Nonlinear Aeroelasticity of a Very FlexibleBlended-Wing-Body Aircraft

Weihua Su∗ and Carlos E. S. Cesnik†

University of Michigan, Ann Arbor, Michigan 48109-2140

DOI: 10.2514/1.47317

This paper presents a study on the coupled aeroelastic/flight dynamic stability and gust response of a blended-

wing-body aircraft that derives from the U.S. Air Force’s High Lift-Over-Drag Active (HiLDA) wing experimental

model. An effective method is used to model very flexible blended-wing-body vehicles based on a low-order

aeroelastic formulation that is capable of capturing the important structural nonlinear effects and couplingswith the

flight dynamic degrees of freedom. A nonlinear strain-based beam finite element formulation is used. Finite state

unsteady subsonic aerodynamic loads are coupled to all lifting surfaces, including the flexible body. Based on the

proposed model, aeroelastic stability is studied and compared with the flutter results with all or partial rigid-body

degrees of freedom constrained. The applicability of wind-tunnel aeroelastic results (where the rigid-body motion is

limited) is discussed in view of the free-flight conditions (with all 6 rigid-body degrees of freedom). Furthermore,

effects of structural and aerodynamic nonlinearities as well as wing bending/torsion rigidity coupling on the aircraft

characteristics are also discussed in this paper.

Nomenclature�A = system matrix of the linearized systemA�s� = beam cross-sectional areaa0 = local aerodynamic frame, with a0y axis aligned

with zero lift line of airfoila1 = local aerodynamic frame, with a1y axis aligned

with airfoil motion velocityB = body reference frameBF, BM = influence matrices for the distributed forces and

momentsb = semichord of airfoil, mb = positions and orientations of the B frame, as

time integral of �bn = expansion coefficients for the inflow velocityCFF, CFB,CBF, CBB

= components of the generalized damping matrix

�CFF, �CFB,�CBF, �CBB

= components of the generalized damping matrixof the linearized system

CBa1 = rotation matrix from the a1 frame to the Bframe

CGB = rotation matrix from the B frame to the Gframe

ci, gi = coefficients based upon geometry of thetrailing-edge control surfaces (i� 1; 2; 3; . . .)

d = distance of midchord in front of beam referenceaxis, m

E1, E2,E3, E4

= influence matrices in the inflow equations withairfoil motion variables

F1, F2, F3 = influence matrices in the inflow equations withindependent variables

Faero,Maero = nodal aerodynamic forces and moments

Fdist, Fpt = distributed and point forcesG = global (inertial) reference frameg = gravity acceleration vectorHhb = matrix consisting of influence from Jacobian

(Jhb) and the body angular velocity (!B)h = absolute positions and orientations of beam

nodeshr = relative positions and orientations of beam

nodes, with respect to the B frameI = identity matrixIij = mass moment of inertia of the beam cross

section about its shear center (i; j� x; y; z)J = Jacobian matrixK�s� = matrix of strain componentsKFF = components of the generalized stiffness matrix�KFF = components of the generalized stiffness matrix

of the linearized systemk�s�, c�s� = beam cross-sectional stiffness and damping

matricesk22, k33, k23 = torsion, out-of-plane bending, and coupled

torsion/out-of-plane bending componentsof k�s�

lmc, mmc, dmc = aerodynamic loads on an airfoil about itsmidchord

lra, mra, dra = aerodynamic loads on an airfoil about beamreference axis

M, C, K = discrete mass, damping, and stiffness matricesof the whole system

M�s� = beam cross-sectional inertia matrixMe, Ce, Ke = element mass, damping, and stiffness matricesMFF,MFB,MBF,MBB

= components of the generalized mass matrix

�MFF, �MFB,�MBF, �MBB

= components of the generalized mass matrix ofthe linearized system

Mdist,Mpt = distributed and point momentsm = mass per unit length, kg=mN = influence matrix for the gravity forceN = number of inflow statesPB = inertial position of the B frame, resolved in the

G framepa = position of an arbitrary point a with respect to

the G framepB, �B = position and orientation of the B frame, as time

integral of vB and !Bpw = position of the w frame with respect to the B

frame

Presented as Paper 2009-2402 at the 50th AIAA/ASME/ASCE/AHS/ASCStructures, Structural Dynamics, and Materials Conference, Palm Springs,CA, 4–7 May 2009; received 22 September 2009; revision received 25 June2010; accepted for publication 29 June 2010. Copyright © 2010 by WeihuaSu and Carlos E. S. Cesnik. Published by the American Institute ofAeronautics andAstronautics, Inc., with permission. Copies of this papermaybe made for personal or internal use, on condition that the copier pay the$10.00 per-copy fee to the Copyright Clearance Center, Inc., 222 RosewoodDrive, Danvers, MA 01923; include the code 0021-8669/10 and $10.00 incorrespondence with the CCC.

∗Postdoctoral Research Fellow, Department of Aerospace Engineering;suw@umich.edu. Senior Member AIAA.

†Professor, Department of Aerospace Engineering; cesnik@umich.edu.Associate Fellow AIAA.

JOURNAL OF AIRCRAFT

Vol. 47, No. 5, September–October 2010

1539

Q1, Q2 = matrices of the state-space system equationsq = independent variables of the equations of

motionRB, RF = rigid-body and flexible components of the

generalized load vectorRaeroB , Raero

F = rigid-body and flexible components of thegeneralized aerodynamic loads

RgravB , Rgrav

F = rigid-body and flexible components of thegeneralized gravity force

rx, ry, rz = position of the cross-sectional mass center inthe w frame

s = beam curvilinear coordinate, mvB, !B = linear and angular velocities of the B frame,

resolved in the B frame itselfWext,W int = external and internal virtual workw = local beam reference frame defined at each

node along beam reference linex = state variablesx, y, z = position of an arbitrary point a in the

corresponding local w frame, m_y, _z = airfoil translational velocity components

resolved in the a0 frame, m=s_� = airfoil angular velocity about the a0x axis,

rad=s� = body velocities, with translational and angular

components, resolved in the B frame� = trailing-edge control surface deflection, rad" = elastic strain vector"x, �x,�y, �z

= element strains corresponding to extension,twist, out-of-plane, and in-plane bending

"0 = initial (prescribed) elastic strain vector� = quaternions defining the orientation of the B

frame� = stiffness-proportional damping coefficient� = rotations of beam nodes, rad� = inflow states, m=s�0 = inflow velocities, m=s = wing material density, kg=m3

1 = air density, kg=m3

= tuning parameter that determines the torsion/out-of-plane bending stiffness coupling

�0 = nonlinear equilibrium state����� = coefficient matrix of the quaternion equations,

a function of body angular velocities

Subscripts

B = reference to the B frameBB, BF = components of a matrix with respect to body/

flexible differential equations of motione = elementF = reference to the flexible degrees of freedomFB, FF = components of a matrix with respect to flexible/

body differential equations of motionhb = h vector with respect to the motion of the B

frameh" = h vector with respect to the strain "pb = nodal position with respect to the motion of the

B framep" = nodal position with respect to the strain "�b = nodal rotation with respect to the motion of the

B frame�" = nodal rotation with respect to the strain "

I. Introduction

H IGH-ALTITUDE long-endurance (HALE) vehicles featurewingswith high-aspect-ratio. These long and slenderwings, by

their inherent nature, can maximize lift-over-drag ratio. On the otherhand, these wings may undergo large deformations during normaloperating loads, exhibiting geometrically nonlinear behavior. Van

