Influence of Transonic Flutter on the Conceptual

Design of Next-Generation Transport Aircraft

Max M. J. Opgenoord∗, Mark Drela†, and Karen E. Willcox‡

Massachusetts Institute of Technology, Cambridge, MA, 02139

Transonic aeroelasticity is an important consideration in the conceptual design of novelaircraft configurations. The influence of aeroelasticity on conceptual design tradeoffs isinvestigated by extending a previously developed low-order physics-based airfoil fluttermodel to swept high aspect ratio wings. This physics-based flutter model uses the flowfield’slowest moments of vorticity and volume-source density perturbations as its states. Themodel is calibrated using off-line 2D unsteady transonic CFD simulations. The resultingaeroelastic system combines this calibrated aerodynamic model with a beam model. Thelow computational cost of the model permits its incorporation in a conceptual design tool fornext-generation transport aircraft. The model’s capabilities are demonstrated by findingtransonic flutter boundaries for different clamped wing configurations, and investigating theinfluence of transonic flutter on the planform design of next-generation transport aircraft.

Nomenclature

A Aspect ratioAa,Ba State-space matrices of aerodynamic

systemAΓ, AΓ, Aκ Coefficients of Γ evolution equationBΓ, Bκ, Bκ Coefficients of κx evolution equationCth (ω) Theodorsen’s functionEI Bending stiffnessGJ Torsional stiffnessIy Mass moment of inertia per span around

elastic axisL Lift per spanM Internal bending momentM∞ Freestream Mach numberMCR Cruise Mach numberMea Aerodynamic moment per span around

elastic axisS Internal shear forceSy Mass unbalance per spanT Internal torsion momentV Velocity fieldVµ Flutter speed indexV∞ Freestream velocityV⊥ Wing-perpendicular velocityb Wing spanc Airfoil chord lengthc`0 Baseline sectional lift coefficientd Distance between quarter-chord and

elastic axis

h Vertical position, positive downwardl Effective wing lengthm Mass per spant Timeua State-space control vector of

aerodynamic systemx, y, z Aircraft coordinate systemx, y, z Coordinates aligned with swept wingxa State-space vector of

aerodynamic systemxcg Position of center of gravity relative to

wing leading edgexea Position of elastic axis relative to wing

leading edge

Symbols

Γ Circulation strengthγ Local dihedral angleθ Pitch angleκx x-doublet strengthΛ Sweep angleλ Taper ratioσ Volume-source strengthτ Thickness ratio of airfoilχfl Maximum real eigenvalue of aeroelastic

systemω Angular velocityω Reduced frequency

∗Graduate student, Department of Aeronautics and Astronautics, mopg@mit.edu, Student Member AIAA†Terry J. Kohler Professor of Aeronautics and Astronautics, Department of Aeronautics and Astronautics, drela@mit.edu,

Fellow AIAA‡Professor of Aeronautics and Astronautics, kwillcox@mit.edu, Associate Fellow AIAA

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I. Introduction

Meeting NASA’s ambitious N+3 goals of reducing aircraft fuel burn by 60% by 2035, requires a movetowards novel aircraft configurations. Such novel aircraft configurations usually have characteristically

high aspect ratios, examples being the Truss-Braced Wing concept1 or D8.x aircraft.2 However, large aspectratios lead to larger flexibility in the wing, and thereby more associated aeroelasticity problems. Fluttertherefore becomes an even more important design consideration for these aircraft concepts, and requiresflutter constraints to be included in the design of these aircraft as early as possible, preferably in theconceptual design stage. This will prevent costly design changes in later design phases, or during flighttesting.

A wealth of literature is available on static aeroelastic optimization of aircraft wings for both conventionaltube-wing configurations,3–5 and next-generation configurations.6 Dynamic aeroelasticity, however, is oftennot included in these studies. Kennedy et al.7 included flutter constraints in their aircraft wing optimization,but the compressibility effects were accounted for using Prandtl-Glauert’s rule, resulting in not being able toaccurately predict the flutter boundary in transonic flow. Moreover, all aforementioned methods use high-fidelity aerodynamic and structural analysis methods, and are therefore too expensive to use in conceptualaircraft design. Conceptual aircraft design tools, such as SUAVE,8 often do not include flutter constraints.Mallik et al.9 did include transonic flutter constraints in an MDO setting for investigating Truss-BracedWing designs, but the flutter model consisted of Theodorsen theory with compressibility corrections, limitingits accuracy for transonic flows.

There are several ways to predict unsteady loads on an aircraft.10 The three most common ones are striptheory (2D unsteady airfoil theory with 3D corrections),11 the doublet-lattice method,12 and the unsteadyvortex lattice method (UVLM).13,14 Considering that our goal is to investigate aircraft concepts with highaspect ratios, the use of strip theory here is appropriate.

