Scaling of the performance of insect-inspired passive-pitching flapping wings

Abstract

Flapping flight using passive pitch regulation is a commonly used mode of thrust and lift generation in insects and has been widely emulated in flying vehicles because it allows for simple implementation of the complex kinematics associated with flapping wing systems. Although robotic flight employing passive pitching to regulate angle of attack has been previously demonstrated, there does not exist a comprehensive understanding of the effectiveness of this mode of aerodynamic force generation, nor a method to accurately predict its performance over a range of relevant scales. Here, we present such scaling laws, incorporating aerodynamic, inertial and structural elements of the flapping-wing system, validating the theoretical considerations using a mechanical model which is tested for a linear elastic hinge and near-sinusoidal stroke kinematics over a range of scales, hinge stiffnesses and flapping frequencies. We find that suitably defined dimensionless parameters, including the Reynolds number, Re, the Cauchy number, Ch, and a newly defined ‘inertial-elastic’ number, IE, can reliably predict the kinematic and aerodynamic performance of the system. Our results also reveal a consistent dependency of pitching kinematics on these dimensionless parameters, providing a connection between lift coefficient and kinematic features such as angle of attack and wing rotation.

1. Background

Animals capable of flapping flight leverage a variety of wing compositions, wing kinematics, and corresponding flight strategies to achieve sustained aerial locomotion, ranging from bats, which use highly articulated compliant membrane wings [1], to insects whose wings resemble rigid flat plates [2]. Flapping flight offers many potential advantages for small-scale engineered flying vehicles including high manoeuvrability and robustness to environmental turbulence. In addition, flapping flight is achieved using comparatively low flapping frequencies, which result in less noise and improved safety when compared to more conventional propeller-driven aircraft.

Among all realizations of flapping flight in nature, the flight of insects is arguably the most accessible for robotic adaptation due to its relative simplicity. Unlike birds and bats, insects lack musculature in the distal regions of their wings such that all input to flight control occurs at the wing root near the abdomen. Although insect wings do flex [2,3], their principal kinematics can be characterized by two main motions: revolution in the dorso-ventral direction, or flapping, and rotation along the span-wise axis, or pitching (figure 1a,b). Despite having relatively simple wing structure and flight kinematics, insects display remarkable agility and manoeuvrability, traits which have led to a strong interest in adopting insect-inspired flight control mechanisms for human-made micro aerial vehicles [4–9]. At the same time, bioinspired aerial robots have also proven to be useful tools in probing the mechanics of insect flight [10].

Figure 1.

Geometry and construction of the passive-pitching wing. The typical insect wing stroke (a,b) is modelled as a 2 d.f. system, described by the wing stroke angle, ϕ, and the wing rotation angle, ψ. The wing rotation angle is passively modulated (c) by an elastic hinge of torsional stiffness, κ, which balances the moments due to aerodynamic lift and to inertia associated with the wing and added (fluid) mass acceleration. The hinge (d) acts as a linear torsion spring whose stiffness is controlled by the elastic modulus of the material and the hinge dimensions. The test apparatus (e) consists of the wing assembly mounted on a six-axis force/torque transducer and driven by a DC motor. The wing angle is recorded by a camera positioned above the apparatus. (Online version in colour.)

Evidence suggests that prominent features of wing pitching behaviour in insects are largely passive phenomena induced by inertial and aerodynamic effects with little active input from the wing hinge joint, which acts like a torsion spring (figure 1c) [11–15]. Despite its elegance and relative simplicity, a detailed understanding of the mechanics of this mode of flight remains elusive; and although the mechanism has been successfully implemented in millimetre-scale flapping robots [4,6,7], tuning the balance of wing mass, torsional stiffness, and aerodynamic forces in the construction of a passive-pitching flapping wing remains largely a trial-and-error challenge for which there are few experimentally verified principles of design.

In particular, it would be of great value to understand how the dynamics of passive-pitching flapping-wing systems are affected by changes in the physical properties of the wing hinge and in the scale of the wing system. This knowledge might be used, not only to better understand insect flight at different scales but also to extrapolate from biological systems and enable the design of engineered passive-pitching flapping-wing systems operating in flight regimes not observed in nature. While there has been significant progress in understanding how the fluid–structure interaction and unsteady fluid dynamic phenomena characteristic of flapping-wing systems contribute to aerodynamic performance [16–18] through biological studies [19–24], quasi-steady blade element method analyses [25–31], two- and three-dimensional computational fluid dynamics simulations [32–38], and experimental realizations [13,21,31,35,39,40], there have yet to be any comprehensive studies regarding how the performance or feasibility of flapping-wing systems with passive pitching kinematics is affected by a change in physical scale.

Much prior work in scaling the performance of biological and bio-inspired flapping wings was limited by available measurement technology [19,20,25–28,41], or was otherwise restricted to cases where the wing motion is completely prescribed [16,39,40,42,43]. Some recent works have addressed systems in which wing pitching kinematics are mediated by a torsion spring, defining scaling parameters which relate fluid forces and elastic restoring torques in a quasi-steady aerodynamic model of a passive-pitching wing. Such models have been used explore the effects of hinge stiffness and aerodynamic loading on lift force generation [13,35,44], as well as to understand the effects of hinge nonlinearity on body yaw torque generation [45]. However, the scaling arguments for passive-pitching wings to date have focused on the translational phase of the wingbeat cycle, with little attention given to the effects of wing inertia and fluid added mass in determining pitching behaviour and the effects on aerodynamic performance during wing rotation at stroke reversals.

This work seeks to further develop tools for understanding the physics of flapping wings with an elastic hinge and for designing bio-inspired passive-pitching flapping-wing systems. Here, we (1) quantify how inertial, aerodynamic, and elastic effects interact to determine the kinematics and aerodynamic performance of a passive-pitching flapping wing and (2) explore how effectively simple analytical models can be used to predict the dynamic characteristics of such systems spanning a broad range of physical parameters. To do this, we describe the design, fabrication, and testing of a centimetre-scale flapping robotic wing with a linear elastic hinge (figure 1) and introduce a scaling parameter, the inertial-elastic number, IE, which captures the essential dynamics associated with the wing rotation. We demonstrate the validity of the scaling analyses for systems of various geometric, kinematic, and dynamic scales, ranging from small insects that operate in Reynolds numbers in the range of Re ∼ 10² to robotic flapping micro aerial vehicles, for which Re ∼ 10³.