Schoor et al. [1] studied aeroelastic characteristics and control ofhighly flexible aircraft. They used linearized modes including rigid-body modes to predict the stability of the aircraft under differentflight conditions. Their results indicated that unsteady aerodynamicsand flexibility of the aircraft should be considered so as to correctlymodel the dynamic system. Patil et al. [2] studied the aeroelasticityand flight dynamics of HALE aircraft. They concluded that the largewing deformations due to the high-aspect-ratio structure may changethe aerodynamic load distribution, compared to the initial shape,which may result in significant changes to the aeroelastic and flightdynamic behaviors of the wings and the overall aircraft. Therefore,the analysis results obtained through a linear analysis based on theundeformed shape may not be valid in this case, since those effectscan only be obtained from a nonlinear analysis. A possible approx-imation to account for the geometric nonlinearity is to determine anonlinear steady state, about which a linear dynamic analysis couldbe conducted. This breaks up, however, when the dynamic motionitself is nonlinear. In a parallel effort, Drela [3] modeled a completeflexible aircraft as an assemblage of nonlinear beams. In hiswork, theaerodynamic model was a compressible vortex/source-lattice withwind-aligned trailing vorticity and Prandtl–Glauert compressibilitycorrection. The nonlinear equation was solved using a full Newtonmethod. Through simplifications of the model, the computationalsize was reduced for iterative preliminary designs. Shearer andCesnik [4] also studied the nonlinear flight dynamics of very flexibleaircraft. The nonlinear flight dynamic responses were governed bythe coupled equations of the 6-DOF rigid-body motions and thenonlinear aeroelastic equations. They compared three sets of solu-tions: rigid, linearized, and fully nonlinear models, highlighting theimportance of using the latter to properly model the very flexiblevehicle motions. Su and Cesnik [5] studied the nonlinear dynamicresponse of a highly flexible flying-wing configuration. An asym-metric distributed gust model was applied to the time-domainsimulations to study the nonlinear behaviors of the flying-wingconfiguration under such perturbations. Bilinear torsional stiffnesschanges due to wrinkling of the skin were addressed as well.Common to all those studies is the fact that the coupled effectsbetween the vehicle flexibility and flight dynamics must be properlyaccounted for in a nonlinear aeroelastic formulation.

For the last several years, the U.S. Air Force has been working onnew platform concepts for intelligence, surveillance, and recon-naissance missions, called sensorcraft. These are large HALE air-craft, with wing span of approximately 60 m. Three basic vehicleconcepts have been considered: blended-wing-body, single-wing,and joined-wing configurations [6]. Among the sensorcraft concepts,this paper will focus on the blended-wing-body configuration. As anexample of studies on this configuration, Beran et al. [7] performedstatic nonlinear aeroelastic analyses of a blended-wing-body. Theyused a high-fidelity computational tool to assess the contributions ofaerodynamic nonlinearities to the transonic air loads sustained by ablended-wing-body with different static aeroelastic deflections. Thestructural deflections prescribed in the nonlinear analyses wereobtained from a linear methodology. On the experimental side, awind-tunnel model [8] was recently created by Northrop Grumann,under the Air Force’s High Lift-Over-Drag Active (HiLDA) wingprogram to study the aeroelastic characteristics of blended-wing-body for a potential sensorcraft concept. In the report, a very lightlydamped pure fore-aft bendingmodewas observed in thewind-tunneltest, which was not correctly modeled in their aeroelastic analysistool, however. They suggested that this phenomena should beanalytically explored further to fully characterize and understand theflexible wing fore-aft mode behavior [8]. The target model used inthis paper is based on the HiLDAwind-tunnel model.

As pointed out in the literature (e.g., [2,4,5,9]), the couplingbetween the low-frequency rigid-body motions of the very flexiblevehicle and its high-aspect-ratio, low-bending-frequency wings isvery important. The natural frequencies of the wings are in the samerange of the rigid-body ones, and they cannot be separated. Suchproximity may lead to direct excitation of wing aeroelastic responsedue to rigid-body motions and vice versa. This can result in adynamic instability known as body-freedom flutter [10]. This paper

1540 SU AND CESNIK

focuses on the study of that aeroelastic instability, but in a nonlinearsense, where the instability boundary is dependent on the trim stateand the size of the disturbance can drive a stable blended-wing-bodyaircraft to flutter.

II. Theoretical Formulation

Because of the interaction between flight dynamics and aero-elastic response, the formulation includes six rigid-body and multi-ple flexible degrees of freedom. The structural members are allowedfully coupled three-dimensional bending, twisting, and extensionaldeformations. Control surfaces are included for maneuver studies. Afinite state unsteady aerodynamicmodel is integrated into the systemequations. The model allows for a relatively low-order set of non-linear equations that can be put into state-space form to facilitatestability analysis and control design. This formulation is imple-mented in MATLAB and is called the University of Michigan’sNonlinear Aeroelastic Simulation Toolbox (UM/NAST). An over-view of the formulation implemented in UM/NAST is describedbelow.

A. Fundamental Descriptions

As shown in Fig. 1, a global (inertial) frameG is defined, which isfixed on the ground. A body frame B is built in the global frame todescribe the vehicle position and orientation, with Bx pointing to theright wing,By pointing forward, andBz being the cross product ofBxandBy. The position and orientation b and the time derivatives _b and�b of the B frame can be defined as

b��pB�B

�_b� ��

�_pB_�B

���vB!B

�

�b� _���

�pB��B

���

_vB_!B

�(1)

where pB and �B are body position and orientation, both resolved inthe body frameB. Note that the origin of the body frame is arbitrary inthe vehicle and it does not have to be the location of the vehicle’scenter of gravity.

As described in Fig. 2, a local beam frame w is built within thebody frame, which is used to define the position and orientation ofeach node along the beam reference line. Vectorswx,wy, andwz arebases of the beam frame, whose directions are pointing along thebeam reference axis, toward the leading edge, and normal to the beamsurface, respectively, resolved in the body frame.

To model the elastic deformation of slender beams, a nonlinearbeam element was developed [11,12]. Each of the elements has threenodes and four local strain degrees of freedom: extension, twist, andtwo bending strains of the beam reference line.

The positions and orientations of each node along the beam aredefined by a vector consisting of 12 components, denoted as

h�s�T � f �pB � pw�s��T; wx�s�T; wy�s�T; wz�s�T g (2)

wherepw is the position of thew frame resolved in the body frame. Insome cases, the nodal position and orientation informationwithin thebody frame is also necessary, defined as

hr�s�T � fpw�s�T; wx�s�T; wy�s�T; wz�s�T g (3)

It is easy to see that hr is the displacement vector due to wingdeformations, while h differs from hr with the position of the bodyreference frame pB.

With the elastic and rigid-body degrees of freedom defined,the complete independent set of variables of the strain-basedformulation is

q��"b

��� "pB�B

�_q�

�_"�

��(

_"vB!B

)

�q���"_�

��(

�"_vB_!B

)(4)

The derivatives and variations of dependent variables h and hr arerelated with those of the independent ones as

�h� Jh"�"� Jhb�b �hr� Jh"�" dh� Jh" d"� Jhb db

dhr� Jh" d" _h� Jh" _"� Jhb _b� Jh" _"� Jhb� _hr� Jh" _"�h� Jh" �"� _Jh" _"� Jhb _�� _Jhb� �hr� Jh" �"� _Jh" _" (5)

where

Jh" �@h

@"Jhb �

@h

@b(6)

are transformation Jacobians obtained from kinematics [4,12,13].

B. Elastic Equations of Motion

The elastic equations of motion are derived by following theprinciple of virtual work. The virtual work of an elastic wing consistsof the contributions of inertia forces, internal strains and strain rates,and external loads. The contribution of each virtual work is derivedseparately and then summed at the end to represent the total virtualwork of the complete vehicle.

1. Inertias

Thevirtualwork due to the inertia of the elasticmembers is derivedstarting from the position of an arbitrary point awithin the airfoil. Asshown in Fig. 2, the position is given as

p a � pB � pw � xwx � ywy � zwz (7)

Gy

Gz

Bx

By

Bz

vB

B

Gx

PB

O

Fig. 1 Global and body reference frames.

Gy

Gz

Gx

wy(0,t)

By

Bz

Bx

PB

wx(0,t)

wz(0,t)

a(s,t)

wy(s,t) w

x(s,t)

wz(s,t)

Pa

Pw

by(s

1,t) b

x(s

1,t)

bz(s

1,t)

Undeformed shape

Deformed shapeO

Fig. 2 Flexible lifting-surface frames and the body frame (for flight

dynamics of the vehicle).

