Flutter of Maneuvering Aircraft

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Flutter of Maneuvering Aircraft

Ilhan Tuzcu∗

California State University, Sacramento, CA 95819, USA

Nhan Nguyen†

NASA Ames Research Center, Moffett Field, CA 94035, USA

February 3, 2014

Abstract

Our objective is to investigate how the aeroelastic stability, particularly flutter, is affected by aircraft

maneuvers. We intend to base our investigation on a comprehensive mathematical model of aircraft, which is

achieved by seamlessly integrating all the disciplines pertinent to flight of aircraft. The aircraft is treated as

an unstrained, flexible multibody system subject to unsteady aerodynamics. The bodies are fuselage, wing,

and horizontal and vertical stabilizers, whose structures are modeled as beams in bending and torsion. The

equations of motion are derived using Lagrange’s equations in quasi-coordinates. The resulting equations are

a set of nonlinear ordinary differential equations of relatively high order. The final model is used to determine

flutter speeds of aircraft at steady level turn and steady climb at various altitudes. These maneuvers are

especially chosen to keep the equations time-invariant. The numerical results are given for the NASA’s

Generic Transport Model (GTM). We show how the stability of GTM is affected by turn radius, climb angle

and altitude. We also show how the results for climbing flight can be extended to address stability of gliding

flight.

∗Associate Professor, Mechanical Engineering Department, [email protected]†Research Scientist, Intelligent Systems Division, Mail Stop 269-1

1

Journal of Aerospace Engineering. Submitted February 10, 2013; accepted February 7, 2014; posted ahead of print February 10, 2014. doi:10.1061/(ASCE)AS.1943-5525.0000415

Copyright 2014 by the American Society of Civil Engineers

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Nomenclature

ab = Distance to elastic axisb = Semi-chordC = Matrix of direction cosines from XY Z to xyzC(k) = Theodorsen functionCi = Matrix of direction cosines from xyz to xiyiziCLαv = Lift-curve slope for vertical stabilizerCLδr = rudder control effectivenessD = Aerodynamic dragdi = Drag per unit length for body i

E = Matrix relating θ to ω

EIi = Bending rigidity for body iF = Force vectorFe = Engine thrust vectorGJi = Torsional rigidity for body ih = Plunge of elastic axisH = Altitudek = Reduced frequencyKi, Ki = Bending and torsional stiffness matrices for body iL = Total Lagrangian�i = Lift per unit length for body iLi = Length of body iLn = Vector of expansion coefficients for liftM = Global mass matrixM = Moment vectorMF = Flutter Mach numberNi = Number of spanwise locations for aerodynamic loads on body io, O = Origins of body and inertial axes, respectivelyP, Q, R = Angular velocity components about x, y, zpi = generalized momenta for body ipv = Linear momentum vectorq = Dynamic pressureqi = Generalized displacement vector for body iri = Radius vector from origin of xiyizi to a typical pointR = Turn radiusR = Position vector from O to osi = Generalized velocity vector for body iTe = Engine thrustT = Total kinetic energyTi = Kinetic energy for body iU, V, W = Translational velocity components in x, y, z directionsui = Elastic bending displacement vector of body iUi = Generalized force vectorUi = Matrix of shape functions for body i in bendingv = Translational velocity vector of xyzV = Global velocity vectorV = Speed of aircraftV = Total potential energyvi = Elastic velocity vector of body ivi = Velocity of a point on body iwi = Bending displacement of body i in zi directionXY Z = Inertial axesxyz = Body axes attached to aircraftxiyizi = Body axes attached to body i

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Greeksα = Local angle of attackαi = Elastic angular velocity vector for body iβ = Control surface deflectionγ = climb angle

δa, δe, δr = Aileron, elevator, rudder anglesδW = Total virtual workδr∗ = Virtual displacement of origin oδr∗i = Virtual displacement of a point on body iδr∗we = Virtual displacement of engineηi = Generalized angular velocity vector for body iδθ∗ = Virtual angular displacement of xyzλ0 = Induced flowρ = Air densityρi = Generalized angular momenta for body iρω = Angular momentum vectorφ, θ, ψ = Euler anglesϕi = Elastic angular displacement of body i about xi

ξi = Generalized angular displacement vector for body iψi = Elastic angular displacement vector for body iΨi = Generalized moment vectorΨi = Matrix of shape functions for body i in torsionω = Frequency of oscillationω = Angular velocity vector of xyzω = Skew symmetric matrix derived from ω

ωF = Flutter frequencyΩ = Turn rate

Subscriptse = Engine, or elevatorf = Fuselageh = Horizontal stabilizeri = body iv = Vertical stabilizerw = Wing

Introduction

Aeroelastic stability analysis of aircraft has received great deal of attention for almost a century, leading to

many important results that enabled the design of modern aircraft (Bisplinghoff and Ashley 1962). The common

practice in the analysis is to address stability about steady level flight at desired altitudes while frequently

ignoring the rigid body degrees of freedom and contributions from components such as fuselage, horizontal and

vertical stabilizers, and engine necelles. Stability statements about maneuvers other than steady level flight can

be meaningful if rigid body degrees of freedom are included in the mathematical model. Also, inclusion of rigid

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body and elastic degrees of freedom of the bodies other than wing is important for the accuracy of the stability

analysis. This paper is motivated by the following questions: How is the stability affected by maneuvers? Do

they tend to improve stability or worsen it? Or does it depend on the maneuvers considered and their parameters,

such as turn radius, climb angle, etc.? This paper intends to offer some answers to these questions. To keep the

governing equations time-invariant, and hence, the analysis simple, we will consider only steady maneuvers such

as steady level turn and steady climb. On the other hand, for the accuracy of the results, we intend to derive

a comprehensive model of aircraft that includes not only all six rigid body degrees of freedom for aircraft as a

whole and elastic degrees of freedom for all aircraft bodies, but also unsteady aerodynamics.

Early studies on aeroelastic stability were limited to simple wing models such as rigid airfoil and cantilevered

beam, which lacked rigid body degrees of freedom, as well as contributions from the other aircraft components.

These classical models were well justified in their times in the absence of powerful computers that can handle

problems involving high-dimensional and highly nonlinear aeroelastic systems. Nonetheless, there were numerous

attempts to include such effects. Among them, we can count such important studies as Bisplinghoff and Ashley

1962, Milne 1962, Taylor and Woodcock 1971 and Dusto et al 1974. However, the impact of these studies was

insufficient to attract interest in more comprehensive models.

