Experimental study on the aerodynamic optimization design of flexible wings for tailless flapping wing micro air vehicles

Abstract

Flexible wings, serving as the key components of tailless flapping wing micro air vehicles (FWMAVs), simultaneously generate lift, thrust, and control torques. Due to the complex unsteady fluid-structure interactions involved in their flapping, accurately predicting their aerodynamic performance, such as mean lift and lift-to-power efficiency, becomes challenging. There is also a lack of widely accepted and rational design methods for flexible wings. To address these, we propose an experimental optimization design method based on response surfaces methodology and investigate the impact of four design parameters—aspect ratio ( $A$ ), slack angle ( $θ$ ), taper ratio ( $λ$ ), and flapping frequency ( $f$ )—on the aerodynamic performance of flexible wings. The results show that the models accurately predict the aerodynamic performance of flexible wings, with an error margin of less than 10% compared to experimental measurements. Utilizing these models, an optimal flexible wing for a tailless FWMAV with a mass of 15 g was designed and manufactured, which can generate 15.24 gf of lift while maintaining a lift-to-power efficiency of 6.07 gf/W. Additionally, the models indicate that the four parameters are nearly equally important for the aerodynamic performance of flexible wings, and the coupling between these parameters also significantly affects the aerodynamic performance. Specifically, $A & λ$ , $A & f$ , $θ & f$ , and $λ & f$ affect mean lift, while $A & λ$ , $θ & λ$ , and $θ & f$ affect lift-to-power efficiency. These coupling effects help explain the contradictions found in previous studies regarding the influence of different parameters. Our research provides clear guidance and practical methods for designing flexible wings in tailless FWMAVs.

Keywords

Flapping wing flexible wing aerodynamic design response surface methodology micro air vehicles

Introduction

Tailless FWMAVs, a category of micro air vehicles that mimic the flight of hummingbirds or insects, have emerged as a prominent research topic^1–4 due to their significant advantages in flight performance, safety, and concealment compared to other micro air vehicles. Numerous teams have successfully developed prototypes of tailless FWMAVs.^5–14 Flexible wings, serving as the crucial elements of tailless FWMAVs, simultaneously generate lift, thrust, and control torques. Designing high-performance flexible wings is essential for enhancing the capabilities of tailless FWMAVs.

When evaluating the aerodynamic performance of flexible wings, the mean lift generated by flexible wings and the ratio of mean lift to power consumed (also known as lift-to-power efficiency or power loading) are commonly used criteria, with higher values indicating superior performance. Despite well-defined design objectives, widely accepted design methods for flexible wings remain scarce, primarily because flexible wing flapping involves complex unsteady fluid-structure interactions, and their performance is influenced by various factors, such as shape, structure, material, and flapping motion profiles.^6,14–28 The existing studies and optimization design methods for flexible wings, particularly simulation-based approaches, are challenging and computationally expensive, often restricted to 2D/3D homogeneous isotropic flexible wings^{19,21,29–34} and biological wings,^35–37 which differ substantially from the actual flexible wings of FWMAVs. The common experimental optimization method, known as the “preferred method”, involves designing flexible wings for FWMAVs by fabricating numerous wings with various shapes, structures, or materials, testing their aerodynamic performance, and selecting the best-performing design.^{5–8,10,18,20,28} The “preferred method” is constrained by unclear theoretical guidance and high experimental costs, and may lead to conclusions that are not widely accepted. For instance, Keennon et al.⁶ found that increasing the taper ratio (the ratio of the wing tip chord length to the wing root chord length) improves lift-power efficiency, making rectangular wings more favorable. In contrast, Nan et al.¹⁶ concluded that the lift-to-power efficiency achieved with trapezoidal wings is superior to that with rectangular wings. Addo-Akoto et al.²² determined that a simple wing-veined rectangular wing with a 5° slack angle (also known as camber angle) exhibits the best lift-to-power efficiency, while Yoon et al.²¹ concluded that the optimal slack angle for maximum lift-to-power efficiency is approximately 15°. Furthermore, the majority of the aforementioned studies have concentrated on examining the impact of individual factors on the performance of the flexible wing while ignoring the coupling effects of these factors.

In response to these issues, we introduce an experimental optimization design method for flexible wings using the response surface methodology. This method establishes polynomial models to evaluate the performance of flexible wings influenced by four parameters (Section 2) and obtains optimal design results based on the model and the design constraints of tailless FWMAVs (Section 3). Furthermore, using the polynomial model (response surface model) derived during the optimal design process, we qualitatively analyze the effects of design parameters on flexible wing performance, considering the effects of individual design parameters as well as the coupling effects of multiple parameters (Section 3).

Experimental setup

Design of the response surface experiment

The flexible wing used in this study comprises a membrane, two straight veins, and two sleeves located respectively at the leading edge and the wing root (see Figure 1), resulting in a simple shape that is easily parameterized. The wing can rotate flexibly around both the leading edge and the wing root simultaneously. Additionally, the membrane has a slack angle ( $θ$ ) and is almost unable to resist bending loads. As a result, the flexible wing undergoes a spanwise negative torsional deformation and a positive camber deformation during flapping, as illustrated in Figure 1 (b) and (c).

Figure 1.

The structure of flexible wings in this study.