SU AND CESNIK 1541

where �x; y; z� is the position of the point within the local beam framew. The infinitesimal virtual work applied on a unit volume is

�Wa � �pa � ��aa dA ds� (8)

where

a a �d2padt2

(9)

The virtual work done by the inertia force along the beam coordinates can be obtained by integrating Eq. (8) over each cross section,which yields

�W int�s� � ��hT�s�

0BBBBB@M�s�

8>>>>><>>>>>:

�pw�s��wx�s��wy�s��wz�s�

9>>>>>=>>>>>;�M�s�

I ~pTw�s�0 ~wTx �s�0 ~wTy �s�0 ~wTz �s�

266664

377775 _�

�M�s�

~!B 0 0 0

0 ~!B 0 0

0 0 ~!B 0

0 0 0 ~!B

266664

377775

I ~pTw�s�0 ~wTx �s�0 ~wTy �s�0 ~wTz �s�

266664

377775�

� 2M�s�

0 _~pTw�s�

0 _~wTx �s�

0 _~wTy �s�

0 _~wTz �s�

2666664

3777775�1CCCCCA (10)

with

M�s��ZA�s�

1 x y zx x2 xy xzy yx y2 yzz zx zy z2

2664

3775dA

�

m mrx mry mrzmrx

12�Iyy� Izz� Ixx� Ixy Ixz

mry Iyx12�Izz� Ixx� Iyy� Iyz

mrz Izx Izy12�Ixx� Iyy� Izz�

2664

3775

(11)

M�s� is the cross-sectional inertial matrix, A�s� is the cross-sectionalarea, m is the mass per unit span at each cross section, �rx; ry; rz� isthe position of the cross-sectional mass center in thew frame, and Iijare the cross-sectional mass moments of inertia about the referenceaxis (shear center in the beamcross section). Note that the operator �~��is defined for a 3 by 1 vector � as

� ~�� 0 ��3 �2�3 0 ��1��2 �1 0

24

35 (12)

In Eq. (10), f �pw�s�T �wx�s�T �wy�s�T �wz�s�T gT is the secondtime derivative of hr�s� given in Eq. (3). It can be written in terms ofthe second time derivative of the independent variables using Eq. (5).In addition, the following relations are defined:

Jhb I ~pTw�s�0 ~wTx �s�0 ~wTy �s�0 ~wTz �s�

2664

3775 _Jhb

0 _~pTw�s�

0 _~wTx �s�

0 _~wTy �s�

0 _~wTz �s�

2664

3775

Hhb

~!B 0 0 0

0 ~!B 0 0

0 0 ~!B 0

0 0 0 ~!B

2664

3775

I ~pTw�s�0 ~wTx �s�0 ~wTy �s�0 ~wTz �s�

2664

3775 (13)

With the above definitions, Eq. (10) can be simplified to

�W int�s�

� �f �"T�s� �bT g

JTh"M�s�Jh" JTh"M�s�JhbJThbM�s�Jh" JThbM�s�Jhb

" #��"�s�_�

�

�JTh"M�s� _Jh" 0

JThbM�s� _Jh" 0

" #�_"�s��

��

0 JTh"M�s�Hhb

0 JThbM�s�Hhb

" #�_"�s��

�

�0 2JTh"M�s� _Jhb0 2JThbM�s� _Jhb

" #�_"�s��

�!(14)

2. Internal Strain and Strain Rate

The virtual work due to the internal strain is

�Wint�s� � ��"�s�Tk�s��"�s� � "0�s�� (15)

where k�s� is the cross-sectional stiffness matrix, and "0�s� is theinitial strain upon the beam initialization.

Internal damping is added to the formulation to better model theactual behavior of the structure. A stiffness-proportional damping isused in the current formulation, given by

c�s� � �k�s� (16)

Thus, the virtual work due to the strain rate is

�W int�s� � ��"�s�Tc�s� _"�s� (17)

3. Internal Virtual Work on Element

Toobtain the total internal virtualwork on each element, one needsto summate Eqs. (14), (15), and (17) and then to integrate thesummation over the length of the element. In practice, the integrationis performed numerically.

As mentioned before, a three-node element is used in the currentimplementation. It is assumed that the strain over each element isconstant. Some of the properties, such as inertias and displacements,are assumed to vary linearly between the three nodes of each element.However, the cross-sectional stiffness k�s� and damping c�s� areevaluated at the middle of each element and are assumed to beconstant over the length of the element. With these assumptions, theinternal virtual work of an element can be written as

�W inte ��f �"Te �bT g

JTh"MeJh" JTh"MeJhb

JThbMeJh" JThbMeJhb

" #��"e

_�

�

�JTh"Me

_Jh" 0

JThbMe_Jh" 0

" #�_"e

�

��

0 JTh"MeHhb

0 JThbMeHhb

" #�_"e

�

�

�0 2JTh"Me

_Jhb

0 2JThbMe_Jhb

" #�_"e

�

��

Ce 0

0 0

" #�_"e

�

�

�Ke 0

0 0

" #�"e

b

���Ke"

0e

0

�!(18)

where

Ke � k�s Ce � c�s

Me � 12�s

14M1 � 1

12M2

112M1 � 1

12M2 0

112M1 � 1

12M2

112M1 � 1

2M2 � 1

12M3

112M2 � 1

12M3

0 112M2 � 1

12M3

112M2 � 1

4M3

24

35

(19)

"e in Eq. (18) is the element strain,�s is the initial element length,Keis the element stiffness matrix,Ce is the element dampingmatrix,Me

is the element inertia matrix, and Mi are the cross-sectional inertiamatrices at each node of an element.

1542 SU AND CESNIK

4. External Virtual Work

In general, the external virtual work applied on a differentialvolume can be written as

�Wext �ZV

�u�x; y; z� � f�x; y; z� dV (20)

where f�x; y; z� represents generalized forces acting on thedifferential volume, which may include gravity forces, externaldistributed forces and moments, external point forces and moments,etc., and �u�x; y; z� is the corresponding virtual displacement. Whenbeam cross-sectional properties are known, the integration ofEq. (20) over the volume is simplified to the integration along thebeam coordinate s. The detailed derivation of the external virtualwork is presented in [12].

5. Elastic Equations of Motion

The total virtual work on the system is obtained by summing up allelements’ internal and external work:

�W �X��W int

e � �Wexte �

� f �"T �bT g �

JTh"MJh" JTh"MJhb

JThbMJh" JThbMJhb

" #��"

_�

�

�JTh"M _Jh" 0

JThbM _Jh" 0

" #�_"

�

��

0 JTh"MHhb

0 JThbMHhb

" #�_"

�

�

�0 2JTh"M _Jhb

0 2JThbM _Jhb

" #�_"

�

��

C 0

0 0

" #�_"

�

��

K 0

0 0

" #�"

b

�

��K"0

0

��

JTh"

JThb

" #Ng�

JTp"

JTpb

" #BFF

dist �JT�"

JT�b

" #BMM

dist

�JTp"

JTpb

" #Fpt �

JT�"

JT�b

" #Mpt

!(21)

whereN,BF, andBM are the influence matrices for the gravity force,distributed forces, and distributed moments, respectively, whichcome from the numerical integration. The equations ofmotion can beobtained by letting the total virtual work be zero. Since the variationf �"T �bT g is arbitrary, this leads to

MFF MFB

MBF MBB

" #��"

_�

��

CFF CFB

CBF CBB

" #�_"

�

��

KFF 0

0 0

" #�"

b

�

��RF

RB

�(22)

where the generalized inertia, damping, and stiffness matrices are

MFF�"� � JTh"MJh" MFB�"� � JTh"MJhbMBF�"� � JThbMJh" MBB�"� � JThbMJhb

CFF�"; _"; �� � C� JTh"M _Jh"

CFB�"; _"; �� � JTh"MHhb � 2JTh"M _Jhb

CBF�"; _"; �� � JThbM _Jh"

CBB�"; _"; �� � JThbMHhb � 2JThbM _Jhb

KFF � K (23)

and the generalized force vector is

�RF

RB

���KFF"

0

0

��

JTh"

JThb

" #Ng�

JTp"

JTpb

" #BFF

dist

�JT�"

JT�b

" #BMM

dist �JTp"

JTpb

" #Fpt �

JT�"

JT�b

" #Mpt (24)

which involves the effects from initial strains "0, gravity field g,distributed forces Fdist, distributed moments Mdist, point forces Fpt,and point moments Mpt. The aerodynamic forces and moments areconsidered as distributed loads.

