The free flapping flight of the dragonfly Sympetrum sanguineum and the damselfly Calopteryx splendens was filmed in a large flight enclosure at 3000 frames s−1. The wingtip kinematics are described for these flights. Despite the two species being similar in size, the damselfly flew with wingbeat frequencies half those of the dragonfly. The damselfly could perform a clap and fling, and the proximity to which the wings approached each other during this manoeuvre correlated with the total force produced during the wingstroke. The dragonfly beat its wings with a set inclination of the stroke planes with respect to the londitudinal body axis; the damselfly, in contrast, showed a greater variation in this angle. Both species aligned their stroke planes to be nearly normal to the direction of the resultant force, the thrust. In order to achieve this, the dragonfly body alignment correlated with the direction of thrust. However, the damselfly body alignment was independent of the thrust direction. Velocities and accelerations were greater for the dragonfly than for the damselfly. However, non-dimensional velocities and accelerations normalised by the wingbeat periods were greater for the damselfly.

Dragonfly flight has fascinated scientists for most of this century. These insects are large, colourful and striking, and thus readily capture one’s attention as they fly past. The dragonfly Order Odonata represents one of the oldest and most primitive forms of insect flight, with two pairs of wings which can beat independently. This primitive flight mode has also been a focus for attention, but it should by no means be considered simple: flight with two pairs of wings may require more complicated aerodynamic modelling than for the more derived insects with one functional pair of wings.

Considerations of dragonfly flight first appeared in the literature in 1921 when Hankin discussed the possible effects that air temperature and the sun have on the prevalence of gliding. High-speed cinematography for a range of dragonflies was first described by Magnan (1934), who took films of tethered dragonflies at 3200 frames s−1 and measured their stroke amplitudes and wingbeat frequencies. Magnan concluded that Zygoptera flew with greater stroke amplitudes than did Anisoptera. Chadwick (1940) also published high-speed kinematic data for a tethered dragonfly, and described its stroke amplitude and wingbeat frequency. In a study of the ventilation of flying insects, Weis-Fogh (1967) published some measurements for the wingbeat frequency and stroke amplitude of two tethered dragonflies.

In a study of the hovering flight of a range of insects, Weis-Fogh (1973) outlined some of the problems that may be experienced by flying dragonflies. Using a quasi-steady analysis, he assumed that the forces acting on an insect wing at each instant were the same as if that wing were in a steady flow with the same relative velocity; these forces were then averaged over the whole wingbeat cycle. Weis-Fogh (1973) showed that the mean lift forces required for flight are greater than quasi-steady values for a variety of insects, including the dragonfly. These insects must therefore rely on the extra lift generated by unsteady wing motion: i.e. wing rotations and accelerations. Renewed interest was thus sparked in the understanding of dragonfly flight to try to explain the high-lift mechanisms involved.

The free hovering flight of Aeshna juncea was filmed in the field by Norberg (1975). He described how this dragonfly hovered with a horizontal body and a steeply inclined stroke plane. This orientation required even greater lift, because weight support is only possible on the downstroke. This is in contrast to insects hovering with a ‘normal’ horizontal stroke plane, where lift on both the morphological up- and downstrokes can be used for weight support. Using a couple of alternative assumptions, Norberg (1975) confirmed that the quasi-steady mechanism is indeed insufficient to explain the lift required by flying dragonflies.

The wingbeat kinematics of Calopteryx splendens were described by Rudolph (1976a,b), although they had previously been investigated by Magnan (1934). Rudolph noted that the wings of this damselfly can perform the ‘clap and fling’, a wing motion originally described by Weis-Fogh (1973) for the wasp Encarsia formosa. As the wings reach the dorsalmost position of their stroke, they clap together with the wing surfaces touching; they then fling apart, rotating about their trailing edges, as they separate on the downstroke. As the wings fling apart, a flow of air into the opening gap creates circulation about each wing. Weis-Fogh (1973) suggested that this mechanism may be a source of lift enhancement. An advantage of the fling-generated circulation is that it occurs before, and completely independently of, the translatory motion of the wing. The fling mechanism has been analysed theoretically (Lighthill, 1973; Edwards and Cheng, 1982; Wu and Hu-Chen, 1984), modelled numerically (Haussling, 1979) and verified experimentally (Bennett, 1977; Maxworthy, 1979; Spedding and Maxworthy, 1986), and it has been shown to produce higher transient circulations and lift forces than are possible from translatory motion alone.

The fling mechanism is not a discrete event, and there is a spectrum of insect wingbeat styles that differ in the dorsal position reached by the wings. Wings that only partially come together perform a partial fling, and those that do not quite touch perform a near fling (Ellington, 1984c). The separation between the wings when they fling apart may provide fine control over the circulation and thus the lift generated. This has been confirmed in model experiments by Sunada et al. (1993) and is seen in the locust (Cooter and Baker, 1977), where the hindwings perform a clap and fling during climbing but not during horizontal flight.

The ability to perform a clap and fling is a difference between the wingbeat kinematics of the damselflies and dragonflies. There is only one cited example of a dragonfly performing a clap and fling: Alexander (1984) studied 91 high-speed film sequences of dragonflies (mainly Libellula luctuosa and Celithemis elisa) in tethered flight and found evidence for this wing motion in only one sequence. In a study of the maximum lift produced by flying insects, Marden (1987) found that insects with clap- and-fling wing motions achieved on average 25 % more muscle mass-specific lift than did animals with other types of wingbeat. In particular, the damselfly muscle mass-specific lift was 44 % greater than that of dragonflies. The clap- and-fling mechanism is clearly a method for insects to generate enhanced lift from their wing strokes and is used by the damselfly.

Dragonflies, unable to use the clap and fling, must rely on other unsteady lift mechanisms. Ellington (1984c) suggests that an isolated rotation of the wings, coupled with flexion of the wing, may generate additional lift in a similar way to the clap and fling. This combined motion causes the wing to rotate about its trailing edge, which should produce a leading-edge vortex of the correct sense for lift on subsequent translation. Verification of this flex mechanism has been hampered by the methods available for determining wing angles during rotation. Angle of rotation estimates for dragonflies are particularly prone to error at supination and pronation; during stroke reversal, the camber, which is generated by aerodynamic force on the wings (Ennos, 1988; Wootton, 1991), is also reversed. Thus, the wings change from being cambered in one direction, through being flat, to being cambered in the other direction, with a resultant change in wing chord. Angles of rotation based on measurements of the wing chord from a single view will thus be prone to errors due to fluctuations in the chord; these errors would be up to 20 ° for dragonflies. To overcome this problem, at least two views of the wing are required so that estimations are not based on chord length alone. Nevertheless, there is some evidence to support the flex idea: wing rotation has been visually estimated for Diptera (Ennos, 1989) showing rotational coefficients within the range required by the flex mechanism, a sharp lift pulse has been recorded when the wing of Drosophila melanogaster is rotated at supination (Zanker and Götz, 1990), and Dickinson (1994) provides evidence that rotation of a wing about its trailing edge can create vorticity that enhances lift during the subsequent stroke.

