AEROELASTICITY AND FLIGHT MECHANICS:STABILITY ANALYSIS USING LAPLACE-DOMAIN

AERODYNAMICS

David EllerAeronautical and Vehicle Engineering

Royal Institute of TechnologySE - 100 44 Stockholm, Sweden

June 2009

IFASD-2009-119

Keywords: Flutter; Flight Dynamics; Eigenvalue Solver; Unsteady Aerodynamics

Abstract

An efficient method for the non-iterative solution of the nonlinear flutter eigenvalueproblem is presented. The properties of the piecewise quadratic decomposition employedmake it particularly suitable for the parallel solution of aeroelastic stability problems whereinteraction of rigid-body and elastic motion is of interest or where the flutter damping mustbe obtained with accuracy, e.g. when comparison to flutter flight testing is intended. Ad-ditionally, the method allows the use of Laplace-domain aerodynamics if available. Thepaper presents a realistically complex test case which exposes some of the advantages andcurrent shortcoming of the proposed solution procedure.

1 IntroductionConventionally, the dynamic stability of flexible aircraft is analyzed twice using specializedapproaches. For rigid-body motion, a first-order in frequency approximation of unsteady aero-dynamic loads is often employed, which tends to be sufficiently accurate for the low rigid bodymotion frequencies typically encountered by conventional aircraft configurations [1]. The anal-ysis of aeroelastic stability, however, needs to take the nonlinear frequency dependence of theaerodynamic loads into account [2]. In many cases, these loads are only available at a discreteset of frequencies, so that some kind of interpolation is needed to solve the flutter problem.

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For realistically complex configurations, unsteady aerodynamic loads are difficult to mea-sure or estimate even for simple rigid-body motion, which is the reason why computationalmethods for frequency-domain aerodynamics often serve to compute unsteady aerodynamicloads for both flight mechanics and aeroelasticity. This raises the question why the stabilityanalyses must be characteristically different, given that unsteady aerodynamic data is obtainedby the same means.

Flutter analysis is primarily concerned with the determination of the stability boundary,while flight mechanic stability solutions are often used to judge handling qualities or designflight control laws, which means that the amount of damping encountered by a certain motionpattern is of considerable interest. Standard methods for the solution of the flutter eigenvalueproblem are tailored towards computing accurate solutions near zero damping and do thereforenot necessarily provide accurate damping information, which makes them less suitable for flightmechanics.

In this paper, an interpolation scheme for aerodynamic loads in the Laplace plane is de-veloped and exploited for a direct eigenvalue solution procedure. The latter is not based oniteration, and formulated such that the real part (damping) of the eigenvalues is determined asaccurately as possible. The formulation is then adapted to allow the solution of rigid-body sta-bility problems by means of the same algorithm and using the same representation of the aero-dynamic loads. It is hoped that this approach can serve to reduce the modeling effort neededfor stability analyses of new configurations and at the same time allow for more consistentinvestigations where coupled rigid-body and elastic motion is of interest.

2 Equations of motionThe time-linearized equations of motion lead to a formulation where the time-dependent motionis expressed as damped harmonic oscillation in terms of a complex frequency

s = σ + iω =u∞b

(g + ik) , (1)

where ω and σ are dimensional frequency and damping, k and g their nondimensional counter-parts obtained by normalizing with the ratio of semichord b and airspeed u∞. In aeroelasticity,s, which can alternatively be seen as the Laplace variable, is usually named p which in thepresent context is too easily confused with the roll rate.

2.1 LinearizationAssuming small disturbances of an equilibrium flight condition, aerodynamic forces are ex-pressed as linear functions of motion and deformation. Using this approximation, the equationof motion in the Laplace domain becomes[

Ms2 + Cs+ K − q∞Q(M, g + ik)]x = F (s), (2)

with the mass, damping and stiffness matrices M , C and K. Aerodynamic loads scale linearlywith dynamic pressure q∞ and depend on Mach number M and reduced complex frequency

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g + ik. Due to linearization, these are written as the product of the aerodynamic load matrix Qand the Laplace domain motion x. The right-hand side F (s) represents external excitation suchas control forces or gust loads, and vanishes for stability analyses. For purely aeroelastic solu-tions, the motion x represents structural deformation, usually expressed in terms of generalized(modal) coordinates.

2.2 Flight mechanicsApplying the same linearization approach to the rigid-body equations of motion, a system oftime-domain equations is obtained. The standard approach then rewrites this system as twouncoupled first-order systems for longitudinal and lateral motion, respectively [1].

