## ABSTRACT

A mean lift coefficient quasi-steady analysis has been applied to the free flight of the dragonfly *Sympetrum sanguineum* and the damselfly *Calopteryx splendens*. The analysis accommodated the yaw and accelerations involved in free flight. For any given velocity or resultant aerodynamic force (thrust), the damselfly mean lift coefficient was higher than that for the dragonfly because of its clap and fling. For both species, the maximum mean lift coefficient was higher than the steady *C*_{L,max}. Both species aligned their strokes planes to be nearly normal to the thrust, a strategy that reduces the required for flight and which is different from the previously published hovering and slow dragonfly flights with stroke planes steeply inclined to the horizontal. Owing to the relatively low costs of accelerating the wing, the aerodynamic power required for flight represents the mechanical power output from the muscles. The maximum muscle mass-specific power was estimated at 156 and 166 W kg^{−1} for *S. sanguineum* and *C. splendens*, respectively. Measurements of heat production immediately after flight resulted in mechanical efficiency estimates of 13 % and 9 % for *S. sanguineum* and *C. splendens* muscles, respectively.

## Introduction

Insects are able to undergo a whole variety of flight manoeuvres, and they may use particular styles for different activities. Migration and foraging over substantial distances for food may entail flight at constant velocities; however, many styles of flight are of an unsteady nature. Accelerations are involved when pursuing prey and for the avoidance of predators, and erratic flight paths are used both for display and for escape. Previous studies of insect aerodynamics have used quasi-steady methods to analyse steady flight (Osborne, 1951; Jensen, 1956; Weis-Fogh, 1973; Ellington, 1984*a*–*d*; Dudley and Ellington, 1990; Cooper, 1993; Willmott, 1995). These studies have considered hovering flight, or flight that is constrained by wind-tunnels. More recently, Dudley and DeVries (1990) have analysed the free migratory flight of the moth *Urania fulgens*, but again the flights were assumed to be at a constant velocity. The limited range of flight styles considered may not be representative of insect flight in general. The present paper extends quasi-steady methods to consider the accelerated flight of the dragonfly *Sympetrum sanguineum* and the damselfly *Calopteryx splendens*.

Aerodynamic studies of dragonfly flight are confounded by the problem of two functional pairs of wings, with flow interactions between the fore- and hindwing pairs. These interactions result in dramatic changes in force production with only minor alterations in wing kinematics (Saharon and Luttges, 1988; Luttges, 1989). To simplify this problem, the effect of all four wings can be modelled as a single actuator disc. The mass flux of air in the far wake must be the reaction to the aerodynamic force acting on the dragonfly. The induced velocity of this air can thus be calculated from the swept area of the wings and aerodynamic force. This approach has previously been taken by Osborne (1951) and Norberg (1975). Previous insect studies have considered the major force to be vertical weight support and have thus made the simplifying assumptions that the induced velocity must also be vertical. For the flights considered here (Wakeling and Ellington, 1997*b*), the aerodynamic force has components to overcome weight support, acceleration and parasite drag. This aerodynamic force, thrust, is thus not necessarily vertical. An imaginary coordinate system is introduced to bring the ‘thrust’ vertical in accordance with the weight support from previous studies; however, the velocity is then inclined to the horizontal. A general form of the Rankine–Froude estimate of induced velocity (Stepniewski and Keys, 1984) is thus used to calculated induced velocity and power.

Profile power is calculated from the relative velocity of the wings. Wing motion is reconstructed both within the stroke plane and from its elevation from that plane. The nature of these accelerated flights results in significant asymmetries in the relative velocities between the left and right wings owing to yaw. Each wing was therefore analysed separately. Mean lift coefficients and profile powers are calculated assuming an appropriate value for the profile drag, which was taken from the lift and drag measurements in Wakeling and Ellington (1997*a*).

The net inertial power for an oscillating wing is zero becauses the energy required to accelerate the wing at the beginning of each half-stroke will be recovered during deceleration at the end of the half-stroke. The insect can usefully recover some of the inertial power from wing deceleration either by storing it in elastic structures in the thorax or by using it to overcome aerodynamic power costs. The inertial cost of accelerating the wings is usually greater than the aerodynamic power requirements (Ellington, 1984*d*). However, a special case for inertial power has recently been highlighted for cases where it is less than the aerodynamic power (Dickinson and Lighton, 1995). If there is no elastic storage, then inertial power recovered during wing deceleration can be totally used to help pay the aerodynamic power cost for that part of the stroke. The net mechanical power during the stroke thus becomes the aerodynamic power. If there is perfect elastic storage of inertial power, then the net mechanical power is similarly the aerodynamic power. Thus, if the inertial power is less than the aerodynamic power, the net mechanical power required from the flight muscles is simply the aerodynamic power regardless of the extent of elastic storage. This case has been shown for some butterflies (Dudley and DeVries, 1990) and some *Drosophila* species (Dickinson and Lighton, 1995). The relatively low wingbeat frequencies for the dragonflies and damselflies in the present study result in them also satisfying this special case where the mechanical power requirements from the muscles can be estimated independently of any assumption as to the degree of elastic energy return.

The power lost by the flight muscles as heat can be calculated from the thermal properties of dragonflies during flight. Corbet (1983) recognised that some dragonflies may thermoregulate, and distinguished between two behavioural types: ‘fliers’ and ‘perchers’. Fliers typically spend most of their time on the wing and they can thermoregulate by shunting blood and thus heat between their thorax and abdomen. Perchers only make short flights to pursue prey, mates and rivals, and there is little evidence for physiologically facilitated heat transfer between the thorax and abdomen; instead, they thermoregulate by controlling their perching posture and flight duration (Heinrich and Casey, 1978; Heinrich, 1993). The dragonfly and damselfly in the present study, *S. sanguineum* and *C. splendens*, are both perchers. May (1976) has shown that for small dragonflies in general there is relatively little effect of blood shunting on the conductance, and so the difference in conductance between live and dead dragonflies is negligible.

During flight, the relative air flow increases the rate of convective cooling (Church, 1960*b*; Casey, 1976; May, 1976), and so the conductance values used for the calculation of heat production should be measured while the dragonfly is cooling in an air stream of an appropriate velocity. In the present study, the conductances are found by cooling experiments in the jet of a wind-tunnel. These are used, in conjunction with measurements of thoracic temperatures immediately after flight, to estimate the power lost as heat from the thoracic muscles. Muscle mechanical efficiencies are calculated for these dragonfly flights from the aerodynamic estimates of the mechanical power requirements, coupled to the measurements of heat production during flight.

## Materials and methods

Lift and drag data for both the bodies and wings of the dragonfly *Sympetrum sanguineum* (Müller) and the damselfly *Calopteryx splendens* (Harris) were taken from Wakeling and Ellington (1997*a*)., and kinematic data for their flights were taken from Wakeling and Ellington (1997*b*). The velocity and acceleration are assumed to be constant for the duration of each wingbeat analysed. A detailed description of the morphological parameters for these and a further five dragonfly species is available elsewhere (Wakeling, 1997) but the parameters relevent to this paper are given in Table 1 and follow the notation in Ellington (1984*a*). Data were processed in Mathematica and MathCad routines on a Macintosh Quadra 650 computer. All calculations assumed values of ρ=1.165 kg m^{−3} for the air density and 𝒱=1.608×10^{−5} m^{2} s^{−1} for the kinematic viscosity, which are appropriate to the 30 °C flight conditions.

