Consistent Structural Linearization in Flexible AircraftDynamics with Large Rigid-Body Motion

Henrik Hesse∗ and Rafael Palacios†

Imperial College, London, England SW7 2AZ, United Kingdom

and

Joseba Murua‡

University of Surrey, Guildford, England GU2 7XH, United Kingdom

DOI: 10.2514/1.J052316

This paper investigates the linearization, using perturbation methods, of the structural deformations in the

nonlinear flight dynamic response of aircraft with slender, flexible wings. The starting point is the coupling of a

displacement-based geometrically nonlinear flexible-body dynamics formulation with a three-dimensional unsteady

vortex latticemethod.This is followedby a linearization of the structural degrees of freedom,which are assumed to be

small in a body-fixed reference frame. The translations and rotations of that reference frame and their time

derivatives, whichdescribe the vehicle flight dynamics, can still be arbitrarily large.The resulting systempreserves all

couplings between rigid and elastic motions and can be projected onto a few vibration modes of the unconstrained

aircraftwith geometrically nonlinear static deflections at a trim condition.Equally, the unsteady aerodynamics canbe

approximated on a fixed lattice defined by the deformed static geometry. Numerical studies on a representative high-

altitude long-endurance aircraft are presented to illustrate the approach. The results show an improvement

compared to those obtained using the mean-axes approximation.

Nomenclature

C = coordinate transformation matrix from frame A to BC = global tangent damping matrixI = unit matrixK = global tangent stiffness matrixM = global tangent mass matrixp = generalized displacements in modal basisQ = global vector of generalized forces, NR = beam local position vector, mr = inertial position of the origin of the body-fixed frame, ms = length of the arc along beam reference line, mT = tangential operatort = physical time, sV = inertial translational velocity at a beam cross section,

m∕sv = inertial translational velocity of the body-fixed frame,

m∕sx = state vectorβ = vector of global translational and rotational velocitiesΓ = circulation strength of a vortex ring, m2∕sγ = beam local force strainζ = quaternions for global orientation of the body-fixed

frameη = vector of nodal displacements and rotationsΘ = Euler angles, radκ = beam local moment strain, m−1

ν = global displacements and rotations as time integral of β

ξ = beam cross-sectional coordinates, mΦ = matrix of mode shapesχ = coordinates of the aerodynamic lattice, mΨ = beam local Cartesian rotation vectorΩ = inertial angular velocity at a beam cross section, rad∕sω = inertial angular velocity of the body-fixed frame, rad∕s

Subscripts

A = body-fixed reference frameB = local reference frame on flexible membersF = fluid degrees of freedomG = inertial reference frameR = rigid-body degrees of freedomS = structural degrees of freedom

Superscripts

�_•� = derivatives with respect to time, t� ~•� = skew-symmetric operator��•� = small perturbations around an equilibrium�•� 0 = derivatives with respect to length, s

I. Introduction

The flight dynamics of flexible aircraft have been traditionallystudied using linear methods. The standard approach in the flightsimulation community is themean-axes approximation [1–4], whichdecouples the structural and rigid-body dynamics degrees of free-dom (DOFs) by assuming a superposition of the “free-free” modalvibrations of the structure to the dynamics of a floating frame linkedto the aircraft center of gravity. The aerodynamic loads are obtainedas the generalized forces corresponding to the mode shapes of thestructure, including their rigid-body component [5]. The resultingequations of motion (EOMs), however, neglect some gyroscopiccoupling terms (whichmay have an important effect on both the flightdynamics and the maneuver loads) and are therefore only valid forperturbations about steady flight of relatively stiff vehicles. A moregeneral solution has been proposed using quasi coordinates [6–8],which directly solve the fully coupled EOMs of a flexible body(in a body-attached reference frame) under the assumption of smallstructural deformations.

Presented as Paper 2012-1402 at the 53rd AIAA/ASME/ASCE/AHS/ASCStructures, Structural Dynamics and Materials Conference, Honolulu, HA,23–26 April 2012; received 31 August 2012; revision received 27 July 2013;accepted for publication 6 August 2013; published online 31 January 2014.Copyright © 2013 by Henrik Hesse, Rafael Palacios, and Joseba Murua.Published by the American Institute of Aeronautics and Astronautics, Inc.,with permission. Copies of this paper may be made for personal or internaluse, on condition that the copier pay the $10.00 per-copy fee to the CopyrightClearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; includethe code 1533-385X/14 and $10.00 in correspondence with the CCC.

*Research Associate, Department of Aeronautics. Member AIAA.†Senior Lecturer, Department of Aeronautics; rpalacio@imperial.ac.uk.

Member AIAA (Corresponding Author).‡Lecturer, Department of Mechanical Engineering Sciences. Member

AIAA.

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The coupled aeroelasticity/flight dynamics problem is often linkedto large structural deformations and most of the recent work [9–14]has been concerned with developing fully coupled models to includethe effect of geometrically nonlinear elastic deformations on theflight dynamics of flexible aircraft. Typically, the displacements androtations along a beam-type structure are the primary variables in thenumerical solution, in which the elastic DOFs are defined in terms ofa global body-fixed reference frame attached to the referenceconfiguration to include the rigid-body motion of the vehicle.Alternative solutions using directly the beam stress resultants asprimary variables [13–16] ormixedmethods based on local velocitiesand strains (the so-called intrinsic description) [17–19] have oftenprovided numerical advantages for aircraft-type geometries.Whereas geometrically nonlinear static aeroelastic effects can

have a significant impact on the vehicle response [20], using fullnonlinear models to compute the flight dynamics of very flexibleaircraft comes at a numerical burden, which makes them less suitablefor efficient control design and optimization studies.Model reductionof the system could solve this problem, but its application is limitedhere because of the tightly coupled, nonlinear nature of the flexible-body dynamics subsystem. Recently, Da Ronch et al. [21] havedemonstrated that third-order Taylor expansions with only a fewDOFs can be obtained directly from very-high-dimensional non-linear aeroelastic models of very flexible wings, but the validity ofthose approximations can be hard to guarantee far from the referenceconditions. In a parallel effort, Su and Cesnik [22] applied a modalapproach to the strain-based, geometrically nonlinear beam equa-tions to solve the dynamics of flexible aircraft. However, the reducedsystem matrices still depended on the modal amplitudes and thecomputational advantage of that approach was small.To compute the dynamic response of structures around a