The theory of flutter behaviour of wings has been investigated thoroughly for decades now, for instancein the seminal work of Theodorsen,15 who considered flutter of a uniform, straight wing in incompressibleflow. Barmby et al.16 extended Theodorsen’s theory to swept wings in incompressible flow. Yates extendedthis theory to subsonic, supersonic and hypersonic flow—the method loses accuracy for transonic flow—witha modified strip theory and validated the theoretical results with experimental results.17

Thus, there exists a need for a flutter model that is cheap enough to allow for potentially thousandsof flutter evaluations of wing designs in the conceptual design phase, but which is also accurate enough tocapture subtle design trade-offs for commercial transport aircraft operating in the transonic flow regime.We therefore extend a previously developed and validated airfoil flutter model to swept aircraft wings andimplement it in a physics-based conceptual aircraft design tool. Section II describes the airfoil flutter modelused, together with its extension to high-aspect ratio swept wings. Section III discusses the aeroelasticbehavior of several wing configurations computed using the flutter model, while Section V concludes thepaper.

II. Methodology

In this section, the 2D transonic flutter model developed in Ref. 18 is extended to swept aircraft wings.Section II.A briefly explains the 2D flutter model, while Section II.B discusses its extension to 3D. Section II.Cpresents the structural model for the aircraft wing. The flutter model is implemented in a conceptual aircraftdesign tool which is explained in Section II.D.

II.A. Airfoil Transonic Flutter Model

We use strip theory to predict flutter for aircraft wings, thereby relying on a locally 2D approximation ofthe aerodynamics and structures of the wing. The seminal work by Theodorsen15 is widely used for flutterprediction of airfoils, but only works for incompressible flows. Instead, we use a physics-based low-ordermodel18 for transonic flutter that is calibrated using high-fidelity Euler/RANS simulations. The approachfor constructing that low-order model is shown in Fig. 1, where the model is derived from first principlesof transonic unsteady flow, and this model’s unknown coefficients are calibrated with high-fidelity unsteadyCFD simulations using Dynamic Mode Decomposition.19

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G = arg min ‖X ′′ − GΩ‖Calibratelow-ordermodel

xκ

Γ

n

V

Physics-based low-order modelwith unknown coefficients

M∞

Vµ

Combine with heave-pitch structural model,extract eigenvalues and flutter speeds

High-fidelity simulations

+

−

−

+− +

ωω( )r,t −−−

∆

V

( )σρ

r

∆

V,t V

∆

ρρ

t+( )1

−−− =−

Vorticity and volume-sourcerepresentation of velocity field

Figure 1. Summary of transonic airfoil low-order flutter model construction, detailed in Ref. 18.

The low-order model uses the circulation and x-doublet perturbation ∆Γ, ∆κx from their steady transonicflow values as its aerodynamic state variables, such that the aerodynamic state-space model can be writtenas,18

d

dt

∆Γ(t)

∆κx(t)

∆κx(t)

=

−AΓ

AΓ−AκAΓ

0

0 0 1

BΓ Bκ Bκ

︸ ︷︷ ︸

Aa

∆Γ(t)

∆κx(t)

∆κx(t)

︸ ︷︷ ︸

xa

+

V∞AΓ

c/4AΓ

1AΓ

0 0 0

0 0 0

︸ ︷︷ ︸

Ba

∆θ(t)

∆ω(t)

∆w(t)

︸ ︷︷ ︸

ua

, (1)

where ∆θ is the pitch angle perturbation, ∆ω is the pitch rate perturbation, ∆w is the vertical velocityperturbation, V∞ is the free-stream velocity, and c is the chord. The coefficients (AΓ, AΓ, Aκ, BΓ, Bκ, andBκ) are calibrated using high-fidelity CFD simulations. For the 2D aerodynamic model, these coefficientsare calibrated as a function of freestream Mach number M∞, baseline lift coefficient c`0 , and thickness ratioτ .

For the airfoil flutter model, the aerodynamic model (1) is combined with the typical-section structuralmodel which has four states – heave perturbation ∆h, vertical velocity perturbation ∆w, pitch angle pertur-bation ∆θ, and pitch rate perturbation ∆ω.20,21 This gives one state-space system with seven states (∆h,∆θ, ∆w, ∆ω for the structural model; ∆Γ, ∆κx, ∆κx for the aerodynamic model). An example of a flutterboundary with transonic dip found using this airfoil flutter model is shown in Fig. 1. This model has beenvalidated for a standard transonic flutter case in Ref. 18.

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II.B. Wing Flutter Model

To assess the flutter characteristics of aircraft wings, we rely on strip theory11,22 using the low-order modeldescribed in Section II.A. Note that we focus on wings with high aspect ratios, therefore the use of striptheory is appropriate. For our model, this means that we use the perpendicular-plane Mach number M⊥and lift coefficient c`⊥ for selecting the appropriate coefficients for the transonic aerodynamics model.

y

x

y

x

l

Elastic axis

Effectivetip

Effectiveroot

Λ

V∞ V⊥

d

θ

h

EA(c/4)

CG

l

Figure 2. Swept wing considered in model. Figure adapted from Ref. 16,18.

Commercial transport aircraft commonly have swept wings to enable higher cruise Mach numbers. Weconsider the wing geometry in Fig. 2 with sweep angle Λ. The effective length of the wing l = l/ cos Λ, andall section parameters such as chord length, moment of inertia, etc., are based on sections perpendicular tothe elastic axis.