2. Methods

We designed and built a test apparatus that allows for the measurement of lift force as a function of geometric parameters: wing dimensions and aspect ratio; dynamical parameters: torsional hinge stiffness; and operational parameters: flapping frequency. Other parameters can be varied—for example, flapping amplitude, wing shape, ambient gas properties, etc.—but these were not explored in the present work.

2.1. Four-bar linkage design and fabrication

The reciprocating motion in the test model was generated using a four-bar linkage mechanism (figure 1e). This method of creating oscillating motion depends solely on geometry. The motion of the wing can be determined by the following relationship [46] between the input angle, θ_i, and the output angle θ_o:

$${\theta }_o({\theta }_i)=2\arctan \left(\frac{A-\sqrt{A^2+B^2-C^2}}{B+C}\right)+{\theta }_{\mathrm{off}},$$ 2.1

where A = sinθ_i, B = a₄/a₁ + cosθ_i, $C=(a_1^2-a_2^2+a_3^2+a_4^2)/(2a_1{a}_3)+(a_4/a_3)\cos {\theta }_i$, and a_i (i = 1 … 4) are the lengths of the four linkage elements. Note that the flapping system used in the present experiments is not a resonant drive as is used in some other robotic passive-pitching flapping wing designs [8,44], and that the stroke amplitude is determined purely by the geometry of the four-bar linkage.

In the present design, a₁ = 9 mm, a₂ = 30 mm, a₃ = 8 mm and a₄ = 30 mm, chosen to generate an approximately sinusoidal flapping profile of amplitude Φ = 126.0° and upstroke-to-downstroke ratio of 0.96. The angle offset, ${\theta }_{\mathrm{off}}$ is included such that the output is centred about θ_o = 0. In this design, ${\theta }_{\mathrm{off}}={86.0}^{\circ }$. The downstroke is defined as the portion of the flapping cycle during which the stroke angle decreases from ϕ = +63° to ϕ = −63° (figure 1a).

Elements of the four-bar linkage were designed in Fusion 360 CAD software (Autodesk, California, USA) and fabricated using a Form 2 SLA resin printer (Formlabs, Massachusetts, USA) in clear photopolymer resin. The linkage mechanism was assembled using stainless steel rods, bearings, and cyanoacrylate adhesive. One of the two stationary points in the four-bar linkage is pinned to a 3D-printed base (Formlabs) such that the wing was elevated approximately 50 cm above the nearest surface. The other stationary point in the four-bar linkage mechanism is fixed to a 2342S006CR DC motor (Micromo, Florida, USA) with a nominal 4 : 1 gear reduction (figure 1e).

2.2. Design, fabrication and characterization of wing hinge

The wing hinge consists of a thin rectangular polymer plate, 0.01 mm thick, clamped at both ends, similar in design to torsion springs used in other passive-pitching flapping robots [4,13,35]. Using Euler–Bernoulli beam theory, it can be shown that the restoring torque of a clamped flat plate undergoing bending deformations is linear with respect to deflection angle: τ = κΔψ. The torsional stiffness, κ, is given by the plate geometry:

$$\kappa =\frac{E{t}^{3}w}{12L},$$ 2.2

where E is the elastic modulus and t, w and L are thickness, width, and length of the plate, respectively (figure 1d).

Elastic wing hinge joints were created with stiffnesses ranging approximately from 1.0 Nmm rad to 2.5 Nmm rad. The torsional stiffness for each spring was determined from a linear fit of the measured reaction torque as a function of deflection angle under static loading. Reaction torque measurements were obtained by averaging a torque signal acquired with a Nano 17 six-axis force/torque transducer (ATI Industrial Automation). Measurements were taken at deflection angles from ψ = −60° to ψ = +60° in increments of 20°.

A set of eight springs was created for the purpose of verifying the elastic properties of this type of hinge under static loading. The mean coefficient of determination for the linear regressions between the reaction torque and deflection angle for those springs was r² = 0.997. The measurements of stiffness for different geometries were also used to verify that κ is linear with the geometrical parameter, t³w/L (equation (2.2)). After the removal of a single outlier, a linear regression between spring stiffness and t³w/L yielded a coefficient of determination of r² = 0.988.

2.3. Wing design and fabrication

The wing frame consists of 0.020 inch diameter carbon fibre rods (DragonPlate, New York, USA) connected by a 3D printed joint which was designed in Fusion 360 (Autodesk) and 3D printed in clear photopolymer resin (Formlabs). A wing membrane cut from 0.01 mm thick plastic sheet was fixed to the frame. Wings were assembled using cyanoacrylate adhesive. The area of individual wings ranged approximately from 18 cm² to 26 cm² with nominal aspect ratios of either 2.7, 2.9 or 3.6.

2.4. Force measurement and data acquisition

The experimental set-up was mounted on an optical table with the wing flapping in air at ambient conditions. Data were acquired using the wings for flapping frequencies ranging approximately from 3 Hz to 11 Hz. Forces and torques were measured using a Nano 17 six-axis force/torque transducer (ATI Industrial Automation, North Carolina, USA) which was calibrated for a force resolution of 0.003125 N. The force sensor was mounted such that the x-axis aligned with the direction of lift and the centre of the force sensor axes was positioned directly below the attachment point of the wing to the four-bar linkage mechanism. Time-resolved force and torque signals were recorded using an analogue–digital converter (USB 6341, National Instruments, Texas, USA) at a sampling rate of 10 kHz. The force sensor was tared each time before changing flapping frequencies, and data were acquired continuously for 30 s at each frequency once the flapping wing had reached a steady oscillating state.

The motor position was recorded using a HEM-16 motor encoder (Micromo). The position of the wing was recorded at 60 fps using a single camera (D7200 Nikon, Japan) positioned above and perpendicular to the stroke plane.

Ambient temperature, pressure and relative humidity values were recorded before each lift force measurement using a WS-2801A electronic weather station (Ambient Weather, Arizona, USA), and these were used to compute fluid density and viscosity.

