Flutter of Telescopic Span Morphing Wings

This paper studies the aeroelastic behavior of telescopic, multi-segment, span morphing wings. The wing is modeled as a linear, multi-segment, stepped, cantilever Euler–Bernoulli beam. It consists of three segments along the axis and each segment has different geometric, mechanical, and inertial properties. The aeroelastic analysis takes into account spanwise out-of-plane bending and torsion only, for which the corresponding shape functions are derived and validated. The use of shape functions allows representing the wing as an equivalent aerofoil whose generalized coordinates are defined at the wingtip according to the Rayleigh–Ritz method. Theodorsen’s unsteady aerodynamic theory is used to estimate the aerodynamic loads. A representative Padé approximation for the Theodorsen’s transfer function is utilized to model the aerodynamic behaviors in state-space form allowing time-domain simulation and analysis. The effect of the segments’ mechanical, geometric, and inertial properties on the aeroelastic behavior of the wing is assessed. Finally, the viability of span morphing as a flutter suppression device is studied.

A large wingspan/aspect ratio improves the aerodynamic e±ciency but reduces maneuvrability, whereas a small wingspan improves maneuvrability but reduces aerodynamic e±ciency. 13][4] Ajaj et al. 3,4 studied the bene¯ts of variable span wings to enhance the aerodynamic e±ciency when actuated symmetrically and to improve roll control when actuated asymmetrically.Span morphing wings have been developed since the start of powered °ight.For instance, the MAK-10 aircraft, develop by Ivan Makhonine, °ew in the 1930s with a telescopic span morphing wing.Makhonine utilized pneumatic actuators to move the telescopic wing to achieve span extensions up to 60%. 5 Recently, there has been some promising work on telescopic span morphing wings.For example, Blondeau and Pines 6 developed a telescopic wing where hollow shells were used to preserve the aerofoil shape and reduce the storage size of the wing.They utilized in°atable actuators to withstand the di®erent loads on the wing.Bae et al. 7 studied wings of a long-range cruise missile and highlighted some of the main challenges associated with the design of a span morphing wing.They achieved drag reduction of 25% and a range increase of 30%.Ajaj and Jankee 8 developed the Transformer Aircraft, a span morphing UAV, capable of symmetric and asymmetric span extensions.A novel actuation system based on a rack and pinion mechanism was utilized.They conducted extensive wind tunnel and °ight testing to assess the e®ect of span morphing on the °ight mechanics and aerodynamic e±ciency.
Similarly, Santos et al., 9 Mestrinho et al., 10 and Felício et al. 11 and developed and tested a variable-span morphing wing (VSW) to be ¯tted on a mini-UAV.They achieved 20% wing drag reduction with symmetric span extension.The roll rate achieved with asymmetric span morphing matched the aileron in terms of roll power.The VSW was constructed from composite materials and was actuated using an electro-mechanical mechanism.Mechanical testing was performed to evaluate the behavior of the wing under various loading scenarios.Flight testing showed full functionality of the VSW and its aerodynamic improvements compared with conventional ¯xed wing.A more extensive review on span morphing wings (applications and concepts) for both ¯xed-wing and rotary-wing aircraft is given in Barbarino et al. 1 Recently, there have been a number of attempts to study the aeroelastic behavior of span morphing wings.For example, Ajaj and Friswell 12 developed a linear, timedomain aeroelastic model that uses Theodorsen's unsteady aerodynamics to study the aeroelastic behavior of compliant span morphing wings.They performed extensive sensitivity studies and concluded that span morphing can be used as an e®ective °utter suppression device.Similarly, Huang and Qiu 13 developed a novel ¯rst-order state-space aeroelastic model based on Euler-Bernoulli beam theory.They assumed time-dependent boundary conditions coupled with a reduced-order unsteady vortex lattice method.Similarly, Li and Jin 14 studied the dynamical behavior and stability of a variable-span wing subjected to supersonic aerodynamic loads.They modeled the span morphing wing as an axially moving cantilever plate and established the governing equations of motion using Kane's method and piston theory.They concluded that a periodically varying (with proper amplitude) morphing law can facilitate °utter suppression.Gamboa et al. 15 studied the aeroelasticity of composite VSW intended for a small UAV.The study concentrated on the °utter critical speed estimation and assessed the e®ect of the interface between ¯xed and moving wing parts.Their aerodynamic solver was based on an unsteady linearized potential theory coupled with three-dimensional lifting surface strip theory approximation for lifting surfaces with high aspect ratio.The results showed that the wing can °y safely within the intended speed envelope.
The aim of this paper is to investigate the aeroelastic behavior of telescopic span morphing wings.The wing is modeled as a stepped, multi-segment Euler-Bernoulli beam consisting of three main segments with di®erent geometric, inertial, and mechanical properties.Theodorsen's aerodynamic theory is utilized for estimating the unsteady aerodynamic loads.The in°uence of the segments' properties on the binary (bending-torsion) aeroelastic behavior of the wing is assessed.Goland wing 16 and the HALE wing 17 are used as the basis of this study.The e®ect of the segments properties on the °utter mode is investigated.Finally, the feasibility of utilizing span morphing as an active °utter suppression device is assessed.

