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: koohi@iaukhsh.ac.ir (R. Koohi), h_shahverdi@aut.ac.ir

(H. Shahverdi), haddadpour@sharif.edu (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