For the aerodynamics model, we must account for the changes to the velocity distribution due to theswept wing. The vertical velocity at any point on the wing is found to be16

Un(x, t) = w + V∞∂h

∂ysin Λ + V∞θ cos Λ + xω + xV∞

∂θ

∂ysin Λ, (2)

where x, y are the coordinates aligned with the swept wing. From Eq. (2), Barmby et al.16 found thecirculatory lift per span for swept wings in incompressible flow to be

Lc = πρV⊥cCth(ω)

[w + V⊥

∂h

∂ytan Λ + V⊥θ +

c

4

(ω + V⊥

∂θ

∂ytan Λ

)], (3)

where V⊥ is the wing-perpendicular velocity and Cth(ω) is Theodorsen’s function15 with ω the reducedfrequency. Following the same approach, here the evolution equation for the circulation strength is definedto be,

∆Γ = −AΓ

AΓ

∆Γ− AκAΓ

∆κx +1

AΓ

(∆w + V⊥

∂∆h

∂ytan Λ

)+V⊥AΓ

∆θ +c/4

AΓ

(∆ω + V⊥

∂∆θ

∂ytan Λ

). (4)

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Lastly, the non-circulatory forces and moments used in the model are16

∆Lnc =1

4πρ∞c

2

[∆h+ V⊥

∂∆h

∂ytan Λ + V⊥∆θ

]− 1

4πρ∞c

2(xea −

c

2

)[∆θ + V⊥

∂∆θ

∂y

](5)

∆Mnc = −1

4πρ∞c

2Vn

[(3

4c− xea

)∆θ +

1

2V⊥∂∆θ

∂ytan Λ

]− ρ∞πc4

128

[∆θ + V⊥

∂∆θ

∂ytan Λ

]

+1

4ρ∞πc

2(xea −

c

2

)[∆h+ V⊥

∂∆h

∂ytan Λ−

(xea −

c

2

)∆θ + V⊥

∂∆θ

∂ytan Λ

]. (6)

Note that this formulation ignores the wing camber effect in the noncirculatory forces and moments, whichis known to have a negligible effect on flutter.16

Finally, note that—using the same assumptions as Barmby et al.16—this unsteady aerodynamics modelignores (1) variations in the trailing vorticity, (2) any variation of the airloads in the spanwise direction, and(3) any three-dimensional wing tip effects. These assumptions are quite reasonable for such high reducedfrequencies, because then the wake is not fully developed. Flutter is typically observed for high reducedfrequencies, making these assumptions appropriate for this work.

II.C. Wing Structural Model

For the structural part of the model, we use Bernoulli-Euler beam theory. Following Bisplinghoff et al.,20

the beam equations are,

m(y)∆h(y, t) + Sy(y)∆θ (y, t) +∂2

∂y2

[EI(y)

∂2∆h(y, t)

∂y2

]= −∆L′(y, t) (7a)

Iy(y)∆θ(y, t) + Sy(y)∆h(y, t)− ∂

∂y

[GJ(y)

∂∆θ(y, t)

∂y

]= ∆M ′

ea(y, t) (7b)

with boundary conditions,

∆h(0, t) = 0,∂∆h

∂y(0, t) = 0,

∂2∆h

∂y2(l, t) = 0,

∂3∆h

∂y3(l, t) = 0, ∆θ(0, t) = 0,

∂∆θ

∂y(l, t) = 0. (7c)

This system can be rewritten as

m(y)∆h(y, t)− Sy(y)∆θ (y, t) +∂∆S (y, t)

∂y= −∆L′(y, t) (8a)

Iy(y)∆θ(y, t) + Sy(y)∆h(y, t)− ∂∆T (y, t)

∂y= ∆M ′

ea(y, t) (8b)

where∂∆M∂y

= ∆S, ∂∆γ

∂y=

∆MEI

,∂∆h

∂y= ∆γ,

∂∆θ

∂y=

∆TGJ

, (8c)

with boundary conditions

∆h(0, t) = 0, ∆γ(0, t) = 0, ∆M(l, t) = 0, ∆S(l, t) = 0, ∆θ(0, t) = 0, ∆T (l, t) = 0. (8d)

To solve the system in Eq. (8), the wing is discretized in several sections, as shown in Fig. 3. The sectionalproperties are all computed using the assumed wingbox shape as shown in Fig. 4. The thickness ratio τ isused to determine the coefficients of the aerodynamic model, as explained in Section II.A. It is assumed thatthe elastic axis is at the center of the wing box. Wing box design variables such as spar and skin thicknessesare obtained from the conceptual design tool in which this flutter model is included.

