Using high-speed cinematography, bumblebees in free flight were filmed over a range of forward airspeeds. A detailed description of the wing tip and body kinematics was obtained from a three-dimensional reconstruction of the twodimensional film image. A technique for determining quantitatively the angle of attack of the wing was developed. Kinematic parameters found to vary consistently with airspeed were body angle, stroke plane angle, geometrical angle of attack, and rotational angles of the wings at the ends of half-strokes. Results of a morphological analysis of the wings and bodies of those insects filmed in free flight are presented for use in later calculations of the lift and power requirements of forward flight.
An accurate description of the wing and body kinematics of flying insects is fundamental to any analysis of the associated mechanics and aerodynamics of forward flight. Traditionally, studies of insect flight kinematics have been performed on tethered insects (Hollick, 1940; Jensen, 1956; Weis-Fogh, 1956; Nachtigall, 1966; Vogel, 1966, 1967; Zarnack, 1972). Kinematic parameters investigated in these studies have included wingbeat frequency, motion of the wing tip and wing profile. Weis-Fogh (1973) initiated a new era in animal flight research with his study of free hovering flight in insects, and this method of analysis has since become increasingly sophisticated (Ellington, 1980, 1984b). Detailed analyses of wingbeat kinematics and wake geometries of free-flying bats and birds have also become increasingly popular (e.g. Norberg, 1976; Rayner et al. 1986; Spedding, 1986, 1987). Particular attention has been paid in these studies to the variation of kinematic parameters with forward airspeed, in order to determine the means by which aerodynamic force is produced and its direction and magnitude regulated.
Comparable analysis of the forward flight of insects has not been possible until recently. The absence of detailed kinematic studies of free forward flight in insects can be attributed to the technical difficulties of eliciting controlled, properly oriented flight for significant periods of time. The alternative method, tethering, introduces a variety of behavioral and mechanical constraints (Ellington, 19846); it is unknown how similar the wing kinematics observed in tethered insects are to those in free flight. Only for one insect species, the migratory locust, are there kinematic data available for a direct comparison of free and tethered flight. Baker et al. (1981) showed that the wingbeat frequency of locusts in free flight was typically 16 % greater than that of tethered insects flying at some preferred speed and that, on average, speeds in free flight were some 39% greater than preferred speeds in tethered flight. It was also shown by Kutsch & Stevenson (1981) that the wingbeat frequency of tethered locusts was significantly less than that of individuals in free flight, for both juvenile and mature individuals.
It is clear that kinematic studies of insects in free flight avoid all the potential limitations imposed by tethering. The best method for controlling free flight in insects is probably to rely upon their optomotor responses; through a judicious, perhaps fortuitous, choice of lighting and moving optical cues, it should be possible to elicit controlled flight (e.g. David, 1979). The present paper describes the experimental methodology and means of analysis used to determine the wing and body kinematics of bumblebees in free forward flight. An optomotor system consisting of moving stripes and fixed lighting is described whereby the orientation and general quality of flight in insects can be controlled. High-speed ciné films were taken of bumblebees flying over a range of forward airspeeds. A projection analysis was used to reconstruct the three-dimensional position of the wing tip and the body from single-view ciné frames. The variation of kinematic parameters with forward airspeed was examined in detail. A detailed morphological analysis was also carried out for those bumblebees filmed in free flight to provide data for calculations of lift and power requirements (Dudley & Ellington, 1990).
A new method is presented for the analysis of the span-wise angle of attack of the wings. This method, based upon the bilateral symmetry of wing motions, allows determination from a single view of the three-dimensional position of arbitrarily chosen points along the wing margin. A detailed description of the orientation of the wing through the wingbeat and at different airspeeds is thereby obtained. Such an approach is computationally and logistically simpler than more traditional methods of stereophotogrammetry, and is more generally applicable to three-dimensional reconstruction from a single view of the position and orientation of symmetrically paired animal appendages. This technique is used to measure for flying bumblebees the angle of attack of the wing through the wingbeat, and to examine in detail the variation of wing orientation with forward airspeed.
