Aeroelastic Modeling of Morphing Aircraft Wings

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manoeuvrability when compared to that of conventional aircraft wings with small scale, discrete high lifting ... This variable-span morphing wing was applied to a long-range ..... Mach number M, reduced frequency k, aspect ratio of the wing (AR) and geometry. (taper ratio .... Mesh generation is performed with the Open-.
Aeroelastic Modeling of Morphing Aircraft Wings Senthil Murugana,, James H. S. Finchama , M. I. Friswella , D. J. Inmanb a College b Department

of Engineering, Swansea University, Singleton Park, Swansea SA2 8PP, UK of Aerospace Engineering, University of Michigan, Ann Arbor, MI, 48109, USA

Abstract An analytical aeroelastic model to investigate the aeroelastic stability of a span morphing aircraft is derived. Structural and aeroelastic models are derived with the parameteric variations introduced by a span morphing wing. Dynamic aeroelastic stability analysis is performed with thin airfoil theory and CFD. A CFD analysis of span morphing shows the lift increases in a non-linear fashion during the morphing process, and after the morphing is completed. The flutter analysis, based on the CFD results, shows a significant reduction in flutter velocity during morphing due to inertial, elastic and aerodynamic variations, and due to the flow evolution after the morphing period. This study shows that the development of low fidelity aeroelastic models of morphing wings, supplemented with high fidelity analysis, can quantify the phenomena associated with the morphing and also serve as precursor to a computationally expensive, fully coupled high fidelity simulation of flutter due to morphing. Keywords: Morphing Aircraft, Morphing Structures, Telescopic Box Beam, Dynamics, Aeroelasticity, Flutter

1. Introduction Morphing aircraft are the class of aircraft that can reconfigure or morph their wing configuration to an optimal shape at each flight condition [1]. Morphing wing structures, small to large scale, have shown significant improvements in the aircraft performance and manoeuvrability when compared to that of conventional aircraft wings with small scale, discrete high lifting devices [2, 3, 4]. In a morphing aircraft, the wing parameters such as the chord length, span and wing camber are morphed to form the multiple optimal shapes. These large scale structural changes or morphing, in flight, have a significant impact on the dynamics and aeroelastic characteristics of the wing. Therefore, considerable research is focused on modeling the dynamics and aeroelastic characteristics of morphing wings [5, 6]. Zhang et al. studied the dynamics of deploying-and-retreating wings in supersonic airflow [12]. Diaconu et al. [11] studied the dynamic transitions of a morphing bi-stable plate. However, very few studies have focused on incorporating the inertial, stiffness and aerodynamic variations introduced by these morphing structures and aerodynamics in the aeroelastic model and flutter analysis[2]. Email address: [email protected] (Senthil Murugan) Preprint submitted to 4th Aircraft Structural Design Conference, Belfast, UK, 2014