Aeroelasticity Model
The wing is modeled as a stepped Euler-Bernoulli beam consisting of three segments.These segments correspond to the ¯xed wing partition, the overlapping region, and the extending partition as shown in Fig. 1.
Across each segment, the mechanical and geometric properties are uniform but they di®er from one segment to another.The properties (considered here) corresponding to each segment are listed in Table 1.Each segment is rectangular, unswept, untapered to minimize geometric and aeroelastic couplings.It is assumed to have a clean wing con¯guration where no control surfaces or engines are attached to it, and there are no fuel tanks embedded within.The continuous, multi-degree-offreedom wing structure is modeled as a two-degree-of-freedom system via the Rayleigh-Ritz method using bending and torsion shape functions.These shape functions correspond to the uncoupled ¯rst bending and ¯rst torsional modes of a stepped cantilever beam.This allows the wing to be modeled as an equivalent two-degree-offreedom aerofoil whose generalized coordinates are de¯ned at the wingtip.The dynamics of the telescopic mechanism are neglected (as the rate of span extension or retraction is low).Using the shape functions, the plunge displacement, speed, and acceleration at any spanwise location ðyÞ and time instant can now be related to those of the wingtip (generalized coordinates) as: where w t ðtÞ and t ðtÞ represent the generalized coordinates coinciding with the wingtip.It should be noted that the datum from which the generalized coordinates are measured is the static position of the wingtip when the wing de°ects under self-weight.

Bending shape functions
To obtain the bending shape functions, the continuity and boundary conditions listed in Table 2 are considered.The bending shape function, h i ðyÞ, corresponding to the ith bending mode is given as: Appendix A details the steps to obtain the bending shape function and natural frequency corresponding to the ith mode.

Torsion shape functions
Similarly, to obtain the torsion shape function, the boundary and continuity conditions listed in Table 3 are considered.
The torsion shape function i ðyÞ corresponding to the ith torsional vibration mode is given as: Table 2. Bending boundary conditions.

Root conditions Continuity conditions
Tip conditions Appendix B details the steps to obtain the torsion shape function and natural frequency corresponding to the ith mode.