C. Kinematics

As discussed in [12], the governing equation, which relates thedependent displacements to the independent strains, is

@h�s�@s�K�s�h�s� (25)

with K�s� being a matrix function of the strains, i.e.,

K �s� �

0 1� "x�s� 0 0

0 0 �z�s� ��y�s�0 ��z�s� 0 �x�s�0 �y�s� ��x�s� 0

2664

3775 (26)

Note that each entry in the above matrix is a 3 by 3 diagonal matrix.With the assumption that each element has a constant strain state, thesolution of Eq. (25) is given by

h�s� � eK��s�s0�h0 � eG�s�h0 (27)

where h0 is the displacement of a fixed or prescribed root node of thebeam (boundary conditions). The nodal displacements can then berecovered from the strain vector by marching Eq. (27) from theprescribed node to the tips of each beam member.

D. Unsteady Aerodynamics

The distributed loads, Fdist andMdist in Eq. (24), are divided intoaerodynamic loads and user-supplied loads. The unsteady aero-dynamic loads used in the current study are based on the 2-D finitestate inflow theory provided in [14]. The theory calculates aero-dynamic loads on a thin-airfoil section undergoing large motions inan incompressible inviscid subsonic flow. The lift, moment, and dragof a thin 2-D airfoil section about its midchord are given by

lmc � 1b2���z� _y _��d ��� � 2 1b _y2

�� _z

_y��1

2b � d

�_�

_y

� �0_y

�� 2 1bc1 _y

2�

mmc � 1b2�� 1

8b2 �� � _y _z�d _y _�� _y�0

�� 2 1b

2c4 _y2�

dmc ��2 1b�_z2 � d2 _�2 � �20 � 2d _z _��2_z�0 � 2d _��0�

� 2 1b

��c1 _y _z � � �dc1 � bg1� _y _� � � c1 _y�0��

1

2bg1 �z�

��1

2bdg1 �

1

4b2g2

����

�(28)

where � is the trailing-edge flap deflection angle, b is the semichord,and d is the distance of the midchord in front of the reference axis.The quantity � _z= _y is the angle of attack that consists of thecontribution from both the pitching angle and the unsteady plungingmotion of the airfoil. (The different velocity components are shownin Fig. 3.) The coefficients ci through gi are based upon the geometryof trailing-edge flaps, and their expressions are provided in [14].

SU AND CESNIK 1543

The inflow velocity �0 is given by

�0 �1

2

XNn�1

bn�n (29)

where �n represents the inflow states determined from

_�� E1�� E2 �z� E3 ��� E4 _� (30)

The coefficient matrices Ei are given in [14], and bn are obtained byleast-squaresmethod [15]. Theoretically,�0 is an infinite summation.However, it can be approximated by letting N be between 4 and 8.Equation (30) can be expressed in terms of the independent systemvariables by using the Jacobians as

_�� F1 �q� F2 _q� F3�� F1

��"_�

�� F2

�_"�

�� F3� (31)

To transfer the loads from the midchord (as defined above) to thewing reference axis, one may use

lra � lmc mra �mmc � dlmc dra � dmc (32)

Furthermore, the loads are rotated to the body reference frame,whichyields

Faero � CBa1(

0

dralra

)Maero � CBa1

(mra

0

0

)(33)

where CBa1 is the transformation matrix from the local aerodynamicframe to the body frame. Note that Prandtl–Glauert compressibilitycorrection is applied to this formulation, and the finite span correc-tions are also included in the force distribution and may come from aCFD solution of the problem or experimental data if available.

E. Coupled Nonlinear Aeroelastic and Flight Dynamic System

Equations of Motion

The coupled nonlinear aeroelastic and flight dynamic systemequations of motion are obtained by augmenting the equations ofrigid-body motion and elastic deformations with the inflow equa-tions, which can be represented as

MFF�"� MFB�"�MBF�"� MBB�"�

" #��"

_�

��

CFF� _"; "; �� CFB� _"; "; ��CBF� _"; "; �� CBB� _"; "; ��

" #�_"

�

�

�KFF 0

0 0

" #�"

b

���RF� �"; _"; "; _�; �; �; ��RB� �"; _"; "; _�; �; �; ��

�

_��� 1

2������ _PB � �CGB��� 0 ��

_�� F1

��"

_�

�� F2

�_"

�

�� F3� (34)

where � is the quaternions describing the orientation of the bodyframe B, PB is the inertial position of the B frame, and CGB is therotation matrix from the body frame to global frame G [4].

F. Linearization of the Nonlinear System Equations of Motion

The coupled nonlinear aeroelastic and flight dynamic systemequations ofmotion are given by Eq. (34).Without loss of generality,the terms associated with the control surface deflection angles in theaerodynamic load formulation are not included from this point on,since the objective of this paper is to study the stability of the trimmedvehicle. The control surface deflections are only used to trim thevehicle before the stability analysis. Therefore, the load vectorsimplifies to contributions from aerodynamic and gravity loads only;i.e.,

�RF� �"; _"; "; _�; �; �; ��RB� �"; _"; "; _�; �; �; ��

���RaeroF � �"; _"; "; _�; �; ��RaeroB � �"; _"; "; _�; �; ��

���RgravF ���RgravB ���

�(35)

and

�RaeroF

RaeroB

�� JTp"

JTpb

� �BFF

aero � JT�"JT�b

� �BMM

aero (36)

where Faero and Maero are the nodal aerodynamic loads. RaeroF and

RaeroB are the flexible and rigid-body components of the generalized

aerodynamic loads, respectively, and RgravF and Rgrav

B are the flexibleand rigid-body components of the generalized gravity force,respectively. The gravity force is transferred from the global frameGto the body frame B. The rotation matrix between the two frames(CGB) is a function of quaternions �, according to Eq. (34).

Linearization is performed about a nonlinear equilibrium state,given by

wz w

y

a0z

a0y

a1z

a1y

lmc

dmc m

mc

By

Bz

O

ra

V

y

z

b bd

Fig. 3 Airfoil coordinate systems and velocity components.

Start

Steady StateSolution

NonlinearDeformation

State-SpaceEquation

Positive RealPart ?

Instability

Yes

No

IncreaseVelocity

AerodynamicJacobians

EigenvalueAnalysis

StructuralJacobians

Trim Module

Re-Trim ? No

YesSubset of

System Matrix

Flutter withConstrainedR. B. Motion

Flutter inFree Flight

Linearization

Flight DynamicStability

Fig. 4 Searching scheme for the stability boundary (R. B. denotes rigid

body).

