Aeroelastic Modeling of Morphing Aircraft Wings

Senthil Murugana,, James H. S. Finchama, M. I. Friswella, D. J. Inmanb

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

Abstract

An analytical aeroelastic model to investigate the aeroelastic stability of a span mor-phing aircraft is derived. Structural and aeroelastic models are derived with the pa-rameteric variations introduced by a span morphing wing. Dynamic aeroelastic stabilityanalysis is performed with thin airfoil theory and CFD. A CFD analysis of span mor-phing shows the lift increases in a non-linear fashion during the morphing process, andafter the morphing is completed. The flutter analysis, based on the CFD results, showsa significant reduction in flutter velocity during morphing due to inertial, elastic andaerodynamic variations, and due to the flow evolution after the morphing period. Thisstudy shows that the development of low fidelity aeroelastic models of morphing wings,supplemented with high fidelity analysis, can quantify the phenomena associated withthe morphing and also serve as precursor to a computationally expensive, fully coupledhigh 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 wingconfiguration to an optimal shape at each flight condition [1]. Morphing wing structures,small to large scale, have shown significant improvements in the aircraft performance andmanoeuvrability 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 asthe 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 onthe dynamics and aeroelastic characteristics of the wing. Therefore, considerable researchis 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 supersonicairflow [12]. Diaconu et al. [11] studied the dynamic transitions of a morphing bi-stableplate. However, very few studies have focused on incorporating the inertial, stiffness andaerodynamic variations introduced by these morphing structures and aerodynamics inthe aeroelastic model and flutter analysis[2].

Email address: s.m.masanam@swansea.ac.uk (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 mor-phing wing (VSMW)[14]. This variable-span morphing wing was applied to a long-rangecruise 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/NASTRANwas used to model the wing-box structure. As the wing span increased, the induced dragdecreased whereas the profile drag increased linearly. The wing-tip deformation of the50% span-extended wing is larger than that of the baseline wing. The results showed thatthe VSMW required larger bending stiffness than the conventional wing. The divergenceboundary considerably decreases as the wing span increases. Hence, the aeroelastic char-acteristics of the VSMW are worse. This static aeroelastic study showed the significanceof aeroelastic phenomenon in the design of a span morphing wing. However, the dynamicaeroelasticity of span morphing was not modeled. Wang and Dowell [15, 16] studied thedynamics and aeroelastic characteristics of multi segmented folding wings. The resultsshowed that increasing the fold angle causes up to 30% increase in flutter speed.

The above studies have shown that the parameterized modeling method for morphingwings is required to perform the aeroelastic stability analysis. That is, the dynamics andaeroelastic models are derived with the parametric variations associated with morphing.However, the aeroelastic modeling of span morphing wings to study flutter problems hasnot been addressed. Further, the aerodynamic effects induced by the morphing processhas not been considered in the flutter analysis. Therefore, the aim of this study is todevelop an aeroelastic model for a span morphing wing in a parameterized modelingframework and perform the flutter analysis.

2. Derivations for Span and Chord Morphing

For a conventional aircraft wing, the aeroelastic equations for stability analysis aregiven as

Aq + (ρVB + D)q + (ρV 2C + E)q = 0 (1)

Here the matrices A, D, E represent the mass, damping and stiffness arising from thestructural aspects of the wing, respectively [17, 18]. The matrices B and C correspondto the damping and stiffness matrices from the aerodynamic loadings on the wing, re-spectively. In a morphing aircraft, the wing undergoes a small to large scale structuralvariation depending on the flight mission or conditions. These structural changes ofthe wing will alter the dynamic and aeroelastic characteristics. In other words, theA, D, E, B, and C matrices are functions of the morphing state denoted by Θ. Ina morphing aircraft, even for a constant free stream velocity, V, the morphing inducedchanges in the aeroelastic system can result in instability. Therefore, these effects haveto be accounted for the aeroelastic equation of motion. The aeroelastic matrices givenin 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

x

y

z

S

(a) 3D elastic wing.

ec

c/4

Flexural axis

V

L

xf

θ

M

c

(b) 2D airfoil section.

Figure 1: Morphing wing models.

stiffnesses (EI(y), GJ(y)) along the span, vary significantly with the morphing. There-distribution of these parameters has to be accounted for in the governing equations ofthe span morphing wing.