geometrically nonlinear static equilibrium, previous work by the firsttwo authors [23] demonstrated the consistent linearization of thestructural DOFs in nonlinear flexible-body dynamics problems. Thisprovides a formulation, which allows projection of the nonlinearEOMs onto the vibration modes of the unconstrained deformedstructure at forward flight with possibly large trim deformations.Although keeping the nonlinearity in the rigid-body dynamics equa-tions and preserving all couplings between rigid-body and structuraldynamics, this approach results in closed-form expressions for all themodal coefficients, which only need to be computed once.The objective of this paper is to exercise this consistent structural

linearization on the dynamic analysis of flexible aircraft. Here, theemphasis is on small elastic deformations around the aircraft trimconfiguration with possibly large static wing deformations. Theflexible-body dynamics formulation is coupled with a three-dimensional unsteady vortex lattice method (UVLM) [24] in theframework for simulation of high-aspect-ratio planes (SHARP)[12,20]. Numerical examples exercise the proposed formulationon the open-loop response of a representative high-altitude long-endurance (HALE) aircraft subject to commanded control surfaceinputs. Comparison to the mean-axes approximation will give aninsight into the effect of neglecting the coupling between the elasticdeformations and aircraft flight dynamics.

II. Flexible-Body Dynamics

The flexible vehiclewill bemodeled using geometrically nonlinearcomposite beam elements on a moving (body-attached) frame ofreference [25,26]. The elastic DOFs are the displacements and theCartesian rotation vector (CRV) at the element nodes, with respectto a moving body-attached frame. The nonlinear EOMs are thenlinearized with respect to the elastic DOFs using perturbationmethods and projected onto a set of elastic modes of the unrestrainedsystem at some reference condition. This will give a compact form ofthe EOMs with large rigid-body motion. The details can be found in[23], and this section will only present the main results.

A. Geometrically Exact Displacement-Based Formulation

As shown in Fig. 1, the deformation of the structure is described interms of a moving, body-fixed reference coordinate system A, which

moves with respect to an inertial (ground) frame G by the inertialtranslational and angular velocities vA�t� and ωA�t� of its origin.Subscripts are used to indicate the coordinate system in which eachvector magnitude is projected. The orientation of the global frameA with respect to the inertial frame G is parameterized usingquaternions ζ�t�. The local orientation along the beam reference lineis defined by a local coordinate systemB and is parameterized by theCRV, Ψ�s; t�. The corresponding coordinate transformation matrixbetween the global body-fixed reference frame A and the localdeformed frame B will be referred to as C�Ψ�.The nodal positions RA�s; t� and the cross-sectional orientations

Ψ�s; t� thus form the set of independent variables. Note that theelastic displacements are given with respect to the body-fixed frameof referenceA, such that the structure is attached to the origin of theAframe (with the corresponding boundary conditions). The rigid-bodyvelocities vA and ωA are used to describe the overall motion of theunconstrained vehicle. Hence, the deformation of the reference linegoing from the undeformed state at t � 0 to the current state at time tis given by the force and moment strains [27], which are defined,respectively,

γ�s; t� � C�Ψ�s; t��R 0A�s; t� − C�Ψ�s; 0��R 0A�s; 0�κ�s; t� � T�Ψ�s; t��Ψ 0�s; t� − T�Ψ�s; 0��Ψ 0�s; 0� (1)

where �•� 0 is the derivative with respect to the length of the arc s,and T�Ψ� is the tangential operator [26]. Analogously, the localtranslational and angular inertial velocities are obtained, respectively, as

VB�s; t� � C�Ψ�s; t�� _RA�s; t� � C�Ψ�s; t���vA�t� � ~ωA�t�RA�s; t��

ΩB�s; t� � T�Ψ�s; t�� _Ψ�s; t� � C�Ψ�s; t��ωA�t� (2)

where �_•� is thederivativewith respect to timeand � ~•� the cross-product(or skew-symmetric) operator [25]. This kinematicdescription serves toexpress the kinetic and potential energy of the system in terms of thebeam displacement and rotations [16,26]. A two-noded finite elementdiscretization is then introduced on both variables, which gives thediscrete form of the nonlinear EOMs. If η is the column vector with allthe nodal displacements and rotations and β⊤ � f v⊤A ω⊤

A g, thedynamic equations are written as [23]

M�η���η_β

���QSgyrQRgyr

���QSstif0

���QSextQRext

�(3)

where structural and rigid-body components (denoted by superscriptsS and R) have been identified in the gyroscopic, elastic, and externalforces. These differential equations couple the nonlinear beamdynamics and the nonlinear rigid-body dynamics through the fullypopulated mass matrix M�η� and the discrete gyroscopic forcesQgyr�η; _η; β�. Because this formulation includes rotational DOFs, themassmatrix depends on the current geometry η to account for a changein rotational inertia on the structural and rigid-body dynamics [26]. In

Fig. 1 Multibeam configuration with the definition of reference frames.

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addition to the inertial coupling, additional couplings can occur throughfollower forces such as aerodynamic loads or thrust. Equation (3) issolved together with the propagation equations that determine theposition and orientation of the body-fixed reference frame [28]. Theequations are time marched using an implicit, constant-accelerationNewmark integration scheme, which wasmodified as in Eq. (7.3.23) in[29] to introduce controlled positive algorithmic damping.

B. Linearization of Elastic Degrees of Freedom in NonlinearFlexible-Body Dynamics

The particular case of small structural deformations about a staticequilibrium condition is considered next. The possibly large elasticdeformations at static equilibrium will be referred to as η as before,whereas �η�t�will be the small elastic deformations that occur togetherwith the (not-necessarily small) rigid-body velocitiesβ�t�.With thoseassumptions, perturbation of the elastic DOFs in Eq. (3) gives linearinertial and elastic terms, which only depend on the deformed shape,whereas the gyroscopic forces require particular attention becauseof the coupling between rigid-body and elastic states. This resultsin [23]

M�η����η_β

�� �C�η; β�

�_�ηβ

�� �K�η; β�

��η0

�� �Qext��η; _�η; β; ζ� (4)

where �C and �K are the damping and stiffness matrices, whichoriginate from the perturbation ofQgyr andQstif in the elasticDOFs. Itis important to note that themassmatrixM only depends on the staticdeformations η, but the damping and stiffness matrices are alsofunctions of the instantaneous rigid-body velocity β.To compute the vibration modes of the unconstrained structure

around a geometrically nonlinear deformed reference condition, wecan further linearize the dynamic system in Eq. (4) with respect to therigid-body velocities around the equilibrium β � 0. Note, however,that the structural linearization is invariant with respect totranslational velocities, and the results can be used for the dynamicsin forward flight. The homogeneous form of the resulting linearsystem of coupled EOMs will also be the basis for the aeroelasticstability analysis presented in Sec. III.B.