Inman and co-workers developed a static aeroelastic model of a variable-span morphing wing (VSMW)[14]. This variable-span morphing wing was applied to a long-range cruise missile in an effort to increase range. A subsonic doublet hybrid method (DHM) was used for the computation of the subsonic aerodynamic forces and MSC/NASTRAN was used to model the wing-box structure. As the wing span increased, the induced drag decreased whereas the profile drag increased linearly. The wing-tip deformation of the 50% span-extended wing is larger than that of the baseline wing. The results showed that the VSMW required larger bending stiffness than the conventional wing. The divergence boundary considerably decreases as the wing span increases. Hence, the aeroelastic characteristics of the VSMW are worse. This static aeroelastic study showed the significance of aeroelastic phenomenon in the design of a span morphing wing. However, the dynamic aeroelasticity of span morphing was not modeled. Wang and Dowell [15, 16] studied the dynamics and aeroelastic characteristics of multi segmented folding wings. The results showed that increasing the fold angle causes up to 30% increase in flutter speed. The above studies have shown that the parameterized modeling method for morphing wings is required to perform the aeroelastic stability analysis. That is, the dynamics and aeroelastic models are derived with the parametric variations associated with morphing. However, the aeroelastic modeling of span morphing wings to study flutter problems has not been addressed. Further, the aerodynamic effects induced by the morphing process has not been considered in the flutter analysis. Therefore, the aim of this study is to develop an aeroelastic model for a span morphing wing in a parameterized modeling framework and perform the flutter analysis. 2. Derivations for Span and Chord Morphing For a conventional aircraft wing, the aeroelastic equations for stability analysis are given as A¨ q + (ρV B + D)q˙ + (ρV 2 C + E)q = 0 (1) Here the matrices A, D, E represent the mass, damping and stiffness arising from the structural aspects of the wing, respectively [17, 18]. The matrices B and C correspond to the damping and stiffness matrices from the aerodynamic loadings on the wing, respectively. In a morphing aircraft, the wing undergoes a small to large scale structural variation depending on the flight mission or conditions. These structural changes of the wing will alter the dynamic and aeroelastic characteristics. In other words, the A, D, E, B, and C matrices are functions of the morphing state denoted by Θ. In a morphing aircraft, even for a constant free stream velocity, V, the morphing induced changes in the aeroelastic system can result in instability. Therefore, these effects have to be accounted for the aeroelastic equation of motion. The aeroelastic matrices given in Eq. (1) are the functions of the wing parameters, given by A = f(c, S, m, xf ), E = f(S, EI, GJ), B = f(c, S, aW , e), C = f(c, S, aW , e) In general, the above wing parameters shown in Fig. 1, vary with the morphing states. Most of the span morphing wing concepts assume the chord length (c), flexural axis (xf ) and aerodynamic centre (c/4) as a constant with the variation in the wing span (S). However, parameters such as the span (S), mass per unit area (m(x, y)) and structural 2

y

S

Flexural axis

L

ec

θ x

V

c/4

M

xf

z

c (a) 3D elastic wing.

(b) 2D airfoil section.

Figure 1: Morphing wing models.

stiffnesses (EI(y), GJ(y)) along the span, vary significantly with the morphing. The re-distribution of these parameters has to be accounted for in the governing equations of the span morphing wing. The effects of re-distribution of the above parameters are considered in the formulation and the aeroelastic equation of motion is re-derived for the span morphing mechanism. To account these effects, the wing is divided into nn sections along the span and mm sections along the chord to capture the morphing aspects of the wing as shown in Fig. 2 [19]. The bending and torsional motion are considered to be uncoupled. The displacement z (downwards +ve) of a general point on the wing is z(x, y, t) = φ1 q1 (t) + φ2 (x − xf )q2 (t)

(2)

where q1 and q2 are generalized coordinates and, φ1 (y) and φ2 (y) are the assumed modes [18]. The aeroelastic equations of motion can be derived using Lagrange’s equations. The large scale morphing concepts such as span morphing are mission based and the rate of morphing is a slow process [3] compared to the wing devices used for high frequency flight control. Therefore, the morphing of the wing is considered as a discrete or quasisteady process. To define the morphing state of the wing, in this study, a variable Θ is introduced. For example, in span morphing, the state Θ can represent the percentage of the span extension compared to the baseline wing span. The variations in the structural properties with the morphing states are shown in the parametric model given in Fig. 2. The subscripts r and s represent the discretisation along the span and chordwise directions of the wing, respectively. For a span morphing wing, the bending and torsional stiffness along the span varies 3

y

Elastic axis (EIr ,

GJr )

Hr Cs+1,r

mass/area

cs,r

(mr,s)

Span, S

a r-1

ar

x

z A1

Figure 2: Parametric structural model and morphing parametrs of the (r, s)th section of wing

depending on the morphed configuration. However, span morphing can result in a mass re-distribution along the wing. The kinetic energy due to the dynamic motion of a morphing wing with (nn × mm) sections can be derived as Z sZ c 1 T = dm(x, y, Θ)z˙ 2 dxdy (3) 2 0 0 where dm(x, y, Θ) is the mass per unit area of the wing and is a function of the position on the wing and the morphing state, Θ, of the wing. The strain energy of the wing due to the bending and torsional stiffness of the morphing wing with the 1-D beam model representation of the wing at the elastic axis, can be written as 1 U= 2

