Optimal flapping wing shape and kinematics are different for different flight velocities: an analysis on local relative wind

Flights of Cinsects, birds and bats have been studied by researchers for many decades. Methods like in vivo study [1–7], experimenting with physical models [8–12], computational fluid dynamic (CFD) analyses [13–15] and analyses with theoretical formulations [14, 16, 17] were employed for qualitative and quantitative understanding. In vivo studies help in understanding the force production mechanism of natural flyers at various flight phases and conditions. However, the effects of variation of wing kinematics cannot be studied thoroughly; since natural flyers always tend to actuate their wings either in performance effective or energy efficient ways, whichever is required at that moment [18]. Experimenting with physical models and CFD would help in this aspect. However, without a proper theoretical model to support, these methods would not be systematic but rather iterative and more costly in terms of time and money. Quasi-steady blade element approach is a well-established and useful theoretical method for flapping wing analyses. Moreover, it is useful in analyzing flows and forces in the wing airfoil section level; which would help in designing the wing and its kinematics for different flight requirements. Hence, we present the formulations in this paper based on this approach.

It is important to manage optimally the aerodynamic force-distribution on the wing of any flying object, in the aerodynamic and structural points of view. In a flapping wing section, the resultant aerodynamic force is the summation of translational force, rotational force, force due to added mass effect, force due to clap and fling effect, and force due to wake capture at stroke reversals [19, 20]; of which translational force is the major contributor. All of these components depend on density of air, reference area, force coefficients and the relative wind velocity. In flapping wings, the possibility of varying density is limited; since mostly they operate at low altitudes. However, the possibility of varying wing planform, force coefficients and relative wind is vast. While the translational aerodynamic force has a linear relation with other parameters, it depends on square of the relative wind velocity. Hence, relative wind plays the major role in producing the aerodynamic forces. Moreover, relative wind influences the wing planform and force coefficients also; since wing planform is decided according to the required force distribution and force coefficients depend on angle of attack, Reynolds number and Mach number, which in turn depend on relative wind.

In fixed-wing flight, managing the force distribution along the span can be done with a little effort in geometric design of the wing; since every section of the wing sees mostly the same relative wind. However, in flapping wings, the relative wind seen by every wing section differs, because of the induced relative wind component due to wing flapping. Hence, in addition to the geometric design of the wing surface, wing kinematics (with degrees of freedom including the movement of the whole wing and also its shape modifications by flexing, twisting etc) should also be optimized. In a flapping wing, there are possibilities for the relative wind to vary in magnitude as well as in direction from wing root to the tip, based on the wing kinematics. Distribution of relative wind magnitude can be managed by the wing actuations at the wing pivot (applicable for non-articulated wings only). However, the variation in relative wind direction along the span may require a wing twist, for enabling the required angle of attack at every wing section. In hovering flight, relative wind on a wing section is strictly a function of the wing kinematic parameters only. However, in other phases, especially in forward flight, the relative wind depends also on the freestream due to flight velocity, in addition to wing kinematic parameters.

Considerable works which studied the effect of wing kinematics on flapping wings have been done previously. Some works were on observing and understanding the wing kinematics of natural flyers [1–7]. Others were aimed at producing optimal simplified functions of kinematic parameters [19, 21–23]. While many works were on hovering flight, there are some studies which focused on wing kinematics effects in forward flight [24]. Some works even modelled instantaneous local velocities along flapping wings [25–27]; and these models were mostly based on specific kinematics of their working models. A mathematic model which is capable of predicting the effects of variations of wing kinematics and flight parameters, would be helpful, especially for experimental studies. In previous experimental works on forward flight [9, 10, 28], the effect of wing kinematics and freestream were studied and related to advance ratio. However, these works were restricted to horizontal stroke plane flapping.

In this paper we produce necessary formulations for taking these experiments to the next level, for the cases of inclined stroke plane flapping. Section 2 contains the mathematical model required to know the instantaneous local relative wind (magnitude and direction) at every wing section, at any position of wing, flapped with any speed, in any stroke plane orientation, flying at any flight speed. It will be helpful in designing the wing planform for required load distribution and also for setting the wing twist for the required angle of attack distribution. Moreover, a discussion on the effect of relative wind distribution on force distribution and local angle of attack is presented; through which the necessity for variation of wing kinematics for varying flight parameters is qualitatively stated. This can be used for further deeper quantitative analyses for different flight purposes like hovering, low-speed flight, high-maneuvering flight, high-speed cruising flight, etc.

