Aeroelastic characterisation of a bio-inspired flapping membrane wing

In this study, we present a novel bio-inspired membrane wing design shown in its flat and undeformed shape in figure 1(a). The leading and trailing edges of the wing are rigid and have brass bearings which allow the edges to rotate around their span-wise axes. The trailing edge has an additional degree of freedom as it can translate in the chord-wise direction in a frictionless manner. No pre-tension is applied to the membrane or the trailing edge sliders. Depending on the aerodynamic pressure difference between the pressure and suction side of the membrane, the wing passively cambers and the distance between the leading and trailing edge shortens (figure 1(b)). The adaptive wing is designed to enhance the aerodynamic performance of the wing in two ways: (1) the camber of the wing grows with increasing aerodynamic pressure, and (2) the leading and trailing edges of the wing rotate and align favourably with the flow. The combined effects of these two mechanisms yield higher lift versus angle of attack slopes, higher maximum values of the lift coefficient, and delayed stall to higher angles of attack due to a reduction in the effective angle of attack at the leading edge.

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The compliant membranes are made of a silicone-based vinylpolysiloxane polymer (Zhermack Elite Double 32 shore A) created by mixing a base and catalyst in a 1:1 ratio and centrifuging the mixture to homogenise it. The mixture is poured into a flat cast to create different silicon sheets of h = 0.3 mm to 1.4 mm thickness. The membrane sheets have a density of ρ = 1160 kg m⁻³ and a Young's modulus of E = 1.22  ± 0.05 kPa [56]. Rectangular wings are cut from the silicon sheets and fixed between two thin carbon fibre plates without pre-stretching the membranes (figure 1(a)). The rigid wings serving as a reference case have a Young's modulus more than three orders of magnitude higher than the membrane wings E = 1.33 MPa and a thickness of h = 1 mm. The reinforced membranes are glued to hollow, cylindrical brass rods which serve as the outer race of the leading and trailing edge bearings. The wings are mounted on the wing frame which itself is connected to the load cell and the flapping wing mechanism [57]. The membrane wings have a chord length c = 55 mm in their undeformed state and a wing span R = 150 mm (figure 1(a)). The root cut-out R₀ is the distance from the stroke-rotation axis to the root of the wing and is constant for all experiments. The pitch rotation axis is at a quarter chord length from the leading edge. The rigid carbon fibre plates and brass bearings make up l_rigid = 8.5 mm or 0.15c at the leading and trailing edges of the wings. A video of the membrane wings in motion and further information about the wing platform can be found in the gallery of fluid motion contribution ³ [58].

The wing kinematics in hovering flight are described in terms of their stroke angle ϕ, the angle of attack α, and the flap or elevation angle θ (figure 2(a)). The flap or elevation angle θ moves the wing up or down in stroke-normal direction (not displayed in figure 2). In this study, the elevation angle is not considered and stays constant at θ = 0°. The stroke plane coincides with the horizontal plane in the laboratory's frame of reference and the stroke angle ϕ defines the motion of the wing in the stroke plane. The stroke angle profile varies sinusoidally from ϕ = −45° to ϕ = 45° and changes in frequency between f = 0.125 Hz and f = 0.4 Hz (table 1). The angle of attack α indicates the wing's rotation relative to the stroke plane and is defined as the angle between the chord length and the horizontal stroke plane as indicated in figures 1(b) and (c). In this study, the wing rotation is symmetric with respect to the stroke motion such that the angle of attack equals 90° at the start and end of the stroke. The angle of attack follows a trapezoidal profile and reaches minimum values ranging from $\hat{\alpha }=15{{}^{\circ}}$ to $\hat{\alpha }=75{{}^{\circ}}$. Throughout this paper, we use $\hat{.}$ to denote amplitudes or minimum or maximum values of different quantities. The angle $\hat{\alpha }$ is kept constant during 68% of the total cycle duration T = 1/f. During the remaining 32% of the cycle, the wing reverses orientation between the symmetric front- and back-strokes, this is often referred to as the flip duration [45, 59].

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Table 1. Experimental parameters of the flexible membrane wings in this study. The experiments are conducted in water with ν_{20 °C} = 1.00 × 10⁻⁶ m² s⁻¹.

Parameters		Model wing
Wing chord	c	55 mm
Wing span	R	150 mm
Membrane thickness	h	0.3 mm to 1.4 mm
Wing stroke frequency	f	0.125 Hz to 0.4 Hz
Angle of attack	$\hat{\alpha }$	15° to 75°
Peak-to-peak stroke amplitude	ϕA	90°
Reduced frequency	k	0.42
Reynolds number	Re	2800 to 9300

Non-dimensional numbers characterising the flow around the hovering wing are the Reynolds number Re and the reduced frequency k. The fluid–structure interactions of the compliant membranes are characterised by the Weber number We and the aeroelastic number Ae defined in equations (3) and (4) [18, 35].