Power terms are calculated as power *P* (mW) and muscle mass-specific power *P** (W kg^{−1}), which is power per unit thoracic muscle mass. The term ‘thrust’, *T*, refers to the total aerodynamic force (as in Wakeling and Ellington, 1997*b*), and this is not necessarily horizontal or vertical.

### Parasite power

Parasite power *P*_{par} is expended to overcome the parasite drag on the dragonfly body. The force required to overcome parasite drag *D*_{par} comes from the thrust generated by the wings, and thus parasite power is implicitly included in the induced power when this is derived from the total thrust.

*D*

_{par}is typically much smaller than the weight, and it can be estimated independently. Parasite power is then simply given by: The overall thrust is nearly vertical, as required by the dominating role of weight support, so the induced velocity

*V*is also approximately vertical. The induced power

*P*

_{ind}is conventionally calculated as if it is indeed vertical, and thus

*P*

_{ind}represents the kinetic energy imparted to the air solely for weight support. The kinetic energy imparted by the horizontal force component is ignored, because it is treated independently by the

*D*

_{par}calculation, but it must be realised that the induced velocity has a horizontal as well as a vertical component. Where

*D*

_{par}is large, for instance at fast flight speeds or where there are large horizontal accelerations, the induced velocity may differ significantly in magnitude and direction from the traditional estimate based on weight support alone. This will lead to errors, albeit probably small, in estimates of both the angles of incidence and the profile drag of the wings. Because of the accelerating flights in this study, the induced velocity was calculated from the total thrust;

*P*

_{par}is therefore implicitly included in

*P*

_{ind}.

### Induced power and induced velocity

*m*is body mass,

**is the acceleration due to gravity and**

*g***is acceleration. Preliminary calculations showed the lift generated by the dragonfly body,**

*a**L*

_{par}, to be less that 1 % of the total thrust, and so it was ignored in the present study. This thrust implicitly includes the acceleration and drag components of the body, and so the induced power estimates also include the power required to accelerate the body and overcome parasite drag.

To bring this thrust more in line with the weight support that is conventionally used for induced velocity calculations, the dragonfly coordinate system was transformed to a thrust-based *X*,*Y*,*Z* coordinate system where the thrust is vertical along the *Z* axis, and the *X* axis is parallel to the sagittal plane (Fig. 1). This thrust-based system is similar to the conventional gravitational coordinate system except that the ‘vertical’ force equals the thrust instead of the weight. The non-dimensional thrust expresses total aerodynamic force as a proportion of the weight:

*x*′,

*y*′,

*z*′ coordinates (Wakeling and Ellington, 1997

*b*), take values of less than 12 ° in thrust-based coordinates for all except one of the flights in the present study; these angles are more typical of results for insects flying at constant horizontal velocities in a wind-tunnel.

The actuator disc is, by definition, ‘horizontal’ in the thrust-based coordinates. It represents the area over which the wings interact with the air to give it a ‘downwards’ impulse and thus can be considered as the projection of the swept area of the wings onto the *X*,*Y* plane. This actuator disc is nearly parallel to the stroke plane, which is approximately normal to the thrust (see Wakeling and Ellington, 1997*b*), so the actuator disc is taken to have an area *A*_{0}=Φ*R*^{2}, where Φ is stroke amplitude and *R* is wing length. This estimation of actuator disc area is never more than 6 % greater than the equivalent areas Φ*R*^{2}cosβ, where β is the stroke plane angle, used by Ellington (1984*b*), Ennos (1989) and Dudley and Ellington (1990).

*w*for the general case of a rotor with velocity

*V*which is inclined at an angle − α′to the actuator disc is: (Stepniewski and Keys, 1984, equation 2.32). It proves convenient to normalise this equation using the Rankine–Froude estimate of induced velocity

*w*

_{0}required for hovering with

*T*=

*m*

**: Equation 4 can then be written in terms of a non-dimensional velocity**

*g**V*′ (=

*V*/

*w*

_{0}) and induced velocity

*ŵ*(=

*w*/

*w*

_{0}) to give: The induced velocity factor

*k*

_{ind}has been added to this equation to correct for the tip losses and non-uniform flow across the actuator disc. Pennycuick (1975) suggests that a value of

*k*

_{ind}=1.2 is appropriate for animal flight, and this has been confirmed for hovering flight with horizontal stroke planes using vortex theories (Ellington, 1984

*c*).

*k*

_{ind}should be lower for two pairs of wings having reduced wake periodicity and thus tip losses, but higher for slight inclinations of the stroke planes (Ellington 1984

*c*); the value of

*k*

_{ind}=1.2 is probably reasonable for dragonflies and has been used in the present study in the absence of a better estimate.

*Sympetrum frequens*(Azuma

*et al*. 1985).

### The quasi-steady model: force coefficients and profile power

Quasi-steady aerodynamic forces depend on the relative velocity of the wings, and this must first be calculated before the force coefficients and profile power are estimated.

This analysis is based on a quasi-steady model developed by C. P. Ellington (unpublished) for steady, level flight without yaw. Ellington assumed a sinusoidal wing motion confined to the stroke plane. Dragonfly wingbeats are neither sinusoidal nor are they restricted to movement in a single plane (see Wakeling and Ellington, 1997*b*). Ellington’s analysis has therefore been modified to include the precise motion of the wing both in the stroke plane and its elevation away from that plane. Yaw has also been added to this model, leading to the profile power costs being asymmetrical between the left and right wings. Profile power is thus calculated for each individual wing for the dragonflies.

The stroke angle ϕ of the wing within the stroke plane and the angle of elevation θ above the stroke plane are calculated using equation 12 and data from Tables 2 and 3 in Wakeling and Ellington (1997*b*), using the first four harmonics of each Fourier series. Non-dimensional forms of the angular velocities are given by:

*ĵ*, where:

*ĵ*is a form of the advance ratio which also includes the induced velocity. Positive values are taken by

*V*

_{X}and

*V*

_{Z}for forward and upward flight. To account for yaw, the sign of

*V*

_{Y}is opposite for calculations of the relative velocity for the left and right wings. For flight with no yaw,

*V*

_{Y}is zero. The calculations in the present study are unusual in that the effects of

*V*

_{Y}are considered. Yaw has been ignored in all previous aerodynamic studies on insect flight, but the dragonflies in thrust-based coordinates show yaws of up to 56 °. Ignoring

*V*

_{Y}here would overestimate by up to 6 % and underestimate

*P*

_{pro}by up to 17 %.

*V*

_{Y}thus makes significant contributions to the force balance and power terms, and so is included in these calculations.