2.5. Calculation of the average and variance of lift forces

For each condition tested, the measured lift force versus time, L(t), is divided into wingbeat cycles using information from the motor encoder. The average lift of the i-th cycle, 〈L〉_i is calculated by integrating the lift force over the cycle:

$${⟨L⟩}_i=\frac{1}{T}{\int }_0^{T}L(t)\hspace{0.17em}\text{d}t.$$ 2.3

The average lift force is calculated as the average of the cycle-averaged lift forces

$$\overline{L}=\frac{1}{N}\sum _{i=1}^N{⟨L⟩}_i,$$ 2.4

where N is the number of cycles recorded (typically approx. 20). The error bars (figure 8) represent the standard error, σ:

$$\sigma =\frac{1}{\sqrt{N}}\left(\frac{1}{N-1}\sum _{i=1}^N{(\overline{L}-{⟨L⟩}_i)}^2\right).$$ 2.5

Figure 8.

The time-averaged lift coefficient, C_L, plotted as a function of the two dimensionless elastic scaling parameters defined in this paper. (a) C_L scales reasonably well with the Cauchy number, Ch. (b) A better collapse of the data is achieved when scaled using the inertial-elastic number, IE. The insets include data from a large-scale oil tank experiment represented by black circles [13] and black triangles [35], and from the millimetre-scale robot, ‘RoboBee’ [8], represented by black squares. Curves showing model predictions for the lift coefficient during translation under different conditions are superimposed on the plot. Aspect ratio is indicated by colour: red, AR = 2.9; blue, AR = 3.6; black, AR = ∞. Reynolds number is indicated by line style: solid, Re = 1.1 × 10²; dashed, Re = 1.4 × 10³; dashdot, Re = 1.4 × 10⁴. For coloured markers, aspect ratio is indicated by marker shape: circles, AR = 3.6; triangles, AR = 2.9; squares, AR = 2.7. (Online version in colour.)

Although the time-resolved lift force measurements are available, they are dominated by repeatable vibrations associated with the motion of the four-bar linkage (which average to zero) and do not give useful insight to aerodynamics. For this reason, we do not present the phased-averaged time histories of the lift forces.

2.6. Image processing and kinematics reconstruction

At each frequency, kinematic trajectories (stroke and pitch angles) were created by superimposing kinematic data acquired from consecutive wingbeats. For each image, the measured stroke angle was used to identify its location in time within the wingbeat cycle, and the pitch angle was computed from the projected image of the wing and the known geometry of the wing planform.

3. Results

We first present more detailed analytical models used to understand the kinematic and aerodynamic behaviour of the pitching wing and then show comparable data from our experiments. The scaling analyses and modelling are split into two parts, reflecting the features of the wingbeat cycle: a quasi-steady aerodynamic model of wing pitch deflection angle during translation and a dynamical, inertia-based, model of the behaviour of the wing during rotation. The aerodynamic and inertial scaling analyses are combined to produce a prediction of the normalized angular velocity during wing rotation. These scaling behaviours are used to organize the experimental measurements of wing kinematics and force production.

3.1. Pitch angle during quasi-steady translation

The ability to predict the performance of flapping flight systems is greatly aided by the appropriate use of scaling analyses, many of which are based on the quasi-steady blade element analysis of a revolving wing [26,27] and have been widely adopted throughout flapping-wing literature [29,39]. Before extending the classical analysis, we review the commonly used non-dimensional parameters.

The input parameters of the problem can be divided into geometric and dynamic contributions. The wing geometry (figure 1b) is governed by the wing length, R, area, A, mean chord, $\overline{c}=A/R$, and aspect ratio, $\text{AR}\hspace{0.17em}=R/\overline{c}$; the dynamic parameters are the flapping frequency, f, the stroke amplitude (in radians), Φ, and the torsional stiffness of the wing hinge, κ. The ambient fluid properties are described by density, ρ, and viscosity, μ.

From these, we can estimate typical forces and torques associated with aerodynamic (dynamic) pressure, viscous fluid stresses, and elastic bending stiffness. By comparing the relative magnitudes of these forces, we can characterize the flapping wing using a few non-dimensional parameters that describe the physical behaviour.

The ratio between aerodynamic and viscous forces yields the Reynolds number which is known to influence fluid behaviour [22,24,39,42]:

$$Re=\frac{2\mathit{\Phi }f(R+x_r)\overline{c}}{\nu },$$ 3.1

where ν = μ/ρ is the kinematic viscosity of the fluid medium (for details, see electronic supplementary material, S-I). x_r is the wing root offset in the span-wise direction, which increases the effective wing velocity. The present experiments were conducted at Reynolds numbers in the range of 10³; this is an order of magnitude greater than that of most insects, which operate at Re in the range of 10², and it is greater than that of many current flapping robotic systems, which do not generally operate at Re exceeding 2000 (table 1).

Table 1.

Representative physical parameters and non-dimensional numbers for three insect species—Tipula obsoleta, Drosophila melanogaster and Apis mellifera—and selected robotic flapping systems. Morphological and kinematic information was obtained from [8,11,13,19,20,24,28,47,48] and approximated when otherwise unavailable.

	Φ [deg]	AR	f [Hz]	R [mm]	Re	Ch	M	IE	CL	β
D. melanogaster	140	2.9	200	2.5	150	0.6	2	1.3	1.4	2.2
T. obsoleta	123	5.2	46	14	450	0.8	2	0.5	1.6	0.6
A. mellifera	90	3.3	240	9.7	1000	—	13	—	0.4	—
Ishihara et al. [13]	123	5.4	0.5	225	420	0.1–1.0	0.006	0.02–0.18	1.3–1.9	0.17
Jafferis et al. [8]	50–70	3.2	200	17	300–1800	0.03–0.5	1.3	0.04–0.5	1.6–3.7	1.0–1.3
present work	126	2.7–3.6	3–11	70–100	2600–9900	0.02–0.2	0.1–0.3	0.008–0.1	0.3–2.3	0.36–0.59

The aerodynamic performance can be characterized by the lift coefficient, C_L, which compares the lift force, F_L, to the aerodynamic force generated by a flapping wing

$$C_L=\frac{F_L}{(1/2)\rho {(2\mathit{\Phi }fR)}^{2}R\overline{c}{\hat{r}}_2^2}.$$ 3.2

Here, ${\hat{r}}_2^2$ is the second radius moment squared, defined as

$${\hat{r}}_2^2={\int }_0^1{(\hat{r}+{\hat{x}}_r)}^2\hat{c}(\hat{r})\hspace{0.17em}\text{d}\hat{r}.$$ 3.3