C. Modal Reduction of the Nonlinear System Equations

It is now possible to write the system of perturbation equations,Eq. (4), in terms of global shape functions by projecting the dynamicsequations on the natural modes of the unconstrained structure at areference condition. These vibration modes are obtained in the body-fixed frame A from the unforced fully linearized version of Eq. (4),such that

��η�ν

�� Φ

�p�ν�

���ΦSS ΦSR

ΦRS ΦRR

��p�ν�

�(5)

where we have introduced the new variable �ν with _�ν � �β, such thatthe vector of rigid-body displacements and rotations, �ν�, describesthe motion of the body-fixed frame A and the displacements of thisframe due to the elastic mode shapes (ΦRS ≠ 0). The vector of theprojected modal coordinates is p, and Φ is the matrix of thecorresponding mode shapes, which include the six zero-frequencyrigid-body modes �Φ⊤

SR Φ⊤RR �⊤.

Without loss of generality, the origin of the body frame A can bedefined to initially coincidewith the center of mass (CM) and its axesto be aligned to the principal axes of the structure. This simplifies therigid-body modes because ΦSR and ΦRR are then a null and a unitmatrix, respectively. Note that this is the same initial condition asfor a mean-axes description, but A is still a body-fixed frame.The remaining elastic mode shapes �Φ⊤

SS Φ⊤RS �⊤ characterize the

vibration modes of the unconstrained structure such that ΦSS arethe deformations with respect to frame A and ΦRS ≠ 0 accounts forthe corresponding rigid-body motion of the body-fixed frame A.However, using this modal basis for the projection of the linear-

flexible/nonlinear-rigid EOMs, Eq. (4), requires a linear trans-formation at each subiteration to obtain the instantaneous rigid-body

velocities needed in the perturbation damping and stiffness matrices.A more convenient basis for projection is

Φ� ��ΦSS ΦSR

ΦRS ΦRR

��Im 0

−ΦRS I6

�(6)

wherem is the number of elasticmodes used for the projection,whichrelieves the inertial coupling between the rigid-body motion of thevehicle and the elastic deformations. This approach is very similar tothat followed by Rodden and Love [5] to obtain the rigid-bodyaccelerations on the structure in a mean-axes framework and willallow us to separate elastic from rigid-body modes in the followingprojection of the perturbation flexible-body EOMs. Note that, in thisformulation, themodal basis is computed for the deformed configura-tion at trimmed flight, which is the same reference condition used forlinearization of the elastic DOFs.

1. Modal Reduction of the Perturbation Equations of Motion

The structural and rigid-body velocities/accelerations can first beexpressed in terms of the modal basis Φ� such that

�_�ηβ

�� �

�_pβ

�and

���η_β

�� �

��p_β

�(7)

which is substituted in Eq. (4) to obtain themodal form of the coupledEOMs for this system, as

Φ�⊤M�η�Φ���p_β

��Φ�⊤ �C�η; β�Φ�

�_pβ

��

Φ�⊤ �K�η; β�Φ��p0

�� Φ�⊤ �Qext��η; _�η; β; ζ�

(8)

The resulting modal EOMs describes the arbitrarily large rigid-bodymotion of the flexible body subject to small elastic deformations,which are captured using the modified vibration modes of theunconstrained structure. Because of the linear mapping of the rigid-body velocities, Eq. (6), the orthogonality property of the resultingvibration modes with respect to the system mass and stiffnessmatrices is lost. However, the system is already coupled due to thecontribution of the gyroscopic forces to the modal damping andstiffness matrices.The modal damping and stiffness matrices remain functions of the

rigid-body DOFs, β. However, it is easy to see that they have,respectively, linear and quadratic dependencies with β and that it ispossible towrite them in terms of third- and fourth-order tensors [23]:

Φ�ij �CjkΦ�kl � cilrβr�t�; Φ�ij �KjkΦ�kl � kstifil � kgyrilrsβr�t�βs�t� (9)

where we sum over repeated indices and have identified thecontributions to the modal stiffness matrix from elastic andgyroscopic forces. The tensors c and k are constant in time and theirdimensions are i, l � f1; : : : ; mg and r, s � f1; : : : ; 6g for mnumber of modes used in the expansion. These tensors are typicallyvery sparse, and this approach generates efficient numerical solutionsthat keep the nonlinearities in the rigid-body DOFs and all couplingswith the linear structure at a low computational cost.

2. Mean-Axes Approximation

In this section, we give a brief description on the implementationof the mean-axes approximation, which will be exercised in thenumerical studies in Sec. IV. As it will be seen, the system size issimilar to the proposed modal formulation, but it neglects thegyroscopic couplings between the elastic and rigid-body dynamics.The mean-axes system is enforced by projecting the linear EOMs

onto free-free modes [4,30]. Here, we obtain the free-free modes inthe inertial frame by solving the eigenvalue problem given by the

unconstrained, undamped elastic EOMs as, MSS �ηf �KSstifη

f � 0,

where ηf is the vector of elastic DOFs without enforcing aconstrained node at the origin of frameA. The corresponding stiffness

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matrix KSstif is obtained through linearization of the elastic forces

QSstif . The relative displacements of the flexible body can then be

described in terms ofmode shapesΦf and generalized displacements

pf, such that ηf �P

iΦfi p

fi . Because of orthogonality of the free-

free modes and if we neglect the contribution of elastic deformationsto the gyroscopic forces in Eq. (3), it is possible to solve the mean-axes EOMs,

�pfi � ω2i p

fi � �Φ

fi �

⊤Qfext��η; β; ζ� (10)

separately from the nonlinear rigid-body dynamics EOMs, extractedfrom Eq. (3), as

MRR�η�_β�QRgyr�η; β� � QRext��η; β; ζ� (11)

where ωi is the vibration frequency of the ith mode and MRR themass matrix of the rigid-body at the CM and principal axes.Neglecting some gyroscopic terms in the damping matrix is anecessary assumption to diagonalize all modal system matrices. Theimplication of this assumption will be investigated in this paper on afull aircraft application. Note that the resulting set of EOMs can stillbe coupled if the external forcesQext, such as aerodynamic loads, notonly depend on rigid-body velocities β and orientation ζ of theaircraft but also on the instantaneous geometry due to deformation ofthe lifting surfaces, �η.

