-
soulame, Am, Te
Keywords:AeroelasticityComposite wingFEMNonlinear utterONERA
aerodynamic
model of a composite wing was considered and improved to
simulate large deformation behavior. Also,
rtant c
the beam-axis coordinate. 1D structural elements (beams) are
sim-pler and computationally more efcient than 2D (plate/shell)
and3D (solid) elements. This feature makes beam theories still
veryattractive for the static, dynamic and aeroelastic analysis of
struc-tures. The famous classical beam models have been
constructedbased on the EulerBernoulli or Timoshenko theory. But,
they have
odges and Dowellnonlinearities forrmations.aindisplac
relations are neglected using an ordering scheme. Rosen andmann
[3] presented more accurate system of equations thanobtained by
HodgesDowell. They considered additional higder nonlinear terms and
therefore their resultswere in better agree-ment with the
experimental results presented by Dowell et al. [4].Crespo and
Glynn [5] applied the extended form of Hamiltons prin-ciple to
develop a set of mathematically consistent nonlinear equa-tions
based on 1D beam model. Cubic terms were not shownexplicitly in
their equations but these equations fully included thecontributions
of nonlinear curvature and inertia terms. They used
Corresponding author. Tel.: +98 3113660011.E-mail addresses:
[email protected] (R. Koohi), [email protected]
(H. Shahverdi), [email protected] (H. Haddadpour).
Composite Structures 113 (2014) 118126
Contents lists availab
Composite S
sevstructural nonlinearity specically for wings.In the
structural dynamic and aeroelastic analyses of wings, for
sake of simplicity, 1D beammodels are always used. In a 1D
model,the 3D problem is reduced to a set of variables that only
depends on
structural models based on the 1D beammodel. H[2] provided
nonlinear equations with quadraticisotropic rotor blades undergoing
moderate defostudy the higher order terms associated with
strhttp://dx.doi.org/10.1016/j.compstruct.2014.03.0120263-8223/
2014 Elsevier Ltd. All rights reserved.In thisementFried-those
her or-design process that may be lead to a catastrophic
failure. Manyaccidents due to this phenomenon have been reported
yet [1].Nowadays, the demands for high maneuverability,
performanceand speed air vehicles as well as agility are increasing
with appli-cation of composite materials in aerospace industries.
To meet theabove characteristics, lightweight and therefore more
exiblestructures have been developed. This will result in
signicant
large deection behavior causes geometrical nonlinearity and
usingnonlinear models are inevitable. For example, in a high aspect
ratiowing with long span, the stiffness and natural frequencies of
thewing may be changed due to large deections. Hence in the
aero-elastic analysis of high aspect ratio wings, nonlinear models
mustbe used to predict the instability boundaries of the wing
precisely.
Several attempts have been made to develop accurate
nonlinearVABS
1. Introduction
Aeroelastic instability is an impoin aerodynamic aspect of view,
a semi-experimental unsteady aerodynamic (ONERA dynamic stall)model
has been incorporated to construct the aeroelastic model. To set up
a utter determination toolbased on the eigenvalue analysis, Finite
Element Method (FEM) has been implemented to discretizethe
aeroelastic equations. Also, a nite element cross-sectional
analysis code VABS (Variational Asymp-totical Beam Sectional
Analysis) has been applied to determine composite cross-sectional
propertiesacross the wing span. Because of the existence of
nonlinear terms in the aeroelastic equations, due tothe large
deformation behavior, the perturbed dynamic equations have been
established about the non-linear static equilibrium to capture the
utter boundaries. The obtained results are in good agreementwith
the available experimental data. It is found that the present
aeroelastic model is appropriate foranalysis of composite wings
with arbitrary cross-sections.