The Reynolds number Re defines the ratio between inertial and viscous forces as a measure for the emergence of flow structures at different length and time scales. In hovering flight, the Reynolds number can be calculated as

$$\begin{equation}\mathrm{R}\mathrm{e}=\frac{\bar{\text{U}}c}{\nu }=\frac{2{\phi }_{\mathrm{A}}fc{R}_{\mathrm{2}}}{\nu },\end{equation} \tag{ 5 }$$

where ν denotes the kinematic viscosity of the fluid and $\bar{\text{U}}=2{\phi }_{\mathrm{A}}f{R}_{\mathrm{2}}$ is the stroke-average wing velocity at the radius of second moment of area R₂ [45, 57, 60]. The radius of second moment of area is the span-wise position where the sum of all forces apply and is ${R}_{\mathrm{2}}=\sqrt{{\int }_{0}^{R}{({R}_{\mathrm{0}}+r)}^{2}\mathrm{d}r/R}$ for a rectangular wing planform [57]. The Reynolds number for all presented experiments in table 1 varies from Re = 2800 to 9300 which is a range where larger flying insects like the Hawkmoth (Manduca sexta), small birds like the Rufous Hummingbird (Selasphorus rufus), and several bat species like the Pallas's long-tongued bat (Glossophaga soricina) take flight [24, 61].

The reduced frequency k compares the spatial wavelength of the flow disturbance to the chord length c and is a metric for the unsteadiness of the flow:

$$\begin{equation}k=\frac{\pi c}{2{\phi }_{\mathrm{A}}{R}_{\mathrm{2}}}.\end{equation} \tag{ 6 }$$

Here, ϕ_A = 90° is the peak-to-peak stroke amplitude. The reduced frequency of all our experiments is k = 0.42 which is similar to many hovering flapping wing fliers in nature [24] and is considered to give raise to highly unsteady aerodynamics.

The aerodynamic performances of the different membrane wings and the rigid reference case are evaluated experimentally with a robotic flapping wing device. The experiments are conducted in an octagonal tank with an outer diameter of 0.75 m filled with water at a temperature of 20 °C. The mechanism is controlled by two servo motors (Maxon motors, type RE35, 90 W, 100 N mm torque, Switzerland) which guide the stroke and pitch axis of the system. The motors are equipped with planetary gear-heads of 35:1 and 19:1 reduction for stroke and pitch respectively and are controlled using a motion controller (DMC-4040, Galil Motion Control, USA). The experimental flapping wing apparatus is designed to be highly repeatable and robust over a large number of experiments. The system allows for complex time-varying kinematics to be executed on both motors and initial tests on all frequencies f and amplitudes $\hat{\alpha }$ show a maximum error of $< 0.1{{}^{\circ}}$ between the motor control signal and the motor response recorded by the encoder throughout the entire stroke.

The aerodynamic loads on the wing are measured using a six-axis IP68 force–torque transducer (Nano17, ATI Industrial Automation, USA) mounted at the wing root with a resolution of 3.13 mN for the force and 0.0156 N mm for torque measurements. The force signals from the load cell are recorded at a sampling frequency of 1000 Hz with a data acquisition module (NI-9220, National Instruments, USA). The instantaneous lift L, drag D and pitch torque T_p are directly retrieved from the load transducer. Here, the lift force L is considered to be the component of the total force vector oriented upwards, perpendicular to the horizontal stroke plane. The drag D is the force component in the stroke plane. The drag component is considered positive when it acts in the direction of the instantaneous velocity U of the flow experienced by wing during its stroke motion (figures 1(b) and (c)). The aerodynamic power P is determined as the sum of pitch power P_p and stroke power P_s. The power expended to rotate the wing around its pitch axis with an angular velocity $\dot{\alpha }$ is determined by ${P}_{\mathrm{p}}=-{T}_{\mathrm{p}}\dot{\alpha }$. Analogously, the power required to rotate the wing around its stroke axis is found with the stroke torque T_s and the stroke velocity $\dot{\phi }$ according to ${P}_{\mathrm{s}}={T}_{\mathrm{s}}\dot{\phi }$. We cannot measure the stroke torque directly but instead determine it from the drag force D across the span T_s = ∫_R D(r)rdr [45]. The stroke torque is calculated as T_s = DR_d and the radial position ${R}_{\mathrm{d}}=\frac{3}{4}\frac{{({R}_{\mathrm{0}}+R)}^{4}-{{R}_{\mathrm{0}}}^{4}}{{({R}_{\mathrm{0}}+R)}^{3}-{{R}_{\mathrm{0}}}^{3}}$ where the sum of the drag force applies assuming a uniform drag coefficient distribution across the span [57].