_{D}, μ

_{D}and ν

_{D}, respectively, for the

*X, Y*and

*Z*directions. These direction numbers are equal to the direction cosines multiplied by : Similar direction numbers, normal to , are resolved for ‘lift’ and take the subscript L: As an intermediate step in calculating the mean aerodynamic forces, non-dimensional integrals sum the velocity components throughout the stroke for all chordwise elements of the wing. Taking a non-dimensional wingbeat period , and the downstroke and upstroke periods as and , respectively, the integrals to describe the drag components in the

*Z*direction are given by: and where

*ĉ*is the non-dimensional chord as calculated from and (the first and second radii of wing area) using a Beta function (Ellington, 1984

*a*) and the subscripts d and u denote downstroke and upstroke, respectively. Similar integrals can be constructed for the drag components along the

*X*and

*Y*directions, and also for all three lift components. The integrals are additionally given subscripts D or L depending on whether they are used for drag of lift calculations, respectively. Values for the ‘lift’ integrals are positive if the relative velocity hits the ventral surface of the wing and negative for the dorsal surface, i.e. positive for the downstroke and negative for the upstroke when the wing flips over. The factor ζ is then used to convert these integrals to force coefficients relative to the non-dimensional thrust : where

*S*is the wing area. Wing areas given in Table 1 are for wing pairs, however, for yawed flight where the forces on the left and right wings are calculated separately and so the appropriate value of

*S*for a single wing must be used.

*Z*direction,

*A*

_{Z}and

*B*

_{Z}, respectively, are given by: and The forces can then be solved for the quasi-steady analysis; the force in the

*Z*direction (both lift and drag) must match the thrust, which by definition is also in the

*Z*direction. This quasi-steady solution is: Where the profile drag coefficient

*C*

_{D,pro}can be estimated, this equation can be solved for a mean lift coefficient at which the wings must be operating throughout their stroke to generate the required total thrust. Dragonflies have two pairs of wings, however, and each wing pair supports only a fraction of the total thrust. If a single value of is assumed for both wing pairs, the forewings generate a fraction of the total thrust: where subscripts f and h denote the fore- and hindwings, respectively. As with

*C*

_{L}, a common value of

*C*

_{D,pro}is assumed for both wing pairs. The minimum at which both wings must operate throughout their stroke to support

*T*is then given by: Profile power is similarly calculated from non-dimensional integrals that describe the cube of throughout the wingstroke. These integrals for the downstroke and the upstroke are: and The power factor σ gives dimensions to these integrals, and is: The profile power costs for each wing pair are then given by: As can be seen from equations 25 and 28, the estimates of both and

*P*

_{pro}are dependent on the choice of

*C*

_{D,pro}. The force measurements from Wakeling and Ellington (1997

*a*) showed that the minimum values for

*C*

_{D,pro}are, on average, 0.13 for

*S. sanguineum*wings and 0.10 for

*C. splendens*wings. These values are for a zero angle of incidence and comprise virtually all skin friction. However, when the wings operate at a non-zero angle of incidence, the pressure drag component of

*C*

_{D,pro}increases. Data from Azuma

*et al*. (1985) show that the mean angle of incidence for

*Sympetrum frequens*during slow climbing flight was 26 °, and data from Rüppell (1989) suggest a mean value of 22 ° for

*C. splendens*during a variety of flights. The profile drag estimates must thus include the pressure drag component for non-zero angles of incidence.

*C*

_{L}, the drag coefficient follows the parabolic law: (von Mises, 1959) where

*b*is a constant and

*C*

_{L}

^{2}/

*b*is the coefficient of induced drag. This induced drag coefficient was calculated for each of the four types of wing from Wakeling and Ellington (1997

*a*) in the linear range of

*C*

_{L}for 0 °<α<20 ° and was subtracted from

*C*

_{D}to obtain

*C*

_{D,pro}(equation 29).

*C*

_{D,pro}was then estimated from linear regression of

*C*

_{D,pro}with α. Assuming mean angles of incidence of 26 ° and 22 °, the mean values are

*C*

_{D,pro}≈0.20 for

*S. sanguineum*and

*C*

_{D,pro}≈0.17 for

*C. splendens*.

*d*) suggests that for the large angles of incidence used for hovering,

*C*

_{D,pro}tends to vary inversely with the square root of Reynolds number

*Re*and is: However, these dragonflies were not hovering, and their angles of attack are smaller than those typical of hovering insects (Ellington, 1984

*b*). The relationship of equation 30 will therefore be used as an upper estimate for

*C*

_{D,pro}for these dragonflies.

*C*

_{D,pro}≈0.20 for

*S. sanguineum*and

*C*

_{D,pro}≈0.23 for

*C. splendens*.

Estimates of *C*_{D,pro} are remarkably similar for the two methods, and a mean profile drag coefficient of 0.20 was therefore chosen for all the flights in the present study. It should be noted that this *C*_{D,pro} is derived from steady-state measurements (Wakeling and Ellington, 1997*a*); however, it is used to estimate what are ultimately unsteady values for . At the Reynolds number and angle of attack used by the odonates, the profile drag is dominated by pressure drag. Unsteady mechanisms that increase the amount of circulation generated by the wings should increase drag as well as lift since the total pressure force on a thin wing must be roughly normal to the surface of that wing. The steady-state estimates for *C*_{D,pro} probably represent a lower limit of those that occur during flight and will thus result in conservative estimates for and *P*_{pro}. When better estimates for *C*_{D,pro} become available, then and *P*pro can be easily recalculated using equations 25 and 28.

*Profile drag and wake inefficiencies*

Owing to the geometry of the wing motion, there is a small ‘downwards’ component to the profile drag *B*_{Z}*C*_{D,pro} (equation 23) during both the morphological up- and downstrokes (Fig. 2). The ‘downwards’ nature of this force is indicated by *B*_{Z} taking a negative sign. This force is caused by air being dragged ‘upwards’ with the wing owing to the ‘upward’ component of , and so it corresponds to a small upwash imparted to the wake as a result of profile drag.

The power cost of creating the upwash by *B*_{Z}*C*_{D,pro} is implicitly included in the *P*_{pro} estimate. However, a downwash must be generated to cancel this upwash, in addition to the net downwash to provide thrust support. Extra energy must thus be imparted into the momentum jet which does not appear as net momentum in the far wake; this is an inefficiency in the momentum jet due to the ‘vertical’ component of profile drag opposing the thrust of the jet.

*D*

_{pro}; this augmented thrust, , is given by: is, on average, 12 % greater than . A revised estimate for induced power,

*P*

_{ind}′, including the power required for these inefficiencies in the wake, can thus be calculated using equation 6 with to calculate a corresponding value for the induced velocity

*w*′. is then calculated from equation 9 with and the new induced velocity

*w*′.

In helicopter flight, the vertical component of profile drag on the rotors is negligible compared with the induced drag, and thus the profile drag contribution to the downwash is also negligible and is ignored. Animal flight studies have drawn their methods from the aeronautics literature and so have also ignored this component. Profile power forms a larger fraction of the mechanical power requirements of animal flight (Ellington, 1991). Dragonflies are unusual animal fliers in that they have two functional pairs of wings; dragonfly profile power is typically 75 % of the induced power, and so the contribution of *B*_{Z}*C*_{D,pro} to the momentum jet may be even greater than for other animals.