The last 10 to 15 years have seen vast number of papers that aim to go beyond the classical models mentioned

above. Among them, we want to first cite the unified formulation of Meirovitch and Tuzcu 2003 and 2004 since

the present paper uses this unified formulation in its model development. The papers regard the aircraft as it is,

namely flying flexible body, and present a mathematical formulation that seamlessly integrates all the necessary

material pertinent to the flight of flexible aircraft. Flight dynamics and aeroelasticity are shown to be special cases

of the unified formulation. Meirovitch and Tuzcu 2005 uses this formulation to simulate the motion of flexible

aircraft executing a time-dependent maneuver, namely a pitch maneuver. The unified formulation is applied to a

high-altitude, long-endurance unmanned aerial vehicle (HALE UAV) in Tuzcu et al 2007, and to a fighter aircraft

in Meirovitch et al 2009. Patil et al 2001 presents a nonlinear aeroelastic study on a complete aircraft model which

is geometrically similar to High-Altitude-Long-Endurance (HALE) aircraft. The focus of the study is on the effect

of the static structural deflections on the aeroelastic behavior. A nonlinear model of a highly flexible flying-wing

is developed in Patil and Hodges 2006. The equations of motion are derived in terms of rather abstract variables,

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namely intrinsic variables. Rigid body degrees of freedom include only the ones corresponding to longitudinal

motion. Results are given for a typical high-aspect-ratio flying-wing configuration and include variations of trim

solutions, and phugoid and short-period modes with respect to payload, and nonlinear simulation of the wing in

response to a flap deflection. Shearer and Cesnik 2007 focuses on the nonlinear response of a very flexible aircraft

in flight. The results include the simulations of the aircraft in various flight conditions such as level gliding

descent, low-pass-filtered square aileron input, rolling/gliding descent, and low-pass square elevator input gliding

descent. The same authors in Shearer and Cesnik 2008 use their model to control six-degree-of-freedom rigid body

motion of an aircraft under large elastic deformations. Tuzcu 2008 studies stability of flexible aircraft in terms

of divergence and flutter. Results from four different models, all derived from the unified formulation as special

cases, are compared: 1) whole flexible aircraft using the full unified formulation, 2) quasi-rigid aircraft (aircraft

treated as rigid), 3) individual flexible components, such as cantilever wing, cantilever horizontal stabilizer, etc.,

and 4) restrained flexible aircraft (aircraft fixed to a point, hence, having no rigid body degrees of freedom).

Baluch and van Tooren 2009 studies the effects of coupling between elastic bending and twist on the dynamics of

whole flexible aircraft. Raghavan and Patil 2009 uses the model developed in Patil and Hodges 2006 to study the

effect of static aeroelastic deformation on the stability of a flexible flying wing. Additional trim cases considered in

the paper are climbing flight, level turn, and climbing turn. However, stability is addressed for only straight and

level trim in terms of root-locus plots for longitudinal and lateral flight dynamic modes in three configurations,

namely, the flexible configuration, a rigid body configuration based on the deformed shape at trim, and a rigid

body configuration based on the undeformed shape. Building on the results of Raghavan and Patil 2009, the same

authors in Raghavan and Patil 2010 use a reduced-order model of a flying wing to design a flight controller for

path following. An analysis and parametric study of the flight dynamics of highly flexible aircraft are presented

in Chang et al 2008. Influences of various design parameters such as wing flexibility and horizontal/vertical

tail aerodynamics are investigated for aeroelasticity and flight dynamics of aircraft. Baghdadi et al 2011 uses

bifurcation and continuation methods to evaluate the effects of flexibility on the dynamics, stability, and control

of elastic aircraft. The paper successfully demonstrates that the significant bifurcation phenomena is present

when the frequencies of the flexible modes are closer to those of the rigid-body modes, even in the absence of

nonlinear elasticity model. The paper also discusses the efficiency of its approach relative to traditional nonlinear

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simulation. Palacios et al 2010 evaluates different computational models for flight dynamics simulations on low-

speed aircraft with very-flexible high-aspect ratio wings. Structural dynamic models include displacement-based,

strain-based, and intrinsic geometrically-nonlinear composite beams, while unsteady aerodynamics models include

thin-strip and vortex lattice methods. Comparisons are made in terms of numerical efficiency and simplicity of

integration of the governing equations. On the structural modeling, it is found that intrinsic models can be several

times faster than conventional ones. For the aerodynamic modeling, thin-strip models are found to perform well

in small amplitudes, while large-amplitude wing dynamics require three-dimensional descriptions such as vortex

lattice. Zhao and Ren 2011 presents a multibody dynamics approach to modeling a flexible aircraft as a feedback-

controlled multibody system. The aircraft is taken as a multibody system consisting of rigid bodies and finite

segment beams, all connected by constraints. The approach is applied to the flexible aircraft in level flight and in

the circling and dive-loop-climb maneuvers. A joint flutter and attitude control is used to suppress flutter response

and stabilize attitude. Another multibody approach is presented by Kruger 2008. Paranjape et al 2012 studies

the performance and stability of a tailless micro aerial vehicle with flexible articulated wings. A linearization

using perturbation methods for flexible slender bodies is considered in Hesse and Palacios 2012. Nonlinear flight

dynamics of a flexible aircraft subjected to aeroelastic and gust loads is presented in Fazelzadeh and Sadat-Hoseini

2012. The results are given in terms of time simulations of aircraft response to aileron and elevator inputs as well

as gust load excitation. Additional studies on the subject, which are less in line with the present paper, can be

found in Seigler 2005, Reschke 2006, Abbas et al 2008, and Su and Cesnik 2010.

Stability statements in the majority of the papers cited above are made for the steady level flight. The

papers that consider maneuvers other than steady level flight are limited to time simulations that require visual

inspection and very large simulation time, and that cannot really address stability with certainty. Three exceptions

are Meirovitch and Tuzcu 2003 and 2004, and Zhao and Ren 2011. The first two address the stability of aircraft

in steady turn maneuver by linearizing the equations and solving the associated eigenvalue problem, although

they do not attempt to determine the flutter speed. The third, on the other hand, addresses flutter stability

about circling and dive-loop-climb maneuvers by simulating the response of aircraft at various flight speeds in

an attempt to capture the flutter speed. Contrary to the maneuvers we consider here, these maneuvers are not

steady and require feedback control, which can alter the dynamics of the aircraft significantly.

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Our earlier attempts toward developing a comprehensive aircraft model for the same aircraft considered in

this paper can be found in Nguyen and Tuzcu 2009, Tuzcu and Nguyen 2010a, 2010b, 2011. However, these

papers had limitations in the sense that the aerodynamics used in them were only quasi-steady aerodynamics.

In the present paper, we aim to improve the model by using an unsteady aerodynamics. Computational fluid

dynamics, or CFD, would potentially yield the most accurate aerodynamics, but for a meaningful accuracy, it

would require extremely large number of aerodynamic degrees of freedom. Aerodynamics by panel methods such

as vortex-lattice are also good candidates (see, for example, Murua et al 2012). However, they also require larger

number of degrees of freedom. CFD and the panel methods would also require an elaborate structural model to go

with them. Considering the level of simplicity of the structural model used in the present paper, an unsteady thin

airfoil theory based on potential flow seems to be much more suitable for the job. Unsteady aerodynamics theory

of Peters et. al. (Peters and Johnson 1994, Peters et al 1995, 2007) seems to be a good choice to complement our

model. This theory is especially chosen also because it is a method of choice in some of the papers cited above; see

for example Patil et al 2001, Patil and Hodges 2006, Shearer and Cesnik 2007, 2008. The theory allows large rigid

body motions and general dynamic deformations including trailing-edge flap motions. Moreover, it is formulated

in terms of generalized deflections so that it can easily be assembled with the rest of the model. However, the

theory is limited to subsonic. For this reason, we will limit our aeroelastic stability analysis to flutter only, since

the investigation of divergence for the aircraft considered here requires supersonic aerodynamics.

The numerical results will be given for the NASA’s Generic Transport Model (GTM), which is a large, com-

mercial, twin-engine aircraft whose length is 44.35 m (145.5 ft) and wingspan 37.80 m (124 ft). An approximate

numerical model of the GTM that includes information pertaining to geometry, and mass and stiffness distri-

butions of the whole aircraft was made available to the authors by NASA to be used in the investigation. The

details of the numerical model will not be presented in this paper since it includes proprietary information. The

numerical model is sufficiently good to capture accurately the dynamic behavior of the aircraft.

