APPLICABILITY OF GEOMETRICALLY-LINEARSTRUCTURAL-DYNAMIC MODELS FOR THE FLIGHTDYNAMICS OF ARBITRARILY-FLEXIBLE AIRCRAFT

Antônio B. Guimarães Neto∗ , Flávio L. Cardoso-Ribeiro∗ , Flávio J. Silvestre∗∗Instituto Tecnológico de Aeronáutica, São José dos Campos, SP, 12228-900, Brazil

Keywords: aeroelasticity, flight dynamics, flexible aircraft, small deformations, large deformations

Abstract

The assumption of small deformations in the for-mulation of the flight dynamics of flexible air-craft can be very convenient as it allows theuse of a reduced number of modes of vibrationto represent the structural dynamics with littleloss of accuracy. However, depending on thelevel of structural flexibility, deformations maybecome large enough to violate this simplifyingassumption, making geometrically-nonlinear for-mulations necessary. Delimiting the range of va-lidity of small deformations is then indispensablefor the flight-mechanics engineer in state-of-the-art aircraft design. In this paper, small- and large-deformation formulations are compared in equi-librium conditions and in time-marching simula-tions for aircraft with different levels of structuralflexibility. The importance of geometrical non-linearities and the range of validity of small de-formations are assessed.

1 Introduction

A methodology to assess the validity of the as-sumption of small deformations in geometrically-linear structural-dynamic models was proposedand applied to an idealized aircraft model byGuimarães Neto et al. [1]. The methodologyis simple and self-contained because it dependsonly on the geometrically-linear model itself,not requiring the availability of geometrically-nonlinear models to compare with.

The methodology is based on the fact that

the linear rigid-body modes of the structural-dynamic model are representative if, and only if,small deformations occur. Hence, consideringthe formulation proposed by Guimarães Neto etal. [1], if two different support locations are con-sidered, one at a time, one near the center of massand the other in the region of maximum deforma-tions with respect to a mean-axis system (in air-craft structures, typically the wing tips), the con-version of the elastic displacement vector fromone condition to the other using linear rigid-bodymodes is possible with minimum error if, andonly if, small deformations occur.

In Ref. [1], this methodology was appliedwithout comparison of the geometrically-linearwith a -nonlinear formulation. The first partof the additional work of comparison betweendissimilar formulations was done by GuimarãesNeto et al. [2], including equilibrium conditionsand small disturbances around such conditions.

Formulations based on the simplifying as-sumption of small deformations have long beenavailable and used in flight dynamics of flexibleaircraft. Quasi-static analysis techniques as pro-posed by Rodden and Love [3] and implementedin MSC Nastran [4] are among the main usesof geometrically-linear aeroelastic finite-elementmethod (FEM) models.

The simplifying assumption can also be con-sidered in dynamically-coupled formulations, inwhich n elastic degrees of freedom (DOFs) areincluded to model the structural dynamics, andthe number of flight-dynamic equations of mo-

1

ANTÔNIO B. GUIMARÃES NETO , FLÁVIO L. CARDOSO-RIBEIRO , FLÁVIO J. SILVESTRE

tion (EOMs) increases from the classical six-degree-of-freedom (6-DOF), rigid-body systemto a 6+n-DOF system. In this case, one of thegreatest advantages of the assumption of smalldeformations is the possibility of using modalsuperposition with a small quantity of normalmodes retained in the model [1, 5, 6, 7], lead-ing to significantly lower computational cost ofnumerical simulations than when compared withgeometrically-nonlinear models.

Slightly-flexible aircraft flight-dynamic mod-els based on small deformations have also beensubjected to experimental validation. Silvestreand Luckner [8] demonstrated the applicabilityof a dynamically-coupled, linearized mean-axisformulation [7] for flight control law and aircraftdesign, based on favorable comparisons betweensimulation and flight-test data obtained with aprototype of the utility aircraft Stemme S15.

A relevant theoretical topic in the flight dy-namics of flexible aircraft regards the set of con-straints that define the body axes. The intro-duction of n additional structural-dynamic DOFswhile keeping the classical six DOFs of the rigidbody implies that constraints must be included toeliminate the rigid-body DOFs of the structural-dynamic model. Such constraints are usually ex-pressed by the choice of body axes. Milne [9]delimited three particular choices of body axes:attached, mean and principal axes. In Ref. [5],Milne’s concept of attached axes was extendedto a more general form named dually-constrainedaxes, in which the origin of the structural axes(the support point, where elastic deformations areassumed null) does not necessarily coincide withthe origin of the flight-dynamic body axes (withrespect to which the EOMs are written). In Ref.[1], the generalization was extended with the sep-aration between the body reference frame (BRF)used for the EOMs and an aerodynamic referenceframe (ARF) used for calculating aerodynamicloads, which made the methodology described inthe initial paragraphs possible.

All the aforementioned dynamically-coupledformulations are adequate whenever large defor-mations do not occur. This is not the case for veryflexible aircraft. An accident with NASA’s Helios

Prototype HP03-2 on June 26, 2003 served as anunfortunate demonstration of the importance ofadequately modeling the nonlinear behavior ofvery flexible aircraft [10]. Actually, theoreticalwork in the field was already being carried outat the time. Patil, Hodges and Cesnik [11] de-veloped a nonlinear formulation for the aeroelas-tic analysis of aircraft in subsonic flow, with ageometrically-exact, nonlinear intrinsic formula-tion for the beam dynamics in a moving frame[12]. The same authors [13] extended the formu-lation for the flight dynamics of HALE aircraft.

Cesnik and Brown [14] developed a strain-based formulation for aeroelastically-tailoredflexible wings. In Ref. [15], the authors re-fined their formulation to consider six rigid-bodyDOFs and fully-coupled three-dimensional bend-ing, twisting, and extensional nonlinear defor-mation in the beam model. Whereas in Refs.[14, 15] the fuselage and the tail were modeledas rigid, in Ref. [16] Cesnik and Su dealt with afully-flexible aircraft with slender bodies.

Shearer and Cesnik [17] coupled the 6-DOFEOMs of a reference point on a very flexibleaircraft with the low-order constant strain-basednonlinear structural model of Ref. [15]. The for-mulation was implemented in the University ofMichigan’s nonlinear aeroelastic simulation tool-box (NAST or UM/NAST).