1544 SU AND CESNIK

�T0 � f �"T0 ; _"T0 ; "T0 ; _�T0 ; �

T0 ; �

T0 ; �

T0 ; P

TB0g (37)

From that, the linearized system equations can be derived as

MFF �"�MFB_�� �CFF � CFF= _"0 _"0 � CFB= _"0�0� _"

� �CFB � CFF=�0 _"0 � CFB=�0�0��� KFF"� RaeroF= �"0

�"

� RaeroF= _"0

_"� RaeroF="0

"� Raero

F= _�0

_�� RaeroF=�0

�� RaeroF=�0

�� RgravF=�0

�

MBF �"�MBB_�� �CBF � CBF= _"0 _"0 � CBB=_"0�0� _"� �CBB

� CBF=�0 _"0 � CBB=�0�0��� RaeroB=�"0

�"� RaeroB=_"0

_"� RaeroB="0

"

� Raero

B= _�0

_�� RaeroB=�0

�� RaeroB=�0

�� RgravB=�0

�

_��� 1

2��� �

1

2���=�0

���0_PB � �CGB 0 ��� � �CGB=�0 �� 0 ��0

_�� F1

��"

_�

�� F2

�_"

�

�� F3� (38)

where ���=z0 denotes @���=@zjz0 , with z representing one of the �0

components.To obtain the state-space form of these equations, the terms on the

right-hand side of Eq. (38) are moved to the left, and the terms withthe same variables are grouped together, yielding

�MFF � RaeroF= �"0� �"� �MFB � Raero

F= _�0� _�� �CFF � CFF= _"0 _"0

� CFB=_"0�0 � RaeroF= _"0� _"� �CFB � CFF=�0 _"0 � CFB=�0�0

� RaeroF=�0��� �KFF � Raero

F="0�"� Raero

F=�0� � Rgrav

F=�0�� 0

�MBF � RaeroB= �"0� �"� �MBB � Raero

B= _�0� _�� �CBF � CBF= _"0 _"0

� CBB=_"0�0 � RaeroB=_"0� _"� �CBB � CBF=�0 _"0 � CBB=�0�0

� RaeroB=�0�� � Raero

B=�0� � Rgrav

B=�0�� 0

_�� 1

2����

1

2���=�0

���0 � 0

_PB � �CGB 0 �� � � �CGB=�0 �� 0 ��0 � 0

_� � F1

��"

_�

�� F2

�_"

�

�� F3�� 0 (39)

According to Eq. (36), the derivatives of the generalized aero-dynamic loads can be expanded, resulting in

@RaeroF

@ �"� JTp"BF

@Faero

@ �"� JT�"BM

@Maero

@ �"

@RaeroF

@ _�� JTp"BF

@Faero

@ _�� JT�"BM

@Maero

@ _�

@RaeroF

@ _"� JTp"BF

@Faero

@ _"� JT�"BM

@Maero

@ _"

@RaeroF

@�� JTp"BF

@Faero

@�� JT�"BM

@Maero

@�

@RaeroF

@"� JTp"BF

@Faero

@"� JT�"BM

@Maero

@"

@RaeroB

@ �"� JTpbBF

@Faero

@ �"� JT�bBM

@Maero

@ �"

@RaeroB

@ _�� JTpbBF

@Faero

@ _�� JT�bBM

@Maero

@ _�

@RaeroB

@ _"� JTpbBF

@Faero

@ _"� JT�bBM

@Maero

@ _"

@RaeroB

@�� JTpbBF

@Faero

@�� JT�bBM

@Maero

@�(40)

Again, one should note that all the derivatives and matrices areevaluated at the equilibrium state�0, and the notation is omitted fromthe equations from now on for simplicity. Using Eq. (40), Eq. (39)can be written as

�MFF �"� �MFB_�� �CFF _"� �CFB�� �KFF"�Raero

F=�0��Rgrav

F=�0��0

�MBF �"� �MBB_�� �CBF _"� �CBB��Raero

B=�0��Rgrav

B=�0��0

_��12����1

2���=�0

���0�0

_PB��CGB 0�����CGB=�0 �� 0��0�0

_��F1

��"_�

��F2

�_"�

��F3��0 (41)

where

Table 1 Properties of the body and wing members of the blended-wing-body model

Body Wing Units

Ref axis location (root/tip) (from leading edge) 64.38%/45.60% chord 45.60%/45.60% chord ——

Center of gravity (root/tip) (from leading edge) 64.38%/45.60% chord 45.60%/45.60% chord ——

Extension stiffness 1:69 108 1:55 108 NOut-of-plane bending stiffness 7:50 105 1:14 104 N �m2

In-plane bending stiffness 3:50 107 1:30 105 N �m2

Torsion stiffness 2:25 106 1:10 104 N �m2

Mass per unit length 50.00 6.20 kg=mFlat bending inertia per unit length 0.70 5:00 10�4 kg �mEdge bending inertia per unit length 22.00 4:63 10�3 kg �mRotational inertia per unit length 4.50 5:08 10�3 kg �m

0.89 m

30o

1.39 m

0.55 m

3.25 m

Body

Fig. 5 Geometry of the blended-wing-body model.

SU AND CESNIK 1545

�MFF �MFF � JTp"BF@Faero

@ �"� JT�"BM

@Maero

@ �"

�MFB �MFB � JTp"BF@Faero

@ _�� JT�"BM

@Maero

@ _�

�CFF � CFF �@CFF@ _"

_"0 �@CFB@ _"

�0 � JTp"BF@Faero

@ _"

� JT�"BM@Maero

@ _"

�CFB � CFB �@CFF@�

_"0 �@CFB@�

�0 � JTp"BF@Faero

@�

� JT�"BM@Maero

@�

�KFF � KFF � JTp"BF@Faero

@"� JT�"BM

@Maero

@"(42)

and

�MBF �MBF � JTpbBF@Faero

@ �"� JT�bBM

@Maero

@ �"

�MBB �MBB � JTpbBF@Faero

@ _�� JT�bBM

@Maero

@ _�

�CBF � CBF �@CBF@ _"

_"0 �@CBB@ _"

�0 � JTpbBF@Faero

@ _"

� JT�bBM@Maero

@ _"

�CBB � CBB �@CBF@�

_"0 �@CBB@�

�0 � JTpbBF@Faero

@�

� JT�bBM@Maero

@�(43)

Finally, Eq. (41) can be put into the state-space form:

f _"T �"T _�T _�

T _PTB _�T gT

�Q�11 Q2f "T _"T �T �T PTB �T gT

� �Af "T _"T �T �T PTB �T gT (44)

or

_x� �Ax (45)

where

xT � f "T _"T �T �T PTB �T g (46)

Q1 �

I 0 0 0 0 0

0 �MFF�MFB 0 0 0

0 �MBF�MBB 0 0 0

0 0 0 I 0 0

0 0 0 0 I 0

0 �F1F �F1B 0 0 I

26666664

37777775

Q2 �

0 I 0 0 0 0

� �KFF � �CFF � �CFB RgravF=�0

0 RaeroF=�0

0 � �CBF � �CBB RgravB=�0

0 RaeroB=�0

0 0 � 12��=�0

�0 � 12�� 0 0

0 0 �CGB 0 � �CGB=�0 0 ��0 0 0

0 F2F F2B 0 0 F3

26666664

37777775

(47)

−0.04 −0.03 −0.02 −0.01 0−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

Fre

quen

cy, H

z

Damping

Altitude: 0 mAltitude: 3048 mAltitude: 6096 mAltitude: 9144 mAltitude: 12192 mAltitude: 15000 m

−25 −20 −15 −10 −5 0−8

−6

−4

−2

0

2

4

6

8

Fre

quen

cy, H

z

Damping

Altitude: 0 mAltitude: 3048 mAltitude: 6096 mAltitude: 9144 mAltitude: 12192 mAltitude: 15000 m

a) Phugoid mode b) Short-period modeFig. 6 Root locus of phugoid and short-periodmodes of the blended-wing-body configuration at different altitudes; speed fromMach 0.2 (square) to 0.7(diamond).

Table 2 Blended-wing-body fundamental

modes and frequencies

No. Mode Frequency, Hz

1 First flatwise bending 3.32 First edgewise bending 10.93 Second flatwise bending 20.44 Third flatwise bending 55.65 Second edgewise bending 69.56 Fourth flatwise bending 82.0

0 1 2 3 4 5 6 7 8115

120

125

130

135

140

145

Root Angle, deg

Flu

tter

Spe

ed, m

/s

0 1 2 3 4 5 6 7 86

6.5

7

7.5

8

8.5

9

Flu

tter

Fre

quen

cy, H

z

Flutter SpeedFlutter Frequency

Fig. 7 Flutter speed and frequency at sea level.

1546 SU AND CESNIK

G. Solution for the Stability Boundary

The nonlinear stability analysis is conducted in an iterativeway, asindicated in Fig. 4. Starting from a predefined flight condition, thesystem is brought to its trimmed nonlinear steady equilibrium stateand linearized about that condition. Eigenvalue analysis of the

resulting system matrix �A in Eq. (44) is performed. The speed isincreased and the process is repeated until eigenvalues with positivereal parts appear, indicating system instability. Onemay use the samesystem matrix for different types of stability analysis, such as flutterof free-flight vehicles, flutter of vehicles with constrained rigid-bodymotions, or the rigid-vehicle flight dynamic stability. To do so, the

appropriate subset of the system matrix must be defined fromEq. (41).

