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12dostrained flexible aircraft (aircraft fixed to a point, hence, having no rigid body degrees of freedom). The paper also presents a method to addresse stability of flexible aircraft when the compressibility correction factor is known only at some discrete Mach numbers.2007 Elsevier Masson SAS. All rights reserved.

ywords: Flight dynamics; Aeroelasticity; Unified formulation; Perturbations

Introduction

Aircraft stability has been the subject of many investiga-ons. In most studies, the subject has been treated as twoparate disciplines, namely, flight dynamics and aeroelasticity,3,6]. Flight dynamics is mainly concerned with the dynamicsquasi-rigid aircraft (aircraft treated as rigid) [3]. The mo-

on is described by nonlinear ordinary differential equations ofder 12, or lower, depending on assumptions. Common prac-

ce is to linearize the equations about a steady trim with thesulting equations admitting an exponential solution, so thatability can be assessed by solving the associated eigenvalueoblem. On the other hand, aeroelasticity is mainly concernedith restrained flexible aircraft [1,6,15]. Most common mod-s include an airfoil section supported by springs, or a wingxed at its root, both acted upon by aerodynamic forces. Theotion is described by either ordinary differential equations fore airfoil section or partial differential equations for the can-lever wing; the equations are linear or nonlinear depending one elasticity and aerodynamics models used. Partial differen-al equations are transformed into sets of ordinary differential

E-mail address: ituzcu@eng.ua.edu.

equations through spatial discretization. If the equations are lin-ear, then the stability can be assessed by solving the associatedeigenvalue problem.

There have been very few investigations treating aircraft asthey are, namely, flying flexible machines. The earliest attemptgoes as far back as 1929 when Frazer and Duncan [4] inves-tigated the effect of mobility of the fuselage on flutter calcu-lations. They showed that the flutter speed is larger when therigid body degrees of freedom (dof) are included. Some im-portant experimental and theoretical results were presented byGaukroger [5], in which stability of two types of modes wereconsidered: symmetric and antisymmetric modes. Accordingto Gaukroger, in the symmetric mode, the flutter characteris-tics depend heavily on fuselage pitching moment of inertia.With sufficiently small pitching moment of inertia, flutter re-sults from the interaction of wing bending and pitch dof, whichis known as body-freedom flutter. The flutter speed is lowerthan the cantilever wing flutter speed and increases as the pitch-ing moment of inertia increases, until it reaches a critical valueabove which the flutter speed starts decreasing and slowly ap-proaching its cantilever wing value. In the antisymmetric mode,the roll degree of freedom is the primary factor in wing flut-ter. The flutter speed might be larger than its cantilever wingAerospace Science and Technolog

On the stability ofIlhan T

Department of Aerospace Engineering and Mechanics, TheReceived 12 January 2007; received in revised fo

Available online 21

bstract

The problem of aircraft stability has been a subject of concern since tithin the confines of two separate disciplines, namely, flight dynamics ad control of flexible aircraft, this investigation uses the system conceidge the gap between stability as understood in flight dynamics and stae following four cases: 1) dynamics of whole flexible aircraft using theeated as rigid), 3) aeroelasticity of flexible components, such as cantile70-9638/$ see front matter 2007 Elsevier Masson SAS. All rights reserved.i:10.1016/j.ast.2007.09.0032 (2008) 376384www.elsevier.com/locate/aescte

exible aircraft

cu

versity of Alabama, Tuscaloosa, AL 35487-0280, USA4 July 2007; accepted 17 September 2007

tember 2007

eginnings of flight. Traditionally, aircraft stability has been treatedaeroelasticity. Based on some recent developments in the dynamicsto provide a broader approach to aircraft stability in an attempt toty as envisioned in aeroelasticity. To this end, stability is studied inified formulation, 2) flight dynamics of quasi-rigid aircraft (aircraftwing, cantilever horizontal stabilizer, etc., and 4) aeroelasticity of

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lue depending on the roll moment of inertia. Also worthymention is the study by Milne [10] who derived the lin-

rized equations of motion about a steady state by assumingall elastic and rigid body motions. The equations are usedstudy the longitudinal stability of the aircraft. An exten-

ve report by Dusto et al. [2] resulting in a computer programown as FLEXSTAB, integrates flexible body mechanics withlow-frequency aerodynamics. The equations are expressedterms of steady perturbations about a reference motion and

e used to determine dynamic stability. The major concern isat the structural dynamics formulation is in terms of meanes while the aerodynamics is in terms of a different set ofes, namely, fluid axes, and there is no indication of anyordinate transformation from one set of axes to the other.fundamental study of the stability of forward swept wing air-aft is presented by Weisshaar and Zeiler [14]. A simple linearodel of three dof including pitch, plunge and wing bendingused to calculate flutter speeds and frequencies for various

rcraft configurations. They concluded that body-freedom flut-r and aircraft aeroelastic divergence, not wing divergence, areimary aeroelastic instabilities experienced by forward swepting aircraft. The instabilities are shown to occur close to theing divergence speed, but depend upon the aircraft geome-y and mass and stiffness distributions. A study by Meirovitchd Seitz [8] is concerned with the aeroelastic tailoring of a