It is possible that high lift forces are generated during the isolated rotation and flexion of dragonfly wings. Also, precise control of the lift generated by the wingstroke may be achieved by small changes in the timing of the wing rotations at supination and pronation. If so, there may be little correlation between the wingbeat frequency, the stroke amplitude and the translatory angle of attack with the velocity and overall aerodynamic force generated during flight. Indeed, the free-flight kinematic studies of Azuma and co-workers (Azuma et al. 1985; Azuma and Watanabe, 1988; Azuma, 1992) found little correlation between velocity and angle of attack, stroke amplitude or wingbeat frequency. The only systematic change that they found was that the wingstroke plane tilted forwards at higher velocities to be more nearly normal to the direction of the velocity.

The most wide-ranging study of the free-flight kinematics of dragonflies has been undertaken by Rüppell (1985, 1989; Rüppell and Hilfert, 1993): angle of attack, stroke amplitude, wingbeat frequency, velocity and stroke plane angle are given for over 20 species covering the three sub-Orders Anisoptera, Zygoptera and the relict sub-Order Anisozygoptera. Damselflies are characterised by lower wingbeat frequencies, higher stroke amplitudes and lower velocities and accelerations than dragonflies. No individual wingbeat has been fully analysed for all its kinematic parameters, however, and so these data have not been used to determine how the individual parameters affect the flight of each dragonfly or damselfly. The lower wingbeat frequencies and higher stroke amplitudes of damselflies have also been described by Newman (1982).

One further way in which the wingbeat kinematics may affect aerodynamic performance is that the phase relationship between the fore- and hindwings may change the flow interactions between the wing pairs. Somps and Luttges (1985) measured the forces generated by a tethered dragonfly and recorded a single large lift peak for each cycle. Flow interactions between the fore- and hindwings produced this single lift peak rather than the double peaks expected from the sum of the forces from two independent wing pairs. Flow-visualisation studies on a flapping model dragonfly showed that the fore- and hindwings generate the same vortical structures as an isolated wing except over a critical range of reduced flapping frequencies, pitching rates and phase relationships, where the vortices become more cohesive and even fuse, with a resultant lift enhancement (Saharon and Luttges, 1988; Luttges, 1989). Changing the kinematic parameters can alter the flow interactions between the wings, resulting in a change of aerodynamic force that could not be predicted solely from the flow around the individual wings. Luttges (1989) noted that the phase relationship between the wings affects the relative proportions of the horizontal and vertical components of the net aerodynamic force.

Dragonflies typically fly with their wings beating out of phase, or counterstroking. It has been suggested that beating their wings in phase, or parallel stroking, may produce higher aerodynamic forces (Alexander, 1984, 1986; Rüppell, 1989) and so may be used for high-lift situations such as rapid accelerations or tandem flying with a mate. In a theoretical study on the effects of the phasing relationship for a dragonfly wingbeat, Lan (1979) plotted the efficiency η and mean thrust coefficient as functions of the phase lag of the forewings. On average, the predicted optimum phase lag was 90 ° for maximum energy extraction by the hindwings from the wake of the forewings; for maximum , the lag was 45 °. Hence, as dragonflies increase the degree to which they beat their wings in parallel, the wings will generate more thrust at the expense of a lower efficiency.

In the present study, the free flights of Sympetrum sanguineum and Calopteryx splendens were filmed using a high-speed cine camera. These species are members of the sub-Orders Anisoptera and Zygoptera; they were chosen as they are individuals of approximately the same mass and wing area. For steady flight, the mean lift required by both species will be similar, but it may be generated by different kinematic mechanisms. However, the flight of Calopteryx splendens may not be representative of that of the Zygoptera as a whole. Calopterygidae have disproportionately large wing areas when compared with other Zygoptera (Grabow and Rüppell, 1995), leading to lower wing loadings. The smaller Zygoptera generally fly with their fore- and hindwings counterstroking with a near half-cycle phase shift (Rüppell, 1989). In contrast, the Calopterygidae can fly with very little phase shift between the wing pairs (Rüppell, 1985). Nonetheless, the Zygoptera, including C. splendens, are characterised by lower wingbeat frequencies and larger stroke amplitudes than the Anisoptera (Rüppell, 1989). Flights from the present study were unrestrained by either tethers or wind-tunnels, and so the insects were free to vary the velocity and acceleration at which they flew. However, the escape nature of the flights resulted in the directions of the velocity and acceleration being roughly correlated. Velocities and accelerations were derived from positional data on the films, and the total aerodynamic force was estimated. The flights were then used to compare the different wingbeat modes between the dragonflies and the damselflies, and also to determine whether any of the kinematic parameters could be correlated with the velocity or aerodynamic force.

Dragonflies of the species Sympetrum sanguineum (Müller) and Calopteryx splendens (Harris) were caught at Quy Fen and Grantchester Meadows, Cambridge, respectively, during the period 3–30 July 1993. Upon capture, each individual was placed in a plastic box, which was stored in the dark inside an ice cooler. Once the dragonflies had cooled down, they ceased to struggle within their boxes. The dragonflies were transferred to a refrigerator in Cambridge and filmed within 24 h of capture. Before filming, the dragonflies in their boxes were left to warm up to ambient temperature for at least 10 min. The mean thermal time constants for S. sanguineum and C. splendens (=1/Newtonian cooling constant, values from Wakeling and Ellington, 1997b) were 143 s and 100 s, respectively, and so they would passively warm up to at least 98 % of the ambient temperature within 10 min. However, the dragonflies were active during this period and would heat up even faster, perhaps to temperatures above ambient. Flights were filmed in the specially adapted greenhouse described in Wakeling and Ellington (1997a). Flight behaviour resembled that seen in the field: both species commonly flew the length of the pond a couple of times before perching on the glass walls, the green netting or the vegetation around the pond.