The effects of re-distribution of the above parameters are considered in the formula-tion and the aeroelastic equation of motion is re-derived for the span morphing mecha-nism. To account these effects, the wing is divided into nn sections along the span andmm sections along the chord to capture the morphing aspects of the wing as shown inFig. 2 [19]. The bending and torsional motion are considered to be uncoupled. Thedisplacement z (downwards +ve) of a general point on the wing is

z(x, y, t) = φ1q1(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. Thelarge scale morphing concepts such as span morphing are mission based and the rate ofmorphing is a slow process [3] compared to the wing devices used for high frequencyflight control. Therefore, the morphing of the wing is considered as a discrete or quasi-steady process. To define the morphing state of the wing, in this study, a variable Θ isintroduced. For example, in span morphing, the state Θ can represent the percentage ofthe span extension compared to the baseline wing span. The variations in the structuralproperties 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 chordwisedirections of the wing, respectively.

For a span morphing wing, the bending and torsional stiffness along the span varies

3

x

y

zA1

a r-1

EIr H r

a r

Elastic axis GJr( )

mass/area

( )

Span, S

cs,r

Cs+1,r

mr,s

,

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 massre-distribution along the wing. The kinetic energy due to the dynamic motion of amorphing wing with (nn×mm) sections can be derived as

T =

∫ s

0

∫ c

0

1

2dm(x, y,Θ)z2dxdy (3)

where dm(x, y,Θ) is the mass per unit area of the wing and is a function of the positionon the wing and the morphing state, Θ, of the wing. The strain energy of the wing dueto the bending and torsional stiffness of the morphing wing with the 1-D beam modelrepresentation of the wing at the elastic axis, can be written as

U =1

2

∫ S

0

EI(y,Θ)

(d2z

dy2

)2

dy +1

2

∫ S

0

GJ(y,Θ)

(dθ

dy

)2

dy (4)

The incremental work done by the aerodynamic lift (L) and moment (M) through incre-mental deflections of the wing are

δW =

∫wing

[dL(Θ)(−δz) + dM(Θ)δθ] (5)

Applying Lagrange’s equations, the full aeroelastic equations of motion with the morph-ing parameters can be written in state-space form as{

qq

}−[

0 I

−A(Θ)−1

(ρV 2C(Θ) + E(Θ)) −A(Θ)−1ρVB(Θ)

]{qq

}=

{00

}(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 methodcan be used to find the eigenvalues after incorporating the necessary modifications neededfor the morphing wing [17].

3. Modeling a Span Morphing Wing

d=nt

b=kt

2t

S

Section I

Section II

Section III

pS

pS

Figure 3: Morphing wing box-beam

In the previous section, aeroelastic equations to simulate the stability of a span andchord morphing wing are developed. As already mentioned, a priori expressions arerequired to account the effects of morphing on the structural and aerodynamic propertiesof the wing. For numerical calculations, a telescopic morphing concept for a typical UAVis designed initially. The corresponding expressions to evaluate the inertial, stiffness andaerodynamic variations are then developed.

3.1. Telescopic morphing wing

The wing is considered to have a box-beam spar that carries the bending and torsionalloads on the wing. To allow the span morphing, the spar is assumed to have two box-beams, an outer box-beam and an inner box-beam, as shown in Fig. 3. The outerbox-beam is fixed at the root of the wing and is a static part of the wing. The innerbox-beam can slide out of the outer box-beam and increase the span of the wing as shownin 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 thebaseline wing. For the telescopic box-beam model considered in this study, the wing canbe represented as three sections (section I, II and III) when the wing span is extended, asshown in Fig. 6. The structural stiffness and mass distribution of the in-board section ofthe morphing wing is primarily governed by the outer box-beam. The out-board sectionof 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’smodulus and I is the second moment of area of the beam cross-section. The height ofthe box-beam is expressed in terms of the thickness to make the derivations simple. Forthe combination of two box-beams, the second moment of area is

Icxx =b(d3 − (d− 4t)3)

12+

4t(d− 4t)3

12(7)

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

Icxx =b((nt)3 − (nt− 4t)3)

12=b(n3 − (n− 4)3)t3

12

Therefore, the bending stiffness of section I and III can be expressed in terms of thebending stiffness of section II, EIc, as

EIc = E

(b(d3 − (d− 4t)3)

12+

4t(d− 4t)3

12

)(8)

EIo = EIc{

(n3 − (n− 2)3)

(n3 − (n− 4)3)

};EIi = EIc

{((n− 2)3 − (n− 4)3)

(n3 − (n− 4)3)

}(9)

where Ioxx corresponds to the outer box-beam section. Similarly, the torsional stiffness ofsections I and III of the sliding box-beam in terms of Section II can be calculated. Thesestiffness expressions are used in the dynamic and aeroelastic analysis of span morphingwing.