The development of the remotely-piloted X-HALE [18] aircraft at the University of Michi-gan deserves mention. As a low-cost platformfor nonlinear aeroelastic flight tests and nonlin-ear control law studies, it aims at capturing inter-actions not easily obtained or not even possiblein wind tunnel tests. The aircraft in its slightly-flexible configuration (aspect ratio of twenty,AR = 20) is currently in operation also at ITA inBrazil, and a more flexible configuration (AR =30) will enter into operation in the near future.

In the present paper, the methodology to as-sess the assumption of small deformations pro-posed in Ref. [1] is applied to the X-HALE inits four-, six-, and eight-meter-span configura-tions (AR = 20, AR = 30, and AR = 40, respec-tively). Results obtained with a linear structural-dynamic, nonlinear flight-dynamic model us-

2

APPLICABILITY OF GEOMETRICALLY-LINEAR STRUCTURAL-DYNAMIC MODELS FOR THEFLIGHT DYNAMICS OF ARBITRARILY-FLEXIBLE AIRCRAFT

ing the described methodology are comparedwith the results obtained with a fully nonlinearmodel, comprising a geometrically-exact beamformulation [14, 17, 19, 20], implemented inthe ITA/AeroFlex program [21, 22]. The aero-dynamic model for the undeformed aircraft isthe same between both models, based on Hed-man’s vortex-lattice method (VLM) [23], butonly in the geometrically-nonlinear case the aero-dynamic mesh is updated to match the instanta-neous bending deformation.

Previously obtained results [2] showed thata geometrically-linear formulation can be usedwith little loss of accuracy for the aircraft withAR = 20, and that a geometrically-nonlinear for-mulation must be used for AR= 40. However, thetransition between small and large deformations,in the configuration with AR = 30, was not suffi-ciently explored with only equilibrium conditionsand small-disturbance analyses. Rather, time-marching simulations become indispensable forthe assessment of the validity of the assumptionof small deformations, as transient conditionsmay attain geometrical nonlinearities in spite ofan initial condition of small deformations.

This paper then aims at extending these pre-vious works [1, 2] with the inclusion of newly-obtained results from time-marching simulations.Qualitatively, the conclusions are analogous tothe ones previously drawn. However, quantitativecriteria for evaluating the assumption of small de-formations are derived.

2 Theoretical Background

In this paper, a geometrically-linear (GL) formu-lation for the flight dynamics of flexible aircraftis compared with a geometrically-nonlinear (GN)one. Both are derived from first principles butindependently from each other. The subsectionsthat follow describe the most relevant details ofthe formulations.

2.1 Geometrically-Linear Formulation

The formulation for small deformations is basedon Ref. [1]. There, the EOMs for the flexible

aircraft were derived using Lagrange’s equations,with the elected set of 6+ n generalized coordi-nates comprising: the components of the posi-tion vector RO,b of the origin O of a body refer-ence frame (BRF) expressed in the body axes, b;the Euler angles providing the orientation of theBRF with respect to the flat-Earth inertial refer-ence frame (IRF): ψ, θ, and φ; and n elastic DOFsof the aircraft structure, constituting the displace-ment vector uG =

{u1 u2 · · · un

}T .A structural-dynamic FEM model of the air-

craft with lumped properties of inertia is consid-ered available. The transformation matrix Cb0from the inertial frame to the body frame is ob-tained by a classical sequence (3-2-1) of Euler ro-tations, ψ, θ and φ [24]. Based on such consider-ations, the kinetic, elastic strain and gravitationalpotential energies can be calculated as shown inRef. [2]. Structural dissipation due to dampingforces of viscous nature is assumed [25], givingrise to Rayleigh’s dissipation function.

The energy expressions and the dissipationfunction are used in Lagrange’s equations forgeneralized coordinates to obtain the left-handside of the EOMs. The right-hand side consists ofthe generalized forces, comprising aerodynamicand propulsive loads. The former are calculatedconsidering the use of the VLM [23], whereasthe latter are modeled as concentrated forces act-ing on the thrust center of each of the engines[2]. The EOMs for the flexible aircraft can be de-rived with the assumption that no change in air-craft mass occurs with time and are given by [2]:

mVb +mωωωbVb−msCG,bωωωb−mωωωbsCG,bωωωb

+m˜ωωωbDCG,buG +2mωωωbDCG,buG (1)

+mωωωbωωωbDCG,buG +mDCG,buG = mgb +Fb +∆Fb,

JOωωωb + ωωωbJOωωωb +msCG,b(Vb + ωωωbVb

)+mDCG,buG

(Vb + ωωωbVb

)+∆J′Oωωωb + ωωωb∆J′Oωωωb +∆J′Oωωωb (2)

+MωωωGuG +MωωωGuG + ωωωbMωωωGuG

= msCG,bgb +mDCG,buGgb +MO,b +∆MO,b,

3

ANTÔNIO B. GUIMARÃES NETO , FLÁVIO L. CARDOSO-RIBEIRO , FLÁVIO J. SILVESTRE

MGGuG +BGGuG +KGGuG

+mDCG,bT (Vb + ωωωbVb

)+MωωωG

Tωωωb

+2MTωωωGωωωb−

12

n

∑g=1

en,gωωωbT ∂∆JO

∂ugωωωb (3)

= mDCG,bT gb +QG.

In Eqs. (1)-(3), ωωωb ={

p q r}T is the angu-

lar velocity vector of the BRF with respect tothe IRF; Vb =

{u v w

}T is the velocity vec-tor of the BRF origin O with respect to the IRF;the skew-symmetric operator, (•) or skew(•), de-notes the matrix-form of the vector cross prod-uct; m is the aircraft total mass; sCG,b refers tothe CG position vector in the undeformed (un-strained) condition; dCG,b = DCG,buG stands forthe change in sCG,b due to structural deforma-tion; JO is the inertia matrix about O; ∆J′O isthe change in the inertia matrix due to structuraldeformation; MGG, BGG, and KGG are the FEMmass, damping and stiffness matrices, respec-tively; Fb and MO,b are the net force and momentvectors, respectively, associated with the rigidairframe; ∆Fb and ∆MO,b are the net incremen-tal force and moment vectors, respectively, due toelastic motion; gb is the gravity vector expressedin the BRF; QG is the column matrix of general-ized aerodynamic and propulsive forces; at last,MωωωG is the inertia coupling matrix between therotational rigid-body and the elastic DOFs. Thetotal number of elastic DOFs is n. The notationeN,i represents a column matrix equal to the ithcolumn of the identity matrix of order N, IN . Alltime derivatives are taken in the BRF.