The design of the flexible wing can be parameterized in the following manner. As illustrated in Figure 2, when the angle between the vein near the wing root and the chord line is fixed at 55°, the trapezoidal flexible wing can be defined by four parameters: wing length ( $R$ ), aspect ratio ( $A = R / c_{r}$ ), slack angle ( $θ$ ), and taper ratio ( $λ = c_{t} / c_{r}$ ), where $c_{t}$ is the tip chord and $c_{r}$ is the root chord. Additionally, as most flapping mechanisms cannot alter their motion profile, once the flapping mechanism is determined, the process of motion and deformation of the flexible wing and its aerodynamic performance, will depend entirely on the wing itself and the flapping frequency ( $f$ ). Considering the constraints on wing length ( $R$ ) commonly encountered in aircraft design, here R is specified as a fixed value (65 mm). Consequently, the flexible wing in Figure 2 could be characterized by only four parameters: A, $θ$ , $λ$ , and f. The parameterized method described above has been employed in several tailless FWMAVs,^6,16,28 and previous studies^{6,7,10,16,18,19,21,22,24} have demonstrated that these parameters significantly affect the aerodynamic performance of flexible wings.

Figure 2.

Geometric shape of trapezoidal flexible wings.

Utilizing above four parameters as optimization design variables, the response surface model to be established could be described as:

\begin{matrix} \bar{L} = \bar{L} (A, θ, λ, f) \\ E_{M} = E_{M} (A, θ, λ, f) \end{matrix}

(1)

The optimization ranges of the four parameters need to be ascertained next. The range of the aspect ratio ( $A$ ) is selected as [2.0,3.0] following most tailless FWMAVs^6,7,9–12; the slack angle ( $θ$ ) is defined within [8°,18°] to achieve optimal aerodynamic performance based on previous studies^16,22,24,25; the range of the taper ratio ( $λ$ ) is [0.5,1.0], grounded in the research by Keennon et al.⁶ and Nan et al.;¹⁶ the range of the flapping frequency ( $f$ ) is specified as [25,30] Hz to guarantee sufficient mean lift generated by the flexible wings. To eliminate the numerical errors of the models, these parameters are normalized to the interval of [–1,1], as shown in Equation (2). Here, the overbar “ $\bar{}$ ” denotes the range size of the variable, “ $\hat{}$ ” indicates the normalized parameter, and the subscript “ $0$ ” denotes the mean value of the range.

\begin{aligned} A = \frac{1}{2} \bar{A} \hat{A} + A_{0} \\ θ = \frac{1}{2} \bar{θ} \hat{θ} + θ_{0} \\ λ = \frac{1}{2} \bar{λ} \hat{λ} + λ_{0} \\ f = \frac{1}{2} \bar{f} \hat{f} + f_{0} \end{aligned}

(2)

It should be noticed that the wing stiffness has a significant influence on the aerodynamic performance, particularly near resonant conditions.^26,31,38 In our study, the flapping frequency is determined by the lift required and wing size, which is designed within 25–30 Hz and far from the resonant frequency of the flexible wings (when the flexible wing is regarded as a homogeneous cantilever beam,³⁹ its first-order natural frequency is approximately 90 Hz). Therefore, the wing stiffness is not considered as a design parameter in this study.

The experimental design for the response surface was carried out utilizing Design-Expert software (Stat-Ease Inc.). In this study, we developed response surface models based on cubic polynomials. The experiment comprised 3 blocks, with a total of 45 test points, corresponding to 45 pairs of flexible wings. Detailed information regarding these test points is provided in Tables S1 and S2 in the supplemental material.

Flexible wing

The membrane of the flexible wing (Figure 2) is polyimide film with a thickness of 25 μm, fabricated via laser cutting. The veins are carbon fiber composite cylindrical rods with a diameter of 0.3 mm, adhered to the membrane using adhesive. The method of fabricating sleeves at the leading edge and wing root is illustrated in Figure 3. The sleeves are used to mount the flexible wing on the flapping mechanism, facilitating the flexible wing flip smoothly at the end and start of each half stroke to enhance the mean lift.

Figure 3.

Method of fabricating sleeves for flexible wings.

Flapping mechanism

As illustrated in Figure 4 (a), the flapping mechanism employed in the experiment consists of a combination of slider-crank mechanisms and two sets of four-bar mechanisms. Each of the four-link mechanisms features a driving rod connected to the rocker. This driving rod is inserted into the leading-edge sleeve of the flexible wing, facilitating its flapping motion. The theoretical movement of the driving rod approximates a sinusoidal pattern with a flapping amplitude ( $ϕ$ ) of 170°, as illustrated in Figure 4 (b). At elevated flapping frequencies, the variation in flapping amplitude caused by bending deformation at the wing's leading edge remains within 5°, which does not significantly impact the experimental outcomes.

Figure 4.

(a) Flapping mechanism and (b) its theoretical flapping motion.

Experimental setup and measurement methods

The measurement system employed in the experiment is illustrated in Figure 5. The experimental setup comprises a force transducer, a measuring amplifier, a current probe, an oscilloscope, a power supply, and a computer.

Figure 5.

Components of the measurement system.

To measure the mean lift ( $\bar{L}$ ), A force transducer (KD34 s 5N, ME-Meßsysteme) and a measuring amplifier (GSV-8DS, ME-Meßsysteme) were employed to measure the instantaneous force in the vertical direction of the flapping mechanism, with a sampling frequency of 2000 Hz. During the experiment, the instantaneous vertical force was measured for 5 s (about 125–130 flapping cycles). The mean value of the measurements collected over 30 flapping cycles was then calculated and designated as the mean lift ( $\bar{L}$ ).

The lift-to-power efficiency ( $E_{m}$ ) could be calculated as:

E_{m} = \frac{\bar{L}}{{\bar{P}}_{m}}

(3)

where

{\bar{P}}_{m}

is the mean mechanical power output of the motor. And the instantaneous mechanical power (

P_{m}

) could be derived as:

P_{m} = (U - I R) (I - I_{0})

(4)

where U is the output voltage of the power supply, I is the instantaneous current of the motor, R is the internal resistance of the motor, and

I_{0}

is the no-load current of the motor. In this experiment, the instantaneous current (

I

) was measured using a current probe (TCP2020, Tektronix) and an oscilloscope (TBS1102C, Tektronix), and the mean value of

P_{m}

from 30 flapping cycles was taken as

{\bar{P}}_{m}

The mean lift and lift-to-power efficiency for each test point were measured three times, and the average of these measurements was taken as the final result.