III. Numerical Studies

A baseline blended-wing-body model is proposed for this study.The wing and body properties are modified from the wall-mountedhalf-vehicle wind-tunnel model described in [8]. In addition to thehalf-vehiclemodel, a full vehicle is also considered. This was createdby symmetrically extending the half-vehicle model. Basic physicalparameters of the models are given in Table 1.

A. Geometry

Figure 5 shows the geometry of the blended-wing-body model.Both the body and the wings are modeled as beams coupled withaerodynamics. The dashed–dotted line indicates the location ofthe beam reference axis. The shear center of the beam varies from thebody’s root (64.38% of the chord) to the wing root (45.60% of thechord) and keeps its relative position unchanged along thewing. Oneconcentrated mass of 80 kg is positioned at the center of the model,0.89 m ahead of the reference line. In addition, nine nonstructuralmasses, 2 kg each, are evenly distributed along thewing from the rootto the tip. Each of the wings contains three independent elevons, asindicated in Fig. 5. These elevons occupy 25% of the chord, runningfrom the wing root to 75% of the wing span.

B. System Modes and Frequencies

To assess the vehicle’s flight characteristic, the full aeroelastic/flight dynamic equations of motion are linearized at differenttrimmed flight conditions. Longitudinal flight modes of the vehicleare then evaluated using the corresponding linearized equations.Figure 6 shows the root loci of these longitudinal modes at differentaltitudes and flight speeds. It can be seen that both the phugoid andshort-periodmodes are stable in the evaluated range (0–15,000mandMach 0.2–0.7). The root locus of the phugoid mode has the tendencyto cross the imaginary axis with the increase of Mach number, whileits frequency is reduced. On the contrary, the root locus of the short-period mode extends away from both the real and imaginary axeswhen the speed is increased. At a given speed (say, Mach 0.4, notedby circles in the figure), the damping of the phugoidmode negativelyincreases with altitude, which indicates a larger stability margin.However, this trend for the short-period mode is reversed.

Table 2 lists the first few fundamental modes of the elastic vehicle.Combining that with Fig. 6, one can see that the frequency of the firstflatwise bendingmode is right within the variation range of the short-periodmode,whichmeans a coupling of these twomodes is possible.

0 1 2 3 4 5 6 7 8230

232

234

236

238

240

242

244

246

Root Angle, deg

Flu

tter

Spe

ed, m

/s

0 1 2 3 4 5 6 7 85.6

5.7

5.8

5.9

6

6.1

6.2

6.3

6.4

Flu

tter

Fre

quen

cy, H

zFlutter SpeedFlutter Frequency

Fig. 9 Flutter speed and frequency at 15,000 m altitude.

0 1 2 3 4 5 6 7 8150

155

160

165

170

175

180

Root Angle, deg

Flu

tter

Spe

ed, m

/s

0 1 2 3 4 5 6 7 86

6.3

6.6

6.9

7.2

7.5

7.8

Flu

tter

Fre

quen

cy, H

z

Flutter SpeedFlutter Frequency

Fig. 8 Flutter speed and frequency at 6096 m altitude.

Fig. 10 Flutter mode shapes at sea level (green: zero root angle, orange: 2� root angle).

SU AND CESNIK 1547

C. Flutter Boundary of Cantilevered Half-Vehicle Model

In this analysis, aeroelastic instabilities of the blended-wing-bodyaircraft are to be identified over a range of flight conditions, using theproposed process described in Fig. 4. The body angle of a half-vehicle model is varied from 0 to 8 deg, without elevon deflections.However, the rigid-body degrees of freedom are all constrained. Thissetup simulates the basic test setup in thewind tunnel. The simulationalso considers different altitudes (from sea level to 15,000 m).Results obtained are summarized in Figs. 7–9.

By observing the plots of flutter boundaries (Figs. 7–9), one mayfind different trends at different altitudes. At sea level, the flutterspeed is initially slightly increased when the root angle is increased.When a critical value of the angle (about 0.92�) is reached, the flutterboundary is significantly reduced with the increase of the root angle.The discontinuity in the flutter boundary is resulted from a change offlutter modes, similar to what was described in [2]. When the rootangle is low, the flutter mode is basically an elastic two-degree-of-freedom flutter: coupled flatwise bending and torsion of the wing.However, when the root angle is larger than the critical value, the in-plane bending participates in the unstable mode, as shown in Fig. 10,resulting in a three-degree-of-freedom flutter. At the higher altitudes(6096 and 15,000 m), the flutter speed decreases with the increase of

the root angle and the flutter mode is always a three-degree-of-freedom one: coupled flatwise bending, in-plane bending, and tor-sion of thewing, although the contribution of the in-plane bending tothe flutter mode is very small when the root angle is about zero.Withhigher root angles, the increasedwingflatwise bending curvature dueto larger aerodynamic loads facilitates the coupling between the in-plane bending and torsion modes [2], making the in-plane bendingcontribution more significant.

To verify the nonlinear flutter boundary calculation presentedabove, two individual nonlinear time-domain simulations are per-formed. With the altitude of 6096 m and the root angle of 8�. one ofthe simulations has a flow velocity (147 m=s) under the flutter speed(154:39 m=s), while the other does at a slightly higher flow velocity(162 m=s) than the flutter speed. The time histories of the tipdisplacements (normalized with respect to the half-vehicle span) areplotted in Fig. 11. From Fig. 11a, the wing deformation of thepreflutter case is stabilized after some initial oscillations. However,the wing oscillation is self-excited for the postflutter case, as seenfrom Fig. 11b. The amplitude of the wing oscillation is increased,until it reaches a limit cycle oscillation.

D. Flutter Boundary of Full Vehicle in Flight

As it has been discussed, the wings of this vehicle are flexible insuch a way that their elastic modes are coupled with the rigid-bodymodes of the whole vehicle. Therefore, the evaluation of the stabilityof the full vehicle should include the rigid-body degrees of freedom,which gives the stability boundary in free flight.

As an example, the full-vehicle stability is evaluated at 6096 maltitude for the level flight condition, i.e., the vehicle is trimmed forevery flight speed at which the stability is evaluated. To assess thecoupling between the wing elastic deformation and the rigid-bodymotion, multiple constraints to the rigid-body degrees of freedom are

0 1 2 3 4 515.18

15.2

15.22

15.24

15.26

15.28

Nor

mal

ized

Win

g T

ip D

ispl

acem

ent,

%

Time, s0 1 2 3 4 5

0

5

10

15

20

25

30

Nor

mal

ized

Win

g T

ip D

ispl

acem

ent,

%

Time, s

a) Speed: 147 m/s b) Speed: 162 m/s

Fig. 11 Normalized wing tip displacements at 6096 m altitude and 8� root angle.

Table 3 Flutter boundaries for different rigid-body

motion constraints

Case Constraint Speed, m=s Frequency, Hz

1 Fully constrained 172.52 7.302 Free plunging only 164.17 7.073 Free pitching/plunging 123.17 3.324 Free flight 123.20 3.32

−50 −40 −30 −20 −10 0 10 20−16

−12

−8

−4

0

4

8

12

16

Damping

Fre

quen

cy, H

z

−50 −40 −30 −20 −10 0 10 20 30 40 50−20

−15

−10

−5

0

5

10

15

20

Damping

Fre

quen

cy, H

z

a) Fully-constrained b) Free-flight

Fig. 12 Aeroelastic root locus of the blended-wing-body configuration at 6096 m altitude; speed from 100 m=s (triangle) to 200 m=s (square).

1548 SU AND CESNIK

used in the calculation of theflutter boundary. First, thevehicle has allrigid-body motions constrained (case 1). Then, the plunging motionis set free (case 2), followed by another case with both plunging andpitching set free (case 3). Finally, all rigid-body degrees of freedomare free and allowed to couple with the wing deformations (case 4).

