## ABSTRACT

The steady slow climbing flight of a dragonfly, *Sympetrum frequens*, was filmed and analysed. By using the observed data, the mechanical characteristics of the beating wings were carefully analysed by a simple method based on the momentum theory and the blade element theory, and with a numerical method modified from the local circulation method (LCM), which has been developed for analysing the aerodynamic characteristics of rotary wings.

The results of calculations based on the observed data show that the dragonfly performs low speed flight with ordinary airfoil characteristics, instead of adopting an abnormally large lift coefficient. The observed phase advance of the hindwing, Δ δ_{1} ≃ 80° can be fully explained by the present theoretical calculation. Similarly, the spanwise variation of the airloading and the time variations of the horizontal force, vertical force, pitching moment and torque or power can be definitely estimated within a reasonable range of accuracy in comparison with the flight data. The distribution of loading between the fore and hind pairs of wings is also clarified by the calculations.

## INTRODUCTION

Dragonflies (Order Odonata) are excellent flyers and are characterized by a large head -much of which is occupied by huge eyes and a relatively large mouth –a robust thorax, two pairs of almost identically shaped wings, and a large, long, slender abdomen. Dragonflies can hover, fly at high speed and manoeuvre skilfully in the air in order to defend their territory, feed on live prey and mate in tandem formation.

The biology of dragonflies has been closely studied but few attempts have been made to analyse their flight mechanics. Norberg (1975) filmed the free hovering flight of *Aeschna juncea* in the field and found that the body is held almost horizontal and that both the fore and hind pairs of wings are beaten at a frequency of 36 Hz in two stroke planes which are almost parallel and tilted 60° relative to the horizontal.

Weis-Fogh (1972), in his analysis of hovering flight, assumed (1) a constant induced velocity distribution over the stroke plane (based on the momentum theory), (2) a simple harmonic motion of the beating wings, and (3) a constant pitch along the span of the wing. Then, following the techniques used in the analysis of helicopter rotor aerodynamics (Gessow & Myers, 1952), he carried out a steady aerodynamic calculation on the beating wings by integrating the elemental forces acting on a blade element over the wings with the aid of strip analysis and by averaging these forces over one stroke cycle. The results required abnormally large lift coefficients, of 3-5 to 6-1.

In order to get a more realistic model of the flow field around the beating wings, Rayner (1979*a*,*b*,*c)* introduced the vortex theory for a pair of wings, in which a chain of coaxial, small-cored, circular vortex rings, simulating the trailing tip-vortices of wings, was brought into the calculation of the induced velocity, as had been developed for rotary wing aerodynamics (Heyson & Katzoff, 1957). However, unlike a helicopter rotor in hovering flight, two pairs of wings arranged in tandem have time varying airloadings and thus, as shown in Fig. 1, generate two corrugated wake sheets filled respectively with a sheet of shed vortices as well as trailing vortices. Thus, the induced velocity, which, unlike that of the hovering rotor, is not always vertical, must be estimated from both the trailing and shed vortices. The wake of the vortices is of considerable importance in and near hovering flight because the downwash generated by a wing, which is retained and developed from the stroke plane, strongly affects the angle of attack of the wing in the subsequent stroke and of another wing operating nearby.

In the case of a helicopter rotor, such effects of the downwash on the airloadings of the succeeding wing have been accurately analysed by simply introducing an attenuation coefficient instead of performing the complex integration over the vortex sheet (Azuma & Kawachi, 1979). This method of calculation, called the ‘Local Momentum Theory’ (LMT), has been extended to the study of the unsteady aerodynamics in the rotary wing of a helicopter (Azuma & Saito, 1982) and also, under the name of the ‘Local Circulation Method’ (LCM), to analyses of the highly skewed flow of propellers (Azuma, Nasu & Hayashi, 1981) and the windmill (Azuma, Hayashi & Ito, 1982).

Since the fundamental aerodynamic characteristics of the beating wing are much like those of the rotary wing, even though wing beating is a reciprocating motion and not a rotational motion like that of a rotary wing, the LCM can similarly be applied to the aerodynamic analysis of the beating wing with a slight modification of the method.

## SYMBOLS

## MODES OF WING MOTION

A dragonfly, *Sympetrum frequens*, flying under a spotlight in a darkened room was filmed with a high speed 16 mm movie camera (HIMAC) at 873 frames per second. The dimensions of the dragonfly are shown in Fig. 2 and Table 1.