Forces and torques are recorded over 16 consecutive cycles. The first five cycles are discarded to account for transient effects. The force and torque measurements are averaged over the remaining 11 cycles to obtain phase-averaged temporal evolutions of the results within the flapping period T and to determine the overall mean values of the aerodynamic coefficients. A minimum tip clearance of 3.5c is found for a tip-to-tip stroke-amplitude of ϕ_A = 90° which has been shown sufficient to avoid wall effects in flapping wing experiments [59, 62]. Initial force measurements at different distances from the wall verify that for the selected stroke amplitude and frequencies no changes in the stroke-average forces are identified for wing tip to wall distances $> 3c$. Initial experiments collecting data for 64 cycles demonstrate that there are some force variations due to recirculation of the tank but these only take effect after the first 16 cycles that are considered here. The influence of the large-scale recirculation in the tank are below 2% compared to the mean force coefficients presented in this study (see appendix A for more details).

For high-frequency flapping wing flight in air, the wing inertial forces can be as strong as the aerodynamic pressure forces [50]. We quantified the effect of the wing inertia on the total forces by conducting additional experiments in air (appendix B). Here, the aerodynamic forces become negligible and the inertial forces dominate the force measurements. The average lift and drag coefficients measured in air are largest for the high frequency cases but remain at least one order of magnitude lower than the results measured in water. For low flapping frequencies the dimensional inertial lift and drag are even lower and drop below the load cell resolution (3.13 mN). The inertial forces are therefore deemed negligible in this study.

In this work, we also perform stereo deformation measurements of the flexible wing platform to quantify the membrane shape and its influence on the aerodynamic forces. Deformation and load measurements are conducted simultaneously for selected experimental conditions to allow for direct comparison of deformation and aerodynamic force response. Two CCD cameras (pco.pixelfly usb, ILA_5150 GmbH/PCO AG, Germany) at a 45° stereo angle and equipped with 12 mm focal length lenses are used to perform marker tracking on the membrane wing throughout the entire flapping wing cycle (figure 2(b)). With a camera resolution of 1392 px × 1040 px, the deformation measurements have a spatial resolution between 0.13 mm px⁻¹, when the wing is closest to the cameras, and 0.23 mm px⁻¹, when the wing is farthest away from the cameras. We record stereo images over 20 cycles. The first five cycles are discarded to remove transient effects. The deformation measurements of the remaining 15 cycles are phase-averaged over one half-cycle, using the symmetry between front- and back-stroke of the flapping motion [57]. The images are recorded at an acquisition frequency of 6.17 Hz uncorrelated to the flapping frequency f. In consecutive cycles the images correspond to different phase-times t/T within one full cycle T. After sorting the frames by their relative phase-time we achieve an increased image acquisition rate of 186 Hz per half-cycle or between 465 and 1488 times the flapping frequency f.

The membrane is covered with black markers and the carbon fibre and structural parts with white markers to ensure high contrast (figure 2(a)). The markers are tracked using the software tool XMALab, an open-source software for marker-based x-ray reconstruction of moving morphologies [63]. The stereo camera configuration is calibrated using a chequerboard recorded at different positions and rotation angles for both cameras. The calibration procedure determines all camera intrinsic parameters, calibration coefficients and the image distortion matrix. The markers are tracked at each time step in the undistorted stereo images and their position in three-dimensional space reconstructed using linear triangulation [64]. The orientation and position of the rigid parts of the wing platform are determined through rigid body transformation. Finally, we determine the membrane shape with a two-dimensional polynomial which fits best the membrane markers and the angle between the membrane and the rigid edges of the wing in a least-square sense.

The flapping wing kinematics in all experiments are defined by the stroke and pitch angle profile presented in figure 3(a). Here, only the first half of the cycle is presented as the front- and back-strokes of the flapping wing motion are symmetric. At the beginning and end of each half-cycle the wing rotates to keep the leading edge ahead of the trailing edge. The duration of the wing rotation is indicated by the grey area in figure 3(a). During the majority of the cycle, the angle of attack remains at a constant value of $\hat{\alpha }$. For brevity, we will refer to the constant minimum angle of attack during the stroke motion simply as the angle of attack. The system's base frequency f and the angle of attack $\hat{\alpha }$ are varied over a wide range shown in table 1 to characterise the aerodynamic performance of the different membrane wings in comparison to rigid reference cases.