ructural model consisting of a low aspect ratio swept wingade of composite materials and a rigid fuselage. The elas-c wing model includes shear deformations and is attached torigid fuselage capable of plunge and pitch. The flutter dy-mic pressure and the frequency are given in terms of tailoringy angle. A comprehensive study of nonlinear aeroelasticity ofrcraft is presented in [11]. The model developed in this pa-r accounts for rigid body degrees of freedom and nonlinearasticity, although, numerical results are given for cantilevering only. The results include the effect of the nonlinearities

the flutter speed and in limiting the amplitude of oscillationsce an instability is encountered, i.e. existence of limit cy-

e oscillations (LCOs). [12] uses the model developed in [11]study nonlinear aeroelasticity and flight dynamics of high-

titude long-endurance aircraft. The paper presents root-locusthe eigenvalues of the complete aircraft for a range of flighteeds. However, the paper does not study the divergence and

utter of the complete aircraft. A fairly inclusive model of flex-le aircraft is given in [13]. The paper focuses on stability oflarge civil transport aircraft about a reference flight, and howe eigenvalues are affected by the changes in reference roll an-e, angle of attack and elevator deflection.A newly developed formulation presented in a series of pa-

rs contains a rigorous and comprehensive study of dynamicsd control of flexible aircraft [7,9]. It integrates seamlessly insingle, consistent mathematical formulation all the necessaryaterials from the pertinent disciplines, namely, analytical dy-mics, structural dynamics, aerodynamics, and controls. Theified formulation is based on fundamental principles and in-rporates in a natural manner both the rigid-body motions and

e elastic deformations, and the couplings thereof. The for-ulation can be used for various purposes, such as stabilityhnology 12 (2008) 376384 377

alysis, simulation of aircraft motion during maneuvers, re-onse to external excitations and real-time control of flexiblercraft.The present paper attempts to paint a complete picture of

e stability of flexible aircraft. This is achieved by compar-g results obtained by means of the unified formulation withose obtained by other approaches. To this end, stability isudied in the following four cases: 1) dynamics of whole flex-le aircraft using the unified formulation, 2) flight dynamicsquasi-rigid aircraft, 3) aeroelasticity of flexible components,ch as cantilever wing, cantilever horizontal stabilizer, etc.,d 4) aeroelasticity of restrained flexible aircraft. The resultstained in each of the four cases are based on the same aircraftnfiguration and structural and aerodynamics models.

The unified formulation

Aircraft can be modeled as flexible multibody systems,here the bodies can be identified as the fuselage (f ), wing) and empennage (e). To describe the motion of the aircraft,

e attach sets of body axes xiyizi , with origins Oi (i = f,w, e)the undeformed bodies and regard the body axes attached to

e fuselage, xf yf zf , as the body axes of the aircraft. Then,e motion can be expressed in terms of three translations andree rotations of the fuselage body axes and elastic deforma-ons of flexible components relative to their respective bodyes. The equations of motion for flexible aircraft of the typeown in Fig. 1 were derived by Meirovitch and Tuzcu [7,9].

he derivation of the equations require only expressions fore Lagrangian and the virtual work of the forces acting one aircraft. The resulting equations include ordinary differen-al equations for the translations and rotations of xf yf zf andrtial differential equations along with appropriate boundarynditions for the bending and torsional deformations of the

exible components relative to the corresponding body axes.r practical reasons, the distributed variables must be dis-

etized in space, which amounts to introducing the expansions

i (ri , t) = Ui (ri )qi (t),i(ri , t) = i(ri ) i (t), i = f,w, e

(1)Fig. 1. Flexible aircraft model.

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here ri (xi, yi, zi) is a radius vector from Oi to a typical pointbody i, ui and i are elastic bending and torsional displace-

ent vectors, respectively, Ui and i are matrices of shapenctions and qi (t) and i (t) are corresponding vectors of gen-alized coordinates. The discrete equations of motion can berived by introducing Eqs. (1) into the expressions for the La-angian and the virtual work directly. From [7,9], the stateuations have the matrix form

= CT V, = E1,i = si , i = i ,V = pV + F, = V pV p + M,i = T /qi Kiqi Cisi + Qi ,i = Ki i Cii +i , i = f,w, e

(2)

here V = [u v w]T and = [p q r]T are the vectors of trans-tional and angular velocities of xf yf zf , V and skew sym-etric matrices derived from V and such that V pV = VpVd p = p, R = [Xf Yf Zf ]T is the position vector ofe origin Of of xf yf zf relative to the inertial axes XIYIZI ,= [ ]T is a symbolic vector of Eulerian angles be-een xf yf zf and XIYIZI , C = C() is a matrix of directionsines between xf yf zf and XIYIZI , E = E() matrix relat-g Eulerian velocities to angular velocities, F = [X Y Z]T and= [L M N ]T are resultants of gravity, aerodynamic, propul-

on and control force and moment vectors acting on the wholercraft in terms of fuselage body axes components, Ki, Ki andi , Ci are stiffness and damping matrices for body i, respec-vely, Qi and i generalized force vectors and, finally,