Equations of motion
The total kinetic energy ðT Þ and total potential energy ðUÞ of the three segments' cantilever, rectangular wing can be expressed as and It should be noted that structural damping is not considered in this study; this assumption is commonplace in aeroelastic analysis as any structural damping will increase the speed at which °utter will occur.Using the expressions of kinetic and potential energies, the full equations of motion of the span morphing wing can developed using Lagrangian mechanics as: where L is the generalized lift force and M ea is the generalized pitching moment.The generalized lift force and pitching moment around the elastic axis can be obtained as: and where L 0 1 , L 0 2 , and L 0 3 are the unsteady lift per unit span on segments 1, 2, and 3, respectively.M 0 ea 1 , M 0 ea 2 , and M 0 ea 3 are the unsteady pitching moments around the elastic axis per unit span on segments 1, 2, and 3, respectively.It should be noted that the expression of lift per unit span and pitching moment per unit span will vary for the di®erent segments due to di®erent chords and di®erent distances between aerodynamic centers and elastic axis of each segment.

Aerodynamics
The aerodynamic loads acting on the wing are modeled according to Theodorsen's unsteady aerodynamics theory.Theodorsen's unsteady aerodynamic model consists of a circulatory component accounting for the e®ect of the wake on the aerofoil (contains the main aerodynamic damping and aerodynamic sti®ness terms) and a noncirculatory component accounting for the acceleration of the °uid surrounding the aerofoil. 18The work of Theodorsen is based on the following assumptions: . Thin aerofoil; . Potential, incompressible °ow; . The °ow remains attached, i.e. the amplitude of oscillations is small and the wake behind the aerofoil is °at.
According to Theodorsen's unsteady aerodynamic theory, L 0 j and M 0 ea j , acting on the jth wing segment can be expressed, respectively, as and where is the air density, c j is the chord of the wing at any jth segment, and âj ¼ 2x ea j c j À 1 is the normalized pitch axis location with respect to half chord of the jth segment.CðkÞ is the frequency-dependent, Theodorsen's transfer function that accounts for attenuation of lift amplitude and phase lag in lift response due to sinusoidal motion.In this paper, unsteady lift per unit span and pitching moment per unit span are expressed in time domain.Therefore, a Pad e approximation for Theodorsen's transfer function was used. 20,21The approximate transfer function CðsÞ in the Laplace domain becomes where The Pad e approximation for Theodorsen function is highly accurate especially at low reduced frequencies which is the case in this paper.For more details on the expression of unsteady lift and moment in state-space form, the reader is advised to consult Ajaj and Friswell. 12A similar analysis was performed by Duan and Zhang 19 in which they used Fourier transform to formulate the aeroelastic equations of motion of a wing in a state-space form.

Validation
The aeroelastic model is validated using Goland wing and HALE wing whose mechanical and geometric properties are listed in Table 4.The wings studied here are of high aspect ratio to be consistent with the Euler-Bernoulli formulation (ignoring the shear deformation of the wing cross-section).
The °utter speed, frequency, and divergence speed estimated for the Goland and HALE wings are presented in Table 5.Table 5 shows a comparison with estimates from various other methods available in literatures.