Modeling Flexible Aircraft

Equations of motion of flexible aircraft can conveniently be derived using the Lagrangian equations of motion

in quasi-coordinates as described in Meirovitch and Tuzcu 2003 and 2004. All we need for the derivation is the

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knowledge of three scalar quantities, namely, kinetic energy, potential energy, and the virtual work due to the

applied forces. The aircraft is regarded as a flexible multibody system where the bodies are the fuselage (f),

wing (w), horizontal stabilizer (h), and vertical stabilizer (v). Then, kinetic energy, potential energy, and the

virtual work for the whole aircraft are the sums of those for individual bodies. The fuselage is treated as the main

body of aircraft and remaining bodies, namely right- and left-half wing, right- and left-half horizontal stabilizer,

and vertical stabilizer, are all assumed to be rigidly connected to the fuselage at respective discrete points. For

consistent kinematics, the motions of each pair of connected bodies must be the same at the point of connection

for both of the bodies. To describe the motion of the aircraft, we first attach a set of body axes xyz to the

undeformed aircraft at a convenient point on the fuselage (not necessarily the center of mass of the aircraft), as

well as similar axes (xiyizi, i = f, w, h, v) to the flexible components at the points of connection, as shown Fig.

1. For convenience, we assume that xfyfzf coincides with xyz. The axes XYZ shown in Fig. 1 are the inertial

axes, which are fixed to the Earth. Position vector from the origin O of XY Z to the origin o of xyz is denoted by

R(t) = [X(t) Y (t) Z(t)]T , in which X(t) is the range, Y (t) the side displacement and Z(t) the altitude of aircraft.

The fuselage, right- and left-half wing, right- and left-half horizontal stabilizer, and vertical stabilizer are all

modeled as hollow beams, each fixed at its respective root, and subject to one flapwise bending displacement wi

in the zi direction and one spanwise torsional displacement ϕi about the xi axis so that linear and angular elastic

displacement vectors of the point located at xi are ui(xi, t) = [0 0 wi(xi, t)]T and ψi(xi, t) = [ϕi(xi, t) 0 0]T ,

respectively. Also, the elastic velocity vectors are denoted by vi(xi, t) = ui(xi, t) and αi(xi, t) = ψi(xi, t). It

is assumed that each xi coincides with the elastic axis of the respective body. Then the motion of a point on

the aircraft can be expressed by rigid-body translations and rotations of the body axes xyz and by the elastic

deformations of the flexible bodies relative to their respective body axes.

The governing equations are obtained using the generic Lagrangian equations of motion in quasi-coordinates

presented in Meirovitch and Tuzcu 2003 and 2004, which require the knowledge of kinetic energy T , potential

energy V , and virtual work δW due to applied forces. The equations include 12 first-order ordinary differential

equations for the rigid-body translations and rotations, and a partial differential equation for each elastic dis-

placement component. The system is hybrid since it includes both ordinary and partial differential equations.

It does not in general admit a closed-form solution, and an approximation requires discretization of the partial

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differential equations. To this end, each elastic displacement is expressed as a matrix of shape functions multiplied

by a vector of generalized coordinates. Hence, we have

ui(xi, t) = Ui(xi)qi(t), ψi(xi, t) = Ψi(xi)ξi(t)vi(xi, t) = Ui(xi)si(t), αi(xi, t) = Ψi(xi)ηi(t), i = f, w, h, v

(1)

where Ui(xi) is a 1 ×m matrix of m shape functions of xi and qi(t) is m-vector of generalized coordinates for

bending displacement of component i while Ψi(xi) is a 1×n matrix of n shape functions of xi and ξi(t) is n-vector

of generalized coordinates for torsional displacement of component i. Moreover, si(t) = qi(t) and ηi(t) = ξi(t)

are generalized velocities.

For good accuracy at low m and n, we choose the shape functions as the eigenfunctions of a uniform can-

tilever beam for the bending displacements, and the eigenfunctions of uniform fixed-free shaft for the torsional

displacements. Rather than deriving first the hybrid equations and then descretizing them in space, it is more

advantageous to carry out the discretization directly in the kinetic energy, potential energy, and virtual work.

Then, the equations are reduced to a set of first-order nonlinear ordinary differential equations. The generic form

of the discrete equations of motion is

d

dt

(∂L

∂v

)+ ω

∂L

∂v= F

d

dt

(∂L

∂ω

)+ v

∂L

∂v+ ω

∂L

∂ω= M (2)

d

dt

(∂L

∂si

)− ∂L

∂qi

= Ui

d

dt

(∂L

∂ηi

)− ∂L

∂ξi= Ψi, i = f, w, h, v

where L = T − V is the Lagrangian for the whole aircraft, v(t) = [U(t) V (t) W (t)]T and ω(t) = [P (t) Q(t) R(t)]T

are the vectors of translational and angular velocities of xyz, θ(t) = [φ(t) θ(t) ψ(t)]T symbolic vector of Eulerian

angles between xyz and XY Z, C = C(φ, θ, ψ) matrix of direction cosines from inertial axes XY Z to xyz,

E = E(φ, θ) matrix relating the vector of Eulerian velocities θ to angular velocity vector ω, F and M resultant

force and moment vectors acting on the whole aircraft in terms of body axes components, Ui and Ψi generalized

force vectors for bending and torsion. Note that the term ω∂L/∂v indicates vector product ω × ∂L/∂v where ω

is a skew symmetric vector derived from ω. In general, for a vector d = [a b c]T , the skew symmetric vector d is

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d =

⎡⎣ 0 −c b

c 0 −a−b a 0

⎤⎦ (3)

The same can also said about v∂L/∂v and ω∂L/∂ω. Note that ∂L/∂R and ∂L/∂θ are omitted in Eqs. (2)

because for an unrestrained aircraft L does not depend on R nor θ. The first of Eqs. (2) is the force equation,

the second the moment equation, and the last two are the generalized equations of motion for the bending and

torsion of the i-th component. For the complete description of the motion of flexible aircraft, these equations

must be considered in conjunction with the kinematical relations

R = CTv, θ = E−1ω, qi = si, ξi = ηi (4)

The body axes xyz are obtained from the inertial axes XY Z through the following sequence of rotations: 1) ψ

about Z to the intermediate axes x1y1z1, 2) θ about y1 to the intermediate axes x2y2z2, and finally 3) φ about x2

to the body axes xyz. For the resulting transformation matrices C and E, the readers are referred to Meirovitch

and Tuzcu 2003.