V = TV

, p = T

,

i = Tsi

, i =T

i, i = f,w, e

(3)

e momentum vectors, in which T is the kinetic energy. Noteat Eqs. (2) include both the velocities and momenta and,nce, must be considered in conjunction with the momenta-locities relation

= T /v = Mv (4)here p = [pTV pT pTf pTw pTe Tf Tw Te ]T , v = [VT T sTf sTwTf

Tw

Te ]T are momenta and velocity vectors, respectively,

d M is the aircraft mass matrix whose expression is given in]. Expressions for F, M, Qi and i in terms of the actualstributed forces are also given in [7].The state equations, Eqs. (2), represent a set of nonlinear

fferential equations whose order depends on the number ofastic dof of the model. The stability of such a system can bedressed by adopting a perturbation approach, which amountsassuming that the quantities associated with the aircraft rigiddy motions can be expressed as the sum of a large part, callede zero-order part, and a small part, called the first order part,follows:

= R + R, = + = V + V, = + (5)

w

bahnology 12 (2008) 376384

which the zero-order quantities, denoted by the overbar, areleast one order of magnitude larger than the first-order quan-

ties, denoted by the overhat. The elastic deformations are as-med to be of first order by definition. Inserting Eqs. (5) into

qs. (4) and separating the different orders of magnitude, thero-order momenta-velocities relation can be written as [7,9]V = mV + S, p = SV + J (6)here m is the total mass and S and J are the first and sec-d moments of inertia of the rigid aircraft, and hence, theye zero-order quantities. Similarly, from [7,9], the first-orderomenta-velocities relation is

= M v + M v (7)here M is the mass matrix for the quasi-rigid aircraft, M is therturbation about it, v and v are the zero-order and first-orderlocity vectors and p is the first-order momentum vector. Thero-order and the first-order equations of motion can be ob-ined by inserting Eqs. (5) into Eqs. (2), separating the differ-t orders of magnitude and using Eqs. (6) and (7) to eliminatee velocities. The zero-order equations of motion can be writ-n in the compact state-space form

t) = f[x(t),u(t)] (8)here x = [RT T pTV pT ]T is the zero-order state vector,= [FE a e r ]T is the control vector, in which FE, a, ed r are the engine thrust and the aileron, elevator and ruddergles and f is a nonlinear function of the state and control vec-rs; they represent quasi-rigid flight dynamics equations [9].oreover, the first-order perturbation state equations of motionve the matrix form

t) = A(t)x(t) (9)here x = [RT T qTf . . . Te pTV pT pTf . . . Te ]T is the first-der state vector and A(t) = A[x(t),u(t)] is a coefficient ma-ix. Eqs. (9) represent expanded aeroelasticity equations [9], ine sense that they include automatically all the aircraft rigid-dy motions and can accommodate any time-varying solutiont) of Eq. (8).The zero-order equations, or the flight dynamics equations,

ven by Eq. (8), are nonlinear and of order 12 at most. The first-der equations, or the perturbation equations, given by Eq. (9)e linear and tend to be of high order. Moreover, we observeat the coefficient matrix A depends on the zero-order statet) and control vector u(t), representing a given maneuver. If xd u are constant, then the system of Eq. (9) is time-invariant,d if x and u depend on time, then the system is time-varying.The control vector u can be set so as to permit any given

rcraft maneuver. For steady cases, such as steady level flightd level steady turn maneuver, this process is known as

imming the aircraft, which involves choosing first the para-eters of a flight, namely, speed (forward velocity), altituded either radius or angular velocity of turn and finding the

ate and the control vectors that satisfy Eq. (8). However,

hen the aircraft executes a chosen maneuver, various distur-nces, such as gust can give rise to the perturbation x(t),

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hich dies out if the first-order system is asymptotically sta-e, or grows without bounds if unstable. Hence, the stabil-y of the flexible aircraft in a given maneuver can be ad-essed by examining the behavior of the first-order system.hen A is time-invariant, the first-order system is asymptot-ally stable if all the eigenvalues of A have negative realrts.Two types of stability of flexible aircraft are of interest,

mely, static and dynamic stability. When the initial state x(0)nonzero, the aircraft will generate initial forces and momentsravitational and aerodynamic). These forces and moments arestoring type if they tend to take x(t) to the equilibrium x = 0d divergent type if they tend to take x(t) away from the equi-

brium. For a sufficiently small flight speed, the forces andoments are restoring type and the aircraft is statically sta-e. However, there may exist a range of larger aircraft speedr which the resulting initial forces and moments cause x(t)diverge from the equilibrium. If this is the case, then there