III. Coupled Aeroelasticity and Flight Dynamic Models

The presented flexible-body dynamics description has beenimplemented in the framework for SHARP [12,20] to study flexibleaircraft, including static aeroelastic analyses, trim, linear stabilityanalyses, and fully nonlinear time-marching simulations. Theunsteady aerodynamics are given by the unsteady vortex latticemethod, which accounts for wing surfaces subject to large deforma-tions. The implementation of the aerodynamics follows Katz andPlotkin [24], and the mapping between the aerodynamic andstructural grid has been presented by Murua et al. [12]. Only asummary is given here.

A. Direct Simulation of Nonlinear Flexible-Aircraft Dynamics

A time-domain simulation of very flexible aircraft dynamics isobtained via a loosely coupled implementation of Eq. (3) with theunsteady aerodynamic loads in the external forcesQext, given by theUVLM. In the UVLM, vortex-ring quadrilateral elements are used todiscretize both lifting surfaces and wakes. Each surface (bound)vortex-ring has an associated circulation strengthΓk and a collocationpoint, at which the impermeability boundary condition is satisfied. Inits general form, the UVLM is a geometrically nonlinear method inwhich the shape of a force-freewake is obtained as part of the solutionprocedure. The wake is therefore formed, shed, convected, andallowed to roll up according to the local flow velocity. The vorticitydistribution of the vortex elements is determined by applying thenonpenetration boundary condition at the discrete time step n� 1,such that

AbΓn�1b � AwΓn�1w � wn�1 (12)

where Γb and Γw are bound (surface) and wake circulation strengths,respectively, and Ab and Aw are the wing-wing and wing-wakeaerodynamic influence coefficient matrices. Elements of thesematrices are obtained by projecting the velocity (computed using theBiot–Savart law) over the vortex-ring normal vector. The vector ofnormal components of the nonvortical velocities at the collocation(points,w, can) include deployment of control surfaces, gust-inducedvelocities, wing deformations and rigid-body motions. Note that, asthe influence coefficients can account for large deflections of theaerodynamic surfaces, the method is geometrically nonlinear.Once the vorticity distribution is computed at each time step, the

aerodynamic pressures can be computed using Bernoulli’s equation.The resulting aerodynamic loads are finally converted into forces and

moments at the beam nodes through amapping procedure as outlinedin Murua et al. [12], which in the current implementation usesspanwise coincident meshes. No effort was done in the direct time-simulations in this paper to simplify the aerodynamic description forsmall wing deformations.

B. Monolithic Description of Linearized Flexible-Aircraft Dynamics

Linearization of the UVLM formulation leads to a compact set ofdiscrete-time state-space equations that, coupled with the fullylinearized form of the flexible-body dynamics EOMs, Eq. (4), can beused for linear stability analysis of very flexible aircraft with possiblylarge deformations at trimmed flight. To obtain the state-space formof the UVLM, the governing equations are linearized on a frozenaerodynamic geometry [20,31]. They are naturally written in adescriptor state-space form as

EFΔxn�1F � FFΔun�1F � AFΔxnF � BFΔunF;ΔynF � CFΔxnF �DFΔunF

(13)

where superscriptsn andn� 1 refer to the current and next time stepsand subscriptF highlights the aerodynamic components. The outputsyF are the aerodynamic loads at panels and the states xF and inputsuFof the linearized aerodynamic system are given by

xF � �ΓTb ΓTw _ΓTb �T and uF � � χTb _χTb �T (14)

where the aerodynamic grid χb and the time derivative _χb can accountfor deployment of control surfaces, gust-induced velocities, andwing deformations. This linearized formulation enables rigid-bodymotions and elastic deformations to be incorporated in a unifiedmonolithic framework. However, as the linear UVLM is written indiscrete time, temporal discretization of the structural equationsis also required before the fluid/structure coupling. A standardNewmark-β discretization is used, which leads to the followingequations for the homogeneous problem [32]:

EsysΔxn�1 � AsysΔxn (15)

where the state vector that completely determines the linear system is

x � �xTFjxTS jxTR�T � �ΓTbΓTw _ΓTb jηT _ηT jβTΘT �T (16)

with the Euler angles Θ used to define the orientation of the bodyframeA. As before, subscriptsS andR refer to the structural and rigid-body states. From Eq. (15), one obtains a discrete-time generalizedeigenvalue problem to determine the dynamic stability of the vehicle,which includes aeroelastic and flight dynamic modes.

IV. Numerical Studies

Previous work [12,16,23] has included extensive verificationstudies of SHARP for static and dynamic solutions of flexible-bodyconfigurations, with and without aerodynamics. The linear-flexible/nonlinear-rigid dynamics formulation, outlined in Sec. II.B, wasalready exercised in [23] on configurations with prescribed loads.Those results served to evaluate the limits of applicability of themean-axes approximation for a multibeam problem in spiral motion.In thiswork,we first return to the spiral dynamics problem to gain a

deeper understanding of the effect of neglecting gyroscopic forces inthe mean-axes approximation. Then, a representative flexible HALEaircraft is introduced to exercise the linear-flexible/nonlinear-rigiddynamics formulation coupled with the unsteady aerodynamicssolver. The L2 relative error norm, defined as ε � kXcom − Xrefk∕kXrefk, will be used throughout this section to compare computedresults Xcom with a reference solution Xref .