2014 Elsevier Ltd. All rights reserved.
oncept in an air vehicle
some restrictions such as warping in and out of plane
deformations.Also, these models are implemented for investigation
of linearstructural dynamic behaviors. It must be noted that
moderate orArticle history:Available online 17 March 2014
The aim of this paper is to develop a modied 1D structural
dynamics model for aeroelastic analysis of acomposite wing under
large deformations. To attain this goal, an accurate available
mechanical beamNonlinear aeroelastic analysis of a compomethod
Reza Koohi a,, Hossein Shahverdi b, Hassan HaddadpaDepartment of
Mechanical and Aerospace Engineering, Science and Research Branch,
IsbDepartment of Aerospace Engineering, Center of Excellence in
Computational AerospaccAerospace Engineering Dept., Sharif
University of Tech., Azadi Ave., PO Box 11155-8639
a r t i c l e i n f o a b s t r a c t
journal homepage: www.elite wing by nite element
r c
ic Azad University, Tehran, Iranirkabir University of
Technology, 424 Hafez Avenue, Tehran 15875-4413, Iran
hran, Iran
le at ScienceDirect
tructures
ier .com/locate /compstruct
-
tructhese equations for nonlinear analysis of a cantilevered
beam [6].Pai and Nayfeh [7] extended these equations to the case of
compos-ite beams. Hodges [8] developed a nonlinear beam model in
whichthe assumption of moderate rotations is removed. This model
issubsequently used as the theoretical basis of the beamelement
usedin the computer program GRASP. Hodges [9,10] presented a
generalbeam theory based on a nonlinear intrinsic formulation for
thedynamics of initially curved and twisted beams in a moving
frame.This beam model is valid for both isotropic and orthotropic
materi-als. Librescu [11] presented a general 1D composite beam
modelthat includes the non-classical effects such as transverse
shearand warping constraint for a thin-walled composite section.
Shiet al. [12] presented a third-order shear deformable
compositebeam element. Also, Tauk et al. [13] developed a linear
compositebeam element with arbitrary cross-section. Lee [14]
presented ananalytical model for exural analysis of I-shaped
laminated com-posite beams based on the rst-order shear deformable
theory.
In recent years, many studies have been performed in
aeroelas-tic analysis of composite wings or blades with the
increasing use ofcomposite materials in aerospace industries. For
example, Cesniket al. [15] investigated aeroelastic instability of
a composite wingbased on geometrically-exact nonlinear structural
equations [9].It must be noted that, their model could not
considered warpingand shear deformations. Xie et al. [16,17]
investigated aeroelasticanalysis of a HALE composite wing with
large deections usingNASTRAN FEM. They perturbed the aeroelastic
equations aboutthe nonlinear static deections and showed the
necessity of usingnonlinear aeroelastic analysis instead of linear
analysis. Haddad-pour et al. [18] and Qin and Librescu [19] studied
the aeroelasticinstability of a single-cell composite box beam
using Librescusthin-walled composite beam model [11]. Flutter
analysis of com-posite wings by the 1D Carrera unied formulation
was conductedby Petrolo [20]. Zhao and Hu [21] studied aeroelastic
analysis ofcomposite wings as thin-walled closed-cross-section
beams. Yuanand Friedmann [22] performed nonlinear aeroelastic
analysis of acomposite rotor blade undergoing moderate deection
usingFEM. Their beam model is similar to Rosen and Friedmann [3]
withsome modications to consider the effects of shear and
warping.The simplication of moderate deection is justied for
compositehelicopter rotor blade analysis since the rotor blades are
designedfor low stress and high-cycle fatigue point of view [22].
Also insome studies, aeroelastic behavior of a wing has been
simulatedby using a composite plate model [23,24].
While composite materials are considered in aeroelastic
analy-ses based on 1D beam models, the computation of the
cross-sec-tional properties are vital and also so complex. One way
toovercome this problem is to utilize the Variational
AsymptoticalBeam Sectional Analysis Code (VABS). This code has been
devel-oped based on the 2D nite element method. VABS can be
appliedto determine all structural stiffness and inertial
coefcients of thewing cross-section with all details of the
cross-sectional geometryand material properties [25]. Thus, using
this software, one can re-duce the dimension of a 3D composite wing
from a 3D elasticityproblem to 1D continuum beam model. VABS is
based on the Hod-ges equations [26] and is suitable for application
of compositematerials. VABS also has been used to calculate the
cross-sectionalproperties needed as inputs for other rotorcraft
analysis codes [27].Friedmann et al. combined their previous work
[22] with VABS andcalled their model as YF/VABS [28]. YF/VABS model
accounts forarbitrary cross-sectional warping, shear strains,
in-plane stresses,and moderate deections.
Another aspect of establishing a starting point for analysis
ofaeroelastic problem is to select a suitable aerodynamic model.
