The influence of aspect ratio and stroke pattern on force generation of a bat-inspired membrane wing

Abstract

Aspect ratio (AR) is one parameter used to predict the flight performance of a bat species based on wing shape. Bats with high AR wings are thought to have superior lift-to-drag ratios and are therefore predicted to be able to fly faster or to sustain longer flights. By contrast, bats with lower AR wings are usually thought to exhibit higher manoeuvrability. However, the half-span ARs of most bat wings fall into a narrow range of about 2.5–4.5. Furthermore, these predictions do not take into account the wide variation in flapping motion observed in bats. To examine the influence of different stroke patterns, we measured lift and drag of highly compliant membrane wings with different bat-relevant ARs. A two degrees of freedom shoulder joint allowed for independent control of flapping amplitude and wing sweep. We tested five models with the same variations of stroke patterns, flapping frequencies and wind speed velocities. Our results suggest that within the relatively small AR range of bat wings, AR has no clear effect on force generation. Instead, the generation of lift by our simple model mostly depends on wingbeat frequency, flapping amplitude and freestream velocity; drag is mostly affected by the flapping amplitude.

1. Introduction

Owing to the size of bats and their capabilities, such as hovering, highly manoeuvrable flight and the ability to carry substantial loads (bat mothers carry their pups until they are almost fully grown [1,2]), these flying animals are a source of inspiration for flapping-wing micro air vehicles (MAVs).

Bats fly with compliant membrane wings, and this feature sets them apart from birds and insects with comparatively rigid wings. Insects control their wingstroke at one joint at the root of the wing, and their wings twist passively due to inertial and aerodynamic forces [3]. Birds have more active control over wing shape and stroke kinematics. For example, a coupled movement of the elbow and wrist executes wing retraction [4], and feathers can be spread to control wing shape and permeability to air.

The control of movement and shape in bat wings is more complex. The skeleton incorporates the entire upper limb and most of the lower limb and, in some species, the tail. Some of the joints of the skeleton move in functional groups, but more than a dozen independent dimensions are needed to describe 95% of the total wing motion [5]. Overall, bat wings possess more degrees of freedom than those of birds or insects.

The surface of the bat wing is composed of a compliant membrane [6]. This membrane is complex; elastin fibres embedded in a predominantly spanwise orientation introduce anisotropy, and muscles, oriented primarily in a chordwise direction, actuate during the wingstroke and might control wing camber during flight [7].

Like those of birds and insects, bat wings exhibit a variety of shapes and sizes. For several decades, biologists have drawn conclusions about flight behaviour and efficiency from easily measured parameters, and connected those conclusions to flight ecology (e.g. [8–10]). Such parameters are, for example, wingspan and wing area, and body weight. In a principal components analysis of wing shape and body size of more than 200 bat species, Norberg & Rayner suggested that aspect ratio (AR, wingspan/wing chord), and wing loading (body weight/wing area) critically determine flight behaviour [9]. They concluded that high AR-winged bats are more likely to fly in open air, whereas low AR wings are more suitable for cluttered habitats. Bats with low wing loading generally fly more slowly than bats with high wing loading. However, the range of ARs among bat species is rather limited compared with the full diversity found in aircraft, birds or insects (figure 1). Indeed, based on the species sample published by Norberg & Rayner [9], more than two-thirds of bats species have half-span ARs between 2.75 and 3.75. Species of the family Molossidae or free-tailed bats are exceptional within this distribution with ARs approximately twice those of other bat species.

Figure 1.

Half-span aspect ratio distribution of 215 bat species (data from Norberg & Rayner [9]). The distribution for the entire species sample is shown in blue. Vespertilionidae, evening bats (n = 75) in green, and Molossidae free-tailed bats (n = 17), in purple.

The idea that flight behaviour can be predicted by AR derives from fixed wing aeromechanics for high Reynolds numbers, specifically the contribution of induced to total drag. Induced drag, D_i, is an inherent consequence of lift generation and the presence of tip vortices; as AR increases, the relative influence of the tip vortex to overall drag production declines

1.1

where is the coefficient of induced drag, C_L the coefficient of lift and e the wingspan efficiency [11].