*P*

_{pro}and

*P*

_{ind}are presented for comparison with previous studies. The aerodynamic power

*P*

_{aero}is the sum of these two powers, but additionally includes this correction to include the wake inefficiency: For cases where there is a net ‘upward’ component to the profile drag, this drag contributes usefully to the momentum jet. The induced thrust required from the momentum jet is thus reduced by

*B*

_{Z}

*C*

_{D,pro}, with a corresponding reduction in the induced power requirements. The take-off flight of the large cabbage white butterfly

*Pieris brassicae*shows such a net upward component to its profile drag (Ellington, 1980). Its stroke plane is approximately vertical, and the wings are moved with the chord perpendicular to the motion during the downstroke; during the upstroke, the wings are strongly supinated, producing an angle of attack near zero. Vertical stroke plane flights have been recorded in other butterflies (Betts and Wootton, 1986; Sunada

*et al*. 1993). Similar kinematics are also seen in G. Rüppell’s films of damselfly flight; during strong backwards accelerations, the damselflies can use a synchronous ‘downstroke’, using large angles of attack, to squeeze air forwards and generate profile drag in a direction useful for their flight.

### Inertial power

The inertial power *P*_{acc} required to accelerate the mass of a wing pair and the added mass of air that moves with it is given by the angular velocity of the wing multiplied by the inertial torque *I*_{w}, where *I*_{w} is the moment of inertia of the wing pair and the added mass (Ellington, 1984*d*). The mean *P*_{acc} is equal to the kinetic energy gained by the wing pair, , divided by the period 1/4*n* of acceleration, and is equal to .

This approach has been used by Ellington (1984*d*), Dudley and Ellington (1990) and Cooper (1993), with the assumption that the wings oscillate with a simple harmonic motion. Dragonflies do not fit this assumption (Wakeling and Ellington, 1997*b*), which would lead to errors in the estimate for . Simple harmonic motion would underestimate for flight CS3.3 where the wings were held virtually motionless for a period at their most dorsal position and then moved relatively quickly during the downstroke; it would also overestimate for cases such as CS2.3, where the angular velocity is reasonably constant throughout the upstroke and the downstroke. For wingbeats where the downstroke-to-upstroke ratio is significantly different from 1, such as CS1.3, the maximum angular velocity will be different for each half-stroke and so must be calculated separately.

*P*

_{acc}is thus given by: where the subscripts u and d denote values for the upstroke and downstroke, respectively.

*I*

_{w}is equal to: where

*ĥ*is the non-dimensional mean wing thickness (Dudley and Ellington, 1990), using the wing parameters and from Table 1 and the wing density ρ

_{w}taken as 1200 kg m

^{−3}, the density of solid cuticle (Wainright

*et al*. 1976).

The contribution of inertial power to the overall costs of flight will be discussed below in connection with the efficiencies of flight.

### Heat production

Owing to inefficiencies in the flight musculature, most of the power expended by the muscles is lost as heat. This power loss can be estimated from measurements of the thermal conductance of the thorax and the thoracic temperature elevation above ambient.

Flight temperatures were measured for *S. sanguineum* and *C. splendens* in the flight enclosure described in Wakeling and Ellington (1997*a*). The dragonflies were encouraged to make ten consecutive flights around the enclosure; each flight consisted of flying one or more lengths of the enclosure and was stimulated by gently squeezing the abdomens of perching dragonflies. Immediately after a dragonfly had finished its ten flights, a thermocouple was inserted laterally into its thorax between the metathoracic and mesothoracic segments. These type K thermocouples were constructed from 0.125 mm wires, joined by silver solder and bonded to the end of a sharp wooden rod with epoxy resin; temperatures were read from an RS 611-234 temperature meter. Weis-Fogh (1964) has shown that the thoracic temperature of the locust, a larger insect than these dragonflies and thus having a greater thermal inertia, rises to a near-maximal value within 2 min of the initiation of flight; the temperatures measured here will similarly be close to maximum. Ambient temperature was measured using a second thermocouple in an open shaded box in the enclosure. Individuals underwent this flight and temperature recording procedure no more than four times, and they were then killed with ethyl acetate vapour, and weighed.

Two sets of conductance measurements were made for each of four individuals each of *S. sanguineum* and *C. splendens*. These insects were killed with ethyl acetate vapour, and their bodies were aligned with the flow of a wind-tunnel. The bodies were mounted on a type K thermocouple inserted laterally between the mesothoracic and metathoracic segments. A second thermocouple, below the tunnel, measured the ambient temperature *T*_{a}. The wind-tunnel was that described in Wakeling and Ellington (1997*a*), and the thermocouples were the same as described above. The dragonfly was heated by a lamp to a thoracic temperature *T*_{b} greater than 50 °C and then allowed to cool while temperatures were recorded for a period of 5 min. When an object is warmed from the outside and then cooled, the surface will initially be warmer than the core (May, 1976). For this reason, specimens were permitted to cool by 5 °C to reverse the temperature gradient before using the data to calculate conductance. Experiments were repeated at air speeds of 0, 1 and 2 m s^{−1}, measured using an Airflow AV-2 rotating-vane anemometer. The room temperature was approximately 28 °C and in sunlight, so the radiant heat losses approximated those during the filmed flights (Wakeling and Ellington, 1997*b*).

*C*is given by the product of the cooling constant, the specific heat

*H*of the dragonfly and its mass

*m*: where the value of the specific heat of dragonfly tissue is taken as 3.37 J kg

^{−1}°C

^{−1}(May, 1979).

### Muscle efficiency

## Results

The flight sequences occurred at varying velocities and thrusts, and hence the calculated results depend simultaneously on both these variables. It must be appreciated that trends against either velocity or thrust cannot be totally isolated.

### Quasi-steady lift coefficients

The quasi-steady analysis predicts that the *S. sanguineum* flights occurred at , and the *C. splendens* flights at (Table 2; Fig. 3). In fact, all except one of the *S. sanguineum* flights occurred at and, in general, the dragonfly flew with lower than the damselfly for any given velocity or thrust.

The *S. sanguineum* values of are within the range of *C*_{L} values which the wings can generate in a steady flow (see Wakeling and Ellington, 1997*a*), and so flights over the range 0.70<*V*<1.66 could be explained by quasi-steady aerodynamics. Dragonfly lift coefficients may reach 1.8 with first-order unsteady mechanisms, i.e. unsteady separated flow and dynamic stall (Azuma *et al*. 1985), and so the extraordinarily high for SSan5.1 may be explained by these mechanisms. This flight was unusual with a small value of α′ (−24 °, Table 2), and so there was a relatively smaller *Z* component of *V* than for the other dragonfly flights. The vertical factors *A*_{Z} and *B*_{Z} were thus small, and subsequently a larger value of *C*_{L} was required to solve the force balance of equation 25. This flight illustrates how the wings must operate at higher to generate the required thrust when the angle between the velocity direction and the stroke planes is small.