In
R. Koohi et al. / Composite Sorder to construct a proper
nonlinear aeroelastic system, anappropriate aerodynamic model is
required as well as a nonlinearbeam model. In this regards some
analytical and semi-empiricalmodels have been developed and
utilized for aeroelastic analyses.For instance, Tang and Dowell
[29,30] studied the aeroelastic re-sponse of a high-aspect ratio
wing. In this study, the beam equa-tions developed by HodgesDowell
[2] were used to model thestructural nonlinearity and the ONERA
stall aerodynamic model[31] was used to describe the nonlinear
aerodynamic loading. Patiland Hodges [3234] investigated the
nonlinear aeroelastic behav-ior of a complete aircraft with high
aspect-ratio wings based ongeometrically-exact nonlinear beam
theory [9] and the nite-stateaerodynamic theory of Peters [35]
along with the ONERA dynamicstall model. Shams et al. [36,37]
studied the aeroelastic response ofslender isotropic wings using a
second order [2] and third-order [5]form of the EulerBernoulli beam
model respectively and an un-steady linear aerodynamic model based
on the Wagner function.There are too much works which are
concerning about nonlinearaeroelastic analyses in the two last
decades.
In nonlinear aeroelastic analysis, the aeroelastic system can
besimulated in time domain or in frequency domain. In the time
do-mainmethod, aeroelastic system is marched in time for various
ini-tial conditions and its response is gained in the form of
time-varying curves or phase planes. However, if the air speed
reachesa critical value, instability occurs. In this case, the
trajectories tendto a limit cycle oscillations (LCO). But, in the
frequency domainmethod, the perturbed dynamic equations of the
aeroelastic systemare linearized about their nonlinear static
equilibrium conditions todetermine the stability boundaries through
an eigenvalue analysis.
In the present study, a nite element code is developed
todetermine the nonlinear utter instability of a composite wingwith
arbitrary sections through eigenvalue analysis. In this regard,a
nonlinear 1D beam model is used to simulate wings
structuraldynamics behavior and cross-sectional properties are
determinedby VABS. Structural model is selected based on the
YF/VABS equa-tions that presented by Friedmann et al. [28], of
course with somemodications. It should be noted that the YF/VABS
equations havebeen developed for rotary wings with moderate
deections. How-ever, in this study, these equations have been
modied for case ofxed wings with large deections. To overcome the
large deforma-tion modeling weakness of the YF/VABS, some important
higherorder terms are incorporated into the original model. Also,
the un-steady aerodynamic model states based on the Joness
approxima-tion and ONERA dynamic stall is implemented for
constructing anappropriate aeroelastic tool. Finally, the
aeroelastic analyses forcertain test cases are performed and the
obtained results are com-pared and validated with those available
in the literature.
2. Structural dynamics simulation
To simulate the structural dynamic behavior of a compositewing
by FEM, the wing must be discretized by utilizing severalbeam type
elements along its elastic axis. It is assumed that
thecross-section of the composite wing has a general shape. The
effectof angle of attack and pre-twist are also included in the
wing struc-tural dynamics model. The nonlinear straindisplacement
relationsare developed from a moderate deection theory (small
strains andmoderate rotations) along with some important large
deectionterms. Nonlinear equations of motion for each beam element
arederived based on the Hamiltons principle.
2.1. Coordinate systems
Several coordinate systems are required to describe deforma-tion
of the wing as shown in Figs. 1 and 2. The rst two systems,e^x;
e^y; e^z and e^x; e^g; e^f, respectively, are used to determine
the
tures 113 (2014) 118126 119position and orientation of each beam
element relative to the wingroot in the unreformed conguration. The
vector e^x is aligned withthe beam element elastic axis, and the
vectors e^y and e^z are dened
-
trucFig. 1. Wing coordinate systems and deections.
120 R. Koohi et al. / Composite Sin the cross-section plane of
the beam. The wing pre-twist angleand angle of attack have been
taken into account by h0 as shownin Fig. 1. This angle is dened as
the change in the orientation ofe^g; e^f with respect to e^y; e^z.
The vectors e^g and e^f are assigned par-allel to the modulus
weighted principal axes of the cross-section.The beam element
straindisplacement relations are derived ine^x; e^g; e^f system.
However, e^0x; e^0g; e^0f coordinate system is usedto state the
orientation of the local wing geometry after deforma-tion. The
orientation of e^0x; e^0g; e^0f is obtained by rotating e^x; e^g;
e^fcoordinate system through three Euler angles in the order of hf,
hg,hx about e^f, rotated e^g and rotated e^x, respectively. This
sequencewas chosen to agree the work of previous authors.