ust exist a flight speed and an initial state x(0) = 0 such thatx(0) = 0, which can be satisfied if and only if the determi-nt of A is zero. Note that this initial state must be equal to angenvector of A corresponding to a zero eigenvalue. The flighteed for which |A| = 0 is known as divergence speed. It is im-rtant to note that for any flight speed the aircraft does notnerate any force in response to the rigid body displacementsand the yaw displacement . These are all reflected in fourro eigenvalues of A, and hence, the determinant of A is zeroen when the speed of the aircraft is lower than the divergenceeed. Therefore, the divergence speed is the lowest speed ofe aircraft at which A has five zero eigenvalues.Static stability does not really imply dynamic stability. In

her words, even when the aircraft generates restoring forcesd moments at t = 0 due to the initial state x(0), the dynamicrces and moments immediately after t = 0 could destabilizee aircraft. One type of dynamic instability is flutter in whiche aerodynamic forces result in negative damping destabilizinge aircraft. Similarly to the case of divergence, flutter occurshen the speed of the aircraft reaches a value known as fluttereed. This is the lowest speed for which A has at least onerely imaginary complex eigenvalue.In conclusion, divergence exists if, as the flight speed in-

eases, any eigenvalue of A crosses the origin of the complexane from left to right. The divergence speed is the speed of thercraft at which the eigenvalue in question is exactly equal toro. Similarly, flutter exists if any eigenvalue crosses the imag-ary axis from left to right. The flutter speed is the speed of thercraft at which the eigenvalue is purely imaginary.

Flight dynamics of the quasi-rigid aircraft

We regard the flexible aircraft shown in Fig. 1 as rigid andfer to it as quasi-rigid. The equations of motion can be ob-

ined from Eqs. (2) by retaining the parts corresponding to thegid-body degrees of freedom and setting all the parts corre-

A

ifhnology 12 (2008) 376384 379

onding to the elastic degrees of freedom identically equal toro. The resulting equations are

= CT V, = E1,V = pV + F, = V pV p + M

(10)

y analogy with Eqs. (6), the relations between the velocitiesd the momenta are

V = mV + ST, p = SV + J (11)here S and J are the first and second moments of inertia fore quasi-rigid aircraft, and are equivalent to S and J definedrlier. Using these relations to eliminate the momenta, we canpress the equations of motion of the quasi-rigid aircraft in thempact state-space form

t) = fQR[x(t),u(t)

] (12)here x = [RT T VT T ]T is the state vector of order 12, u ise control vector defined earlier and fQR is a nonlinear functionthe state and control vectors.The control vector u can be chosen so that the aircraft exe-

tes a desired maneuver. However, various disturbances causee aircraft to undergo perturbations from the desired flightth. To address the stability of the quasi-rigid aircraft, we againploy a perturbation approach and write the state as x = x + x

here x is one order of magnitude smaller than x. Substitut-g this into Eq. (12), separating terms of different orders ofagnitude and ignoring the higher-order terms, we obtain thero-order state equation

t) = fQR[x(t),u(t)

] (13)d the first-order perturbation state equation

t) = AQR(t)x(t) (14)here

QR(t) = AQR[x(t),u(t)

] = fQRx

x=x

(15)

he zero-order equation, Eq. (13), is in general nonlinear. Noteat Eqs. (8) and (13) are equivalent. The only difference is thatq. (8) is in terms of momenta while Eq. (13) is in terms oflocities. Because x and u enter into the first-order equation asputs, Eq. (14) is linear and time-invariant if both x and u arenstant, and time-varying otherwise. As soon as a zero-order

ate representing a desired maneuver is chosen, the requiredntrol input enabling the maneuver can be obtained by solving

q. (13) for u(t). Note that the quasi-rigid aircraft representsunderactuated system in the sense that the number of control

puts, which is typically four, is smaller than the number off. However, four control inputs are sufficient for designingost common maneuvers, such as steady level flight, steadyrn and pitch maneuver.As in the case of whole flexible aircraft, the stability ofasi-rigid aircraft for a given maneuver can be treated by con-

dering the perturbation dynamics given by Eq. (14). When

QR is constant, the first-order system is asymptotically stablethe eigenvalues of AQR have negative real parts. However,

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e recall from the preceding section that AQR also has four zerogenvalues. The rest are the eight eigenvalues representing typ-ally the phugoid, short-period, spiral, rolling convergence andutch roll modes [3]. Note also that the existence of any posi-ve real eigenvalue clearly indicates static instability [3].