Results and discussion

Model and validation

The experimental data were analyzed using Design-Expert software. After eliminating terms with p-values exceeding 0.1, we formulated response surface models for mean lift ( $\bar{L}$ ) and lift-to-power efficiency ( $E_{m}$ ), as shown in Equations (5) and (6). It is worth noting that although all the design parameters are normalized and dimensionless on the right side of Equations (5) and (6), these equations are still dimensional because the dimensions of the design parameters are transferred to their coefficients. In both models, single-parameter terms indicate the impact of individual parameters on the mean lift or lift-to-power efficiency, and multi-parameter terms show the interactions among multiple factors. The coefficients of the single-parameter terms have the same order of magnitude, which indicates that their effects on the aerodynamic performance of the flexible wing are almost equally important. The absolute values of the coefficients of the linear terms in Equations (5) and (6) are nearly an order of magnitude larger than those of the higher-order terms, so the effects of individual parameters on the aerodynamic performance are more pronounced than those of multi-parameter coupling effects. Among these coupling effects, the coupling effects of two parameters are the most significant, while the effects involving three or more parameters can be considered negligible.

The analyses of variance (ANOVA) for the two models are presented in Table S3 and Table S4, respectively. The p-value for each item in the models is less than 0.1, indicating that both models are statistically significant. Figure S1 (a) and Figure S2 (a) present the normal probability plots of the residuals for the two models, respectively. The data points in the two plots are approximately linearly distributed, indicating that the two models satisfy the normality assumption of error. Figure S1 (b)-(d) and Figure S2 (b)-(d) show the residual plots for the two models, illustrating their behavior with the trial order, the fitted values, and the experimental blocks, respectively. None of these plots reveal a discernible pattern, indicating that both models satisfy the assumptions of independence and homogeneity of error. Figure 6 compares the measured values with the model-predicted values at each test point. It is evident that both models closely align with the experimental results, with the maximum relative error for the mean lift ( $\bar{L}$ ) being 5.3% and for the lift-to-power ( $E_{m}$ ) efficiency being 9.1%.

\begin{aligned} \bar{L} = & 12.21 - 2.13 \hat{A} - 0.85 \hat{θ} + 2.58 \hat{λ} + 2.37 \hat{f} + 0.17 \hat{A} \hat{θ} \\ - 0.88 \hat{A} \hat{λ} - 0.75 \hat{A} \hat{f} - 0.44 \hat{θ} \hat{f} + 0.69 \hat{λ} \hat{f} + 0.58 {\hat{A}}^{2} \\ - 0.50 {\hat{θ}}^{2} - 0.22 {\hat{f}}^{2} - 0.35 \hat{A} \hat{θ} \hat{f} - 0.29 \hat{A} \hat{λ} \hat{f} \\ + 0.61 {\hat{A}}^{2} \hat{λ} + 0.44 {\hat{λ}}^{2} \hat{f} - 0.52 {\hat{A}}^{3} \end{aligned}

(5)

\begin{aligned} E_{m} = & 4.85 - 0.21 \hat{A} + 0.39 \hat{θ} + 0.26 \hat{λ} - 0.16 \hat{f} + 0.13 \hat{A} \hat{λ} \\ + 0.13 \hat{θ} \hat{λ} - 0.13 \hat{θ} \hat{f} - 0.19 {\hat{λ}}^{2} - 0.16 {\hat{f}}^{2} \end{aligned}

(6)

Figure 6.

Plot of measured values versus model-predicted values, (a) lift model, (b) lift-to-power efficiency model.

Both experimental and theoretical methods were employed to validate the accuracy of the two models. For the experimental validation, four additional test points were chosen to assess the precision of both models. Among the four test points, one was particularly notable with the condition $\hat{A} = \hat{θ} = \hat{λ} = \hat{f} = 0$ , while the others were selected at random. The results of experimental tests, detailed in Table 1, indicate that both models can effectively predict the mean lift and lift-to-power efficiency of the flexible wings.

Table 1.

Experimental validation results of the lift model and lift-to-power efficiency model.

Test point	1	2	3	4
$\hat{A}$	0.00	0.26	−0.44	−1.00
$\hat{θ}$	0.00	0.91	0.60	1.00
$\hat{λ}$	0.00	−0.16	0.58	1.00
$\hat{f}$	0.00	0.36	0.49	0.46
Predicted $\bar{L}$ (gf)	12.21	10.71	15.92	19.99
Measured $\bar{L}$ (gf)	12.24	10.36	15.73	19.88
Relative error for $\bar{L}$	0.2%	3.2%	1.2%	0.6%
Predicted $E_{m}$ (gf/W)	4.85	4.96	5.12	5.36
Measured $E_{m}$ (gf/W)	4.91	4.88	5.23	5.21
Relative error for $E_{m}$	1.1%	1.5%	2.1%	2.8%

The theoretical validation of the lift model was conducted by comparing the partial derivatives of the model (Equation (5)) with their corresponding theoretical expressions. Since both the response surface model and the theoretical model represent the mean lift, their partial derivatives should, in theory, be equal. We will begin by presenting the derivation of the theoretical expression. The mean lift ( $\bar{L}$ ) can be expressed as

\bar{L} = {\bar{C}}_{L} \frac{1}{2} ρ U_{R_{2}}^{2} (2 S)

(7)

where

{\bar{C}}_{L}

is the mean lift coefficient; S is the wing area;