2.2. Strain relations
In this study, the nonlinear kinematics of deformation is
basedon the mechanics of curved rods [22]. The kinematical
assumptionsused in [22] are: (1) the deformations of the
cross-section in its ownplane are neglected and (2) the strain
components are small com-pared to unity. But in the present study,
besides of the mentionedassumptions, the axial and warping terms
are also neglected.
The strain components after applying the ordering scheme
become
exx exx v ;xxg cosh0 / f sinh0 / gcxg;x s0cxf fcxf;x s0cxg w;xxg
sinh0 / f cosh0 /
12g2 f2/;x2
cxg cxg f/;x /0cxf cxf g/;x /0 1
Fig. 2. Wing cross-section before and after deections.where v,
w, / are out of plane and in-plane deection and twist atthe elastic
axis, respectively (Figs. 1 and 2) and exx; cxg; cxf can beshown to
be the axial and the transverse shear strains, respectively,at the
elastic axis.
2.3. Constitutive relations
The constitutive relations are dened based on the assumptionsof
the linear elastic orthotropic model and the zero stress
compo-nents within the cross-section (rgg = rff = rgf = 0). Using
theseassumptions the constitutive relations will be
r Qe )rxxrxfrxg
264
375
Q11 Q15 Q16Q15 Q55 Q56Q16 Q56 Q66
264
375
exxcxfcxg
264
375 2
where Q is the reduced beam material stiffness matrix.
2.4. Strain energy
The variational form of the strain energy is dened by
dU Z le0
ZZA
dexxdcxfdcxg
8>:
9>=>;
T Q11 Q15 Q16Q15 Q55 Q56Q16 Q56 Q66
264
375
exxcxfcxg
264
375dgdfdx: 3
The small angle assumption for / yields:
cosh0 / cosh0 / sinh0sinh0 / sinh0 / cosh0
4
Thus, the variation of the left hand side of Eq. (4) is:
dcosh0/d/sinh0/d/sinh0/cosh0dsinh0/ d/cosh0/ d/cosh0/sinh0
5
However, the variation of the right hand side of Eq. (4) is
dcosh0 / sinh0 d/ sinh0;dsinh0 / cosh0 d/ cosh0
6
The main deference between the present study and Ref. [22], is
thatin the present study, Eq. (4) is implemented after taking
variation ofaxial strain, exx, that results in Eq. (5) and keeps
higher order terms,which is important in large deection
computations, but in Ref. [22]these terms did not appear because
Eq. (4) has been used beforetaking variation of exx that yields to
Eq. (6).
Integrating Eq. (3) over the cross-section gives the
modulusweighted section constants, which are presented in Ref [22].
Thesesection constants can be calculated using a separate,
two-dimen-sional linear FEM analysis of an arbitrarily shaped
composite cross-sectionwhich isdecoupled fromthenonlinear,
one-dimensional glo-bal analysis for the beam. However, in this
study, an improved niteelement cross-sectional analysis code (VABS)
[25] is used. Hereshow to use it to provide the required
cross-sectional properties forthebeamanalysis.
TheproperusageofVABSoutputs in thestructuralmodel is explained here
concisely (for more details see Ref. [28]).