For compatibility with (2), the original variables for position x, y, z (in the body frame) andorientation φ, θ, ψ (with respect to the inertial frame) are retained here, so that the second-ordertime-domain system becomes

mx−mgθ cos θ0 = Fx(t) (3)

my +mgφ cos θ0 −mu∞ψ cos θ0 = Fy(t) (4)

mz −mgθ sin θ0 +mu∞θ = Fz(t) (5)

Ixx(θ − ψ sin θ0) (6)

−Ixzψ cos θ0 = Mx(t) (7)

Iyyθ = My(t) (8)

−Ixz(θ − ψ sin θ0) + Izzψ = Mz(t). (9)

Here, the mass m and rotational inertias Iij are assumed not to depend on deformation. Fur-thermore, a flat earth with gravitational acceleration g pointing toward the negative z directionis assumed. Time-domain aerodynamic and control forces Fi(t) and moments Mi(t) are meantto be the deviations from level-flight values, as are all location variables.

Aeroelastic analysis models are usually closely related to the dynamic structural model used,which in turn tends to employ the global coordinate system chosen by structural design. Often,this is not the same system as the convention used in flight mechanics. To avoid confusion, therigid-body equations (3) - (9) are expressed in the usual structural design coordinate system,where x is the body coordinate pointing backwards, y is positive towards starboard and z up-wards (as shown in Figure 4). Rotational directions are adjusted accordingly. In order to arriveat (3) - (9), the conventional assumptions of small disturbances from level flight are used, andsteady rotational rates p, q, r taken zero in equilibrium, so that

φ0 = 0, ψ0 = 0 (10)p0 = q0 = r0 = 0 (11)

p = φ (12)

q = θ − ψ cos θ0 (13)

r = ψ cos θ0. (14)

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Using the substitution√mx = ξ0,

√my = ξ1,

√mz = ξ2, (15)√

Ixxφ = ξ3,√Iyyθ = ξ4,

√Izzψ = ξ5, (16)

Equations (3) - (9) can be written in mass-normalized form as

ξ0 −√

m

Iyyg cos θ0ξ4 =

Fx√m

(17)

ξ1 +

√m

Ixxg cos θ0ξ3 −

√m

Izzu∞ cos θ0ξ5 =

Fy√m

(18)

ξ2 −√

m

Iyyg sin θ0ξ4 +

√m

Iyyu∞ξ4 =

Fz√m

(19)

ξ3 −

(√IxxIzz

sin θ0 +Ixz cos θ0

IxxIzz

)ξ5 =

Mx√Ixx

(20)

ξ4 =My√Iyy

(21)(cos θ0 +

IxzIzz

sin θ0

)ξ5 −

Ixz√IxxIzz

ξ3 =Mz√Izz

, (22)

which immediately yield mass, damping and stiffness matrix terms which can be included in (2).Note that in purely aeroelastic solutions, the viscous damping term Cs vanishes as structuraldamping tends to be introduced by means of an imaginary stiffness term, so that K ′ = (1 +igs)K for relative structural damping gs. Coupling with flight mechanics does however lead tononzero ’viscous’ damping terms due to the rotation rate effect in (4) and (5).

2.3 Structural modelFor the elastic structure, a linear finite-element model is assumed. The number of degrees offreedom is reduced by means of modal subspace projection where a limited set of low-frequencyeigenmodes are retained. Provided that the rigid-body equations are written with respect to thecenter of mass of the structure, this approach eliminates inertial coupling between the rigid-body and structural degrees of freedom. A disadvantage is that different modal analyses needto be performed for changes in mass distribution.

3 Unsteady aerodynamicsAlthough the original Doublet-Lattice Method (DLM) is restricted to purely harmonic motion[3], it has later been extended to treat damped motion as well [4, 5]. Even Stark’s AEREL

system used at SAAB has this capability [6]. For the method discussed here, the aerodynamicload matrix Q(s) in (2) is computed by means of a boundary element method for unsteady

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potential flow, which uses a wetted-surface discretization instead of the lifting surface approachof the methods mentioned previously [7].

To simplify the implementation of the stability analyses, all motion is represented by mode-shapes. The required rigid-body modes can either be generated internally or extracted from themodal analysis results of a finite-element method.