Z

S

 EI(y, Θ)

0

d2 z dy 2

2

1 dy + 2

Z

S

 GJ(y, Θ)

0

dθ dy

2 dy

(4)

The incremental work done by the aerodynamic lift (L) and moment (M ) through incremental deflections of the wing are Z δW = [dL(Θ)(−δz) + dM (Θ)δθ] (5) wing

Applying Lagrange’s equations, the full aeroelastic equations of motion with the morphing parameters can be written in state-space form as        0 I q˙ q 0 − = −1 −1 q ¨ q˙ 0 −A(Θ) (ρV 2 C(Θ) + E(Θ)) −A(Θ) ρV B(Θ) (6) The eigenvalues of the above equation determine the system frequencies and damping 4

ratios at a particular flight condition and morphed state, Θ. A traditional p-k method can be used to find the eigenvalues after incorporating the necessary modifications needed for the morphing wing [17]. 3. Modeling a Span Morphing Wing Section III

Section II

Section I

pS

d=nt

2t

pS

S

b=kt

Figure 3: Morphing wing box-beam

In the previous section, aeroelastic equations to simulate the stability of a span and chord morphing wing are developed. As already mentioned, a priori expressions are required to account the effects of morphing on the structural and aerodynamic properties of the wing. For numerical calculations, a telescopic morphing concept for a typical UAV is designed initially. The corresponding expressions to evaluate the inertial, stiffness and aerodynamic variations are then developed. 3.1. Telescopic morphing wing The wing is considered to have a box-beam spar that carries the bending and torsional loads on the wing. To allow the span morphing, the spar is assumed to have two boxbeams, an outer box-beam and an inner box-beam, as shown in Fig. 3. The outer box-beam is fixed at the root of the wing and is a static part of the wing. The inner box-beam can slide out of the outer box-beam and increase the span of the wing as shown in Fig. 3. Here, the percentage of span morphing, p, represents the morphing state, Θ. The inner box-beam is assumed to be attached to the outer box-beam. The stiffness of the wing for various morphed states is expressed in terms of the baseline wing. For the telescopic box-beam model considered in this study, the wing can be represented as three sections (section I, II and III) when the wing span is extended, as shown in Fig. 6. The structural stiffness and mass distribution of the in-board section of the morphing wing is primarily governed by the outer box-beam. The out-board section of the wing is governed by the inner box-beam. A mid overlapping section, section II, has contributions from both of the box-beams.

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3.2. Stiffness redistribution with morphing For a box-beam, the bending stiffness can be given as EI where E is the Young’s modulus and I is the second moment of area of the beam cross-section. The height of the box-beam is expressed in terms of the thickness to make the derivations simple. For the combination of two box-beams, the second moment of area is c Ixx =

b(d3 − (d − 4t)3 ) 4t(d − 4t)3 + 12 12

(7)

For a typical airfoil, the maximum thickness is almost 10-20 % of the chord. That is the box-beam spar is designed as rectangular box with a breadth much larger than the depth. Therefore, the bending stiffness contributions from the vertical walls of the boxbeam are ignored. The structural parameters corresponding to the sections I, II and III are denoted with superscripts ‘o’, ‘c’ and ‘i’ representing the outer box-beam, combined box-beam and inner box-beam, respectively. Further, expressing the depth of the boxc beam as d = nt, the second moment of area Ixx of the box-beam corresponding to section II, can be given as c Ixx =

b(n3 − (n − 4)3 )t3 b((nt)3 − (nt − 4t)3 ) = 12 12

Therefore, the bending stiffness of section I and III can be expressed in terms of the bending stiffness of section II, EI c , as  3  b(d − (d − 4t)3 ) 4t(d − 4t)3 EI c = E + (8) 12 12     3 ((n − 2)3 − (n − 4)3 ) (n − (n − 2)3 ) i c o c ; EI = EI (9) EI = EI (n3 − (n − 4)3 ) (n3 − (n − 4)3 ) o where Ixx corresponds to the outer box-beam section. Similarly, the torsional stiffness of sections I and III of the sliding box-beam in terms of Section II can be calculated. These stiffness expressions are used in the dynamic and aeroelastic analysis of span morphing wing.