From Ref. [26], the VABS strain energy is given by
2UV Z le0
exxcxgcxfjxjgjf
8>>>>>>>>>:
9>>>>>=>>>>>;
T S11 S12 S13 S14 S15 S16S21 S22 S23 S24 S25 S26S31 S32 S33 S34
S35 S36S41 S42 S43 S44 S45 S46S51 S52 S53 S54 S55 S56S61 S62 S63
S64 S65 S66
2666664
3777775
exxcxgcxfjxjgjf
8>>>>>>>>>:
9>>>>>=>>>>>;dx
2Z le0
exxjxjgj
8>:
9>=>;
T
exxA jxB jgC jfDexxjxjgj
8>:
9>=>;dx
tures 113 (2014) 118126f f
7
-
the H, A, B, C, and D matrices and so this hybrid strain energy
will
dT e
0 AqV dVdgdfdx 8
where the velocity vector, V, is obtained by
V _R 9The position vector, R, of a point on the deformed beam is
written inthe following form
R he xe^x v e^y we^z ge^0g fe^0f 10All the terms in the velocity
vector were transformed to thee^x; e^y; e^z coordinate system
by
2.7. Aerodynamic modeling
For a two-dimensional airfoil undergoing sinusoidal motion
inpulsating incompressible ow, Based on the Greenbergs extensionof
Theodorsens theory and using the Jones approximation unstea-dy
aerodynamics theory [35], the unsteady aerodynamic lift (L)
andpitching moment (M) per unit span (Fig. 3) about the elastic
axiscan be expressed as
L 0:5aqAb2 _Uf0 xA 0:5bhn o
aqAbUg0
Uf0h 0:5Uf0 b xA _h Xni1
ciBi
( )
M 0:5aqAb2(xA 0:5b _Uf0 0:5bUg0 Uf0h _h
1=8b2 xA 0:5bh) aqAbxAUg0
Uf0h 0:5Uf0 b xA _h Xni1
ciBi
( )15
Also, the prole drag per unit span is dened as
D CdqAbU2R CdqAbU2g0 U2f0 16where UR is the resultant airfoil
velocity relative to air (Fig. 3), a isthe lift curve slope of the
wing section; b is the semi-chord; qA is air
0
tructures 113 (2014) 118126 121e^0x e^0g e^
0f
T Tde e^x e^y e^z T 11where the transformation matrix Tde is
expressed as
where
s0c v ;x sin h0 w;x cos h0v ;x cos h0 w;x sin h0 13
Integrating Eq. (8) over the cross-section provides mass
weightedsection constants about the shear center and are taken
directly fromthe VABS outputs (for more details see Refs.
[22,28]).
2.6. External work contributions
Using the principle of virtual work, the effects of the
non-con-servative distributed loads are involved. The virtual work
on eachbeam element is dened as
dWe Z le0P du Q d~hdx 14
where P and Q are the distributed aerodynamic force and
moment~
Tde 1 v ;x
v ;x cosh0 / w;x sinh0 / cosh0 /v ;x sinh0 / w;x cosh0 / sinh0 /
s0c cosh
264vectors along the elastic axis; du and dh are the virtual
displacementand rotation vectors, respectively, of a point on the
deformed elasticaxis.be accurate for modeling of composite beams
(for more details seeRef. [28]).
2.5. Kinetic energy
The variation of the kinetic energy for each beam element isZ l
ZZwhere the elastic twist is given by jx, while jg and jf are the
mo-ment strains corresponding to bending. The S, A, B, C, and D
matri-ces are in the output list of VABS.
Using the strain energy relations given in Eq. (3) for the
presentformulation, and the corresponding Eq. (7) for VABS, a
direct com-parison of the cross-sectional constants associated with
both equa-tions can be conducted. In order to couple VABS to the
presentmodel, the cross-sectional parameters in the present strain
energyformulation are replaced with their VABS counterparts.
InsteadVABS accounts for in-plane stresses and out-of-plane warping
in
R. Koohi et al. / Composite Sdensity; h is the pitch angle with
respect to free-stream and xA is thenon-dimensional distance
between the aerodynamic center andelastic axis of the airfoil
cross-section, positive for aerodynamiccenter ahead of the elastic
axis. The velocity vector of a point onthe wing elastic axis
relative to the air is
U VEA VA Ux0 e^x0 Ug0 e^g0 Uf0 e^f0 17
w;xsinh0 /
cosh0 / s0c sinh0
375 12Fig. 3. Components of aerodynamic force acting on the
wing.
-
VEAxVEAyVEAz
8>>>:
9>>=>>;
0_v_w
8>:
9>=>;;
VAxVAyVAz
8>>>:
9>>=>>;
0VF0
8>:
9>=>; and
Ux0
Ug0
Uf0
8>:
9>=>; Tde
VEAx VAxVEAy VAyVEAz VAz
8>>>:
9>>=>>;
18
where VF is the free-stream velocity. Also, Bi is the
aerodynamicstate according to the Jones approximate unsteady
aerodynamicstheory [35] which satises
_Bi biVF=bBi Uf0 b xA _h 19where ci = VFbiai/b.