For fixed wings, efficiency has some dependency on AR [12–16]. The wake of a fixed wing in steady flow can be described as a horseshoe vortex system with the bound vortex around the aerofoil and the tip vortices being the only prominent vortex structures. However, in flapping flight, wing motion during the wingbeat cycle and pressure and velocity gradients along the wingspan can cause highly unsteady flow conditions. Limited validity of quasi-steady assumptions to explain force generation in flapping flight was first demonstrated over 30 years ago [17]. Several unsteady aerodynamic effects that cannot be predicted with traditional aeromechanics have been identified, such as the presence of stable leading edge vortices, wing-wake interactions, and clap-and-fling [18–20]. Wakes of flapping wings are considerably more complex than those of fixed wings in steady flow, and factors other than the induced drag of the tip vortices influence a wing's efficiency. AR thus might have less effect on force generation and flight power than does the pattern of flapping motion. Although few experimental studies have investigated flapping, highly compliant membrane wings, available data suggest that force generation and power demands depend strongly on stroke pattern [21,22].

Here, we explore the relative roles of AR, wingbeat frequency, wingbeat amplitude, sweep angle and downstroke ratio on aerodynamic force generation. We hypothesize that wingstroke kinematics have a stronger effect on overall wing performance than AR. We test this idea with a robotic flapper employing mechanical wings with ARs in a bat-relevant range, 2.5 < AR < 4.5. We designed the wings based on the vespertilionid Eptesicus fuscus, the big brown bat. The shoulder joint of the model allowed for independent control of flapping and sweep amplitude, and thus testing of a wide range of kinematic parameters.

2. Material and methods

2.1. Models

The principal design and driving mechanism of our mechanical flapping wing was adopted directly from previous work [23] in which a full-scale wind tunnel model was designed based on the wing geometry of the dog-faced fruit bat, Cynopterus brachyotis, and fabricated using three-dimensional-printed parts. That model had three degrees of freedom: flapping (up-down), sweep (fore-aft) and folding, and was actuated by means of push–pull cables that connected the skeletal joints to servo motors mounted outside the wind tunnel test section.

We simplified the mechanics of our model by omitting wing folding and keeping only the two motors that allow for flapping and sweep motion of the wing. We built five models that encompass wing shape variation. The baseline model, AR3.5bl, is based on E. fuscus in size and general shape. It has an AR of 3.5, a half-span of b = 13 cm and a wing area of S = 48 cm². For ease of comparison and to reduce possible influences on force generation from other sources such as scalloping of the trailing edge and tapering of the wing, we simplified the geometry of the wing planform. The armwing consists of a rectangle in which the membrane runs between the body and what would be digit V in a bat. The membrane is supported by a simple upper and forearm skeleton. The handwing consists of two triangles that are formed by a total of three digits (figure 2a). We used a ratio of handwing to armwing area of 0.6, and of handwing to armwing span of 1.15, both characteristic of E. fuscus [9]. The ratios of handwing to armwing area and span were preserved.

Figure 2.

(a) Skeleton of the five wings tested. The baseline model, AR3.5bl is shown in the middle. The aspect ratio of the models to its left and right is varied by changing the wingspan, b, keeping the area, S, constant. The aspect ratio of the models below and above it is changed by changing the wing area, keeping the span constant. All wings have a built-in angle of attack of 6°, and 9% camber at 1/4 chord. (b) Perspective view of the wing assembly. The shoulder part rotated along the long axis of the model to allow for the flapping motion, the arm rotated about its pivot point at the centre of the shoulder piece to allow for the sweep motion. (c) Top view of the model to illustrate the sweep motion of the arm rotation. (d) Back view of the model to illustrate the flapping motion of the shoulder rotation.

The AR for non-rectangular wings is usually described as AR = b²/S. AR can therefore be easily modified by changing wing area while keeping wingspan constant. The local spanwise velocities introduced along the wingspan by flapping motion remain constant among all models of a given wingspan. Alternatively, wing area may be kept constant, and wingspan was changed. Using these considerations, we built five models with three ARs of 2.5, 3.5 and 4.5. Three of the models share the same wing area, and three models share the same wingspan (figure 2a). All five wings have a built-in static angle of attack of α_o = 6°, and 9% camber at 1/4 chord.

The wing skeletons were designed in SolidWorks^® 2012 x64 (Dassault Systèmes SOLIDWORKS Corp., Waltham, MA, USA), and three-dimensionally-printed (Dimension 1200es, Stratasys, Eden Prairie, MN, USA) with ABS plastic (acrylonitrile butadiene styrene). We applied a coating of superglue (M60 Advanced performance instant adhesive, Adhesive Systems Inc., Frankfort, IL, USA) to all digits to strengthen them. To reduce some of the strain on the membrane, the attachment site on the body was modified from previous versions of the flapper [23]. The membrane was glued to an appendix of the shoulder piece and follows the shoulder rotation during the flapping motion (figure 2b–d). Our membranes were made of a highly elastic silicone rubber, Dragon Skin^® (Smooth-On Inc., Macungie, PA, USA). This material has a reported shore-A durometer value of 10 (Young's modulus of about 0.7 MPa). We adjusted the component A : B mix ratio to 3 : 1, which enabled us to produce thinner membranes.