$\hat{r}$ and $\hat{c}$ represent the radial dimension normalized by the wing length, R, and the chord normalized by the mean chord length, $\overline{c}$, respectively. ${\hat{x}}_r$ represents the wing root offset in the span-wise direction normalized by the wing length, and ${\hat{y}}_r$ represents the wing root offset in the chord-wise direction normalized by the mean chord length. The second radius moment term acts as a correcting factor to account for the greater translational velocity (and greater lift force) at the tip of a revolving wing than at its base. A similar definition can be written for the coefficient of drag

$$C_D=\frac{F_D}{(1/2)\rho {(2\mathit{\Phi }fR)}^{2}R\overline{c}{\hat{r}}_2^2}.$$ 3.4

Comparing the aerodynamic torque to the elastic restoring torque of the hinge yields the Cauchy number, Ch:

$$Ch=\frac{2\rho {\overline{c}}^2{R}^3{\mathit{\Phi }}^2{f}^2{\hat{r}}_2^2(1+{\hat{y}}_r)}{\kappa }\hspace{0.17em}$$ 3.5

(for details, see electronic supplementary material, S-I). The Cauchy number bears resemblance to parameters defined in prior works. Whereas some studies use the instantaneous wing velocity [44,45], we choose a stroke-averaged aerodynamic torque to scale the hinge stiffness. Note that the definition of the Cauchy number used here differs by a purely geometric factor from the definitions described in [13,35].

The pitch angle during translation can be estimated by assuming an equilibrium balance between aerodynamic and elastic forces. Here, we follow the approach described in prior works [44,45], with slight differences in the definition of the aerodynamic scaling parameter and in aerodynamic coefficient approximations.

The total aerodynamic force acting on the wing is the sum of lift and drag forces, F = F_Lsin (ψ₀) + F_Dcos (ψ₀), acting at the centre of pressure of the wing, $\overline{c}(1+{\hat{y}}_r)$. Substituting this into the definition of the Cauchy number (equation (3.5)) we find

$$Ch=\frac{{\psi }_0}{\sin ({\psi }_0)C_L({\psi }_0)+\cos ({\psi }_0)C_D({\psi }_0)}.$$ 3.6

To obtain quantitative estimates of the pitch angle, we have fitted sinusoidal approximations to the experimental data of Lentink & Dickinson [39], which span a range of single wing aspect ratios and Reynolds numbers (see electronic supplementary material, S-II), and numerically solved equation (3.6) (figure 2a), finding that ψ₀ is a convex, monotonically increasing function of Ch. The slope of the curve is close to 1 for low values of ψ₀, but tapers off to 0 as ψ₀ approaches 90°. A weak dependence of ψ₀(Ch) on both AR and Reynolds number, Re, is observed. The lift coefficient, C_L, as a function of Ch is shown in figure 2b and indicates that there is a maximum achieved at an intermediate value of Ch.

Figure 2.

Analytic models of the quasi-steady portion of the wingbeat cycle. (a) The pitch deflection angle, ψ₀, during translation as a function of the Cauchy number, Ch, obtained using the quasi-steady aerodynamic model of a passive-pitching flapping wing during translation. (b) The lift coefficient during translation as a function of the Cauchy number. Curves representing results for different wings of aspect ratio, AR, and Reynolds number, Re, are are differentiated by colour and line style, respectively. (Online version in colour.)

3.2. Inertial effects during wing rotation

While the Reynolds and Cauchy numbers accurately characterize the balance between fluid inertia, viscous stresses, and elastic forces, a key component of the dynamics of passively pitching wings is the role of the wing inertia and added (fluid) mass that must be accelerated each time the wing undergoes pitch rotation. Recognition of the importance of these dynamics is not new [24,40,49–51], but they have not yet been incorporated into a scaling analysis.

To assess these effects, we introduce a parameter called the inertial-elastic number, IE, to capture the dynamics during wing rotation when the translational velocity of the wing is close to zero but the wing experiences significant acceleration as it undergoes stroke reversal. IE represents a balance between inertial and elastic stresses, and can be written as

$$\text{IE}\hspace{0.17em}=\frac{I(2\mathit{\Phi }{\pi }^2{f}^2)}{\kappa },$$ 3.7

where the total inertia of the system, I, is the sum of inertial contributions from the wing structure and fluid added mass, I_w and I_f, respectively. Substituting the inertia terms with first-order approximations based on wing geometry yields the expression

$$\text{IE}\hspace{0.17em}=\frac{2{\pi }^2\rho {\overline{c}}^{4}R\mathit{\Phi }{f}^2{\hat{c}}_3^3}{\kappa }(1+M),$$ 3.8

where ${\hat{c}}_3$, the third chord moment, is a non-dimensional description of the geometry (see electronic supplementary material, S-I) and M, the mass number, is a dimensionless parameter which describes the ratio of the wing inertia to fluid inertia (i.e. added mass):

$$M=\frac{I_w}{I_{\hspace{0.17em}f}}\approx \frac{m_w{y}_{\mathrm{ci}}^2+m_w{(y_{\mathrm{cm}}+y_r)}^2}{(\rho {\overline{c}}^{2}R){\overline{c}}^2{\hat{c}}_3^3}.$$ 3.9

Here, m_w is the mass of the wing, y_cm is the location of the center of mass of the wing, and y_ci is the location of the center of inertia of the wing with respect to the rotational axis associated with the pitching motion (see electronic supplementary material, S-I).

The mass number can span several orders of magnitude depending on the physical and geometric properties of the system. When M ≪ 1, the inertial contribution of the added fluid mass dominates over that of the wing mass. Conversely, the added fluid mass is negligible in the case of larger mass numbers. Representative values of M, IE, Ch, Re for a selection of biological and robotic flyers are shown in table 1.

Finally, it is useful to define the ratio of the Cauchy and inertial-elastic numbers, β:

$$\beta =\frac{\text{IE}}{Ch}=\frac{(1+M){\pi }^2{\hat{c}}_3^3}{{(\text{AR})}^2\mathit{\Phi }{\hat{r}}_2^2(1+{\hat{y}}_r)},$$ 3.10

which is a parameter that compares the relative roles of inertial to aerodynamic forces, independently of flapping frequency and hinge stiffness. Higher values of β may correspond to cases in which wing rotation plays a more important role in the overall dynamics of force production. The parameter β determines how the values of Ch are mapped to values of IE; as β increases, a given range of Ch will span a wider range of IE.