Kinetic Energy

The kinetic energy for the whole aircraft requires the knowledge of the velocity and mass distribution over the

aircraft. Velocity of a typical point on the fuselage is the velocity of the origin o of the axes xyz plus the relative

velocity of the point in question with respect to o, which can be expressed as

vf (rf , t) = v + (rf + uf )Tω + rTf αf + vf (5)

where rf = [xf yf zf ]T is the radius vector from o to the point in question. Similarly, the velocity of a point on

the wing is the velocity of the origin ow of xwywzw, which is equal to the velocity of the point on the fuselage

at which the wing is attached, plus the relative velocity of the point in question with respect to ow. The same

argument can also be made about the velocity of a point on the horizontal and vertical stabilizers. Hence, the

velocity of a typical point on any of these three components will have the form

vi(ri, t) = Civ + [Ci(rfi + ufi)T + (ri + ui)

TCi]ω + rTi Ci(Ωfi +αfi) (6)

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+Ci(vfi + rTfiαfi) + rTi αi + vi, i = w, h, v

where ri = [xi yi zi]T , Ci is matrix of direction cosines from xyz to xiyizi, rfi = [xfi yfi zfi]

T is the radius

vector from the origin of xyz to the origin of xiyizi, ufi(t) = uf (xfi, t) is the elastic displacement of the fuselage

at xf = xfi, Ωfi(t) = [0 (−∂uf/∂xf )(xfi, t) 0]T is the angular velocity of xiyizi due to the bending of the

fuselage at xf = xfi, vfi(t) = vf (xfi, t) is the velocity of the origin of xiyizi due to the elastic velocity of the

fuselage, and αfi(t) = αf (xfi, t) is the angular velocity of xiyizi due to the elastic torsional velocity of the

fuselage at xf = xfi. The engines are assumed to be lumped masses and attached to the wing. Their velocities

are determined by evaluating wing velocity at the respective point of attachment, i.e. ve(t) = vw(rwe, t) where

rwe is the position vector from ow to the engine location. The total kinetic energy is equal to the summation of

the individual kinetic energies:

T =∑i

Ti, Ti = 1

2

∫vTi vidmi, i = f, w, h, v, e (7)

where dmi is the mass differential element for the component i. Introducing Eqs. (1) into Eqs. (5) and (6), and

the resulting expressions into Eqs. (7), the total kinetic energy can be written in the compact form

T = 1

2VTMV (8)

where

V = [vT ωT sTf sTw sTh sTv ηTf ηT

w ηTh ηT

v ]T (9)

is the global velocity vector and M is the global mass matrix. Once T is known the ij entry of the mass matrix

can then be determined from

Mij =∂2T

∂Vi∂Vj

(10)

where Vi and Vj are the i-th and j-th entries of the velocity vector V. It is important to mention that the

mass matrix M is a nonlinear function of the generalized displacements qi, and hence, it changes as the aircraft

deforms.

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Using the kinetic energy expression, we can also construct the momenta associated with the velocities:

pv =∂T∂v

, ρω =∂T∂ω

; pi =∂T∂si

, ρi =∂T∂ηi

, i = f, w, h, v (11)

Then, it can be shown that

p =∂T∂V

= MV (12)

where

p = [pTv ρT

ω pTf pT

w pTh pT

v ρTf ρT

w ρTh ρT

v ]T (13)

is the global momentum vector.

Potential Energy

Potential energy is due to the strain energy. Similarly to the kinetic energy, the total potential energy is the

summation of strain energies of the individual bodies, which can easily be expressed in terms of bending and

torsional elastic displacements:

V =∑i

Vi, Vi =1

2

∫ Li

0

[EIi

(∂2ui

∂x2

i

)2

+GJi

(∂ψi

∂xi

)2]dxi, i = f, w, h, v (14)

where Li is the length, EIi flexural rigidity, and GJi torsional rigidity of the component i. Introducing the first

two of (1) into Eqs. (14), the total potential energy can be written as

V = 1

2

∑i

(qTi Kiqi + ξTi Kiξi) (15)

where

Ki =

∫ Li

0

EIi∂2UT

i

∂x2

i

∂2Ui

∂x2

i

dxi, Ki =

∫ Li

0

GJi∂ΨT

i

∂xi

∂Ψi

∂xi

dxi, i = f, w, h, v (16)

are the bending and torsional stiffness matrices, respectively.

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Virtual Work

The virtual work can be expressed in terms of the applied forces and virtual displacements:

δW =∑i

∫fTi δr∗i dxi + FT

e δr∗

we (17)

where fi is the distributed force due to aerodynamics and gravity, Fe is the engine thrust, δr∗i is the virtual

displacement of a typical point on the component i, and δr∗we is the virtual displacement δr∗w evaluated at the

engine location. We assume in Eq. (17) that the engines are mounted on the wing. Virtual displacements δr∗i

have similar expressions as vi:

δr∗f = δr∗ + (rf + ˜Ufqf )T δθ∗ +Ufδqf + rTf Ψfδξf

δr∗i = Ciδr∗ + [Ci(rfi + ˜Ufiqf )

T + (ri + ˜Uiqi)TCi]δθ

∗ + (rTi CiΔUfi + CiUfi)δqf (18)

+Uiδqi + (rTi Ci + CirTfi)Ψfiδξf + rTi Uiδξi, i = w, h, v

where δr∗ is the virtual displacement due to the quasi-displacement r∗ whose time derivative is equal to the quasi-

velocity v, i.e. r∗ = v. Similarly, δθ∗ is the virtual angular displacement due to the angular quasi-displacement θ∗

whose time derivative is equal to the quasi velocity ω, i.e. θ∗

= ω. Moreover, Ufi = Uf (xfi), Ψfi = Ψf(xfi), and

ΔUfi are constant matrices such that Ωfi = ΔUfiqf . Now, substituting Eqs. (18) into Eq. (17) and separating

the terms, we get

δW = FT δr∗ +MT δθ∗ +∑

i(UTi δqi +ΨT

i δξi) (19)

where

F =∑i

CTi

∫fidxi + Fe (20)

is the resultant force on the aircraft in which summation is carried out for i = f, w, h, v;

M =

∫(rf + ˜Ufqf )ffdxf +

∑i

∫[(rfi + ˜Ufiqf )C

Ti + CT

i (ri + ˜Uiqi)]fidxi (21)

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+[(rfw + ˜Ufwqf )C

Tw + CT

w (rwe + ˜Uweqw)]Fe (22)

is the resultant moment on the aircraft in which rw = rwe is the radius vector from the origin of xwywzw to the

engine location, Uwe = U(xwe), and summation is carried out for i = w, h, v;

Uf =∫UT

f ffdxf +∑

i

∫(rTi CiΔUfi + CiUfi)

T fidxi + (rTweCwΔUfw + CwUfw)TFe

Ψf =∫ΨT

f rf ffdxf +∑

i

∫ΨT

fi(rTi Ci + Cir

Tfi)

T fidxi +ΨTfw(r

TweCw + Cw r

Tfw)

TFe(23)

is the generalized force and moment for the fuselage in which the summations are for i = w, h, v; and finally

Uw =∫UT

wfwdxw +UTw(xwe)Fe

Ψw =∫ΨT

wrwfwdxw +ΨTw(xwe)rweFe

Ui =∫UT

i fidxi, Ψi =∫ΨT

i rifidxi, i = h, v(24)

are the generalized force and moment for the wing, horizontal stabilizer and vertical stabilizer.