To fabricate the thin membrane, we poured the uncured mixture onto a Teflon-covered aluminium plate. This was covered with a second plate that was treated with mould-release and weighted, for a combined load of about 8 kg. The resulting membranes had a thickness of about 0.2 mm. The membrane was glued directly to the skeleton and the reinforcement structures with silicone epoxy (Sil-poxy, Smooth-On, Inc., Easton, PA, USA) with the wing in its neutral gliding position (figure 2b).

We reinforced the entire leading edge and trailing edge between the fingers with 0.25 mm thin elastic (Stretchrite sewing thread, Jo-Ann Stores, Inc.) [23]. The remainder of the trailing edge was reinforced with a 0.5 cm wide strip of Dragon Skin membrane glued to the skeleton at the trailing edge, and a second strip added at a distance of about 1 cm further from the trailing edge. Those two strips were glued on with the wing placed in its most swept-back position.

The flapping and sweep motions were driven by two brushless servo motors with integrated encoders (BE163CJ-NFON, Parker Hannfin Corp., Rohnert Park, CA, USA) and controlled by a servo controller (Accelera DMC-4060, Galil Motion Control, Rocklin, CA, USA) with integrated amplifiers (AMP-43040, Galil Motion Control). A Matlab script translated the inputs for flapping frequency, flapping amplitude angle, sweep angle and downstroke-to-upstroke ratio into PVT (position, velocity and time) commands that were sent to the servo controller.

2.2. Experimental set-up

A custom-made force plate was mounted to the floor panel of the test section of a closed-circuit wind tunnel at Brown University (test section dimensions: 3.8 × 0.6 × 0.82 m³). The force plate is a flexure based system that allows for independent displacement in two perpendicular directions (lift and drag axes) of the centre section which serves as mount for the experimental model. A rack to mount the motors and models was attached to the force plate, allowing the wings to extend into the test section through the floor of the wind tunnel, and the motors remained accessible from outside the test section (figure 3). Displacement of the test plate was measured using two optical displacement sensors (D64, 20 kHz resolution, Philtec Inc., Annapolis, MD, USA) and recorded using a data acquisition board (NI USB-6210, National Instruments Corporation, Austin, TX, USA) at 1024 Hz using a customized Matlab script. The Scope tool of the GalilTools software (GalilTools 1.6.4.576, Galil Motion Control Inc., Rocklin, CA, USA) was used to record the voltage, current and position of the servo motors, along with the trigger signal, all at 512 Hz.

Figure 3.

Computer-aided-design rendering of the experimental set-up. The force plate is mounted below the test section with the wing extending vertically into the air flow. Two motors outside the test section control the sweep and flapping motion of the wing.

2.3. Wing stroke kinematics

To investigate the effect of wing stroke kinematics, we chose two baseline stroke patterns about which to carry out a series of variations. The first pattern was based on kinematics of E. fuscus [24]. A second stroke pattern was based on the kinematics of Tadarida brasiliensis (Brazilian free-tailed bat) [25], a member of the Molossidae, a family of bats characterized by high AR wings (figure 1). The E. fuscus pattern is characterized by higher flapping and sweep amplitudes, and an equal duration of up- and downstroke (table 1). For all stroke patterns the flapping and sweep motion are in phase resulting in a straight-line trajectory input signal. In general, the stroke plane is close to vertical when the sweep angle is small, and the angle between stroke plane and the horizontal decreases with increasing sweep motion.

Table 1.

Range of stroke kinematics tested. For comparison, the baseline values are also given, derived from Eptesicus fuscus, the bat species on which the physical model is based [24], as well as for a second species with a markedly different wing stroke, Tadarida brasiliensis [25].

parameter	E. fuscus	T. brasiliensis	min.	max.	increment
frequency, f (Hz)	9	9	2	10	2
flapping amplitude, θ (°)	110	80	20	110	15
sweep amplitude, φ (°)	55	15	15	55	10
downstroke ratio, DR	0.55	0.44	0.44	0.56	0.06

We performed parameter sweeps over all variables based on these two baseline stroke patterns (table 1). The two baseline strokes and the 17 combination strokes that arise from varying one parameter at a time were tested at wind speeds of U_∞ = 5.0 and 7.5 m s⁻¹ for all five wings (Reynolds number range based on mean chord 10 000 < Re_c < 26 000).