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The stroke-average lift coefficient ${\bar{C}}_{\mathrm{L}}$ and hovering efficiency $\bar{\eta }$ for all tested flapping frequencies f, wing thicknesses h, and angles of attack $\hat{\alpha }$ are presented in figures 3(b) and (c). The black curve connects the results for the rigid reference case averaged over all stroke frequencies f. As the rigid wings do not deform, low variance in ${\bar{C}}_{\mathrm{L}}$ and $\bar{\eta }$ is observed for each angle of attack $\hat{\alpha }$ compared to the membrane wings. The results of the best performing membrane wings are highlighted by the large green markers. The small markers indicate individual test cases. In general, the membrane wings reach maximum stroke-average lift ${\bar{C}}_{\mathrm{L}}$ and efficiency $\bar{\eta }$ values in a range around the values attained by the rigid reference cases for the same angle attack $\hat{\alpha }$. By varying the thickness of the membranes, we cover a relevant parameter range in terms of membrane stiffness. Our set of tested membrane wings include membranes that perform better and some that perform worse than their rigid counterparts. For the lowest angle of attack $\hat{\alpha }=15{{}^{\circ}}$, all membrane wings tested at best match the performance of the rigid wing. For the highest angle of attack $\hat{\alpha }=75{{}^{\circ}}$, all membrane wings perform equally well or better than the rigid wing.

The highest stroke-average lift is found at $\hat{\alpha }=55{{}^{\circ}}$ with ${\bar{C}}_{\mathrm{L},\mathrm{m}\mathrm{a}\mathrm{x}}=2.43$ for the membrane wings and at $\hat{\alpha }=50{{}^{\circ}}$ with ${\bar{C}}_{\mathrm{L},\mathrm{m}\mathrm{a}\mathrm{x}}=2.06$ for the rigid wing.

The variance in aerodynamic performance of the membrane at each angle of attack $\hat{\alpha }$ in figures 3(a) and (b) is due to the deformation of the membrane which depends on the ratio between the dynamic pressure on the wing as a function of the flapping frequency f and the rigidity or bending stiffness of the membrane which varies with membrane thickness h. The aeroelastic number Ae (equation (4)) represents the ratio between membrane compliance and the dynamic pressure and is used to further characterise the influence of the fluid–membrane interaction on the aerodynamic performance of the membrane wings. Note that no additional pre-tension is applied to the membrane or the trailing edge sliders that could influence the effective stiffness of the system.

The lift produced in hovering flight is presented in figure 4 as a function of the aeroelastic number Ae (equation (4)) for all conducted experiments. Overall, the stroke-average lift of the membrane wings increases with increasing angle of attack for $\hat{\alpha }\leqslant 50{{}^{\circ}}$ and decreases for $\hat{\alpha }\geqslant 55{{}^{\circ}}$. For visual clarity, the results are split in different panels in figure 4. The two panels to the left (figures 4(a) and (b)) include all experiments with angles of attack ranging from $\hat{\alpha }=15{{}^{\circ}}$ to 50° for the flexible membrane wings (Ae = 0.25 to 12 in figure 4(a)) and the rigid wings (Ae = 870 to 8900 in figure 4(b)). The markers represent individual experiments. The solid lines indicate the average variation of ${\bar{C}}_{\mathrm{L}}$ as a function of Ae and are obtained by training a Gaussian process regression model at each angle of attack $\hat{\alpha }$ and predicting the response over the Ae range of the experiments. The stroke-average lift coefficient ${\bar{C}}_{\mathrm{L}}$ for $20{{}^{\circ}}\leqslant \hat{\alpha }\leqslant 50{{}^{\circ}}$ increases with increasing Ae until it reaches a maximum between Ae = 1 and Ae = 5. The maximum stroke-average lift coefficient ${\bar{C}}_{\mathrm{L}}$ is reached at slightly lower values of the aeroelastic number at higher angles of attack. For Ae > 5 and $\hat{\alpha } > 20{{}^{\circ}}$, the stroke-average lift decreases with increasing Ae and asymptotically converges to the average lift produced by the rigid wings (figure 4(b)).

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At lower angles of attack $\hat{\alpha }\leqslant 20{{}^{\circ}}$, the maximum stroke-average lift values are obtained for the stiff wings and the flexible wings with the highest values of Ae. For the lowest angle of attack $\hat{\alpha }=15{{}^{\circ}}$, the stroke-average lift initially decreases with increasing Ae for Ae < 1 and we notice larger variations among individual experiments. Observations of the experiments showed that the cambering of the membrane wings started later into the cycle compared to higher angles of attack and that the amount of camber showed more variability between consecutive strokes. Additional experiments at the lower angles of attacks are required to determine if this effect is caused by unsteady aeroelastic effects or limitations of the experimental system. For higher angles of attack $\hat{\alpha } > 50{{}^{\circ}}$, the overall stroke-average lift decreases with increasing angles of attack (figures 4(c) and (d)). At these high angles of attack, the flexible wings attain values of the stroke-average lift above those attained by their solid counterparts for all values of the aeroelastic number. The aeroelastic number at which the overall maximum stroke-average lift coefficient is reached remains around Ae ≈ 1 and continues to decreases with increasing angles of attack.