Aerodynamic and Gravity Forces

We assume that only the wing, and the stabilizers are subject to aerodynamics, and ignore contributions from

the fuselage and the engine nacelles, which mostly contribute in the aerodynamic drag. Aerodynamic forces and

moments on the wing are approximated by the unsteady theory of Peters et al 1994, 1995 and 2007. The theory

actually yields aerodynamic loads over 2-D airfoil section. However, we will approximate 3-D loads by using the

2-D theory at a finite number of spanwise locations of the wing. Once aerodynamic lift, drag and moment per unit

span are computed at these spanwise locations, they can be determined at any arbitrary location by interpolating

their values at the two locations neighboring the location in question. Using similar notations used in Peters et al

1994 and 1995, we first express the loads for an airfoil section. We then use these expression to obtain the loads

at N number of spanwise locations. The aerodynamic loads are given in terms of the following:

Ln

2πρ= −b2M

(hn + vn

)− bu0C

(hn + vn − λ0

)− u2

0Khn − bG (u0hn − u0vn + u0λ0)

D

2πρ= −b

(hn + vn − λ0

)T

S(hn + vn − λ0

)+ b

(hn + vn

)T

Ghn (25)

−u0

(hn + vn − λ0

)T

(K − H)hn + (u0hn − u0vn + u0λ0)T Hhn

where ρ is the air density, b the semi-chord as shown in Fig. 2, Ln = [L0 L1 . . . Ln]T in which L0, . . . , Ln are the

expansion coefficients for the lift L, hn = [h0 h1 . . . hn]T in which h0, . . . , hn are the expansion coefficients for the

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chordwise elastic deformation h of the airfoil section, vn = [v0 v1 0 . . . 0]T , D is the drag, and λ0 = [λ0 0 . . . 0]T

in which λ0 is the weighted average of the induced flow from the trailed circulation. The matrices M , C, K, G,

S, and H are as given in Peters and Johnson 1994. Also, u0 and v0 are linear velocities and v1 is the velocity

gradient of the airfoil section before the elastic deformation, whose positive directions are shown in Fig. 2. We

assumed that the chordwise deformation is due to the plunge h, the pitch α, and the control surface deflection β,

as shown in Fig. 2, so that the expansion coefficients for h are expressed as

h0 = h− abα+βb

π(sinϕm − ϕm cosϕm)

h1 = bα+βb

π(ϕm − sinϕm cosϕm) (26)

hn =βb

π

[1

n+ 1sin[(n+ 1)ϕm] +

1

n− 1sin[(n− 1)ϕm]− 2

ncosϕm sin(nϕm)

], n ≥ 2

where ϕm = cos−1 d.

Using the relations given in Eqs. (26), we can establish a transformation matrix T so that hn = T [h α β]T .

Then the actual aerodynamic loads can be written as [L Ma Mβ ]T = T TLn where L is the lift per unit span, Ma

is the pitching moment per unit span about the elastic axis, and Mβ is the hinge moment per unit span of the

control surface. To determine λ0, we need to have a wake model. For a flat wake with steady free stream and

harmonically oscillating airfoil, we can use the Theodorsen theory (Peters et al 1994 and 1995):

λ0 =[u0α+ h+ b(1

2− a)α

][1− C(k)] (27)

where C(k) is the Theodorsen function, k = ωb/u0 is the reduced frequency in which ω is the frequency of the

oscillation of the airfoil. The aerodynamics reduces to the quasi-steady one if λ0 = 0.

The theory described above is applied to the each half of the wing and the horizontal stabilizer at Nw and Nh

numbers of spanwise locations, respectively, from the root to the tip. For the airfoil section located at xi = xij

(i = w, h; j = 1, 2, . . . , Ni), h(t) = wi(xij , t), α(t) = ϕi(xij , t) cosΛi where Λi is the swept angle for the elastic

axis. u0(t) = vix(rij , t), v0(t) = viz(rij , t) with wi and ϕi set equal to zero, where vix(rij , t) and viz(rij , t) are the

x− and z−components of the velocity vi(rij , t) in which rij = [xij 0 0]T , and v1(t) � bQ(t). Also, β = δa for the

wing, and β = δe for the horizontal stabilizer. When the theory is used to obtain the lift �i and drag di per unit

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span for i = w, h, the resultant aerodynamic force per unit span can be written as

fia =

⎡⎣ �i sinαi0 − di cosαi0

0−�i cosαi0 − di sinαi0

⎤⎦ � �i

⎡⎣ αi0

0−1

⎤⎦− di

⎡⎣ 1

0αi0

⎤⎦ (28)

where αi0 = W/U is the approximate angle of attack.

For the vertical stabilizer, it will be sufficient to use a quasi-steady theory. The local angle of attack can be

approximated as αv � vvz/U + ϕv where vvz is the zv component of vv. Using this angle of attack, the lift per

unit length can be expressed as

lv = qcvCLv = qcv(CLαvαv + CLδr δr) (29)

where cv is the chord, CLv the lift coefficient, CLαv the slope of the lift curve, δr the rudder angle, CLδr is the

rudder effectiveness, and q � ρ(U2 + V 2 +W 2)/2 is the dynamic pressure. In the absence of CLδr, we will treat

δr = CLδr δr as the control input. Similarly, the drag per unit span is

dv = qcvCDv = qcvCDαvαv (30)

where CDv is the drag coefficient, and CDαv is the slope of the drag curve. Finally, distributed aerodynamic force

on the vertical stabilizer can be written as

fva =

⎡⎣ 0

lv sinαv0 − dv cosαv0

lv cosαv0 + dv sinαv0

⎤⎦ � lv

⎡⎣ 0

αv0

1

⎤⎦+ dv

⎡⎣ 0

−1αv0

⎤⎦ (31)

where we use αv0 = V/U .

The gravity forces per unit volume are simply

fig = CiC

⎡⎣ 0

0ρig

⎤⎦ , i = f, w, h, v (32)

where ρi is the mass density for the component i and g is the gravitational acceleration.

Discrete State Equations

Recognizing that the potential energy V does not depend on the velocities, we can cast the discrete equations of

motion in the following first-order form

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R = CTv

θ = E−1ω

qi = si

ξi = ηi

pv = −ωpv + F (33)

ρω = −V pv − ωρω +M

pi = ∂T /∂qi −Kiqi +Ui

ρi = −Kiξi +Ψi, i = f, w, h, v

The first half of Eqs. (33) are kinematical relations, the first two of the second half are the force and moment

equations, and the remaining equations are generalized equations for bending and torsion, respectively. Note that

both velocities and momenta appear in Eqs. (33) so that the equations must be considered in conjunction with

the momenta-velocities relation given in Eq. (12). ∂T /∂qi terms appearing in the equations are responsible for

the centripetal and Coriolis effects on the deformations of the flexible beams.

Control inputs enter into the equations through the engine thrust and the aerodynamic forces. We assume

that the engine thrust has a component only in x direction so that Fe = [Te 0 0]T , and that aileron angles on the

right- and left-half wing have the same magnitudes, but opposite directions. Hence, the control input vector can

be written as u = [Te δa δe δr]T .

Note that the aircraft model includes six cantilever beams and each beam has one bending displacement and

one torsional displacement. Since we use m shape functions to represent each bending displacement and n for each

torsional displacement, the number of total elastic degrees of freedom for the whole aircraft is 6m+6n. With the

addition of six rigid body degrees of freedom, the number of total degrees of freedom is 6+6m+6n = 6(1+m+n),

and hence, the order of Eqs. (33) is 12(1 +m+ n).

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Trim Solutions

Trim (or equilibrium) solution of the equations can be obtained by postulating the states corresponding to a

desired maneuver and solving the equations of motion for control inputs that will allow the realization of the

maneuver. To keep the equations time-invariant, we will consider the following steady flights.