2.4. Data collection

Each flapping trial started with the wing in a gliding position (figure 3). The wing was then moved to a position representing the top of the downstroke and through 50 complete wingbeat cycles before moving back to the neutral gliding position. Force data were collected throughout the flapping and terminal gliding phase.

2.5. Data processing

To ensure that a steady state condition was achieved, the first 10 and last five wingbeat cycles of the 50 recorded cycles were excluded from analysis. The flexure-supported force plate acts as a mass-spring-damper subjected to the unsteady periodic forcing due to the flapping wing, F(t). Note that this force comprises both aerodynamic and inertial forces associated with the wing motion. For simplicity, we describe motion in the y direction, although the following procedure applied to both the x and y directions. When the plate is forced dynamically, the displacement of the plate sections can be written as

2.1

where y, and are the position, velocity and acceleration of the system, respectively. The mass, m, spring constant, k, and damping coefficient, β, are known characteristics of the force plate, measured previously from a dynamic calibration [24] (table 2).

Table 2.

Characteristics of the force plate in the lift and drag directions. The stiffness, k, was determined using a static calibration procedure, measuring the displacement versus applied force. The natural frequency, mass and damping were determined using a dynamic calibration, or ‘ring-down’ test. The sensing plate was moved away from its equilibrium position and then released. The oscillatory decay of the plate position was used, in conjunction with the stiffness, to determine the natural frequency, f_o; mass, m; and damping coefficient, β [24].

	spring constant	damping coefficient	mass	natural frequency
	k (N m−1)	b (Ns m−1)	m (kg)	f0 (Hz)
lift axis	224 500	6.5	3.3	35
drag axis	89 200	1.2	2.7	27

The measured displacement, y(t), was fit to a Fourier series, retaining terms associated with the driving frequency and higher harmonics, up to the natural frequency of the force plate. Velocity, , and acceleration, , were computed from the Fourier series, and using the known characteristics of the force plate, m, k and β, the driving force, F(t), was calculated using equation (2.1). The inertial contribution to the force, measured by recording the forces resulting from flapping the wing in still air and without an attached membrane, was then subtracted, leaving only the aerodynamic force for a particular kinematic parameter combination.

Some kinematic parameter combinations seemed to be more prone to measurement noise, leading to poor repeatability of force measurements at these settings. Measurements that obviously differed from all other traces of a parameter combination set were excluded from further analysis. Parameter combinations with fewer than two valid trials were also excluded from further analysis.

2.6. Velocity scaling and dimensionless numbers

The force generated by each set of parameters depends on seven input variables: wing half-span, b, wing area, S, flapping frequency, f, flapping amplitude angle, θ, sweep amplitude angle, φ, downstroke ratio, DR, and freestream velocity, U_∞. To compare the force generation for different stroke patterns and different wing models, we characterized each trial by the dimensionless ratio of average relative horizontal to average vertical wind speed at mid span: U_v/U_h. Note that the velocity ratio differs between downstroke and upstroke. The time-resolved vertical velocity, U_v, can be expressed as

2.2

Similarly the horizontal velocity, U_h, is defined as

2.3

where ω = fπ/DR and n = 0 for the downstroke, and ω = fπ/(1 − DR) and n = 1 for the upstroke.

The combination of freestream velocity, flapping and sweeping motion also affects the effective angle of attack, α_eff, and the effective air speed, U_eff, experienced by the wing during the wingbeat cycle

and

(figure 4). Effective instantaneous angles of attack are more extreme at the lower freestream velocities, especially during the upstroke, largely because the relative contribution of the vertical velocity component, induced by the flapping motion, decreases as the wind speed increases. Flapping frequency and amplitude produce the greatest changes in local flow conditions, leading to substantial changes in angle of attack. The coefficients of lift and drag are defined by the magnitude of the time-resolved velocity vector described by U_v and U_h

2.4

and

2.5

Figure 4.

Summary of relative flow direction and strength during flapping. Effective angles of attack (a) and magnitude of flow (b) over the wingbeat cycle at the centre of the handwing for different flapping frequencies, f, flapping and sweep amplitudes, θ and φ, and downstroke ratios, DR, for both freestream velocities. The black arrows show the direction of increasing parameter value. Changes in flapping frequency have the strongest effect on effective angle of attack and relative flow velocity, whereas changes in sweep amplitude and downstroke ratio have only small effects.