Materials and methods
Bumblebees, because of their size, general availability in the summer months, and noted ability to hover and to sustain forward flight, were chosen as the model insect for the present study. All experimental work in the present paper was performed on the buff-tailed bumblebee Bombus terrestris (L.) sensuAlford (1975), and all future text references to bumblebees refer to this species. Insects were collected in the Cambridge University Botanic Garden and were usually filmed within a day. They were maintained in large well-ventilated cages and were fed a dilute mixture of honey and water (50% v/v). Only the most vigorous individuals were selected for filming.
The wind tunnel used in all experiments was the same open-jet tunnel as used by Weis-Fogh & Jensen (1956), wherein a detailed description of the tunnel design may be found. The original motor was replaced with a variable-speed d.c. motor for better speed control. A bead thermistor anemometer (Prosser AVM 502) was used to determine velocity profiles in the region of the working section where insects were filmed. Mean air velocities at each of six different points, all located in the general region of the working section where flight was filmed, were found typically to be within 1 % of the designated free-stream value, over a range of 1–5ms−1. At any one point, measured velocity fluctuations were typically within 3% of the mean value, with a dominant frequency in the range 6–10 Hz at an airspeed of 3ms−1.
The working section of the open-jet tunnel was enclosed by a cylindrical section of 2 mm acetate sheet (Fig. 1A); the enclosed working section was 280 mm in length and 140 mm in diameter. Insects were kept within the working section by a circular piece of coarse-weave fabric at the downstream end, and by a line plastic netting fitted between the front of the working section and the opening of the wind tunnel upstream. A 60° sector of the acetate sheet was removed and replaced with a 75 mm × 75 mm window of plate glass to reduce optical distortion in filming. Air velocity measurements with and without the window in place suggested that its effect on the airflow in the general region of flight was negligible.
Controlling the free flight of the insects proved to be difficult. A variety of schemes was experimented with, and the most successful method to date is presented here: ultraviolet light proved to be a strong stimulus for initiating and sustaining flight, and a moving stripe pattern provided directional orientation. An 8 W Atlas ultraviolet fluorescent light was positioned above and perpendicular to the working section; most of the light was masked out such that only a small central section (approximately 3 cm in length) was visible (Fig. IB). Above and on either side of the working section were positioned two rotating ‘barber pole’ drums 60 mm in diameter and 160 mm long. The exterior of each drum was covered with alternating black and white stripes at a pitch of 45°. The drum axles were linked mechanically such that the direction of stripe motion was the same for each drum. The drums were driven at 1–3 revss−1 by a small d.c. motor; the speed of rotation was controlled remotely by the observer. Room temperature at the time of filming was always within 3° of 25 °C.
A Hyspeed 16 mm ciné camera [John Hadland (P.I.), UK], fitted with a 75 mm Bolex lens at f/8, was used for all high-speed photography. The camera was operated at 5000 frames s−1, which, for a typical bumblebee wingbeat frequency of 150Hz, resulted in approximately 33 film frames per wingbeat. The camera arrangement can be seen in Fig. 1A. The insect image was reflected to the camera by a front surface mirror, which was positioned above and to the right of the working section. By virtue of the mirror tilt, the angle between the object plane and horizontal was 70°, while the angle between the longitudinal axis of the tunnel and the optical axis of the camera, when projected onto a horizontal plane, was 25°.
Backlighting of the insect provided a silhouette image for filming: sufficient contrast could not be obtained with front lighting, even when using 5 kW of illumination. Two 1 kW quartz-iodine lights were placed approximately 5 cm behind and underneath the far side of the working section (Fig. 1A). A sheet of translucent drafting film taped around the bottom of the working section acted as a light diffuser, giving a uniformly lit background against which the silhouette image of the flying insect could be seen.
The high-speed camera required approximately 0·5 s to reach a running speed of 5000framess−1. To prevent film wastage, therefore, a 15·3 m film leader was spliced to the length of film to be exposed, the latter usually being 3·7 m in length and representing approximately 0·1 s of actual filming. A relay box turned on the camera lights approximately 0·5 s after the camera was triggered, at the same time that the beginning of the film to be exposed passed through the shutter mechanism. Any potentially adverse effects of the intense lighting on flight behavior were thus avoided in the 0·5 s prior to actual film exposure.
