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AIAA-98-1955

NONLINEAR AEROELASTIC ANALYSIS OF

AIRCRAFT WITH HIGH-ASPECT-RATIO WINGS

Mayuresh J. Patil, Dewey H. Hodges

Georgia Institute of Technology, Atlanta, Georgia

and

Carlos E. S. Cesnik

Massachusetts Institute of Technology, Cambridge, Massachusetts

Abstract

The paper describes a formulation for aeroelasticanalysis of aircraft with high-aspect-ratio wings. Theanalysis is a combination of a geometrically-exactmixed formulation for dynamics of moving beams andfinite-state unsteady aerodynamics with the ability tomodel dynamic stall. The analysis also takes into ac-count gravitational forces, propulsive forces and rigid-body dynamics. The code has been validated againstthe exact flutter speed of the Goland wing. Furtherresults have been obtained which give insight into theeffects of the structural and aerodynamic nonlineari-ties on the trim solution and flutter speed.

Introduction

High-aspect-ratio wings have come into promi-nence lately due to the interest in High Altitude LongEndurance (HALE) aircrafts for future military aswell as civilian missions. Military missions include,i)reconnaissance: requirements are of long range andsubstantial loiter period to be able to gather requiredinformation and to help in targeting, ii) massive air-lifter: the million pound airlifter is required to havea long range ( 12, 000 NM), capable of returning to

Graduate Research Assistant, School of Aerospace Engi-

neering. Member, AIAA.Professor, School of Aerospace Engineering. Fellow,

AIAA. Member, AHS.Boeing Assistant Professor of Aeronautics and Astronau-

tics. Senior Member, AIAA. Member, AHS.

Copyrightc1998 by Mayuresh J. Patil, Dewey H. Hodges,

and Carlos E. S. Cesnik. Published by the American Institute

of Aeronautics and Astronautics, Inc., with permission.

base after cargo delivery. HALE aircraft have to have

very high lift-to-drag ratio wings, thus leading to highaspect ratios.1

The Uninhabited Reconnaissance Aerial Vehicle(URAV) will be essentially autonomous with as fewmanual inputs as possible.2 These would require anaccurate model which is also amenable to controldesign.3 Also, in the future there is a possibility of Un-inhabited Combat Aerial Vehicles (UCAV) that willpush for high maneuverability, extending the flightenvelope into higher angles of attack.

Long wings tend to have large deflections. Thus,an accurate modeling of non-linear phenomena in

aeroelasticity would be very much a part of next gen-eration aircraft design. Such a model would alsobe required in designing a control system for theentire expected flight regime. This paper presentsthe ongoing research toward development of such amodel. High-fidelity computational techniques are al-ready available for both the structural analysis andaerodynamics,4 but the emphasis in this research istowards using a far less computationally expensivemodel. The reasons lie in the goals of this exercise,viz.,i) to develop an inexpensive but reasonably highfidelity model which might help get more insight intothe true nonlinear aeroelastic behavior and its con-trol, ii) to build an analysis tool for preliminary de-sign and control system design.

Background

Aeroelastic analysis of composite wings is a sub-ject of an ever increasing body of literature. The

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interest stems from the possibility of using direc-tional properties of composites to optimize a wing(i.e., aeroelastic tailoring). Ref. 5 presents a histor-

ical background of aeroelastic tailoring and the the-ory underlying the technology. The paper provideshistorical perspective on the development of codesand the activities of various research groups up tothat time. It is still true that in many studies, thestructural deformation model used is that a beam-likewing, since tailoring focuses on bend-twist deforma-tion coupling. Restraining the freedom of the chord-wise bending mode, which is often done, can result insubstantially different natural frequencies and modeshapes for highly coupled laminates. Also, the pru-dence of retaining rigid-body modes in flutter analysisduring design iterations was pointed out.