The model presented here is compatible with quasi-steady blade-element approach for flapping wing analyses. All wing motion are defined in wind axis coordinate system XYZ (shown in figure 1(a)) with origin fixed to the wing pivot, such that the freestream due to flight is always in the negative X direction. Hence, this model can be used for all flight phases including hovering, takeoff, climbing, level flight, descend and landing; except for gliding. The wing motion is restricted to the stroke plane and out-of-plane deviations are considered zero; to study the effect of stroke plane orientation substantially with ease. Wing flexure in its normal direction is considered zero; hence the wing axis is always a straight line. The axis of the wing is placed passing through the quarter chord locations of all sections. The effect of wing rotation about its axis on the relative wind is neglected; hence the relative wind of a wing section defined here is what seen by the wing axis at that section.

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Using proper Euler angle rotations, the position of any wing section at a distance r from the wing pivot can be derived in XYZ axes as

$$\begin{eqnarray}&&{\bf{P}}=-\left(r\,\sin \,\phi \,\cos \,\gamma \right)\widehat{{\bf{i}}}+\left(r\,\cos \,\phi \right)\widehat{{\bf{j}}}+\left(r\,\sin \,\phi \,\sin \,\gamma \right)\hat{{\bf{k}}}\end{eqnarray} \tag{ 1 }$$

where, $\widehat{{\bf{i}}},$ $\widehat{{\bf{j}}}$ and $\widehat{{\bf{k}}}$ represent unit vectors in X, Y and Z directions respectively.

The unit vectors along the wing section local coordinate axes X_s, Y_s and Z_s (see figure 1(a))—which also represent the chordwise, lateral and normal directions of an un-twisted wing section respectively—at any instant can be derived as

$$\begin{eqnarray}&&\begin{array}{l}{\widehat{{\bf{X}}}}_{s}=\left(\sin \,\gamma \,\cos \,\theta +\,\cos \,\gamma \,\cos \,\phi \,\sin \,\theta \right)\widehat{{\bf{i}}}+\left(\sin \,\phi \,\sin \,\theta \right)\widehat{{\bf{j}}}\\ \,\,\,+\,\left(\cos \,\gamma \,\cos \,\theta -\,\sin \,\gamma \,\cos \,\phi \,\sin \,\theta \right)\widehat{{\bf{k}}}\\ {\widehat{{\bf{Y}}}}_{s}=-\left(\cos \,\gamma \,\sin \,\phi \right)\widehat{{\bf{i}}}+\left(\cos \,\phi \right)\widehat{{\bf{j}}}+\left(\sin \,\gamma \,\sin \,\phi \right)\widehat{{\bf{k}}}\\ {\widehat{{\bf{Z}}}}_{s}=\left(\sin \,\gamma \,\sin \,\theta -\,\cos \,\gamma \,\cos \,\phi \,\cos \,\theta \right)\widehat{{\bf{i}}}-\left(\sin \,\phi \,\cos \,\theta \right)\hat{{\bf{j}}}\\ \,\,\,+\,\left(\cos \,\gamma \,\sin \,\theta +\,\sin \,\gamma \,\cos \,\phi \,\cos \,\theta \right)\widehat{{\bf{k}}}\end{array}\end{eqnarray} \tag{ 2 }$$

The relative wind seen by a wing section is the summation of the contributions by the freestream and that due to wing motion; and is represented as ${{\bf{V}}}_{r}={{\bf{V}}}_{\infty }-d{\bf{P}}/dt;$ where ${{\bf{V}}}_{\infty }=-{V}_{\infty }\widehat{{\bf{i}}}.$ This relative wind is not always in the plane of wing airfoil section and generally have an additional component which is in the lateral direction, which negligibly contribute to force production by that section. This component of V_r along Y_s is usually neglected and the component in the plane of wing section (X_s Z_s plane) is generally considered for quasi-steady blade-element analyses [10]; and is found by ${{\bf{V}}}_{s}={{\bf{V}}}_{r}-({{\bf{V}}}_{r}\cdot {\hat{{\bf{Y}}}}_{s}){\hat{{\bf{Y}}}}_{s}$ as

$$\begin{eqnarray}&&\begin{array}{l}{{\bf{V}}}_{s}=\left({V}_{\infty }({\cos }^{2}\gamma {\sin }^{2}\phi -1)+r\dot{\phi }\,\cos \,\gamma \,\cos \,\phi \right)\widehat{{\bf{i}}}\\ \,\,-\,\left({V}_{\infty }\,\cos \,\gamma \,\sin \,\phi \,\cos \,\phi -r\dot{\phi }\,\sin \,\phi \right)\widehat{{\bf{j}}}\\ \,\,-\,\left({V}_{\infty }\,\cos \,\gamma \,\sin \,\gamma {\sin }^{2}\phi +r\dot{\phi }\,\sin \,\gamma \,\cos \,\phi \right)\widehat{{\bf{k}}}\end{array}\end{eqnarray} \tag{ 3 }$$