The system in Eq. (8) is solved using finite differences, specifically the trapezoidal rule. However, Eq. (8)has 6 × nbeam structural parameters in the aeroelastic system, of which only two have inertial terms (∆h

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xz

y

∆θ(y)

meng, ¯Ieng

TH

−∆h(y)

∆reng

mi, EIi, GJi,Iθi , xcgi , xeai

i

Figure 3. Discretized beam model used in structural part of the flutter model.

and ∆θ). Such a large system slows down the eigenvalue computation. Instead, we can reduce the size ofthe system through Schur complements. Consider the discretized system of Eq. (8),

Ehh 0 0 0 Ehθ 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

Eθh 0 0 0 Eθθ 0

0 0 0 0 0 0

∆h

∆γ

∆M∆S∆θ

∆T

+

0 0 0 AhS 0 0

Aγh Aγγ 0 0 0 0

0 AMγ AMM 0 0 0

0 0 ASM ASS 0 0

0 0 0 0 0 AθT

0 0 0 0 AT θ AT T

∆h

∆γ

∆M∆S∆θ

∆T

=

−∆L′

0

0

0

∆M′ea

0

(9)

The matrices Aγh, AMγ , ASM, and AT θ are all tightly banded and therefore cheap to invert. This allowsfor taking the Schur complement of the system in Eq. (9) to obtain a system with only h, θ,[

Ehh Ehθ

Eθh Eθθ

][∆h

∆θ

]+

[Ahh 0

0 Aθθ

][∆h

∆θ

]=

[−∆L′

∆M′ea

]. (10)

This structural model is then coupled to aerodynamic model described in Section II.B through theaerodynamic lift and moment, yielding a 7× nbeam descriptor state-space system.

The engine adds a large amount of inertia and mass to the wing, and it is therefore important to representin a flutter model. The engine is modeled as a point mass meng with inertia ¯I eng and angular momentumHeng, which are rotated into the swept coordinate system. As with the rest of the model, we only consider

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c⊥

τc⊥ EA

x

z

Figure 4. Wing box cross section.

perturbations from the steady state. The thrust and moment generated by the engine are thus not takeninto account.

The influence of the engine on the structural model comes through a discontinuity in ∆S, ∆M, and ∆Tat the section to which the engine is attached, as shown in Fig. 5. These discontinuities are captured witha zero-length grid interval at the engine attachment point.

y

x

∆reng

i

∆yeng

∆xeng

∆Feng · k

∆Si+

∆Si−

∆S

y

∆Meng · y

∆Mi+

∆Mi−

−∆M

y

Figure 5. Inclusion of engine in beam model. Example curves for ∆S, ∆M are shown for h < 0, notethat a similar discontinuity is observed in ∆T .

The inertial-reaction forces and moments on the ith section on the wing due to the engine are,23

∆Feng = −meng ∆aeng (11a)

∆Meng = ∆reng ×∆Feng −∆ωi ×Heng − ¯I eng ∆ωi −

:' 0

∆ωi ×(

¯I eng ∆ωi

)(11b)

where ∆reng = [∆xeng,∆yeng,∆zeng]T

is the distance vector between the ith beam section and the engine, and∆Feng and ∆Meng are the force and moment, respectively, on the ith beam section. ∆ωi is the rotation rate

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perturbation of the ith beam section with respect to the aircraft body axes, defined as ∆ωi = [−∆γ,∆θ, 0]T .We observe that ∆ωi×Heng yields only an in-plane bending moment for unswept wings, while in-plane motionis not modeled here. Note that the last term in Eq. (11b) is a quadratic term in the perturbation rates, andcan therefore be neglected. Finally, ∆aeng is the inertial acceleration perturbation of the engine, defined as

∆aeng = ∆ai + ∆ωi ×∆reng +

:' 0

∆ωi × (∆ωi ×∆reng) (11c)

where ∆ai is the acceleration perturbation of the ith beam section. Again, the last term in Eq. (11c) is ahigher-order term in the perturbation rates and hence is omitted. Note that in this formulation, the aircraftfuselage is assumed to not participate significantly in the flutter modes.

The discontinuity in the shear force and bending moments results in additional terms in Ehh, Ehθ, Eθh,and Eθθ in Eq. (10).

II.D. Conceptual Aircraft Design Tool

We use the Transport Aircraft System OPTimization (TASOPT)24 tool for the conceptual design of next-generation aircraft concepts. TASOPT was developed at MIT as part of a NASA project to design aircraftto meet aggressive fuel burn, noise, and emission reduction goals for the 2035 timeframe.25,26 To assess theperformance of next-generation aircraft concepts, this conceptual design tool is primarily developed fromfirst principles rather than using historical correlations. It uses low-order physical models implementingfundamental structural, aerodynamic, and thermodynamic theory, and relies on historical correlations onlyfor the weight of secondary structure and aircraft equipment.

TASOPT can both size the aircraft for a particular mission (range and payload) and optimize the aircraftby varying, for instance, cruise altitude, cruise lift coefficient, aspect ratio, wing sweep, etc., to minimizemission fuel burn. TASOPT can therefore be used to model existing aircraft, assess their off-design perfor-mance, and perform a sensitivity analysis for an aircraft. TASOPT is also used to assess the influence of newtechnologies, such as advanced materials, on an airframe design. Finally, this tool is used to design entirelynew aircraft for a set of missions.

As with most aircraft conceptual design tools, dynamic aeroelasticity is currently not a design consider-ation in TASOPT. To assess the influence of transonic flutter on the conceptual design of next-generationaircraft, the developed transonic flutter model is implemented in TASOPT here.