In Ref. 6, static aeroelastic problems such asspanwise lift redistribution, lift effectiveness, andaileron effectiveness are discussed. Two theoreticalmodels are commonly used: (1) laminated plate the-ory with elementary strip theory airloads and (2) amore general representation of the laminated wingstructure, in matrix form, with discrete-element aero-dynamics. In the latter case the box beam is char-acterized by bending stiffness EI, torsional stiffnessGJ, and bending-twist coupling K. These are de-rived from classical plate theory applied to the topand bottom flanges.

While it is true that some high-fidelity modelsexist,4 in the present research effort the use of beammodels is emphasized. When the aspect ratio is suffi-ciently high, the main concern ceases to be the plate-like deformation and turns instead to geometricallynonlinear beam-like behavior. In this case it is alsoimportant to accurately characterize the equivalentbeam properties of the wing. There are two philoso-phies concerning this. One is to assume a simpli-fied geometry, such as a box beam. Librescu and hisco-workers were among the first to use a box beammodel made up of various composite laminates for the

wing7, 8

as opposed to laminated plates. This typeof model was analyzed for static aeroelastic instabil-ities. The focus was on potentially advantageous useof bending-torsion structural coupling. Higher-ordereffects such as transverse shear flexibility and warp-ing restraint were also considered.

Aeroelastic characteristics of highly flexible air-craft is investigated by van Schoor and von Flotow.9

The complete aircraft was modeled using a few modesof vibration, including rigid-body modes. Banerjeeand his co-workers have been working on a simi-

lar problem and have investigated the optimizationof a composite cantilevered beam with frequencyand aeroelastic constraints.10, 11 The dynamic stiff-ness matrix method is used for modal analysis of thebeam. In another study, parametric analysis of boxbeams with thick walls is being conducted by Chat-topadhyay and co-workers.12 Here, higher-order lam-inate theory is used for each wall of the box beam.The authors of the present paper have also investi-gated the influence of ply angle layup on the aeroe-lastic characteristics of high-aspect-ratio wings.13, 14

The aerodynamics used in all the above inves-tigations was based upon simple Theodorsens striptheory. The drawback of k-type aerodynamics is thatresults for true aerodynamic damping at a given flightcondition cannot be obtained. Also, important com-pressibility and finite aspect ratio effects are not ac-counted for. Structurally the wing was clamped atone end thus neglecting the rigid-body modes. Theanalyses were useful in indicating the strong influenceof structural coupling, but were insufficient in termsof accuracy for practical configurations to be used forpreliminary design or control synthesis.

Nonlinear aeroelastic analysis has gathered a lotof momentum in the last decade due to understand-

ing of nonlinear dynamics as applied to complex sys-tems and the availability of the required mathemat-ical tools. Boyd investigated the effect of chordwisedeformation and steady-state lift on the flutter ofcantilevered wings.15 He used a simplified nonlinearstructural coupling between torsion and bending aris-ing due to deformation. Theodorsens strip theorywas used for aerodynamic analysis. The studies con-ducted by Dugundji and his co-workers are a com-bination of analysis and experimental validation ofthe effects of structural couplings on aeroelastic in-stabilities for simple cantilevered laminated plate-likewings.16, 17, 18 The studies include the effects of rigid-body motion and aerodynamic nonlinearities due tostall. Stall flutter is investigated using the ONERAstall model. Virgin and Dowell have looked intothe nonlinear behavior of airfoils with control surfacefree-play and investigated the limit-cycle oscillationsand chaotic motion of airfoils.19, 20 On the other handmore insight into nonlinear dynamics itself using non-linear aeroelastic behavior of an airfoil as an example

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is investigated by Strganac and co-workers.21 The im-portance of this work stems from the theoretical andexperimental work being conducted simultaneously.

Present Work

The present work has the goal of developing alow-order ( 500 degrees of freedom) high-fidelityaeroelastic model for preliminary design and controlsynthesis. In the course of the next couple of yearsthe modeling and analysis should have achieved thefollowing capabilities:

Geometrically nonlinear modeling of the wingstructures as beams

Cross-sectional analysis of arbitrary, anisotropiccross sections

Finite-state airloads modeling with compressibil-ity

Finite-state unsteady induced flow modeling

ONERA stall model

Rigid-body modes in the analysis

Control surface aerodynamics

Actuator dynamics

Reduced-order control design for aircraft stabi-lization in the flight regime, gust load alleviationand flutter suppression.