122 R. Koohi et al. / Composite StrucThe constants ai and bi are
the coefcients used in the quasi-polynomial approximation of the
Wagner function that for the rstand second states are
a1 0:165; a2 0:335; b1 0:0455; b2 0:3The aerodynamics can be
extended to include dynamic stall effectsby complementation with
the ONERA stall model [38]. So that
LT L Lstall; Lstall bqu2CL2MT M Mstall; Mstall 2b2qu2CM2
LstallxA
20
where CL2 and CM2 are additional 2-dimensional lift and
momentcoefcients due to stall which satisfy
t2s CL2 ats _CL2 rCL2 r DCL tse@DCL@a
_a
CM2 DCM21
where ts = b/U.The parameters DCL and DCM are the deviation from
the ex-
tended linear force curve (Fig. 4). Nonlinearity in the ONERA
modelarises from Eq. (21) due to the dependence of its
coefcients(a, r, e) on DCL. These parameters must be identied for a
specialairfoil.
3. Solution methodology
As it mentioned before the nite element method is imple-mented
in this study for solving the system of aeroelastic equa-tions.
Therefore, the wing is divided into several beam elements.The
discretized form of the Hamiltons principle is written asZ t2t1
Xni1
dUi dTi dWeidt 0 22
where n is the total number of beam elements and dU, dT and
dWeare the variation of strain energy, kinetic energy, and virtual
work ofexternal loads, respectively. The Hermitian shape functions
are usedto discretize the space dependence: cubic polynomials for v
and w;Fig. 4. Schematic of DCL.quadratic polynomials for / and the
transverse shears at the elasticaxis.
v fUcgTfVg; w fUcgTfWg; / fUqgTfUgcxg fUqgTfCgg; cxf
fUqgTfCfg
23
Each beam element consists of two end nodes and one internal
nodeat its mid-point, which results in 17 nodal degrees of freedom,
asshown in Fig. 5. Thus,
fVg V1 V1;x V2 V2;x T ; fWg W1 W1;x W2 W2;x T ; fUg /1 /2 /3
T
fCgg cxg1 cxg2 cxg3 ; fCfg cxf1 cxf2 cxf3 24The vector of
element nodal degrees of freedom, q, can be dened as
q fVgT fWgT fUgT fCggT fCfgTh iT 25
Since the variation of the generalized coordinatesdv ; dw; d/;
dcxg; dcxf are arbitrary over the time interval, thereforedq is
also arbitrary; and this results in the nite element equationsof
motion for the ith beam element, which is written as
Mifqg Kifqg fFig 0 26
where [M] is the structural mass matrix, [K] is the stiffness
matrixincluding linear structural stiffness matrix, nonlinear
structuralstiffness matrix and the nonlinear aerodynamic stiffness
matrix thatalso is a function of the aerodynamic states. Also, the
applied aero-dynamic force vector, {F} is a nonlinear function of
deections andtheirs derivatives with respect to time. So, it
includes the aerody-namic damping terms.
After computing and assembling the mass, stiffness matricesand
force vector, the natural frequencies and related mode shapesof the
wing are rstly calculated. Hence, for the free vibration anal-ysis,
the equations of motion for total elements are
Mq KSq 0 27The superscript s denotes the linear structural
matrix used in thefree vibration analysis. After imposing the
boundary conditions, astandard eigenvalue procedure is implemented
to nd the naturalfrequencies and related mode shapes of the wing.
In order to reducethe computational size of the problem, a modal
coordinate transfor-mation is then applied. For the ith element,
the modal coordinatetransformation has the following form
qi Q iy 28The new unknowns of the problem, y, is the vector of
the general-ized modal coordinates and has a size of Nm, where Nm
is the num-ber of modes used to perform the modal coordinate
transformation.The columns of [Qi] correspond to the portions of
the normal modeeigenvectors for the ith element. The assembled
matrices and loadvector of the wing are obtained as follows:
K Xni1
Q iT KiQ i; C Xni1
Q iT CiQ i; 29
M Xni1
Q iT MiQ i; F Xni1
Q iT Fi;
After applying this transformation to Eq. (26) and introducing
theaerodynamic states, a set of nonlinear, coupled, ordinary
differentialequations containing multiple variables is obtained as
follows
tures 113 (2014) 118126f MeqfXg KeqfXg fFeqg 0 30
where
-
and
R. Koohi et al. / Composite Structures 113 (2014) 118126 123Fig.
5. Wing nite element modelMeq My 00 0
; Keq Ky;
_y; y;B; _B 00 0
" #;
1emfFeqg Fy;_y; y;B; _B
FBy; _y;B; _B;C; _C; C
( ) 31
The new unknowns generalized modal coordinate vector is
fXg f y B C gT 32Here, {B} is the Jones approximate unsteady
aerodynamic statesthat has a size of 2n and is dened as
fBg fB11B12B21B22 . . .Bn1Bn2gT
and {C} is the ONERA stall aerodynamic states that has a size of
nand is dened as
Fig. 6. Wing construction and specimen dimensions.