3.3. Mass re-distribution with morphing The mass per unit area or length of the wing is a key parameter in the dynamics of a morphing wing. In this study, the mass re-distributions are approximately expressed as a constraint. For the span morphing wing mechanism, the mass per unit length corresponding to the three sections are donoted as mI , mII and mIII , respectively. The mass per unit length of the baseline wing is denoted as MB . For the span morphing wing mechanism, the mass re-distribution can be modeled as a constraint and given as mI pS + mII (1 − p)S + mIII pS = MB ∗ S

(10)

For the telescopic box-beam considered in this study, mII = MB and mI and mIII are taken as 0.5MB .

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Table 1: Box-beam properties

Parameter Material Ex Gx Breadth, b Depth, d Thickness of single box Torsion, GJ Bending, EI Wing mass, MB

Value Boron-UD 200 GPa 5 GPa 200 mm 30 mm 3.8mm 6.5708e+03 Pa 80000 Pa 17.65 kg/m

3.4. Unsteady aerodynamics of morphing The aerodynamic loads have to be computed for aeroelastic analysis. Applying strip theory, together with Theoderson’s model of unsteady aerodynamics, the lift and pitching moment acting along the flexural axis of each elemental strip dy is given as zo + (Lθ + ikLθ˙ )θ0 ]eiωt b zo M = ρV 2 b2 [Mz + ikMz˙ + (Mθ + ikMθ˙ )θ0 ]eiωt b

L = ρV 2 b[Lz + ikLz˙

(11)

where Lz , Lz˙ , Mz , etc. are the non-dimensional oscillatory aerodynamic derivatives and are functions of reduced frequencies [20]. For a three dimensional wing, the above aerodynamic derivatives are functions of Mach number M, reduced frequency k, aspect ratio of the wing (AR) and geometry (taper ratio, sweep angle, etc.). For a morphing wing, the geometry and aspect ratio of the wing changes in flight. The corresponding changes in the aspect ratio and lift curve slope are introduced in the aeroelastic analysis. 4. Dynamic Aeroelastic Stability of Morphing Wing A baseline UAV wing is designed to study the aeroelastic analysis of a morphing wing. Initially, a telescopic wing model (TMW) is designed to match the stiffness properties of the UAV given in Ref. [21]. The chord of the base airfoil is fixed as 300 mm. The dimensions of the designed telescopic composite box-beam are given in Table 1. A dynamic aeroelastic stability analysis of the morphing wing is performed with the aeroelastic models and morphing structural expressions developed in the previous sections. Flutter analysis is performed with the p-k method. However, the p-k method is modified to include the morphing induced changes. The modified p-k method is described below: 1. Select the morphing state, Θ. 7

−15

150

Flutter velocity reduction (%)

100

ζ (%)

50

0

−50

−100

−150 0

10

20

30

40

50

V (m/s)

(a) V-ω and V-ζ of the baseline

−20

−25

−30

−35

−40 0

20

40 Span morphing (%)

60

80

(b) Flutter velocity reduction

Figure 4: Flutter velocities of the baseline and span morphing wings (elastic axis = 0.25c)