With the wind tunnel already up to speed, an insect was introduced into the working section. The drums were set rotating, and the ultraviolet light was then flickered on and off until the insect began to fly, at which point the light was left on. Several seconds of non-oriented and unsteady flight occurred before the insect ‘locked on’ to the moving stripes and began to fly steadily in the filming area of the working section. Flight duration ranged from less than Is to 1·5min; a detailed account of flight performance is given below in the Results section. After steady flight had been attained and the insect was positioned properly for filming, the high-speed camera was triggered. Immediately after the film had finished (approximately 0·75 s), the camera lights, ultraviolet light and rotating drums were all turned off. While the insect walked about inside the working section, a new film was put into the camera, and then the entire process was repeated. For a particular insect flying at one wind velocity, three filming attempts were usually necessary to guarantee at least one successful film in which the insect was properly oriented and the focus sufficient for analysis. Failure to obtain a good film was typically due to the unpredictable departure of the insect from the filming area in the 0·5 s before the camera reached full running speed.
Insects were filmed flying at three wind velocities (1, 2·5, and 4·5ms−1), and while hovering with the wind tunnel turned off. A video camera was also used to record all filming sessions, to identify those films in which the insect was in the field of view during filming, and also to ensure for films of hovering flight that in the seconds prior to actual filming the insect was in normal unaccelerated flight. The flight speeds used in this study correspond approximately to those typically found in nature. For example, Heinrich (1979) states that bumblebees fly to and from foraging sites at 3–5ms−1, while J. H. Mayberry (personal communication) has found that bumblebees foraging between artificial flowers spaced less than Im apart fly at speeds of less than 2ms−1.
Towards the end of filming sessions, insects displayed an increasing reluctance to fly at the higher velocities, but were almost always capable of hovering beneath the ultraviolet light. Filming was thus begun with flight at 4·5 m s−1, and continued with successively lower velocities, culminating in the filming of hovering flight. An entire film sequence of flights over the indicated range of velocities could be obtained in less than lh, given a cooperative animal. Insects were weighed both before and after a filming sequence. Weight loss over the filming period was typically less than 10% of the initial value, and therefore the mean of the two measurements was taken to be representative of the body mass during filmed flight.
Films were developed at 25 °C in a 1:9 solution of Ilford PQ developer and were fixed in a 1:3 solution of FX-Universal fixer. All films were analyzed using the projection equipment described by Ellington (1984b). A 12 bit A-D converter (CUBAN-12, Control Universal Ltd) was used to interface the digitizer to a Model B BBC microcomputer with a 6502 second processor. Computer programs for the kinematic analysis were written in HI-BASIC, an extended version of BBC BASIC.
Wing tip kinematics and body angle
The analysis of wing tip kinematics was generally the same as that used by Ellington (1984b), to which the reader is referred for further details. To analyze the motion of the wing tip through the course of a wingbeat, it was necessary to digitize for each film frame three points on the film image: the near wing base P1 (which was always the base of the left wing of the insect), the corresponding wing tip P2, and the opposite or far wing tip P3 (Fig. 2). Additional points of interest which will be considered later include the anterior tip of the body P4, the far wing base P5, and the point of intersection of the wing base axis with the median longitudinal plane of the body, P6.
By virtue of the symmetry condition, there exists for any point P7 on the near wing margin a corresponding point P8 on the opposite wing margin (Fig. 3). This can be expressed in the (x′,y′,z′) coordinate system by means of the following identities:
The origin of the filming coordinate system was always taken to be the near wing base Pi. The maximum projected wing length for the wingbeat sequence was determined and used to normalize all coordinates. To transform the digitized coordinates of the near and far wing tips to the (x,y,z) system, it is first necessary to calculate values of x*. Since the near wing base is defined as the origin of the (x*,y*,z*) system, the x** coordinate of the near wing tip for each frame can be easily determined from the projected wing length. A more involved approach is, however, necessary to calculate the coordinate of the far wing tip. In the (x*,y*,z*) system, the direction of the line connecting the two wing tips can be given as (a*,l,c*), which is entirely analogous to the direction (a,l,c) in the (x,y,z) system, as given by Ellington (19846), and is parallel to the y′ axis for any viewing coordinate system. The direction number is the slope on the film image of the line connecting the near and far wing tips; a mean slope c*, the value of c* in each frame being weighted by the square of the projected line length, was calculated for a particular wingbeat sequence. The direction number a* was calculated using positions P(l) and P(2) of both wing tips at either ends of the half-stroke (Fig. 2D; see Ellington, 19846). A mean value of a* was determined from several combinations of frames at either end of the wingbeat. Given knowledge of and , values of for the opposite wing tip can be calculated for each frame of the sequence. Coordinates of the near and far wing tips were then transformed to the (x,y,z) system using equations 1–3.