Variation of the uncoupled shape functions with wing properties (aerodynamics OFF)
In this section, it is assumed that Goland wing is extended by 50% so that the wing semi-span is 9.144 m.The wing consists of three main segments.Segment 1 represents the main wing (without overlapping region), whereas Segment 2 represents the overlapping region and Segment 3 represents the extension.The mechanical and geometric properties of Segment 1 are exactly the same as those of Goland wing (listed in Table 3) except the length of Segment 1 is l 1 ¼ 5 m.Segment 2 has a length l 2 ¼ 1:096 m, whereas Segment 3 has a length l 3 ¼ 3:048 m.Two scenarios are studied here, and in both scenarios it is assumed that the locus of the center of gravity (CG) of the three segments is a continuous line starting at the wing root and ending at its tip.Similarly, the locus of the shear center of the three segments is a continuous line starting at the wing root and ending at its tip.The Transformer Aircraft 8 is a good example where the loci of the CG and shear entre are continuous straight lines.
i. Scenario 1 In Scenario 1, Segments 1 and 2 are assumed to have the same chords, bending rigidity, and torsional rigidity.The properties of Segment 3 are varied, and the bending and torsion shape functions for the ¯rst bending and torsion modes associated with changes in Segment 3 chord are presented in Figs.2(a) and 2(b).The chord of Segment 3 is varied as a fraction of the Segment 1 chord.A change in the chord of Segment 3 results in changes in the bending rigidity, torsion rigidity, mass per unit span, inertia per unit span according to the expressions in Table 6.
ii. Scenario 2 In Scenario 2, Segments 1 and 3 are assumed to have the same mechanical and inertial properties.The chord of Segment 2 is varied as fraction of the chord of Segment 1.The properties of Segment 2 vary according to the expressions listed in Table 6.The variations of the uncoupled ¯rst bending and ¯rst torsion shape functions with Segment 2 chord are shown in Figs.2(c) and 2(d).
It can be seen from Figs. 2(a) and 2(c) that as the bending sti®ness of Segment 3 or of Segment 2 reduces, the contribution of Segment 1 to the mode shape will be much smaller than the contributions of Segments 2 and 3. On the contrary, the ¯rst torsional mode shape is more sensitive to variations in properties (mainly torsional rigidity) of Segment 2 than of Segment 3.This can be clearly noticed when comparing Figs.2(b) and 2(d).Large variations in the torsional rigidity of Segment 2 severely distort the ¯rst torsion mode shape.Depending on the length of Segment 2, this can signi¯cantly a®ect the aeroelastic stability of the wing.
Figure 3 shows the variation of the uncoupled ¯rst bending and ¯rst torsion natural frequencies for each scenario.Increasing the chord of Segment 3    (while keeping the properties of the other segments constant) reduces the bending and torsion natural frequencies.On the contrary, increasing the chord of Segment 2 (while keeping the properties of the other segments constant) increases the ¯rst bending and ¯rst torsion natural frequencies.It can be noticed that the change in the natural frequencies is negligible when the chord of Segment 2 is above 75% of chord of Segment 1.

E®ect of Segment 3 on °utter
In this section, the wingspan is extended quasi-statically to determine the e®ect of span extension on the °utter speed, frequency, and divergence speed.The HALE wing, whose properties are listed in Table 4, is used as the basis for this study.The properties of Segment 3 are varied to assess the e®ect of its mechanical, inertial, and geometric properties on the °utter speed.Two wing models are used in this study.The ¯rst is a two-segment model in which the wing consists of two segments (Segments 1 and 3) and the overlapping region is not considered.The second is a three-segment model in which the wing consists of three segments: Segments 1, 2, and 3. Segment 2 represents the overlapping section whose length varies as the wingspan extends.In the three-segment model, the properties of Segment 2 (overlapping region) such as mass per unit span, torsional, and bending rigidity depend on those of Segments 1 and 3. Figure 4 shows the variation of the aeroelastic behavior of the multi-segment span morphing wing for di®erent con¯gurations of Segment's 3. It should be noted that in Fig. 4 the °utter speed, frequency, and divergence speed are normalized by the corresponding values associated with the baseline (nonmorphing) Hale wing (listed in Table 5).Three di®erent con¯gurations of Segment 3 are studied here.They correspond to the chord of Segment 3 being equal to 40%, 70%, and 100% of Segment's 1 chord.For each con¯guration of Segment 3, its properties change according to the relationships expressed in Table 6.
It can be seen that for a given span extension, con¯gurations with smaller chords (of Segment 3) have higher °utter speed and frequency.This is true for both models (two-segments and three-segments).In general, span extension results in reduction in °utter speed, which reduces the °ight envelope of the aircraft.As the chord of Segment 3 (and the associated properties) gets smaller, the sensitivity of °utter and divergence speeds with span extension reduce signi¯cantly.It can be seen from Fig. 4(b) that when the chord of Segment 3 is the same as the chord of Segment 1, increasing the wingspan by 50% reduces the °utter speed by 35%.In contrast, when the chord of Segment 3 is 40% of the chord of Segment 1, the °utter speed reduces by 10% at 50% span extension.
The results from the two di®erent models show the in°uence of the overlapping region on the aeroelastic behavior.In Figs.4(b), 4(d), and 4(f), the normalized °utter speed, divergence speed, and frequency do not start from unity as with the two-segment model.As the wingspan increases, the length of Segment 2 reduces and hence its in°uence on the aeroelastic behavior of the wing.It should be noted that the three-segment model treats the overlapping region as an idealized joint and does not take into account any form of localized sti®ness, damping, and/or freeplay that may exist in real telescopic joints.As the wingspan extends by 50%, the length of the overlapping segment shrinks to become 10% of the baseline semi-span (16 m). Figure 4(b) shows that at zero span extension, the °utter speed is higher than that of the uniform baseline wing (nonmorphing) by around 8%.