Aeroelasticity of flexible components

We consider cantilever models of the flexible componentscluding the fuselage, wing and empennage. The wing and therizontal stabilizer are assumed to be fixed at the mid-span, re-lting in right-half and left-half cantilever wing and horizontalabilizer, respectively. The fuselage is assumed to be fixed atf resulting in fore and aft cantilever parts of the fuselage. Thertical stabilizer is assumed to be fixed at the root. Equationsmotion for each component can be obtained from Eqs. (2)retaining the elastic equations of motion for a given compo-

nt and setting the elastic variables corresponding to the othermponents equal to zero, where the control vector u(t) and the

gid body variables are included as inputs into the equations.ence, the equations of motion for component i take the form

i = si , i = i ,i = T /qi Kiqi Cisi + Qi ,i = Ki i Cii +i

(16)

here all quantities are as defined in Eqs. (2). Eqs. (16) are lin-r and of order 2ni where ni is the number of elastic degreesfreedom of component i. Using the momenta-velocities rela-

ons pi = Misi , i =Mii to eliminate the momenta, wherei and Mi are the corresponding mass matrices, the resultinguations of motion can be cast in the compact matrix form(t) = Aixi (t) (17)here xi = [qTi Ti sTi Ti ]T is the corresponding state vectord Ai is a constant coefficient matrix.The flexible component i is stable if the eigenvalue of the

efficient matrix Ai have negative real parts. In the case ofeady level flight with sufficiently small flight speed, all of thegenvalues of Ai (i = f,w, e) have negative real parts. Largeright speeds may result in negative stiffness, and hence in di-rgence, and/or in negative damping, and hence in flutter, asplained earlier.

Aeroelasticity of restrained flexible aircraft

In this case, the flexible aircraft is assumed to be fixed atf , so that it is prevented from undergoing rigid-body mo-ons, except for steady longitudinal translation. The equations

motion for this case can again be obtained from Eqs. (2) byiminating the rigid-body dof; the result is

i = si , i = i ,i = T /qi Kiqi Cisi + Qi ,i = Ki i Cii +i , i = f,w, e

(18)

here all quantities are as defined in Eqs. (2). Note that, evenough Eqs. (18) do not contain any rigid-body dof, the rigid-

dy variables and the control vector for a given steady ma-uver enter into the equations as inputs. Eqs. (18) are linearhnology 12 (2008) 376384

d of order 2n, where n is the number of elastic dof of thehole aircraft. Considering again the relations pi = Misi andi =Mii , the equations of motion can be cast in the compactatrix form

t) = ARAx(t) (19)here x = [qTf . . . Te sTf . . . Te ]T is the corresponding statector and ARA is a constant coefficient matrix.The motion of the restrained aircraft consists of stable elastic

bration for the most part. However, under certain circum-ances the aerodynamic forces can cause instability in the formdivergence or flutter. The restrained aircraft is asymptotically

able if all the eigenvalues of ARA have negative real parts.

Application to a business jet

As a numerical example, we consider the aircraft shown inig. 1 and assume a structural model as depicted in Fig. 2 of [9],which all the flexible members consists of beams in bendingd torsion. In particular, the fuselage is modeled as a fore partd an aft part, both cantilevered at Of and undergoing bend-g vibrations in the yf and zf directions and torsion about. The wing consist of a right-half wing and a left-half wing

odeled as cantilever beams undergoing bending in the zw di-ction and torsion about xw . Finally, the empennage consistsa vertical stabilizer and a right-half and left-half horizontal

abilizer, all modeled as cantilever beams undergoing bendingd torsion. Each cantilever beam is assumed to be discretized

the Galerkin method using two shape functions per dis-acement component. For bending the shape functions are thegenfunctions of a uniform cantilever beam and for torsion thegenfunctions of a uniform clamped-free shaft.The model considered here is of an actual business jet made

ailable by an aircraft manufacturer. The data regarding theometry, the mass and stiffness distributions and the aero-namic coefficients of the aircraft are as given in [7]. Theatrices C and E and the computed stiffness matrices for thedividual components, Ki and Ki (i = f,w, e), are also given[7]. We assume that the structural damping matrices are pro-rtional to the stiffness matrices with the each proportionalitynstant being equal to 2/ where is the damping factord is the lowest natural frequency of the respective compo-nt. For this numerical example, we assume that = 0.5% forch flexible component.Fig. 2. Compressibility correction factor.

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The maneuver of interest in this numerical example is theeady level flight, which is by far the most common case. Therst task is to obtain a trim state for a chosen steady levelight, which can be done by considering Eq. (8) (or equiva-ntly Eq. (13)). Note that the quasi-rigid aircraft has 6 rigiddy dof, so that Eq. (8) is of order 12. In this case, the zero-der variables have the form

=[ 0

0

], R =

[V

00

], = 0

= C[V

00

]= V

[cos

0sin

], = 0

(20)

here V is the forward velocity of the aircraft (in the XI direc-on). The trim state can be obtained by substituting Eqs. (20)to Eq. (8) and solving the resulting equations for the existingknowns (whether they are states or control inputs). Of course,is requires that the aerodynamics be explicitly known, whichturn requires that we choose the flight speed V and the alti-

de h. The numerical model also includes the lift-curve slopesd the control effectiveness coefficient for some number of

rips on the fuselage, wing and empennage lifting surfaces [7].he compressibility effect is included through a compressibilityrrection factor, fc, given for some discrete values of subsonicach numbers ranging from M= 0.1 to M= 0.85, as shownFig. 2. fc is assumed to vary linearly between two consec-

ive discrete Mach numbers. Also, note that fc for a giventitude h can be expressed as a function of V . In the remainderthe paper, we refer to the curve fc versus V as correctionrve.