2.1.1 Dually-Constrained Axes

The FEM stiffness matrix, KGG, refers to an un-restrained 3D structure and hence it is a positivesemi-definite matrix, with a null space spannedby six linearly-independent vectors [26]. In otherwords, six linearly-independent rigid-body mo-tions are allowed by this FEM formulation. Thisrigid-body freedom of the FEM model is not de-sired, since the coordinates of the origin O andthe Euler angles were already considered to bethe rigid-body DOFs of the flexible aircraft. This

means that six constraints are needed to eliminatethe rigid-body motion from the elastic DOFs.

In the present paper, the so-called dually-constrained axes (DCA) [1] are used to constrainthe six redundant DOFs. In the DCA, the ori-gin S of the structural axes (the support point,with no elastic displacement) is a material point(and structural node) that can be non-coincidentwith the origin O of the body axes. Details of theconstraint equations of the DCA can be found inRefs. [1, 2]. Any structural node can have its dis-placements assumed null in the formulation, andthe origin O keeps its position constant with re-spect to the undeformed aircraft (first constraint)and the structural node S is the point where theundeformed and the deformed airframes coincideat any time instant (second constraint).

2.1.2 Aerodynamic Model

The aerodynamic loads acting on the flexible air-craft are the superposition of loads that would beobtained were the airframe rigid with incrementalloads due to structural deformation. In this paper,the generalized aerodynamic loads are calculatedwith the VLM [23], which provides the followinglinear system of equations:

A−1∆Cp = w, (4)

where w ∈ RNP is the vector of non-dimensionalnormalwashes at the NP panel (box) controlpoints; ∆Cp ∈ RNP is the vector of panel pres-sure coefficient differences; and A ∈ RNP×NP isthe AIC (aerodynamic influence coefficient) ma-trix. The VLM AIC matrix depends on the geom-etry and discretization of the aerodynamic liftingsurfaces in the model. Dependence on the Machnumber, M, is neglected in this paper.

The body frame of reference used to calcu-late the aerodynamic loads is defined as an aero-dynamic reference frame (ARF) [1]. Its inertialangular rates are written in the ARF coordinatesystem as pa, qa, and ra, and its inertial velocityhas the components ua, va, and wa in the samesystem [1]. The rigid-body motion of the aircraftthen contributes to the generalized aerodynamicforces (GAFs) in the elastic DOFs in terms of pa,

4

APPLICABILITY OF GEOMETRICALLY-LINEAR STRUCTURAL-DYNAMIC MODELS FOR THEFLIGHT DYNAMICS OF ARBITRARILY-FLEXIBLE AIRCRAFT

qa, ra, ua, va, wa, control surface deflections andother possible rigid-body variables. The elasticdeformation of the structure with respect to theARF, given by uG/A, contributes to the incremen-tal GAFs. The total GAFs are then given by [1]:

QG = qGAGT SAP (∆Cp,u +∆Cp,e) , (5)

where q is the dynamic pressure; GAG ∈RNA×n isthe matrix that interpolates elastic displacementsfrom the structural nodes to the centroids of theVLM boxes (aerodynamic grid points); SAP ∈RNA×NP transforms panel pressure coefficient dif-ferences to forces and moments at the aerody-namic grid points, and is usually called an inte-gration matrix; ∆Cp,u is the vector of panel pres-sure coefficient differences related to the rigid-body state and control variables; and ∆Cp,e is thevector of panel incremental pressure coefficientdifferences, due to the elastic DOFs, given by [1]:

∆Cp,e = A(DPA,0GAGuG/A

+(bw/Va)DPA,1GAGuG/A), (6)

where DPA,0,DPA,1 ∈ RNP×NA are the differenti-ation matrices that allow the calculation of con-trol point normalwashes at three quarters of theboxes’ mean chords from the displacements atthe aerodynamic grid points, respectively; bw isthe reference wing semi-chord; and NA = 2NPis the total number of aerodynamic degrees offreedom (each panel has two DOFs, plunge andpitch). The integration and differentiation matri-ces in Eqs. (5)-(6) can be found in Ref. [2].

In this paper, the ARF is modeled with at-tached axes [1]. Hence, their origin A coincideswith or is rigidly connected to a material point Cthat remains fixed when elastic deformation oc-curs. References [1, 2] present the equations forthe aerodynamic loads based on the ARF DOFs.Induced drag effects due to both the rigid-bodymotion and the elastic deformation are included,calculated with the methodology of Ref. [27].

2.1.3 Geometrically-linear beam elements

The adopted geometrically-linear beam elementin three dimensions has two nodes and twelve

DOFs. The element shape functions are given by:

ue (xe,ye,ze) = a0 +a1xe−∂we

∂xeze−

∂ve

∂xeye, (7)

ve (xe,ye,ze) =3

∑i=0

bixei−Θe (xe,ye,ze)ze, (8)

we (xe,ye,ze) =3

∑i=0

cixei +Θe (xe,ye,ze)ye, (9)

where ue (xe,ye,ze) is the axial displacement andve (xe,ye,ze) and we (xe,ye,ze) are the edgewiseand flatwise displacements’ shape functions, re-spectively. The set becomes complete with thetwist angle shape function:

Θe (xe,ye,ze) = d0 +d1xe. (10)

The twelve DOFs of the beam element are,at xe = ye = ze = 0: ue = ue1, ve = ve1, we =we1, Θe = φe1, ∂we

∂xe= −θe1, ∂ve

∂xe= ψe1; at xe =

Le,ye = ze = 0: ue = ue2, ve = ve2, we = we2,Θe = φe2, ∂we

∂xe= −θe2, and ∂ve

∂xe= ψe2, where Le

is the element length. Strains are given by:

εx =∂ue

xe, (11)

εy =−νεx, (12)εz =−νεx, (13)

γxy =∂ve

xe+

∂ue

ye, (14)

γxz =∂we

xe+

∂ue

ze, (15)

γyz =∂we

ye+

∂ve

ze, (16)

where ν is the material Poisson’s ratio. Us-ing Voigt’s notation, the element strains may becollected in a six-dimensional column matrix,εεε =

{εx εy εz γxy γxz γyz

}T , which is itselfa linear function of the twelve element DOFs.Stresses can be calculated with consideration ofthe isotropic linear elastic material stiffness ma-trix, C, presented in Ref. [2], so that:

σσσ = Cεεε. (17)

5

ANTÔNIO B. GUIMARÃES NETO , FLÁVIO L. CARDOSO-RIBEIRO , FLÁVIO J. SILVESTRE

The element stiffness matrix Kee is such thatthe element strain energy satisfies:

Ue =12

∫∫∫element

σσσT

εεεdV =12

ueT Keeue, (18)

with ue ={

ue1 ve1 we1 φe1 θe1 ψe1

ue2 ve2 we2 φe2 θe2 ψe2}T . The final

equation for the stiffness matrix is presented inRef. [2] and is omitted here for brevity.

Although a consistent mass matrix can alsobe obtained for the beam element, the flight-dynamic formulation is not prepared to deal withconsistent masses, due to resulting inertial cou-pling between the two element nodes. Rather, alumped mass matrix is generated by transferringto each node half of the mass of the element, aswell as the first and second moments of inertiaand the products of inertia due to each half.

2.2 Strain-based geometrically-nonlinearbeam formulation

In this paper, the strain-based geometrically-nonlinear (GN) formulation is based on Refs.[19, 20, 28, 29]. A toolbox named ITA/AeroFlexwas developed with this formulation, usingMATLAB R© [22]. The formulation considersthe following fundamental kinematic relationshipthat relates the displacements h(s, t) at a pointalong the beam to the strains ε(s, t):

∂h∂s

(s, t) = K (s, t)h(s, t) , (19)

K (s, t) = (20)

=

0 1+ εx(s, t) 0 00 0 κz(s, t) −κy(s, t)0 −κz(s, t) 0 κx(s, t)0 κy(s, t) −κx(s, t) 0

,

where εx(s, t) is the extensional strain, andκx(s, t), κy(s, t) and κz(s, t) are the curvatures atpoint s and time t.

The flexible structure is split into elementsand the strains are assumed to be spatially-constant but time-dependent along each element,

so that Eq. 19 has an analytical solution:

h(s, t) = eK (s−s0)h0(t) , (21)

where h0(t) is the displacement of a fixed nodeat s = s0. The matrix exponential eK (s−s0) has aclosed-form expression as presented in Ref. [20].

In the ITA/AeroFlex computer program, flex-ible elements with three nodes were imple-mented, as well as rigid elements with time-independent, null strain. Such rigid elements areused to model rigid components and do not intro-duce new states to the model.

Using Eq. 21, it is possible to compute thedisplacement vector for each structural node as afunction of the strains:

h(t) = h(ε(t)) . (22)

The time derivative of the displacement vec-tor due to both the strain rates ε and the rigid-body motion β (linear and angular velocity com-ponents) is given by:

h(t) = Jhεε(t)+ Jhbβ(t) . (23)

where Jhε(ε(t)) is a Jacobian that relates the ele-ment strains to nodal displacements and Jhb is theequivalent but for the rigid-body motion.

The kinetic energy is computed as:

T =12

hT Mh , (24)

where M is the structure mass matrix, computedassuming a linear variation of the nodal speedsbetween the nodes. The kinetic energy can berewritten as a function of the strain rates andrigid-body velocities using Eq. 23:

T =12[ε β

][MFF MFBMBF MBB

][ε

β

], (25)

where:

MFF = JThεMJhε , MFB = JT

hεMJhb ,

MBF = JThbMJhε , MBB = JT

hbMJhb .(26)

The elastic strain energy is given by:

U =12

εT Kε , (27)

6

APPLICABILITY OF GEOMETRICALLY-LINEAR STRUCTURAL-DYNAMIC MODELS FOR THEFLIGHT DYNAMICS OF ARBITRARILY-FLEXIBLE AIRCRAFT

where K is a block-diagonal matrix, composed ofthe stiffness matrices of each element, Ke:

Ke =

k11 k12 k13 k14k21 k22 k23 k24k31 k32 k33 k34k41 k42 k43 k44

. (28)

From the Euler-Lagrange equations, theequations of motion are computed as:[

MFF MFBMBF MBB

][ε

β

]+

[CFF CFBCBF CBB

][ε

β

](29)

+

[K0

]ε =

[RFRB

]The matrices CFF , CBF , CFB and CBB in-

clude the gyroscopic terms (due to rotation of therigid body and of the structural elements) and thestructural damping term. In this paper, the struc-tural damping matrix is assumed proportional tothe stiffness matrix: CFF = αK.

RF and RB are the generalized forces thatare applied to the airplane. They are obtainedfrom the aerodynamic, gravitational and propul-sive forces applied to each node of the structure:[

RFRB

]=

[JT

pε

JTpb

]F pt +

[JT

θε

JTθb

]Mpt (30)

+

[JT

pε

JTpb

]BFFdist +

[JT

θε

JTθb

]BFMdist +

[JT

pε

JTpb

]N~g

The Jacobian matrices Jpε and Jθε represent therelationship between structural strains (ε) andnodal displacements and rotations. Jpb and Jθbrepresent the relationship between rigid-bodyDOFs and nodal displacements and rotations.The Jacobian matrices are nonlinear functions ofthe strain vector ε. Closed-form expressions forthe Jacobians are presented in [20].