At this stage, the direction in the (x,y,z) system of the y′ axis was determined; mean values of a and c, weighted by the square of the appropriate line length in each frame, were calculated for the (x,y,z) coordinate system. From these direction numbers, the direction cosines relating the (x,y,z) coordinate system to the (x′,y′,z′) coordinate system fixed in the insect body could be determined. The near wing tip and far wing tip in the (x,y,z) system were then transformed, using the direction cosines, to the (x′,y′,z′) system. A mean wing base separation for the sequence was calculated from the y′ values of the near and far wing tips using equation 8.
The stroke plane angle ft (Fig. 4) was determined from the slope of a linear regression of the x′ and z′ coordinates of the near wing tip. The wing tip position is defined by spherical coordinates, the positional angle ϕ and the angle of elevation θ, which are derived from a coordinate system based upon the stroke plane. Roll r/ of the insect is given by the angle between the z and z′ axes, while yaw (deviation from the longitudinal axis of the tunnel) is given by the difference between the horizontal projection of the angle between the tunnel axis and the optical axis of the camera, 25°, and the angle between the x and x′ axes. When the left wing is below horizontal, roll is negative; negative yaw corresponds to a yaw towards the right of the insect. Wingbeat frequency was determined from 1 ms timing marks on the film, account being taken of the film acceleration during a sequence.
Span-wise angle of attack
For a particular frame of a wingbeat sequence previously analyzed for wing tip kinematics, the three-dimensional configuration of the wing margins was determined as follows. Initially, the wing margins were digitized from the film image and stored in the computer. The x* coordinates of all points on the margins were then calculated, and all coordinates in the filming coordinate system were transformed first to the (x,y,z) coordinate system and then to the insect coordinate system (x′,y′,z′). Finally, two rotations of the wing such that the longitudinal axis of the wing was coincident with the x′ axis permitted the determination of the endpoints of a series of chords equally spaced along the wing span. When the wing was rotated back to its original position, these chords were then used to calculate the chord angle relative to horizontal at span-wise locations.
The span-wise angle of attack of wing chords was determined as follows. The near wing base P1 was first digitized, and served as the origin for all points subsequently entered. The near and far wing tips P2 and P3 were then digitized, as was the far wing base P5, if it was visible. Points were then entered along the wing margins, from wing tip to wing base, for each of four cases: near leading edge, near trailing edge, far leading edge, and far trailing edge. The maximum projected length R’, as determined for the sequence by the previous wing tip analysis, was used to normalize all coordinates. The x* coordinate of the near wing tip P2 was also determined from the maximum projected wing length.
From the film image, the y* and z* coordinates of any point P7 on the near wing margin are known. The y* and z* coordinates of the corresponding point P8 on the opposite margin are, in general, unknown. It is readily apparent, however, that the line connecting these two points must be parallel to the line connecting the wing tips P2 and P3, given the assumption of bilateral symmetry of wing motions (see Fig. 3). In any two-dimensional projection the slope s of the line connecting P7 and P8 is therefore equal to the slope of the line connecting the wing tips. In the (x*,y*,z*) system of the film image, the y* and z* coordinates of the point P8 can be found from the intersection with the far wing of a line with slope s passing through point P7.
For each point P7 on the near leading or near trailing edge, therefore, the equation of a line with slope 5 passing through that point was determined. The point of intersection of this line with the far leading or far trailing edge, respectively, was then determined as follows. The opposite wing margin was represented by a series of line segments connecting consecutive points entered along the wing margin, the last point being the far wing base, or the last entered point along the wing margin, if the far wing base was not visible. The slope and the z*-intercept of the lines represented by each line segment were calculated. Beginning at the wing tip, the y* value of the point of intersection of the line with slope 5 passing through P7 and each consecutive line segment on the opposite wing margin was determined. If this value of y* was intermediate to the two values of y* of the points defining the line segment in question, this indicated a possible point of intersection P8, with being determined from the equation of the line connecting the two endpoints of the line segment. This procedure was carried out for each consecutive line segment on the opposite wing margin. For a particular point on the near wing margin, there were occasionally multiple points of intersection with the opposite wing margin. In these cases, the corresponding point P8 was taken to be that point least distant in the y*z* plane from the last point of intersection that was found. This procedure for finding the corresponding point on the opposite wing margin was repeated for each point entered on the leading and trailing edges of the near wing.