Fig. 7. Meshed cross-sectiofCg fC1L2C2L2 . . .CnL2gT 33
{FB} is the additional force vector for modeling the unsteady
aerody-namic (Eq. (19)) and stall aerodynamic (Eq. (21)) .The
solutions ofEq. (30) can be expressed in the form
X X0 DX 34where X0 denotes steady-state condition and DX denotes
the smallperturbation on it. The static equilibrium position, X0,
is obtainedfrom Eq. (30) by setting _X X 0 and solving the
resulting nonlin-ear algebraic equations using the iterative
NewtonRaphson meth-od. Subsequently, Eq. (30) can be linearized
about the nonlinearstatic equilibrium position X0, to yield:
MX0DX CX0D _X KX0DXH:O:T 0 35
where
M @f=@ XX0 ;0;0 C @f=@ _XX0 ;0;0 K @f=@XX0 ;0;0 36Eq. (35), can
be expressed in the rst order state variable form afterneglecting
the higher order terms by
related nodal degrees of freedom._z Az 37where the state vector
z is dened as
z DXD _X
38
and the system matrix A has the following form
A 0 I
M1K M1C
" #39
n wing as VABS input.
-
The stability of the system can be investigated through the
eigen-value analysis of A. Of course, these eigenvalues are complex
conju-gate pairs
kj fj ixj; j 1; . . . ;Nm 40
The wing is stable if all eigenvalues have the negative real
parts.
4. Results and discussion
Two test cases including [03/90]S and [152/02]S
Graphite/Epoxylaminates with NACA 0012 Styrofoam fairings from Ref.
[39] areconsidered here to validate the present aeroelastic model.
The rel-ative wing characteristics are shown in Figs. 6 and 7 and
Table 1. Toobtain the cross- sectional stiffness and mass
properties of thewing by using the VABS, the wing cross-section is
meshed by 2Delements as shown in Fig. 7. For numerical simulation,
the wingis discretized using 11 spanwise beam elements and the rst
20structural eigenmodes are retained in the aeroelastic analysis(nm
= 20).
4.1. Linear results
In this section the linear aeroelastic behavior of the
presentmodel is validated. Table 2 presents the computed [03/90]S
and[152/02]S wings natural frequencies by neglecting all
nonlineareffects. They are compared against the reported numerical
and
Table 1Material Properties [39].
Parameter Graphite/Epoxy Styrofoam
EL, longitudinal modulus 97.3 Gpa 15 MPaET, transverse modulus
6.3 Gpa 15 MPaGLT, shear modulus 5.3 Gpa 8 MPamLT, Poissons ratio
0.28 0.28q, density 1540 kg/m3 35 kg/m3
t, ply thickness 0.135 mm
Table 2Comparison of the linear modal frequencies (Hz).
Composite layup Mode number Experiment [39] Present analysis
Ref. [39] FEM (NASTRAN)
Value % Error with experiment Value % Error with experiment
[03/90]S 1st Bending 4.0 4.2 5.0 4.3 7.5 4.22nd Bending 27.1
27.3 0.7 27.2 0.4 26.71st Torsion 21.4 20.1 6.1 24.6 15 21.8
[152/02]S 1st Bending 3.6 3.8 5.5 3.9 8.3 3.82nd Bending 27.1
27.4 1.1 28.6 5.5 26.01st Torsion 22.7 21.8 4.0 23.5 3.5 22.0
Table 3Comparison of linear aeroelastic results.
Composite layup Instability speed and frequency Present analysis
Ref. [39] % Error
[03/90]S Flutter speed (m/s) 28.40 28.2 0.5Flutter frequency
(Hz) 11.21 11.86 5.5Divergence speed (m/s) 28.45 28.20 0.9
[152/02]S Flutter speed (m/s) 28.32 26.89 5.3Flutter frequency
(Hz) 12.52 11.64 7.5
124 R. Koohi et al. / Composite Structures 113 (2014) 118126Fig.
8. [03/90]S Wing static deection results.