The values for *C. splendens* are generally greater than those for *S. sanguineum*. The main kinematic differences between the wingstrokes of these two species are the lower wingbeat frequency and greater stroke amplitude for *C. splendens*, and the fact that the damselfly performs a clap and fling at the dorsal end of its stroke (Wakeling and Ellington, 1997*b*). The wingbeat frequency and stroke amplitude both affect the flapping velocity and hence the lift of the wing. The enhancement of for *C. splendens* may therefore be due to the clap-and-fling mechanism, which generates more lift for a given mean wing velocity.

The analysis assumes that both wing pairs operate at the same , and then calculates the thrust partitioning between the fore- and hindwings. For sequences in which the fore- and hindwing kinematics differ, particularly for the damselfly where they perform different degrees of partial fling, this assumption may be inadequate. Nevertheless, it does provide a value for the minimum at which all the wings must be operating and an indication of the thrust partitioning. For all *S. sanguineum* flights except one, the forewing thrust was less than 0.5, and hence the hindwings provided the most aerodynamic force (Table 2). Values for *C. splendens* show , with the fore- and hindwing pairs sharing the aerodynamic load nearly equally. The difference in thrust partitioning between *S. sanguineum* and *C. splendens* may be attributed to their wing areas. The dragonfly hindwings have a larger area than the forewings (Table 1) and so would generate more aerodynamic force for the same kinematics and ; in contrast, the damselfly fore- and hindwings are of similar area and so would generate ^{–}similar aerodynamic forces for the same kinematics and .

### Induced power

Induced power is similar for both *S. sanguineum* and *C. splendens* and shows a general increase with both velocity and thrust (Table 3; Fig. 4); there is a stronger relationship for the increase with thrust. For horizontal flights at nearly constant thrusts equal to body weight, Pennycuick (1975) predicts that *P*_{ind}∝1/*V* and hence decreases with increasing velocity. The reason for the observed increase of *P*_{ind} with *V* can be attributed to the correlation between thrust and velocity for these flights (Wakeling and Ellington, 1997*b*). *P*_{ind} is more strongly dependent on *T* than on *V*, and hence the relationship with *V* is partially obscured by the effect of *T*.

### Profile power

Profile power is also similar for both species of dragonfly (Table 3) and shows an increase with both *V* and *T* (Fig. 4). The increase with *V* is expected because, at higher velocities, the relative velocity and thus *D*_{pro} increase. The increase with *T* is because additional aerodynamic force is generated by the wings beating at larger stroke amplitudes for the dragonfly and at higher frequencies for the damselfly (Wakeling and Ellington, 1997*b*), both of which increase the relative velocity.

### Inertial power

*P*_{acc} shows a general increase with both *V* and *T* (Fig. 4), again an expected result because at higher *V* and *T* the wings must beat at higher velocities to generate the required forces, and this entails higher *P*_{acc} costs. The inertial power costs for *S. sanguineum* are typically double those for *C. splendens*. The dragonfly wings have lower moments of inertia than those of the damselfly because their mass and area are distributed closer to the wing bases; however, their higher wingbeat frequencies and maximum wing velocities lead to the higher values for *P*_{acc}. For all except one of the flights of both species, *P*_{acc} is less than *P*_{aero} (Table 3).

### Aerodynamic power

Graphs for aerodynamic power *P*_{aero} as functions of *V* and *T* are shown in Fig. 5, and for muscle mass-specific in Fig. 6. *P*_{aero} is generally lower for *C. splendens* than for *S. sanguineum* at any given *V*; this again can be attributed to the clap-and-fling mechanism, which is the major kinematic difference between the two, as being a more efficient way of generating aerodynamic force. The damselfly has a lower proportion of muscle mass than the dragonfly (Table 3) so, despite the fact that both species were of similar mass, the muscle mass-specific is greater for *C. splendens* than for *S. sanguineum*.

*S. sanguineum* shows a monotonic increase in *P*_{aero} with speed, but it slightly increases for *C. splendens* at low speeds.

The damselfly flights extend to lower values of *V* and advance ratio *J* than those of the dragonfly, so the reason for the dragonfly not showing an increase in *P*_{aero} at the lower velocities may be the more restricted speed range. Again, the true shape of the relationship between *P*_{aero} and *V* may be obscured by correlations with *T*.

Both species showed an increase in *P*_{aero} with *T*; this is to be expected as high thrusts necessarily require high *P*_{ind}.

The maximum values were 156 W kg^{−1} and 166 W kg^{−1} for *S. sanguineum* and *C. splendens*, respectively. Because *P*_{acc} is smaller than *P*_{aero}, these are also the maximum mechanical power outputs from the muscle and are unaffected by the extent of elastic storage.

### Heat production

Values for the Newtonian cooling constant *k* are shown in Fig. 7. As a result of convective cooling, *k* almost doubles when the airspeed increases from 0 to 1 m s^{−1}, and then shows a lesser increase as speed increases to 2 m s^{−1}. The relative flight velocities *V*_{r} were 1.13<*V*_{r}<2.05 m s^{−1} and 0.92<*V*_{r}<1.91 m s^{−1} for *S. sanguineum* and *C. splendens*, and so values of *k* during flight are well represented by the respective means of *k*=−0.017 s^{−1} and *k*=−0.020 s^{−1} from cooling in an air flow of 1–2 m s^{−1}.

Thoracic temperatures immediately after flight are shown in Fig. 8. The mean values for muscle mass-specific heat production were 663 W kg^{−1} and 838 W kg^{−1} for *S. sanguineum* and *C. splendens*, respectively.

## Discussion

Current understanding of dragonfly aerodynamics is still limited by our poor knowledge of the nature of the flow interactions between the fore- and hindwings. Dragonflies can beat all their wings independently, and they can alter the phase relationships between the fore- and hindwings; near synchronous wingbeats may be used to generate more aerodynamic force than the usual dragonfly mode of ‘counterstroking’ (Alexander, 1984, 1986; Rüppell, 1989). Mechanical models have shown that changing the phase relationship between the fore- and hindwings can, indeed, alter the force production (Luttges, 1989), and similar conclusions have been predicted from theory (Lan, 1979; Azuma *et al*. 1985). Vortices shed from the wings interact, and the resultant force cannot be predicted from the kinematics of the individual wings alone (Saharon and Luttges, 1988). Until the nature of the wing interactions is better understood, the effects of kinematic variations for individual wings cannot be predicted with confidence.

The approach taken in the present analysis uses a single actuator disc for all four wings in the induced power calculations and investigates the net mass flux of air into the far wake. Problems created by flow interactions between the wings are thus circumvented. A more detailed aerodynamic description of dragonfly flight will only be possible when the precise flows around each wing can be accurately modelled. Nonetheless, the results from the present study are useful for assessing dragonfly flight performance and for comparisons with other insects. Additionally, the methods used here for considering the effects of accelerations are applicable to insects with one functional pair of wings.