With such a modeling capability various prob-lems could be solved, and insights into higher-ordereffects gained. They include:

Aircraft flight dynamics including aeroelastic de-formation

Aeroelastic instabilities (and the damping of var-

ious modes) of a complete aircraft Limit-cycle oscillations of the complete aircraft

Parametric studies of composite wings

Effects of steady-state lift and drag on aeroelas-tic characteristics

Effect of control surface free-play

Flight dynamics of a flexible aircraft

Control system design and evaluation

Aeroservoelastic tailoring of the wing/aircraft/-control system

A nonlinear aeroelastic model of a complete air-craft has been formulated. The formulation is thesubject of most of the remaining paper. Preliminaryresults have been obtained and are presented in thefinal section, including trim results, flutter results forcomplete aircraft (flight dynamics interactions) andresults which illustrate the effects of structural andaerodynamic nonlinearities. An extended analysis ofthe effects of nonlinearities on the flight dynamics ofthe complete aircraft and control system design shall

be analyzed in a later paper.

Formulation

The theory is based on two separate works, viz.i) mixed variational formulation based on exact in-trinsic equations for dynamics of moving beams,22

and,ii) finite-state airloads for deformable airfoils onfixed and rotating wings.23, 24 The former theory isa nonlinear intrinsic formulation for the dynamics ofinitially curved and twisted beams in a moving frame.There are no approximations to the geometry of the

reference line of the deformed beam or to the ori-entation of the cross-sectional reference frame of thedeformed beam. A compact mixed variational formu-lation can be derived from these equations which iswell-suited for low-order beam finite element analysisbased in part on Ref. 22. The latter work presentsa state-space theory for the lift, drag, and all gen-eralized forces of a deformable airfoil. Trailing edgeflap deflections are included implicitly as a specialcase of generalized deformation. The theory allowsfor a thin airfoil performing arbitrary small motionswith respect to a reference frame which can performarbitrary large motions.

The formulation presented is an extension of themixed variational formulation for dynamics of movingbeams. It includes global frame motion as variable,gravitational potential and control surface dynamics.To generate the equations for this problem the onlychanges are the inclusion of the appropriate energiesin the formulation. The equations of motion are ob-tained by mere application of calculus of variation.

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First, a few words have to be said about the notationand the reference frames used.

There is no difference, per se, between the sym-

bols for scalars and those used for column matri-ces (the elements of which are measure numbers ofvectors in a certain basis). The subscript in boththe cases would indicate the object or the point re-ferred to. In the case of a column matrix, a super-script might be added to indicate with which refer-ence frame the measure numbers are associated. Therotation matrix would be denoted byCwith the su-perscripts indicating the two reference frames in be-tween which it transforms; e.g., xa =Cabxb.

There are various reference frames used in theformulation: i is the inertial reference frame, with

i3 vertically upward (needed to define the directionof gravitational forces); a is a frame attached to theaircraft, with a2 pointing towards the nose, and a3pointing upward; b is a series of frames attachedto the undeformed beam (wing) reference line, b1 isalong the reference line; B is the deformed beam ref-erence frame; finally, c and C are the undeformedand deformed reference frames of the control surfaceat the point of attachment of the control surface tothe wing.

The subscripts f, w, cs will be used to refer toproperties related to the fuselage, wing, control sur-face respectively. The subscripts 0, b,crefer to points

on the aircraft, beam reference line, control surfacereference line respectively. Note that the formulationis presented here assuming just one wing for clarity.In actual implementation an user-defined number ofwings would be allowed, thus accounting for twowings, tail wings, vertical stabilizer, canard, or anyother wing-like surfaces. Similarly the various con-trol surfaces can be attached to the various wing-likesurfaces.

The variational formulation is derived from theHamiltons principle, and can be written as,

t2t1

(K U) + W

dt= A, (1)

whereKis the kinetic energy of the system, Uis thepotential energy of the system, W represents thevirtual work done on the system, A is the virtualaction at the end of the time interval, is the vari-ational operator, and t1, t2 specify the time intervalover which the solution is required. Additional terms

from the ends of the wings will be generated once thebeam energies are represented as spatial integrations.