In order to get a clear image of the feathering motion of the wings, a 2·5 mm width of each wing tip was painted with ‘Pentel White’ 03-667-3331EX237 (Fig. 3). The mass of the paint was about 0·03 mg, which was about 2% of the mass of the wing (1·5 mg) and was somewhat less heavy than the pterostigma. The additional moment of inertia caused by the paint about the flapping hinge was 3·0 × 10^{−4}gcm^{2}, which was about 5% of that of the wing (5·6 × 10^{−3} gem^{2}). The additional moment of inertia of the paint about the feathering axis was 2·2 × 10^{−6}gcm^{2}, which was about 3·5% of that of the wing (6·3 × 10^{−5}gcm^{2}). The local backward shift of the centre of mass at the painted portion of the wing was about 4·6% of the chord.

The mass of paint therefore has a small effect on the beating motion of the wing. However, an experimental check of the beating frequency by means of stroboscope and video camera did not reveal any difference between a painted dragonfly and a naked or unpainted one, although the beating frequency was estimated to be reduced by about 2·5%. In comparison with Norberg’s result (1972) on the inertial effects of the pterostigma on the wing motion it can be considered that the mode of beating was not significantly altered by the painting and the analysed data gave useful information about the flight mechanics of the dragonfly.

Analysis of a series of frames of almost steady flight of the dragonfly in very slow climbing flight revealed the following (Fig. 3).

The body axis, which is a straight line connecting the tip of the head and the tip of the tail, is tipped about 10°head-up with respect to the horizontal.

The orbit of the wing tips of the hind pair is almost completely in a single plane but the profile of the orbit of the fore pair is a very thin ellipse, the major axis of which may be considered to be the stroke plane. Both stroke planes are tilted about 50° with respect to the body axis, or γ= 40° with respect to the horizontal as shown in Fig. 4.

The beating frequency of all wings is 41’5 Hz.

The flapping angles or the azimuth angles of the observed data in the respective stroke planes are shown in Fig. 5. The Fourier expansion series of the observed data,

*=2Jπ*× 41·5rads

^{−1}≃ 15000°s

^{−1}and the phase difference between fore and hind pairs of wings to be Δ δ = δ

_{1,h}— δ

_{1,f}= 77° by which the fore pair followed the hind pair.

(v) The flight velocity, V, is 0·54ms^{−1}, in a direction nearly normal to the stroke plane.

(vii) The maximum values of the Reynolds number and the reduced frequency at three-quarter radius point are *Re* = 3·2 × 10^{3} and k = ωc/2U = 0·12 respectively.

## SIMPLE ANALYSIS

*ψ*in a stroke plane which is tilted by y with respect to the X-axis directed backward in the horizontal plane. Thus, in order to make an orthogonal Cartesian coordinate system (X, Y, Z), the Y-axis is directed to the right in the horizontal plane and the Z-axis is directed vertically upward. A local coordinate system (x, y, z) is defined as follows. The x-axis or feathering axis is directed outward along the aerodynamic centre or quarter chord line of the wing and is assumed to be common for all wing sections. The y-axis is directed in the direction of the inflow at the wing section in consideration. The z-axis is directed to make an orthogonal Cartesian coordinate system together with the other two axes. Then the transformation matrix between the (X, Y, Z) axes and the (x, y, z) axes can be given by:

In the calculation of the aerodynamic forces, the following assumptions are introduced:

*x*= r/R where R is half of the wing span.

_{e}is the area swept by one pair of beating wings and C

_{hf}is an interference coefficient specifying the effect of the induced velocity of the fore pair on the hind pair, whereas the effect of the hind pair on the fore pair has been neglected because it is so small.

(iv) By neglecting the outflow along the span, the aerodynamic forces acting on a blade element are given by those of two-dimensional airfoils and can be integrated along the span without any interference among the elements.

(v) The aerodynamic characteristics of the wing section or airfoil at low Reynolds number (Fig. 8) were modelled by referring to those given by Vogel (1967) and Jensen (1956), and modified to have the maximum lift coefficient of C_{L,max}=1 · 8 by considering the effects of the unsteady separated flow and of the dynamic stall on the maximum lift, as presented, for example, by Izumi & Kuwahara (1983), Conner, Willey & Twomey (1965) and Ericsson & Reding (1971).