Since plain positions (x, y, z, φ, θ, ψ) are chosen as primary variables for rigid-body motion,aerodynamic loads for these modes vanish for the steady case g = k = 0. Considering that thestiffness terms are zero as well for these modes, it is expected that there are six eigenvaluesolutions at s exactly zero. Aerodynamic loads in x are very small (but nonzero) even fornonzero complex frequency, so that this type of motion would lead to another solution withextremely small s unless a model for the change in drag due to change in x (and thereforevelocity when s 6= 0) is included. For the present analysis, a very simple quadratic drag modeland the assumption of constant-power propulsion of a propeller-driven aircraft is used in orderto arrive at a speed damping term C0,0 according to

Fx =

(D0 +Di −

P

u∞

)+∂Fx∂u

∆u (23)

Fx = 0 + (3D0 −Di)∆u

u∞(24)

C0,0 =Di − 3D0

mu∞, (25)

where Di is the equilibrium induced drag (linear in u−2), D0 the parasitic drag (linear in u2)and P the constant propulsion power which balances the drag in the reference state.

4 Eigenvalue solverSince Q depends nonlinearly on the complex reduced frequency g+ ik, the eigenvalue problem(2) cannot simply be solved directly. In the standard p− k method, (2) is written as a standard(linear) eigenvalue problem in λ = s2, depending on a parameter k, so that

[Mλ+ Cik + K − q∞Q(M, ik)] x = 0. (26)

A solution of (26) is an approximate solution to (2) only when the parameter k matches theimaginary part of s. Therefore, k must be repeatedly updated, Q(M, ik) interpolated and thelinear eigenvalue problem (26) solved until the alignment condition is met for each mode inde-pendently. Therefore, the computational complexity of the p − k method applied to a problemwith n modes is O(ni · n4), where ni is the number of iterations needed to arrive at a con-verged solution in each mode. Traditionally, simple fixed-point iteration was used to arrive atIm(s) = k [8]. Besides being rather inefficient, this approach has unfavorable convergenceproperties. It can be improved substantially by treating (26) as an optimization problem in oneparameter, which can be solved robustly by means of a safeguarded Newton iteration [9]. Inthis way, complex mode-tracking logic is dispensable as well. The algorithmic complexity,however, remains unchanged even if much lower ni can be achieved.

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The p − k solution strategy is intended for use with frequency-domain aerodynamics, thatis, Q(g + ik) ≈ Q(ik) is assumed. The g-method implemented in ZAERO [10] employs afirst-order approximation of the effect of damping by setting

Q(g + ik) ≈ Q(ik)− ig ∂Q∂k

∣∣∣∣g=0

. (27)

In this way, an approximate expression for aerodynamic loads on the complex plane is obtainedwith Q(ik) computed on the imaginary axis only. The solution procedure that makes use of thisfirst-order expansion is however relatively intricate and requires to keep track of modes whilechanging a solution parameter.

None of the iterative solution methods known to the author can handle multiple eigenvalueseasily. Although the steady translation and rotation solutions mentioned above (six-fold eigen-value at s = 0) may not be of interest from an engineering point of view, they are still a solutionof (2) and should therefore at least be allowed to occur.

4.1 Piecewise quadratic problemIn order to exploit the availability of Laplace-domain aerodynamic loads, a piecewise quadraticeigenvalue solution procedure is adopted. Based on an approach first proposed by Goodman[11], it extends the method developed in [12] to the entire complex plane. The underlyingobservation is that a piecewise quadratic approximation

Q(s) = Q0 + Q1s+ Q2s2 ∀ s ∈ Sij (28)

can be constructed to cast (2) as a quadratic eigenvalue problem[(M − q∞Q2)s2 + (C − q∞Q1)s+ (K − q∞Q0)

]x = 0, (29)

which can then be solved using standard methods for polynomial eigenvalue problems withoutiteration [13]. Since the approximation (28) is only valid for a certain region

Sij : g ∈ [gi, gi+1] ∩ k ∈ [ki, ki+1] (30)

of the complex plane, only solutions of (29) falling inside this region are considered.Since solution segments Sij are independent of each other and no continuation along a

parameter is needed, the eigenvalue solution step can be easily parallelized. Furthermore, thereis no need to start the procedure from a certain point (e.g. q∞ = 0), so that new solutions can beobtained anywhere at low cost. In the remaining part of the paper, this solution procedure willbe referred to as the p2 method.

4.2 Load interpolationGoodman arrives at a quadratic approximation (28) by means of discontinuous local interpo-lation. The resulting eigenvalue solutions are therefore not continuous, which is remedied byinterpolating computed eigenvalues between segments. Unfortunately, the latter step requires

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reliable mode tracking again. In [12], another interpolation is constructed with slope continuityalong g = 0, which at least guarantees extremely small jumps in eigenvalue solution near theimaginary axis. The latter approach is however only applicable to frequency-domain aerody-namics.