### Quasi-steady lift coefficients

The quasi-steady for *S. sanguineum* appears surprisingly low from the present study. For all except one of its flights, , despite the fact that the advance ratio was consistently low (*J*<0.5, Wakeling and Ellington, 1997*b*) and so the flights were approaching hovering. Previous authors who found high values for the dragonfly (Osborne, 1951; Weis-Fogh, 1967; Norberg, 1975) all studied flights in which the stroke planes were steeply inclined, i.e. where they were not nearly normal to the thrust. The flight SSan6.1 had a reasonably inclined stroke plane with β′=−18 °, but all the other *S. sanguineum* flights were in the range −12 °<β′<8 ° and thus were nearly normal to the thrust. *S. sanguineum* can generate unsteady lift from its wings: flight SSan5.1 required equal to 1.8, a value which is incompatible with quasi-steady assumptions, and in the field *S. sanguineum* hovers with a horizontal body and inclined stroke planes. However, it tends to ‘hover’ facing into the wind whenever there is any breeze, and this effectively gives it a forward velocity. For accelerated flight, as in the present study, dragonflies may adopt kinematics with stroke planes nearly normal to the thrust more often than has previously been thought. The upstroke is thus recruited for an equal share of the thrust production, halving compared with a strongly inclined stroke that relies on the downstroke alone for thrust.

The quasi-steady for *C. splendens* is generally higher than for *S. sanguineum*: . The mean wing velocity is lower for the damselfly, mainly because its wingbeat frequency is half that of the dragonfly. However, the body mass and wing area are similar for the two species. For a given level of thrust generation, the damselfly must operate its wings at higher , and this is indeed what happens. The extra lift for the damselfly wing may be derived from the clap-and-fling mechanism, and has already been discussed in Wakeling and Ellington (1997*b*).

The values for have been calculated on the assumption that useful lift is produced on both the up- and downstrokes. For damselfly flights where there is a strong clap and fling, this assumption may not be adequate. As the wings are flung apart, a flow of air into the opening gap creates a circulation about each wing; these circulations are created prior to, and completely independently of, the translatory motion of the wings. Downstroke circulation is thus enhanced by circulation generated at the preceding fling. At the start of the upstroke, however, the isolated wings must fight the Wagner effect which could be exacerbated by the shed vorticity from the downstroke. Upstroke circulation may thus be lower than from quasi-steady predictions. In the extreme case, all the lift may be generated during the downstroke, with none being generated during the upstroke. can be calculated for such cases by taking a value of zero for *I*_{L,Z(u)} in equation 22: profile drag still occurs during each half-stroke, but lift is only produced on the downstroke. Such estimates of based on lift from the downstroke alone represent the upper limits of and are given in Table 4. It should be noted that, while there may be some justification for these values for some of the damselfly flights, the original assumption of both upstroke and downstroke lift is probably more appropriate to the dragonfly and the rest of the damselfly flights.

The damselfly utilises its highest during its lowest-velocity flights. This may underlie differences in its flight behaviour when compared with the dragonflies. Field observations show that dragonflies typically fly quickly through the large airspaces above a body of water whilst damselflies fly more slowly, manoeuvring around the vegetation at the water’s edge. The clap-and-fling mechanism may be useful during the slow precision flight of damselflies. Dragonflies, in contrast, have become adapted to faster flights in more open airspaces and have done so with modifications to their wing shapes (Wakeling, 1997) and thoracic structures that include a reduction of the stroke amplitude available to the forewings (Pfau, 1986, 1991). These modifications make the dragonfly less suited to performing flings, but flings may not be necessary during higher-velocity flights. Nonetheless, dragonflies are adept at hovering (Norberg, 1975), and they may achieve this by exploiting circulation generated during isolated wing rotations.

### Inefficiencies in the momentum jet

The ‘vertical’ component of the profile drag results in an effective increase in thrust and induced power of 12 %, on average, for the momentum jet. This additional thrust required to overcome the profile drag can be considered an inefficiency in the wake. This effect may be accommodated by a correction factor *k*_{ind} to the Rankine–Froude estimate for induced power that is greater than 1.2. The precise value of *k*_{ind} may depend on the ratio of the profile to the induced drag, and so may be different for dragonflies than for insects with one functional pair of wings. Justification for such a modification to *k*_{ind} will have to await confirmation from studies on other insects.

### Local circulation method

Azuma *et al*. (1985), Azuma and Watanabe (1988) and Azuma (1992) state that dragonfly flight can be explained using quasi-steady assumptions alone. They used a modified local circulation method from helicopter analysis to analyse dragonfly flight and obtained no unusually high lift coefficients. This is in contrast to the conclusions from all other studies on dragonfly flight. Their methods were originally developed for helicopter flight where the rotors beat in a nearly horizontal plane, with the thrust perpendicular to this plane and thus nearly vertical. In adapting their first ‘simple analysis’ to dragonfly flight, Azuma *et al*. (1985) maintained the assumption that the thrust is perpendicular to the stroke plane, even though the flights involved inclined stroke planes. If the major force acting on the dragonfly is its weight, which is vertical, then the net aerodynamic force must also be vertical and is generated by a horizontal actuator disc. The stroke planes for their dragonflies were not horizontal, however, and their actuator disc lies in the stroke plane rather than in the horizontal; this is where their simple analysis departs from the methods used here and leads to erroneous conclusions.

Azuma *et al*. (1985) filmed *Sympetrum frequens* during steady, slow climbing flight, i.e. without acceleration. The low parasite drag was neglected, which is a reasonable assumption, so the only net force is the vertical weight support. However, the thrust is calculated normal to the stroke planes with a vertical component to match *m*** g**; as the stroke planes are inclined to the horizontal, there would be a corresponding horizontal force component of 0.81

*m*

**. The induced velocity was calculated using a form of equation 8, i.e. for a flight velocity normal to the actuator disc;**

*g**V*was indeed normal to the disc, but this equation gives an induced velocity also normal to the disc. Because the only net force on the dragonfly is the vertical

*m*

**, the induced velocity should be vertical instead. Using equation 6 with their data yields an induced velocity over six times their value and greater than**

*g**V*

_{X}.

They calculate the net forces over the wingstroke using a modified blade-element analysis. With *V* and *w* being normal to the stroke planes, the relative velocity *V*_{r} is also normal to the stroke planes. It is implicitly assumed that the upstroke and downstroke contribute equally to thrust generation. Because *w* should be vertical, however, *V*_{r} should not be normal to the stroke planes. The wings actually beat with an asymmetry between the up- and downstrokes because they are inclined to the relative velocity, and in such cases must be higher to support the required thrust. An attenuation coefficient *C*_{hf} equal to 0.3 was used to model the effect of the forewing’s induced velocity on the hindwings; i.e. 30 % of that induced velocity acted on the hindwing. This value seems very low and should probably be closer to 2 (Stepniewski and Keys, 1984). Finally, they solve simultaneously the conditions that the net horizontal force is zero and that the vertical force balances the weight. However, as stated above, this analysis implicitly creates a large horizontal force which is subsequently ignored. Additionally, instantaneous lift forces were calculated from a *C*_{L}(α) function with *C*_{L,max}=1.8; this clearly does not describe a dragonfly wing under steady-state conditions (Wakeling and Ellington, 1997*a*).