E®ect of overlapping segment on °utter
It is essential to assess the e®ect of the overlapping segment (Segment 2) on the aeroelastic behavior of the wing.Three main parameters of Segment 2, including mass per unit span, bending rigidity, and torsional rigidity, are studied at di®erent wingspans (corresponding to 0%, 25%, and 50% extensions).The chord of Segment 3 is set at 80% of the chord of Segment 1.During this sensitivity study, only one parameter is varied at a time.For instance, when the torsional rigidity of Segment 2 varies between 0.1 and 2 times the torsional rigidity of Segment 1, the bending rigidity and mass per unit span of Segment 2 are kept constants.
It can be seen that regardless of the parameter investigated, the sensitivity of °utter speed and frequency reduce as the wingspan is extended and the size of the overlapping region drops.However, when the HALE wing is fully retracted, the e®ect of Segment 2 parameters on the aeroelastic behavior of the wing is signi¯cant.It can be seen from Fig. 5 that °utter is most sensitive to torsional rigidity followed by the mass per unit span.The bending rigidity of Segment 2 has minor e®ect on °utter speed and frequency, irrespective of the span extension.It can also be seen that when the wing is fully retracted (and the length of Segment 2 is maximum), the °utter speed and frequency are very sensitive to mass per unit span especially when the ratio of the mass of Segment 2 to mass of Segment 1 is less than 0.5.It should be noted that it is impractical to have the ratio of the masses less than 1 at least for wing studied here.

Multimode Flutter Analysis
Section 3 of this paper focussed on the e®ect of the segment properties on the binary (bending-torsion) aeroelastic behavior of the wing.However, it is essential to determine the e®ect of span morphing on the °utter mode.Therefore, the three-segment binary aeroelastic model (discussed in Sec. 3) is extended to include high-order vibration modes, mainly the ¯rst, second, third, and fourth bending modes and the ¯rst, second, third, and fourth torsion modes.For the baseline (nonmorphing) Goland and HALE wings, the °utter speed and °utter frequency and divergence speed obtained using the multimode aeroelastic model are listed in Table 5. Goland wing is used as the baseline wing in this section.The variation of the di®erent modes with airspeed for the baseline (nonmorphing) Goland wing and for the telescopic span morphing Goland wing at 50% extension is shown in Fig. 6.For the results in Fig. 6, the chord of Segment 3 (extension) is set equal to the chord of Segment 1.
The morphing wing at 50% span extension °utters at around 100 m/s compared with 137 m/s for the nonmorphing baseline Goland wing.It should be noted that the modal damping of the ¯rst bending mode (mode 1) increases with airspeed at a higher rate for the extended morphing wing compared with the baseline wing.The same is true for the ¯rst torsion mode in which the modal damping initially increases at a higher rate; then once a critical speed is reached, it reduces at a higher rate when compared with the baseline nonmorphing wing.Figure 6 shows that span extension does not change the °utter mode for the clean rectangular wing considered here (without engines, control surfaces, and fuel tanks).For both the morphing wing and the nonmorphing, the ¯rst torsion mode (mode 2) is the ¯rst to go unstable.