The kinetic energy of the system comes from

the three subsystems which have mass, viz., fuselage,wing and control surface. The kinetic energies can berepresented in the respective deformed frames as,

Kf = 1

2

MfV

a0TVa0 2Mf

a0

Va0 f+ a0TIf

a0

Kw = 1

2

0

mwV

Bb

TVBb 2mw

Bb V

Bb w

+BbT

iwBb

dx1,

Kcs = 1

2

0 mcsVCcT

VCc 2mcsCc V

Cc cs

+CcT

icsCc

dx1,

(2)where Mf, f, If are, respectively, the mass, massoffset from the aircraft center of mass, and inertiamatrix of the fuselage; m, , i are the wing inertialproperties per unit length, x1 is the running axialcoordinate along the wing and is the wing length;and column matrices V and represent the totalvelocity and total angular velocity, respectively.

Mass also leads to gravitational potential energy.Representing it by G, the expressions for the gravi-tational potential energies can be written as

Gf=MfgeT3C

ia (ua0+ f)

Gw=0

mwgeT3C

ia

ua0+ rab + u

ab + C

aBw

dx1

Gcs=0

mcsgeT3C

ia (ua0+ rab +

uab + CaB

bc+ C

BCcs

dx1(3)

whereCdenotes the rotation matrices, r denotes the

position vector, bc denotes the vector from pointb to point c. eis are the unit vectors in the i

th-direction.

The strain energy due to elastic deformation ofthe wing is given by,

Uw=1

2

0

T[S]

dx1 (4)

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where and are one-dimensional strain measures,and [S] is the 6 6 cross-sectional stiffness matrix.The stiffness matrix for arbitrary cross-section can be

obtained from VABS (Variational Asymptotic BeamSectional Analysis).25 Note that the strain energy as-sociated with the continuous deformation of the con-trol surface, which is constrained to move with thewing is also a part of Uw. This contribution is ob-tained by calculating the cross-sectional stiffness forthe complete cross section, i.e., including the controlsurface. Thus the cross-sectional stiffness will be afunction of the control surface deflection.

The control surface is assumed to have just onedegree of freedom relative to the wing. It is allowedto rotate about its reference line which is fixed in the

wing. The control surface/actuator dynamics is as-sumed here to be due to a flexible connection. Thus,an elastic strain energy associated with this flexibilitycan be written as,

Ucs =1

2

0

K(c 0)2dx1 (5)

where 0 is the control command, c is the actualcontrol surface deflection, and K is the effective ac-tuator flexibility.

Now taking the variation of individual energies,

we get,

K= Va0TPa0 +

a0THa0 +

0

VBb

TPBb +

BbT

HBb + VCc

TPCc +

Cc

THCc

dx1

(6)

where the Ps are the linear momenta, and Hs arethe angular momenta. The expressions for P and Hhave been derived in Ref. 26 and are given by,

P =

KV

T

=m

V

H=K

T =i + mV

(7)

Also,

U=G +0

TFBb +

TMBb + cM

dx1(8)

where F represents the internal force, and M repre-sents the internal moments. The expressions for F

and Mare obtained as

F

M = U

T

U

T = [ S]

M=

Ucc

=Kcc

(9)

The virtual work done on the system can be writ-ten as,

W =ua0Tfa0 +

a

0

Tma0 +

0

uab

Tfab

+a

b

Tmab + cm

dx1

(10)

where f, m are the external force and moment vec-

tors. Propulsive forces and aerodynamic lift and dragare a few examples of external forces.

