Subsonic Wind Tunneland Flight Flutter Tests,” Journal of Aircraft, Vol. 6,No. 1, pp.9-17, 1969.[16] York, D.L., “Analysis of Flutter and Flutter Sup-pression via an Energy Method”, MS Thesis, Depart-ment of Aerospace and Ocean Engineering, VirginiaPolytechnic Institute and State University, Blacksburg,VA, May 1980.[17] Leishman, J.G., ”Unsteady Lift of an Airfoilwith a Trailing-Edge Flap based on Indicial Concepts,”Journal of Aircraft, Vol. 31, No, 2, March-April, 1994.[18] Marzocca, P., Librescu, L., Chiocchia, G. “Aeroe-lastic Response of a 2-D Airfoil in Compressible FlightSpeed Regimes Exposed to Blast Loadings,” AerospaceScience and Technology, Vol. 6, No. 4, June 2002, pp.259-272.[19] Qin, Z., Marzocca, P., Librescu, L., “AeroelasticInstability and Response of Advanced Aircraft Wingsat Subsonic Flight Speeds,” Aerospace Science andTechnology, Vol. 6, No. 3, March 2002, pp. 195-208.[20] Marzocca, P., Librescu, L., Silva, W.A., “Flutter,Post-Flutter and Control of a Supersonic 2-D LiftingSurface,” Journal of Guidance, Control, and Dynam-ics, Vol. 25, No. 5, 2002, pp. 962-970.[21] Djayapertapa, L., Allen, C.B., “Numerical Simu-lation of Active Control of Transonic Flutter,” Paper4.1.1, Proceedings of the 23rd ICAS Congress, Toronto,2002, pp.411.1-411.10.

AppendixIn order to be reasonably self-contained, a few steps

enabling one to obtain the flutter and aeroelastic re-sponse are provided here. Detailed developments aresupplied in Refs. 14,16,17.The ten first-order simultaneous differential equa-

tions of the aeroelastic airfoil-flap system can be rep-resented in the form

.

X (t) =d

dt

.

Y (t)Y (t)xA (t)

=

..Y (t).

Y (t).xA (t)

= A

.Y (t)Y (t)xA (t)

+Bu (t) +Gw (t) (31)

Y (t) =

·h (t)

bα (t) β (t)

¸T(32)

where xA (t) contain the four aerodynamic state vari-ables:

xA (t) = [B1 (t) B2 (t) A1 (t) A2 (t)]T (33)

The aerodynamic matrix is given as:

10x10A =

A11 A12 A13

A21 A22 A23

A31 A32 A33

(34)

Notice that Aij are the submatrix of A. Collecting theterms, the open-loop governing equation reduces to:£

M+ πρb2Z1¤ ..Y (t) = −KY (t)

− πρb2hZ2

.

Y (t) +Z3Y (t) +Z4xA (t)i

− πρb3

1 0 00 1 00 0 1

wL (t)wM (t)wf (t)

(35)

This system of equations can be solved for..

Y (t):

..Y (t) = − £M+ πρb2Z1

¤−1 ³πρb2Z2

.Y (t)

+¡K+ πρb2Z3

¢Y (t) + πρb2Z4xA (t)

¢−πρb3 £M+ πρb2Z1

¤−1 1 0 00 1 00 0 1

wL (t)wM (t)wf (t)

(36)

The model of the controlled lifting surface with flapunder gust loads is given by

..Y (t).

Y (t).xA (t)

=A .

Y (t)Y (t)xA (t)

−πρb3 £M+ πρb2Z1

¤−11 0 00 1 00 0 107x3

wL (t)wM (t)wf (t)

+Bu (t) (37)

where the disturbance-input matrix, G, and the exter-nal disturbance vector, w(t), are defined as

G = −πρb3 £M+ πρb2Z1¤−1

1 0 00 1 00 0 107x3

(38)

w (t) =£wL (t) wM (t) wf (t)

¤T(39)

7

In addition, the following matrices are needed:

r1 = [R1 R2 R3] ; r2 = [0 R4 R5] (40)

r3 = [R6 R7 R8] ; r4 = [0 R9 R10] (41)

3x3A11 = −

£M+ πρb2Z1

¤−1πρb2Z2 (42)

3x3A12 = −

£M+ πρb2Z1

¤−1 ¡K+ πρb2Z3

¢(43)

3x4A13 = −

£M+ πρb2Z1

¤−1πρb2Z4 (44)

A21 = I3x3; A22 = 03x3; A23 = 03x4 (45)

4x3A31 =

r1A11 + r2r1A11 + r2r3A11 + r4r3A11 + r4

(46)

4x3A32 =

r1A12

r1A12

r3A12

r3A12

(47)

4x4A33 =

−β1Vb 0 0 0

0 −β2Vb 0 0

0 0 −β1Vb 0

0 0 0 −β2Vb

+

r1A13

r1A13

r3A13

r3A13

(48)

In Eqs. (42)-(44),

Z1 =

L..h

L..α L..

β

M..hM..

α M..β

T..h

T..α T..β

(49)

Z2 =

L .h

L .α L .

β

0 M .α M .

β

T .h

T .α T .