In general, the distributed aerodynamic forces include circu-tory and noncirculatory terms and the circulatory terms aree more significant of the two [6]. According to the Modified-trip theory by Yates [15], the effect of the unsteadiness ofe oscillatory motion on the circulatory terms can be approx-ated by modifying the lift-curve slope Cl by ClC(k) forch strip where C(k) is a complex-valued function, known ase Theodorsen function, of k = b/V , the nondimensionalduced frequency, in which V is the freestream velocity, equency of oscillation and b the semicord [1,6]. Note thatrodynamics for C(k) = 1 is known as quasi-steady aerody-mics [6]. In this numerical example, we assume that onlye circulatory terms are significant and ignore the noncircula-ry ones. As indicated in [15], the Modified-Strip theory yieldsore accurate results for wing of higher aspect ratios and lowerept angles. The wing of the aircraft considered in this pa-r has a large aspect ratio (AR = 7.3) and low swept angle1.41) and the theory is certainly suitable for this wing. In

e remainder of the paper quasi-steady and unsteady aerody-mics are labeled as QS and US, respectively.The trim equations Y = 0, L = 0 and N = 0 are satisfied if= 0 and r = 0. The remaining equations X = 0, Z = 0 and= 0 can be satisfied by choosing , FE and e. The nonzero

riables entering as inputs into the perturbation equations,qs. (9), (14), (17) and (19), are , u = V cos , w = V sin ,

E and e, and they are all constant. On the other hand, the onlyme-varying state, namely, the forward displacement Xf = V t

clVhnology 12 (2008) 376384 381

Fig. 3. Eigenvalues of AQR, h = 0 m,M= 0.5 to 0.85.

es not appear explicitly in the equations. Hence, the perturba-on equations, Eqs. (9), (14), (17) and (19), are time-invariant,d their stability can be assessed by examining the systemgenvalues, as discussed earlier.To determine the stability of the quasi-rigid aircraft, we com-te the eigenvalues i (i = 1,2, . . . ,12) of AQR for various

titudes and discrete values of M from 0.5 to 0.85. The locusthe eigenvalues for h = 0 m (sea level) is shown in Fig. 3.

s can be seen in the figure, AQR is unstable because it hasur zero and one positive real eigenvalues, although the pos-

ive real eigenvalue drops from = 0.00197 to = 0.00036M increases from 0.5 to 0.85. The locus of the eigenvalues

r h = 1524 m (= 5000 ft), h = 3048 m and h = 4572 m aremilar to the one shown in Fig. 3; they are not given here forevity.With the structural model defined in the beginning of this

ction, the aircraft has 32 elastic dof. Hence, the restrainedrcraft has 32 dof and ARA is 64 64. With the addition of thex rigid body dof, the unrestrained aircraft has 38 dof and A is76. Similarly, Af for both fore and aft fuselage are 1212ch, and Aw for the right- and left-half wing, Ae for the right-d left-half horizontal stabilizer and the vertical stabilizer arel 8 8 each.The first 12 eigenvalues of unrestrained aircraft are very

milar to those of quasi-rigid aircraft. Hence, the unrestrainedrcraft has also four zero and one positive real eigenvalues forlM= 0.50.85 and the four altitudes. On the other hand, Themaining 64 eigenvalues are similar to those of the restrainedrcraft and they are stable as long as the flight speed is less thanth the divergence and the flutter speeds.The divergence speed VD of the unrestrained aircraft for a

ven altitude h can be estimated by using a trial and errorproach, which starts with a sufficiently small initial guessr V = VD . A corresponding compressibility correction factorom the correction curve is used to generate the aerodynamicrces and moments; a trim state is obtained by solving the trimuations mentioned above and, finally, the eigenvalues of Ae computed. If A has a positive real eigenvalue, other thane positive real eigenvalue mentioned in the preceding para-aph, then V > VD and V must be slightly reduced. However,it has no positive real eigenvalue, then V < VD and V mustslightly increased. The procedure is repeated until V is asose to VD as desired. In this manner, the divergence speedD , for which A has a zero eigenvalue in addition to the four

38 Tec

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Fig. 4. Divergence at h = 0 m.

Fig. 5. Divergence at h = 1524 m.

ro eigenvalues mentioned earlier, can be estimated with rea-nable accuracy. However, the functional dependence of fcV introduces some complexity into this process. Perhaps, it

ould be easier to use a slightly different approach, as follows:irst choose a compressibility correction factor fc and computee trim states and A. Next, use the trial and error approach totimate the lowest V such that A has an additional zero eigen-lue and repeat this procedure a number of times for differentoices of fc. Then, map the resulting (fc,V ) combinationsd connect the neighboring points to obtain a curve repre-nting potential points of divergence. We refer to this curvepotential divergence curve. Next, we overlay the potential

vergence curve and the correction curve for the altitude cho-n. The true VD is the point of intersection of these two curves.f course, an intersection may not exist, which would clearlyean that divergence does not exist.The method outlined in the above paragraph can be used to