The free flights of the dragonflies were filmed using a John Hadland P.I. Hyspeed cine camera. Ilford Pan F 16 mm black- and-white film was shot at 3000 frames s−1; timing lights within the camera strobed at 1000 Hz to give a time reference on the films. A Cinor 50 mm fixed focal length lens was used at an aperture of f/11. The films were developed in a John Hadland P.I. 16/35B R2 negative film processor set to 30 °C and the slowest speed to enhance contrast. A 300 W halogen lamp was used to backlight the dragonflies to provide a silhouette image. The light was partially diffused through a frosted glass plate and was then focused through a Fresnel screen into the cine camera. This arrangement minimised the light intensity needed for filming and thus reduced the heat dissipated from the lamp. The Fresnel screen measured 0.2 m×0.3 m and was placed on the opposite side of the pond 2.2 m away from the cine camera. Dragonflies were released 1.8 m from the camera. Approximately 0.5 s after starting the camera, the top of the box was removed; this corresponds to the time taken for the film to accelerate up to speed. Characteristic flights for both species involved flying steeply out of the box and then taking a shallower course along the pond. Dragonflies flying out of the box necessarily entered the filming area, and this was recorded on 36 of the 95 sequences filmed. Ten S. sanguineum and four C. splendens individuals were filmed with no more than 10 sequences for each individual.

The dragonflies were filmed approximately 60 wing lengths away from the camera, and the frame covered a horizontal distance of eight wing lengths. Analysis of the projection errors for this filming geometry (Ellington, 1984b) shows that the reconstructed angles will be less than 1 ° in error over the entire filming area and that the assumed horizontal plane will deviate by less than 1.6 ° from the true horizontal. Perspective errors can thus be ignored during the kinematic analysis.

Dragonflies were weighed before and after each filming session, and the mean was assumed to be the mass during the flight sequences. Dragonflies were killed with ethyl acetate immediately after filming. Measurements of wing and body morphology were made within 6 h of filming using methods described elsewhere (Wakeling, 1997).

Films were analysed using a Vanguard Instrument Corporation model M-16C 16 mm projection head as described by Ellington (1984b). Analogue output from the digitising device was read by a National Instruments MIO16L-9 12-bit analogue-to-digital converter in a Macintosh Quadra 650 computer. The coordinates of any point on the screen were accurate to 0.1 mm, or 0.2 % of the maximum projected wing length. This corresponds to a single bit from the analogue-to-digital converter. The accuracy of digitised sequences is thus limited by the image quality and the clarity of the wing outlines rather than by any inherent inaccuracies in the digitiser. The computerised three-dimensional reconstruction of the wing kinematics, as described by Ellington (1984b), had previously been coded by A. P. Willmott; this routine was modified to use a time-varying value for the maximum projected wing lengths as explained below. All other custom software was specifically written for this dragonfly study, mainly in LabView and Mathematica for Macintosh computers.

Velocities and accelerations

The films of the flying dragonflies consist of two-dimensional images taken from a single viewing direction. These images were digitised in a y,z coordinate system in which both the y and z axes are parallel to the film plane, with the y axis horizontal and the z axis vertical. All the information required to reconstruct the three-dimensional flight path must be taken from these images. The velocity of the dragonfly away from the camera is calculated along a horizontal x axis, which is perpendicular to both the y and z axes.

This study only investigated flights in which the acceleration during each sequence is linear and constant. Each film was initially tested for this condition as follows. The near forewing base position was digitised from every tenth frame, and the y and z coordinates were plotted against time. Least-squares linear, quadratic and cubic regression lines were fitted to these data. For flight at a constant velocity, the linear fit will be no worse than the quadratic or cubic fit; and for flight at a constant acceleration, the quadratic fit will be no worse than the cubic fit. The quality of each fitted regression line is judged by the root mean square (RMS) residual values, giving the variation of the data from that fitted line. Where the quadratic fit showed substantially larger RMS residual values than the cubic fit, the flight sequence was not consistent with a constant acceleration; in such a case, if a shorter section of that flight appeared to be at a constant acceleration, then that section was isolated and retested. Flight sequences were only considered for further analysis if they showed constant acceleration for several wingbeats. It is assumed that, if acceleration is constant along the y and z axes, then it will also be constant along the x axis. By definition, linear acceleration cannot have any rotatory component. Hence, the flight sequences considered here have negligible angular acceleration and this component is thus ignored.

Movement along the x axis was determined from changes in the image size. The distance S of an object from the optical centre of the camera lens can be calculated using simple lens theory:
formula
where f is the focal length and M is the magnification (=image length/object length). The beating wings were used to act as a reference length for the dragonfly. During each wingbeat, the wings continually change orientation with respect to the film plane. In some cases, the wings pass through a position where they are parallel to the film plane; this gives the maximum projected wing length for that particular distance of the wing from the camera. The projected wing lengths are calculated from the digitised wingtip and wingbase positions from the y,z film coordinate system as:
formula
The true maximum projected wing length occurs twice for each each wingbeat. However, if the beating wing never passes parallel to the film plane, the projected length still reaches a maximum but there is only a single maximum per wingbeat. These two cases are distinguished in Fig. 1.

Whenever a dragonfly flight sequence generates a true maximum projected wing length for either the forewings or the hindwings, the distance from the wings to the optical centre of the lens can be calculated, giving the additional positional information along the x axis required for velocity and acceleration estimates.

The film sequences which showed flight at a constant acceleration were redigitised, this time for the base and tip positions of the near wings in each frame; the projected wing lengths were then calculated for each frame. Where maximum projected wing lengths occurred, the non-dimensional distance in wing lengths, Ŝ, from the camera was calculated:
formula
where Rf is the forewing length. Maximum projected wing lengths can occur for both the fore- and hindwings, and so the maximum hindwing lengths were normalised to the forewing length, and a pooled data set was used containing the maximum values from both wing pairs. Values for Ŝ for the frames in which there was no maximum projected wing length were taken as linear interpolations of the distances at which there was a maximum. These interpolated values will only be an estimate of the distance of the dragonfly from the camera, however, as the dragonflies accelerated and decelerated slightly within each wingbeat. Finally, a quadratic function was fitted to the non-dimensional x position from the camera against time. This least-squares quadratic fit follows the assumption that the flight occurred at constant acceleration. The function describing the x position was differentiated to obtain the non-dimensional x velocity and acceleration. Sample positional data are shown in Fig. 2. The maximum projected wing lengths were then also described by a least-squares quadratic fit. However, a cubic spline fit proved more useful for the kinematic reconstruction and is discussed more fully below.

The motion of the dragonfly along the y axis was calculated as a non-dimensional velocity in wing lengths s−1, relative to the estimated maximum projected wing length for each frame. The mean y velocity during the wingbeat was the mean value from a linear least-squares regression of the non-dimensional y velocity against time, and the gradient of this line gave the non-dimensional y acceleration. Displacements in the z axis were treated in the same way as those in the y axis.