Now using the kinematic relationships derived inRef. 22 and the transformed representation presentedin Ref. 26, the expressions for the velocities and thegeneralized strains can be written as

Va0 = ua0

a0 = CaiCai

T

VB

b

=CBa Va0 +a0(rab

+ uab

) + uab

Bb = CBaCBa

T+ CBaa0C

BaT

VCc =CCB

VBb +

Bb bc

Cc = CCBCCB

T+ CCB Bb C

CBT

= CBa

Cabe1+ uab e1

=

CbBCBa

Cba

Cab

(11)

where ( ) denotes derivative with respect to time, and( )

denotes derivative with respect to x1. Before pro-ceeding any further, it is necessary to define some ro-tational variables to represent the orientation of theaircraft, wing sections, and control surface. The ori-entation ofCframe with respect to B frame can rep-resented in terms of one angle, c. The orientation ofB frame with respect to a frame can be represented

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in terms of Rodrigues parameters. Rodrigues param-eters have been applied to nonlinear beam problemswith success. For the orientation of the aircraft, i.e.,a

frame, the regular use of the Rodrigues parametersare insufficient because of a singularity at rotation of180. Thus, the rotation matrix will be used as ro-tational variable. The expression for the angular ve-locity will automatically constrain the six additionalunknowns. Thus we have,

CabCBa =

1

abTab4

ab +

ab abT

2

1 +ab

Tab4

CCB = 1 0 00 cos c sin c

0 sin c cos c

(12)

where ab are the Rodrigues parameters.

Now, the expressions for the angular velocitiesand moment strain can be simplified in terms of theorientation variables as

Bb =Cba

a

b2

1 +ab

Tab4

ab + CBaa0

Cc =ce1+ C

CBBb

= Cba

ab2

1 +ab

Tab4

ab

(13)

where is the 3 3 identity matrix.

Given above are the variational forms of all theenergies, and the expressions for all the variables usedtherein. In the mixed formulation, the variable ex-pressions are enforced as additional constraints usingLagrange multipliers. Denoting the expressions of all

the variables by ()

, Hamiltons equation becomes,

t2

t1 Va0TPa0 +

a0THa0 + u

a0Tfa0 +

a

0

Tma0

G Va0T(Pa0 P

a0) a0

T(Ha0 Ha0)

Pa0T(Va0 V

a0) Ha0

T(a0 a0)

+0

VBb

TPBb +

Bb

THBb + V

Cc

TPCc

+CcT

HCc TFBb

TMBb cU

cM+ uabTfab +

a

b

Tmab + cm

+T(FBb

FBb

) + T(MBb

MBb

)

VBbT

(PBb PBb

) BbT

(HBb HBb

)

+FBbT

( ) + MBbT

( )

PBbT

(VBb VBb

) HBbT

(Bb Bb

)

+c(M M) VCc

T(PCc P

Cc

)

CcT

(HCc HCc

) PCcT

(VCc VCc

)

HCc

T

(Cc

Cc

)

dx1

dt= A(14)

The expressions for various quantities and their varia-tions can be substituted in the above equations to geta complete expression for the Hamiltons equation.

The external forces and moments in the aboveexpressions are the various loads acting on the air-craft, including aerodynamic and propulsive loads.Propulsive loads will be assumed as given. The aero-dynamic loads will be calculated as described in thefollowing section.

Aerodynamic Loads

The aerodynamic loads used are as described indetail in Ref. 23. The results are presented here. Thetheory calculates loads on a deformable airfoil under-going large deformation in a subsonic flow. Certainaerodynamic parameters for the particular airfoil are

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required and are assumed to be known empirically orthrough a CFD analysis.

Let the mean chord line deformation of the air-foil cross section be described by h(x2, t) where x2 isalong the mean chord line. The frame motion alongx2, x3-directions are, respectively, u0, v(x2, t). Let denote the induced flow due to free vorticity. LetL(x2, t) denote the distribution of force perpendic-ular to the mean chord line. Let D be the drag onthe airfoil. The integro-differential airloads equationscan be converted into ordinary differential equations(ODE) through a Glauert expansion. The ODEs arein terms of the expansion coefficients which are rep-resented by a subscript n.