The choice of the instantaneous velocity for normalization imposed certain limitations because it is based on quasi-steady assumptions, but it also has considerable utility, particularly if the wing speed is comparable to or even larger than the forward flight speed.

2.7. Statistics

Statistical tests were performed using Matlab's Statistics and Machine Learning toolbox. To compare force generation among models, we performed an analysis of covariance (ANCOVA) using aoctool on the wingstroke-averaged coefficients of lift and drag. Data from separate data collection events were treated as independent data points. We independently tested C_L and C_D of up- and downstroke using linear regression. We grouped the data into 15 bins, with a bin width of 1/15 the total velocity ratio range of each test. This number of bins ensured a minimum of two data points for every non-empty bin. Prior to the statistical analysis, we excluded outliers in each bin. Data points that were more than the mean ± s.d. different from all data in the bin were excluded from the statistical analysis (electronic supplementary material).

After identifying significant differences between the linear regression of force versus velocity ratio, we compared the results of the previous analysis using multcompare to determine statistically significant differences among the models.

3. Results and discussion

3.1. Change of forces with velocity ratio

During gliding, lift and drag vary among models and, for each model, vary with velocity. Gliding forces show no trend in relation to AR (table 3). The coefficient of drag for model AR2.5ca for the U_∞ = 5.0 m s⁻¹ seems suspiciously low, and requires future confirmation, but is included here for completeness.

Table 3.

Summary of the coefficients of lift and drag for all models at both freestream velocities.

	wing area	half-span	coefficient of lift		coefficient of drag
	S (cm2)	b (cm)	5.0 m s−1	7.5 m s−1	5.0 m s−1	7.5 m s−1
AR3.5bl	48.0	13.0	0.75	0.81	0.50	0.36
AR2.5ca	48.0	11.0	0.86	0.78	0.15	0.41
AR4.5ca	48.0	14.7	0.86	0.82	0.40	0.34
AR2.5cs	67.2	13.0	0.75	0.58	0.30	0.26
AR4.5cs	37.3	13.0	0.79	0.88	0.42	0.50

The range of velocity ratios, U_v/U_h, is smaller for downstroke than upstroke (figure 5). The sweep motion of the wing increases the horizontal velocity (denominator) during downstroke, when the wing moves forward, whereas the backwards motion during upstroke decreases the relative horizontal velocity.

Figure 5.

Summary of the mean coefficients of lift and drag for the baseline model, AR3.5bl, separated into downstroke and upstroke. The freestream velocity is designated by marker size. The marker style designates the downstroke ratio. The colour coding represents the wing movement: more green colours indicate little wing movement, and purple colours large wing movement. Blue is an indicator for wing movement dominated by the sweep motion, and red for an almost pure flapping motion. The speed ratio is the ratio of the relative horizontal to the relative vertical wind velocity over the wing.

All wings generate positive lift during downstroke and negative lift during upstroke, but the magnitude of lift depends strongly on the velocity ratio (figure 5). High velocity ratios, associated with larger wing motions, result in higher forces. At low velocity ratios, the coefficient of lift is lowest for stroke patterns with large flapping and sweep angles during downstroke (figure 5, purple markers).

During the downstroke, the wing sweeps forward and the combined motion results in increased effective angle of attack and airspeed (figure 4). The opposite is true during the upstroke. In addition, the compliance of the wing membrane allows for auto-camber, which is influenced by flow direction. During upstroke, angle of attack is usually negative (figure 4) and is likely to decrease the 9% built-in camber, or even lead to reversed camber of the skin membrane in the armwing region, where the membrane shape is not reinforced by the digits. In such a case, the wing would deflect airflow up, instead of downward, which may enhance the opposing trends in lift coefficients of down- and upstroke. With few exceptions, the magnitude of C_L is larger during the downstroke than during the upstroke and thus the wing generates net lift over a complete cycle (figure 7 and §3.2).

Figure 7.

Summary of the effect of flapping frequency, downstroke ratio, flapping amplitude, and sweep amplitude on lift and drag during the downstroke. In each plot, one parameter of the respective baseline stroke varies over the given range (table 1) and the other parameters are kept constant. Circles: E. fuscus baseline stroke pattern; crosses: T. brasiliensis stroke pattern; blue: 5 m s⁻¹ freestream velocity; orange: 7.5 m s⁻¹. Error bars indicate standard deviation of the means of independent trials. Baseline data thicker in all panels. (a) Flapping frequency, (b) downstroke ratio, (c) flapping amplitude and (d) sweep amplitude.

