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ENOC 2011 , 24-29 July 2011, Rome, Italy

Nonlinear Aeroelastic Behavior of Flexible High-Aspect Ratio Wings

Andrea Arena , Walter Lacarbonara and Pier Marzocca Dipartimento di Ingegneria Strutturale e Geotecnica, Sapienza University of Rome, Rome, Italy Mechanical and Aeronautical Engineering Department, Clarkson University, Potsdam, NY USA

Summary . The nonlinear aeroelastic behavior of High-Altitude, Long-Endurance (HALE) wings is investigated in the regime of large

displacements, rotations, and deformations. To this end, a geometrically exact structural model for the wing deforming in 3D spaceis coupled with the Beddoes-Leishman (B-L) incompressible unsteady aerodynamic model, modied to account for dynamic stall andow separation. Flutter and post-utter behaviors are investigated. The post-critical ight regime simulations employ path-followingand bifurcation tools applied to a Galerkin reduced-order version of the aeroelastic governing equations describing the motion of theHALE wing and augmented by the aerodynamic lag states to account for unsteady ow and stall.

Introduction

The exibility of HALE wings is such that large deections occur during normal ight operation. Operations at highangles of attack may lead to aeroelastic instabilities that can be primarily associated with dynamic stall. These aircraftsare susceptible to dynamic instabilities which give rise to large-amplitude bounded oscillations, commonly known asLimit-Cycle-Oscillations (LCO). Structural nonlinearities, coupled with aerodynamic nonlinear effects, can induce LCOsin the ight envelope. An extensive review of the state of the art on nonlinear aeroelasticity is provided in the monograph

by Dowell et al. [1] and the references therein. The importance of accounting for large-amplitude deformations from thestable equilibrium state is documented in [2]. In the present work, while the nonlinear structural model is geometricallyexact, the aerodynamic nonlinearities are associated with the presence of aerodynamic stall. One of the most criticalaspects in nonlinear aeroelasticity is the understanding of how nonlinearities affect the system dynamics. HALE wingsmight exhibit structural and aerodynamic nonlinearities that can potentially initiate aeroelastic instabilities both above andbelow the utter speed predicted by the linear theory [3]. The partial differential equations of motion are reduced to a setof ordinary differential equations using a Galerkin projection which are then solved using path-following procedures.

A Fully Nonlinear Aeroelastic Model for HALE Wings Undergoing Dynamic Stall

The three-dimensional parametric model is based on a geometrically exact semi-intrinsic theory which yields the equationsof motion, here obtained within the context of an Updated Lagrangian Formulation (ULF) [4, 5]. The ULF is adopted todescribe the wing kinematics and dynamics induced by the static and the unsteady aerodynamic loads. The wing exactstrain parameters are dened for both the static and dynamic congurations ( B 0, B ). By enforcing the balance of linearand angular momentum, the associated static aeroelastic (steady-state equilibrium) and dynamic aeroelastic governingequations are obtained. The stress and moment resultants ( n , m ) are dened on the wing cross section at the referencespanwise position x along the reference line and time t. The balance equations in the congurations ( B 0, B ) are expressedin terms of incremental kinematic parameters, [u 0(x), 0i (x)] and [u (x, t ), i (x, t )], respectively. The equations of motionin vector notation can be expressed as

x n (x, t ) + f (x, t ) = Au tt + t i + i

x m (x, t ) + x r (x, t ) n (x, t ) + c (x, t ) = J E t + (J E ) + i u tt(1)

where subscript t denotes differentiation with respect to time t; i is the vector listing the area static moments of inertia

with respect to the local frame C E

,b1 ,

b2 ,

b3 and is the mass density; J

E

is the inertia tensor of the wing cross sectionsper unit length referred to the elastic center C E (see Fig. 1); is the incremental angular velocity vector of the wing crosssections dened in the local reference frame such that t bk = bk . In addition, f (x, t ) and c (x, t ) denote the totalexternal forces and couples per unit length, respectively, including the aerodynamic loads, acting on the wing in the current

b2

b1

e 1

e 2

fI c

I

3

C EC AC

d E

d Ab /2

2 b

f Ac A

d W

U

W

b2

b1

C E

e2

e1

e3

b3

C E

r S

r S b 0 3

b 01

W + 3 + 03(x )

W + 3 + 3(x ,t )b 0

2

r S,0

Figure 1: 2D lifting surface, reference frames and 3D wing congurations: stress-free B , static B 0 and dynamic B .