Flutter is indicated using the eigenvalues of the aeroelastic system—the overall system that consists ofthe wing aerodynamics model and beam model. If the largest real part of the eigenvalues of the aeroe-lastic system—denoted here as χfl—becomes positive, flutter occurs. Flutter constraints are included in aconceptual design tool by constraining the value of χfl for several different points in the flight envelope.

The design routine in this conceptual design tool wraps an optimizer around a weight-convergence rou-tine. During the optimization several design parameters, such as cruise altitude, wing sweep, aspect ratio,etc., are varied to minimize the fleet-wide fuel energy consumption per payload-range. Constraints can beincorporated in this optimization loop, such as minimum balanced field length and maximum wing span.Here, the requirement for no flutter occuring for K operating points is included in the design process as asum of constraints in this optimization loop via a penalty term in the objective function,

f(x) = f(x) +

K∑k=1

ν max [0, χfl,k − χfl,max]2 , (12)

where x are the design variables, f(x) is the overall objective function, f(x) is a function that includesmission fuel burn with penalty terms for, e.g., balanced field length and maximum wing span, and ν is asuitable weighting factor. χfl,k is largest real part of the eigenvalues of the aeroelastic system for the kthoperating point, and χfl,max is the largest value tolerated, typically −0.005/s.

The flutter model is included in TASOPT using a database of flutter coefficients with c`0 , M⊥, τ asthe database independent variables. We calibrate 2D low-order models, using the methodology described inSection II.A, for several thickness ratios τ of an airfoil family (Fig. 6b) at different baseline lift coefficientsc`0 and Mach numbers M⊥ and store these in a database. An example of how such a flutter coefficient varieswith c`0 and M⊥ is shown in Fig. 6a. Whenever TASOPT queries a flutter evaluation, the appropriate fluttercoefficients are found through spline interpolation using the flutter coefficient database.

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0.15

1

0.2

0.8

0.25

0.6 1

0.3

0.80.6

0.4

0.35

0.20.40

(a) Example of flutter coefficient stored in database. This exampleis for the 11% thick C-airfoil.

(b) C-airfoil family used for the D8.

Figure 6. A database of flutter coefficients is generated for an airfoil family as a function of M⊥, c`0 ,and τ .

TASOPT uses a parameterized spanwise loading p(y), whose shape is either specified, or optimized forthe best tradeoff between structural weight and induced drag (computed via a Trefftz-Plane analysis). Here,this same loading is used to obtain the local lift coefficient,

c`⊥(y) =2p(y) cos Λ

ρ∞V 2⊥ c⊥(y)

, (13)

where c⊥(y) is the local chord length perpendicular to the elastic axis. This c`⊥ is then used to get theappropriate flutter model for each section of the wing using the flutter coefficient database.

III. Influence of Wing Geometry on Transonic Flutter Boundaries

The flutter model explained in Section II is used here to quantify the influence of several parameters of anaircraft wing on the flutter boundary. First, we compute the flutter boundary of a clamped wing in a windtunnel-like set-up in Section III.A. Second, we use the flutter model coupled with TASOPT to evaluate theinfluence of the wing parameters on the flutter damping values in Section III.B where each aircraft on theflutter boundary is a weight-converged aircraft design.a

III.A. Clamped Wing

We consider a wind tunnel-like set-up for a straight, swept, tapered wing as shown in Fig. 2. This problemhas twelve nondimensional parameters, which are listed in Table 1.

Table 1. Nondimensional parameters for straight, swept, tapered wing.

Parameter Parameter

A Aspect ratio Λ Sweep angle

M∞ Freestream Mach number (xea/c) Relative elastic axis position

c`0 Baseline lift coefficient (xcg/c) Relative center of gravity position

τ Thickness ratio (ωθ/ωh) Ratio of uncoupled pitch and heave frequencies

λ Taper ratio µ Apparent-mass ratio

Vµ Flutter speed index r2θ Radius of gyration

aNote that we are not making any claims on the general applicability of these trends; the onset of flutter is sensitive to smallchanges in the wing parameters and the results described herein only apply to the particular nondimensional parameters of themodels described.

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The apparent-mass ratio µ, flutter speed index Vµ, and radius of gyration r2θ are defined as,

µ =4mmac

πρ∞c2mac

(14a)

Vµ =2 V∞|χfl=0√µωθcmac

(14b)

r2θ =

4Iy,mac

mmacc2mac

(14c)

where,

ωθ =

√4 (GJ)mac

mmacb2c2mac

(15a)

ωh =

√(EI)mac

mmacb4. (15b)

These non-dimensional parameters determine the planform shape and sectional properties of the meanaerodynamic chord (MAC). To obtain the sectional properties over the entire wing, we scale the sectionalproperties with the local chord length: m ∼ c2, Iy ∼ c4, EI ∼ c4, and GJ ∼ c4.

MAC

A, Λ, λµ, r2θ , (ωθ/ωh)

(xea/c), (xcg/c), c`0

M∞, Vµ

Figure 7. Geometry for clamped straight swept tapered wing, including the nondimensional parame-ters of the problem.