U_{R_{2}} (= 2 ϕ f R_{2})

is the mean flapping velocity at the

R_{2}

position (reference velocity), and

R_{2}

is the radius of the second moment of wing area. Disregarding the effect of the clipped region of the wing root (due to its trivial contribution to lift),

R_{2}

could be expressed as:

R_{2} = \sqrt{\frac{\int_{0}^{R} c r^{2} d r}{S}} = R \sqrt{\frac{1 + 3 λ}{6 + 6 λ}}

(8)

and S can be expressed as:

S = \frac{(1 + λ) R^{2}}{2 A}

(9)

The mean lift coefficient ( ${\bar{C}}_{L}$ ) is affected by the wing geometry, Reynolds number (Re), and the angle of attack (AoA). Since the experimental Reynolds number remains relatively constant, as mentioned in the corresponding study, the mean lift coefficient ( ${\bar{C}}_{L}$ ) is minimally affected by wing geometry when we choose $U_{R_{2}}$ as the reference velocity.⁴⁰ In other words, the ${\bar{C}}_{L}$ is hardly affected by the aspect ratio ( $A$ ) and the taper ratio ( $λ$ ). For real flexible wings, the wing deformation varies slightly as the flapping frequency ( $f$ ), thus it also affects the ${\bar{C}}_{L}$ , but this effect is weaker than that of slack angle ( $θ$ ). In summary, it can be temporarily assumed that ${\bar{C}}_{L}$ is solely the function of $θ$ :

{\bar{C}}_{L} = {\bar{C}}_{L} (A, θ, λ, f) \approx {\bar{C}}_{L} (θ)

(10)

substituting Equations (8)-(10) into (7), then

\bar{L}

could be expressed as:

\bar{L} = \frac{1}{3} ρ ϕ^{2} R^{4} {\bar{C}}_{L} (θ) f^{2} \frac{1 + 3 λ}{A}

(11)

If Equation (11) is expressed in terms of the normalized parameters in Equation (2), then we obtain Equation (12), an approximate theoretical expression for the mean lift ( $\bar{L}$ ).

\bar{L} = \bar{L} (\hat{A}, \hat{θ}, \hat{λ}, \hat{f}) = \frac{1}{3} ρ ϕ^{2} R^{4} {\bar{C}}_{L} (\hat{θ}) \frac{{(\frac{1}{2} \bar{f} \hat{f} + f_{0})}^{2} (2 + 3 \bar{λ} \hat{λ} + 6 λ_{0})}{\bar{A} \hat{A} + 2 A_{0}}

(12)

By substituting the mean lift of test point 1 (from Table 1) into Equation (12), ${\bar{C}}_{L} (\hat{θ}) |_{0}$ could be calculated. The expression of partial derivatives of $\bar{L}$ with respect to $\hat{A}$ , $\hat{λ}$ , and $\hat{f}$ could be derived from Equations (5) and (12), respectively. Then, we could obtain the values of the partial derivatives of the two equations at $\hat{A} = \hat{θ} = \hat{λ} = \hat{f} = 0$ (Table 2). However, since the expression of ${\bar{C}}_{L} (\hat{θ})$ remains undetermined, the partial derivative of $\bar{L}$ with respect to $\hat{θ}$ in Equation (12) cannot be obtained. As evidenced in Table 2, the first-order partial derivatives are numerically close to their counterparts, and the higher-order partial derivative is consistent with their counterparts in both sign and order of magnitude, thereby indicating the validity of the lift model. Two exceptions are ${\frac{\partial^{2} \bar{L}}{\partial {\hat{f}}^{2}} |}_{0, 0, 0, 0}$ and ${\frac{\partial^{3} \bar{L}}{\partial {\hat{λ}}^{2} \partial \hat{f}} |}_{0, 0, 0, 0}$ , and the probable reason is that in Equation (10) it is simply assumed that ${\bar{C}}_{L}$ is a function of $θ$ only, and the effects of $λ$ and f are neglected.

Table 2.

Comparison of the values of partial derivatives obtained from equation (5) and (12).

Partial derivative	Value from Equation (12)	Value from Equation (5)	Partial derivative	Value from Equation (12)	Value from Equation (5)
$\frac{\partial \bar{L}}{\partial \hat{A}}$	−2.45	−2.13	$\frac{\partial^{2} \bar{L}}{\partial {\hat{A}}^{2}}$	0.98	1.16
$\frac{\partial \bar{L}}{\partial \hat{θ}}$		−0.85	$\frac{\partial^{2} \bar{L}}{\partial {\hat{θ}}^{2}}$		−1.00
$\frac{\partial \bar{L}}{\partial \hat{λ}}$	2.81	2.58	$\frac{\partial^{2} \bar{L}}{\partial {\hat{f}}^{2}}$	0.20	−0.44
$\frac{\partial \bar{L}}{\partial \hat{f}}$	2.22	2.37	$\frac{\partial^{3} \bar{L}}{\partial \hat{A} \partial \hat{θ} \partial \hat{f}}$		−0.35
$\frac{\partial^{2} \bar{L}}{\partial \hat{A} \partial \hat{θ}}$		0.17	$\frac{\partial^{3} \bar{L}}{\partial \hat{A} \partial \hat{λ} \partial \hat{f}}$	−0.10	−0.29
$\frac{\partial^{2} \bar{L}}{\partial \hat{A} \partial \hat{λ}}$	−0.56	−0.88	$\frac{\partial^{3} \bar{L}}{\partial {\hat{A}}^{2} \partial \hat{λ}}$	0.14	1.22
$\frac{\partial^{2} \bar{L}}{\partial \hat{A} \partial \hat{f}}$	−0.44	−0.75	$\frac{\partial^{3} \bar{L}}{\partial {\hat{λ}}^{2} \partial \hat{f}}$	0	0.88
$\frac{\partial^{2} \bar{L}}{\partial \hat{θ} \partial \hat{f}}$		−0.44	$\frac{\partial^{3} \bar{L}}{\partial {\hat{A}}^{3}}$	−0.59	−3.12
$\frac{\partial^{2} \bar{L}}{\partial \hat{λ} \partial \hat{f}}$	0.30	0.69

In contrast to the lift model, the lift-to-power efficiency model (Equation (6)) is difficult to validate using this method. This is because $P_{m}$ encompasses the mechanical power consumed by the flapping mechanism, the aerodynamic power, and the inertial power consumed by the flexible wings. Consequently, the dimensionless coefficients of ${\bar{P}}_{m}$ (or $E_{m}$ ) is difficult to be reduced to a function of a few parameters like Equation (10).