From the representation of the wing outline in the (x′,y′,z′) system (Fig. 5A), the wing was then rotated to determine the three-dimensional location of chords equally spaced along the wing span. This system involves two sets of rotations from the insect coordinate system, such that the longitudinal axis of the wing is coincident with the x′ axis. The points along the leading and trailing edges corresponding to the chords spaced at 0·1R intervals, where R is wing length, along the wing span were then determined. For each such chord, the two closest data points D and E along the leading edge and F and G on the trailing edge of the wing were found (Fig. 5B). They′ and z′ values of the two points of intersection of the chord with the leading and trailing edges were then determined by linear interpolation between the endpoints of the line segments and , respectively. The nine pairs of span-wise points were then retransformed to the (x′,y′,z′) system.
Errors in the determination of the span-wise angle of attack were investigated by testing the analysis procedure on outlines of wing pairs drawn on a digital plotter (Houston Instruments DMP-6). The outlines were of approximately the same size as the magnified film image. The orientation of the insect relative to the viewing system was approximately the same as that in the high-speed filming. The principal variables were investigated for the possible introduction of errors: positional angle of the wing, span-wise position of the chord, and projected chord length. Figures were drawn with constant span-wise angle of attack but differing positional angles of the wing, varying span-wise chord but constant span-wise angle of attack, and variable angle of attack along a wing of constant chord length. Typically 10 points were digitized along each wing margin, and the procedure was repeated five times to ascertain the precision of the analysis procedure. In general, the computer program returned the actual angle of attack to within 5°, with values being repeatable to within 3°. Systematic errors were least when the projected chord was smallest, and when chord lengths were relatively small. Therefore, no direct correlation existed between angle of attack or positional angle per se and the systematic errors associated with measuring the angle of attack. The sensitivity of the analysis program to errors in the calculated values of the direction cosines and the maximum projected wing length was also examined. Errors of up to 5 % in these parameters introduced errors in the measurement of angle of attack that were within the general limits of the systematic errors imposed by the procedure.
To carry out an aerodynamic and energetic analysis of forward flight, it was necessary to determine a number of morphological parameters for those insects previously filmed. A detailed description of those morphological parameters relevant to flight has been given in Ellington (1984a). In particular, he has drawn attention to the significance of the moments of wing mass and wing area about the wing base, which reflect the distribution of mass and area along the wing span. Such moments, and their associated radii, are of immediate mechanical significance and, moreover, serve to characterize quantitatively the shape of the wing. A quasi-steady aerodynamic analysis of hovering flight (Weis-Fogh, 1973) demonstrated that the mean lift force and mean profile power are proportional to the second and third moments of wing area, respectively. Inertial forces acting on the wings are proportional to the first moment of mass; the associated radius through which the resultant force acts is the centre of mass. The second moment of mass is equivalent to the moment of inertia, its associated radius being the radius of gyration. Moments and radii analogous to those of wing mass can also be derived for the wing virtual mass, the mass of air accelerated and decelerated by the motion of the wing. Moments and radii of moments for wing area, mass and virtual mass will figure prominently in later calculations of lift and power requirements. The current morphological study follows for the most part Ellington’s previously published methodology, which may be consulted for a more detailed description of the techniques and mathematical calculations involved.
The total mass m of the insect was first measured to ±0·1mg, and both wing pairs were then cut from the body. Wing length R and body length lb were measured to within ±0·1 mm using Vernier calipers. The non-dimensional body length is given by the ratio of lb to R. The legs were then cut from the body, and the thorax was weighed. The ratio of thoracic mass to body mass is designated . For the insects filmed in free flight, gross morphological parameters as well as the aspect ratio of the wings and the wing loading pw are given in Table 1. The aspect ratio is the ratio of the wing span squared to the wing area, and the wing loading is the ratio of body weight to wing area.