3 Numerical Models

The X-HALE aircraft in its four- (‘XH4’), six-(‘XH6’) and eight-meter-span (‘XH8’) configu-rations is analyzed. The XH4 configuration con-tains four wing sections with span of 1.0 m andchord of 0.2 m each, as well as three pods at the

connections between wing sections. The aircraftengines, landing gears, electronics and sensorsare installed at the pods. Booms are connectedto the pods and, at the tip of each boom, a hor-izontal tail is mounted. The two side tails areall-moving control surfaces that can be used forboth longitudinal and lateral-directional control,and are then termed elevons. The central tail hasa flipping-up capability that alters the aircraft fly-ing qualities as desired in operation. For groundclearance during take-off, the central tail has ap-proximately 33% less span in its right (bottom)part than in the left (top) part. Anyway, all con-figurations analyzed in this paper have the centraltail in the horizontal position. The wing-tip sec-tions have a dihedral angle of 10◦. The wing isbuilt with an incidence of 5◦.

The XH6 configuration has two more wingsections, pods, booms and tails than the XH4.The XH8 configuration has only two wing sec-tions added with respect to the XH6 configura-tion, without additional pods, booms or tails.

The numerical models consider exactly thesame stiffness properties previously adopted inRef. [2], and these properties are employed inboth the GL and the GN formulations. All air-craft components except the wing are assumedrigid. The distributed mass properties of the air-craft components are also identical to those pre-sented in Table 1 of Ref. [2]. The concentratedinertias match those listed in Table 7 of Ref. [18].

3.1 Verification of the models

The focus of this paper is on time-marching sim-ulations using both the GL and the GN formula-tions. Because all structural DOFs are kept inthe EOMs for both formulations, the resultingdynamical systems have hundreds of states andthe differential equations are stiff. Having this inmind, the criterion used to create the structural-dynamic models in this paper is different fromthe one used previously in Ref. [2], where the au-thors were concerned almost only with accuracyof the models in static conditions, with modelsize having less computational impact becausedifferential equations were not to be solved. Con-

7

ANTÔNIO B. GUIMARÃES NETO , FLÁVIO L. CARDOSO-RIBEIRO , FLÁVIO J. SILVESTRE

vergence analyses led to 10 elements per wingsection in the GN formulation and of 20 in theGL one in Ref. [2]. With such choices, the winghad the same quantity of nodes in both models.

However, numerical tests demonstrated thatsimulations using such refinement would be un-feasible using the current routines implementedin the ITA/AeroFlex program. Because the differ-ential equations are stiff, common solvers do notwork for the GL or the GN simulations. Rather,it was found that the generalized-α algorithm[30, 31, 20] was able to solve the equations. Cur-rently, the nonlinear balance equation of this im-plicit algorithm is solved at each time step withthe trust-region dogleg algorithm implemented inthe MATLAB R© ‘fsolve’ function. A reduction inthe number of structural DOFs was then manda-tory to make the simulations less costly and thepresent work feasible.

It was found that using 4 elements per wingsection in the GN model and 8 elements in theGL yields no more than 3.5% difference betweenthem in modal frequencies less than 20 Hz for theXH6 configuration. This refinement can be con-sidered enough for the purpose of the present pa-per, which is to identify when nonlinearities startto play an important physical role in the flight dy-namics of flexible aircraft, in phenomena that aremuch likely to occur at lower frequencies. More-over, a time step of 10 milliseconds is used inthe generalized-α method, rendering differencesin higher-frequency modes progressively less im-portant the farther they are from 20 Hz.

The VLM mesh is the same for both formula-tions and, for the same reasons as the aforemen-tioned ones, is coarser than the one adopted inRef. [2]. For simplicity, it was built to match thespanwise divisions of the wing sections in the GNstructural-dynamic model, that is, it comprises 4uniformly distributed boxes spanwise per wingsection. Chordwise, 4 boxes are used in the wing,also uniformly distributed. The tails are dividedinto 2 boxes chordwise and 2 boxes spanwise. Novertical surface representing each pod is includedin the aerodynamic model, also for simplicity.

The wing incidence of 5◦ and the wing re-flexed EMX07 airfoil [18] camber are approxi-

mately represented by invariant normalwash vec-tors, given by the local effective camber line in-clination at 75% of each box chord.

In the GN formulation, the VLM mesh is up-dated with structural deformation, such that thewing and tails’ boxes’ side edges are displacedboth laterally and vertically by exactly the sameamount as the structural node with which they co-incide spanwise. Hence, the mesh update makesthe aerodynamic forces consistently behave asfollower forces in the GN formulation.

The displacements and displacement rates ofthe beam elements’ central nodes in the GN for-mulation are directly used to calculate the nor-malwashes. No camber deformation is consid-ered, and hence the displacement transferal fromthe structural nodes to the boxes’ control points isstraightforward, assuming rigid arms. The aero-dynamic loading is considered as distributed, andappropriate matrices transfer the distributed loadsto nodal loads at each element, as in Eq. (30).

In the GL formulation, to preserve linearity,the VLM mesh is never updated. Linear splineinterpolation matrices, as derived in Ref. [4],are calculated and used. Guimarães Neto et al.[2] did already validate the aerodynamic modelimplementations. The GL structural-dynamicmodel and the VLM mesh are plotted in Figs. 1-2 for the XH6 configuration, with the XH4 andXH8 models being analogous.

4 Numerical Results

Three different support locations are used in theGL formulation. One is at a central locationat the wing structural node in the wing sym-metry plane, and is denoted ‘C’. The two oth-ers are at the structural nodes at the wing tips,named ‘L’ for the left one and ‘R’ for the rightone. Therefore, the plots presented hereinafterwill contain three data sets for the GL formula-tion: GLC, GLL, and GLR. These shall be com-pared with the single data set obtained with thegeometrically-nonlinear implementation: GN.

Stiffness-proportional structural damping isconsidered, with a constant of proportionalitysuch that the first free-free mode of vibration has

8

APPLICABILITY OF GEOMETRICALLY-LINEAR STRUCTURAL-DYNAMIC MODELS FOR THEFLIGHT DYNAMICS OF ARBITRARILY-FLEXIBLE AIRCRAFT

Fig. 1 XH6 GL structural-dynamic modelin ITA/AeroFlex. GRIDs: structural nodes;CBARNs: beam elements; RBARs: rigid bar el-ements; CONM2 CGs: CG locations of lumped-mass elements; CONM2 Offsets: offsets betweensuch CG locations and the structural node towhich the lumped-mass element is attached; andSUPORT: support location.