To explain the differences observed among the stroke-average lift and efficiency measured of the flapping membrane and rigid wings, we have extracted the membrane shape deformation for selected wings. The membrane wings passively deform when flapping. The main characteristics of the passive deformation are the cambering of the membrane, the rotation of the leading edge and the rotation of the trailing edge (figure 5(a)). The stroke-maximum camber ${\hat{z}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ and the stroke-maximum rotation angles of the leading and trailing edges with respect to the chord, denoted by γ_LE,max and γ_TE,max, respectively, are presented as a function of the aeroelastic number in figures 5(b) and (c) for $\hat{\alpha }=55{{}^{\circ}}$. Positive angles of γ are associated with a positive camber $\hat{z}$ following the convention indicated in figure 5(a). The stroke-maximum values reveal a clear variation with the aeroelastic number which relates the stiffness of the membrane and the aerodynamic pressure. At low values of the aeroelastic number, a relatively smaller amount of aerodynamic pressure is required to deform the wing. This results in large values of the stroke-maximum camber and the stroke-maximum leading and trailing edge rotation angles. With increasing values of the aeroelastic number, the effective stiffness of the wing increases and the maximum camber and maximum leading and trailing edge rotation angles decrease towards zero for values of Ae ≫ 1.

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The decrease of the stroke-maximum camber ${\hat{z}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ with increasing aeroelastic number follows an exponential decay as evidenced by the linear evolution in the semi-log space. Exponential fits in form of straight lines are included in the semi-logarithmic plots as a reference. The decrease of the leading and trailing edge rotation angles also appear close to linear in the semi-log space at first glance, but a closer look reveals a more complicated variation. The leading and trailing edge rotation angles are identical for Ae ⩾ 1.68, indicating symmetric cambering at higher values of the aeroelastic number. For lower values of the aeroelastic number and thus reduced effective membrane stiffness, the fluid–membrane interaction leads to asymmetric bending of the membrane and a shift of the maximum camber position towards the leading edge where the rotation angle is largest. The asymmetric bending is the result of a more complex fluid structure interaction between the aerodynamic pressure due to the stroke motion and a prominent leading edge vortex. The change between symmetric and asymmetric camber lines occurs between Ae = 1 and Ae = 2 which corresponds to the range of aeroelastic numbers where we observe a maximum in the stroke-average lift coefficient for this angle of attack $(\hat{\alpha }=55{{}^{\circ}})$. The lift-optimal stroke-maximum camber in this case is ${\hat{z}}_{\mathrm{m}\mathrm{a}\mathrm{x}}\approx 0.2c$. An increase in the maximum camber ${\hat{z}}_{\mathrm{m}\mathrm{a}\mathrm{x}}/c$ beyond 0.2 results in a decrease of the stroke-average lift. To better understand the inertial force response on the membrane wing, we will now look at the orientation of the leading and trailing edges with respect to the stroke plane.

The passive cambering of the membrane wings and the rotation of the leading and trailing edges lead to a decrease in the leading edge angle of attack with respect to the stroke plane and a decrease in the angle between the trailing edge and the stroke plane as defined in figure 6(a). The stroke-minimum of the leading angle α_LE,min and the stroke-maximum of the trailing edge angle α_TE,max are presented as a function of the aeroelastic number in figures 6(b) and (c) for $\hat{\alpha }=55{{}^{\circ}}$. As expected based on the measured decay of the maximum camber, the minimum leading and trailing edge angles both increase with increasing aeroelastic number in an approximately exponential manner. For the higher values of the aeroelastic number, where the deformation is symmetric, the angles lie above the fitted approximations. For the lower values of the aeroelastic number, where the deformation is no longer symmetric, we find angles below the fitted approximation.

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The maximum stroke-average lift coefficient and the onset of asymmetric membrane deformation occur at the same aeroelastic number of Ae ≈ 1.7. The measured membrane shape at these lift-optimal aeroelastic conditions is shown in figure 6(a1). The rotation of the leading edge has reduced the effective angle of attack at the leading edge to a more moderate angle and the trailing edge is oriented vertically with respect to the stroke plane. The reduced leading edge angle of attack allows the flow to accelerate smoothly around the leading edge and the trailing edge orientation directs the flow straight downwards in the near wake increasing the vertically upward force on the wing itself.