Flutter Suppression
The aim of this section is to assess the feasibility of using span morphing as an e®ective °utter suppression device.Two main scenarios are studied to demonstrate the °utter suppression capability.A telescopic span morphing wing whose baseline dimensions (before span extension) are similar to those of the Goland wing is considered.Di®erent °ight conditions are used to assess the viability of the device at a range of operating conditions.i. Scenario 1: Span retraction at °utter speed The telescopic span morphing Goland wing is set at 5 angle of attack with an airspeed equal to the °utter speed at 25% span extension.Once released into the air°ow, the wing starts a series of undamped oscillations in pitch and plunge.At t ¼ 2 s, the wingspan is retracted by 25% (new wing semi-span is 6.096 m).Two retraction speeds are studied (1.5240 m/s and 15.240 m/s).The behavior of the wing for the di®erent retraction speeds can be seen in Fig. 7. Figure 7 shows that span retraction damps the oscillations in pitch and plunge.As the span retraction rate increases, the wing oscillations decay faster.This is mainly due to the fact that as the wingspan is reduced, its bending and torsional sti®ness increases, resulting in an increase in the °utter speed.
ii. Scenario 2: Span retraction above °utter speed The span morphing Goland wing is set at 1 angle of attack and the airspeed is set at 5 m/s above the °utter speed with 25% span extension.Initially, the wing starts diverging in pitch and plunge until at t ¼ 1 s, where the n is retracted by 25%.Two actuation speeds are considered (1.5240 m/s and 15.240 m/s).
It is evident from Fig. 8 that span morphing can suppress °utter allowing the aircraft to operate over a wide range of airspeeds.Figure 8  wingtip's oscillations damp out faster for higher span retraction rates.Span morphing is able to signi¯cantly shift the stability of the wing.It should be noted that the choice of the actuation speeds in both scenarios is done manually without the use of a feedback control system to determine optimum retraction speeds.

Conclusion
This paper presented a linear aeroelastic model to study the behavior of telescopic span morphing wings in time domain.The wing was modeled as a stepped, three segment, Euler-Bernoulli beam.Rayleigh-Ritz energy method was used to derive the generalized equations of motion.Theodorsen's unsteady aerodynamic theory was used for aerodynamic predictions.A representative Pad e approximation for the Theodorsen's transfer function was utilized to model the aerodynamics in statespace form, allowing time-domain simulation and analysis.The e®ect of the mechanical, geometric, and inertial properties of the overlapping segment and the extending segment on the aeroelastic behavior of the wing was assessed.The span extension segment (Segment 3) has signi¯cant e®ects on the aeroelastic behavior of the wing; however as its chord, bending rigidity, and torsional rigidity reduce, its e®ect on °utter signi¯cantly diminishes.In contrast, the overlapping region has higher e®ect on °utter speed at low span extensions, and the sensitivity of °utter to the properties of the overlapping region reduces as the wingspan increases.The e®ect of span morphing on the °utter mode is assessed for rectangular span morphing wing.It was found that mode 2 (¯rst torsion) is the ¯rst to go unstable.Finally, the feasibility of utilizing span morphing as a °utter suppression device was assessed.It was found out that span morphing can act as a °utter suppression device especially if high actuation/retraction rates are used to prevent large amplitudes oscillation.