Velocity V and acceleration A were calculated from their x, y and z components using trigonometry. Non-dimensional velocity is the velocity in wing lengths per wingbeat:
formula
where R is the forewing length and n is the mean wingbeat frequency. Non-dimensional acceleration  is the acceleration in wing lengths per wingbeat per wingbeat:
formula
The inclinations of the velocity and acceleration relative to the horizontal axis are given by ξV and ξA, respectively.
The total aerodynamic force generated by the dragonfly during a wingbeat must overcome its weight, acceleration and parasite drag Dpar. The thrust T is the vector sum of these three forces and is discussed in greater detail in Wakeling and Ellington (1997b). In the present study, the thrust T is the total aerodynamic force, as in the helicopter literature; it is not the horizontal component of the aerodynamic force. Non-dimensional thrust is thrust per body weight:
formula
where m is mass and g is the acceleration due to gravity.

Values for Rf and m are given in Table 1; full morphological data for these insects are given in Wakeling and Ellington (1997b).

Kinematic reconstruction

The wingtip and body kinematics were reconstructed according to the protocol described in Ellington (1984b). This method takes the two-dimensional coordinates of the wings and body and transforms them through a series of coordinate systems as outlined below. The kinematic reconstruction assumes that the left and right wings beat symmetrically about the body, and so is able to use positional information from the far wings to calculate the body orientation relative to the camera. This assumption is most robust for straight flight and fully banked turns.

The kinematic analysis treats a single pair of wings, and so each wing pair from the dragonfly was analysed separately. For clarity, the methods outlined here describe analysis of the forewing only, but the method for the hindwing was identical. The following eight points around the dragonfly outline were digitised in the y,z system for each frame (Fig. 3): the most anterior and most posterior positions on the longitudinal body axis (head and anus), the base and tip positions for the ‘near’ fore- and hindwings (those where the base was visible throughout the stroke), and the tip positions for the ‘far’ fore- and hindwings. The origin of the y,z axes was taken to be the near forewing base (see Fig. 3). Discrete jumps in the wingtip position sometimes occurred between frames owing to light shining differently through the wing corrugations as the wings rotated with respect to the light source; this was most noticeable at supination and pronation.

The original analysis (Ellington, 1984b) used the maximum projected wing length for the wingbeat as a reference length with which to normalise the calculations. This is only suitable, however, when the maximum projected wing length and hence the distance from the camera does not change between wingbeats: i.e. for insects which are hovering, tethered or flying at a fixed position in a wind-tunnel. In the present study, the dragonflies were free-flying and frequently changed their distance from the camera. The maximum projected wing length increases by 4 % during a typical dragonfly wingbeat starting 60 wing lengths away from the camera and moving towards the camera at ; it is thus important to incorporate a time-varying value for the maximum projected wing length. Within a wingbeat, the dragonfly’s body will accelerate and decelerate, although the precise function of this motion is not known. Initially, the maximum projected wing length was estimated by a least-squares quadratic fit to the measured values. However, frames where projected wing lengths were greater than the predicted maximum led to a breakdown in the analysis. A cubic spline function was used instead to predict the maximum projected wing length for each frame, and this length was never smaller than the individual projected lengths. The cubic spline produced a smooth change in lengths between frames and was probably as good as any method for estimating the actual value. An inherent problem in this analysis is that the reconstructed angles are sensitive to small measurement errors when the wing is close to maximum projection (Ellington, 1984b). This effect is sharply peaked, with errors dropping to ±3 ° for positions 10 ° on either side of the object plane. Significant errors can also occur around maximum projection if the value used for the maximum projected wing length is itself subject to digitising error. With the very high sampling rate per wingbeat used in this study, it is likely that values very close to the maximum projected wing length were measured several times. The maximum projected wing length may thus be a biased overestimate due to digitising error at both the wingbase and the wingtip. This is revealed by a characteristic discontinuity in the reconstructed angles around the position of maximum projection. In such cases, the value for the maximum projected wing length was decreased by up to 1 % to remove the local discontinuity.

The time-varying maximum projected wing length was then used in Ellington’s (1984b) protocol for wingtip kinematics. Yaw Ψ is the horizontal component of the angle between the longitudinal body axis and each motion vector. Yaw was calculated with respect to both velocity, ΨV, and acceleration, ΨA. Roll of each wing pair η, body angle χ and stroke plane angle β are all calculated relative to the horizontal. The wingtip positions are described by spherical coordinates (ϕ,θ) (Fig. 3). The positional angle of the wing ϕ within the stroke plane takes positive values when the wing is dorsal, and negative values when it is when ventral. The angle of elevation θ of the wing from the stroke plane is positive when the wing lies above the stroke plane. The stroke plane angle β is calculated with the implicit assumption that no pitching occurs during the wingstroke. A mean body angle x̄ is similarly taken to be the mean value for χ during the stroke.

Wingbeat frequency

Wingbeat frequency n was estimated directly from the films. A characteristic wing position was traced at either supination or pronation when a chord near the wingtip was vertical. The wingbeat period was the time until the wing returned to the same position after one wingbeat, with the tracing superimposed on the new wing image; time was measured relative to the timing markers on the film. Supination and pronation are rapid events with the wing quickly rotating through a large angle in a few frames, and so this method is accurate to within a single frame per wingbeat: typically 1.3 % for S. sanguineum and 0.7 % for C. splendens.

Fourier series analysis of the wingtip kinematics

Fourier series were fitted to describe ϕ and θ with time. Up to six terms were used to assess the importance of higher harmonics within the wingbeat. The standard error (S.E.M.) of each Fourier series was calculated so that the quality of the fits could be compared. The Fourier series applied to the data were:
formula
with the standard errors:
formula
where a, ci and oi are the Fourier coefficients, i is the harmonic number, t is time, and N is the number of data points within the wingbeat.
The films were recorded at very high sample rates per wingbeat, and so there are many data points around the maxima and minima of the ϕ curve. Each point is subject to digitising and analysis error, and so it is likely that the values for ϕmax will be overestimated and those for ϕmin underestimated. The Fourier series, however, smoothes through these regions and more accurately estimates ϕmax and ϕmin. The Fourier series using four harmonics predicts values for ϕmax to be 1.9 ° smaller and ϕmin to be 2.4 ° greater than the unsmoothed estimates. These discrepancies are consistent with the errors introduced during digitising and analysis, and so the maximum and minimum positional angles, ϕmax and ϕmin, were calculated from reconstructed wingbeats using four harmonics in the Fourier series for ϕ. The mean positional angle within the stroke plane, , is thus given by:
formula
and the total stroke amplitude is Φ (=ϕmax−ϕmin). The ratio of the downstroke to upstroke duration, d/u, is given by the ratio of the period from ϕmax to ϕmin to the period from ϕmin to ϕmax.