By decreasing the aeroelastic number Ae for the same angle of attack $\hat{\alpha }$, the deformation of the membrane increases but the stroke-average lift decreases. This is attributed to the trailing edge angle decreasing below α_TE,min < 90° (dashed line in figure 6(c)) and the leading edge angle decreases to values below the static stall angle of a flat plate which delays flow separation and the formation of a strong leading edge vortex. The stroke-maximum membrane deformation at Ae = 0.61 is presented in figure 6(a2). At this low value of the aeroelastic number, the chord-normalised camber becomes as high as 30% and the leading edge angle even becomes negative which causes the lift coefficient to drop below the values of their stiff counterparts. The stroke-average lift of our flapping wings is increased when they are deformed up to a point when they over-camber. Here, we consider a membrane to be over-cambered when its minimum leading edge angle α_LE,min becomes negative or its minimum trailing edge angle α_TE,min rotates below 90° as marked by the dashed lines in figures 6(b) and (c). The performance of over-cambering wings rapidly falls with decreasing aeroelastic number.

We further explore the relationship between the aeroelastic properties and their contribution to the force production on the wing by analysing the temporal evolution of the lift and power coefficients for the highest lift producing angle of attack $\hat{\alpha }=55{{}^{\circ}}$ in figures 7(c) and (d). The colours indicate the range of aeroelastic numbers from Ae = 0.61 to 4.96. Here, all quantities are phase-averaged over one half-cycle (t/T = 0 to 0.5) exploiting the symmetry between the front- and back-stroke of the hovering cycle [57]. The results of the rigid wing are presented by the black dashed line. The phase-averaged temporal evolutions of the lift and power were filtered using a 5th-order Butterworth filter with no phase delay at a cut-off frequency 12 times higher than the flapping frequency f.

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The overall temporal evolution of the lift and the power coefficient for all wings is dominated by the evolution of the stroke velocity $\dot{\phi }$ (figure 7(b)). Lift and power increase or decrease with increasing or decreasing stroke velocity and reach a maximum value around t/T = 0.25. Interesting differences are observed when the aeroelastic number varies. The main differences in the lift coefficient evolution occur towards the end of the first half and during the second half of the stroke (t/T > 0.2) (figure 7(c)). The flexible wings all reach a higher maximum lift coefficient than their rigid counterpart. The highest value of C_L is measured for an aeroelastic number of Ae = 1.64. The higher lift values for the flexible wings are maintained during the second half of the stroke. The competition for the top rankings among different aeroelastic numbers is decided in the second half of the stroke. This holds true for all angles of attack as evidenced by the contour plots of the average lift coefficient across the first and second half of the stroke (figures 7(e1) and (e2)). The average lift coefficient is consistently lower during the first half of the stroke than during the second half for all values of the aeroelastic number and all angles of attack. The average lift coefficient during the first half does not vary substantially with the aeroelastic number for $\hat{\alpha }< 45{{}^{\circ}}$. For $45{{}^{\circ}}< \hat{\alpha }< 70{{}^{\circ}}$, the average lift increases with increasing aeroelastic number for Ae < 1 and reaches a plateau for Ae > 1. The average lift coefficient during the second half shows a clear global maximum for $\hat{\alpha }\approx 50{{}^{\circ}}$ and Ae ≈ 1.7. This corresponds to conditions that yield the overall best performance.

The analysis of the power coefficient C_P leads to the same conclusion that the performance difference by the deformable wings is made in the second half of the stroke (figures 7(d) and (f)). All power curves follow a similar increase in power with increase in the stroke velocity in the first part of the stroke (figure 7(d)). A maximum in C_P is reached slightly before mid-stroke. The highest peak in power belongs to the rigid wing. In the second half of the cycle, the aerodynamic power curves associated with the deformable wings remain below the rigid wing curve for most of the second part of the stroke. The least amount of power is required by the lowest value of the aeroelastic number which is the most deformable wing. The least deformable wing with the highest value of Ae requires approximately the same amount of power as the rigid wing. For all tested angles of attack $\hat{\alpha }$ and aeroelastic numbers, the first half of the stroke requires more power than the second (figures 7(f1) and (f2)). The strongest dependence on the aeroelastic number is observed for angles $\hat{\alpha } > 45{{}^{\circ}}$ in the second half of the stroke (figure 7(f2)). Here, the power is drops significantly with decreasing values of Ae. At low Ae, the stronger cambering of the membrane leads to a decrease in the frontal area which leads to reduced drag and power coefficients.