3.3. Mass re-distribution with morphing

The mass per unit area or length of the wing is a key parameter in the dynamics ofa morphing wing. In this study, the mass re-distributions are approximately expressedas a constraint. For the span morphing wing mechanism, the mass per unit lengthcorresponding to the three sections are donoted as mI , mII and mIII , respectively. Themass per unit length of the baseline wing is denoted as MB . For the span morphing wingmechanism, the mass re-distribution can be modeled as a constraint and given as

mIpS +mII(1 − p)S +mIIIpS = MB ∗ S (10)

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

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Table 1: Box-beam properties

Parameter ValueMaterial Boron-UDEx 200 GPaGx 5 GPaBreadth, b 200 mmDepth, d 30 mmThickness of single box 3.8mmTorsion, GJ 6.5708e+03 PaBending, EI 80000 PaWing mass, MB 17.65 kg/m

3.4. Unsteady aerodynamics of morphing

The aerodynamic loads have to be computed for aeroelastic analysis. Applying striptheory, together with Theoderson’s model of unsteady aerodynamics, the lift and pitchingmoment acting along the flexural axis of each elemental strip dy is given as

L = ρV 2b[Lz + ikLzzob

+ (Lθ + ikLθ)θ0]eiωt

M = ρV 2b2[Mz + ikMzzob

+ (Mθ + ikMθ)θ0]eiωt (11)

where Lz, Lz, Mz, etc. are the non-dimensional oscillatory aerodynamic derivatives andare functions of reduced frequencies [20].

For a three dimensional wing, the above aerodynamic derivatives are functions ofMach 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 ofthe wing changes in flight. The corresponding changes in the aspect ratio and lift curveslope 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 propertiesof the UAV given in Ref. [21]. The chord of the base airfoil is fixed as 300 mm. Thedimensions of the designed telescopic composite box-beam are given in Table 1.

A dynamic aeroelastic stability analysis of the morphing wing is performed withthe aeroelastic models and morphing structural expressions developed in the previoussections. Flutter analysis is performed with the p-k method. However, the p-k method ismodified to include the morphing induced changes. The modified p-k method is describedbelow:

1. Select the morphing state, Θ.

7

0 10 20 30 40 50−150

−100

−50

0

50

100

150

V (m/s)

ζ (

%)

(a) V-ω and V-ζ of the baseline

0 20 40 60 80−40

−35

−30

−25

−20

−15

Span morphing (%)

Flu

tter

velo

city r

eduction (

%)

(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 withthe morphing as given in Eqns. (8), (10) and (11).

3. The flutter analysis is performed with standard procedure of the p-k method. Thefrequency 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 structuralmodels and the thin airfoil theory discussed in Sec. 3. The wing span is morphed from100 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 27m/s, as shown in Fig. 4. A continuous aeroelastic simulation of the span morphing wingis performed with the equivalent models and procedure developed. The flutter velocityis calculated with the morphing parameter, Θ varying from 0 to 90% with an incrementof 1%. The V −ω and V −ζ plots for the baseline is shown in Fig. 4(a). The variation influtter velocity with span morphing, with the elastic axis located at a distance of 0.25cfrom the leading edge, is shown in Fig. 4(b). The results show that the reduction influtter speed is relatively high at the initial stages of span morphing. A span extension of40 % shows an almost 30 % in the reduction in the baseline flutter speed. Therefore, theflutter instability forms an important constraint on the maximum cruise speed of spanmorphing wing designs.