Flight sequences had to pass two criteria before they were fully analysed. The sequences had to show at least one complete wingbeat in which all four wingtips were visible and in which the near wings passed through the film plane, yielding true maximum projected wing lengths. Second, each sequence had to be for a flight in which the mean acceleration of the body was constant. From a total of 95 films, 36 had flight sequences on them, with seven for S. sanguineum and nine for C. splendens passing the criteria for analysis. These data were from four S. sanguineum and four C. splendens individuals. Examples of these sequences are presented in Figs 4 and 5. Examples of the time series for ϕ and θ are shown in Fig. 6 for both S. sanguineum and C. splendens. Reconstructions of the wingbeats are shown in Fig. 7 for S. sanguineum and Fig. 8 for C. splendens. Sequences are identified by a prefix SSan or CS, referring to S. sanguineum and C. splendens, respectively, followed by a number, referring to the individual’s identification number, and finally a decimal point and a number categorising that flight.

The kinematic diagrams

The upper drawings in Figs 7 and 8 show the dragonfly body with its velocity and acceleration vectors. For clarity, each drawing has been reconstructed in perspective. Around the dragonfly body is a circle with radius equal to the forewing length Rf. The circle is horizontal, but is viewed from above for this drawing. Four complete spokes are drawn within the circle at 90 ° intervals. The spokes pass through the centre of mass of the dragonfly. The thicker spokes lie in the sagittal plane, with the arrow facing in the forward direction. A further eight marks around the inside of the circle indicate 30 ° divisions. The body is drawn to scale, and the angle between the body and the forward direction spoke is the mean body angle x̄. The non-dimensional velocity is drawn as a thick brown line with a solid arrowhead, whilst the non-dimensional acceleration  is drawn as a thick green line with an open arrowhead. The velocity and acceleration vectors start at the centre of mass and are drawn along the hypotenuses of right-angled triangles, with the other two sides being horizontal and vertical. Both vectors are drawn to scale, and at the correct inclination ξ and yaw Ψ; angles ξ and Ψ are illustrated in Fig. 7A. Values for , Â, ξ and Ψ are given for each sequence. The velocity V and acceleration A can be calculated from and  using the mean forewing length Rf, the mean wingbeat frequency n [=(nf+nh)/2] and equations 4 and 5.

The lower drawings in Figs 7 and 8 show a side view of the dragonfly with the respective wingtip positions during a wing stroke. Points for the forewing tip are marked as blue circles, while points for the hindwing tip are marked as red diamonds. The stroke planes for the fore- and hindwing pairs are at angles βf and βh to the horizontal, respectively. The stroke planes are indicated by the pairs of coloured arrows and pass through their respective wingbases, each denoted by a cross. The distance from the wingbase to the tip of each stroke plane arrow denotes the wing length R for that particular wing pair. The length of each stroke plane arrow denotes the mean chord for the respective wings. The body is shown at the mean body angle x̄ and is also drawn to scale.

The wingtip path is given by the coloured symbols. These points represent the projected positions of the wingtip onto the x′,z′ plane; i.e. they indicate where the wing tips would be seen if viewed from the side. Black arrows show the direction of travel for the wingtips. Values are given for the stroke plane angle β, the stroke amplitude Φ, the mean positional angle of the stroke, and the roll angle η for each wing pair. The angles between the velocity and acceleration vectors and the normal to the stroke plane, κV and κA, respectively, are also given for each wing pair. Values presented for Φ and are calculated from the Fourier series, which are discussed below.

The period for the forewing beat is scaled to non-dimensional time , where:
formula
and so for the forewings . In this same scale, the stroke period for the hindwings is not necessarily 1. The start of the hindwing cycle relative to that of the forewing is and is in the scale. The fore- and hindwing beat cycles were not always digitised from equivalent positions, so does not necessarily represent a hindwing phase shift. This phase shift can be visualised by reconstructing the Fourier series for both wing pairs with the hindwing cycle commencing at . The advance ratio J is given by:
formula
The respective fore- and hindwing beat frequencies, nf and nh, and the ratio of the downstroke to upstroke duration d/u are also given in Figs 7, 8.

Fourier series analysis of the kinematics

Coefficients a, ci and oi, as fitted by the Fourier series to the ϕ and θ data (see equation 7), are presented in Tables 2 and 3. The standard errors for the predicted curves with each additional level of harmonics (as calculated by equation 8) are also given. The Fourier series for ϕ show that the fundamental (i=1) dominates the curve, with the second harmonic typically more important than the higher harmonics.

An inherent problem with this type of analysis is a subjective element in digitising the wingtips. The wingtip is not well-defined for some wing orientations, and its perceived position may change as the wings move. Furthermore, as the wingtips reappear on the far side of the body silhouette, a different position for the tip may be chosen. These problems were especially noticeable for the θ data as it was often harder to judge the chordwise location of the wingtip than the spanwise location. The higher harmonics of the Fourier series give a better fit to the data where these discontinuities occur, but smaller standard errors do not necessarily reflect a better goodness of fit. Curves using the first four harmonics describe the data well without producing unrealistic kinks, whereas by the fifth harmonic kinks are introduced into the fits around the regions of digitising uncertainty.

Flight descriptions

When describing the flights of S. sanguineum or C. splendens, it must be remembered that the data for each species contain measurements from a number of different individuals.

There were no cases in which two individuals flew with the same velocity and acceleration, and so there is no way of assessing the variation between individuals. The problem of a comparative description of the flight of each species is confounded by the fact that the flights were at different velocities and accelerations. Past studies have concentrated on steady flight, in which there is no acceleration and in which the thrust is predominantly used for weight support and thus is reasonably constant. The flights filmed in the present study were totally unrestrained and showed a threefold range of thrust and velocity.

The precise nature of any changes in the kinematic parameters with velocity and thrust is unknown. Indeed, a major objective of this study was to discover such trends. Preliminary examination of the results showed that relationships more complicated than linear could not be justified. It was therefore decided to test whether each kinematic parameter changed with velocity and thrust using simple least-squares linear regression, i.e. whether the slope was significantly different from zero. The purpose of fitting linear regression lines to the data is only to identify trends and not to estimate a functional relationship; as such, it was deemed inappropriate to record the equations for the regression lines. For the purposes of discussion, regressions are deemed significant at a 95 % confidence level. However, to clarify trends in Figs 9 and 10, regression lines are distinguished at a 90 % confidence level, and this confidence level is deemed ‘good’ in the text.