We note that the inertial-elastic number also serves as a scaled flapping frequency. Specifically, it compares the flapping frequency, f, to the natural frequency of the wing rotation, ω_n. To explain this relationship, we model the wing pitching dynamics during stroke reversals as a harmonic oscillator:

$$I\hspace{0.17em}\ddot{\psi }=-\kappa \psi -C\hspace{0.17em}\dot{\psi }.$$ 3.11

Aerodynamic contributions are neglected because stroke angle velocity is close to zero at stroke reversal.

The (undamped) natural frequency of the this system is ${\omega }_n=\sqrt{I/\kappa }$, and substituting this into the definition of the inertial-elastic number (equation (3.7)), we obtain

$$\text{IE}\hspace{0.17em}=2\mathit{\Phi }{\pi }^2\frac{\hspace{0.17em}{f}^2}{{\omega }_n^2}.$$ 3.12

Using this, IE can be used to assess the fraction of the flapping period, T = 1/f, that is occupied by the wing rotation, or the normalized duration of wing rotation, Δt/T. The time between peaks in a system oscillating at its damped natural frequency, ω_d, is Δt = π/ω_d; thus the normalized wing rotation duration is

$$\frac{\mathrm{\Delta }t}{T}=\frac{\pi f}{{\omega }_d}.$$ 3.13

The damped natural frequency of the system is given by

$${\omega }_d=\sqrt{{\omega }_n^2-{\gamma }^2},$$ 3.14

where γ = C/2I is the damping coefficient. We can combine equations (3.12) and (3.14) to rewrite equation (3.13) in terms of IE:

$$\frac{\mathrm{\Delta }t}{T}=\sqrt{\frac{\text{IE}}{2\mathit{\Phi }}}\sqrt{\frac{{\omega }_n^2}{{\omega }_n^2-{\gamma }^2}}.$$ 3.15

The presence of damping (either structural or fluid) would generally increase the normalized duration rotation time. However, detailed investigation of the effects of damping is beyond the scope of the present work.

The predicted wing rotation time (equation (3.15)) for the case of γ = 0 is shown in figure 3 for representative values of the stroke amplitude, Φ. As Φ increases, the rotation occupies a smaller fraction of the stroke for a given IE. It should be noted that the maximum range of validity for this simple model is IE = Φ/2, since it is impossible for a single wing rotation to occupy more than half of the wingbeat cycle.

Figure 3.

The normalized rotation duration as a function of the inertial-elastic number, obtained by modelling the wing during the rotation as a simple harmonic oscillator. Different line styles are used to demonstrate the effect of varying wing stroke angle, Φ.

3.3. Angular velocity during wing rotation

The results of the two previous sections can be combined to predict the angular velocity during wing rotation. We assume that the pitch angle at the start of wing rotation, ψ(0), is determined by the quasi-steady aerodynamic loading, ψ(0) = ψ₀(Ch), and that the wing is initially at rest, ψ′(t) = 0. Using these as initial conditions in our model for the wing rotation (equation (3.11)) we find the normalized maximum angular velocity, ω*, to be (in the absence of damping)

$${\omega }^{\ast }=\pi \sqrt{2\mathit{\Phi }}\hspace{0.17em}\frac{{\psi }_0(Ch)}{\sqrt{\text{IE}}}.$$ 3.16

Solutions to equation (3.16) for Φ = 120° and β = 0.5 using different models of the lift and drag coefficient are shown in figure 4a. Similarly to the trends observed in figure 2a, increasing aspect ratio has the effect of decreasing the angular velocity and shifting the curve to the right; increasing the Reynolds number leads to similar changes, though to a lesser degree. These effects of aspect ratio and Reynolds number are also reflected in the values of ψ₀ at which ω* reaches a maximum. Note that the value of ψ₀ at maximum ω* is determined by the functional relationships between ψ₀ and Ch and between Δt/T and IE, but it is independent of stroke amplitude and β.

Figure 4.

The normalized angular velocity during wing rotation, ω*, as a function of the inertial-elastic number, IE. (a) The effects of changing aspect ratio, AR, and Reynolds number, Re, are shown for β = 0.5 and Φ = 120°. (b) The effects of changing β are shown for AR = 2.9, Re = 1.4 × 10⁴, and Φ = 120°. Empty circles correspond to different values of the quasi-steady pitch deflection angle, ψ₀, in 15° intervals, starting at ψ₀ = 15°. The location of the maximum value of ω* is marked by an asterisk. (Online version in colour.)

Representative solutions to equation (3.16) for AR = 2.9, Re = 1.4 × 10⁴, and Φ = 120° for a range of β are shown in figure 4b. As β increases, the locations of the maximum normalized angular velocity and lift coefficient during translation (ψ₀ = 45°) migrate to higher values of IE because β acts as the scaling factor between Ch and IE. Due to the square root dependence of Δt/T on β, the normalized angular velocity has the inverse relationship: ω* ∝ β^−1/2.

3.4. Measured wing kinematics

We now turn our attention to the experimental measurements, and in the following sections we will assess how our robotic wing behaves in the context of the previously described scaling laws.

Since the trajectory of the wing stroke is determined by a kinematic relationship with the position of the driving motor through a four-bar linkage mechanism, the wing stroke angle, ϕ, follows its prescribed path independently of the wing geometry, wingbeat frequency, and hinge stiffness (figure 5a). Note that the motion is neither strictly sinusoidal nor perfectly symmetric on the up- and down-stroke.

Figure 5.

Kinematics of the wing motion during the flapping cycle for three different characteristic wings. (a) The stroke angle, ϕ, varies over ± 63°, tracing out a quasi-sinusoidal trajectory defined by the geometry of the four-bar linkage. (b) The wing pitch angle, ψ, changes passively in response to the wing motion as well as to aerodynamic and inertial forces. Key features of the pitching trajectory include the buildup of aerodynamic forces (I), the wing rotation (II) and the elastic recoil (III). The legend in (b) shows the values of the Cauchy number, Ch, and inertial-elastic Number, IE, for each of the traces. Aspect ratio is indicated by marker shape: circles, AR = 3.6; triangles, AR = 2.9. (Online version in colour.)