2. Calculate the stiffness, mass and aerodynamic parametric changes associated with the morphing as given in Eqns. (8), (10) and (11). 3. The flutter analysis is performed with standard procedure of the p-k method. The frequency and damping are calculated for various air speeds of interest. 4. The next morphing state is chosen and the steps (2) and (3) are repeated. Now, the aeroelastic stability of span morphing wing is studied with the structural models and the thin airfoil theory discussed in Sec. 3. The wing span is morphed from 100 to180% of the baseline span, S and thin airfoil theory is used for the aerodynamics. The flutter speed of the TMW model with the baseline design properties is found to be 27 m/s, as shown in Fig. 4. A continuous aeroelastic simulation of the span morphing wing is performed with the equivalent models and procedure developed. The flutter velocity is calculated with the morphing parameter, Θ varying from 0 to 90% with an increment of 1%. The V − ω and V − ζ plots for the baseline is shown in Fig. 4(a). The variation in flutter velocity with span morphing, with the elastic axis located at a distance of 0.25c from the leading edge, is shown in Fig. 4(b). The results show that the reduction in flutter speed is relatively high at the initial stages of span morphing. A span extension of 40 % shows an almost 30 % in the reduction in the baseline flutter speed. Therefore, the flutter instability forms an important constraint on the maximum cruise speed of span morphing wing designs. 5. Morphing Flutter Analysis with CFD In this section, a high fidelity CFD analysis is performed to evaluate the aerodynamic characteristics of a span morphing wing. The flutter analysis is then performed with the CFD results and structural models discussed in Sec. 3. 5.1. CFD simulation of a morphing wing The OpenFOAM CFD package is used to produce the numerical results [22]. Different solvers are included for various types of flow. The simpleFoam solver is used to 8

(a) Undeformed (left) and deformed (right) wing surfaces. The colourmap on the deformed wing represents the velocity of the surface in the y direction

(b) Streamlines at the tip during a span extension.

Figure 5: CFD analysis of span morphing.

produce steady-state results. Unsteady results are produced using pimpleFoam and pimpleDyMFoam solvers. Although PIMPLE solvers are used, they are set up to use the well known PISO algorithm for solving unsteady flows. At the starting of the morphing process, simpleFoam is used to initialise the solution. Once the solution is converged, the pimpleDyMFoam solver is used to solve the unsteady flow during deformation of the wing. Mesh motion is solved via the velocityLaplacian solver, which is based on Laplacian smoothing. Variable diffusivity is utilised to maintain the mesh quality near to the wing surface during the deformation. An inverse-quadratic profile is chosen for this, as its high diffusivity near to the surface helps to preserve the mesh quality in this location [23]. Morphed and unmorphed configurations of the wings are shown in Fig. 5(a) with the velocity of the cells along the spanwise direction on the wing surface. During the motion, the wing tip is assumed to move as a single rigid body, whilst the rest of the wing stretches, such that it remains fixed at the root, and stays attached to the wing tip. The tip moves with a constant velocity throughout the morph. This implies infinite acceleration at the beginning and end of the morph. Once the motion has been completed, the wing is held in a fixed position, and the pimpleFoam solver is used to simulate the period of time over which the flow settles. This involved a significant portion of the flow simulation. The OpenFOAM implementation of the Spalart-Allmaras turbulence model is used to provide closure for the RANS equations. This model is known to provide good stability and accuracy, despite its relatively simple formulation. At the freestream boundary, a value of ν˜ = 5ν is used, which effectively causes fully turbulent boundary layers over the wing. Mesh generation is performed with the OpenFOAM snappyHexMesh tool, which produces unstructured hexahedral meshes with layer 9

110 105 Lift (total) on morphed wing (N)

Lift (total) on morphing wing (N)

78 76 74 72 70 68

95 90 85 80 75 70

66 64 1.5

100

1.6

1.7

1.8 1.9 2 2.1 Span of morphing wing (m)

2.2

65 0.5

2.3

1

1.5

2

2.5

3

Time (s)

(a) Lift variation during the span morphing.

(b) Lift variation after the span morphing.