Table 4 gives the mean values and ranges for the kinematic parameters. Some of the regressions against velocity and thrust are shown in Figs 9 and 10. The significance of the trends are indicated in Table 5.

Flight of Sympetrum sanguineum

Sympetrum sanguineum faces its flight direction, with its body at a nose-up attitude to the horizontal. All flights were ascending, and the mean body angle shows a significant linear regression with the angle of inclination of the velocity (Fig. 9), giving a mean body angle 31 ° shallower than the inclination of the flight velocity.

The stroke planes are approximately perpendicular to the direction of the velocity; the mean angle between the normals to the stroke planes and the direction of the velocity was 16 ° for both fore- and hindwings. Significant linear regressions for these angles, κfV and κhV, with the flight velocity show that as V increases the stroke planes tilt to face the direction of the velocity more closely. The normals to the stroke planes are also kept close to the direction of the thrust T, with the angles κfT and κhT between the normals to the stroke planes and the thrust direction taking mean values of 19 ° (Table 4).

The stroke planes are at fixed angles relative to the thorax, with the angles between the stroke planes and the longitudinal body axis being 48 ° and 50 ° for the fore- and hindwings, respectively; a matched-pair t-test showed that there was no significant difference between these angles. As the body inclination increases for climbing flight, the stroke planes flatten out relative to the horizontal, facing a more vertical direction.

The hindwing shows a greater stroke amplitude Φ than the forewing; this difference is due to the hindwing reaching a lower position ϕmin than the forewing. The upper position of the stroke, ϕmax, is similar for the two wings. The midpoint of the forewing stroke, , is approximately 6 ° higher on average than for the hindwing, , and so the forewings beat around a more dorsal position within their stroke plane than do the hindwings. Both the fore- and hindwings show a general increase of Φ with V. Only the hindwing shows a good regression of Φh with T (Fig. 10), mainly due to increases in ϕh,max; the only good trend for the forewings with T, in contrast, is the decrease of ϕf,min with increasing T.

The wingbeat frequency n remained reasonably constant for all the flights, with a mean value of 39 Hz for both the fore- and hindwings. There was no significant regression of n with either V or T. The downstroke-to-upstroke ratios d/u were 0.98 and 0.93 for the fore- and hindwings, respectively, but a matched-pair t-test showed there to be no significant difference between these values. The forewing motion lagged behind that of the hindwing in all sequences by approximately 26 % of the forewing period, i.e. a 94 ° phase lag (Fig. 6A).

The mean flapping velocity 2nΦR was 14 % higher for the hindwings than for the forewings. There was a greater variance in Φf than in Φh, and less variance in nf than in nh, but the variance for the mean flapping velocity was greatest for the hindwings. Changes in the aerodynamic output of the wings may be due more to changes in hindwing motion than to changes in forewing motion.

Flight of Calopteryx splendens

Calopteryx splendens faces its flight direction and does not raise its body inclination to meet the inclination of the velocity. Nearly half the flights had a horizontal body posture, whilst the body varied between head-down and head-up inclinations for the others. The mean body angle did not show any significant linear regression with the inclination of velocity (Table 5).

The wing motion is more variable than for S. sanguineum; the angle between the stroke plane and the body axis is more variable and θ deviated more from the stroke plane. Angles between the normals to the stroke planes and the direction of velocity are greater than for S. sanguineum, but the angles to the thrust vectors were similar. The normals to the stroke planes were at mean angles of 31 ° and 22 ° from the velocity direction for the fore- and hindwings, respectively, and 21 ° from the thrust direction for both. βh shows a significant linear relationship with the inclination of velocity but not with the mean body angle. There were also no significant regressions for κfV and κhV with velocity, nor were there significant regressions for κfT and κhT with thrust. The stroke planes do not tilt to face V or T more closely as these parameters increase. Both the fore- and hindwings beat through a similar amplitude Φ of approximately 120 °, and this amplitude is considerably greater than that for S. sanguineum. A matched-pair t-test showed that was systematically approximately 12 ° greater than , with the forewings beating around a more dorsal position, similar to the dragonfly. The forewings normally come close together at the top of their stroke, with a mean of 83 ° for ϕf,max, and they show little variation in ϕf,max.

The hindwings beat to a lower position ϕh,min than the forewings, but most of the variation of the stroke occurs dorsally. The lower mean ϕh,max of 72 ° gives the hindwings more room for manoeuvre at their dorsal extreme, and they do indeed show a greater range of ϕh,max. The hindwing stroke amplitude Φ shows a good regression (P<0.01) with V, but there are no notable trends linking any parameter for stroke position with T.

Flight CS3.3 (Fig. 8G) was atypical of damselfly flights and is an example of ‘threatening’ flight (Rüppell, 1985), where all four wings are briefly held together motionless at the top of the stroke. If this unusual, more ballistic, flight of CS3.3 (at T=0.96×10−3 N) were to be ignored, then there is a significant increase in ϕh,max with increasing thrust (P=0.029).

The mean wingbeat frequency n was 20 Hz for both the fore- and hindwings, half the frequency of the S. sanguineum wingbeats. There was no significant regression of n with V. There was, however, a good regression of n with T. The downstroke-to-upstroke ratios d/u were 1.09 and 1.01 for the forewings and hindwings, respectively, but a matched-pair t-test showed there to be no significant difference in these values. The downstroke duration was slightly longer than that of the upstroke, which is opposite to the situation for S. sanguineum. The forewing motion was advanced by 8 % of its period relative to the hindwing, i.e. a 29 ° phase advance (Fig. 6B); again, this is in contrast to the phase lag of S. sanguineum.

A matched-pair t-test showed that the mean flapping velocities of the fore- and hindwings, 2nΦR, were not significantly different from each other. If the clap- and-fling mechanism enhances the lift generated by a wing pair, and the magnitude of this enhancement is a function of the proximity of the wings when they fling apart, then the forewings of C. splendens must generate more lift than the hindwings. The hindwings, however, reach a more variable position at the top of their stroke and are used for controlling the total thrust from the wing system.