1

2{Ln} = b2[M]

hn+ vn

bu0[C]

hn+ vn 0 u02[K] {hn}

b[G] { u0hn+ u0n u0vn+ u00}

1

2{D} = b

hn+ vn 0

T[S]

hn+ vn 0

+b

hn+ vn

T[G] {hn}

u0

hn+ vn 0T

[KH] {hn}

+ { u0hn+ u0n uovn+ u00}T

[H] {hn}(15)

where , b are the air density and semichord respec-tively. The matrices denoted by [K], [C], [G], [S],[H], [M] are constant matrices whose expressions aregiven in Ref. 23

The required airloads (viz., lift, moment aboutmid chord, and hinge moment) are obtained as a lin-ear combination ofLn. The theory described so far isbasically a linear, thin-airfoil theory. But the theorylends itself to corrections and modifications from ex-perimental data. Thus, corrections such as thicknessand Mach number can be incorporated very easily asdescribed in Ref. 24

Inflow Theory The inflow is obtained throughthe finite-state inflow theory.27 The inflow (a) is rep-

resented in terms ofN states 1, 2, . . ., N as

a1

2

N

n=1

bnn (16)

where the bn are found by least square method, andthe n are obtained by solving a set ofNfirst-orderdifferential equations27

a 1

22+

uTb

1 = 2

1

2n

n1 n+1

+ uT

b n =

2

n

(17)

Where, is the normalized circulation 2b

. The ex-pression for the normalized circulation is calculatedbased on the deformable airfoil model as

= {1}T[CG]{hn+ vn 1} +u0

b{1}T[K]{hn}

(18)

Stall Model The airloads and inflow model canbe modified to include the effects of dynamic stallaccording to the ONERA approach.

LTn =Ln+ uTn n 1

T= +

(19)

where

uT =

u02 + (v0+ h0 0)2

n+ uTb

n+uTb

22n =

2uT

3cn

b 2euT

ddt

(uTcn)

(20)

The parameters cn, , 2, and e must be identified

for a particular airfoil. is the correction to the cir-culation obtained for c. To calculate the correctionto Lift (L0) and drag (D) the following equationsare used, which also include the effect of skin friction

drag.

LT0 =L0 u0 cduT(v0+ h0 0)b

DT =D (v0+ h0 0)+ cduTu0b(21)

The airloads are inserted into the Hamiltonsprinciple to complete the aeroelastic model.

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Solution of the Aeroelastic System

Two kinds of solutions are to be found. i)

Nonlinear steady-state solution and stability analysisabout the steady state. ii) Nonlinear time marchingsolution for nonlinear dynamics of the system.

For stability analysis, the formulation is con-verted to it weakest form in space, while letting thetime derivatives of variable remain. This is achievedby transferring the spatial derivatives of variables tothe corresponding variation by integration by parts.Due to the formulations weakest form, the simplestshape functions can be used.22 Thus,

u = ui(1 ) + uj u= ui

= i(1 ) + j = i

F =Fi(1 ) + Fj F =Fi

M=Mi(1 ) + Mj M=Mi

P =Pi P=Pi

H=Hi H= Hi

(22)

With these shape functions, the spatial integration inEq. (14) can be performed explicitly to give a set ofnonlinear equations as described in Ref. 28. Theseequations can be separated into structural (FS) andaerodynamic (FL) terms and written as

FS(X, X) FL(X,Y, X) = 0 (23)

whereXis the column matrix of structural variablesand Yis a column matrix of inflow states. Similarlywe can separate the inflow equations into an inflowcomponent (FI) and a downwash component (FW) as

FW( X) + FI(Y, Y) = 0 (24)

The solutions of interest for the two coupled sets ofequations (Eqs. 23 and 24) can be expressed in theform

X

Y

=

X

Y

+

X(t)

Y(t)

(25)

where ( ) denotes steady-state solution and ( ) de-notes the small perturbation on it.

For the steady-state solution one gets Y identi-cally equal to zero (from Eq. 24). Thus, one has tosolve a set of nonlinear equations given by

FS(X, 0) FL(X, 0, 0) = 0 (26)

The Jacobian matrix of the above set of non-linear equations can be obtained analytically and isfound to be very sparse.28 The steady-state solutioncan be found very efficiently using Newton-Raphsonmethod. Once a steady-state solution is obtained,stability analysis about the steady state is conducted.