Optimization design of flexible wings

As illustrated in Equations (5) and (6), the significant coupling effects present make it nearly impossible to obtain the optimal design for flexible wings by merely analyzing the variation of aerodynamic performance when one design parameter changes or by applying the frequently-used “preferred method”. However, once the response surface models of the flexible wing's aerodynamic performance are established, numerical methods can be employed to achieve optimal design results within given constraints.

If we choose maximum endurance as the design goal, the optimization design of flexible wings can be described as:

{\begin{matrix} max {E_{m} (\hat{A}, \hat{θ}, \hat{λ}, \hat{f})}; \\ s . t . \bar{L} \geq m \end{matrix}

(13)

where m is the total mass of the tailless FWMAV.

The numerical solution of Equation (13) was obtained from Design-Expert (assuming m =15 gf): $\hat{A}$ =−1.00, $\hat{θ}$ =−1.00, $\hat{λ}$ =0.92, $\hat{f}$ =−0.58, i.e., A =2.0, $θ$ =18°, $λ$ =0.98, f =26.06 Hz. The predicted and actual mean lift of optimal flexible wings are 15.12 and 15.24 gf, respectively, with a relative error of 0.8%. The predicted and actual lift-to-power efficiency is 5.65 and 6.07 gf/W, respectively, with a relative error of 6.9% (Table 3). Although the actual lift-to-power efficiency ( $E_{m}$ ) is slightly higher than the predicted value, it is still within the error band of the model and does not have any negative effect on practical applications.

Table 3.

Predicted and measured values of the aerodynamic performance, as well as of the relative errors of optimized flexible wings.

Model predictions for $\bar{L}$ (gf)	Measured values for $\bar{L}$ (gf)	Relative error for $\bar{L}$	Model predictions for $E_{m}$ (gf/W)	Measured values for $E_{m}$ (gf/W)	Relative error for $E_{m}$
15.12	15.24	0.8%	5.65	6.07	6.9%

Beyond aerodynamic performance, it is important to verify that flexible wings meet other requirements. For tailless Flexible Wing Micro Air Vehicles (FWMAVs), it is essential to ensure that the wing design complies with the control system's specifications. The verification is associated with the vehicle's control mode. For instance, Nano Hummingbird,⁶ COLIBRI,¹⁰ KUbeetle-S,¹⁴ etc., alter the AoA during flapping by deflecting the wing root position to generate control moments. This control method is equivalent to changing the slack angle ( $θ$ ) of the flexible wing. Therefore, it is necessary to confirm the effect on mean lift when the slack angle changes around the design point. If the mean lift varies monotonically with the slack angle and the slope of the variation is substantial, flexible wings are capable of generating control moments efficiently using this control approach. Conversely, if the change in mean lift is insignificant or exhibits non-monotonic variations, it will lead to reduced control efficiency or even control reversal. As evident in Figure 7, the optimized flexible wing fulfills the vehicle's control requirements.

Figure 7.

Changes in the mean lift ( $\bar{L}$ ) of the optimized flexible wings when solely adjusting the slack angle ( $θ$ ).

If the design objective of a tailless FWMAV is altered (e.g., with maximum payload as the design objective), the design results will vary accordingly. The designer only needs to establish design constraints akin to Equation (13) based on the vehicle's operating state and requirements, and then obtain the optimized wing design results utilizing the models. Methods for determining design constraints are beyond the scope of this study and thus will not be further discussed.

Effects of design parameters on mean lift and lift-to-power efficiency in flexible wings

Effects of individual variable parameters on mean lift and lift-to-power efficiency

Before this section, for ease of discussion, we perform a simple transformation of equation (6) to obtain two approximate expressions of lift-to-power efficiency ( $E_{m}$ ) as detailed below. Since mechanical power ( ${\bar{P}}_{m}$ ) is primarily employed to overcome aerodynamic drag, the product of the mean drag ( $\bar{D}$ ) and the reference velocity ( $U_{R_{2}}$ ) can be used instead of ${\bar{P}}_{m}$ :

E_{m} \sim \frac{\bar{L}}{\bar{D} \cdot U_{R_{2}}}

(14)

Mean drag ( $\bar{D}$ ) can be expressed as:

\bar{D} = {\bar{C}}_{D} \frac{1}{2} ρ U_{R_{2}}^{2} (2 S)

(15)

where

{\bar{C}}_{D}

is the mean drag coefficient. Substituting Equation (7) and (15) into (14) yields two approximate expressions for

E_{m}

(Equation (16) and (17)), where

p_{w} (= \bar{L} / 2 S)

is the wing loading, and

{\bar{C}}_{L} / {\bar{C}}_{D}

is the lift-to-drag ratio.