General flight remarks

The mean velocity for the dragonflies was faster than that for the damselflies: however, the maximum velocity of 1.66 m s−1 was the same for both species. During gliding (Wakeling and Ellington, 1997a), a higher maximum velocity of 2.6 m s−1 was recorded for S. sanguineum, which was faster than the maximum for C. splendens. Neither experimental set-up was designed for eliciting maximum velocities, and it is unlikely that these values are the maximum possible by these insects. However, the values are similar to the maximum reported velocities of dragonflies in the field. Field velocities of dragonflies show a tendency to increase with increasing body mass (data from Rüppell, 1989); for the smaller dragonflies the velocities are: Epitheca cynosura (175 mg) 2 m s−1, Micrathyria atra (214 mg) 3 m s−1 (May, 1991), Sympetrum danae (105 mg) 4 m s−1, and C. splendens (118 mg) 1.9 m s−1 (Rüppell, 1989).

Both the mean and the maximum accelerations of S. sanguineum were higher than those for C. splendens. The respective maximum accelerations, 18.33 m s−2 and 8.28 m s−2, again fall into the range of those previously measured for free-flying dragonflies: E. cynosura 11.8 m s−2, M. atra 15.9 m s−2 (May, 1991), S. danae 25 m s−2, and for damselflies C. splendens 13 m s−2 (Rüppell, 1989). The maximum non-dimensional thrust generated by S. sanguineum was greater than that for C. splendens: 2.65 and 1.59, respectively. Again, these values are similar to those already published for insects during take-off (Marden, 1987) with the equivalent values for being 3.0 for a 125 mg Sympetrum dragonfly and 2.2 for a 149 mg Mecistogaster damselfly.

Although the dragonfly may typically fly faster and with greater acceleration than the damselfly, the non-dimensional velocities and accelerations were greater for C. splendens. The damselfly flew relatively further per wingbeat and accelerated relatively more per wingbeat than did the dragonfly. The main reason for this difference is that the damselfly flew at similar velocities but at wingbeat frequencies of half those of the dragonfly. Nevertheless, these two species are of similar mass, wing area and thus wing loading, and so the difference in the effectiveness of the wingstrokes must be due to the different wing kinematics. The stroke amplitude Φ was smaller and the wingbeat frequency n was greater for the dragonfly than for the damselfly, and it also had higher mean flapping velocities. Perhaps more importantly, the damselfly performed some degree of clap and fling, and beat its wings more synchronously, whereas the dragonfly wings did not touch at the top of the stroke and the forewings lagged by approximately 90 °.

The wing motions recorded here match some of those predicted from the morphology of the odonatan wing joint (Pfau, 1986, 1991). Pfau describes how the mechanism of odonatan wing motion is fundamentally different from that for the other Pterygota. In the Odonata, there is no tergal-arching mechanism for the downstroke, which is powered instead by the direct action of the basalar and subalar muscles; by contrast, the dorsolongitudinal muscle in other pterygotes causes the tergal plate to arch, which in turn leads to a depression of the wings. Because the wing motion is uncoupled from tergal arching, the Odonata are able to move all their wings relatively independently. The dorsolongitudinal muscle takes a different role in the Odonata; by tilting the tergal plate and thus the wing hinge, it allows the stroke plane to be altered relative to the body axis. The Anisoptera show further evolutionary modifications to the ancient flight mechanics of the Zygoptera/Anisozygoptera; they have almost completely reduced the dorsolongitudinal muscles in the metathorax and have completely reduced the pleuro-alary (stroke plane) muscles in both thoracic segments. The Anisoptera have thus simplified the motor system, and a steeper plane of wing movement has been achieved in both segments by developing a more vertical alignment of these segments.

Pfau (1986, 1991) suggests that the wing motion of the Anisoptera should have fewer degrees of freedom than that of the Zygoptera, with the anisopteran stroke plane at a relatively fixed orientation to the body. Differences in the wing joints also suggest that the Anisoptera must beat their hindwings through a smaller amplitude than their forewings. The kinematics reconstructed from the present study show that the dragonfly indeed has fewer degrees of freedom in its wing motion. In S. sanguineum, the stroke plane is almost rigidly fixed at inclinations of 48 ° and 50 ° from the longitudinal body axis for the fore- and hindwings, respectively. C. splendens, in contrast, varies its stroke plane from −5 ° to 53 ° relative to the body axis. Also, within each wingstroke, C. splendens had a greater freedom of wing motion from the stroke plane, as shown by a greater variation in θ; the wing motion of S. sanguineum was much more confined to the stroke plane. The hindwings of S. sanguineum actually beat through greater stroke amplitudes than did the forewings, which is in contrast to Pfau’s (1986, 1991) predictions.

Stroke plane angles and mean body angles

The orientation of the dragonfly during flight will be a compromise between the orientation of the stroke plane and that of the body, and this has been shown for Anisoptera in general (Rüppell and Hilfert, 1993). The wings are more effective at producing thrust when the stroke plane is normal to the direction of that thrust (Wakeling and Ellington, 1997b). The body experiences least drag Dpar when it is aligned to face the direction of the relative velocity, as Dpar is minimised at small body angles (Wakeling and Ellington, 1997a). Indeed, during the horizontal forward flight of insects in general, body angles decrease with increasing velocity to minimise the cost of Dpar. Correlations have been found between the body angle and flight velocity for the dragonfly Anax parthenope (Azuma and Watanabe, 1988), Drosophila spp. (Vogel, 1966; Götz, 1968; David, 1979), the honeybee (Esch et al. 1975), the bumblebee Bombus terrestris (Dudley and Ellington, 1990; Cooper, 1993) and neotropical butterflies (Dudley, 1990). For the damselfly, where the alignment of the stroke plane is relatively independent of the body direction, it might be expected that the stroke plane can be normal to the direction of the thrust at the same time as the body faces the relative velocity. The dragonfly, in contrast, has a more restricted wing motion relative to the body; its orientation may be more of a compromise.

The mean body angle for S. sanguineum increases as the inclination of the flight velocity increases, suggesting that it is maintaining a low Dpar orientation. There are no significant regressions between the mean body angle and the stroke plane angle with the direction of thrust. The angles between the normals to the stroke planes and the direction of the velocity, κV, are smaller than the angles to the direction of the thrust κT, and there is less variance in κV than in κT. The orientation which S. sanguineum adopted during these sequences seems to have been dominated more by aligning the body towards the relative flight direction and thus minimising Dpar than by positioning the stroke planes to face the thrust. It should be noted that the flights recorded here were all at a reasonable forward velocity, but at low flight speeds the parasite drag will be a negligible concern. If the mean values of κT of approximately 19 ° are considered typical for S. sanguineum, then there is little evidence to suggest that it would hover with steeply inclined stroke planes, as suggested for Odonata in general (Weis-Fogh, 1973) and Aeshna juncea in particular (Norberg, 1975).