The flutter speed and frequency corresponding to each of the fourdifferent cases are listed in Table 3. These results are obtained usingthe proposed process described in Fig. 4. The root loci of cases 1 and4 are plotted in Fig. 12. Figure 13 illustrates the flutter mode shape ofcase 4, with the vehicle in wire frame as the reference shape andposition. From this figure, onemay find that the rigid-bodymode is acoupled pitching/plunging compared to the reference, while thewing

elastic mode consists of both flatwise bending and twist. The swept-wing platform favors the coupling between the pitching/plunging ofthe rigid body and the bending/twist of the wings. Since the rigid-body components of the unstable mode for case 4 consist only of thepitching and plunging motions, the flutter speed of this case is nearlyidentical to case 3, with only pitching and plunging degrees offreedom. For the casewhen only plunging is free, there is no couplingbetween wing deformation and the pitching of the body, whichresults in a weak coupling between the wing elastic deformation andthe free rigid-body motion. That flutter boundary is the closest to thecase with the rigid-body degrees of freedom fully constrained.

In the time domain, two individual nonlinear simulations arecarried out for verification purposes, one of which simulates the levelflight of the vehicle with a flight velocity lower than the flutter speed(120 m=s), while the other flies with a slightly higher velocity(125 m=s) than the flutter speed. A sinusoidal deflection of all thethree elevons is used as a small perturbation to the system equi-librium. The elevon input is given by

��(0 t < 0:5 s

�0:1 sin 2 �t � 0:5� deg 0:5 s � t � 1:5 s

0 t > 1:5 s

(48)

The time histories of the tip displacement and body-frame pitchingangle are plotted in Figs. 14 and 15. For the preflutter case (seeFig. 14), the responses converge after initial oscillations. However,the responses of the postflutter case diverge, exhibiting instability, asshown in Fig. 15.As one can see fromFig. 15b, the pitchingmotion isnot stable, which is correctly predicted by the eigenvalue analysis.One more observation from the time-domain simulation is that the

0 2 4 6 8 10 120.98

1

1.02

1.04

1.06

1.08

1.1

Nor

mal

ized

Win

g T

ip D

ispl

acem

ent,

%

Time, s0 2 4 6 8 10 12

1.85

1.9

1.95

2

2.05

2.1

Bod

y F

ram

e P

itchi

ng A

ngle

, deg

Time, s

a) Wing tip displacement b) Body frame pitching angle

Fig. 14 Normalized wing tip displacement and body-frame pitching angle of the preflutter case for the blended-wing-body vehicle at 6096 m altitude;

speed 120 m=s.

Fig. 13 Flutter mode shape of the free-flight case at 6096 m altitude.

0 2 4 6 8 10 120

0.5

1

1.5

2

Nor

mal

ized

Win

g T

ip D

ispl

acem

ent,

%

Time, s0 2 4 6 8 10 12

0.5

1

1.5

2

2.5

3

Bod

y F

ram

e P

itchi

ng A

ngle

, deg

Time, s

a) Wing tip displacement b) Body frame pitching angle

Fig. 15 Normalized wing tip displacement and body-frame pitching angle of the postflutter case for the blended-wing-body vehicle at 6096 m altitude;

speed 125 m=s.

SU AND CESNIK 1549

frequency of the unstable oscillation is approximately 3.33 Hz,which agrees with the eigenvalue analysis within 0.3%.

E. Response of Full Vehicle to Gust Perturbations

To better understand the nonlinear nature of the aeroelasticstability, the vehicle is simulated under gust perturbations. The simu-lation is conducted at 6096 m with the flight speed of 110:64 m=s(Mach 0.35). A discrete gust model is used, whose spatial distri-bution is governed by a 1-cosine function, with the maximum gustspeed of 15:24 m=s (50 ft=s) and the gust region along the flight pathextending for 13.72 m, which is approximately 25 times the wingchord length. The gust is symmetrically applied to the vehicle. Note

that stall effects are considered through a simplified stall model [5]when the local angle of attack at an airfoil reaches the critical stallangle.

There are three types of simulation considered: 1) nonlinearsimulation with follower aerodynamic loads (sim 1), 2) linearizedsimulationwith follower aerodynamic loads (sim2), and3) linearizedsimulation with fixed-direction aerodynamic loads (sim 3). Thesimulation results are compared in Figs. 16–20. It can be seen that theapplied gust perturbation does not excite any instability of thevehicle. The phugoid mode can be clearly observed from the plots oflongitudinal velocity (Fig. 16) and altitude (Fig. 19). This mode isstable and slightly damped. One may measure the period of thephugoid mode, and only the period from nonlinear simulation withfollower aerodynamic loads (sim 1) agrees with what has beenpredicted from the eigenvalue analysis (see Fig. 6). The other twosimulations give results that are slightly off, with significant differ-ence in oscillation amplitudes. For the linearized simulation withfixed-direction aerodynamic loads (the dashed–dotted line inFig. 19), the vehicle altitude is not recovered after one cycle.Therefore, it does not accurately simulate the phugoid motion.

The high-frequency response at the initial stages is associated withthe short-period mode excited by the gust perturbation. The ampli-tude of this oscillation decays quickly with a high negative damping.Onemay find that the short-period mode frequency (Figs. 17 and 18)is quite close to that of the wing elastic oscillation (Fig. 20), whichindicates coupling between the two motions.

To study the effects of the gust on the vehicle response atsubcritical conditions, the flight speed of the vehicle is increased to122 m=s, which is approximately 1% lower than the flutter bound-ary. The same nominal gust is used for the nonlinear simulation withfollower aerodynamic loading. The results are compared withanother simulation, in which the vehicle flies in calm air (seeFigs. 21–26). One may find that the flight in calm air is stable, as the

0 10 20 30 40 50 60 70 80 90104

106

108

110

112

114

116

Long

itudi

nal V

eloc

ity, m

/s

Time, s

Sim 1Sim 2Sim 3

0 1 2 3 4 5 6 7 8 9 10104

105

106

107

108

109

110

111

Long

itudi

nal V

eloc

ity, m

/s

Time, s

Sim 1Sim 2Sim 3

a) Time range: 0 - 90 s b) Time range: 0 - 10 s

Fig. 16 B-frame longitudinal velocity with gust perturbation for different simulations at 6096 m altitude.

0 1 2 3 4 5 6 7 8 9 10−4

−3

−2

−1

0

1

2

3

Pitc

hing

Rat

e, r

ad/s

Time, s

Sim 1Sim 2Sim 3

Fig. 17 B-frame pitching rate with gust perturbation for different

simulations at 6096 m altitude.

0 10 20 30 40 50 60 70 80 90−8

−6

−4

−2

0

2

4

6

8

10

12

Pitc

hing

Ang

le, d

eg

Time, s

Sim 1Sim 2Sim 3

0 1 2 3 4 5 6 7 8 9 10−8

−6

−4

−2

0

2

4

6

8

10

12

Pitc

hing

Ang

le, d

eg

Time, s

Sim 1Sim 2Sim 3

a) Time range: 0 - 90 s b) Time range: 0 - 10 s

Fig. 18 B-frame pitching angle with gust perturbation for different simulations at 6096 m altitude.

1550 SU AND CESNIK

speed is lower than the flutter boundary. However, the finite gustperturbation brings instability to the whole system even after theperturbation is over. The vehicle sinks due to the gust perturbationand the flight velocity is increased to be greater than the flutterboundary after 15 s, where the short-period mode coupled with thewing elastic bending deformation grows to be unstable. Eventually,this motion develops into a beating oscillation (see Figs. 23, 24, and26).