First, we show the influence of wing taper on the flutter boundary. Those results are shown in Fig. 8.It is observed that the flutter boundary as a function of Mach number is almost independent of M∞ andthe taper ratio only determines the offset, with higher taper ratios being more susceptible to flutter. This isexpected because for lower taper ratios the mass is concentrated closer to the fuselage.

The aspect ratio of the wing has a large influence on the flutter speed of the wing, which is a concernfor next-generation aircraft designs. In order to look at a realistic variation of aspect ratio for an aluminumtransport aircraft wing, we scale the properties of the wing as b ∼

√A, c ∼ 1/

√A, h ∼ 1/

√A, m ∼ A,

GJ ∼ 1, EI ∼ 1, and Iy ∼ 1. This ensures that the wing has a fixed wing area, fixed lift, and fixed wingboxstress. Note that the nondimensional parameters are not kept constant with such a sweep over aspect ratio.Therefore we use the nondimensional parameters of A = 15 to scale the results.

The flutter boundary as a function of aspect ratio is shown in Fig. 9. We see that a higher aspect ratioreduces the flutter speed substantially, as expected. This influence does, however, plateau for high aspectratios. Furthermore, the characteristic transonic dip is again observed.

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0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

3

4

5

6

7

8

(a) Flutter speed index versus Mach number

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

3

4

5

6

7

8

(b) Flutter speed index versus taper ratio

λ = 1

λ = 0.2

Vµ, M∞

(c) Planform geometry for varyingtaper ratios.

Figure 8. Influence of taper ratio on the flutter boundary for A = 16, Λ = 0, µ = 30, ωθ/ωh = 35,r2θ = 0.15, (xea/c) = 0.400, (xcg/c) = 0.401, τ = 0.13, and c`0 = 0.6 (C-airfoil family).

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.96

7

8

9

10

11

12

(a) Flutter speed index versus Mach number

10 12 14 16 18 20 22 24 26 28 306

8

10

12

14

16

(b) Flutter speed index versus taper ratio

A = 10

A = 20

A = 30

Vµ, M∞

(c) Planform geometry forvarying aspect ratios.

Figure 9. Influence of aspect ratio and freestream Mach number on the flutter boundary. Thenondimensional parameters for A = 15 are Λ = 0, λ = 0.75, µ = 33, ωθ/ωh = 40, r2

θ = 0.125,(xea/c) = 0.300, (xcg/c) = 0.301, τ = 0.13, and c`0 = 0.6 (C-airfoil family).

The sweep of the wing has a profound effect on the flutter boundary, particularly for transonic flowsbecause it lowers the Mach number seen by the section perpendicular to the elastic axis. Flutter boundaries

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for various sweep angles and Mach numbers are shown in Fig. 10. We observe that for low subsonic Machnumbers the flutter boundary follows a shifted parabola for incompressible flows, similar to what was foundby Barmby et al.16 That behavior changes completely with transonic flow, though. Whereas for low subsonicflows the sweep angle generally increases the flutter speed, for transonic flows the sweep angle decreases theflutter speed. Fig. 10a clearly shows that wing sweep delays the transonic dip to higher freestream Machnumbers, which is partly the reason for a decrease in flutter speed for transonic Mach numbers.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

5.5

6

6.5

7

7.5

8

(a) Flutter speed index versus Mach number

0 5 10 15 20 25 30

5.5

6

6.5

7

7.5

8

(b) Flutter speed index versus sweep angle

Λ = 0

Λ = 30

Λ = 15

Λ

Vµ, M∞

(c) For these results, the wing planformis constant and the sweep angle onlychanges the orientation with respect to thefreestream.

Figure 10. Influence of wing sweep angle on the flutter boundary for A = 15, λ = 0.3, µ = 40,ωθ/ωh = 35, r2

θ = 0.125, (xea/c) = 0.400, (xcg/c) = 0.401, τ = 0.14, and c`0 = 0.6 (C-airfoil family).

When the engine is included in the model, additional nondimensional parameters need to be considered.Specifically, we consider the mass ratio of the engine µeng, the radius of gyration of the engine ¯reng, therelative spanwise location of the engine ηeng, the relative chord-wise location of the engine ξeng,x, and therelative normal location of the engine ξeng,z. These parameters are defined as,

µeng =meng

mmacb(16a)

¯r2eng =

4¯Ieng

mengc2mac

(16b)

ηeng =yeng

b(16c)

ξeng,x =∆xeng

cmac(16d)

ξeng,z =∆zeng

cmac. (16e)

The influence of the mass of the engine on the flutter boundary is shown in Fig. 11. We see a similarreversal of trends between low subsonic and transonic flows; for low subsonic flows the additional mass ofthe engine increases the flutter speed, but for transonic conditions the additional mass suddenly decreasesthe flutter speed.