Fig. 2 XH6 configuration VLM model inITA/AeroFlex. Control surfaces in orange.

2% damping ratio.The obtained results for equilibrium condi-

tions with the coarser structural-dynamic andaerodynamic models are similar to the resultspreviously obtained with the more refined onesin Ref. [2]. However, it was now possible to cal-culate trimmed conditions in a longitudinal ma-neuver with vertical load factor of 2. Table 1 and

Figs. 3-5 show the results in terms of structuraldeformations. The deformations for the GL for-mulation are with respect to a common ARF.

Wing Displacements [m]Tip XH4 XH6 XH8

GLLLeft

0.192 0.578 2.015(−2.5%) (−8.0%) (−21.8%)

Right0.196 0.599 2.275

(−1.5%) (−5.4%) (−11.8%)

GLCLeft 0.197 0.628 2.577

Right 0.199 0.633 2.580

GLRLeft

0.194 0.594 2.273(−1.5%) (−5.4%) (−11.8%)

Right0.194 0.583 2.017

(−2.5%) (−7.9%) (−21.8%)

GNLeft

0.221 1.018 2.345(+12.2%) (+62.1%) (−9.0%)

Right0.223 1.029 2.356

(+12.1%) (+62.6%) (−8.7%)

Table 1 Wing tip vertical displacements (posi-tive upwards) for the X-HALE configurations intrimmed longitudinal flight with a vertical loadfactor of 2 at 20 m/s. (Percentages with respectto GLC left and right wing tip displacements.)

Fig. 3 XH4 configuration in trimmed longitudi-nal flight with a vertical load factor of 2 at 20 m/s.

Two factors shall be considered when analyz-ing Table 1 and Figs. 3-5: first, the differences in

9

ANTÔNIO B. GUIMARÃES NETO , FLÁVIO L. CARDOSO-RIBEIRO , FLÁVIO J. SILVESTRE

Fig. 4 XH6 configuration in trimmed longitudi-nal flight with a vertical load factor of 2 at 20 m/s.

Fig. 5 XH8 configuration in trimmed longitudi-nal flight with a vertical load factor of 2 at 20 m/s.

structural deformations for different support lo-cations in the GL formulation; second, the factthat the aerodynamic mesh is updated only inthe GN formulation. Hence, for the XH4 con-figuration, one sees that the difference betweendifferent support locations is practically negli-gible; however, the GN result shows a slightlygreater deformation, possibly due to the follower-force effect enabled by the mesh update. Forthe XH6 configuration, differences of up to 8.0%in wing tip displacements are already observedin the GL formulation between the central andwing tip support locations, suggesting the pres-

ence of geometrical nonlinearity. This is con-firmed by the GN result, which predicts signif-icantly greater structural deformations. It is in-teresting to observe that the GL formulation pre-dicts wing tip deformation of up to 21.1% of thenominal wing semi-span in this case. The XH8is clearly highly flexible, with significant differ-ences between the support locations in the GLformulation and strong nonlinearities in the de-formed U shape seen in Fig. 5.

Time-marching simulations were performedwith commanded elevon doublets. The elevondoublet can be either symmetrical, lasting 0.6second and applied to all the elevons, or anti-symmetrical, lasting 1.4 second and applied onlyto the outboard elevons. The doublets have 5.0◦

amplitude and are C1-continuous with cubic tran-sitions lasting 0.1, 0.2 and 0.1 second, respec-tively. The commands begin at t = 0.2 s, withthe aircraft initially in a trimmed level-flight con-dition. The first pulse is positive for the rightelevon(s), and the second is negative.

Figure 6 shows that the pitch rate response ofthe XH4 configuration to the symmetrical elevondoublet is practically the same for all formula-tions considered, and the wing tip vertical dis-placement does not exceed 12% of the wingsemi-span. Small deformations can be assumed.

The same is not observed for the XH6 config-uration. As seen in Fig. 7, there is disagreementin the pitch rate between the GLL, GLR and GLCformulations, particularly when the wing defor-mation is near or at its maximum. According tothe methodology proposed in Ref. [1], this in-dicates that geometrical nonlinearities are poten-tially present. This is confirmed by the plots ofthe GN formulation, with discrepancies observedin how fast the wing tip recovers its equilibriumdeformation. Figure 7 also shows that the wingtip vertical displacement reaches almost 14.7%of the wing semi-span in the GL formulation.

The anti-symmetrical elevon doublet alsoshows interesting results for the XH6 configura-tion in Fig. 8. First, elevon roll reversal is ob-served, as the elevons are initially deflected ina sense that would result in a roll to the left ina rigid aircraft, with the flexible aircraft rolling

10

APPLICABILITY OF GEOMETRICALLY-LINEAR STRUCTURAL-DYNAMIC MODELS FOR THEFLIGHT DYNAMICS OF ARBITRARILY-FLEXIBLE AIRCRAFT

t [s]

Pitc

h ra

te [d

eg/s

]

0 1 2 3 4 5−60

−40

−20

0

20

40

60

GLCGLLGLRGN

t [s]

R. w

ing

tip v

ert.

disp

lace

men

t [m

]

0 1 2 3 4 5−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

GLCGLLGLRGN

Fig. 6 XH4 response to symmetrical elevon dou-blet at 20 m/s.

to the right. The wing tip pitch rates are ini-tially positive in the left wing tip and negativein the right one, confirming that significant wingtwist produces the reversal. Moreover, geomet-rical nonlinearities occur, but another importantphenomenon is also present: because the lift-ing surfaces have their boxes’ dihedral anglesconstantly updated in the GN formulation, theaircraft lateral-directional characteristics do notonly start different from those of the GL formula-tion but are constantly changing during the sim-ulation. This explains why the GN formulationattains smaller roll rate in response to the anti-symmetrical elevon doublet.