Steady Level Flight

In a steady level flight, aircraft nominally maintains 1) a constant velocity V = VX , 2) a constant altitude H , 3)

zero roll and yaw angles, and constant pitch angle, 4) constant elastic displacements, so that

R =

⎡⎣ VXt

0H

⎤⎦ , θ =

⎡⎣ 0

θ0

⎤⎦ , qi = constant, ξi = constant, i = f, w, h, v (34)

From the first three of Eqs. (33), we have

v = CR =

⎡⎣ cos θ 0 − sin θ

0 1 0sin θ 0 cos θ

⎤⎦⎡⎣ VX

00

⎤⎦ = VX

⎡⎣ cos θ

0sin θ

⎤⎦ , ω = Eθ = 0, qi = 0, ξi = 0 (35)

To keep these nominal velocities constant, we must require v = 0, ω = 0, si = 0, ηi = 0 for i = f, w, h, v, from

which we also have pv = 0, ρω = 0, pi = 0, ρi = 0 for i = f, w, h, v. Furthermore, the centripetal and Coriolis

forces depend on ω so that ω = 0 implies that ∂T/∂qi = 0 for all i = f, w, h, v. Once a forward velocity and a

cruise altitude are chosen for the steady level flight, the corresponding trim solution can be obtained by solving

Eqs. (33) for the nominal pitch angle θ, static deformations qi and ξi for all i = f, w, h, v as well as nominal

control input vector u. Note that in steady level flight all of the states are constant except the range, which is

X(t) = VXt. However, since X(t) does not explicitly appear in the equations of motion, the equations linearized

about the steady level flight is time-invariant.

Steady Level Turn

For a turn of radius R and angular velocity Ω, we have

R =

⎡⎣ R sinΩt

R(1− cosΩt)H

⎤⎦ , θ =

⎡⎣ φ

θΩt


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v = CR = V

⎡⎣ cos θ

sin θ sinφcosφ sin θ

⎤⎦ , ω = Eθ = Ω

⎡⎣ − sin θ

cos θ sinφcos θ cosφ

⎤⎦ , qi = 0, ξi = 0 (37)

where the speed of the aircraft is V = RΩ. For given R, Ω and H , the trim solution is obtained by solving Eqs.

(33) for the pitch angle θ, bank angle φ, static deformations qi and ξi for all i = f, w, h, v, and control input

vector u. Notice that the range X(t) = R sinΩt, the side displacement Y (t) = R(1 − cosΩt), and yaw angle

ψ(t) = Ωt are time-varying, but this poses no problem since X(t), Y (t), and ψ(t) do not appear in the equations

explicitly. Hence, the linearized equations about the steady turn are also time-invariant.

Steady Climb

R =

⎡⎣ Vt cos γ

0H +Vt sin γ

⎤⎦ , θ =

⎡⎣ 0

θ0


v = CR = V

⎡⎣ cos(θ − γ)

0sin(θ − γ)

⎤⎦ , ω = Eθ = 0, qi = 0, ξi = 0 (39)

where γ is the climb angle. For given V, γ, and H , the remaining states and u are determined similar to the

earlier two cases. The linearized equations about the steady climb are also time-invariant.

Numerical Model

Numerical results will be presented for the NASA’s Generic Transport Model (GTM), which is a notional twin-

engine transport aircraft. The structures of fuselage, wing, and horizontal and vertical stabilizers are modeled as

hollow beams with constant thicknesses. To approximate the mass and stiffness distributions of the bodies, we

use the cross-sectional properties of the model, such as geometric center, cross-sectional area, area moments of

inertia etc., at some number of stations on the respective components. A distribution of the geometric centers of

the cross-sections of the individual components are shown in Fig. 3. For each component i, we fit the respective

geometric centers to a straight line to obtain the line of geometric centers. xi is chosen to be the line of geometric

centers for the fuselage and the vertical stabilizer. On the other hand, for the wing and horizontal stabilizer xi

is ab = −0.25b behind the respective line of geometric centers. We assume that xi coincides with the elastic axis

for all i = f, w, h, v.

The orientation of the component body axes xiyizi can be obtained from xyz by a sequence of rotations. The

same rotation sequence used in the construction of C will be used for the individual Ci. The Euler angles for each

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of the flexible components are listed in Table 1. Bending and torsional rigidity distributions of the individual

flexible components with respect to their xi axes are shown in Fig. 4.

As mentioned earlier, the 2-D aerodynamics theory is used at finite number of spanwise locations to approxi-

mate the 3-D aerodynamic loads. The spanwise locations are chosen to be the horizontal lines on the wing and

the horizontal stabilizer shown in Fig. 5. The value of the lift-curve slope Clα used in Eqs. (25) is 2π. We will

obtain a more accurate value for Clα at the same spanwise locations using the Vortex Lattice Method (VLM).

Our VLM model includes only the combination of the wing and the horizontal stabilizer. The VLM panels are

also shown in Fig. 5. Clα distribution over the wing and the stabilizer is shown in Fig. 6. Notice from the figure

that Clα for the finite wing and the stabilizer is significantly lower than 2π. The Prandtl-Glauert correction,

1/√1−M2, is applied to Clα to account for the compressibility effect.

In this numerical example, we aim to use the constructed model of the GTM to study its stability in terms

of its flutter speed in various steady flight conditions, namely, steady level flight, steady turn and steady climb

at four different altitudes: sea level, 3048 m (10, 000 ft), 6096 m (20, 000 ft) and 9144 m (30, 000 ft). Since the

equations of motion are time-invariant in each of these maneuvers, the stability can be addressed by linearizing

the equations about the nominal flight and solving the associated eigenvalue problem for the linearized equations.

The flutter speed is the lowest speed of the nominal flight at which one of the eigenvalues of the system crosses the

imaginary axis. For best accuracy, we use the frequency-domain version of the wake model, and choosem = n = 2

so that the order of the equations is 12(1 + m + n) = 60. Note that this wake model is valid only at flutter.

Since the equations are highly nonlinear and determination of the flutter condition requires iterative methods,

choosing m and n higher than 2 makes the calculations extremely more time-consuming, with very little gain in

the accuracy of the results. For this reason, choosing both m and n as 2 seems to be a good compromise between

the accuracy and attainability of the results. The system, hence, has 60 eigenvalues and the first 12 of them

are dominated by the rigid body motion of the aircraft, and are similar to the eigenvalues encountered in flight

dynamics. The remaining eigenvalues are dominated by the elastic motion and are similar to the eigenvalues

encountered in aeroelasticity. For the steady level flight, the flutter Mach number and frequency, as well as the

values of the pitch angle (θ), engine thrust (Te), and elevator angle (δe) for the flight are all listed in Table 2 and 3

for quasi-steady (QS) and unsteady models (US), respectively. The resulting bending and torsional displacements

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at the tip of the wing are also listed in the tables.

Stability of the same aircraft using the same numerical model, but a different formulation is also studied in

Nguyen et al 2012, which uses a quasi-steady aerodynamics and a structural model that assumes flexibility for

only the wing. Nguyen et al 2012 estimates the flutter speed of a steady level flight at 9144 m (30,000 ft) as

MF = 0.63, which is closed to our estimate of MF = 0.7329 for the same altitude. Since our model uses degrees

of freedom for not only wing, but also fuselage and stabilizers, its estimate is expected to be larger.

Note that unsteady model includes more accurate aerodynamics of the two models. This means that the end

results such as flutter speed and frequency from the unsteady model are also more accurate than those from the

quasi-steady model. Comparing the results from the quasi-steady and the unsteady models, we conclude that the

results from the quasi-steady theory show significant errors in the flutter conditions. At lower altitudes, Flutter

Mach number (MF ) from US model is lower than MF from QS model. On the other hand, at higher altitudes,

it is just the opposites. In other words, MF from US model is higher than MF from QS model. This means that

there exist an altitude at which the both models yields the same MF .