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A. Assessment of the Mean-Axes Approximation on a FlexibleMultibeam Configuration

A flexible multibeam configuration [23] (FMB) in vacuumwill beused to investigate the effect of the gyroscopic forces on the mean-axes approximation. The geometry, material properties, and loadhistory of the FMBproblem are defined in Fig. 2, where x-z is a planeof symmetry for the model and the cross-sectional properties aregiven in the local material frame with local x axis acting along thebeam reference line and local y axis is normal to y-z plane of body-fixed frame A. A set of follower loads is applied at the origin of thebody-fixed frame A, such that the rigid structure follows a circulartrajectory with radius RS. The parameter pR is introduced in thisstudy to vary the radius of the trajectory, which is prescribed by thetangential and centripetal forces FT and FC, respectively, and themomentMC. The total mass of the frame ism and Jz is themoment ofinertia around the z axis at the reference configuration. The body-

fixed frameA is rigidly linked to the FMB at pointPG, and the CMofthe undeformed geometry initially coincides with the origin of frameA to simplify comparison with themean-axes solution. At last, a deadload is applied at PG to prescribe the spiral motion. Note that nogravity forces are acting on the structure in this problem. Amesh sizeof 60 two-noded elements and a time step ofΔt � 0.1 swas found tobe sufficient to capture the predominantly rigid-body dynamics for800 time steps. The constant-acceleration Newmark scheme withalgorithmic damping of 0.01 was used to time-march the solution.The stiffness properties of the structure, shown in Fig. 2, result in

maximum tip displacements below 1% relative to the arm lengthL. Itwas shown before [23] that for pR � 1 the relative error (of about12%) in tip deformations using the mean-axes approximation isindependent of the stiffness even for sufficiently stiff wings. Tocomplete the analysis, this study will investigate the effect of thevehicle kinematics (characterized by radius parameter pR) on theerror given by the mean-axes approximation.The trajectory of the centroid is shown in Fig. 3 for four different

load cases. For this very stiff problem, the prescribed forces result inzero angular velocity components around the x and y axes, and themaximum ωGz component at t � 80 s is inversely proportional topR. The translational velocity components vGx and vGy are oscillatorydue to the circular motion, but the norm of vG remains independent ofthe prescribed spiral radius. This shows that only the inverseproportionality ofpR with the angular velocityωGz dictates the effectof gyroscopic forces on the tip deflections.The centrifugal forces result in predominantly antisymmetric in-

plane bending of the frame, whereas the resulting normalized tipdisplacements, shown in Fig. 4 for pR � 1 and 5, remain below 1%relative to the arm lengthL defined in Fig. 2. Comparison between thenonlinear, Eq. (3); linearized, Eq. (4); and mean-axes solutions,Eq. (10), in Fig. 4 demonstrates that the elastic response is linear forthis very stiff problem. However, it is clear that application of themean-axes approximation results in smaller deflections, whichconverge to the nonlinear solution for larger radii and hence smallerangular velocities.This behavior is demonstrated in detail in Fig. 5, which shows the

correlation between the maximum relative error norm of the right tipdeflection and its componentswith the radius parameterpR. The errorin in-plane bending (RAy and RAz) contributes the most to the overallrelative error norm. It is evident that there is a direct correlation

0 20 40 60 80−10

−8

−6

−4

−2

0

x 10−3

∆RA

z / L

time

nonlinearlinearizedmean−axes

a) pR = 1 b) pR = 5

0 20 40 60 80−10

−5

0

5x 10

−4

∆RA

z / L

time

nonlinearlinearizedmean−axes

Fig. 4 Vertical displacement of point P, projected in the A frame, for the FMB problem with different radii.

1 2 3 4 5

10−2

10−1

max

. rel

. err

or n

orm

ε

radius parameter pR

∆RA

∆RAx

∆RAy

∆RAz

Fig. 5 Maximum error norm of displacements at point P for the FMBproblem, comparing mean-axes approximation to the nonlinear solutionfor different radii.

Fig. 2 Definition of the FMB problem.

−50

0

50

100

−150

−100

−50

0−90

−60

−30

0

rGx

/ LrGy

/ L

r Gz /

L

pR=1

pR=2

pR=3

pR=5

Fig. 3 Trajectory of the origin of the body-fixed frame for the FMBproblem with different radii. Markers are plotted every 20 time steps.

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between the angular velocity ωGz, which is inversely proportional topR, and the relative error of the mean-axes approximation to predictthe elastic response. The contribution of the gyroscopic forces todamping and stiffness matrices are neglected under the mean-axesapproximation in order to decouple the modal system equations. Theerror of this approximation is dominated by the angular velocityωGz,as the norm of the translational velocity vG is constant for all radii.This shows that the gyroscopic forces can have a noticeable impact onthe structural dynamics.

B. Flight Dynamics of a Flexible Aircraft

A similar study, this time also including aerodynamic effects, willbe carried out on a full aircraft configuration. The representativeHALE aircraft, shown in Fig. 6, consists of large-aspect-ratio flexiblewings, a rigid fuselage, and a rigid empennage. It is similar to those ofPatil et al. [10] andMurua et al. [20], but it includes dihedralmembersin the main wing for lateral stability and the point mass of 50 kg(referred to as payload, even though it is just a first approximation ofthe nonstructural mass) was moved to the center of the main wing toguarantee static pitch stability. The dihedral members are rigidlylinked to the main wing at both ends at an angle of 20 deg. The lengthfraction of each dihedral member over the semispan of the main wingis defined as λ, whichwill be used in this work as a parameter to studythe impact of wing dihedral on the vehicle dynamics. The horizontalstabilizer and the vertical fin are modeled as lifting surfaces withoutcamber or pretwist, and the former includes an elevator modeled as aquarter-chord-length control surface. The empennage is raised by1.25 m with respect to the main wing to avoid wake–tail collisions[32], and it is rigidly linked to themain wing by a nonlifting fuselage.This basic aircraft configuration also includes two massless

propellers, which are modeled as point forces rigidly linked to themainwing. Themass per unit length of the fuselage is the same as thatof the horizontal and vertical tail planes, and thus, the total mass ofthis aircraft, including payload and structural mass, is 75.4 kg. The

flight conditions are V∞ � 30 m∕s at an altitude of 20 km (airdensity 0.0889 kg∕m3). The relevant properties of the members arelisted in Table 1. The stiffness parameter σ will be used in thefollowing studies to vary the stiffness properties of the main wing(decreasing σ results in a more flexible wing).