Figure 6: Lift variation.

addition for boundary layers. An example of the streamlines at the tip during a span extension is shown in Figure 5(b). A detailed description of the unsteady aerodynamics of a span-morphing wing is given in Reference [24]. 5.2. Flutter analysis The aerodynamic parameters obtained from the CFD are used in the flutter analysis. The span is morphed from 1.5 m to 2.0 m. Here, the chord length of the wing is given as 0.6m. The span extension is carried out in 0.8 seconds. The variation in lift during the span morphing is shown in Fig. 6(a). The lift variation shows a sudden increase, followed by a decrease, and finally increases during the span morphing. At the end of morphing, the lift is increased from 69 N to 76 N. However, after the span is fully morphed, the lift is further increased over a period of time as the flow settles as shown in Fig. 6(b). The lift curve slope (CL )of the wing is calculated by non-dimensionalizing the lift variations [20]. The flutter analysis is carried out at two stages. In the first stage, the variations in structural and aerodynamic parameters are considered in the analysis. In the second stage, once the span is fully morphed, only the aerodynamic variations are considered. The results of flutter analysis are shown in Fig. 7. For comparison, the flutter analysis is carried out with the thin airfoil theory also. The aeroelastic stability analysis based on thin airfoil theory shows the flutter velocity reduces from 42 m/s to 31 m/s. However, this analysis does not capture the aerodynamic effects induced at the end of morphing. The flutter analysis with CFD based aerodynamic parameters show the flutter velocity reduces from 36 m/s to 25m/s during span morphing. After the span is fully extended, the flutter velocity reduces further from 25 m/s to 22 m/s due to the tip vortex induced flow variations. These results clearly show a significant reduction in the flutter velocities with span morphing and aeroelasticity has to be evaluated during the conceptual design stages of morphing aircraft.

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42 CFD (Morphing stage) Thin airfoil (Morphing stage) CFD (Post morphing stage

40

Flutter speed (m/s)

38 36 34 32 30 28 26 24 22 0

0.5

1 1.5 2 Span morphing time (s)

2.5

3

(a) Figure 7: Flutter variation during the span morphing and after the morphing.

6. Conclusion The aeroelastic stability of a span morphing wing is studied. Initially, a span morphing wing mechanism with a telescopic composite box-beam is designed. The analytical expressions to evaluate the changes in the structural and aerodynamic parameters due to morphing are derived. The aeroelastic equations incorporate the parametric changes associated with the span morphing wing using the assumed mode shapes method. The aeroelastic models in terms of a single morphing parameter allows continuous aeroelastic simulation to be performed for various morphing states. The following conclusions are drawn. • An aeroelastic stability analysis is performed with the beam models coupled with thin airfoil theory. Numerical results show a considerable reduction in the flutter velocity with the increase in span. A span extension of 40 % shows a reduction of 30% in the flutter velocity. • A CFD analysis is performed to capture the aerodynamic effects of morphing. The results show the lift increases in a non-linear fashion during the morphing and the flow evolves for a time period after the wing is fully extended. These aerodynamic effects can be captured only by the high fidelity analysis, such as CFD, and has to be considered in the aeroelastic analysis. • An aeroelastic stability analysis is carried out with the CFD results coupled with the structural models. Numerical results show a considerable reduction in the flutter velocity during the span morphing process. The flutter velocity is reduced further due to the flow evolution at the wing tips after the morphing is completed. This study clearly shows the effect of the morphing process on the dynamic aeroelastic stability and the computational advantages of developing theoretical models of morphing wings supplemented with high fidelity simulations.