The coefficient of drag shows only a weak dependence on velocity ratio (figure 5). Many high sweep cases (blue and purple symbols) generate relatively high drag. This phenomenon arises from the mechanical behaviour of the membrane during flapping. At the beginning of the downstroke, when the wing is in its most swept-back position, the membrane bulges (figure 6, red arrow). As the wing sweeps forward, the membrane stretches and forms a smooth surface until mid-downstroke, when the wing starts to fold in towards the body. The trailing edge reinforcement is too compliant to take up the slack from the membrane and the bulge re-appears. This effect is most pronounced when the freestream velocity and the flapping amplitude angle are low, corresponding to the parameter combinations in which the higher drag values are observed. At higher freestream velocities and flapping amplitudes, the increased aerodynamic pressure on the membrane subjects the membrane to greater tension and probably reduces this bulging effect.

Figure 6.

Sequence of snapshots of one wingbeat cycle in quiescent air. The flapping amplitude is θ = 110°, sweep is β = 55°, and flapping frequency, f = 9 Hz. Camera 1 is located in front and above the wing, looking at the top side of the wing in a mostly front view, camera 2 is slightly elevated as well showing the bottom side of the wing. The upper row shows the downstroke, the lower row the upstroke sequence. The red arrows show the membrane bulging when the wing is in a swept-back position. Our data indicate that the membrane bulging affects the force generation more when the freestream velocity is low and the flapping amplitude is small.

3.2. Influence of kinematics on force generation

Because changes in effective instantaneous angles of attack are more extreme at the lower freestream velocities, especially during the upstroke (figure 4), and because the flapping frequency and amplitude produce the greatest changes in the local flow conditions, the flapping frequency and amplitude affect lift more than do sweep and downstroke ratio (figure 7a,c), which had little net effect (figure 7b,d). Drag decreases with flapping amplitude, but flapping frequency has no strong effect other than a slight increase in lift at the highest flapping frequency (figure 7a,c). Drag increases with sweep amplitude (figure 7d), probably because of membrane bulging (see also above). By contrast, previous work, using a similar model with a more robustly reinforced armwing trailing edge, tested in the same wind tunnel, [22], demonstrated a positive correlation between flapping angle, flapping frequency and stroke plane angle for mean lift and thrust. Differences in model design, specifically trailing edge flutter and deformation of the compliant membrane due to aerodynamic pressure may underlie these divergent results. Our results agree well with those of Hu et al. [21], who found a positive correlation between flapping frequency and net lift with a membrane model tested over a range of 1 to 10 Hz in flapping frequency and 0 to 10 m s⁻¹. For freestream velocities lower than 4 m s⁻¹, the compliant membrane wing used in that study generated thrust as the frequency increased; however, like our wings, their compliant membrane wing generated net drag instead of thrust at the higher freestream velocities comparable with the velocities tested in our experiments, and drag increased when the flapping frequency was greater than 6 Hz.

3.3. Effects of aspect ratio

AR does not influence lift or drag over the range of stroke patterns tested (figure 8; electronic supplemental material). Although the force coefficients, C_L and C_D, clearly increase or decrease in relation to velocity ratio in many cases, they exhibit substantial scatter, particularly in the case of C_D during upstroke, and we observe no statistically significant trend that depends on the wing AR. Two wings, AR3.5bl and AR4.5cs show no decrease in the coefficient of drag with increasing velocity ratio, unlike the other three wings, where C_D decreased with increasing velocity ratio. However, the slope of these five regression lines are statistically not significantly different (probability α = 5%). The slopes of the C_L versus velocity ratio regressions for the two models with the same area but the highest and lowest AR, AR2.5ca and AR4.5ca, differ only during the upstroke (p < 0.05). The dependence on velocity ratio of lift during the downstroke and of drag during the entire wingbeat does not differ significantly between those two wings (p > 0.05) (electronic supplemental material). The wing that differs the most from all other wings with respect to lift generation is model AR2.5cs; in relation to velocity ratio, this wing shows an increase in lift during downstroke, and a decrease during upstroke greater than that of other wings. However, its drag regression slopes are most similar to the AR2.5ca and AR4.5ca wings during downstroke (p = 1.00 and p = 0.96, respectively).

Figure 8.

Summary of results from statistical analysis. Separate linear regression line fits for all five models for coefficients of lift and drag, separated into down- and upstroke.