The wing pitch angle, ψ, follows a more complex trajectory, increasing as the wing accelerates and reaching a maximum (figure 5b-I) at approximately t/T = 0.4. At this time, the wing decelerates and changes direction, causing the wing to rotate. The pitch angle rapidly changes sign, eventually reaching another peak in magnitude, ψ_II (figure 5b-II). A ‘recoil’ is observed with the pitch angle magnitude decreasing and reaching a minimum (figure 5b-III) shortly after stroke reversal but before the stroke angle accelerates again. The process I–II–III repeats during the second half of the flapping cycle. This general pattern is observed for all the configurations tested. However, the features described here are less pronounced for lower values of the Cauchy number and the inertial-elastic number.

3.5. Scaling of wing kinematics

The trends in wing pitching behaviour are illustrated more clearly in figure 6. The amplitude of the pitch angle just prior to wing reversal, ψ_I (figure 5b-I), is shown for a range of wing geometries, spring stiffnesses, and flapping frequencies, and the data are observed to collapse well when plotted against the Cauchy number, Ch (figure 6). The experiment data, which span Reynolds numbers from Re = 2.6 × 10³ to Re = 9.9 × 10³, appear to be well-approximated by the quasi-steady model (equation (3.6)). Predictions are shown for AR = 2.9 and 3.6, and for Re = 1.4 × 10³ and 1.4 × 10⁴. The inset in figure 6, which uses log–log axes, replots the present data, but also includes data from the oil tank experiments of Ishihara et al. [13] for wings of AR = 5.4 conducted at approximately Re = 4.2 × 10². These data extend to larger values of the Cauchy number and the pitch angles lie between the model predictions for AR = 2.9 and Re = 1.4 × 10³ and for AR = ∞ and Re = 1.1 × 10².

Figure 6.

Maximum amplitude of the pitch angle at the end of each stroke before wing rotation, ψ_I, as a function of the Cauchy number, Ch. The predictions of the quasi-steady pitch angle, ψ₀, for wings of different aspect ratio, AR, and Reynolds number, Re, are superimposed on experimental data. The scaling of the experimental data reflects the importance of the dynamic pressure during this phase of the flapping cycle and supports the validity of the quasi-steady model. The inset re-plots the same data on a log–log scale and includes the results of Ishihara et al. [13] represented by filled black circles, and the prediction for AR = ∞ and Re = 1.1 × 10² represented by a solid black curve. For coloured markers, aspect ratio is indicated by marker shape: circles, AR = 3.6; triangles, AR = 2.9. (Online version in colour.)

The normalized duration of the wing rotation, Δt/T = (t_II − t_I)/T, is plotted against IE in figure 7a. Δt/T is observed to increase monotonically with IE. Although the magnitude of Δt/T from the experimental data is approximately matched by the results from the simplified model, collapsing well when plotted against the inertial-elastic parameter, the measured duration of wing rotation is higher and the rate of increase slightly smaller than predicted.

Figure 7.

Normalized properties of the wing rotation—duration and mean angular velocity—as a function of the inertial-elastic number, IE. (a) Normalized duration of the wing rotation, Δt/T, calculated as the time between peaks I and II in figure 5b divided by the cycle period. The solid black curve shows the prediction obtained by modelling the wing during rotation as a simple harmonic oscillator for Φ = 120° (figure 3). (b) Normalized mean angular velocity during wing rotation, 2ω*/π. These data are bounded by the model predictions for Re = 1.4 × 10³, β = 0.3, and AR = 3.6, shown as a dashed cyan curve; and for Re = 1.4 × 10³, β = 0.6, and AR = 2.9, shown as a dashed magenta curve. Aspect ratio is indicated by marker shape: circles, AR = 3.6; triangles, AR = 2.9. (Online version in colour.)

Finally, the mean angular velocity during the wing rotation normalized by the cycle period, calculated as (ψ_II − ψ_I)T/(t_II − t_I), is shown as a function of IE in figure 7b. As with the rotation duration, the mean normalized angular velocity generally scales well with IE, and is bounded by two curves representing the model predictions for Re = 1.4 × 10³ at the two extremes of the parameter range used in the experiment: β = 0.6 and AR = 2.9; and β = 0.3 and AR = 3.6. The model predictions of ω* were scaled by 2/π to obtain mean, and not maximum, normalized angular velocities for this comparison.

3.6. Scaling of the time-averaged lift forces

The scaling of the cycle-averaged coefficient of lift, C_L, with both the Cauchy number and the inertial-elastic number are shown in figure 8a,b, respectively. Here, we see that the lift coefficient increases from C_L = 0 to approximately C_L = 2.3 as either Ch or IE increase. At the upper limit of the Ch or IE number range covered by experiments, there is a decrease in the slope of the data indicating that the lift coefficient is approaching a maximum. The collapse of data in both cases is relatively good, although close inspection of the C_L data suggests that the data scale better with IE due to the elimination of scatter between data points of different values of β. The inset replots the current data, but also includes the data of Ishihara et al. [13,35] and the ‘RoboBee’ data of Jafferis et al. [8].

4. Discussion

The present work, unlike past experimental and numerical studies of flapping wings [33,34,39], seeks to identify patterns in lift generation where pitching kinematics are passively controlled by an elastic wing hinge. Instead of taking the approach often used in which the dynamic scale of insects is matched by tuning the parameters of the larger model to that of the biological system [13,31,35,47], we aim to characterize trends in kinematic behaviour and aerodynamic performance across a wider range of physical scales, material properties, and wing shapes so as to identify limits in the generalization of passive pitching mechanics.

4.1. Scaling of pitching kinematics

The three features of the passive pitching kinematics (figure 5b) can be understood in terms of the primary forces at work during each part of the wingbeat cycle. The first peak, ψ_I, occurs towards the end of the stroke when the translational velocity is high. Here, the wing is affected primarily by aerodynamic forces, generated by the wing translation, and the elastic restoring torque from the wing hinge. The observed scaling of ψ_I with Cauchy number Ch (figure 6) thus makes physical sense and agrees with other experiments [13]. The values of the maximum pitch angle before rotation, ψ_I, are also close to the quasi-steady pitch angle, ψ₀, predicted by the simple theory (equation (3.6)). However, ψ_I is consistently greater than ψ₀. This is likely explained by the fact that our definition of the Cauchy number uses an average stroke velocity, which is less than the maximum stroke velocity by a factor of 2/π, thus not providing an accurate estimate of the instantaneous aerodynamic loading when the velocity and pitch deflection angle is at a maximum. Nevertheless, the agreement between the model prediction and experiment is reasonable.