For Laplace-domain aerodynamics, an extension of the approach taken in [12] is to cover thecomplex plane with rectangular regions which extend over a limited range of damping values.The quadratic coefficients Q0..2 for each region are determined from a global least-squares prob-lem such that the error between the approximation and computed discrete values of Q(g + ik)is as small as possible. Additionally, imposing that the jump in matrix element values acrossregion boundaries shall vanish yields further equations. By using a larger number of regions,smaller residuals can be achieved, resulting in smaller discontinuities across segment bound-aries and better fit to prescribed values. Obviously, this simple global least-squares strategy canbe further refined.

Figure 1 shows the result of this approximation scheme for one generalized aerodynamicforce coefficient taken from the test case described below. The particular entry Q7,7 chosenhere corresponds to a structural eigenmode strongly involved in the critical flutter mechanism.The left part of Figure 1 illustrates the result of using computed loads at g = 0 only to construct

0 0.5 1 1.5

−4

−2

0

2

4

6

Reduced frequency

Gen

eral

ized

forc

e co

effic

ient

Slice at constant g

g = −0.2

g = 0.0

g = 0.2

−0.2 −0.1 0 0.1 0.2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

Reduced damping

Gen

eral

ized

forc

e co

effic

ient

Slice at constant k

k = 0.05

k = 0.2

Figure 1: Approximation of aerodynamic loads. Values computed by the aerodynamic solverare marked with cross (real part) or circle (imaginary part); solid lines represent evaluations ofthe approximation scheme (28).

an approximation over the complex plane. Evaluating the approximation model along the imag-inary axis (solid black line) yields a very close fit of the computed points and extremely smalldiscontinuities. At large |g| (colored solid lines), some discontinuities occur but are limited toless than 0.5% for this particular case.

On the right side, a more demanding case is shown. For the same configuration and eigen-mode, loads are now computed for a grid of damping values and frequencies; these values areplotted with markers only. The piecewise quadratic approximation (solid red and blue lines)follows these values closely for k = 0.2, but less so for the lower frequency. Relative ap-proximation errors for k = 0.05 reach about 4% at g = −0.15. The apparently unnecessary

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additional wiggle of the approximated curve near the lower border of the interval results fromthe requirement to fit neighboring segments in both the k and g-direction under the restrictionof a quadratic function of a single complex variable.

5 Case studyTo investigate how the proposed flutter solution procedure performs for a reasonably complexcase, the tailless low-speed UAV configuration shown in Figure 2 was analyzed. Dimensionsand mass were selected to match public data of the ScanEagle surveillance aircraft developedby Insitu and Boeing. Due to the lack of an aftbody and tail surfaces, this configuration is char-acterized by comparatively small pitch inertia and weak aerodynamic pitch damping. Couplingbetween structural eigenmodes involving substantial wing bending and rigid-body rotation istherefore more likely than with conventional tail configurations.

5.1 ModelingThe main wing is swept back 22◦ and twisted -4.5◦ towards the tip in order to achieve staticstability about the pitch axis for reasonable locations of the center of gravity and elevon trimsettings. Laterally, the rigid aircraft is only marginally stable.

Figure 2: Aerodynamic mesh. Figure 3: Finite-element shell mesh.

The aerodynamic mesh shown in Figure 2 consists of 34 000 triangles on the surface, anda further 134 000 wake elements are automatically created by the solver. This mesh is used toobtain the aerodynamic loads for flutter analysis.

For the structural data, typical light aircraft construction is assumed. Most surfaces arefoam-cored sandwich shells with thin glass-fiber/epoxy face sheets, but the main wingbox andcontrol surfaces are reinforced with layers of unidirectional carbon fiber composite. A smallnumber of internal ribs support control surface hinges (four for each elevon and rudder) and

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carry the localized forces of the wing-fuselage connection. Payload, engine and fuel are mod-eled as non-structural mass distributed on the fuselage shell. Control surfaces are separate bod-ies which are attached to the rear spar by means of rigid elements and multi-point constraintsso that the rotation about the hinge axis is restrained by a single discrete scalar spring elementonly.

Figures 4 to 7 show a few eigenmode shapes with low frequency. These modes were selectedbecause they participate in aeroelastic modes of interest discussed later. In all of the figures,the global axis system is shown to illustrate the motion of the body with respect to the center ofmass.