General flight performance
Bumblebees in the working section of the tunnel usually took off and flew upwards within seconds of the ultraviolet light being turned on. The strength of this response could be varied by adjusting the light intensity. If the striped drums were not rotating, the flight was quite erratic with no consistent orientation, the insect frequently flying into the sides of the working section. When the drums were rotating, however, insects generally took 1–2 s to ‘lock on’ to the moving stripes and begin steady forward flight. In such flight, the insect rarely moved more than three body lengths from some mean position in the working section. Control of flight was excellent, and only rarely was steady flight interrupted by periods of instability from which the insect generally recovered within 1 s. Continuous steady flights sometimes lasted up to 1·5min. Surprisingly, no strong correlation was observed between the rate of rotation of the striped drums and the flight velocity of the insect; flight at 4·5 ms−1 could sometimes be maintained at a very low drum rotation rate. At a particular flight speed, however, it was possible suddenly to increase the drum speed, and thereby elicit short-term movement of the insect forward in the tunnel. Simply by reversing the apparent stripe motion on the drums relative to the direction of the wind, it was also possible to induce backwards flight over the entire speed range. Bumblebees upon occasion exhibited remarkable feats of maneuverability. In one instance a bumblebee worker took off backwards from the back netting of the working section, flew backwards for several seconds, then rotated precisely about the vertical axis by 180°, and flew forwards. This feat was accomplished at an airspeed of 4·5 ms−1.
When the lights for filming were turned on, insects generally continued flying for up to Is before becoming disoriented by the intense lights and subsequently landing. Visual observation indicated that flight in the fraction of a second the insects were actually being filmed was stable, unaccelerated flight. Only those films which showed properly oriented flight, as shown by minimal roll from the vertical and yaw from the oncoming wind, were chosen for the kinematic analysis. Three series of bumblebee films were analyzed in detail. Each series consisted of four films of a particular bumblebee flying at the three aforementioned forward airspeeds and while hovering. For one insect, the bumblebee queen (BB03), no good quality flight was observed at 4·5 m s−1, and this speed is therefore omitted in the results for this insect.
Table 4 shows for each analyzed film the roll from vertical and yaw from the tunnel of the insect during the analyzed wingbeat. The yaw value in hovering flight is, of course, of no particular importance. Also given in Table 4 is the advance ratio, J, at each flight speed. The advance ratio is the ratio of the flight velocity to the mean flapping velocity of the wings, and can be defined as (Ellington, 1984b):
Wing tip and body kinematics
Fig. 6 shows the variation with airspeed of the body angle χ and the stroke plane angle β. Body angle invariably decreased with forward airspeed; the stroke plane concomitantly increased, with the angle between the stroke plane and the longitudinal body axis remaining approximately constant for any particular insect. Fig. 7 shows the variation with airspeed of the following kinematic parameters: the minimum positional angle, ϕmin, mean positional angle, , maximum positional angle, ϕmax, stroke amplitude, ϕ (=ϕmax–ϕmin), and the wingbeat frequency, n. There was no statistically significant change with airspeed in minimum positional angle, mean positional angle, maximal positional angle or stroke amplitude; slopes of linear regressions relating these quantities to airspeed were in no case significantly different from zero (P>0·05). Similarly, no consistent trend characterized the relationship between wingbeat frequency and airspeed (P>0·05). Frequency measurements were also made on a large number of films that were not analyzed for detailed kinematics; wingbeat frequency was independent of air speed, for insects ranging in body mass from 150 to 595 mg. No consistent relationship was found between wingbeat frequency and body mass at any forward airspeed or for hovering flight. Insects of a similar mass often differed in wingbeat frequency by 20–30 Hz.
Detailed wing tip kinematics
Detailed wing tip kinematics for each airspeed and hovering flight are given in Figs 8–10. On the right side of each figure is depicted the variation of the positional angle ϕ with non-dimensional time (=nt); the wingbeat frequency is also given. The open circle on each graph corresponds to the frame of maximum projected length, where errors in the projection analysis will be greatest. The diagrams on the left-hand side of Figs 8–10 depict the wing tip path relative to the stroke plane, the wing base being indicated by a cross. The positional angle ϕ equals 0° at the wing base, and ±90° at the tips of the large arrowheads. The elevational angle θ can be read off an identically scaled axis perpendicular to the stroke plane and passing through the wing base. The small arrowheads indicate the separate paths of the downstroke and upstroke. For each wing tip path, the stroke plane angle, β, stroke amplitude, Φ, and mean positional angle, , are also given in degrees.