Comparison with real insect wing systems is difficult because, while wing pitch angles have been measured in flight [11,52], there are no reliable biomechanical data on insect wing hinge stiffness. Note that Ennos reported torsional compliance of the entire wing under torsion [2], which was later used by Ishihara et al. [13,35] while Bergou et al. [11] estimated the elastic properties of the wing hinge by fitting a model to their observations of insect flight. However, we note that the values of Ch for insect systems calculated using these approximate values of hinge stiffness (table 1) lie within the range expected for normal operation based on the quasi-steady aerodynamic model (figure 2). In particular, using morphological parameters from Ellington [28] and the hinge stiffness reported by Bergou et al. [11], we can estimate the Cauchy number for Drosophila melanogaster to be approximately Ch = 0.6; applying the quasi-steady analysis (figure 2a) using appropriate parameters (Re = 1.1 × 10², AR = 2.9) yields a pitch deflection angle ψ = 60°, which is similar to what is observed in flight [11].

The second feature of the stroke cycle, wing rotation, is primarily a dynamic event. We see that the normalized duration of the wing rotation, Δt/T, scales well with IE (figure 7a). As IE increases, the duration of the wing rotation occupies a larger fraction of the total wingbeat cycle. This trend appears to be the result of the increase in the relative magnitude of the flapping frequency, f, compared to the natural frequency of wing rotation, ω_n. The values of Δt/T measured for the present system are in agreement with observations of insect flight. Ellington [52] reports that each wing rotation in an insect wingbeat typically accounts for between 10% and 20% of the whole wingbeat duration, depending on the insect species. The values of Δt/T from the experiment are greater than those predicted by the simple harmonic oscillator model of wing rotation. The inclusion of damping effects may improve the accuracy, but it will not address a weakness in the model, specifically that Δt/T as a normalized property cannot increase indefinitely as the model implies.

The timing of the wing rotation also shifts as IE changes. The wing rotation starts at approximately the same time during the flapping cycle, t/T ≈ 0.4 (figure 5b-I), due to the prescribed stroke kinematics. As IE increases, the normalized rotation duration increases, delaying the relative timing of the wing rotation with the stroke reversal such that it transitions from being advanced to being symmetric with respect to the stroke reversal. The rate of increase of the amplitude of wing rotation is greater than that of the normalized wing rotation duration, so the normalized angular velocity of the wing increases with IE, which matches the results of the modelling analysis described earlier (figure 7b). Despite the limitations of the estimates of different features of wing rotation, ψ_I and Δt/T, the model provides a good estimate of the normalized angular frequency.

The third feature of the pitch angle history, the recoil (figure 5b-III), also reflects a balance between wing inertia and elastic restoring torque. At this stage in the cycle, the wing is not moving very fast and aerodynamic forces are relatively small. However, the inertia of the wing has caused it to over-rotate during stroke reversal, and here the elastic hinge pulls it back towards the neutral position. This overshoot is also observed in insect flight kinematics [11,52].

4.2. Scaling of the lift coefficient with Ch versus IE

Over the range of parameters tested here, the lift coefficient, C_L, seems to scale reasonably well with our definition of the quasi-steady aerodynamic scaling parameter, the Cauchy number, Ch. In particular, examining the inset in figure 8a reveals that the location of maximum lift coefficient follows the trends in maximum quasi-steady lift and normalized angular rotation with aspect ratio discussed in the previous section. The data from the model tested by Ishihara et al., which had an aspect ratio of AR = 5.4, reach a maximum at higher values of Ch than the wings in the present experiment, which have aspect ratios ranging from AR = 2.7 to AR = 3.6. The data from the present experiment also separate by aspect ratio, and the value of Ch at maximum C_L for AR = 2.7 and AR = 2.9 is lower than that for AR = 3.6. The location of the maximum of the data from the RoboBee, for which AR = 3.3, appears to coincide with that of the data from the present experiment for AR = 3.6.

An improvement in the collapse of measured lift coefficients is observed when scaling with the inertial-elastic number, IE (figure 8b), indicating that the balance of hinge stiffness and inertia (both fluid and structural) is the dominating factor in determining lift production. This can be understood by considering the two main contributions to lift in a flapping-wing system: (1) quasi-steady lift during the mid-stroke and (2) rotational lift during stroke reversals. The parameter β = IE/Ch is a measure of the relative importance of the rotational and quasi-steady contributions to total lift. When the lift contribution from wing rotation is negligible (β ≪ 1), the time-averaged lift of a flapping wing would be expected to collapse onto the quasi-steady model prediction. Examination of the main axes in figure 8a shows that lift coefficients measured in the present experiments, which were conducted at intermediate values of β, are significantly greater than the model prediction. The contribution from the wing rotation, then, is clearly critical to the generation of lift in this system. Furthermore, we can infer that the mean lift generation of the wing rotation phase of the wingbeat cycle is greater than that of the quasi-steady translation phase, a conclusion supported by observations of a rotational lift ‘peak’ in various flapping-wing systems [20,29,31]. Experimental time-averaged lift coefficients for flapping are consistently greater than those observed for steadily revolving wings [34,39,40]. We, therefore, expect systems with larger values of β to exhibit higher lift coefficients.

Data from prior experiments provide further support for this argument and help to distinguish the effects of β and AR. The data from the RoboBee [8] (figure 8 inset, squares), for which β is approximately 1, show a maximum lift coefficient of C_L = 3.7, which greatly exceeds the quasi-steady prediction. The difference is assumed to come from the rotational lift contribution. On the other hand, the model tested by Ishihara et al. [13,35] (figure 8 inset, circles and triangles) has a β of approximately 0.17 and shows a maximum lift coefficient close to C_L = 2. Again, the magnitude of β explains the smaller departure from the quasi-steady lift prediction. The data from the present experiment span β = 0.3 to β = 0.6, and, indeed, the maximum lift coefficients in the present results appear to lie between the bounds set by these two other experimental systems.