Figure 4: Eigenmode 1 at 7.7 Hz Figure 5: Eigenmode 4 at 19 Hz.

Most of the elastic eigenmodes couple with control surface motion since the control surfacesare not statically mass-balanced. Therefore, even modeshapes such as the in-plane bendingmode 4 shown in Figure 5 generate significant aerodynamic loads.

Figure 6: Eigenmode 5 at 22 Hz. Figure 7: Eigenmode 6 at 32 Hz

5.2 Flutter analysisEquation (2) is solved for sea-level density and a velocity range of 20 to 75 m/s, using loadsfrom an incompressible aerodynamic analysis. For the first comparison, both a simplified im-

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plementation of the modified p−k method of Back and Ringertz [9] and the piecewise quadraticapproach discussed above are used. Note that, in this case, both methods use the same aerody-namic data evaluated on the imaginary axis only, so that the only difference is in the formulationof the eigenvalue problem and in the type of interpolation of the same aerodynamic loads. Thep− k method employs a cubic line interpolation through the 21 discrete values of Q(ik) alongthe imaginary axis, while the p2 procedure generates 42 rectangular regions. For each region,another set of polynomial coefficients Q0..2 is determined by means of a least-squares fit.

In Figure 8, a selected subset of the solutions is shown for the full range of reduced frequen-cies used. The colored solid lines are solutions found by the piecewise quadratic eigenvaluesolver while crosses indicate eigenvalues for which the p − k solution converged to within atolerance of ∆k < 10−7.

−0.1 −0.08 −0.06 −0.04 −0.02 0 0.020

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

g [−]

k [−

]

20 30 40 50 60 70−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

Airspeed [m/s]

g [−

]

Figure 8: Root-locus and velocity-damping plot comparison

As expected, the eigenvalue solutions near the imaginary axis match almost exactly, andthe predicted flutter crossing at 46.5 m/s and reduced frequency 0.87 (corresponding to 56Hz) is identical for both methods. Even the critical modeshape is the same : A combinationof eigenmode 4 (Figure 5), second symmetric wing bending, substantial winglet and rudderdeflection, and some elevon motion.

Differences occur however for the more strongly damped modes shown in blue and greenat around g ≈ −0.06. These two modes are dominated by first wing bending and counteractingelevon rotation, that is, upward bending is combined with negative elevon deflection whichprobably causes the strong damping.1 The mode marked with the solid blue line even turnstoward the imaginary axis in the piecewise quadratic solution while it does not when using thep− k method.

As it appears that differences between the methods tend to increase with lower reducedfrequency and higher damping, Figure 9 shows a comparison of aeroelastic eigenvalues related

1Similar aeroelastic modes containing destabilizing elevon rotation exist as well and become unstable at 55m/s. The red line in Figure 8 is the symmetric version; the antisymmetric mode is located close nearby and is notshown to improve clarity.

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to rigid-body motion. Here, the p − k solution is plotted with crosses and the p2 results basedon the same aerodynamic data are shown as solid lines.

Additionally, empty circles indicate eigenvalue solutions obtained from the p2 procedurewhen using aerodynamic loads computed for several damping values as well. The differencesbetween solid lines and circle markers can hence be seen as the effect of Laplace-domain aero-dynamics on the solution characteristics. The modeshapes plotted with black symbols and lines

−0.1 −0.08 −0.06 −0.04 −0.02 0 0.020

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

g [−]

k [−

]

−0.01 −0.005 0 0.005 0.010

0.005

0.01

0.015

0.02

0.025

g [−]

k [−

]

Figure 9: Comparison at low frequencies

are related in that they are dominated by rigid-body roll coupled to elevon motion. However,the p2 solution shows a component of antisymmetric wing bending as well, which becomesmore important with higher dynamic pressure. This particular mode is interesting as it changesfrom a convergent roll mode at 20 m/s to a coupled roll/bending mode oscillating with 3.6 Hzat 75 m/s. Observe that, at low speed, where this mode is overcritically damped, neglectingdamping effects in aerodynamics and eigenvalue solution has a significant impact even on thepredicted frequency of oscillation.