F. Aeroelastic Tailoring

In the nominal blended-wing-body model, there is no elasticcoupling in the stiffness matrix between the torsion and the out-of-

plane bending of the wing. A tuning parameter is used for thetailoring of the wing stiffness, such that the cross-sectional torsion/out-of-plane bending coupling stiffness k23 is determined by

k23 � ����������������k22 � k33

p � 0; 0:25; 0:50; 0:75 (49)

The change of torsion/out-of-plane bending couplingmay impact thevehicle’s stability boundary. To evaluate the impact, the flutterboundary of the vehicle in free-flight condition (levelflight at 6096maltitude, no constraints in rigid-body degrees of freedom) iscalculated for configurations with different coupling levels. Thevariation of the flutter boundary is plotted in Fig. 27. By looking atthe unstable mode shapes for these configurations, one may find that

0 10 20 30 40 50 60 70 80 905700

5750

5800

5850

5900

5950

6000

6050

6100

Alti

tude

, m

Time, s

Sim 1Sim 2Sim 3

0 0.5 1 1.5 2 2.5 36092.5

6093

6093.5

6094

6094.5

6095

6095.5

6096

6096.5

Alti

tude

, m

Time, s

Sim 1Sim 2Sim 3

a) Time range: 0 - 90 s b) Time range: 0 - 3 s

Fig. 19 Altitude change with gust perturbation for different simulations at 6096 m altitude.

0 1 2 3 4 5 6 7 8 9 10−6

−4

−2

0

2

4

6

8

10

Nor

mal

ized

Win

g T

ip D

ispl

acem

ent,

%

Time, s

Sim 1Sim 2Sim 3

Fig. 20 Normalized wing tip displacement (with respect to the half-vehicle span) with gust perturbation for different simulations at 6096 m

altitude.

0 5 10 15 20 25 30 35 40 45100

105

110

115

120

125

130

135

Long

itudi

nal V

eloc

ity, m

/s

Time, s

Sim with GustSim in Calm Air

Fig. 21 B frame longitudinal velocity with gust perturbation at the

subcritical condition; altitude 6096 m.

0 5 10 15 20 25 30 35 40 45−40

−30

−20

−10

0

10

20

30

40

Ver

tical

Vel

ocity

, m/s

Time, s

Sim with GustSim in Calm Air

Fig. 22 B frame vertical velocity with gust perturbation at the

subcritical condition; altitude 6096 m.

0 5 10 15 20 25 30 35 40 45−6

−4

−2

0

2

4

6

Pitc

hing

Rat

e, r

ad/s

Time, s

Sim with GustSim in Calm Air

Fig. 23 B frame pitching rate with gust perturbation at the subcritical

condition; altitude 6096 m.

SU AND CESNIK 1551

the shift of flutter mode when the tuning parameter is changed frompositive to negative. If the coupling between the torsion and out-of-plane bending stiffness is positive, the flutter modes are the same asthe nominal configuration, as shown in Fig. 13. However, if thecoupling is negative, the flutter modes are switched to the roll motionof the body with complex in-plane/out-of-plane bending and twist ofthe wing, as described in Fig. 28. It is also noticeable that the flutterboundary is more sensitive to the positive coupling coefficient (washin) than the negative one (wash out). When the tuning parameter is�0:75, the flutter speed is reduced to less than half of the nominalconfiguration. Also, no higher flutter speed was found than the onewith equal to zero.

IV. Conclusions

In this paper, the coupled nonlinear flight dynamic and aeroelasticstability characteristics of a blended-wing-body aircraft weredescribed by a set of nonlinear equations. The geometrically non-linear structural analysis was performed using a strain-basedformulation, which was able to capture the large deformations inslender structures effectively. Finite state unsteady subsonic aero-dynamic loads were coupled with the structural formulation, whichcompleted the aeroelastic formulation. Nonlinear equations ofmotion for a frame attached to the vehicle (not necessarily at its c.g.point) were used to complete the coupled flight dynamic/aeroelasticformulation. With these equations, fully nonlinear time-marchinganalyses were performed. The nonlinear equations were alsolinearized about a given nonlinear state and put into the state-spaceform, such that stability boundary andflight dynamicmodes could beassessed.

Longitudinal flight dynamic modes were studied for a samplemodel vehicle with the Mach number ranging from 0.2 to 0.7 atdifferent altitudes. In the studied range, the phugoid and short-periodmodes of this vehiclewere both stable. However, the frequency of theelastic wing-body bendingmodewas low enough that it could couplewith the short-period mode of the rigid body. That means the rigid-body motion could excite the instability of the wing-body elasticsystem,which introduces a special type of dynamic instability: body-freedom flutter.

Enlightened by the above analyses, the flutter boundary of thewhole vehicle was studied with different rigid-body constraints. Forthe particular vehicle studied here, the body-freedom flutter was theresult of the coupled short-period mode with the wing elasticbending/torsion mode. As the wing oscillations were coupled withthe rigid-body motion of the entire vehicle, the flutter boundary infree-flight condition differed from that of a constrained vehicle. Boththe flutter boundary and the unstable modes obtained in free flight

0 5 10 15 20 25 30 35 40 45−25

−20

−15

−10

−5

0

5

10

15

Pitc

hing

Ang

le, d

eg

Time, s

Sim with GustSim in Calm Air

Fig. 24 B frame pitching angle with gust perturbation at the subcriticalcondition; altitude 6096 m.

0 5 10 15 20 25 30 35 40 455300

5400

5500

5600

5700

5800

5900

6000

6100

6200

Alti

tude

, m

Time, s

Sim with GustSim in Calm Air

Fig. 25 Altitude change with gust perturbation at the subcritical

condition; altitude 6096 m.

0 5 10 15 20 25 30 35 40 45−15

−10

−5

0

5

10

15

20

Nor

mal

ized

Win

g T

ip D

ispl

acem

ent,

%

Time, s

Sim with GustSim in Calm Air

Fig. 26 Normalized wing tip displacement (with respect to the half-vehicle span) with gust perturbation at the subcritical condition; altitude

6096 m.

−1 −0.75 −0.5 −0.25 0 0.25 0.5 0.75 150

60

70

80

90

100

110

120

130

Torsion/Out−of−plane Bending Coupling Parameter

Flu

tter

Spe

ed, m

/s

−1 −0.75 −0.5 −0.25 0 0.25 0.5 0.75 10

2

4

6

8

10

12

14

16

Flu

tter

Freq

uenc

y, H

z

Flutter SpeedFlutter Frequency

Fig. 27 Change of flutter speed and frequency in free flight with

different torsion/out-of-plane bending elastic coupling effects.

Fig. 28 Antisymmetric mode shape with �0:75 coupling parameter.

1552 SU AND CESNIK

were different from the one in wind-tunnel tests when the rigid-bodymotions were fully constrained. This indicated that for the vehicleswith very flexiblewingmembers, the flutter analysis should considerthe whole vehicle’s degrees of freedom, including the rigid-bodymodes. The traditional method of performing wind-tunnel tests in aconstrained model to evaluate its flutter boundary may not beappropriate.

To study the gust response, a spatially distributed discrete gustmodel was seamlessly incorporated into the time simulation scheme.The gust perturbation was symmetrically applied to the vehicle.Three types of simulations were performed: 1) nonlinear simulationwith follower aerodynamic loads, 2) linearized simulation withfollower aerodynamic loads, and 3) linearized simulation with fixed-direction aerodynamic loads. Only the nonlinear simulation withfollower aerodynamic loads could accurately simulate the phugoidmode predicted in the eigenvalue analysis. Moreover, the phugoidmode was not observed from the simulation with fixed-directionaerodynamic loads. This indicated the importance of the nonlinearanalysis to the flight dynamic responses, especially the followernature of the aerodynamics, even though the overall elastic defor-mation of the wings was small. The finite gust perturbation couldbring instability to the vehicle at subcritical flight conditions. As thesimulation performed in this studywas only open-loop, future effortscan bemade to design an appropriate control schemes to alleviate theinstability under finite gust perturbations.

Finally, the nominal configuration was elastically tailored withdifferent levels of bending/twist elastic coupling stiffness of thewings. For the example configuration, both the positive and negativecouplings resulted in reduction of the flutter boundary. However, thenegative coupling introduced a different antisymmetric flutter mode.

Acknowledgments

This work was supported in part by the U.S. Air Force ResearchLaboratory (AFRL) under theMichigan/AFRLCollaborativeCenterin Aeronautical Sciences (MACCAS), with Gregory Brooks as thetask Technical Monitor.

References

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