1. Trim for Steady Flight

As a first step, the vehicle is trimmed for steady level flight throughthree inputs, namely angle of attack, elevator deflection δtrime , andthrust per propeller. The trimmingmethod is a tightly coupled processthat is described in detail inMurua et al. [20]. The trim characteristicsof the HALE aircraft are presented in Fig. 7 for dihedral memberratio, 0 ≤ λ ≤ 1∕2, and stiffness parameter of the main wing,1.1 ≤ σ ≤ 1000. The tip deflection of the main wing, non-dimensionalized with the overall semispan of the main wing, B, isalso included in Fig. 7. Parameter σ � 1 corresponds to the originalstiffness properties from [10], but it gives tip deflections over 40% atcurrent flight conditions, which were deemed unrealistic. It can beseen that the angle of attack initially drops with increasing flexibilitydue to favorable twisting of the wing. However, this is opposed bywing bending for very flexible configurations (σ ≥ 2) resulting inloss of vertical forces because of the additional wing dihedral andillustrates the importance of geometrically nonlinear effects on theaircraft trim characteristics. Similarly, the increasing span ratio of thedihedral members, λ, is also compensated by a larger angle of attackand elevator deflection. The wing tip deflection decreases forincreasing values of λ due to the smaller bending moment from the

Fig. 6 Undeformed HALE aircraft geometry with front and top views (not to scale).

Table 1 HALE aircraft properties

Property Main wing Tail plane

Chord 1 m 0.5 mSemispan for λ � 0, B 16 m 2.5 mElastic axis 50% chord 50% chordCenter of gravity 50% chord 50% chordMass per unit length 0.75 kg∕m 0.08 kg∕mMoment of inertia 0.1 kg · m 0.01 kg · mTorsional stiffness 1σ × 104 N · m2 ∞Bending stiffness 2σ × 104 N · m2 ∞In-plane bending stiffness 4σ × 106 N · m2 ∞

Fig. 7 Trim angle of attack, elevator deflection, thrust per propeller,and vertical tip deflection of the main wing, ΔRAz∕B, as a function ofwing flexibility σ and dihedral member ratio λ.

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dihedral members, as the lift vector is increasingly pointing inward.Figure 8 shows the trim deformations of the HALE configuration fora range of stiffness values σ and for λ � 1∕2 and demonstrates thelarge effective wing dihedral for the very flexible aircraft.

2. Linear Stability Analysis

The dynamic stability of this HALE configuration is studied nextto investigate the effect of dihedralmember ratio λ and flexibility σ onthe vehicle flight dynamics. The stability characteristics will besubsequently compared to the open-loop response of the aircraft. Foreach trim solution, we can directly solve the generalized eigenvalueproblem posed by the linear aeroelastic system coupled with theflight dynamics of the flexible aircraft, derived from Eq. (15).The aerodynamic mesh used to solve this problem is shown in Fig. 8.The wake is cut off after 20 m with a time step of 0.01 s. The beamelements coincide with the aerodynamic panels in the spanwisedirection. The resulting total system size is 1581 states, whichincludes 924 aerodynamic states, 648 structural states, and the 9rigid-body states.Figure 9 presents the root loci as a function of the wing stiffness σ

for a dihedral ratio of λ � 1∕4. Note that the discrete-time eigenvalueanalysis derived from Eq. (15) yields as many roots as the number ofstates in the problem, but only the dominant modes are presentedhere. The first few coupled modes of the flexible vehicle have beenidentified and are designated by the names of their rigid-aircraftcounterparts, e.g., phugoid, short period, etc., although they alsoinclude vehicle deformations. The aeroelastic modes are labeled withnumbers in which modes 1, 2, and 5 have been identified as coupledsymmetric bending/torsion modes, andmodes 3, 4, and 6 correspond

to the antisymmetric modes. Aerodynamic modes can also becaptured with this stability analysis, but they are highly damped andmostly fall beyond the boundaries of Fig. 9a. The stiffness valueshave been unevenly spaced, as defined in Fig. 9a, in which the twoextreme cases are representative of a very flexible (σ � 1) and a stiff(σ � 1000) wing. Note that the discrete-time eigenvalues have beenconverted to continuous time for an easier interpretation.Figure 9a demonstrates that for very flexible aircraft the frequency

separation between flight dynamic and aeroelastic modes candisappear. As the stiffness of the main wing is increased, the higher-frequency aeroelastic modes become less stable, whereas the low-frequency flight dynamic modes show the opposite trend. Thenonoscillatory spiral divergence on the other hand, becomes lessstable with increasing stiffness. This can be observed in detail inFig. 9b, which highlights two flight-dynamics lateral modes: Dutchroll and spiral divergence. As flexibility increases, the less stableDutch roll illustrates the detrimental impact flexibility might have onthe handling qualities of the vehicle. In contrast, the spiral modestability margin substantially increases with flexibility due to thelarger static deformations of the main wing at trim conditions.Figure 9c zooms further in the root loci in order to visualize the

phugoid mode. For a rigid aircraft, it is a lightly damped mode.As flexibility is increased, this mode demonstrates the complexinteraction between the elastic and rigid-body DOFs: as the aircraftpitches up and climbs first, the wings undergo a downwardflappinglike motion. As the vehicle pitches down and descends, thewing bends upward, reaching themaximum tip deflections before thecycle is finished. Whereas rigid and stiff aircraft present stablephugoid modes, damping drops with increasing wing flexibilityresulting in an instability for 1.2 ≤ σ ≤ 1.7. However, for highlyflexible cases (σ < 1.2), this trend is reversed, which results in stableconfigurations. To investigate this reversal in stability behavior, wehave included the extremely flexible case of σ � 1 in Fig. 9 and willexamine the effect of the outboard dihedral in the following.Figure 10 shows the root loci for a very stiff (σ � 1000) and a very

flexible (σ � 1.1) aircraft for varying dihedral ratio λ. Here, we onlyfocus on the dominant flight dynamic modes: Dutch roll, spiraldivergence, and phugoid. Figure 10a shows that an increase of thedihedral member ratio leads to a less stable Dutch roll, whereas it

Fig. 8 Trim deformation of HALE aircraft with λ � 1∕2 and differentstiffness parameters σ. (Figure shows actual deformations.)

a)

b)

Fig. 10 Root loci for σ � 1.1 (“flexible”) andσ � 1000 (“rigid”)with thedihedral ratio λ as parameter: a) dominant roots and b) zoom near to theorigin.