On the right side, the region near the origin is enlarged. There are two unstable rigid-bodymodes, the first of which is a Dutch roll mode with parameters shown in Table 1 below. Allthree flutter solutions provide very similar results for this case, plotted in magenta in Figure 9.The other unstable rigid-body mode (cyan in Figure 9) is much harder to identify. According

Period [s] Time-to-double ψ : φp− k 1.74 1.60 0.288p2 1.70 1.29 0.289

Table 1: Properties of the Dutch roll mode at 30 m/s

to the shape characteristics shown in Table 2, it is neither a divergent phugoid mode nor apure short-period motion, as it combines large angle of attack variation with significant speedchanges. Refining the aerodynamic model by computing loads for damped motion makes thisparticular root change into a longitudinal divergence mode. Considering the large speed change

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Period [s] Time-to-double u : α θ : αp− k 10.2 0.67 0.37 0.22p2 4.8 0.83 0.53 0.42

Table 2: Unstable longitudinal mode at 45 m/s

u = ∆uu∞

= − sxu∞

, it remains uncertain if these solutions are accurate since the coupled equationsof motion described in Section 2 do not yet account for the dynamic pressure derivative, that is,q∞ is taken independent of x, which is obviously not the case for modes involving significantchanges in velocity.

Stable modes shown in the right part of Figure 9 differ relatively little between eigenvaluesolution methods: The trajectories of the stable short-period (red) and spiral convergence (yel-low) are fairly close and share similar development with dynamic pressure.

5.3 Dependency on stateIn aeroelasticity, it is common to assume that the generalized aerodynamic force matrices Q areindependent of state variables such as the angle of attack and sideslip angle, at least for subsonicairspeeds. For the coupled rigid-body/aeroelastic problem discussed here, Q contains termscorresponding to, for example, Cl,β , the derivative of the roll moment with respect to sideslip.This particular derivative depends strongly on the lift coefficient even in inviscid subsonic flow.Therefore, Q(M, s) should now at least theoretically not only depend on complex frequencyand Mach number, but also on other state variables such as angle of attack and possibly sideslipand steady rotation rates. This would mean that load matrices Q(M, s, α, β, ...) need to becomputed for many more combinations of frequencies and flight states.

Fortunately, that is not the case. Figure 10 illustrates the influence of the angle of attackα on the β-derivatives of the roll moment coefficient (left) and an elastic force component(right). First of all, only the real part, that is, the component in phase with β, is affected at

0 0.5 1 1.5−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Reduced frequency

Rol

l mom

ent w

rt s

ides

lip

0 0.5 1 1.5−0.25

−0.2

−0.15

−0.1

−0.05

0

Reduced frequency

Gen

eral

ized

forc

e co

effic

ient

Real part, −2 degree

Imaginary part, −2 degree

Real part, 6 degree

Imaginary part, 6 degree

Figure 10: Load dependency on angle of attack

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all, while the out-of-phase component shows very little variation with α. Furthermore, thevariation with angle of attack is quite accurately described by a constant shift independent offrequency. Finally, only the loads caused by rigid-body motion (i.e. the first six columns of Q)are affected by α. Hence, Q(M, s) needs only be computed once for a set of frequencies, anda constant real term can be added to (2) to account for the dependence of the load derivativeson flight condition. The eigenvalue analysis, however, needs to be repeated for different flightconditions. Fortunately, this is typically about two orders of magnitude less costly than thecomputation of the unsteady aerodynamic loads. If matched-point solutions for level flight aredesired, there is no computational overhead since all state variables take a fixed value for eachsweep parameter.

5.4 Computational costFor the 34 000-panel mesh shown in 2, the computation of Q(s) for 21 discrete frequencies and20 eigenmodes requires about 17 minutes on four 3 GHz processor cores (Intel Xeon 5160). Onthe same computer, a flutter solution sweep using the p2 algorithm with 200 (speed or altitude)points completes in less than 20 seconds.

6 ConclusionsThe proposed flutter solution method yields results which are identical to standard p− k eigen-value solutions at the flutter boundary. For aeroelastic problems without rigid-body coupling,the present approach offers the advantage of much improved computational efficiency and betterrobustness because of its non-iterative nature.

A more important advantage is the possibility to treat the rigid-body stability problem usingthe same method, so that configurations with coupled modes can be analyzed without recourseto specialized modeling procedures. The present implementation does not yet correctly accountfor all terms which tend to affect such eigenvalue solution as shown by the test case, the speedderivatives being an example. This deficiency will be addressed in the near future.

The use of Laplace-domain aerodynamics has a significant effect on the predicted dampingof eigenmodes far from the imaginary axis. This aspect may improve correlation of flutteranalysis results with flight test data, especially if aeroelastic modes of interest are stronglydamped.

7 AcknowledgmentsThis work is financially supported by the Swedish Defense Materiel Establishment (FMV),which the author gratefully acknowledges.

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