Using this equation, it can be proved that V_s is not always in the stroke plane, except for the cases of hovering (J = 0) and horizontal stroke plane flapping (γ = 0) which were mostly taken for investigation [9, 10, 19, 23] (see figures 2 and 3).

For the purpose to be used in aerodynamic force calculations, the magnitude of V_s can be obtained as

$$\begin{eqnarray}&&{V}_{s}=\sqrt{{V}_{\infty }^{2}(1-{\cos }^{2}\gamma {\sin }^{2}\phi )-2{V}_{\infty }r\dot{\phi }\,\cos \,\gamma \,\cos \,\phi +{(r\dot{\phi })}^{2}}\end{eqnarray} \tag{ 4 }$$

For generalization this can be further non-dimensionalized by dividing with reference wing tip velocity as ${\hat{V}}_{s}={V}_{s}/\left|R{\dot{\phi }}_{{\rm{ref}}}\right|$ and obtained as

$$\begin{eqnarray}&&{\hat{V}}_{s}=\sqrt{{J}^{2}(1-{\cos }^{2}\gamma {\sin }^{2}\phi )-2J\hat{r}\hat{\dot{\phi }}\,\cos \,\gamma \,\cos \,\phi +{(\hat{r}\hat{\dot{\phi }})}^{2}}\end{eqnarray} \tag{ 5 }$$

where, J is advance ratio given by $J={V}_{\infty }/\left|R{\dot{\phi }}_{{\rm{ref}}}\right|,$ $\hat{r}$ is the normalized radial location of wing section given by $\hat{r}=r/R,$ $\hat{\dot{\phi }}$ is the non-dimensional wing angular velocity given by $\hat{\dot{\phi }}=\dot{\phi }/\left|{\dot{\phi }}_{{\rm{ref}}}\right|.$ ${\dot{\phi }}_{{\rm{ref}}}$ can be taken to be the maximum angular velocity ${\dot{\phi }}_{\max }$ in a flapping cycle (useful for effectiveness studies) or the average angular velocity $\bar{\dot{\phi }}$(useful when cycle-average parameters are analyzed). Since ${V}_{s}$ is ${\hat{V}}_{s}$ times $\left|R{\dot{\phi }}_{{\rm{ref}}}\right|,$ ${\hat{V}}_{s}$ quantifies how effectively the wing movement is utilized for relative wind generation for force production.

Equation (3) can be used to find the direction of the relative wind in any wing section. However, for simplifying the process we introduce an angular parameter λ_s defined as the angular displacement of V_s from the chordwise direction (${\hat{{\bf{X}}}}_{{s}_{0}}=\sin \,\gamma \hat{{\bf{i}}}+\cos \,\gamma \hat{{\bf{k}}}$) of an unrotated, untwisted wing section. Accordingly, λ_s can be obtained by

$$\begin{eqnarray}&&{\lambda }_{s}=\mathrm{sgn}\left(({{\bf{V}}}_{s}\times {\hat{{\bf{X}}}}_{{s}_{0}})\cdot {\hat{{\bf{Y}}}}_{s}\right){\cos }^{-1}\left(-\displaystyle \frac{{{\bf{V}}}_{s}\cdot {\hat{{\bf{X}}}}_{{s}_{0}}}{{V}_{s}}\right),\end{eqnarray} \tag{ 6 }$$

which can be further reduced to

$$\begin{eqnarray}&&{\lambda }_{s}=\mathrm{sgn}(J\,\cos \,\gamma \,\cos \,\phi -\hat{r}\hat{\dot{\phi }}){\cos }^{-1}\left(\displaystyle \frac{J\,\sin \,\gamma }{{\hat{V}}_{s}}\right)\end{eqnarray} \tag{ 7 }$$