Flutter Mach number, MF , and flutter frequency, ωF , versus turn radius, R, in steady turn at sea level,

3048 m (10, 000 ft), 6096 m (20, 000 ft) and 9144 m (30, 000 ft) altitudes for both quasi-steady and unsteady

aerodynamics are shown in Figs. 7-10. Due to the space limitations, the nominal values of roll angle (φ), pitch

angle (θ), the engine thrust (Te), control surface angles (δa, δe, δr) as well as the tip displacements of the wing

(ww(Lw, t), ψw(Lw, t)) versus R are presented for only 9144 m altitude, which is also in Figure 10. From these

figures, we notice that MF decreases while ωF increases as R increases for all altitudes except the sea level

altitude at which MF increases while ωF decreases with R for the US model, and both MF and ωF remains

almost unchanged with R for the QS model. From these results, we can conclude that there likely exists a critical

altitude at which both MF and ωF remain unchanged with R for also the US model. For both QS and US models

and all of the altitudes, both MF and ωF approach their values at steady level flight as R approaches infinity.

The magnitudes of the slopes of the curves for both MF and ωF , as well as QS and US models, are larger for

higher altitudes. Since the flutter speed is a margin for aeroelastic stability, we can conclude that the aircraft

considered here has better stability margin in a steady turn than a steady level flight at the altitudes higher than

the critical altitude mentioned above, and just the opposite at the altitudes below the critical altitude.

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We also present the flutter mode shapes using the US model at three turn radiuses in Fig. 11. Only some of

the degrees of freedom that have the largest contributes are included in the figure. Quite surprisingly, the first

bending mode of the fuselage (qf1) has the largest contribution among the all linear displacements. That is why

we normalize the flutter mode so that qf1 = 1 m. The second largest contributions are from the second bending

modes of the right- and the left-half wing, which are slightly out of phase with each other. The magnitudes of

these bending modes increases as the turn radius R decreases (or as the turn gets tighter). Among all the angles,

the first and the second torsional modes of the left- and right-half wing have the largest contributions. The first

modes of the left- and right-half wing are 180◦ out of phase with each other. The same can also be said about

the second modes of the left- and right-half wing. Among the rigid body displacements, the side displacement Y ,

and the Euler angles φ and ψ have significant contribution in the flutter mode. Their contributions decreases as

the turn radius R increases.

Results in steady climb are somewhat similar to the ones in steady turn. MF and ωF versus the climb angle,

γ, at sea level, 3048 m (10, 000 ft), 6096 m (20, 000 ft) and 9144 m (30, 000 ft) altitudes for both quasi-steady

and unsteady aerodynamics are shown in Figs. 11-14. The nominal values of pitch angle (θ), engine thrust (Te),

elevator angle (δe) as well as the tip displacements of the wing (ww(Lw, t), ψw(Lw, t)) versus γ are presented

for only 9144 m altitude, which is also in Figure 14. From the figures, we observe that MF increases while ωF

decreases as γ increases for all of the altitudes. Notice that MF and ωF are equal to their values at steady level

flight for γ = 0. The magnitudes of the slopes of the curves for both MF and ωF , as well as QS and US models,

are somewhat larger for higher altitudes. Also notice that flutter speed of gliding flight can be approximately

determined from these figures. For the 9144 m altitude, we see that Te = 0 for approximately γ = −3.12◦, and

the corresponding flutter Mach number and frequency are MF = 0.7787 and ωF = 34.939 rad/s. As a result, we

conclude that the aircraft, at all of the altitudes, has better stability margin in a steady climb (positive γ) and

worse stability margin in a descending flight (negative γ), including a gliding flight, than in a steady level flight.

The flutter mode shapes are presented for three climb angles in Fig. 16. Similar to the steady turn case, the

first bending mode of the fuselage has the largest contribution among the all linear displacements. The second

bending modes of the right- and the left-half wing come next. In angles, the first and the second torsional modes

of the left- and right-half wing have the largest contributions. The plunge displacement Z, and the pitch angle θ

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also have significant contributions. As seen in the figure, the flutter mode shapes practically remain unchanged

as the climb angle changes.

We at this point want to stress that the above results are for only the type of aircraft considered in this

investigation. The results might be different for a different type of aircraft. Aircraft has complex dynamics and

any effort in predicting these results in advanced will be an exercise in futility. In fact, predicting the stability

of aircraft requires development of an accurate mathematical model of its dynamics and stability analysis, as

described in this paper. Anything short of that will just be speculation.

To the best of our knowledge, this is the first study that addresses the flutter in the maneuvers considered. For

that reason, there is no study in the open literature to which the present paper can be compared. However, to at

least validate the results for steady level flight, we compare it to Nguyen et al 2012 which uses the same numerical

model, but a different formulation to study the flutter of the GTM with only wing flexibility. Our result is in

good agreement with that of Nguyen 2012. The most fundamental conclusion we can draw from this investigation

is that maneuvers affect the stability of aircraft and the degree of this effect depends on maneuver parameters

such as turn radius, climb angle, etc., as well as flight altitude and aircraft parameters such as geometry, and

mass and stiffness distributions.

Conclusions

In this paper, we investigate how the aeroelastic stability, particularly flutter, is affected by aircraft maneuvers.

For the accuracy of the results, we base our investigation to a comprehensive mathematical model of aircraft,

which is described by a set of ordinary differential equations that accounts for all rigid body degrees of freedom

for aircraft as a whole, elastic degrees of freedom for each of the flexible components (fuselage, wing, horizontal

stabilizer, and vertical stabilizer), and the circulatory and noncirculatory unsteady aerodynamics, all in coupled

form. We use the model to address the stability of the NASA’s Generic Transport Model (GTM) by computing

its flutter speed in not only steady level flight, but also steady turn and steady climb maneuvers at various flight

altitudes. For the steady turn maneuver above a critical altitude, we conclude that the flutter Mach number

decreases while flutter frequency increases as the turn radius increases. It is just the opposite below the critical

altitude. Flutter speed and frequency approach to their values in steady level flight as the turn radius goes to

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infinity, without depending on the altitude. For the steady climb, the flutter Mach number increases while flutter

frequency decreases as the climb angle increases. This shows climbing flight somewhat improves stability while

descending flight somewhat worsens it. We also show how the results of steady climb can be extended to cover

the stability of gliding flight. Since gliding flight is also a descending flight, it also worsens the stability. All of

these effects mentioned here are larger for larger flight altitudes. The results are given for both quasi-steady and

unsteady aerodynamics, which demonstrates the order of error in the results when the aerodynamics is limited

to only quasi-steady theory.

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Figure Caption List

Figure 1: Aircraft Model

Figure 2: Airfoil Section

Figure 3: xi axes and sectional geometric centers for the bodies.

Figure 4: EI(xi) and GJ(xi) distributions with respect to xi.

Figure 5: The VLM Panels.

Figure 6: Clα distribution from the VLM model.

Figure 7: Flutter Mach number and frequency versus turn radius at sea level.

Figure 8: Flutter Mach number and frequency versus turn radius at 3048 m (10,000 ft).

Figure 9: Flutter Mach number and frequency versus turn radius at 6096 m (20,000 ft).

Figure 10: Flutter Mach number, frequency, trim conditions, and tip displacements versus turn radius at 9144 m

(30,000 ft).

Figure 11: Flutter Displacements and Angles Corresponding to qf1 = 1 m at Various Turn Radius (From US

Model at 9144 m Altitude).

Figure 12: Flutter Mach number and frequency versus climb angle at sea level.

Figure 13: Flutter Mach number and frequency versus climb angle at 3048 m (10,000 ft).

Figure 14: Flutter Mach number and frequency versus climb angle at 6096 m (20,000 ft).