At last, the results for the symmetrical elevondoublet for the XH8 configuration are presentedin Fig. 9. The disagreement in the pitch rate be-

t [s]

Pitc

h ra

te [d

eg/s

]

0 1 2 3 4 5−80

−60

−40

−20

0

20

40

60

80

GLCGLLGLRGN

t [s]

R. w

ing

tip v

ert.

disp

lace

men

t [m

]

0 1 2 3 4 50

0.1

0.2

0.3

0.4

0.5

GLCGLLGLRGN

Fig. 7 XH6 response to symmetrical elevon dou-blet at 20 m/s.

tween the GLL, GLR and GLC formulations ismuch more pronounced during and after the ap-plication of the doublet. There is no agreementat all with the GN formulation, demonstratingthat geometrically-linear structural dynamics isnot applicable to such a highly-flexible aircraft.

5 Conclusions

A self-contained methodology to assess the as-sumption of small deformations in geometrically-linear structural-dynamic models was applied toaircraft with different levels of structural flex-ibility. The methodology allegedly does notdepend on the availability of higher-fidelity,geometrically-nonlinear models, because it isbased on the validity of the linear rigid-body

11

ANTÔNIO B. GUIMARÃES NETO , FLÁVIO L. CARDOSO-RIBEIRO , FLÁVIO J. SILVESTRE

t [s]

Rol

l rat

e [d

eg/s

]

0 1 2 3 4 5−60

−40

−20

0

20

40

GLCGLLGLRGN

t [s]

Left

win

g tip

pitc

h ra

te [d

eg/s

]

0 1 2 3 4 5−60

−40

−20

0

20

40

60

GLCGLLGLRGN

t [s]

Rig

ht w

ing

tip p

itch

rate

[deg

/s]

0 1 2 3 4 5−60

−40

−20

0

20

40

60

80

GLCGLLGLRGN

Fig. 8 XH6 response to anti-symmetrical elevondoublet at 20 m/s.

modes of the structure, which, once violated, alsoimplies the violation of the assumption of smalldeformations.

In order to validate the methodology, how-

t [s]

Pitc

h ra

te [d

eg/s

]

0 0.5 1 1.5 2−80

−60

−40

−20

0

20

40

GLCGLLGLRGN

t [s]

R. w

ing

tip v

ert.

disp

lace

men

t [m

]

0 0.5 1 1.5 21.2

1.4

1.6

1.8

2

2.2

2.4

Fig. 9 XH8 response to symmetrical elevon dou-blet at 14 m/s.

ever, geometrically-linear models were comparedwith a geometrically-nonlinear one in this paper.For any given aircraft, the same number of wingnodes and the same undeformed aerodynamicmodel, based on the quasisteady VLM, were em-ployed for both the geometrically-linear and thegeometrically-nonlinear formulations. In the lat-ter, the VLM mesh was consistently updated withstructural deformation. It remains to be investi-gated whether including the aerodynamic meshupdate in the former would be worth the highercomputing cost implied in recalculating the AICand spline matrices in each time step.

Numerical results indicate thatgeometrically-linear models do progressivelylose fidelity as wing tip displacements exceed12% of the wing semi-span or relative differences

12

APPLICABILITY OF GEOMETRICALLY-LINEAR STRUCTURAL-DYNAMIC MODELS FOR THEFLIGHT DYNAMICS OF ARBITRARILY-FLEXIBLE AIRCRAFT

of more than 5% in any wing tip displacementoccur for different support locations. Time-marching simulations also allow us to infer that,if large deformations are attained at some pointin time, the aircraft response from that instant onmay become unrealistic in geometrically-linearmodels.

The self-contained methodology has provedits value in that discrepancies in the results fordifferent support locations correspond to dis-agreements between the geometrically-linear and-nonlinear formulations. Despite structural nodesat the two wing tips were considered as supportlocations in this paper, the procedure can be di-rectly simplified to consider only one of them.Moreover, aircraft responses in time-marchingsimulations show that the central support loca-tion tends to have the highest correlation with thegeometrically-nonlinear formulation.

Contact Author Email Address

antoniobgn@gmail.com

Acknowledgement

This work has been funded by FINEP and EMBRAERunder the research project Advanced Studies in FlightPhysics, contract number 01.14.0185.00.

Copyright Statement

The authors confirm that they, and/or their companyor organization, hold copyright on all of the origi-nal material included in this paper. The authors alsoconfirm that they have obtained permission, from thecopyright holder of any third party material includedin this paper, to publish it as part of their paper. Theauthors confirm that they give permission, or have ob-tained permission from the copyright holder of thispaper, for the publication and distribution of this pa-per as part of the ICAS proceedings or as individualoff-prints from the proceedings.

References

[1] Guimarães Neto, A. B., Silva, R. G. A.,Paglione, P. and Silvestre, F. J. Formula-

tion of the flight dynamics of flexible air-craft using general body axes. AIAA Journal,Vol. 54, No. 11, pp. 3516-3534, 2016. DOI:10.2514/1.J054752.

[2] Guimarães Neto, A. B., Silvestre, F. J., Cardoso-Ribeiro, F. L., Bussamra, F. L. S., Silva, R. G.A. and Cesnik. C. E. S. Validity of the assump-tion of small deformations in aircraft with dif-ferent levels of structural flexibility. Proceed-ings of the International Forum on Aeroelas-ticity and Structural Dynamics (IFASD), Como,Italy, June 25-28, 2017. IFASD-2017-080.

[3] Rodden, W. P. and Love, J. R. Equations of mo-tion of a quasisteady flight vehicle utilizing re-strained static aeroelastic characteristics. Jour-nal of Aircraft, Vol. 22, No. 9, pp. 802-809,1985.

[4] Rodden, W. P. and Johnson, E. H.MSC.NASTRAN Aeroelastic Analysis User’sGuide. MacNeal-Schwendler Corporation, LosAngeles, 1994, Chap. 2.

[5] Guimarães Neto, A. B. Flight dynamics of flex-ible aircraft using general body axes: a the-oretical and computational study. Thesis (PhDin Aeronautical and Mechanical Engineering) –Instituto Tecnológico de Aeronáutica, São Josédos Campos, Brazil, 2014.

[6] Etkin, B. Dynamics of flight: stability and con-trol. John Wiley & Sons, Inc., 1959.

[7] Waszak, M. R. and Schmidt, D. K. Flight dy-namics of aeroelastic vehicles. Journal of Air-craft, Vol. 25, No. 6, pp. 563-571, 1988.