The lack of a significant change in C_L and C_D with AR suggests that AR is a poor predictor of a species' flight behaviour and ecology over the range of values observed in bats, with the exception of molossids. In this study, we did not test models with ARs as high as those common in this family, although we did use one baseline stroke pattern that was based on the wingbeat kinematics of the molossid, T. brasiliensis. T. brasiliensis can fly at high altitudes and achieve high peak speeds [26,27] and their higher AR might allow for a higher lift to drag ratio compared with the low AR models tested here. We did not investigate the effect of AR on power consumption, but predict wings with the largest area or wing span require more power, because their angular momentum is higher in the first case due to greater mass overall, and because of the larger radius (span) that places some of the mass further away of the body, which requires more torque from the motors to reverse the stroke direction in the second. This effect might be balanced by lower induced drag and therefore lower induced power due to the larger wingspan, but because we did not observe a significant influence of AR on drag, we do not expect a strong impact.

We noted the greatest amount of wear on models AR2.5cs, the wing with the largest wing area, and AR4.5ca, the wing with the longest wingspan. We believe that this arose directly from the loads they experienced. Overall, the aerodynamic force experienced by the AR2.5cs model is higher than for the other wings (comparable C_L as the other wings with a larger area indicates higher lift force). For the AR4.5ca wing, inertial forces are higher than for the models with shorter wingspans, which causes higher strain on the skeleton at the rapid reversal of the wing stroke direction.

4. Implications for robotic wing design

4.1. The role of sweep on wing tension

Unlike many robotic models that use only wing flapping, the model we employed here used both flapping and sweeping motion. This additional degree of freedom provides some operational advantages; forward sweep at low forward flight speeds can be used to maintain the magnitude of the effective velocity while still controlling the effective angle of attack. This might be useful for robotic devices in which low speed operation with heavy payloads and high coefficients of lift are required. However, the kinematics of combined flapping and sweeping has its challenges: the compliant membrane follows flapping actuation closely, particularly where it attaches to the model's body, but it deviates more substantially from the theoretically dictated sweep motion; specifically the area of the armwing decreases when the wing is swept back and increases with forward sweep. The membrane ‘bulging’ observed at high sweep angles is symptomatic of this lack of precise control over the membrane area and it leads to undesirable increases in drag coefficients.

Bats have evolved active mechanisms to control the tension of the wing membrane, such as leg motion [28] and active modulation of skin stiffness by muscles that attach to the collagen and elastin connective tissue elements of the skin [7,29]. In C. brachyotis, the length of the trailing edge between the ankle and fifth digit increases during downstroke and decreases during upstroke due to this dynamic control of the wing geometry [28]. Inspired by this, we are developing a new version of our mechanical flapper with a ‘leg’, in which the wing attachment site on the body moves in synchrony with the sweep motion of the arm skeleton. When the wing sweeps fore and aft, the leg rotates with the arm segment, and the body attachment plus arm, combined, follows the rotation of the shoulder. This ensures that the wing area does not vary as dramatically, and thus helps to maintain tension in the armwing membrane. A second improvement would be to develop a more hyperelastic wing membrane material that, like the biological bat wing membrane, can better accommodate a substantial range of wing motion without excessive wrinkling or buckling.

4.2. Importance of wing twist

At high angles of attack, the airflow over a wing no longer follows the wing's profile and separates, leading to increased drag and decreased lift, ultimately producing stall. Although compliant wings are able to tolerate higher angles of attack before stall than rigid wings because the wing's camber can self-adjust to balance the pressure difference between the upper and lower surfaces of the wing [30,31], airflow still separates when the angle of attack becomes too extreme. Kinematic patterns that encompass rapid and high amplitude flapping motion produce unfavourably large effective angles of attack, particularly at the outer portion of the wing, and at mid-downstroke and mid-upstroke when the vertical speed is greatest (figure 4a). These portions of the stroke cycle are thus susceptible to separation and stall. Wing twist, in which the effective angle of attack at the distal portions of the wing can be reduced by reorienting the wing relative to the flow, adjusting the pitch downward during downstroke and upward during upstroke, can mitigate these problems.

Birds use of wing twist has long been recognized [32] and there is evidence to suggest that bats also control the effective angle of attack locally along the wingspan. The effective angle of attack changes over the wingbeat cycle at different flight speeds in Glossophaga soricina (Pallas' long-tongued bat) [33]. The relative angle of attack remains relatively constant throughout the downstroke at higher flight speeds, but becomes moderately negative during upstroke. This suggests pitching of the handwing for better alignment with the main flow direction [33]. Similar results have been reported for Leptonycteris yerbabuenae (lesser long-nosed bat) [34]. Similarly, pteropodid bats maintain a very small, or even negative chord line at mid-downstroke [35]. In the big brown bat, E. fuscus, we observe a clear pitch motion about the wrist [24]. At mid-downstroke, the wing is oriented with a very shallow or possibly slightly negative angle with respect to the flight direction, but at mid-upstroke, the handwing is pitched up, with the digits in an almost vertical position (figure 9). Long-axis rotation of the forearm to allow for pitch of the handwing is possible for the mechanical wings. Numerical studies also demonstrate the importance of wing twist, which appears to be necessary to increase lift and generate thrust instead of drag in a compliant bat-like wing [36,37].