Appendix A. Bending Shape Functions
The bending shape functions for the ith bending mode for wing Segments 1, 2, and 3 can be expressed as: These shape functions can be rearranged as: where The constants b 2 i and b 3 i for the ith bending mode can be expressed as: The dimension of the conditions matrix is n-by-n where n is equal to 4 multiplied by the number of wing's segments.To obtain the nontrivial solution (Eq.(A.7)), the determinate of the matrix is set to zero and the value of b 1 i is obtained numerically.Each time the determinate of the matrix becomes zero provides a new value for b 1 i corresponding to a vibration mode.The clamping condition at the wing root allows simplifying the ¯rst shape function to where The matrices S 1 , S 2 , and S 3 expressed in Table A.1, represent the continuity boundary conditions for each wing segment at its ends.The bending shape function of Segment 2 can be expressed as: ðA:10Þ This can be rearranged further such as and therefore Similarly, the bending shape function of Segment 3 can be expressed as where and therefore The free boundary condition (no bending moment and no shear force) at the tip of Segment 3 can be represented as ðA:16Þ The tip boundary condition can be rearranged as:   This allows expressing the shape functions as follows. ðA:18Þ The value of A 1 can be obtained through normalization such that the shape function is unity at the wingtip.The natural frequency of the ith bending mode can be expressed as: The torsion shape functions for the ith vibration mode for wing Segments 1, 2, and 3 can be expressed as: ðB:1Þ These shape functions can be rearranged as: The constants k 2 i and k 3 i for the ith torsional vibration mode can be expressed as To obtain the nontrivial solution of the above equations, the determinate of the lefthand side matrix must be set to zero.This is solved iteratively to ¯nd the values of k 1 i that make the determinate of the matrix zero.Each time the determinate becomes zero represents a new vibration mode (1st, 2nd, etc.).The clamping conditions at the root result in b 1 ¼ 0. Using the continuity boundary conditions, the coe±cients a 2 , a 3 , b 2 , and b 3 can be expressed as: It should be noted that a 2 , a 3 , b 2 , and b 3 depend on a 1 .The value of a 1 can be obtained through normalization such that the shape function is unity at the tip.The natural frequency of the ith torsion mode can be expressed as:

c
= chord of the aerofoil/wing " EI = bending rigidity " GJ = torsional rigidity hðyÞ = bending shape function I 0 ea = mass moment of inertia around the elastic axis l = length L 0 = lift per unit span L = equivalent lift force LE = leading edge m 0 = mass per unit span M 0 ea = pitching moment per unit span around the elastic axis M ea = equivalent pitching moment around the elastic axis s = Laplace variable t = time T = total kinetic energy U = total potential energy V = true airspeed x = distance between elastic axis and center of gravity w = plunge displacement at elastic axis y = spanwise location measured relative to the wing root = pitch angle ðyÞ = torsion shape function = 3 i = ith vibration mode j = jth wing segment Superscripts .= ¯rst time derivative .. = second time derivative , = ¯rst spatial derivative ,, = second spatial derivative 1. Introduction

Fig. 1 .
Fig. 1.A top view of the telescopic span morphing wing.

Fig. 5 .
Fig. 5. In°uence of the overlapping segment on the aeroelastic behavior of the Hale wing.

Fig. 6 .
Fig. 6.Frequency and damping trends for the Goland wing at di®erent span extensions.
Fig. 7. (Color online) Goland wing at the °utter speed and 5 AOA.At t ¼ 2 s, the semi-span is retracted by 25%.Two retraction speeds are considered (red thin curve for 1.5240 m/s and black thick curve for 15.240 m/s).

1 m 0 2 ðEIÞ 2 3 i ¼ b 1 i ðEIÞ 1 m 0 3 ðEIÞ 3
The root, continuity, and tip boundary conditions can be rearranged in the matrix format

1 I 0 ea 2 ðGJÞ 2
The dimension of the coe±cients matrix is m-by-m where m is equal to 2 multiplied by the number of wing segments.The coe±cient matrix can be expressed as

Table 1 .
Geometric and mechanical properties of the multi-segment wing.

Table 4 .
Geometric and mechanical properties of Goland and HALE wings.

Table 5 .
Validation using Goland and the HALE wings.Int.J. Str.Stab.Dyn.2019.19.Downloaded from www.worldscientific.comby UNIVERSITY OF BRISTOL on 06/20/19.Re-use and distribution is strictly not permitted, except for Open Access articles.

Table A .
1. Continuity boundary conditions across the wing's segments.