timate the divergence speed using the other models, includinge restrained aircraft (RA) and cantilever flexible componentsch as cantilever wing (CW), cantilever horizontal stabilizerHS), etc. The method can also be used for estimating fluttereed VF . The only difference is that, instead of finding the po-ntial divergence curve, we find the potential flutter curve,hich is the curve fc versus lowest V such that A has at leaste purely imaginary complex eigenvalue. The true flutter oc-rs where the potential flutter curve intersects the correctionrve. Flutter does not exist if the two curves do not inter-ct.The correction curve and potential divergence curves for

e unrestrained aircraft (UA), RA and CW models at h = 0

d h = 1524 m are shown in Figs. 4 and 5. The curves for= 3048 m and h = 4572 are similar to the ones in Figs. 4

so

fohnology 12 (2008) 376384

Fig. 6. Flutter at h = 0 m.

Fig. 7. Flutter at h = 1524 m.

Fig. 8. Flutter at h = 3048 m.

Fig. 9. Flutter at h = 4572 m.

d 5 and are not presented in the paper. Divergence of thentilever wing is the most critical among the individual can-

lever flexible components discussed in Section 4. For this rea-n, only the potential divergence curve for the cantilever wingodel is shown in the figure. As seen in the figure, none ofe potential divergence curves intersects the correction curve,

that all three models predict that divergence does not exist

r h = 0 m altitude. Divergence of the aircraft may be possi-

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ble 1utter speed and frequency, and trim values

UA RA CWh = 0 m

[m/s] QS 235.68 228.98 174.38US 216.81 210.0 174.30

[r/s] QS 231.50 234.28 242.88US 250.48 250.59 248.25

[] QS 0.4680 0.5154 0.9440US 0.5944 0.6396 0.9473

[N] QS 11714 11064 6481.7US 9931.5 9325.5 6475.6

[] QS 4.2731 4.5690 6.5545US 4.9928 5.2122 6.5747

h = 1524 m[m/s] QS 244.68 237.90 190.53

US 225.42 219.29 185.03[r/s] QS 231.16 234.04 242.39

US 250.84 250.97 248.32[] QS 0.4614 0.5097 0.9067

US 0.6114 0.6622 0.9633[N] QS 10887 10299 6662.3

US 9260.1 8771.3 6294.1[] QS 4.0629 4.3641 6.3688

US 4.9699 5.2465 6.6120h = 3048 m

[m/s] QS 251.15 246.21 207.21US 232.12 226.47 196.77

[r/s] QS 230.91 233.81 241.90US 251.11 251.25 248.39

[] QS 0.4558 0.5042 0.8726US 0.6237 0.6787 0.9815

[N] QS 9843.1 9465.3 6751.4US 8428.7 8030.7 6107.4

[] QS 3.8248 4.1499 6.1657US 4.8596 5.1721 6.6588

h = 4572 m[m/s] QS 251.99 220.21

US 238.82 233.40 208.11[] QS 233.66 241.54

US 251.36 251.50 248.46[] QS 0.4995 0.8476

US 0.6359 0.6944 0.9990[N] QS 8465.0 6504.8

US 7619.1 7284.6 5832.1[] QS 3.9107 5.9071

US 4.7442 5.0794 6.6620

e for a more flexible aircraft. In fact, all three curves move toe left, and closer to the correction curve, as the flexibility in-eases. Note that the potential divergence velocity for UA isgnificantly lower than that for both the RA and CW models.e expect this trend to persist also in the case of divergence.ence, if divergence exists, VD for the UA model is likely tosignificantly smaller than VD for both the RA and CW mod-

s. From Figs. 4 and 5 and the potential divergence curves for= 3048 m and h = 4572 m altitudes (not shown), it can bencluded that the onset of divergence becomes less likely ase altitude increases.

The potential flutter curves using both QS and US aerody-

mics and the correction curve for h = 0, 1524, 3048 and 4572tiflhnology 12 (2008) 376384 383

altitudes are shown in Figs. 69. The exact values of the flut-r speed and frequency, and the corresponding trim variablesr each model and aerodynamics used are listed in Table 1.he gain in the accuracy of the results due to inclusion of moregid body and elastic dof, and unsteady aerodynamics is clearlyen in the figures.All six potential flutter curves (three for QS and three for

S) intersect the correction curve for the first three altitudes,nce predicting flutter, with the flutter speed for UA being thergest of the three for the each of the cases of QS and US aero-namics. For h = 4572 m, the potential flutter curve of the UAodel with QS aerodynamics does not intersect the correctionrve, so that this model fails to indicate the existence of flutterthis altitude. The flutter frequencies, on the other hand, are

ose to one another for the all three models, but they are higherr the US aerodynamics than QS aerodynamics.From Fig. 6 and Table 1, it is easy to see that the flutter

eeds obtained from the CW model with the US and the QSrodynamics are close to one another, but this does not meane nature of the flutter in both cases are the same. In fact, flutterequencies (Table 1) and the modes (Table 2) are significantlyfferent in these two cases. Hence, it can be concluded for thisodel that the unsteadiness in the aerodynamics has more effectthe flutter frequency and the mode than the flutter speed.Among the all models, the UA model with US aerodynam-