_{T}and Up, are given by:

*a*, and the inflow angle

*ϕ*are respectively given by: Since the lift and drag of the wing element are directed to the z- and y-axes respectively, the total forces of one pair of wings along the X- and Z-axes and torque, Q, about the joint (positive for flapping up) can respectively be given by: INSimilarly, the thrust, T, pitching moment (head-up positive), M, and the required aerodynamic power, P, are respectively given by: where the aerodynamic power for feathering motion, being very small, is neglected.

_{hf}≃ 0 · 3 for = W = 2 · 60 × 10

^{− 4}kgf = 2 · 54 × 10

^{− 3}N, and was very much smaller than that expected. In the momentum theory for an isolated single wing the induced velocity must be twice that at the stroke plane in the fully developed wake (Gessow & Myers, 1952). This is probably because the simple analysis, based on the momentum theory and blade element theory is inadequate.

## LOCAL CIRCULATION METHOD (LCM)

Since this method is fully described in a paper presented by Azuma *et al*. (1981), only a brief explanation of the method of calculation and the modification made in this method for the beating wing will be presented here.

As shown in Fig. 10, a beating wing can be decomposed into n imaginary wings arranged one-sidedly in diminishing size of span, each of which has an elliptical circulation distribution and thus may be called an elliptical wing, and operates in a twisted flow. Fig. 11 illustrates the flow profile at an arbitrary section as a control point of the flow. Since longitudinal vortex filaments trailing from an elliptical wing do not lie on a flat plate, the induced velocities at that section caused by the respective vortex filaments consequently do not point in the same direction either.

*l*, the circulation, Γ, and the lift coefficient as a function of angle of attack C

_{λ}(α) can be related as follows: where S is a spanwise unit vector. The angle of attack

*α*is, here, considered to be given by: where U

_{T}and U

_{p}are respectively tangential and perpendicular components of the total inflow velocity

*U*, with respect to a stroke plane, at a control point under consideration.

*U*can be given by the summation of the components induced by the respective wings, and are related to the circulation ΔΓ

_{i}as follows: Then, the tangential and normal components of the induced velocity and of the total inflow velocity with respect to the stroke plane can be given by: where v

_{n 0}is the normal component of the induced velocity with respect to the stroke plane, generated and left by the preceding wings at the control point, whereas v

_{n}is that generated by the present wing at the present time.

If the induced velocity left on the control point, V_{n},_{0}, is specified as a known value or related to other known variables, then by combining equations (20) to (27), the spanwise distribution of the airloading, circulation and the perpendicular and tangential components of the induced velocity can be solved numerically. If the induced velocity outside the elliptical wings is neglected, then the solution can be obtained successively (Azuma & Kawachi, 1979).

_{n}o- In the local circulation method for a rotary wing (Azuma

*et al*. 1981) the induced velocity left at the control point at time step t

_{j-1}is considered to be given by multiplying the induced velocity generated at the control point at a time t

_{j-1}, v

^{j-1}, by an attenuation coefficient C

^{−1}as C

^{j-1}v

^{j-1}. In the case of two pairs of wings, the induced velocities left at the control points (P

_{f}and P

_{h}) for the fore and hind pairs of wings, can respectively be given by: where and v are respectively the induced velocities of the fore and hind pairs of wings at the control points at the time t

_{j-i}, and where and C

_{fh}, C

_{hf}are direct or time-related attenuation coefficients and cross-or space-related interference coefficients respectively.

_{fh}and C

_{hf}may be given by the ratio of the induced velocity generated by the respective system of trailing vortex as follows:

where ( )_{f} and ( )_{h} show the quantity ( ) specified by the trailing vortex of the fore and hind wings, respectively. Since the effect of shed vortices on the downwash at the most active part of the respective wing, i.e. at spanwise position of r= (3/4)R and azimuth angle of *ψ = 0* °, can be neglected in comparison with that of the trailing vortex, only the trailing vortex has been considerated for the determination of the interference coefficients and the attenuation coefficients. Usually, C_{fh}, can further be neglected as being a small quantity (Azuma & Saito, 1979).

_{f}and C

_{h}may be given by the ratio of the induced velocities generated by the trailing vortices of both pairs of wings at time t

_{j}and t

_{j}+ Δt as follows:

where S is the area of the respective stroke plane.