Fig. 11 Control inputs for ailerons δ�a and elevator δ�e . Transients aresine functions.

a)

b)

c)

Fig. 9 Root loci for λ � 1∕4 and stiffness of the main wing, σ, asparameter: a) flight dynamic (solid), aeroelastic (dashed), andaerodynamic (dotted) modes; b–c) zooms near to the origin.

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increases the resistance against spiral divergence. This coincideswiththe trends observed in Fig. 9a with increased flexibility; that is, staticdeformations and dihedral ratio have the same effect on the lateralmodes. As the dihedral ratio is varied, these trends are consistent forboth the rigid and flexible vehicles. As expected, wings with verysmall or no dihedral lead to unstable spiral modes.Figure 10b focuses on the region closer to the origin in order to

expose the characteristics of the phugoid mode. Interestingly, thetrends are opposite for the rigid and flexible vehicles: the increase ofdihedral ratio leads to a more stable phugoid in the flexible aircraft(except for λ � 0), but the damping of the rigid phugoid decreaseswith λ. From the trends of the phugoid mode in Figs. 9 and 10, it isclear that static and dynamic aeroelastic effects have opposingcontributions, such that an increase of effective wing dihedral due towing bending (static effect) decreases phugoid stability, as deducedfrom Fig. 10, which explains the trend for 1.2 ≤ σ ≤ 1.7 in Fig. 9.For more flexible configurations, the dynamic properties result inincreased damping, which dominate over the static effects and tend tostabilize the aircraft.

3. Open-Loop Response of HALE Aircraft with Large Rigid-Body

Motions

Finally, we can investigate the effect of the consistent linearizationof the structural DOFs on the dynamic response of the vehicle definedin Fig. 6 and Table 1. The aircraft is subject to various excitations inthe formof commanded inputs on ailerons and elevator, and the open-loop response is computed using the linearized and mean-axessolutions, as well as the full nonlinear solver. To isolate the effectof linearization of the structural DOFs only, the aerodynamicswere computed using the geometrically nonlinear time-marchingdescription of the UVLM for all results presented here. The stiffness

parameter σ is also used in this study to explore the effect of flexibilityon the nonlinear flight dynamic response.Wewill limit ourselves to the configuration with dihedral member

ratio λ � 1∕4, which was shown to be stable for the whole range ofstiffness values with lightly damped spiral and phugoid modes (seeFigs. 9 and 10). For the dynamic analysis, the control surface inputsof ailerons and elevator (denoted by subscripts a and e, respectively)are prescribed such that δa � �δ�a and δe � δtrime � δ�e with the timehistories of δ�a;e shown in Fig. 11. The elevator inputs are prescribedaround the trim inputs δtrime , whereas the ailerons, as defined inFig. 6, are not used for trimming the aircraft but are deployedantisymmetrically with �δ�a . The open-loop response of the aircraftis computed for 50 s using a time step of 0.025 s.Figure 12a shows the angular velocity components of the rigid

vehicle expressed in the body-fixed frame A. The time history of theangular velocity norm jωAj indicates that the aircraft responseremains lightly damped even long after the command (t ≥ 20 s). Thisfinding is in agreement with the preceding stability analysis in Fig. 9,which suggests that the rigid aircraft is only marginally stable. It wasalso shown, however, that the spiral mode becomes more stable withincreasing flexibility due to wing dihedral. To demonstrate thisbehavior, Fig. 12b compares the maximum norm of the angularvelocity ωA, and its components for different values of the flexibilityparameter σ. The yaw rate ωAz is damped much faster for the flexibleplatforms (σ < 5) as the response is dominated by the less stablephugoid mode in this flexibility range.In addition to the stability characteristics, flexibility impacts the

controllability of the aircraft. Figure 13a shows its trajectory in thex-y plane (top view) for different stiffness values σ and demonstratescontrol reversal with increasing flexibility of the main wing. For theflexible configuration with σ � 2, wing twisting results in minimal

0 10 20 30 40 50−0.1

0

0.1

0.2

0.3

0.4

time [s]

angu

lar

velo

city

[rad

/s]

ωAx

ωAy

ωAz

|ωA|

100

101

102

103

0

0.1

0.2

0.3

0.4

stiffness parameter σ

ωAx

ωAy

ωAz

ωA

a) b)

max

imum

ang

ular

vel

ocity

no

rm [r

ad/s

]

Fig. 12 Rigid-body angular velocity components of theHALEaircraft: a) time history of the rigid vehicle and b)maximumnormofωA and its individualcomponents for varying stiffness computed using the geometrically nonlinear model.

−1500 −1000 −500 0 500

−1000

−800

−600

−400

−200

0

200

400

600

800

r Gy [m

]

rGx

[m]

nonlinearlinearizedmean−axes

σ=10

σ=5

σ=1000

σ=1

σ=1.5 σ=1.1

σ=2

0 10 20 30 40 50−100

−80

−60

−40

−20

0

20

40

r Gz [m

]

time [s]

σ=1000σ=10σ=5σ=2σ=1.5σ=1.1σ=1

a) x-y plane using different models b) Time history of altitude, z, using the geometricallynonlinear model

Fig. 13 Trajectory of the HALE aircraft for λ � 1∕4 with stiffness of the main wing as parameter.

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control authority of the ailerons. Controllability can be furtherexplored on the linear system by evaluating the controllabilityGramians [33], but this will not be investigated here. Figure 13bshows the effect of flexibility on the altitude response of the HALEaircraft. The frequency of the resulting oscillatory motion matchesclosely the phugoid mode presented for this configuration in Fig. 9.The reduction in frequencywith increasing flexibility is also capturedin this response.To compare the effect of linearization of the elastic DOFs,