From the graphs of positional angle versus non-dimensional time, non-dimensional angular velocities were calculated according to Ellington (1984b). Root mean square (rms) and cube root mean cube (crmc) values were calculated for each wingbeat. No relationship was found between the rms or crmc values and forward airspeed (r typically 0·1, P>0·05). Additionally, no relationship was found between forward airspeed and the ratio of the rms value during the downstroke to that during the upstroke (r= -0·21, P>0·05). Values of rms and crmc were only 1–2 % smaller than the values characterizing simple harmonic motion. Corresponding estimates of quasi-steady aerodynamic forces and mean profile power are about 5% lower than estimates based on the assumption of simple harmonic motion for the wings. Durations of the downstroke and of the upstroke were determined from the graphs of positional angle versus non-dimensional time; no correlation was found between the ratio of the downstroke to upstroke and forward airspeed (r = -0·28, P>0·05). This ratio averaged 1·06 (S.D. 0·08) for the bumblebee sequences analyzed.
Wing tip paths generally were open loops with the downstroke dorsal to the upstroke. Several sequences showed single crossovers, either in the middle or at the end of the wingbeat. No double crossovers were observed. Curves were drawn by eye through the plots of positional angle versus elevational angle; mean values of the elevational angle were calculated for the downstroke and upstroke. The difference between these values, , was determined, as was the maximum difference |Δθmax| between the mean elevational angle of the downstroke and that of the upstroke at any particular value of ϕ. The two parameters and |Δθmax| give an indication of the divergence of the wing tip path from the stroke plane. Both the quantities and |Δθmax| tended to be positively correlated with airspeed (r = 0·06, P<0·05). The maximum observed value of |Δθmax| was approximately 16°, indicating that, in spite of the aforementioned trend of a more dorsal downstroke, wing motion is generally confined to the stroke plane at all airspeeds.
Span-wise angle of attack
Figs 11–13 depict geometrical angles of attack a relative to the stroke plane as a function of non-dimensional time for the analyzed bumblebee sequences. Although the initial analysis of angle of attack determined chord angles relative to horizontal, angles of attack relative to the stroke plane are, in general, more informative and are presented here. Use of the phrase ‘angle of attack’ henceforth refers to the geometrical angle of attack. Values of a are given for three consecutive sections along the wing span, and represent the mean angle of the three chords comprising each section. Each data point is the mean of three separate determinations from a film image of the span-wise angle of attack. The analysis program proved to be particularly inaccurate when the margins of opposite wings were parallel to one another; the determination of corresponding points on the wing margins was then very sensitive to digitizing errors. Hence frames in the middle of a half-stroke (ϕ near zero) were difficult to analyze, and most frames analyzed were near the onset or finish of rotation at the ends of half strokes. Particularly near the wing base, the resolution of the wing margins was poor, and angles of attack for the proximal section are therefore not as accurate as those for the two more distal sections.
Angles of attack during both the downstroke and the upstroke tended to increase with increased airspeed, although there were a few exceptions to this trend. Angles of attack were generally constant to within 15° during the periods between wing rotation, although, particularly during the upstroke, a number of sequences displayed continuously increasing or continuously decreasing angles of attack. During the downstroke, bumblebee wings were typically twisted along the wing span by 10–20°, approximately the same amount as is observed under static conditions. Wings were cambered during the downstroke, and for flight at the higher airspeeds remained so during the upstroke; the extent of wing cambering during the upstroke for flight at 1ms−1 and for hovering flight was difficult to ascertain.
At the ends of half-strokes the wings increased their angle of attack, rotating about the longitudinal axis to the angle of attack characteristic of the following half-stroke. Nose-down rotation of the wing is pronation, whereas nose-up rotation is supination. The exact timing of wing rotation was difficult to determine. Particularly at the end of either pronation or supination the termination of wing rotation was difficult to specify precisely, and the time chosen to indicate the end of wing rotation was that point after which no more elastic recoil was apparent. Similarly, the onset of wing rotation usually occurred between film frames, with consequent loss of temporal resolution. Measurements of the timing of wing rotations can therefore only be regarded as approximate. The durations of pronation and supination were approximately equal, averaging about 22 % of the wingbeat period. No relationship was found between forward airspeed and the duration of pronation or supination (r typically –0·35, P>0·05), and no consistent relationship was found between the onset of pronation or supination and the top or bottom, respectively, of the wingbeat (r typically –0·2, P > 0·05).