The signum function in the above equation gives sign to λ_s, which in turn represents whether V_s is displaced counter-clockwise (+ve) or clockwise (−ve) from ${\hat{{\bf{X}}}}_{{s}_{0}}$ when viewed from the wing tip. The value of λ_s at the wing root (i.e., at $\hat{r}=0$)—hereafter designated as ${\lambda }_{{s}_{0}}$—decides the rotation of the wing which in turn decides the angle of attack at the wing root. The variation of λ_s (i.e., ${\lambda }_{s}-{\lambda }_{{s}_{0}}$) along the span decides the twist required to achieve the required angle of attack distribution along the span of the wing. Accordingly, for obtaining a required angle of attack ${\alpha }_{{s}_{0}}$ at the root, the wing rotation to be made is

$$\begin{eqnarray}&&\theta ={\lambda }_{{s}_{0}}-{\alpha }_{{s}_{0}}\end{eqnarray} \tag{ 8 }$$

Then, for obtaining an angle of attack ${\alpha }_{s}$ at any wing section, the wing twist required at that section is given by

$$\begin{eqnarray}&&{\tau }_{s}=({\lambda }_{s}-{\lambda }_{{s}_{0}})-({\alpha }_{s}-{\alpha }_{{s}_{0}})\end{eqnarray} \tag{ 9 }$$

The wing twist ${\tau }_{s}$ denotes the rotation of a wing section about the wing lateral axis after θ is applied at the wing root (see figure 1(b)). ${\tau }_{s}$ follows the same sign convention as θ.

3.1. General characteristics of relative wind distribution in hovering and forward flight

Hovering is the simplest case in which the relative wind on the wing depends only on the wing motion; since the flight velocity is zero. As shown in figure 2, the relative wind for all wing sections from root to tip, lie in the stroke plane; hence there is no variation in direction along the span. The magnitude of the relative wind increases linearly from zero at the root to a maximum equal to the wing tip velocity at the wing tip. This distribution exists common for both downstroke and upstroke of flapping irrespective of the stroke plane orientation ¹ . Since the distribution is planar and always at one side of the wing, adjusting the wing rotation at the wing pivot would be just sufficient for enabling the required angle of attack at the wing sections. Hence a rigid flapping wing would be sufficient for the case of hovering.

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The complication starts when the freestream velocity comes into picture as the flyer starts moving. In forward flapping flight, it is unrealistic to have zero stroke plane orientation. However, it is a simplified case for understanding the effect of advance ratio; hence it was chosen for experimentation in previous works related to forward flight [9, 10]. In those works, rigid flat wings were used and the angle of attack was directly related to the wing rotation as α = (π/2–θ), for simplicity. If referred to the velocity distribution (figure 3(a)), this works fine for downstroke of flapping. However, this assumption is not applicable for all the wing sections in upstroke of the wing. Although the relative wind distribution is linear and planar, it is not at the same side of the wing as in the case of downstroke. At wing sections nearer to the wing pivot, there is a possibility for the magnitude of the wing motion contribution to be lesser than that of freestream. Hence, although the wing is moved backward, the relative wind will not oppose the wing and will be in the same direction of wing motion. Because of this kind of relative wind distribution, a rigid flat wing with a given rotation angle will not enable a constant angle of attack throughout the span. Since there is a change in direction of the relative wind, there is a necessity for changing orientation of the wing section along the span; in other words, a wing twist is required. The relative wind distribution in the forward flight with inclined stroke plane (the realistic flapping forward flight case) is not even planar (figure 3). It has non-linear variation in magnitude and continuous change in direction in the 3D space. It is notable that these variations are lesser in downstroke compared to that of upstroke.

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3.2. Scope of the analysis

A membranous wing—as used in many flapping wing flying models—will passively (may be partially) adapt its shape, generating the twist to accommodate the changing relative wind directions along the span. However, for deeper knowledge on this relative wind velocity distribution—which would aid in effective design of wing and its kinematics—we analyzed it for variations in magnitude and direction with respect to the variations in flight speed and flapping speed (in terms of the advance ratio J), stroke plane orientation γ, and the wing's flapping position ϕ. For studying the effect of magnitude of relative wind velocity, ${{\hat{V}}_{s}}^{2}$ is plotted across the wing span for various wing flapping positions at different combinations of advance ratio and stroke plane orientation (figures 4 and 6); since the translational aerodynamic force depends on square of the relative wind velocity. Moreover, ${{\hat{V}}_{s}}^{2}$ non-dimensionally represent the local dynamic pressure available at a wing section for force production; since flapping flight mostly happens in incompressible flow regime where density change is negligible. This is also useful in determining the effectiveness in utilizing the freestream and flapping action towards force production; as ${{\hat{V}}_{s}}^{2}$ represents the magnitude of the local dynamic pressure $\tfrac{1}{2}\rho {{V}_{s}}^{2}$ compared to the maximum contribution possible by the wing's flapping motion $\tfrac{1}{2}\rho {(R\dot{\phi })}^{2}.$ Further, for studying the effect of direction of relative wind of a wing section, λ_s is plotted across the wing span for various wing flapping positions at different combinations of advance ratio and stroke plane orientation (figures 5 and 7).