-
trucR. Koohi et al. / Composite Sexperimental results in Ref.
[39] and the nite element results ob-tained by NASTRAN. The
obtained results, including the naturalfrequencies of the rst two
out of plane bending modes and therst torsion mode, show good
agreement in comparison with avail-able experimental data and FEM
(NASTRAN). Table 3 compares theobtained results for linear
aeroelastic analysis including utterspeed, utter frequency and
divergence speed with those exist inRef. [39]. It should be noted
that the reported results in Ref. [39]were obtained by applying the
HodgesDowell structural modelwith three bending and three torsion
beam mode shapes andimplementation of the unsteady ONERA
aerodynamic model.
4.2. Nonlinear results
To construct the eigenvalue analysis of the considered
nonlin-ear aeroelastic model, in the rst stage a concentrated force
andmoment is applied to the tip of the each aforementioned
composite
Fig. 9. [152/02]S Wing sta
Fig. 10. [03/90]S Wing torsional natural frequencies versus tip
displacement.tures 113 (2014) 118126 125wing which causes
deformation of the wing. The deformation re-sults have been shown
in Figs. 8 and 9.
Fig. 10 reveals that the rst three torsional natural
frequenciesobtained at these deformations for [03/90]S wing. In the
secondstage, a steady angle of attack is added to the root of
[03/90]S and[152/02]S wings which causes deformation of the wings
due toaerodynamic loads. Thus, the aeroelastic system can be
expressedin the perturbed form about this deformation state
(linearizationmethod). The favorite results, including nonlinear
utter speed ofaeroelastic model obtained by the solution of the
perturbed eigen-value problem, are shown in Figs. 11 and 12. In
order to compari-son, the reported results by Dunn and Dugundji
[39] are alsopresented in these gures. It should be noted that Dunn
andDugundji [39] used the HodgesDowell [2] equations that didnot
include transverse shear deformations and some anisotropicmaterial
coupling terms. These gures show good agreementbetween present
analysis and experiment results by [39]. Figs. 9and 10 also show a
better agreement between the present resultand the experimental
data than that for Ref. [39]. It can be notedthat the effects of
transverse shear deformations and composite
tic deection results.
Fig. 11. Variation of utter speed with root angle of attack for
[03/90]S wing.
-
[5] Crespo da Silva M, Glynn C. Nonlinear exuralexuraltorsional
dynamics ofinextensional bea ms-I. Equations of motions. J Struct
Mech 1978;6(4):43748.
[6] Crespo da Silva M, Glynn C. Nonlinear exuralexuraltorsional
dynamics ofinextensional beams-I. Forced motions. J Struct Mech
1978;6(4):44961.
126 R. Koohi et al. / Composite Structures 113 (2014)
118126material coupling terms (except for 0/90 lay-up) in
aeroelasticanalysis play an important role.
5. Concluding remarks
A modied aeroelastic model with the capability of calculatingthe
stability of a composite wing was developed based on theHamiltons
principle and using a nite element formulation. Theobtained results
including the natural frequencies and aeroelasticstability of the
selected wing congurations were presented andcompared with those
available in the literature. This study revealsthat the present
method has better agreement in accordance with
Fig. 12. Variation of utter speed with root angle of attack for
[152/02]S wing.the experimental data.The following remarks are also
obtained:
Incorporating Jones approximate unsteady aerodynamic alongwith
ONERA stall model with the modied YF/VABS structuralmodel leads to
an alternative applicable aeroelastic model forreal composite wing
analysis with arbitrary cross-section.
It is important to consider shear deformation and
compositematerial coupling terms in the structural equations of
motionfor composite wings specically except for 0/90 lay-ups.
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Nonlinear aeroelastic analysis of a composite wing by finite
element method1 Introduction2 Structural dynamics simulation2.1
Coordinate systems2.2 Strain relations2.3 Constitutive relations2.4
Strain energy2.5 Kinetic energy2.6 External work contributions2.7
Aerodynamic modeling
3 Solution methodology4 Results and discussion4.1 Linear
results4.2 Nonlinear results
5 Concluding remarksReferences