Figure 9.

Two snapshots of E. fuscus in steady flight [24]; digit III highlighted in red. (a) Mid-downstroke; handwing is oriented in a shallow or slightly negative angle with respect to the flight direction. (b) Mid-upstroke; handwing is pitched up, resulting in steep positive angle between wing and flight direction.

These observations, combined with an analysis of the differences between upstroke and downstroke, suggest that, in a robotic wing model, force generation would benefit significantly from an additional degree of freedom in pitch. In a model of this kind, high wing pitch would be a key mechanism by which to reduce both drag and negative lift during upstroke.

4.3. Wing area and membrane properties

In addition to wing twist, modulating wing area during upstroke is an effective means for controlling drag and reducing inertial costs during the upstroke, and has been observed in both small birds [38,39] and pteropodid bats [40]. Birds are able to reduce wing area easily without incurring negative side effects. Feathers can slide over each other as they overlap with no disruption to the smooth lifting surface. Bats, however, face a problem similar to that observed with our model during sweep: reduction of the wing surface can reduce the tension that keeps the membrane smooth, leading to bulging or wrinkling. For bats, the solution to this problem lies in skin composition: the membrane skin of the bat wing is a fibre composite composed of a collagenous matrix with an imbedded network of pre-strained elastin fibres [29]. This unique membrane construction serves to corrugate the wing membrane as it folds, taking out the excess length and preventing the wing from flapping or bulging.

The benefit of varying wing area has been successfully demonstrated in a previous version of the mechanical flapper employed here [22,23]. However, the challenge of accommodating excess membrane length during the retraction of isotropic wings remains challenging. Attaching the membrane to the wing at its most swept-back configuration would prevent wrinkling, and keep tension in the membrane as the wing sweeps forward during the downstroke. However, this solution imposes high demands on actuators (servo motors in this case) and high stresses on the wing skeleton and body attachment site when ‘conventional’ elastic membrane materials such as silicone are employed. This stress leads to fatigue and a reduced operational life. Mimicking the anisotropic, hyperelastic behaviour of the biological membrane material would be a preferable approach, and research in this direction is promising [41].

5. Concluding remarks

The design, fabrication and testing of a robotic wing that captures several important characteristics of bats and also allows experimental manipulation has demonstrated several valuable features of bioinspired wing design, and dispelled some initial expectations regarding the role of AR. We conclude that wing stroke pattern has a stronger effect on aerodynamic force generation than geometric AR. The generation of lift depends strongly on the ratio of relative vertical to relative horizontal wind speed, with higher vertical velocities resulting in more lift. The dependence of drag is less clear, partly because drag values are lower and subject to greater experimental uncertainty. Nevertheless, higher velocity ratios decrease drag during the downstroke.

Whether our findings apply to larger bats that fly at a higher Reynolds number regime, where quasi-steady aerodynamics become more applicable, remains uncertain. Furthermore, we did not investigate the very high AR range that is relatively rare among bats, but is characteristic of Molossidae.

Based on insights gained during these experiments using relatively simple models, we propose some desirable directions for wing design in future robotic wing experiments: (i) incorporation of a wing-area reduction mechanism, using elbow and wrist flexion during upstroke [23], (ii) mitigation of membrane bulging at the swept-back position of the wing, either by the use of a ‘leg’ attachment that follows the sweep motion or by the use of anisotropic hyperelastic materials that allow extreme spanwise stretch without substantial chordwise elongation, and last, (iii) introduction of long-axis rotation of the forearm to enable pitch of the handwing and thus control the local angle of attack.

Data accessibility

Data are supplied as part of the electronic supplemental material.

Authors' contributions

C.S. contributed to the experimental design, data collection, data analysis and manuscript preparation; K.S.B. and S.M.S. contributed to the experimental design, data analysis and manuscript preparation.

Competing interests

We declare we have no competing interests.

Funding

This work was supported by AFOSR grant FA9550-12-1-0210, monitored by Doug Smith, and NSF-NRI grant CMMI 1426338. The support of the Ostrach Graduate Fellowship (C.S.) is gratefully acknowledged.