s is the most accurate one because it includes more rigid bodyd elastic degrees of freedom and more accurate aerodynamicsan the rest. On the other hand, the RA and CW models withS aerodynamics yields smaller flutter speeds, and hence, theye more conservative. The CW model with US aerodynamicsust be used if the interest is to obtain the most conserva-e flutter speed. The models with QS aerodynamics appear toeld larger flutter speeds than the corresponding models withS aerodynamics. Perhaps, one might conclude that the actualutter speed appears to be in between the speeds from the CWodel with US and the UA model with QS aerodynamics.If the interest is to design a feedback control, the most accu-

te model, i.e. UA with US aerodynamics, must be used. Thisay, the controller is less likely to spillover into the unmodelednamics.

Conclusions

Recent developments in the dynamics and control of flexiblercraft permit a broader approach to the concept of aircraft sta-lity. The state equations for the flight of flexible aircraft arenlinear and generally of high order. A perturbation approachrmits the separation of the problem into two problems, a non-

near one for the flight dynamics of a quasi-rigid aircraft andlinear one for the small perturbations from the flight path,here the latter includes the elastic vibration and perturbationsthe rigid-body variables. The first can be used to define a trim

ate and the second to study aircraft stability, whether for flyinggid aircraft alone, for nonflying elastic aircraft, or in generalr flying flexible aircraft. They can also be used to produceme simulations of the rigid-body and elastic variables duringight.

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S 1.0 90 0.0176 48.7S 1.0 90 0.0163 73.8

This paper attempts to reveal the fundamental trends in theability of the flexible aircraft. To this end, it examines the fol-wing cases: 1) flying flexible aircraft, 2) quasi-rigid aircrafts in flight dynamics), 3) cantilevered flexible components (asaeroelasticity) and 4) restrained (nonflying) flexible aircraft.

xcept for the case in which the aircraft executes time-varyinganeuvers, the stability can be reduced to a study of the eigen-lues of the perturbation equations. For flutter prediction, twopes of aerodynamics are used, namely, quasi-steady and un-eady aerodynamics. The paper also presents a method for theediction of divergence and flutter, which makes it easier toach a conclusion about the existence or nonexistence of di-rgence and flutter. The results presented can be improved bycluding more elastic degrees of freedom and refining the aero-namics.

cknowledgements

The author would like to thank Prof. Leonard MeirovitchVirginia Polytechnic Institute and State University for his

tensive review of the manuscript and his useful suggestions.

eferences

1] R.L. Bisplinghoff, H. Ashley, Principles of Aeroelasticity, John Wiley &Sons, Inc., New York, 1962. Reprinted by Dover Publications, Inc., NewYork, 1975.hnology 12 (2008) 376384

w [rad]1 2

0.2065 97.1 0.0329 83.40.4076 102.8 0.0629 77.7

2] A.R. Dusto, et al., A method for predicting the stability characteristics ofan elastic airplane, vol. 1 FLEXSTAB Theoretical Description, NASACR-114712, October 1974.

3] B. Etkin, Dynamics of Flight, third ed., John Wiley & Sons, New York,1996.

4] R.A. Frazer, W.J. Duncan, Wing Flutter as Influenced by the Mobility ofthe Fuselage, British Aeronautical Research Council, 1929, R&M 1207.

5] D.R. Gaukroger, Wing flutter, AGARD Manual on Aeroelasticity, Part V,Chapter 2, 1960.

6] D.H. Hodges, G.A. Pierce, Introduction to Structural Dynamics andAeroelasticity, Cambridge University Press, New York, 2002.

7] L. Meirovitch, I. Tuzcu, Integrated approach to the dynamics and controlof maneuvering flexible aircraft, NASA CR-2003-211748, June 2003.

8] L. Meirovitch, T.J. Seitz, Structural modeling for optimization of low-aspect-ratio wings, Journal of Aircraft 32 (5) (1995) 11141123.

9] L. Meirovitch, I. Tuzcu, Unified theory for the dynamics and control ofmaneuvering flexible aircraft, AIAA Journal 42 (4) (2004) 714727.

0] R.D. Milne, Dynamics of the deformable aeroplane, Parts I and II, HerMajestys Stationery Office, Reports and Memoranda 3345, London, Sep-tember 1962.

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2] M.J. Patil, D.H. Hodges, C.E.S. Cesnik, Nonlinear aeroelasticity andflight dynamics of high-altitude long-endurance aircraft, Journal of Air-craft 38 (1) (2001) 8894.

3] N. Siepenktter, W. Alles, Stability analysis of the nonlinear dynamics offlexible aircraft, Aerospace Science and Technology 9 (2) (2005) 135141.

4] T.A. Weisshaar, T.A. Zeiler, Dynamic stability of flexible forward sweptwing aircraft, Journal of Aircraft 20 (12) (1983) 10141020.

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