E_{m} \sim \frac{\sqrt{\frac{ρ}{2}} \frac{{\bar{C}}_{L}^{\frac{3}{2}}}{{\bar{C}}_{D}}}{\sqrt{p_{w}}}

(16)

E_{m} \sim \frac{{\bar{C}}_{L}}{{\bar{C}}_{D}} \frac{1}{2 ϕ R_{2} f}

(17)

Figure 8 illustrates the effects of individual parameters on the mean lift ( $\bar{L}$ ) and lift-to-power efficiency ( $E_{m}$ ) when the wing length is fixed. In Figure 8, when a certain normalized parameter changes, all others are set to zero. As the aspect ratio ( $A$ ) increases, mean lift decreases significantly (see Figure 8 (a)); this is because, when the wing length is constant, the increase of A causes wing area decreases, while the mean lift coefficient ( ${\bar{C}}_{L}$ ) remains almost constant (see Figure 9), thus $\bar{L}$ decreases (Equation (7)). Lift-to-power efficiency ( $E_{m}$ ) decreases slightly as the aspect ratio ( $A$ ) increases (see Figure 8 (b)), and this phenomenon may be explained in the following way. When A is small, the area of the outboard wing is large, and the flexible wing takes a heavy wing load when flapping, so the torsion deformation of the flexible wing is significant, and the mean AoA ( $\bar{α}$ ) is small. From our previous study,⁴¹ the lift-to-drag ratio ( ${\bar{C}}_{L} / {\bar{C}}_{D}$ ) is relatively large when $\bar{α}$ is small (see Figure 10), so $E_{m}$ is high (Equation (17)). As A decreases, the torsional deformation of the flexible wing weakens, and $\bar{α}$ increases, so $E_{m}$ decreases (Equation (17)).

Figure 8.

(a) Mean lift ( $\bar{L}$ ) and (b) lift-to-power efficiency ( $E_{m}$ ) versus four individual parameters. (All other normalized parameters are zero when a single parameter is changed).

Figure 9.

The mean lift coefficient ( ${\bar{C}}_{L}$ ) versus four individual parameters. (All other normalized parameters are zero when a single parameter is changed).

Figure 10.

The mean lift coefficient ( ${\bar{C}}_{L}$ ), drag coefficient ( ${\bar{C}}_{D}$ ), lift-to-drag ratio ( ${\bar{C}}_{L} / {\bar{C}}_{D}$ ) and ( ${\bar{C}}_{L}^{\frac{3}{2}} / {\bar{C}}_{D}$ ) versus mean AoA ( $\bar{α}$ ).⁴¹

As the slake angle ( $θ$ ) increases, the mean lift ( $\bar{L}$ ) decreases while the lift-to-power efficiency ( $E_{m}$ ) increases (see Figure 8 (a) and (b)). This is because an increase in $θ$ leads to a decrease in mean AoA ( $\bar{α}$ ), and it is known from Figure 10 that the mean lift coefficient ( ${\bar{C}}_{L}$ ) decreases, therefore $\bar{L}$ decreases; whereas the lift-to-drag ratio ( ${\bar{C}}_{L} / {\bar{C}}_{D}$ ) increases (see Figure 10), thus causing the increasing of $E_{m}$ (Equation (17)).

As the taper ratio ( $λ$ ) increases, the mean lift ( $\bar{L}$ ) increases (see Figure 8 (a)). This is because the wing area ( $S$ ) increases as $λ$ increases, and thus $\bar{L}$ increases (Equation (7)). Also, as $λ$ increases, lift-to-power efficiency ( $E_{m}$ ) increases (see Figure 8 (b)). This is because increasing $λ$ causes an increase in the area of the outboard wing, as the outboard wing takes stronger loads, the negative torsional deformation is more significant, resulting in a decrease in mean AoA ( $\bar{α}$ ) and an increase in ${\bar{C}}_{L}^{\frac{3}{2}} / {\bar{C}}_{D}$ (see Figure 10), thus $E_{m}$ increases (Equation (16)). In addition, we note that when $λ$ itself has a large value, its variation does not have a significant effect on $E_{m}$ . This phenomenon may be attributed to another opposite effect of $λ$ on $E_{m}$ : the wing loading ( $p_{w}$ ) increases as $λ$ increases (Figure 11), and this leads to a decrease in $E_{m}$ (Equation (16)).

Figure 11.

Wing loading ( $p_{w}$ ) versus aspect ratio ( $A$ ) and taper ratio ( $λ$ ). (All other normalized parameters are zero when a single parameter is changed).

As the flapping frequency ( $f$ ) increases, the mean lift ( $\bar{L}$ ) increases (Figure 8 (a)). This is because increasing f causes the increase of reference velocity; thus $\bar{L}$ increases (Equation (7)). Lift-to-power efficiency ( $E_{m}$ ) decreases as f increases, and this result is obtained easily from Equation (17). At the same time, we note that the variation of $E_{m}$ is not obvious at the beginning of the increase of f, which may be attributed to another effect of f on $E_{m}$ : the increase of f leads to the increase of wing loading (Figure 11), which causes the enhancement of negative torsional deformation and the decrease of the mean AoA ( $\bar{α}$ ) of the flexible wings, hence the lift-to-drag ratio ( ${\bar{C}}_{L} / {\bar{C}}_{D}$ ) increases (see Figure 10)), thus $E_{m}$ increases (Equation (17)).

Overall, in the optimization space, when all other normalized parameters are set to zero, increasing the aspect ratio ( $A$ ) decreases both the mean lift ( $\bar{L}$ ) and lift-to-power efficiency ( $E_{m}$ ); increasing the slack angle ( $θ$ ) decreases the $\bar{L}$ and increases the $E_{m}$ ; increasing the taper ratio ( $λ$ ) increases both the $\bar{L}$ and $E_{m}$ ; and increasing the flapping frequency ( $f$ ) increases the $\bar{L}$ and decreases the $E_{m}$ .