Figure 15: Flutter Mach number, frequency trim conditions, and tip displacements versus climb angle at 9144 m

(30,000 ft).

Figure 16: Flutter Displacements and Angles for qf1 = 1 m at Various Climb Angles (From US Model at 9144 m

Altitude).

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Tables

Table 1: Rotations to Construct Ci

Flexible Euler angles [ ◦ ]Component φ θ ψFuselage 0 0 0

Right-Half Wing 0 5.16 110.55Left-Half Wing 0 5.16 -110.55

Right-Half Stabilizer 0 6.39 115.51Left-Half Stabilizer 0 6.39 -115.51Vertical Stabilizer 90 125.53 0

29



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Table 2: Flutter Using the Quasi-Steady Theory (QS)

h [m] MF ωF [rad/s] θ [ ◦ ] Te [kN] δe [ ◦ ] ww(Lw, t) [m] ψw(Lw , t) [◦ ]

0 0.6851 15.604 1.1461 17.791 −3.6194 −1.0794 1.1745

3048 0.7451 35.228 1.3449 20.876 −4.0749 −1.0370 1.1497

6096 0.7227 35.305 2.4763 38.430 −6.6427 −0.9138 1.0809

9144 0.7329 35.321 3.8298 59.408 −9.6013 −0.8549 1.0518

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Table 3: Flutter Using the Unsteady Theory (US)

h [m] MF ωF [rad/s] θ [ ◦ ] Te [kN] δe [ ◦ ] ww(Lw, t) [m] ψw(Lw , t) [◦ ]

0 0.6742 17.773 1.2209 18.951 −3.7906 −1.0621 1.1643

3048 0.7433 34.923 1.3589 21.093 −4.1069 −1.0344 1.1483

6096 0.7565 34.930 2.0867 32.386 −5.7663 −0.9427 1.0965

9144 0.7791 34.936 3.0625 47.519 −7.9412 −0.8827 1.0650

31




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X

Y

Z

O

x

y

zR

y

zxw

w

w

rfw oy

zxhh

h

xv

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zv




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β

h

β

α

b b

ab

db

Elastic axis

Undeformed

Deformed

v

u

0

0

v1




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0

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0.2 0.4 0.6 0.8 1.0

5

10

15

20

25EI

GJ

[GN-m ]

x /L

2

f f

EI

GJ

[GN-m ]2

x /Lw w

0.2 0.4 0.6 0.8 1.0

1

2

3

4

EI

GJ

[GN-m ]

x /L

2

h h

0.2 0.4 0.6 0.8 1.0

1

2

3

4

5

6

7

EI

GJ

[GN-m ]2

x /Lv v

0.2 0.4 0.6 0.8 1.0

0.2

0.4

0.6

0.8

1.0




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-1.0 -0.5 0.5 1.0

1

2

3

4

5

wing

horizotal stabilizer

x/L




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5 6 7 8 9 10 11 12

0.674

0.676

0.678

0.680

0.682

0.684

5 6 7 8 9 10 11 12

16.0

16.5

17.0

17.5

R [km]

ωFF

R [km]

QS

US

�

QS

US




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5 6 7 8 9 10 11 12

0.745

0.750

0.755

0.760

0.765

0.770

0.775

5 6 7 8 9 10 11 12

34.9

35.0

35.1

35.2

R [km]

ωFF

R [km]

�

QSUS

QS

US




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0.73

0.74

0.75

0.76

0.77

0.78

5 6 7 8 9 10 11 12

34.8

34.9

35.0

35.1

35.2ωF

F

QS

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R [km] R [km]




J. Aerosp. Eng.

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5 6 7 8 9 10 11 12

0.74

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0.78

0.80

0.82

0.84

5 6 7 8 9 10 11 12

34.6

34.8

35.0

35.2

5 6 7 8 9 10 11 12

0

10

20

30

40

50

5 6 7 8 9 10 11 12

0

1

2

3

4

5

6

7

5 6 7 8 9 10 11 12

0

20

40

60

80

100

120

140

5 6 7 8 9 10 11 12

0.0

0.5

1.0

1.5

5 6 7 8 9 10 11 12

-11.0

-10.5

-10.0

-9.5

-9.0

5 6 7 8 9 10 11 12

0.0

0.1

0.2

0.3

0.4

0.5

5 6 7 8 9 10 11 12

-1.4

-1.3

-1.2

-1.1

-1.0

-0.9

5 6 7 8 9 10 11 12

1.2

1.4

1.6

1.8

2.0

R [km] R [km]

R [km] R [km]

R [km] R [km]

R [km] R [km]

R [km]R [km]

φ[ ] oθ[ ]

oδ [ ]δr

e

oδ [ ]a

oψ (L ) [ ]w wu (L ) [m]ww

F

ωF

T [kN]e

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J. Aerosp. Eng.

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by

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L o

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-0.1 0.1 0.2 0.3 0.4 0.5 0.6

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-0.2

0.2

qwR1

qwR2

qwL2

Y

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ξ wL2

ξ wR1

ξ f1

ξ wL1

ξ wR2

φψ

R=3658 m

R=5486 m

R=9144 m

Displacements [m] Angles [Degree]

q =1 mf1

Im

Im

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J. Aerosp. Eng.

Dow

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-15 -10 -5 0 5 10 15

0.674

0.676

0.678

0.680

0.682

0.684

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16.0

16.5

17.0

17.5

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ωFF

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0.740

0.742

0.744

0.746

0.748

0.750

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34.90

34.95

35.00

35.05

35.10

35.15

35.20

35.25

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0.725

0.730

0.735

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0.755

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35.0

35.1

35.2

35.3

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by

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0.74

0.75

0.76

0.77

0.78

-15 -10 -5 0 5 10 15

35.0

35.1

35.2

35.3

2.8

3.0

3.2

3.4

3.6

3.8

4.0

-100

0

100

200

-12

-10

-8

-6

-4

-2

0

-0.90

-0.88

-0.86

-0.84

-0.82

-0.80

-0.78

0.95

1.00

1.05

1.10

FωF

oθ[ ] T [kN]e

oδ [ ]e

oψ (L ) [ ]w w

u (L ) [m]ww

oγ [ ]

oγ [ ]

oγ [ ]oγ [ ]

oγ [ ]oγ [ ]

oγ [ ]

-15 -10 -5 0 5 10 15-15 -10 -5 0 5 10 15

-15 -10 -5 0 5 10 15-15 -10 -5 0 5 10 15

-15 -10 -5 0 5 10 15

QS

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J. Aerosp. Eng.

Dow

nloa

ded

from

asc

elib

rary

.org

by

UN

IVE

RSI

TE

LA

VA

L o

n 07

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14. C

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ight

ASC

E. F

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nal u

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righ

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-0.1 0.1 0.2 0.3 0.4 0.5

-0.5

-0.4

-0.3

-0.2

-0.1

0.1

0.2

-10 -5 5 10

-15

-10

-5

5

10

15

γ = 0

γ = -15

γ = 15

q =1 mf1o

o

o

qwR2

ξ wL1

θ

Im

Re

Im

Re

, qwL2

Z

ξ wR1

ξ wL2

ξ wR2

Displacements [m] Angles [Degree]




J. Aerosp. Eng.

Dow

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ded

from

asc

elib

rary

.org

by

UN

IVE

RSI

TE

LA

VA

L o

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/07/

14. C

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ASC

E. F

or p

erso

nal u

se o

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all

righ

ts r

eser

ved.

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Flutter of Maneuvering Aircraft

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