Fig. 13a also presents the dynamic behavior obtained using the linear-flexible/nonlinear-rigid formulation, Eq. (4), which suggests thatgeometrically nonlinear deformations have a negligible effect on theaircraft motion in the x-y plane. The effect of geometric nonlinearityis very different in the altitude response of the aircraft. This isdemonstrated in Fig. 14, which shows the maximum error of thelinearized formulation to compute the vertical position for varyingflexibility of the main wing. The maximum norm of the HALEaircraft altitude was used to normalize the error for each σ. Thelinearized solution accurately captures the rigid-body response formildly flexible vehicles (σ ≥ 5 corresponding to 7% wing tipdeflections), but the geometrically nonlinear deformations dominatefor more flexible configurations.Next, the open-loop response of the HALE aircraft was computed

using the mean-axes approximation, Eq. (10), but the aerodynamicsin our implementation are still computed on the actual instantaneousgeometry. For small deformations, this converges to the classicalmean-axes approximation as it is commonly found in flight dynamicsimulations [4]. Themean-axes solution to the rigid-body response ofthe HALE aircraft is included in Figs. 13 and 14 and compared withthe fully coupled, nonlinear reference solution. The error norm inFig. 14 gives an insight into the effect of neglecting the gyroscopicforces due to elastic deformations. One would expect the error toincreasewith flexibility, as is the case for σ < 2, but a local maximumoccurs around σ � 10, which represents a rather stiff configurationwith 5% maximum wing deformations. We saw from Fig. 12b thatthe angular velocity of the aircraft, and hence the contribution ofthe gyroscopic forces, reduces with increasing flexibility. This com-petition between the opposing effects of gyroscopic forces andflexibility results in the peak at around σ � 10.

The right wing tip deflection is presented in Fig. 15 for a stiff(σ � 10) and a very flexible (σ � 1.1) aircraft. Whereas the flexiblecase exhibits static equilibrium deformations of 37%of the semispan,the maximum amplitude in the transient around this equilibrium isbelow 10% of the semispan. As a result, the linearized formulation isable to capture the amplitudes of the large deformations well even forthe very flexible problem,which is presented in Fig. 15b.However, asthe mass matrix is assumed constant in the linearized approach, thechange in geometry due to the deformations and the resulting changein stability behavior of the aircraft cannot be captured, which resultsin a frequency shift. Figure 16 shows the maximum error norm of thevertical wing tip displacement, which were normalized with themaximum wing deformations for each σ. The linearized approachpredicts the displacements well even for very flexible aircraft withσ � 2 andwing deformations of up to 26%of the semispan. Formoreflexible configurations, the resulting phase shift due to geometricnonlinearity effects dominate.At last, it is shown that the mean-axes approximation tends to

overpredict the elastic deformations in the open-loop dynamics.Figure 15a indicates that, even for relatively stiff problems (σ � 10),corresponding to maximum deflections below 5% of the semispan,the mean-axes approximations results in errors around 9% comparedto the nonlinear reference solution. Similar to the FMB problem,Fig. 16 shows that even for near-rigid vehicles (σ ≥ 100) the relativeerror remains constant around 1% for the commanded maneuver.Whereas it was shown in Fig. 14 that the mean-axes approximationcan give a good estimate of the open-loop flight dynamic response forvery stiff structures, it results in a poor prediction of the elasticdeformations even for very stiff aircraft. Thismakes it less suitable fordynamic maneuver and gust load analyses.

V. Conclusions

This paper has presented a coupled aeroelastic and flight dynamicsframework for flexible aircraft with a consistent linearization ofthe structural dynamics around a nonlinear static trim equilibrium.Starting from a displacement-based geometrically nonlinear com-posite beam model, coupled with an unsteady vortex lattice method,linearization in only the structural degrees of freedom has resultedin flexible-body dynamics equations written in matrix form, with

100

101

102

103

0

0.2

0.4

0.6

0.8

1

stiffness parameter σ

linearizedmean−axes

max

imum

rel

ativ

e er

ror

norm

ε

Fig. 14 Maximum relative error norm of aircraft altitude using

linearized and mean-axes approximations compared to the nonlinearsolution for λ � 1∕4 and varying stiffness σ.

0 10 20 30 40 500.025

0.03

0.035

0.04

0.045

0.05

0.055

norm

aliz

ed ti

p de

flect

ion

time [s]

nonlinearlinearizedmean−axes

a) = 10σ b) = 1.1σ

0 10 20 30 40 500.25

0.3

0.35

0.4

0.45

0.5

0.55

time [s]

norm

aliz

ed ti

p de

flect

ion

Fig. 15 Vertical tip deflection of the right main wing normalized with the semispan of B � 16 s for dihedral ratio of λ � 1∕4.

100

101

102

103

0

0.05

0.1

0.15

0.2

stiffness parameter σ

linearizedmean−axes

max

imum

rel

ativ

e er

ror

norm

ε

Fig. 16 Maximum relative error norm of vertical wing tip deflectionsfor λ � 1∕4 and varying σ.

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coefficients that vary only with the vehicle rigid-body velocities.Approximating the finite element discretization in terms of a fewvibration modes of the unconstrained structure has been shown toresult in closed-form expressions for the equations of motion withconstant coefficients that capture the gyroscopic couplings betweenthe rigid body and the structural dynamics. The size of the resultingsystem is thus comparable to that inmean-axes approximations, but itincludes the gyroscopic couplings and vehicle dynamics, which arewritten around a nonlinear aeroelastic equilibrium, including adeformed aerodynamic lattice. Moreover, it is also not necessary toprecompute frequency-domain aerodynamics, as it is done insolutions based on the doublet lattice method.Numerical results on a free-flying flexible multibeam con-

figuration have shown that the mean-axes approximation results in aconstant relative error independent of the extent of deformations.More importantly, this error correlates directly to the angular velocityof the flexible body. Neglecting the gyroscopic couplings maytherefore have an important effect in the dynamicmaneuver loads. Totest that, the proposed linearized framework has also been exercisedon a representative HALE configuration subject to commandedlateral control inputs. A linear stability analysis and the open-loopresponse of the aircraft have confirmed the dramatic effect offlexibility on the vehicle dynamics, and this was shown to beadequately captured even for rather flexible configurations using thelinear-flexible/nonlinear-rigid model. It has finally been shown thatthemean-axes approximation fails to predict appropriately the elasticdeformations in this open-loop response even for stiff aircraft.In summary, the linearized flexible-aircraft formulation provides a

powerful tool to accurately describe the flight dynamics of flexiblehigh-aspect-ratio aircraft. The proposed model captures all couplingterms due to gyroscopic forces, which implies that the resulting linearelastic deformations can be captured accurately. This may be criticalfor the prediction ofmaneuver loads on flexible vehicles, and becausea modal projection of the structural degrees of freedom was alsointroduced, it has only a small computational burden.

Acknowledgments

The work of Henrik Hesse is sponsored by the U.K. Engineeringand Physical Sciences Research Council. This support is gratefullyacknowledged.

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