Using the methods presented in the present study, their data for the *S. frequens* flight result in and . This slow flight involves a mean lift coefficient that certainly cannot be explained by quasi-steady assumptions, in contradiction to their conclusion.

A local circulation method was then presented by these authors for analysing the flight of *S. frequens* (Azuma *et al*. 1985) and *Anax parthenope* (Azuma and Watanabe, 1988). To overcome their main objection to the ‘simple analysis’ (that the induced velocity was constant across the actuator disc in both time and space), a time-varying induced velocity is introduced; the analysis is computationally more complex and only a skeleton of the methods is presented in the two papers. Nonetheless, the results and conclusions from the two methods are similar. The time-varying induced velocity in the local circulation method gives mean horizontal and vertical forces that are small and balance weight support, respectively. However, without further information on how the initial kinematic and force data were incorporated into the local circulation method, its suitability for the analysed dragonfly flight cannot be assessed.

Both methods discussed above require that each wing pair supports nearly half the thrust; this was an imposed restriction on the simple analysis, but it emerged as a solution in the local circulation method (certainly for *Anax parthenope* case 4, for which graphical data are presented). For both methods, the interference from the hindwing on the forewing was considered negligible and was ignored. Thus, calculations for the forewings supporting half the thrust will be irrespective of any wing interactions and can be treated as an insect beating one pair of wings. The methods of the present study predict that the forewings alone of *Anax parthenope* would have operated at if they supported half the thrust, and thus quasi-steady lift generation is unable to account for the aerodynamics of these flights.

### Muscle power output

For all except one of the flights in the present study, inertial power was lower than aerodynamic power. The mechanical power output of the flight muscles is thus equal to the aerodynamic power regardless of the degree of elastic storage (Dickinson and Lighton, 1995). Elastic elements within the flight motor system are certainly able to store and return most of the inertial energy. Various elastic systems have been found in the thoraces of insects (Alexander, 1988); indeed, Weis-Fogh (1960) found an elastic tendon in dragonflies, including *Sympetrum* and *Calopteryx* species, which was a cylinder of pure resilin interpolated in the apodemes of some wing muscles. Elastic torque measurements for *Aeshna grandis* show that approximately 75 % of the inertial energy can be stored and released by both the wing ligaments and the muscle fibres themselves (Weis-Fogh, 1972). Non-fibrillar muscle fibres can store up to 2.5 J kg^{−1} of elastic energy (Alexander and Bennet-Clark, 1977), and this would correspond to 50 and 100 W kg^{−1} at the respective wingbeat frequencies of *S. sanguineum* and *C. splendens*, making up 93 % and 78 %, respectively, of the maximum inertial power. Even if elastic elements in the thorax returned none of the inertial power, the inertial power expended at the beginning of each half-stroke would be recovered as the wing decelerates during the second half-stroke, contributing to the aerodynamic power costs at the end of each half-stroke. The net mechanical power output from the muscles thus equals the aerodynamic power, regardless of elasticity assumptions, and is 156 and 166 W kg^{−1} for *S. sanguineum* and *C. splendens*, respectively. It should be noted that during a complete clap and fling, in which the left and right wings come to a halt as they touch, much of the kinetic energy might be lost as heat and sound from the wings and could not be stored in thoracic elements. However, the damselfly flights typically involve partial flings in which the wings do not halt at the top, and so there is still scope for elastic storage.

These maximum muscle power outputs of around 160 W kg^{−1} are at the upper end of power outputs, both measured and modelled, for insect synchronous flight muscle. Weis-Fogh and Alexander (1977) proposed that for non-fibrillar and vertebrate striated muscle the maximum power output is governed by the intrinsic contraction frequency of the muscle; with a high intrinsic contraction frequency of 25 s^{−1} and contraction over 15 % of its resting length, the maximum power is approximately 250 W kg^{−1}. Pennycuick and Rezende (1984) simplified this model and argued that power was the product of the frequency imposed by the biomechanics and the work done per unit muscle mass. The simplified model suggests that maximum power would asymptotically approach a limit of 860 W kg^{−1} as contraction frequency increases, but estimates powers of 400 W kg^{−1} for synchronous muscle at the wingbeat frequencies typical of Odonata. Ellington (1985) further modified this model to account for the volume of sarcoplasmic reticulum and, assuming that locust muscle contracted by 5 % of its length at the measured intrinsic contraction frequency of 9 s^{−1}, calculated a revised value of 80 W kg^{−1} for this classic synchronous flight muscle.

Josephson (1985) studied the power output of tettigoniid synchronous flight muscle using a work loop method first developed by Machin and Pringle (1959) for asynchronous muscle. By measuring muscle tension while an oscillating length change is imposed on it, *in vitro* results should more closely approximate *in vivo* performance. Maximum power outputs for this and similar studies have recorded 76 W kg^{−1} for tettigoniid flight muscle (Josephson, 1985), 33 W kg^{−1} for locust flight muscle (Mizisin and Josephson, 1987) and 90 W kg^{−1} for hawkmoth flight muscle (Stevenson and Josephson, 1990). In their study of hawkmoth muscle, Stevenson and Josephson (1990) found that maximum power output increased with temperature, and they recorded maxima of 130 W kg^{−1} for two preparations at 40 °C. They also showed that the values from other studies were comparable with the hawkmoth results when temperature was taken into account. Their value of 130 W kg^{−1} is the highest measured from an insect synchronous flight muscle. Willmott (1995) has recently estimated a value of 150 W kg^{−1} for hawkmoth flight muscle using aerodynamic arguments.

Marden (1987) has estimated the maximum forces produced by insects during take-off by attaching weights to them. Ellington (1991) reanalysed his data, correcting the induced power calculations and adding a further 30 % as a rough estimate for the profile power. The present study has shown that profile power for two pairs of wings is typically 75 % of the induced power, and so the maximum aerodynamic power recalculated from Marden’s data would be 150 W kg^{−1} for Odonata.

### Aerodynamic costs of flight

For flight at constant velocities, the aerodynamic power changes with velocity. There is some debate about whether the power curves are U-shaped or J-shaped (see Ellington, 1991). Whatever the precise shape, however, the curves share several common features: is smallest at some intermediate speed, often with a very broad minimum, and increases rapidly at higher velocities, reaching the maximum mechanical power output at the maximum velocity possible for the animal. Finding this maximum velocity is experimentally difficult, because animals are reluctant to cooperate at such high performance levels. Hence, this velocity is commonly estimated by extrapolation of to the maximum power output expected from the muscle.