5. Morphing Flutter Analysis with CFD

In this section, a high fidelity CFD analysis is performed to evaluate the aerodynamiccharacteristics of a span morphing wing. The flutter analysis is then performed with theCFD 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]. Dif-ferent solvers are included for various types of flow. The simpleFoam solver is used to

8

(a) Undeformed (left) and deformed (right) wingsurfaces. The colourmap on the deformed wingrepresents the velocity of the surface in the y di-rection

(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 pim-pleDyMFoam solvers. Although PIMPLE solvers are used, they are set up to use thewell 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 unsteadyflow during deformation of the wing. Mesh motion is solved via the velocityLaplaciansolver, which is based on Laplacian smoothing. Variable diffusivity is utilised to maintainthe mesh quality near to the wing surface during the deformation. An inverse-quadraticprofile is chosen for this, as its high diffusivity near to the surface helps to preserve themesh quality in this location [23]. Morphed and unmorphed configurations of the wingsare shown in Fig. 5(a) with the velocity of the cells along the spanwise direction on thewing surface.

During the motion, the wing tip is assumed to move as a single rigid body, whilstthe rest of the wing stretches, such that it remains fixed at the root, and stays attachedto the wing tip. The tip moves with a constant velocity throughout the morph. Thisimplies infinite acceleration at the beginning and end of the morph. Once the motion hasbeen completed, the wing is held in a fixed position, and the pimpleFoam solver is usedto simulate the period of time over which the flow settles. This involved a significant por-tion of the flow simulation. The OpenFOAM implementation of the Spalart-Allmarasturbulence model is used to provide closure for the RANS equations. This model isknown 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 fullyturbulent boundary layers over the wing. Mesh generation is performed with the Open-FOAM snappyHexMesh tool, which produces unstructured hexahedral meshes with layer

9

1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.364

66

68

70

72

74

76

78

Span of morphing wing (m)

Lift (t

ota

l) o

n m

orp

hin

g w

ing (

N)

(a) Lift variation during the span morphing.

0.5 1 1.5 2 2.5 3

65

70

75

80

85

90

95

100

105

110

Time (s)

Lift (t

ota

l) o

n m

orp

hed w

ing (

N)

(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 spanextension is shown in Figure 5(b). A detailed description of the unsteady aerodynamicsof 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 as0.6m. The span extension is carried out in 0.8 seconds. The variation in lift during thespan morphing is shown in Fig. 6(a). The lift variation shows a sudden increase, followedby 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 liftis further increased over a period of time as the flow settles as shown in Fig. 6(b). Thelift 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 variationsin structural and aerodynamic parameters are considered in the analysis. In the secondstage, 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 iscarried out with the thin airfoil theory also. The aeroelastic stability analysis based onthin 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 velocityreduces 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 inducedflow variations. These results clearly show a significant reduction in the flutter velocitieswith span morphing and aeroelasticity has to be evaluated during the conceptual designstages of morphing aircraft.

10

0 0.5 1 1.5 2 2.5 322

24

26

28

30

32

34

36

38

40

42

Span morphing time (s)

Flu

tter

speed (

m/s

)

CFD (Morphing stage)

Thin airfoil (Morphing stage)

CFD (Post morphing stage

(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 morph-ing wing mechanism with a telescopic composite box-beam is designed. The analyticalexpressions to evaluate the changes in the structural and aerodynamic parameters dueto morphing are derived. The aeroelastic equations incorporate the parametric changesassociated with the span morphing wing using the assumed mode shapes method. Theaeroelastic models in terms of a single morphing parameter allows continuous aeroelasticsimulation to be performed for various morphing states. The following conclusions aredrawn.

• An aeroelastic stability analysis is performed with the beam models coupled withthin airfoil theory. Numerical results show a considerable reduction in the fluttervelocity with the increase in span. A span extension of 40 % shows a reduction of30% in the flutter velocity.

• A CFD analysis is performed to capture the aerodynamic effects of morphing. Theresults show the lift increases in a non-linear fashion during the morphing and theflow evolves for a time period after the wing is fully extended. These aerodynamiceffects can be captured only by the high fidelity analysis, such as CFD, and has tobe considered in the aeroelastic analysis.

• An aeroelastic stability analysis is carried out with the CFD results coupled withthe structural models. Numerical results show a considerable reduction in theflutter velocity during the span morphing process. The flutter velocity is reducedfurther 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 aeroe-lastic stability and the computational advantages of developing theoretical modelsof morphing wings supplemented with high fidelity simulations.

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Acknowledgments

The authors acknowledge the support of the European Research Council throughproject 247045 entitled “Optimization of Multi-scale Structures with Applicationsto Morphing Aircraft”.

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