To explain the higher aerodynamic performance at moderate Ae and the variations during the second half of the stroke, we analyse the temporal evolution of the passive membrane deformation characterised by the camber $\hat{z}$ and the leading and trailing edge angles α_LE and α_TE in figure 8. The data is presented for the lift-optimal angle of attack $\hat{\alpha }=55{{}^{\circ}}$ over one stroke or half the flapping cycle. The results of the rigid wing are presented by the black dashed line. The line colour of the solid lines indicates the value of the aeroelastic number.

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The temporal evolutions of the membrane camber $\hat{z}$ for the deformable wings roughly follow a trapezoidal shape in response to the trapezoidal variation in their geometric angle of attack α (figure 8(a)). The camber for the lowest values of Ae reaches the highest camber of $\hat{z}/c=0.28$ just before mid-stroke and keeps a steady camber over a larger portion of the second half of the stroke. We refer to the time interval during which the camber remains at its maximum value as the camber plateau time. The time at which the maximum camber is reached does not vary significantly with effective membrane stiffness expressed by the aeroelastic number. The deformation increases as long as the stroke velocity increases. Yet, the rate of change of the camber during the first part of the stroke does not vary consistently with the stroke velocity and the response strongly depends on the value of Ae. For the lowest values of Ae, we measure an immediate increase of the camber already during the rotation phase. For the higher values of Ae, the increase in camber becomes more prominent after the wing rotation has ended. This suggests that the rotational acceleration affects the initial response of the membrane, but it does not affect the maximum camber which was shown to scale well with the aeroelastic number (figure 5(a)). The maximum camber decreases with increasing values of Ae and this holds try for any value of the camber $\hat{z}$ at all times during the stroke. The camber plateau time also reduces with increasing values of Ae. At the highest presented aeroelastic number, the membrane cambers up to $\hat{z}/c=0.08$ and immediately reduces again during the second part of the stroke. A plateau of constant camber is no longer observed. The persistence of the higher camber during the second part of the stroke at lower aeroelastic numbers leads to higher lift and a reduction of the power during that time (figure 7).

The temporal evolution of the phase-average leading and trailing edge angles with respect to the stroke plane are presented in figures 8(b) and (c). When the membrane cambers more at lower Ae, the leading and trailing edges angles drop below the corresponding reference values imposed on the rigid wing. All leading edge angles of attack α_LE in figure 8(b) reach a minimum angle during the wing's camber plateau time figure 8(a). With increasing deformation at lower Ae, the leading edge angle of attack minima decrease and are attained later in the second half of the stroke. At the lowest Ae, the leading edge angle of attack becomes negative α_LE = −3° at t/T = 0.35 in figure 8(b). The excessive rotation of the leading edge due to over-cambering is accompanied by a decrease in lift coefficient C_L but also by reduction in power C_P (figures 7(c) and (d)).

The trailing edge angle α_TE evolves differently than the leading edge angle (figure 8(c)). The trailing edge orientation appears symmetric around mid-stroke and all curves of α_TE follow an M-shape evolution. Two peaks emerge at the end and the start of the wing rotations. After the first peak, the trailing edge angle decreases and the trailing edge rotates in the direction of the motion. For moderate to high values of Ae, the angle remains constant for a part of the stroke and the duration of this constant angle phase increases again with decreasing aeroelastic number, similar to the behaviour of the camber plateau time. At lift-optimal aeroelastic conditions, the trailing edge orientation is perpendicular to the flow with α_TE = 90°. Further increase in the camber $\hat{z}$, by lowering Ae, causes the trailing edge to rotate into the flow at angles down to α_TE = 70°. The trailing edge rotation α_TE < 90° indicates over-cambering and coincides with a decrease in lift production by the membrane wings.

In the previous sections, we analysed the relationship between lift coefficient C_L, aeroelastic number Ae for the lift-optimal angle of attack $\hat{\alpha }=55{{}^{\circ}}$. In figure 9, we summarise the results of the stroke-average lift coefficient ${\bar{C}}_{\mathrm{L}}$ and hovering efficiency $\bar{\eta }$ (equation (7)) for the entire experimental parameter range covered in this study. The input parameter space is spanned by the angle of attack $\hat{\alpha }$ and the aeroelastic number Ae. The two markers in the contour plots indicate the optimal-lift (×) and optimal-efficiency (◦) points which lie at different locations.