According to the definition of the advance ratio ($J={V}_{\infty }/\left|R{\dot{\phi }}_{{\rm{ref}}}\right|$), an increase in it can be visualized as an increase in flight velocity and/or a decrease in flapping angular velocity and/or a decrease in span of wing from root to tip. However, generally, it is more often used in the context of flight velocity. Generally, flapping flyers fly with advance ratios ranging from 0 to 1.3 [9, 10, 29, 30]. Hence, we restricted our analysis in the advance ratio range 0 to 1.5. As the flight advances in velocity, the flapping changes from a horizontal stroke plane to a vertical stroke plane [1, 3, 30–32]; hence, a stroke plane orientations in the range 0 to 90° were taken for analysis. A full flapping range (ϕ = 90° to ϕ = −90°) with a constant flapping velocity ($\left|\hat{\dot{\phi }}\right|=1$) was considered; since it would simplify the analysis like a revolving wing experiment [9, 12, 33], moreover, would represent a maximum effective flapping case [23]. Accordingly, hereafter, $\dot{\phi }$ will be used in place of ${\dot{\phi }}_{{\rm{ref}}}.$

The distribution of magnitude of the relative wind will determine the aerodynamic force distribution and also the structural stresses along the wing span. A cantilever wing design would prefer the loads to reduce as moving towards the free end. This can be partially achieved by the wing planform design; however, proper management of relative wind distribution is also required.

Whereas, the change in orientation of the relative wind along the wing span influences the angle of attack of every wing section; which in turn will affect the force distribution. Hence, there is a necessity for dynamic variation of wing twist throughout a flapping cycle, for enabling the required angle of attack distribution along the span. For example, if a flapping wing is expected to have constant angle of attack α_s throughout its span, a flat plate wing with a given rotation angle, would not work except for the case of a hovering flight (J = 0) and a hypothetical forward flight with zero stroke plane orientation flapping (only in downstroke). Instead, the wing should first be rotated by an angle equal to ${\lambda }_{{s}_{0}}-{\alpha }_{s}$ and then twisted such that the twist angle ${\tau }_{s}$ of any section with respect to the root section should be equal to ${\lambda }_{s}-{\lambda }_{{s}_{0}}.$ Other angle of attack distributions can be obtained by proper manipulation of the wing twist using equation (9) as required. However, the wing structural constraints may not permit the required dynamic wing twist. Hence, there is a requirement for proper design of wing kinematics, such that the dynamic variations of distributions of relative wind magnitude and direction are minimized in a flapping cycle; which would reduce the complexities imposed on the flapping mechanism and enable an effective flapping flight with minimum vibration and fatigue. With these basic understandings, figures 4–7 can be consulted for deciding the design of wing and its flapping kinematics for specific flight requirements.

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In hovering flight, the advance ratio J is zero. For J = 0, ${{\hat{V}}_{s}}^{2}$ distribution does not change with respect to wing position or stroke plane (see figures 4 and 6). Moreover, λ_s is constant throughout the span. Hence, hovering does not require a dynamic wing twisting. However, ${{\hat{V}}_{s}}^{2}$ has lower values compared to that of non-zero advance ratios. Hence, a high value of $\left|R\dot{\phi }\right|$ is required for obtaining the necessary dynamic pressure for producing the necessary lift. Hovering flyers are meant for flying through small confined spaces with exceptional maneuverability; which would require a low wing span R. Hence, the only possible way to get the necessary dynamic pressure for sustained flight is by increasing $\dot{\phi }$ with a high frequency flapping. Considering the upstroke, it too has the same ${{\hat{V}}_{s}}^{2}$ distribution as the downstroke (see figures 4 and 6). Hence, just a reversing of the wing pitch at the wing pivot would make the upstroke work very similar to the downstroke, in hovering.