[8] Silvestre, F. J. and Luckner, R. Experimen-tal validation of a flight simulation modelfor slightly flexible aircraft. AIAA Journal,Vol. 53, No. 12, pp. 3620-3636, 2015. DOI:10.2514/1.J054023.

[9] Milne, R. D. Dynamics of the derformable aero-plane. Her Majesty’s Stationary Office TR RM3345, London, 1964.

[10] Noll, T. E. et al. Investigation of the Heliosprototype aircraft mishap. Washington, DC:NASA, 2004.

[11] Patil, M. J., Hodges, D. H. and Cesnik, C. E. S.Nonlinear aeroelastic analysis of complete air-craft in subsonic flow. Journal of Aircraft, Vol.37, No. 5, pp. 753-760, 2000.

[12] Hodges, D. H. A mixed variational formulation

13

ANTÔNIO B. GUIMARÃES NETO , FLÁVIO L. CARDOSO-RIBEIRO , FLÁVIO J. SILVESTRE

based on exact intrinsic equations for dynam-ics of moving beams. International Journal ofSolids and Structures, Vol. 26, No. 11, pp. 1253-1273, 1990.

[13] Patil, M. J., Hodges, D. H. and Cesnik, C. E.S. Nonlinear aeroelasticity and flight dynamicsof high-altitude long-endurance aircraft. Jour-nal of Aircraft, Vol. 38, No. 1, pp. 88-94, 2001.

[14] Cesnik, C. E. S. and Brown, E. L. Model-ing of high aspect ratio active flexible wingsfor roll control. Proceedings of the 43rdAIAA/ASME/ASCE/AHS/ASC Structures, Struc-tural Dynamics, and materials conference andexhibit, Denver, 2002. AIAA Paper 2002-1719.

[15] Cesnik, C. E. S. and Brown, E. L. Activewing warping control of a joined-wing air-plane configuration. Proceedings of the 44thAIAA/ASME/ASCE/AHS/ASC Structures, Struc-tural Dynamics, and materials conference andexhibit, Norfolk, 2003. AIAA Paper 2003-1715.

[16] Cesnik, C. E. S. and Su, W. Nonlin-ear aeroelastic modeling and analysis offully flexible aircraft. Proceedings of theAIAA/ASME/ASCE/AHS/ASC 46th Structures,Structural Dynamics and Materials Conference,2005, Austin. AIAA Paper 2005-2169.

[17] Shearer, C. M. and Cesnik, C. E. S. Nonlinearflight dynamics of very flexible aircraft. Journalof Aircraft, Vol. 44, No. 5, pp. 1528-1545, 2007.

[18] Cesnik, C. E. S., Senatore, P. J., Su, W., Atkins,E. M. and Shearer, C. M. X-HALE: a very flexi-ble unmanned aerial vehicle for nonlinear aeroe-lastic aests. AIAA Journal, Vol. 50, No. 12, pp.2820-2833, 2012.

[19] Brown, E. L. Integrated strain actuation in air-craft with highly flexible composite wings. The-sis (PhD) – Massachusetts Institute of Technol-ogy, 2003.

[20] Shearer, C. M. Coupled nonlinear flight dy-namics, aeroelasticity and control of very flex-ible aircraft. Thesis (PhD in Aerospace Engi-neering) – University of Michigan, Ann Arbor,2006.

[21] Cardoso-Ribeiro, F. L. Dinâmica de voo deaeronaves muito flexíveis. Dissertation (MScdegree in Aeronautical and Mechanical Engi-neering), Instituto Tecnológico de Aeronáutica,São José dos Campos, Brazil, 2011.

[22] Cardoso-Ribeiro, F. L., Paglione, P., Silva, R.G. A. and Sousa, M. S. AeroFlex: a toolbox forstudying the flight dynamics of highly flexibleairplanes. VII Congresso Nacional de Engen-haria Mecânica (CONEM), São Luís, Maran-hão, Brazil, 2012.

[23] Hedman, S. G. Vortex lattice method for cal-culation of quasi steady state loadings on thinelastic wings. Aeronautical Research Inst. ofSweden Rept. 105, Stockholm, 1965.

[24] Stevens, B. L. and Lewis, F. L. Aircraft controland simulation. Wiley, Hoboken, NJ, 2003, pp.25-29; 116-138; 263-265.

[25] Bismarck-Nasr, M. N. Structural Dynamics inAeronautical Engineering. AIAA Education Se-ries, AIAA, Reston, VA, 1999, pp. 53-91.

[26] Mantegazza, P. Tutorial on Attached-MeanAxes and Their Use in the Calculation of De-formable Static and Damped-Undamped Vibra-tion Modes of a Free-Free Structure. Journal ofAeroelasticity and Structural Dynamics, Vol. 2,No. 1, pp. 81-98, 2011.

[27] Kálmán, T. P., Giesing, J. P. and Rodden, W. P.Spanwise distribution of induced drag in sub-sonic flow by the vortex lattice method. Journalof Aircraft, Vol. 7, No. 6, pp. 574-576, 1970.

[28] Su, W. Coupled nonlinear aeroelasticity andflight dynamics of fully flexible aircraft. Thesis(PhD in Aerospace Engineering) – University ofMichigan, Ann Arbor, 2008.

[29] Su, W. and Cesnik, C. E. S. Strain-basedgeometrically nonlinear beam formulationfor modeling very flexible aircraft. Interna-tional Journal of Solids and Structures, Vol.48, No. 16-17, pp. 2349-2360, 2011. DOI:10.1016/j.ijsolstr.2011.04.012.

[30] Chung, J. and Hulbert, G. M. A time integra-tion algorithm for structural dynamics with im-proved numerical dissipation: the generalized-α method. Journal of Applied Mechanics,Vol. 60, No. 2, pp. 371-375, 1993. DOI:10.1115/1.2900803.

[31] Jansen, K. E., Whiting, C. H. and Hulbert, G. M.A generalized-α method for integrating the fil-tered Navier-Stokes equations with a stabilizedfinite element method. Computer Methods inApplied Mechanics and Engineering, Vol. 190,No. 3-4, pp. 305-319, 2000.

14