Coupling effects of two parameters on mean lift and lift-to-power efficiency

In Equation (5), it is evident that the absolute values of the coefficients of the terms $\hat{A} \hat{λ}$ , $\hat{A} \hat{f}$ , $\hat{θ} \hat{f}$ and $\hat{λ} \hat{f}$ are more substantial compared with the term $\hat{A} \hat{θ}$ , indicating significant coupling effects of aspect ratio ( $A$ ) and taper ratio ( $λ$ ), aspect ratio ( $A$ ) and flapping frequency ( $f$ ), slack angle ( $θ$ ) and flapping frequency ( $f$ ), taper ratio ( $λ$ ) and flapping frequency ( $f$ ), whereas the coupling effect of aspect ratio ( $A$ ) and slack angle ( $θ$ ) could be disregarded here.

The four significant coupling effects on mean lift ( $\bar{L}$ ) are illustrated in Figure 12. When two parameters in the figure change, the remaining two parameters are set to zero. The coupling effect of the aspect ratio ( $A$ ) and the taper ratio ( $λ$ ) manifests as: when $λ$ is large, the increasing of A causes a more significant decrease in $\bar{L}$ , compared with when $λ$ is small (Figure 12 (b)). The coupling effect of the aspect ratio ( $A$ ) and the flapping frequency ( $f$ ) manifests as: when A is small, the increasing of f causes a more significant increase in $\bar{L}$ , compared with when A is large (Figure 12 (d)). The coupling effect of the taper ratio ( $λ$ ) and the flapping frequency ( $f$ ) manifests as: when $λ$ is large, the increasing of f causes a more significant increase in $\bar{L}$ , compared with when $λ$ is small (Figure 12 (e)). These three coupling effects are straightforward to explain. The aspect ratio and the taper ratio directly determine the flexible wing area ( $S$ ) and the position of the second moment of wing area ( $R_{2}$ ), and the flapping frequency is positively correlated with the reference velocity ( $U_{R_{2}}$ ). Because the $\bar{L}$ is positively correlated with the $U_{R_{2}}^{2}$ S (Equation (7)), the two-parameters coupling effects of A, $λ$ , and f on $\bar{L}$ are inevitable.

Figure 12.

Coupling effect of (a, b) aspect ratio ( $A$ ) and taper ratio ( $λ$ ), (c, d) aspect ratio ( $A$ ) and flapping frequency ( $f$ ), (e, f) taper ratio ( $λ$ ) and flapping frequency ( $f$ ), (g, h) slake angle $(θ$ ) and flapping frequency ( $f$ ) on the mean lift ( $\bar{L}$ ). (In each graph, the other two unrelated normalized parameters are set to zero).

Owing to the wing's flexibility, the coupling effect of the slack angle ( $θ$ ) and the flapping frequency ( $f$ ) on the mean lift ( $\bar{L}$ ), differs from the other three. As illustrated in Figure 12 (h), when f increases, the slack angle ( $θ^{'}$ ), corresponding to the maximum mean lift ( ${\bar{L}}_{\max}$ ), decreases. This coupling effect is also confirmed by Yoon et al.¹⁴ and can be elucidated as follows. The flexible wing generates the maximum mean lift when the mean AoA ( $\bar{α}$ ) is approximately 50° (Figure 13); as the f increases, the wing loading ( $p_{w}$ ) increases, the negative torsional deformation of the wing is enhanced, and $\bar{α}$ decreases subsequently. Due to the $\bar{α}$ being inversely related to the $θ$ , at low flapping frequency f, a large $θ$ situates the $\bar{α}$ approximately at 50°; whereas, a small $θ$ is necessary to achieve the maximum mean lift at high f because of significant torsional deformation.

Figure 13.

Coupling effect of (a, b) aspect ratio ( $A$ ) and taper ratio ( $λ$ ), (c, d) slack angle ( $θ$ ) and taper ratio ( $λ$ ), (e, f) slack angle ( $θ$ ) and flapping frequency ( $f$ ) on the lift-to-power efficiency ( $E_{m}$ ). (In each graph, the other two unrelated normalized parameters are set to zero).

In Equation (6), the terms $\hat{A} \hat{λ}$ , $\hat{θ} \hat{λ}$ , and $\hat{θ} \hat{f}$ represent the coupling effects of aspect ratio ( $A$ ) and taper ratio ( $λ$ ), slack angle ( $θ$ ) and taper ratio ( $λ$ ), as well as slack angle ( $θ$ ) and flapping frequency ( $f$ ) on the lift-to-power efficiency ( $E_{m}$ ), respectively. Since the absolute values of the coefficients of the three terms do not differ substantially, none of them can be disregarded here. These coupling effects are discussed later.

The aforementioned three significant coupling effects on lift-to-power efficiency ( $E_{m}$ ) are illustrated in Figure 13. When two parameters in the figure change, the remaining two parameters are set to zero. The coupling effect of aspect ratio ( $A$ ) and taper ratio ( $λ$ ) on the $E_{m}$ manifests as: the taper ratio ( $λ^{'}$ ), corresponding to the maximum lift-to-power efficiency ( $E_{m, \max}$ ), rises as A increases (Figure 13 (b)). The explanation of the above coupling effect is as follows. When the A of the flexible wing is small, the wing root chord ( $c_{r}$ ) is long, and the wing root has a strong restraining effect on the torsional deformation of the flexible wing, so the mean AoA of the outboard wing (this part of flexible wing generates most of the lift) is large, thus the lift-to-drag ratio ( ${\bar{C}}_{L} / {\bar{C}}_{D}$ ) is low. In this case, choosing a small $λ$ is beneficial to reduce drag and improve $E_{m}$ . When the A is large, the $c_{r}$ is short, the constraint effect of the wing root is weak, and the mean AoA of the outboard wing is small, thus the lift-to-drag ratio ( ${\bar{C}}_{L} / {\bar{C}}_{D}$ ) is high. At this time, the flexible wing with a large $λ$ is beneficial to generate high lift with low aerodynamic power consumption and high $E_{m}$ . In summary, as A increases, $λ^{'}$ also increases. This coupling effect also clarifies the contradiction of the effect of the taper ratio on lift-to-force efficiency in the studies of Keennon et al.⁶ and Nan et al.¹⁶