Recall, as discussed when the parameter β was defined, that a larger value of β implies that given value of Ch corresponds to a larger value of IE. With this in mind, we can understand how the separation of C_L according to β observed in figure 8a is removed when C_L is scaled by IE (figure 8b).

As IE rises, we observe changes in three features of wing rotation which affect lift generation: the normalized duration of the wing rotation increases (figure 7a), the normalized angular velocity increases (figure 7b), and the timing of the wing rotation is delayed such that it becomes more symmetric (figure 5b). The first two of these trends increase the lift contribution from the rotational dynamics and, consequently, the overall lift coefficient. However, the third trend has a detrimental effect on the overall lift coefficient because it creates a less favourable angle of attack for generating lift at the start the subsequent stroke [34,40,53].

Based on the experimental lift coefficient data (figure 8b inset), it appears that the effects of the increase in ω* and Δt/T on lift coefficient outweigh the detrimental effect of the change in wing rotation timing as IE increases. The data further suggest that the strongest factor in determining the lift coefficient magnitude is Δt/T. With a flapping amplitude of approximately Φ = 60° and IE = 0.3 at maximum C_L, the RoboBee would be expected to operate with a normalized rotation duration time close to the theoretical maximum, Δt/T = 0.5. On the other hand, the models used in the present experiment (figure 7a) and by Ishihara et al. [13] both have stroke amplitudes close to Φ = 120° and attain maximum C_L when IE = 0.1, operating at roughly Δt/T = 0.17. The higher maximum C_L of the RoboBee may be explained by the fact that a greater fraction of each wingbeat cycle is occupied by wing rotation—the phase of the cycle with greater lift-generating potential. The magnitude of ω* has an inverse square root dependence on β (figure 4), but the corresponding increase in IE, and therefore Δt/T, appears to have a greater effect on lift production.

This conclusion provides some insight to the relative merits of scaling of C_L with either Ch or IE (figure 8). Using Ch as the scaling factor aligns data points which share the same quasi-steady pitch angle, ψ₀. Therefore, for a wing of given aspect ratio, (i) deviations in lift coefficient from the quasi-steady model reveal the relative contributions of rotational lift and (ii) it is possible to tune system parameters to maximize C_L (figure 2a). On the other hand, scaling with IE aligns data points which share the same normalized rotation duration and improves the collapse of C_L values, making it possible to predict C_L for a given system and operating point.

It is also worth noting that the scaling of C_L with IE holds even though the Reynolds numbers of the present experiments and referenced studies span approximately two orders of magnitude (10²–10⁴). This weak dependence is perhaps not surprising; although including viscous effects in the sinusoidal approximations of lift and drag coefficients does lead to slight improvements in the accuracy of the quasi-steady model (figure 5), the overall trends are relatively insensitive to this parameter.

5. Conclusion

The simplicity of under-actuated torsion spring-mediated wing pitch rotation, widely observed in insect flight, has tremendous appeal for the design of small, engineered aerial vehicles. The use of elastic wing hinges to reversibly store the energy during flapping flight and passively maintain angles of attack favourable for generating lift is efficient, elegant, and easy to implement. Using this lift generation mechanism, the emulation of insect flight at insect scales continues to make advances [7,9].

Prior to this work, the physical principles with which one could understand the performance of passive-pitching flapping wings and develop bio-inspired wing designs were somewhat limited. However, understanding the interplay between inertial, aerodynamic, and elastic torques and recognizing the relative contributions of quasi-steady and rotational lift will enable the association of different flapping flight strategies with the morphological and dynamic properties of a flapping-wing system. The identification of physics-based scaling laws for wing design using passive pitching kinematics will not only open up a wide range of design options, but also facilitate the communication of results between research on passive-pitching flapping-wing systems conducted from biological and engineering perspectives.

This work shows that aerodynamic performance and kinematic behaviour for passive-pitching flapping wings can be reliably predicted using elastic scaling parameters for a wide range of specifications. The scaling presented here has been experimentally validated using a robotic flapping model, and measurements show reasonable agreement with similar flapping-wing systems found in nature.

While the Cauchy number, the inertial-elastic number, and their ratio, β, are evidently powerful tools in characterizing the performance of a passive-pitching flapping wing, there remain inconsistencies which imply the existence of dependencies which have not been explored in the present analysis. The prediction of the maximum normalized angular velocities (figure 4) might be improved by developing a more refined model of wing rotation. In addition, sinusoidal approximations do not capture all of the features of the relationship between the aerodynamic force coefficients and pitch angle, which also may change with wing geometry and surface roughness. More accurate estimates of the centre of pressure, added mass, and damping would also yield more precise predictions of kinematics and lift forces.

Of course, wing performance is still a complex dynamical process, and these results need to be considered with other factors which play a role in flapping-wing sysems. Compliance of the wing surface at various scales affects the dynamic conformation of the wing and the generation of lift [2,3,54–56], as do interactions between wing surfaces, such as in the ‘clap-and-fling’ behaviour observed in Drosophila melanogaster [24,37,52].

Further experiments which span a wider range of parameters will also prove useful in defining the limits of the scaling analysis presented here with respect to mass number, Reynolds number, wing geometry, and stroke amplitude. The results presented here could also be extended by the consideration of nonlinear hinges and of biased or non-sinusoidal stroke profiles, which may more accurately model actual insect wing systems. Moreover, it is difficult to discuss the importance of rotational lift in passive pitching flapping-wing systems without understanding how the mean lift generated during the wing rotation phase of the wingbeat is affected by changes in geometric, kinematic, and dynamic scales—a topic of continued investigation in flapping wing aeromechanics. Nevertheless, the work presented here demonstrates the utility of generalized scaling laws as tools for understanding of biological and engineered passive-pitching flapping-wing systems alike.

Data accessibility

Data and software code can be made available upon request.

Authors' contributions

All authors contributed to the design of the experimental approach. J.N. designed and fabricated initial versions of the experimental hardware. K.W. designed and fabricated the final version of the experimental apparatus, acquired and analysed the data. K.B. directed the research and contributed to experimental design and data analysis. K.W. and K.B. wrote the manuscript. All authors gave final approval for publication and agree to be held accountable for the work performed therein.

Competing interests

We declare we have no competing interest.

Funding

This work was supported by the National Science Foundation, NRI: Collaborative Research Program (award no. 1426338). K.W. was supported by the Brown University School of Engineering Carl Nielson ’56 Summer Research Award.