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dynamic conguration and the damping forces assumed proportional to the section velocity u t and angular velocity . Anappropriate unsteady aerodynamic model, described within an indicial formulation, and modied to account for the effectof a nonstationary trailing edge separation point, is adopted. This model is obtained from a modied B-L formulation[6], and takes into account considerations and recommendations from RIS [7]. The unsteady lift, expressed in terms of dynamic variables, is provided in its noncirculatory and its circulatory components by the indicial formulation and denedby two state variables W i (x, t ):

E3 (x, t ) = W3 1 2

i =1 Ai + 1T U

2

i =1 Bi Ai W i (x, t ) with W i (x, t ) + 1T U B

i W i (x, t ) = W3 (x, t ) i = 1 , 2 (2)

where B i and A i are the Wagner function coefcients, W3 (x, t ) = w(x, t )/U , w is the downwash at the three-quarterchord and the time constant T U is dened as T U := b/U . In order to dene the dynamics of the ow separation atthe trailing edge, as suggested in the B-L model [7], two additional auxiliary state variables are introduced. The rststate variable W 3 (x, t ) describes the separation as a function of the pressure distribution over the wing section in termsof the lift on the airfoil, while the state variable W 4 (x, t ) is introduced to describe the dynamics of the boundary layerdetermining the separation to lag behind the quasi-steady value f FS(FS3 ). The state equations governing the phenomenadescribed above can be written as W 3 (x, t ) + 1T P W 3 (x, t ) =

1T P

C FAL (x, t ) and W 4 (x, t ) + 1T F

W 4 (x, t ) = 1T F f FS(FS3 ), where

T P is the time constant for the pressure lag and T F is a time constant for the lag in the boundary layer; such constantsare experimentally evaluated and depend on the considered wing section prole. A NACA 6315 is assumed as liftingsurface and the values suggested in [6] are adopted, T P = 1 .7 s and T F = 3 s. The unsteady aerodynamic resultant loads

accounting for the separation at trailing edge and associated dynamic stall are written in terms of the four state variablesW i (x, t ) and are introduced in the equations of motion (1) by the vectors f and c .It is worth mentioning that the steady-state equilibrium for a given ight condition is evaluated rst to account for prestresseffects; the aeroelastic governing equations are then solved by employing the Galerkin method whereby the projectionis performed via the Simpson rule over a grid of N subdomains. Figure 2a shows the utter condition through variationof the real part of the eigenvalues of the lowest three modes with the ow velocity while Fig. 2b shows the LCO in thetwisting mode and its frequency content past the utter condition.

0 5 10 15 20 25 30 35 40 45

U [m/s]

-1.75

-1.5

-1.25

-1

-0.75

-0.5

-0.25

0

0.25

0.5

R

Flapping

Lagging

Twisting

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5t [s]

-20

-10

0

10

20

30

40

3 ( l

, t )

[ d

e g

] 0 5 10 15 20 25 30 35 40 45 50 f [Hz]

0.01

0.1

1

10

Figure 2: Regions of utter (gray zone) and of stability (white zone): variation of the real part of the twisting lowest eigenvalue withthe free-stream velocity U (left). Twisting rotation at the wing tip, time history for U = 40 .2 m/s and FFT plot (right).

Conclusions

The proposed geometrically exact model of high-aspect ratio wings, coupled with an incompressible unsteady aerody-namic formulation accounting for the nonstationary trailing edge separation and associated dynamic stall, is employedto study the utter and the post-utter behavior which is characterized by large-amplitude limit cycles. Particular effortsare devoted to investigate the sub-critical, softening or hardening, behavior of such wings in the proximity of the utterboundary.

References

[1] Dowell E. H. et al. (2004) A Modern Course in Aeroelasticity. 4th Ed., Kluwer Academic Publishers.[2] Frulla G., Cestino E., Marzocca P. (2009) Critical behavior of slender wing congurations. J. Aerosp. Engrg Part G, Proceedings of the Institution of

Mechanical Engineers, 224 :587600.[3] Tang D. M., Dowell E. H. (2001) Experimental and theoretical study on aeroelastic response of high-aspect-ratio wings. AIAA Journal 39 :1430

1441.[4] Antman S. S. (2005) Nonlinear Problems of Elasticity, 2nd ed., Springer-Verlag, NY.

[5] Lacarbonara W., Arena A. (2011) Flutter of an Arch Bridge via a Fully Nonlinear Continuum Formulation. J. Aerosp. Engrg. 24:112123.[6] Leishman J. G., Beddoes T. S. (1989) A semi-empirical model for dynamic stall. J. Am. Helicopter 7:317.[7] Hansen M. H., Gaunaa M. and Madsen H. A. A. (2004) A Beddoes-Leishman type dynamic stall model in state-space and indicial formulations.

em Ris National Laboratory.