β

(50)

Z3 =

0 Lα Lβ0 0 Mβ

0 Tα Tβ

(51)

Z4 =

LB1LB2 0 0

0 0 0 00 0 TA1 TA2

(52)

[−L (t) −M (t) − T (t)]T =−πρb2

hZ1

..Y (t) +Z2

.Y (t) +Z3Y (t) +Z4xA (t)

i(53)

As concern the coefficients Ri’s appearing in Eqs. (40)-(41), these are defined in terms of the Theodorsen’scoefficients, flight and geometrical parameters as:

R1 =b2

V; R2 =

b2

V

µ1

2− xEA

¶; (54)

R3 =Φ2b

2

2πV; R4 = V ; (55)

R5 =Φ1V

π; R6 =

Φ8b3

πV; (56)

R7 =Φ8b

3

πV

µ1

2− xEA

¶; (57)

R8 =Φ2Φ8b

3

2V π2; R9 =

Φ8V b

π; (58)

R10 =Φ1Φ8V b

π2(59)

. The first order differential equation for the aerody-namic lag states is written as16

.xA (t) = PxA (t) +

.Q (60)

or in a detailed form:

.xA (t) =

h .

B1 (t).

B2 (t).

A1 (t).

A2 (t)iT

=

−β1Vb 0 0 0

0 −β2Vb 0 0

0 0 −β1Vb 0

0 0 0 −β2Vb

B1 (t)B2 (t)A1 (t)A2 (t)

+

.Q1 (t).

Q1 (t).

Q2 (t).Q2 (t)

(61)

.Q1 (t) = [R1 R2 R3]

..Y (t)

+ [0 R4 R5].

Y (t) = r1..

Y (t) + r2.

Y (t) (62)

.Q2 (t) = [R6 R7 R8]

..Y (t)

+ [0 R9 R10].

Y (t) = r3..

Y (t) + r4.

Y (t) (63)

In the time domain the aerodynamic loads have theform presented in Eqs. (18)-(20). As stated in Ref.16, the formulation presented here eliminates the needof evaluating the Duhamel integrals. In this sense, theaerodynamic governing equations reduces to a systemof first-order differential equation.

8

x 10-'.2 10"' 2 10'

1-510-7 110'

h 110-7 h 0

510.. -110'V =600 WG 20 ..": '

f 10

0 -210'0 10 20 ~ 40 0 10 15 20 25 30

T ime(sec) x 10-1. T ime(sec)

F ig. 5 Open/ closed-loop plunging time-history of the 1-510'

aeroelastic system under a s tep pulse for selected flight V f=V F110'speeds . E ffect of the control. 880

x1rr'. 510'210'

V K V a 0I F,1510 V =880

-510'

110'-110'

h

5 10. -1510'0 6-25 125 1675 25

T ime(sec)

F ig. 7a,b Open-loop plunging/pitching time-

histories of the aeroelastic system for selected flight

-510'0 12 15 speeds under a sonic-boom pulse (tp = 10sec,r = 2).

T ime(sec) 210..X 10-'.

1 10. v.v 1510..V =600 880 I F

, . ! 1 1 ! II 'I ii !'(i i !! , i! 110..

510. i~I~~,~I, h 510..

a i~~§!~ii~~i:~~~§))ii~~\~

' ~;.""f' I".!";j'r l'. '~: -510..~1!!11.11 j!j' liijI11 ijri . .-510. : ! 1 ! i ! I! 1i . i! !: -110

' I : I ' , . , , : : :; , : : ; : ; : ,: ! ! i i. . ! -1.510..

0 10 20 30 40 50

-110' 10 T ime(sec)0

T . ( ) F ig. 8 Open / closed-loop plunging time-history of theIme sec

-,. aeroelastic system under a sonic-boom pulse (tp =310. 10sec,r = 2). P lunging/pitching and combined ve-

210' locity feedback controllaw applied.1500

x1rr'. ~ g.K 3 110. h-g.=101000 h

J 3

500-110. a .

-210' 0

-3 10' -500 w

0

T ime(sec) T ime(sec)-1000

0 10 20 30 40 50F ig. 6a,b,c Open-loop plunging/pitching/flapping F ig. 9 Open/closed-loop pitching time-history of the

time-histories of the aeroelastic system under an ex- aeroelastic system under a graded gust. P lunging ve-

plos ive blast ( tp = 10 sec, r = 1) , for selected flight locity feedback controllaw applied.

speeds . 10

-300

-200

-100

0

100

200

300

0 10 20 30 40 50

β

Time(sec)

x 10-10

uncontrolledw

G

gα =3

gα =10..

Fig. 10 Open/closed-loop flapping time-history of theaeroelastic system under a graded gust. Pitching ve-locity feedback control law applied.

-2 10-8

0

2 10-8

4 10-8

6 10-8

0 10 20 30 40

uncontrolledcontrolled

Time(sec)

α

Vf=600 700 800

Vf=V

F

wG

(Vf=V

F)C

Fig. 11 Open/closed-loop (LQR) pitching time-history of the aeroelastic system under a step pulsefor selected flight speeds.

0

2

4

6

8

10

12

14

0 10 20 30 40 50

MBBFLC

h

Time(sec)

~

wG

uncontrolled

FLCMBB

Fig. 12 Open/closed-loop (MBB and FLC) plung-ing time-history of the aeroelastic system under a steppulse.

0

1

2

3

4

0 4 8 12 16 20

MBBLQR

α

wG

x10-8

uncontrolled

0

1

2

3

4

0 4 8 12 16 20

MBBLQR

α

Time(sec)

wG

x10-8

uncontrolled

0

1

2

3

4

0 4 8 12 16 20

MBBLQR

α

Time(sec)

wG

x10-8

uncontrolled

Fig. 13a,b,c Open/closed-loop (MBB and LQR)plunging/pitching/flapping time-histories of the aeroe-lastic system under a step pulse for three selected val-ues of umax.11