By perturbing Eqs. (23) and (24) about the cal-culated steady state using Eq. (25), the transient so-lution is obtained from

FSX FL

X FL

Y

0 FIY

X=X

Y=0

X

Y

+

FSX FL

X 0

FWX

FIY

X=X

Y=0

X

Y

=

0

0

(27)

Now assuming the dynamic modes to be of theform est, the above equations can be solved as aneigenvalue problem to get the modal damping, fre-quency and mode shape of the various modes. Thestability condition of the aeroelastic system at variousoperating conditions (steady states) is thus obtained.

Nonlinear dynamics of aeroelastic system

To investigate the nonlinear dynamics of the air-craft a time history of aircraft motion and deforma-tion has to be obtained. To get such a solution space-time finite elements are used. This requires thatthe formulation be converted into its weakest formin space as well as time. Thus, the spatial and tem-poral derivatives are transferred to the variations. Toget the space-time finite elements the following shape

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functions have to be used:29

u = ui(1 )(1 ) + uj(1 )

+uk+ ul(1 ) u= ui

= i(1 )(1 ) + j(1 )

+k+ l(1 ) = i

F =Fi(1 ) + Fj F =Fi

M=Mi(1 ) + Mj M=Mi

P=Pi(1 ) + Pj P =Pi

H= Hi(1 ) + Hj H= Hi(28)where and are dimensionless elemental temporaland spatial co-ordinates. With these shape functionthe equations (Eqs 23 and 24) take the form,

FS(Xi, X , X f) FL(Xi, X , X f, Y) = 0

FW(Xi, X , X f) + FI(Yi, Y , Y f) = 0(29)

where subscripts i, f, represent the variable valuesat the initial and final time. If the initial conditionsand time interval are specified, the variable values

during the time interval, and the final condition isobtained by solving the set of nonlinear equations.Thus, the solution history is obtained and the non-linear dynamic behavior can be interpreted.

Preliminary Results

Flutter and divergence results have been ob-tained for a metallic wing of Ref. 30, and comparedwith published results (available for the linear case).The results obtained indicate that the steady statesolution and the eigenvalues can be computed effi-

ciently and are accurate. The Goland metallic wingwas used as a test case since it is based on a realwing and thus gives a realistic idea of the effect ofnon-linearities in wings. Aeroelastic tailoring of com-posite box beam wing was conducted in an earlierpaper13 and will not be repeated here.

Test Case Data

0

0.5

1

1.5

2

2.5

0 5 10 15 20 25

Liftcoefficient

angle of attack (deg)

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0 5 10 15 20 25

MomentCoefficient

angle of attack (deg)

LinearStall

LinearStall

Figure 1: Linear and stall data for c and cm

The Goland wing data is given in Ref. 30. Otheraircraft data (tail data, fuselage data) was created

and added to the wing model to get a complete air-craft model as described below:

Wing half span = 20 ftWing chord = 6 ftMass per unit length = 0.746 slugs per ftRadius of gyration of wing about mass cen-

ter = 25 % of chordSpanwise elastic axis of wing = 33 % of

chord (from l.e. )Center of gravity of wing = 43 % of chord

(from l.e. )

Bending rigidity (EI

b) = 23.65 10

6

lb ft

2

Torsional rigidity (GJ) = 2.39 106 lb ft2

Tail position = 10 ft behind wing elastic axisTail half span = 4 ftTail chord = 3 ftControl surface = 50 % of chordMass per unit length (tail) = 0.373 slugs per

ftRadius of gyration of the tail = 25 % of

chordCenter of gravity of tail = 50 % of chord

(from l.e. )Mass of fuselage = 149.2 slugs

Radius of gyration of fuselage = 6 ftMass offset = 1 ft in ahead of wing elasticaxis

The aerodynamic data for the airfoil is obtainedby curve-fitting the c and cm data given in Ref. 31.Fig. 1 shows the plot of the assumed linear and stalldata. The coefficients for the dynamic stall model,

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Flutter Vel. Flutter Freq.(fps) (rad/s)

Present Analysis 445 70.2Exact Solution 450 70.7Galerkin Solution 445 70.7

Table 1: Comparison of flutter results for Golandwing

i.e. , and e for a symmetrical airfoil are given asa function of c in Ref. 32 as,

= 0.25 + 0.10(c)2

= 0.20 + 0.10(c)2

e= 3.3 0.3(c)2

(30)

As shown in Table 1, the current analysis givesthe flutter speed and flutter frequency results towithin 1 % of the exact linear flutter speed of thecantilevered wing.