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Acknowledgments The authors acknowledge the support of the European Research Council through project 247045 entitled “Optimization of Multi-scale Structures with Applications to Morphing Aircraft”. References [1] J. Valasek, Morphing aerospace vehicles and structures, Vol. 56, John Wiley & Sons, 2012. [2] Y. Zhao, H. Hu, Parameterized aeroelastic modeling and flutter analysis for a folding wing, Journal of Sound and Vibration 331 (2) (2012) 308–324. [3] S. Barbarino, O. Bilgen, R. M. Ajaj, M. I. Friswell, D. J. Inman, A review of morphing aircraft, Journal of Intelligent Material Systems and Structures 22 (9) (2011) 823–877. [4] T. A. Weisshaar, Morphing aircraft systems: Historical perspectives and future challenges, Journal of Aircraft 50 (2) (2013) 337–353. [5] E. Selitrennik, M. Karpel, Y. Levy, Computational aeroelastic simulation of rapidly morphing air vehicles, Journal of Aircraft 49 (6) (2012) 1675–1686. [6] S. Liska, E. H. Dowell, Continuum aeroelastic model for a folding-wing configuration, AIAA Journal 47 (10) (2009) 2350–2358. [7] J. N. Scarlett, R. A. Canfield, B. Sanders, Multibody dynamic aeroelastic simulation of a folding wing aircraft, in: Proceedings of the 47th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, 2006. [8] D. Tang, E. H. Dowell, Theoretical and experimental aeroelastic study for folding wing structures, Journal of Aircraft 45 (4) (2008) 1136–1147. [9] S. Murugan, M. Friswell, Morphing wing flexible skins with curvilinear fiber composites, Composite Structures 99 (2013) 69–75. [10] S. Murugan, E. I. Saavedra Flores, S. Adhikari, M. Friswell, Optimal design of variable fiber spacing composites for morphing aircraft skins, Composite Structures 94 (5) (2012) 1626–1633. [11] C. G. Diaconu, P. M. Weaver, A. F. Arrieta, Dynamic analysis of bi-stable composite plates, Journal of Sound and Vibration 322 (4) (2009) 987–1004. [12] W. Zhang, L. Sun, X. Yang, P. Jia, Nonlinear dynamic behaviors of a deploying-andretreating wing with varying velocity, Journal of Sound and Vibration 332 (25) (2013) 6785– 6797. [13] F. Gosselin, M. Paidoussis, A. Misra, Stability of a deploying/extruding beam in dense fluid, Journal of sound and vibration 299 (1) (2007) 123–142. [14] J.-S. Bae, T. M. Seigler, D. J. Inman, Aerodynamic and static aeroelastic characteristics of a variable-span morphing wing, Journal of aircraft 42 (2) (2005) 528–534. [15] I. Wang, S. C. Gibbs, E. H. Dowell, Aeroelastic model of multisegmented folding wings: Theory and experiment, Journal of Aircraft 49 (3) (2012) 911–921. [16] I. Wang, E. H. Dowell, Structural dynamics model of multisegmented folding wings: Theory and experiment, Journal of Aircraft 48 (6) (2011) 2149–2160. [17] D. H. Hodges, G. A. Pierce, Introduction to structural dynamics and aeroelasticity, Vol. 15, Cambridge University Press, 2002. [18] J. R. Wright, J. E. Cooper, Introduction to aircraft aeroelasticity and loads, Vol. 20, Wiley. com, 2008. [19] Y. Zhao, Flutter suppression of a high aspect-ratio wing with multiple control surfaces, Journal of Sound and Vibration 324 (3) (2009) 490–513. [20] Y.-c. Fung, An introduction to the theory of aeroelasticity, Courier Dover Publications, 2002. [21] F. Sabri, S. Meguid, Flutter boundary prediction of an adaptive morphing wing for unmanned aerial vehicle, International Journal of Mechanics and Materials in Design 7 (4) (2011) 307– 312. [22] H. Jasak, A. Jemcov, Z. Tukovic, Openfoam: A c++ library for complex physics simulations, in: International workshop on coupled methods in numerical dynamics, Vol. 1000, 2007, pp. 1–20. [23] H. Jasak, Z. Tukovic, Automatic mesh motion for the unstructured finite volume method, Transactions of FAMENA 30 (2).

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[24] J. H. S. Fincham, C. S. Beaverstock, A. B. Coles, L. L. Parsons, M. I. Friswell, Aerodynamic forces on morphing wings during span extension, in: Proceedings of the RAeS Advanced Aero Concepts Design and Operations Conference, Royal Aeronautical Society, Bristol, UK, 2014.

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