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0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

7

8

9

10

11

(a) Flutter speed index versus Mach number

0 0.05 0.1 0.15 0.2 0.25

7.5

8

8.5

9

9.5

10

10.5

11

(b) Flutter speed index versus engine weight

µeng

Vµ, M∞

(c) Planform geometry

Figure 11. Influence of engine weight and freestream Mach number on the flutter boundary forA = 15, λ = 0.3, Λ = 25, µ = 25, ωθ/ωh = 35, r2

θ = 0.125, (xea/c) = 0.350, (xcg/c) = 0.351, τ = 0.13,c`0 = 0.6 (C-airfoil family), ηeng = 1/4, ξeng,x = −1/4, ξeng,z = −1/8, r2

eng,xx = 0.5, r2eng,yy = 0.3, and

r2eng,zz = 0.3 (off-diagonal terms zero).

III.B. Weight-converged aircraft

The results in Section III.A showed the influence of wing parameters on the flutter boundary for a wingclamped to the wall. However, for an aircraft design, any change in parameter of the wing cascades throughthe entire aircraft design, because the aircraft has to be weight converged. In other words, the lift thewing generates has to be equal to the weight of the aircraft. When this is taken into account, the fluttercharacteristics as a function of wing parameters also change. This section shows the influence of wingparameters on the flutter damping values while ensuring that the aircraft design is weight-converged at eachflutter evaluation. These aircraft designs are generated using TASOPT.

D8.2

D8.0

5 m

Figure 12. Comparison of D8.0 and D8.2 configurations.2

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We consider two different D8.x aircraft configurations, as described in Ref. 2. The D8.0 and D8.2configurations are compared in Fig. 12. The D8.0 is a “fuselage-only” modification to the Boeing 737,keeping the wing the same as for the 737 – the engines therefore hang under the wing. The D8.2 has twochanges compared to the D8.0: the engines are moved to the back of the fuselage to enable boundary layeringestion and the cruise Mach number is reduced to allow for a lower-sweep wing.

Flutter is evaluated for several worst-case points on the (M∞, ha) envelope, as shown in Fig. 13. Allof the flutter evaluation points are for the never-exceed dynamic pressure, because an increase in dynamicpressure always results in the aircraft being closer to flutter, even in transonic flight. Due to the transonicdip behaviour, it is not clear which freestream Mach number is the worst-case for flutter. Therefore severalMach numbers between the cruise Mach number and 10% above the cruise Mach number are sampled. Foreach flutter evaluation point we also consider empty wing tanks, half-full wing tanks, and full wing tanks,because these change the mass and center of gravity of the wing substantially.

Start cruise phase

End cruise phase

Cru

ise

Mac

hnum

ber

Nominal mission

Flutterevaluations

1.10MCR

Mach number, M∞

Alt

itud

e,ha

0

climb

descen

t q∞smalllarge

Figure 13. (M∞, ha) envelope for the D8.2 aircraft.

Fig. 14 shows the aeroelastic eigenvalues as function of wing aspect ratio A and cruise Mach numberM∞. Note that χfl > 0 indicates flutter. These results consider a freestream Mach number 10% higher thanMCR and full wing tanks as the flutter evaluation point. Note that the trends differ between the differentflutter evaluation points shown in Fig. 13, but the one shown in Fig. 14 is one of worst cases. We see thatthe D8.0 and D8.2 have wildly different values for χfl, while the trends are also dissimilar. For the D8.2 theincrease in aspect ratio leads to an increase in χfl (i.e., closer to instability), whereas for the D8.0 the aspectratio at first decreases χfl but for higher aspect ratios this trend reverses. The trend for the D8.2 is thereforesimilar to the results for the clamped wing configuration in Fig. 9, where a higher aspect ratio decreasesthe flutter speed because the wing is more flexible for higher aspect ratios. For the D8.0, the wing also getsmore flexible for larger aspect ratios, but the engine also moves further outboard. For lower aspect ratios,putting the engine further outward moves the wing towards stability. However, for higher aspect ratios theeffect of the larger flexibility in the wing again moves the wing away from stability. An increase in designcruise Mach number puts the D8.2 closer to instability. However, for the D8.0 for lower aspect ratios thecruise Mach number decreases χfl, whereas for aspect ratios around A = 11 higher cruise Mach numbersincrease χfl.

Fig. 15 shows the aeroelastic eigenvalues of weight-converged aircraft designs for different sweep anglesof the wing Λ and different design cruise Mach numbers MCR. These results consider a freestream Machnumber 10% higher than MCR and empty wing tanks as the flutter evaluation point. The results for the D8.2show that an increase in sweep angle moves the design away from instability, as does a decrease in cruiseMach number. For the D8.0, however, these trends are mostly reversed, showing the large influence of theinertia and mass of the engine on wing flutter. As explained in Section II.C, with an increase in sweep angle,∆xeng decreases and ∆yeng increases. The additional stability from changing the engine position thereforeoutweighs the push towards instability by the increase in sweep angle.

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8 8.5 9 9.5 10 10.5 11 11.5 12 12.5 13-1

-0.5

0

0.5

1

(a) D8.0

12 13 14 15 16 17 18-3

-2.5

-2

-1.5

-1

(b) D8.2

Figure 14. Influence of aspect ratioA on maximum aeroelastic eigenvalue χfl for two different aircraft;the D8.0 has a wing with a large sweep angle and wing-mounted engines, the D8.2 has a lower-sweepwing and fuselage-mounted engines. These results are for a flutter evaluation at M∞ = 1.10MCR withfull wing tanks.