_{n,0}at time t

_{j}for both the fore and hind pairs of wings successively.

## RESULTS CALCULATED BY THE LCM

By applying the LCM for a steady, slow-climbing flight of a dragonfly and by using the observed data, the following results have been obtained.

The time variations of the angle of attack *α*, and the interference coefficient Chf at the three-quarter radius point of the wing for the fore and hind pairs are shown in Fig. 15. It is remarkable that the angles of attack remain in the linear range of the lift coefficient during the effective phase of the respective strokes and the interference coefficient varies appreciably. The C_{hf} is less effective near the switching of the beating strokes of the hindwings, from up to down and down to up, where the induced velocity generated by the forewing is predominant. This results from adequate selection of the phase lead of the hindwing δ _{1} = 77 °. The mean value of the interference coefficient is about C_{hf} =1 · 2, which is more reasonable and much larger than that given in the Simple Analysis section (p. 88).

Fig. 16 shows the time variations of the horizontal force, vertical force, pitching moment and power. The negative horizontal force is generated mainly by the forewing in the latter half of its downstroke, whereas the positive horizontal force is generated by the hind wing in the last half of its upstroke. The mean horizontal force is a very small positive value because the dragonfly is climbing only slowly.

The vertical force is mostly generated in the downstroke of the respective wings. The total mean vertical force is equal to the weight of the dragonfly. The higher harmonics of pitch changes are important to keep this balance.

The moment about the centre of gravity, which was not always accurately observed but only roughly measured, varies appreciably during the wing cycle, from positive in the downstroke of the forewing to negative in the upstroke of the forewing. The mean value is, however, almost zero for a steady flight.

The power is always positive and its variation is very similar to that of the vertical force. That is to say, the power is mainly consumed to sustain the weight of the body in this example. The mean value of this required power is equivalent to a ‘specific power’ of 160 W kg^{− 1}, which is the required power per unit mass of muscle. This value falls in a reasonable range (70–260 Wkg^{−1}) as estimated by Weis-Fogh (1975, 1977).

The effects of the phase difference Δ δ_{1} of the beating motion between the fore and hind pairs of wings on the mean values of horizontal force, vertical force, pitching moment and power are shown in Fig. 17. The horizontal force and pitching moment are strongly dependent on the phase difference or phase lead of the hindwing δ_{1}, whereas the vertical force and power are almost invariant or slightly dependent on the phase difference. There are two phase differences for zero pitching moment, δ_{1} = 80 ° and 150 °. As shown by dotted lines in Fig. 17, by adjusting the CG position from −10% (forward) to +10% (backward) of the mean wing chord the above trimmed phase difference varies as 60 ° <αδ _{1}<95 ° and 140 ° < α δ_{1} < 190 °. The former phase difference gives a small mean horizontal force for low-speed flight whereas the latter one generates a large mean horizontal force probably for high-speed flight. The observed value of Δ δ_{1} ≃ 80 ° clearly corresponds to the above low-speed flight. Contrary to expectation, the power was not small at the former trimmed point.

In the above calculations, the unsteady flow effects on the aerodynamic forces and moments acting on the beating wings have not been introduced. In the linear unsteady wing theory the effects of periodically shed vortices are introduced by simply multiplying either the Theodorsen function for the wing motion or the Sear’s function for the change of oncoming flow by the lift slope as precisely explained in textbooks (e.g. Bisplinghoff, Ashley & Halfman, 1955). In the present calculation, however, the effects of shed vortices have been introduced as the spanwise variation of the induced velocity generated and left by the preceding blades by the term of or h as given by equation (31). Since the chordwise gradient of this term is nearly equivalent to the shed vortices (Azuma *et al*. 1982), other unsteady effects have been omitted.

## CONTRIBUTION OF INERTIAL FORCE

^{− 11}kgs

^{− 2}.

Shown in Fig. 18 are the time variations in the inertial torque Q_{i} and the aerodynamic torque Q_{a} of a forewing beating in the same steady flight. In the latter half of both the down- and upstrokes the inertial torque assists to compensate the aerodynamic torque. It is, however, known that the inertial torque can be cancelled or reduced by adopting either an elastic property into the wing itself or a resilient material such as an apodeme at the flapping hinge of the wing (Weis-Fogh, 1972; Alexander, 1975).

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