Effect of Nonlinearities on Flutter

Structural as well as aerodynamic nonlinearitiesare known to affect flutter. One of the goals of thisresearch is to be able to determine up front thosecases for which nonlinear models are essential for ac-curacy. As a first step towards that goal, flutter anal-

ysis is conducted on the Goland cantilevered wing.The gravitational forces and skin friction drag areneglected in these results. Fig. 2 shows the variationof the flutter speed with increasing angle of attack.The results show the effect of structural nonlineari-ties, nonlinear static experimental aerodynamic data,and, dynamic stall model on the flutter speed.

As the angle of attack is increased, the aerody-namic load on the wing increases and so do the bend-ing and torsional displacements. The flutter speed isseen to increase due to geometric stiffening. If exper-imental static aerodynamic data are included in theanalysis, then the flutter speed increases even moredue to the lower lift-curve slope in the experimentaldata (i.e., 5.5 as compared to the theoretical valueof 2). Again there is a slight increase in flutterspeed with angle of attack due to geometric stiffen-ing. The results including dynamic stall model aremarkedly different from those without. This is dueto coupling between the structural states and the stallstates. The stall delay frequency of around 25 rad/sec

200

250

300

350

400

450

500

0 5 10 15

FlutterSpeed(ft/s)

Angle of Attack (deg)

Linear

Structural Nonlinearities

Strc. NL + Static Stall

Strc. NL + Dynamic Stall

Figure 2: Variation of flutter speed with angle of at-tack

interacts with the first two structural modes, leadsto additional coupling and coalesence, and change influtter mode. The flutter mode frequency shifts fromaround 70 rad/sec at 7 to 55 rad/sec at 12. Also asthe angle of attack is increased wing stall occurs atlower speeds and thus the lower stall flutter speed.

The effects of structural nonlinearities seem tobe small in the above test case which is a low aspectratio conventional wing. The effects would be con-siderably higher for a flexible high aspect ratio wingsused in UAVs. More work on different models will bepresented in a later paper.

Effect of Rigid-Body Motion

To quantify the effects of rigid body motion andflight dynamic interactions, the Goland wing modelwas extended ad hoc to create an aircraft model.Fig. 3 shows the root loci plot of some eigenvaluesof the system with velocity. There are two sets ofcurves, one for the cantilevered wing and another forthe complete aircraft. Again for the test case onedoes not see much differece between the two results,except that the phugoid mode is captured if the anal-ysis includes rigid-body modes. One can expect moreflight dynamic interactions for a flexible wing where

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-100

-50

0

50

100

-30 -20 -10 0 10

imaginarypartofeigenvalue(frequ

ency)

real part of eigenvalue (damping)

Cantilevered Wing

Complete Aircraft

Figure 3: Root loci plot of the aeroelastic aircraftmodel

the structural frequencies will be lower and closer tothe rigid body frequencies.16

Conclusions

An aeroelastic analysis tool has been developedwith a capability to model the complete aircraft as aset of beams. A geometrically exact mixed formula-tion for the structural dynamics analysis is coupled toa finite-state airloads model which includes both aninflow model and the ONERA dynamic stall model.Validation studies were conducted on the Golandmetallic wing and for a complete aircraft model ex-trapolated from it. The results indicate the necessityof including higher-order nonlinear effects for accu-rate aeroelastic analysis.

The aeroelasticity and flight dynamics code willbe used to investigate in depth the effects of nonlin-earities on aeroelastic behavior by studying variouskinds of aircraft. Results from that study will bepresented in a later paper.

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