21 22 23 24 25 26 27 28 29 30-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

(a) D8.0

10 11 12 13 14 15 16 17 18 19-0.8

-0.75

-0.7

-0.65

-0.6

-0.55

-0.5

-0.45

(b) D8.2

Figure 15. Influence of sweep angle Λ on maximum aeroelastic eigenvalue χfl for two different aircraft;the D8.0 has a wing with a large sweep angle and wing-mounted engines, the D8.2 has a lower-sweepwing and fuselage-mounted engines. These results are for a flutter evaluation at M∞ = 1.10MCR withempty wing tanks.

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IV. Influence of Transonic Flutter on Aircraft Designs

This section discusses the influence of transonic flutter on novel aircraft designs. We will focus here onseveral variants of the D8.0. All results in this section are generated using TASOPT and with the flutterevaluation points as described in Fig. 13.

First, we show the influence of transonic flutter on the planform design of an advanced-technology versionof the D8.0. For this design, better engine technology, laminar-bottom wings, and carbon-fiber structures areused, allowing for higher aspect ratios leading to a higher susceptibility to flutter. The optimized planformswith and without flutter constraints are shown in Fig. 16. The flutter constraints essentially serve as a spanconstraint, as we have already seen that higher aspect ratios lead to lower flutter speeds in Fig. 9. Due tothe lower aspect ratio, the flutter-constrained design has worse aerodynamic performance, and therefore a3.3% higher fuel burn. The lower span does result in a lower maximum take-off weight, but that does notoffset the higher fuel burn due to poorer aerodynamic performance.

5 m Without flutter constraint

With flutter constraint3.3% higher fuel burn2% lower WMTO

Figure 16. Advanced-technology D8.0 designed with and without flutter constraints.

Finally, we investigate specifically the influence of newer material technology on the performance of theD8.0 and describe the influence of transonic flutter constraints on this design. The specific allowable stress isused as a surrogate for new material technology, which is quantified here by a factor multiplying the baselinevalue, corresponding to aluminum in this case. The varying planform designs and influence on fuel burnand maximum take-off weight are shown in Fig. 17 for specific allowable stress up to 50% higher values thanaluminum.

As expected, an increase in specific allowable stress results in a lower fuel burn and maximum take-offweight, which was already observed by Drela27 for Boeing 737-class aircraft. The inclusion of the flutterconstraints limits the efficiency gains seen by an increase in specific allowable stress, because the aspectratio is limited by the flutter constraints. As already observed in Fig. 16, the lower aspect ratio does resultin a lower maximum take-off weight, but the fuel burn is still higher than for the design without flutterconstraints. The trends in Fig. 17 are similar to those described by Drela27 for the Boeing 737-class aircraftwith and without a span constraint.

V. Conclusion

A transonic flutter model applicable to high-aspect ratio swept wings was presented in this paper. This workextends a previously developed 2D transonic flutter model to 3D and implements it in a conceptual designtool. This model discretizes the wing in several sections and builds an aeroelastic system for the wholewing using calibrated flutter coefficients for transonic flow. Such a system is relatively low-dimensional,

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0% 10% 20% 30% 40% 50%

−10%

−5%

0%

%chan

gein

fuel

burn

Without flutter constraint

With flutter constraint

0% 10% 20% 30% 40% 50%

−10%

−5%

0%

% change in specific allowable stress

With flutter constraint

Without flutter constraint

%chan

gein

WM

TO

Figure 17. Variations of optimum fuel burn and maximum take-off weight with specific allowablestress.

allowing for fast computation of the eigenvalues, making it applicable for use in a conceptual design tool.The approach is demonstrated to find transonic flutter boundaries for several wing configurations, showingthe influences of Mach number, taper ratio, and sweep.

The model is also used to investigate the aeroelastic characteristics of different aircraft designs. The flutterbehaviour for two different aircraft configurations—one with a high-sweep wing with an engine attachedbelow, one low-sweep wing with no engines attached to the wing—is quite different, and several trends offlutter damping values with respect to wing design parameters are reversed between the two configurations.For example, a larger sweep angle pushes the aicraft with wing-mounted engines towards instability, whereasfor the aircraft with fuselage-mounted engines a larger sweep angle moves the design away from instability.Furthermore, it was found that the inclusion of flutter constraints in the aircraft design optimization limitsthe aspect ratio of the wings, resulting in higher fuel burn. Transonic flutter therefore limits the performancegains seen by using more advanced materials in the wing. Note that these trends are not generally applicableto any aircraft design; the onset of flutter is quite subtle and is sensitive to small changes in wing geometry,engine parameters, or aerodynamic forces. It is therefore crucial to include an accurate, fast aeroelasticanalysis early in the conceptual design phase.

Acknowledgements

This work was supported in part by the NASA LEARN program grant number NNX14AC73A and theSUTD-MIT International Design Center.

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