Another notable property of hovering is: negligibly small ${{\hat{V}}_{s}}^{2}$ near the wing root, which reaches zero at the root section. Hence, whatever may be the speed of flapping, the wing region near the root section will receive a very small dynamic pressure and will contribute negligibly towards the aerodynamic forces of the wing. This means that, there will not be a considerable loss in terms of aerodynamic force production, even if this portion of the wing does not exist. Hence for hovering dominant flyers, the portion of the wing near the wing pivot can be trimmed off to a minimum chord thickness which is required for structural integrity. Doing so would remove that useless mass and also its inertial moment which would cost on the flapping effort. This is one of the reasons why hovering insects have minimum chord length near the wing pivot and the wing width increases as moving along the span towards the tip [34].

Figures 4–7 show all the possible combinations of advance ratio J and stroke plane orientation γ. However, in practical conditions, forward flapping flight with all the combinations is not possible. For a given configuration of flyer, the drag variation with flight velocity is fixed; hence the thrust to lift ratios of the flyer, to be maintained for different flight velocities, are also fixed. Hence, the liberty to change the stroke plane orientation corresponding to a flight velocity is very limited. However, the possibility for changing advance ratio ($J={V}_{\infty }/\left|R\dot{\phi }\right|$) for a given flight velocity is not as stringent as that for the stroke plane orientation; since advance ratio has two parameters in the denominator viz wing span R and flapping angular velocity $\dot{\phi },$ to work with. Unlike hovering, in forward flight, the downstroke of flapping contributes mostly towards the force production [20, 35, 36]. Hence, for any specific flight purpose, firstly the wing and its kinematic function need to be optimized for downstroke.

5.1. Downstroke

5.2. Upstroke

The above analyses and formulations were made focused on understanding the local relative wind and hence to understand and optimize the translational component of aerodynamic force. However, these can also be used for understanding the effects of local relative wind on other quasi-steady minor force components due to wing rotational effect and added mass effect; and on force components due to unsteady effects due to wake capture and clap-and-fling mechanism.

6.1. Rotational and added mass effects

6.2. Wing-wake interaction effects

The key conclusions arrived through the study on local relative wind of a flapping wing are as follows: Low speed high-maneuvering flyers should have low aspect ratio wings (for ensuring structural integrity) with broader surface near the tip (for producing more force near the tip which in turn create more moment required for quick maneuvers) operated at high frequency (to obtain necessary relative wind for force production; since the availability of freestream is insufficient at low speeds) for effective maneuverability. High speed long range cruising flyers should have high aspect ratio wings with narrow wing tips (for minimizing the induced drag), highly cambered section near the wing root (for effective utilization of lower relative wind magnitude near root and ensuring decreasing lift distribution towards the tip), operated at low frequency (since freestream supplies sufficient relative wind for force production) and short amplitude (for operating wing in effective wind utilization region and avoiding wastage of resultant force contributing towards side-force). Quick accelerating flight needs more thrust and hence needs more forward twist and hence needs to be operated in low advance ratio (i.e., high frequency flapping). Upstroke of flapping is generally more complicated than downstroke and requires more understanding of local relative wind for avoiding negative lift. In most cases it is not advisable to use upstroke for force production, for minimizing the compulsory drag contribution and simplifying the flapping kinematics for comfortable flapping. Perpendicular flapping (flapping with stroke plane perpendicular to flight direction) is found to be the easiest, simple and effective way of flapping in all cases. Hence, it is advisable to accelerate quickly to the speed at which perpendicular flapping will produce the necessary thrust to lift ratio. Otherwise, the flight path can be made inclined (that can be a climbing path which will require more tilt of stroke plane towards perpendicular flapping or a gliding path which actually does not require flapping). If the perpendicular flapping is accompanied with a high advance ratio (high speed flight with low frequency flapping), the wing structural deformations (in terms of twist) required can also be minimized and hence the structural fatigue can be reduced.

All these inferences based on this study on local relative wind on flapping wing prove that natural flyers have their wing and its motion unique and custom made, optimized for their own flight purposes. Moreover, the formulations will be helpful for systematically experimenting flapping wings for optimum planform and wing kinematics for various applications. Further combining the effects of various angle of attack distributions and the resulting force distributions, would give a deeper understanding on flapping wings, leading to development of flapping wing vehicles with exceptional performance and efficiency.

All data that support the findings of this study are included within the article (and any supplementary files).