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The maximum lift ${\bar{C}}_{\mathrm{L},\mathrm{m}\mathrm{a}\mathrm{x}}=2.43$ is found at Ae = 1.68 and $\hat{\alpha }=55{{}^{\circ}}$ (figure 9(a)). The most efficient hovering $\bar{\eta }=0.858$ occurs at a higher aeroelastic number Ae = 2.44 and a lower angle of attack $\hat{\alpha }=33{{}^{\circ}}$ than the lift optimum (figure 9(b)). In literature, we found an experimental study using compliant flat plate wings with a stiff leading edge that reports an enhancement of the aerodynamic performance of these flexible flapping wings in hover compared to rigid wings for a range of effective stiffness from Π₁ = 0.5 to Π₁ = 10 [66]. The maximum lift production in [66] was found at Π₁ = 3.5. We find maximum lift production at a lower value of Π₁ = 1.17 using $\mathrm{A}\mathrm{e}={{{\Pi}}_{\mathrm{1}}}^{3}$. The highest efficiency for our wings occurs at Π₁ = 1.34. Both optima still fall within the range of effective stiffness values that are reported to enhance aerodynamic performance by [66]. Compared to different insect species, the optimum chord-wise effective stiffness in our study matches with the tarantula-hawk wasp (Pepsis grossa) which has similar aspect ratio wings than the membrane wings in our study [66, 67]. A recent study on membrane wings for energy harvesting applications discovered highest energy extraction at their lowest tested aeroelastic number Ae = 5(Π₁ = 1.71) [54].

Rigid flapping wings at lower angles of attack promote the stable growth of a leading edge vortex and a lift favouring orientation of the normal force vector leading to more efficient hovering flight [57]. A shift of the most efficient hovering conditions to larger value of Ae indicates a higher relative stiffness of the membrane and lower camber. The same leading edge deflection of up to γ_LE,max = 45° observed at $\hat{\alpha }=55{{}^{\circ}}$ and lift-optimal aeroelastic number Ae = 1.68 would lead to over-cambering and negative leading edge angles of attack α_LE at a geometric angle of attack $\hat{\alpha }=33{{}^{\circ}}$ of most efficient hovering.

To quantify the relationship between aerodynamic performance and membrane deformation over the entire experimental parameter range, a Weber number is calculated, We = C_N /Ae = N/(EhR), with N the chord-normal force (equation (3)) [18, 35]. The Weber number represents the ratio between inertial forces, e.g. lift and drag on the wing, and the membrane tension. Previous work on membranes at a steady free-stream velocity and angle of attack demonstrated that the maximum membrane camber ${\hat{z}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ increases with increasing Weber number for a pre-strain parameter ₀ (equation (3)) [18, 35]. The trailing edge of the membrane wing in this study can move freely in chord-wise direction and a pre-strain of ₀ = 0 can be assumed. The Weber number is presented in figure 10(a) for all our experiments in the parameter space spanned by Ae and $\hat{\alpha }$. The two markers indicate the global lift-optimum (×) and efficiency-optimum (◦) found in figure 9. The Weber number decreases from We = 2.20 to 0.94 between the lift and the efficiency optimum. With a reduction in Weber number the pressure and lift pulling on the membrane decrease relative to the stiffness of the membrane and less deformation is expected. Assuming the membrane camber is a direct function of the Weber number [18, 35], we can infer the change in maximum membrane camber ${\hat{z}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ and deflection angle γ_max from the deformation measurements at $\hat{\alpha }=55{{}^{\circ}}$ in figures 10(b) and (c). For most efficient hovering at We = 0.94 the camber is predicted to drop below ${\hat{z}}_{\mathrm{m}\mathrm{a}\mathrm{x}}< 0.15c$ and limiting γ_max < 25° preventing over-cambering and negative angles of attack at the leading edge. The relationship between the increase in the maximum camber $\hat{z}$ and the Weber number in figure 10(b) agrees qualitatively with experiments conducted for fixed membrane wings without pre-strain [18, 35]. Compared to the results under steady flow conditions, the Weber number in our study is one order of magnitude higher. This difference in Weber number is attributed to the change to unsteady flapping wing flight and to a difference in the mechanism that causes the membrane deformation. In [18, 35], the membrane deformation is dominated by stretching. In our study it is dominated by bending.

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Two distinct regions of maximum lift production and hovering efficiency are identified in the full experimental parameter range. The ability to move from high lift production to efficient hovering by modifying the angle of attack or the aeroelastic properties makes the membrane wing platform a promising model for the design of novel micro air vehicles. Many natural fliers already modify their wings' angle of attack and stiffness in flight to adjust to different flight situations, for example when performing manoeuvres or to alleviate a gust encounter. Different types of flapping wing vehicles have already incorporated angle of attack variation in their designs and progress has been made recently in developing membrane wings with variable stiffness [68, 69]. Our results show that combining both effects, variable stiffness and angle of attack variation, enhances the aerodynamic performance and has the potential of improved control capabilities of micro air vehicles.