The curves from the present study show to be less than 60 W kg^{−1} for 0.5<*V*<1.0 m s^{−1} and then to increase at higher velocities for both *S. sanguineum* and *C. splendens*. The maximum velocities of 1.7 m s^{−1} occur at power outputs which are likely to be near the maximum possible for odonatan flight muscle. These maximum velocities, however, do not represent the maximum velocities possible for these species. The costs increase with both velocity and thrust, and the flights at the highest *V* also occur at some of the highest values of *T*. Were these flights to have been horizontal at a constant velocity, then would probably have been smaller. During one of the gliding sequences from Wakeling and Ellington (1997*a*), a velocity of 2.6 m s^{−1} was recorded for *S. sanguineum*, showing that they can indeed fly at velocities greater than 1.7 m s^{−1} if they are not simultaneously accelerating.

increases with thrust for both *S. sanguineum* and *C. splendens*. It is important for both of these species to generate thrusts of at least so that males can support heavier females while flying in tandem during mating (relative masses from Grabow and Rüppell, 1995). The maximum values recorded in the present studies were 2.65 and 1.59 for *S. sanguineum* and *C. splendens*, respectively. More data points are desirable at the higher thrusts, but the data reported here suggest that *S. sanguineum* can comfortably achieve mating flight, whereas *C. splendens* will be struggling to achieve for much less than 160 W kg^{−1} muscle power output.

### Heat production

Cooling constants have been measured for a number of dragonfly species by May (1976); the interpolated value for a temperate species with a 68 mg thorax (typical of *S. sanguineum*) is *k*=−0.0056 in still air, with the spread of the data covering the value of *k*=−0.0070 reported here. The thorax of *C. splendens* is smaller than that of *S. sanguineum* and, because of the scaling of conductance, the larger value found for *k* (−0.0100) for *C. splendens* is to be expected. The rate of heat loss increases in an air flow (Church, 1960*b*; May, 1976; Casey, 1976, 1980, 1981), and this can lead to two- or threefold increases in the cooling constant for the speeds at which these flight sequences occurred (Fig. 7). Heat loss due to evaporative cooling is low in small flying insects (Church, 1960*a*); *Sympetrum* species have 85 % of their thoracic wall insulated by air sacs (Church, 1960*b*); and percher species in general have little physiologically facilitated heat transfer between the thorax and abdomen (Heinrich and Casey, 1978; Heinrich, 1993). Heat production estimates based on conductance values from dead dragonflies should thus be fairly reliable. Indeed, the mean cooling constants for live dragonflies cooling to 30 °C differ by no more than 3 % from those for dead dragonflies (May, 1976).

The mean muscle mass-specific heat production during the flights recorded in the present study are 663 and 838 W kg^{−1} for *S. sanguineum* and *C. splendens*, respectively. The mean muscle mass-specific metabolic power is the sum of these mean heat productions and the mean aerodynamic powers (Table 3) and is 759 and 918 W kg^{−1} for *S. sanguineum* and *C. splendens*, respectively. May (1995) presents metabolic rates during flight for Anisoptera in the body mass range 170–3160 mg; the data include his own estimates from heat production and Polcyn’s (1988) respirometry measurements. The data show considerable scatter, but there is a general increase in mass-specific metabolic rate with decreasing body size. *S. sanguineum* falls below the range of May’s data, but its rate falls within the 95 % confidence limits for the extrapolated trend. The metabolic rates also compare favourably with those measured for moths: Casey (1981) found thoracic mass-specific metabolic rates during hovering of 551 W kg^{−1} and 1591 W kg^{−1} for two moth species of body mass 90–100 mg.

Resting metabolic rates of a 120 mg dragonfly may typically be 11 W kg^{−1} muscle mass (May, 1979). Compared with the estimated metabolic rates of *S. sanguineum* and *C. splendens*, this would give a metabolic scope of 70–80, which lies comfortably within the range of 50–100 typically found for flying insects (Ellington, 1984*d*).

### Muscle efficiency during flight

Muscle efficiencies scale with animal size for locomotory activities. With increasing size, the mass-specific metabolic rate of animals decreases, but the mass-specific cost of transport is reasonably independent of size; hence, larger animals perform the same mass-specific work for a lower metabolic cost and are thus more efficient. Muscle efficiencies in vertebrate locomotion decrease from 70 % for humans and kangaroos to 7 % for a 30 g quail (Heglund and Cavana, 1985); the high values are largely due to substantial elastic energy return from the muscle–tendon complex. The highest recorded muscle efficiency from an isolated vertebrate muscle is 37 % (DeHaan *et al*. 1989). Decreasing muscle efficiency with size has been shown for a variety of moth species (Casey, 1989). For a range of euglossine bees, efficiency decreased from 16 % to a mere 4 % over a corresponding mass decrease from 1000 to 80 mg (Casey and Ellington, 1989), assuming perfect elastic return of inertial power. The muscle efficiency during hovering (also assuming perfect elastic storage) has been estimated at 6 % for *Bombus*, 5 % for *Apis* and 8 % for *Eristalis* (Ellington, 1984*d*), and at 11 % for *Drosophila* flight regardless of elastic storage mechanisms (Dickinson and Lighton, 1995). Muscle efficiencies during insect flight are typically much lower than those found for vertebrate muscle, but this can be attributed to their small size. The values depend on the assumed extent of elastic storage; with no elastic return, the mechanical power output is greater, and so the efficiency is higher. However, it is currently thought that substantial elastic storage occurs during insect flight. In order to resolve this dilemma, Josephson and Stevenson (1991) measured the O_{2} consumption and the power output simultaneously during cyclic work loop experiments on locust flight muscle. This direct measurement of muscle efficiency gave a value of 6 %, confirming that insect flight muscle does indeed operate at very low efficiencies.

The mechanical power outputs from the present study, and thus also the muscle efficiencies, are unaffected by assumptions of elastic storage because the inertial power is less than the aerodynamic power. The mean muscle efficiencies are 12.6 % and 8.7 % for *S. sanguineum* and *C. splendens*, respectively. These efficiencies are both reasonable given the other values for insects, and they are at the upper limit of what can be expected from insect flight muscle.

Dragonfly ancestors, the Protodonata, are amongst the earliest winged insect fossils, and the dragonfly mode of flight has persisted for 300 million years (Wootton, 1974; May, 1982). The evolution of the more modern, neopteran, insects has superseded the odonates, and their modern mode of flight with one functional pair of wings can be considered to be evolutionarily more advanced. Dragonflies are still a major aerial insect predator, however, and they have by no means been ousted by their younger relatives. Indeed, modern dragonflies can out-manoeuvre and prey upon neopteran insects. The dragonflies’ success can be attributed to the exceptional aerodynamic properties of their wings, their powerful and efficient flight muscles, and one of the highest ratios of flight muscle to body mass for any animal (Marden, 1989). The two sub-orders, Anisoptera and Zygoptera, have become adapted to different styles of flight: dragonflies fly rapidly in large airspaces, whereas damselflies manoeuvre through vegetation at lower speeds. These differences can be explained in terms of their different wing shapes (Wakeling, 1997), thoracic structure and the degrees of freedom available to the wings (Pfau, 1986), wingbeat kinematics (Wakeling and Ellington, 1997*b*) and their aerodynamics, with the damselfly utilising the clap-and-fling mechanism which is useful for its low speed flight.

## ACKNOWLEDGEMENTS

We thank A. P. Willmott for stimulating discussions, an anonymous referee for helpful comments on the manuscripts and the BBSRC for financial support.

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