C. splendens adopted a mixed strategy for its body orientation during these flight sequences. During half the sequences, the body was nearly horizontal despite its inclined velocity direction, and for the other half it showed an increasing body angle with increasing flight inclination. The flights with horizontal bodies were not exclusively those at the lowest velocities, where Dpar would be smallest. The stroke planes faced the direction of the thrust more closely than the direction of the velocity, with the means and variances of κT for the forewings being smaller than those of κV. This highlights the extra freedom that the damselfly has in its wing motion: it can align its body to the flight direction at the same time that its stroke planes are facing the thrust. In cases where the thrust is rapidly changing, the damselfly does not necessarily need to realign its body in order to face its stroke planes in the new thrust direction. Evidence from the present study suggests that C. splendens would hover with horizontal stroke planes in a manner categorised as ‘normal hovering’ (Weis-Fogh, 1972, 1973; Ellington, 1984a).

Wingstroke amplitudes and wingbeat frequency

Increases in the stroke amplitude Φ will increase the relative velocity of the wing and hence the forces produced during a wingstroke. However, there have been few clear correlations of Φ with flight velocity in insects. The dragonfly Anax parthenope (Azuma and Watanabe, 1988) showed no correlation of Φ with velocity. Sotavalta (1947) suggested that the amplitude for the sphingid Sphinx ligustri may be higher during slow forward flight than during hovering, but Cooper (1993) reported a general decrease in Φ with velocity for the bumblebee Bombus terrestris. Magnan (1934) noted a general decrease in Φ for free-flying insects with increased velocity. Studies on tethered flight have consistently shown a decrease in Φ with velocity for tethered Calliphora erythrocephala, Locusta migratoria, the dragonfly Orthetrum cancellatum (Gewecke, 1967, 1970, 1974, respectively) and the fly Muscina stabulans (Hollick, 1940). It is debatable whether such tethered results are applicable to free flight (Rüppell, 1989), especially when the insects are in fixed tethers and are unable to regulate their body angles. The tethered flight of locusts (Baker et al. 1981) and Heteroptera (Betts, 1986) is known to differ from that in the field.

In the present study, there were no significant regressions between any of the positional parameters and velocity (Table 5). However, some trends were apparent and can be seen more easily in Figs 9 and 10. There is a slight increase in Φ with velocity for both S. sanguineum and C. splendens. In general, these increases in Φ are generated by both increases in ϕmax and decreases in ϕmin, with the mean positional angle of the wing remaining reasonably invariant. The only significant regression between the positional parameters and thrust was for ϕmax for the damselfly, and only when flight CS3.3 was ignored. This regression does show how the proximity of the hindwings during a partial fling affects the forces produced during that fling.

Increases in the wingbeat frequency n will also increase the relative velocity of the wing and hence the forces produced during a wingstroke. Different trends of n with velocity have been shown for different insects. In the dragonfly Anax parthenope, no correlation was found (Azuma and Watanabe, 1988), and wingbeat frequency was observed to be remarkably constant for C. splendens under ‘all experimental conditions’ (Rudolph, 1976b). Lack of correlation between n and velocity has also been reported for Drosophila virilis (Vogel, 1967) and bumblebees (Dudley and Ellington, 1990; Cooper, 1993).

Positive correlations, however, have been found between n and velocity for the locust both while free-flying (Baker et al. 1981) and tethered (Gewecke, 1975). The wingbeat frequency of dragonflies during free flight has been shown to be correlated with body temperature (Sotavalta, 1954; May, 1981).

In the present study, there were no significant regressions between n and either velocity or thrust (Table 5) except that the wingbeat frequency of C. splendens increases with thrust (Fig. 10).

Wing interactions; phasing relationships and clap and fling

The damselflies flew with higher non-dimensional velocities and accelerations than the dragonflies, showing that they were moving relatively further and accelerating relatively more per wingbeat. The mean non-dimensional damselfly thrusts were 82 % of the dragonfly values despite flying at half the wingbeat frequency. The damselfly wingbeat must therefore be more effective at producing aerodynamic force than the dragonfly wingbeat.

The main differences between the wingbeat kinematics of C. splendens and S. sanguineum are that the damselfly performs clap and flings or partial flings, its fore- and hindwings beat almost exactly in phase, and its wingbeat frequency is half that of the dragonfly. S. sanguineum wings, in contrast, do not come close together at the top of their stroke and they beat with the forewings lagging by approximately 90 °. The precise interactions between the fore- and hindwing pairs of the dragonfly are not fully understood, but the present study shows (see also Alexander, 1984, 1986; Lan, 1979; Rüppell, 1989) that wingbeats producing the greatest thrusts are those where the fore- and hindwings beat almost synchronously.

Some of Rüppell’s films show Calopterygidae manoeuvres in which the damselfly rapidly accelerates backwards while using a synchronous fore- and hindwing downstroke. Such a wingstroke will effectively squeeze air out from between the wings, and a large aerodynamic force is produced by the profile drag; this is discussed in more detail by Wakeling and Ellington (1997b). The geometry of the wing motion from the flights in the present study, however, would not lead to useful force production by profile drag (Wakeling and Ellington, 1997b). The clap and fling is a recognised mechanism for lift enhancement and is employed by the damselfly but not the dragonfly; this is probably the main cause for the greater thrust produced by the damselfly wingbeat.

Aerodynamic mechanisms of dragonfly flight

The flights described in this study covered a threefold range of thrust for S. sanguineum and a twofold range for C. splendens. If the aerodynamic force were to be produced by entirely quasi-steady mechanisms, then it should be expected that there would be a correlation between thrust and the kinematic factors governing force production, i.e. stroke amplitude, wingbeat frequency and angle of attack. The angle of attack was not measured in this study; however, the lift-to-drag polars for these two species demonstrate little change in lift coefficient over a wide range of angles (Wakeling and Ellington, 1997a), showing that there would be little effect on the quasi-steady lift of changes in angle of attack. Additionally, Azuma and Watanabe (1988) failed to find any trend in angle of attack over a fourfold range of velocity for Anax parthenope. The lack of correlation between any of these kinematic parameters and thrust suggests that aerodynamic force must be produced by mechanisms that are not entirely quasi-steady in nature. Wing flex during supination and pronation was not quantified in this study, but it may be a possible method for additional force generation during the stroke. The present study thus adds to the growing body of evidence that unsteady lift-producing mechanisms must be involved during insect flight. The aerodynamics of the flights from this study are considered in further detail in Wakeling and Ellington (1997b).

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