The coupling effect of slack angle ( $θ$ ) and taper ratio ( $λ$ ) on the lift-to-power efficiency ( $E_{m}$ ) manifests as: the taper ratio ( $λ^{'}$ ) corresponding to maximum lift-to-power efficiency ( $E_{m, \max}$ ) rises as $θ$ increases (see Figure 13(d)). This coupling effect can be explained as follows: when $θ$ is small (when $\bar{α}$ is large), the outboard wing experience intense wing loading, and the small $λ$ could moderately decrease wing loading to enhance $E_{m}$ ; as $θ$ increases (when $\bar{α}$ decreases), the mean drag coefficient ${\bar{C}}_{D}$ declines, and then the flexible wing with large $λ$ could generate higher $\bar{L}$ , concurrently augmenting $E_{m}$ . In summary, as $θ$ increases, $f^{'}$ also increases.

The coupling effect of slack angle ( $θ$ ) and flapping frequency ( $f$ ) on the lift-to-power efficiency ( $E_{m}$ ) manifests as: the flapping frequency $f^{'}$ , corresponding to maximum lift-to-power efficiency ( $E_{m, \max}$ ), decreases as $θ$ increases (see Figure 13(f)). This coupling effect can be explained as follows. When $θ$ is small (the mean AoA ( $\bar{α}$ ) is large), the lift-to-drag ratio ( ${\bar{C}}_{L} / {\bar{C}}_{D}$ ) of the wing is large. In this case, the flexible wing will be subject to intense wing loading when flapping at high f, resulting in significant torsional deformation, which helps to reduce the $\bar{α}$ (improve ${\bar{C}}_{L} / {\bar{C}}_{D}$ ) to improve the $E_{m}$ of the wing. However, the above method to improve $E_{m}$ is limited. When f reaches $f^{'}$ , if the f continues to increase, the $E_{m}$ will decrease (Equation (17)). On the contrary, when the $θ$ is large ( $\bar{α}$ is small), the ${\bar{C}}_{L} / {\bar{C}}_{D}$ is small, and the flexible wing no longer requires significant torsional deformation, so the corresponding $f^{'}$ is lower than its counterpart when $θ$ is small. In summary, as $θ$ increases, $f^{'}$ decreases.

The coupling effects of $θ$ and $λ$ , as well as $θ$ and f on $E_{m}$ show that the slake angle corresponding to the maximum lift-to-power efficiency is not constant, which may explain the difference of the previous study.^21,22

In summary, the interplay among these parameters significantly influences the aerodynamic performance of flexible wings. It may be impossible to acquire optimal design results by considering only the influence of individual parameters on the aerodynamic performance of flexible wings. Some coupling effects can be qualitatively explained by the negative torsional deformation of the wing, implying that the variations of the AoA induced by the deformation of the wings have significant effects on the aerodynamic performance.

Conclusion

As there is a lack of established flexible wing design methods for tailless FWMAVs, we proposed an experimental optimization wing design method based on the response surface methodology. In this method, the design parameters and optimization space are determined first. Then, the response surface models for predicting the aerodynamic performance of the flexible wing are obtained through experiments. Finally, according to the actual requirements of the vehicle, the design constraints are given and incorporated into the models, and the final optimal design results are obtained. We designed flexible wings using this method for a tailless FWMAV weighing 15 grams. The optimal design was validated through experiments to ensure it met the anticipated performance and requirements of the aircraft. Our proposed design method addresses the problem of achieving optimal design due to the complex interactions between parameters. The response surface models generated during the optimization process accurately predict the mean lift and lift-to-power efficiency of the flexible wings within the optimization space, with a maximum prediction error of under 10% when compared to experimental.

We discussed the effects of four parameters—aspect ratio, slack angle, taper ratio, and flapping frequency—on the aerodynamic performance of flexible wings based on response surface models. We found that the individual changes of these four parameters have significant effects on the aerodynamic performance of the flexible wing, and their effects are almost equally important. In addition, the coupling effects between these four parameters, especially in pairs, are also significant. These coupling effects explain the opposite influence laws of the same parameter in different past studies. At the same time, our study also shows that it is difficult to achieve the optimal design of the flexible wing when these coupling effects are ignored.

Compared with the “preferred method”, the experimental optimization design method based on the response surface methodology not only provides explicit theoretical guidance for flexible wing design but also yields the optimal solution under the design space and constraints. Furthermore, this method requires less time to predict the performance of flexible wings compared to numerical simulation. Additionally, this method can be applied not only to optimize aerodynamic performance but also to enhance the design of wing structures and reduce flapping noise in flexible wings. This approach provides substantial support for the design of FWMAVs.

Supplemental Material

sj-docx-1-mav-10.1177_17568293251348206 - Supplemental material for Experimental study on the aerodynamic optimization design of flexible wings for tailless flapping wing micro air vehicles

Supplemental material, sj-docx-1-mav-10.1177_17568293251348206 for Experimental study on the aerodynamic optimization design of flexible wings for tailless flapping wing micro air vehicles by Jianghao Wu, Shijie Sheng, Ziming Liu, Jin Cui, Chenyang Wang, Long Chen and Yanlai Zhang in International Journal of Micro Air Vehicles

Footnotes

ORCID iDs

Jianghao Wu

Shijie Sheng

Ziming Liu

Jin Cui

Chenyang Wang

Long Chen

Yanlai Zhang

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China, (grant number 12472231).

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Supplemental material

Supplemental material for this article is available online.

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