Mean angles of wing rotation at the ends of half-strokes were determined by taking the mean of the angle of rotation for the nine equally spaced span-wise chords. These angles tended to decrease with increased airspeed for both pronation (r=–0·67, P<0·05) and supination (r=—0·75, P<0·05). A non-dimensional form of wing rotation , equal to the mean angular velocity of rotation divided by the wingbeat frequency, represents the angle through which the wing would rotate over an entire wingbeat (Ellington, 1984/?). The variation with airspeed of this quantity is given in Fig. 14; for both pronation and supination, tended to decrease with increased airspeed (pronation: r= –0·50, P<0·05; supination: r= –0·76, P<0-05).
During wing rotation, significant changes in wing profile were observed. The anterior region of the wing first began to rotate as a unit, and was followed by rotation of the posterior region, as described by Ellington (1984/?) for insects in hovering flight. Rotation of the posterior region was always manifested as a torsional wave moving towards the wing base and never towards the wing tip. Wings usually rotated past the angle of attack characterizing the following half stroke, and then recoiled back to this angle. Profile flexion of the wings was most pronounced in hovering flight, and decreased progressively as flight speed increased; recoil of the wing after rotation was similarly reduced at higher airspeeds.
Forward flight in animals requires, in addition to the support of body weight, the generation of thrust to overcome body drag. Thrust is the net horizontal component of the total forces generated by the wing during the wingbeat. As forward airspeed increases, the magnitude of this horizontal component must increase to offset the additional body drag, while the net vertical force produced must continue to equal the body mass (less any lift on the body). Typically, in flying insects the ratio of drag to body weight is small, even at high airspeeds, and the resultant force vector tilts forwards only by several degrees. This small change can be contrasted with the large change in body angle and stroke plane angle over the same airspeed range. A strong correlation between flight speed and body angle has been observed in flying insects (Drosophila’. Vogel, 1966; Götz, 1968; David, 1979; honeybee: Nachtigall et al. 1971; Esch et al. 1975; bumblebees: present study) and birds (Brown, 1953; Greenewalt, 1960; Pennycuick, 1968; Rüppell, 1971). The angle between the stroke plane and the longitudinal body axis is relatively constant, and the gradual decrease in body angle with forward airspeed is paralleled by a concomitant increase in stroke plane angle (Ellington, 1984b).
What are the consequences for the magnitude and direction of aerodynamic forces of the increase in airspeed and the tilt of the stroke plane? By using the parameter of the advance ratio J (= V/2ΦnR), it is possible to evaluate the consequences for forward flight. For acute stroke plane angles, an increase in the advance ratio will result in the forces generated by the wing during the downstroke and the upstroke becoming increasingly asymmetrical (Ellington, 1984b). The relative velocity during the downstroke becomes greater than that during the upstroke, with a concomitant increase in the magnitude of aerodynamic forces, and the downstroke becomes progressively more horizontal while the upstroke becomes more vertical, with a corresponding change in the direction of the resultant aerodynamic force. These changes are mitigated only partially by the increase in stroke plane angle, which tends to decrease the velocity differences between the half-strokes. In bats and birds the asymmetry between half-strokes can be enhanced by changes in wing profile which considerably reduce the magnitude of aerodynamic forces produced during the upstroke. Insects, however, cannot drastically alter wing shape during the wingbeat, apart from transverse flexion at the ends of half-strokes and changes in wing camber.
Because the advance ratio is inversely proportional to the wingbeat frequency and to the stroke amplitude, variation in these kinematic parameters will alter the magnitude and direction of aerodynamic forces produced by the wings. An increase in either the wingbeat frequency or the stroke amplitude while the other parameter is constant will increase the flapping velocity of the wings, with consequent changes in the relative velocity experienced by a wing section. Because lift and drag forces are, in general, proportional to the square of the relative velocity, the magnitudes of the resulting aerodynamic forces will change disproportionately. For this reason, considerable attention has been given to the variation with airspeed of the wingbeat frequency and the stroke amplitude.
For the buff-tailed bumblebees of the present study, no systematic change in wingbeat frequency with airspeed was observed. For various insects, bats and birds, a number of studies have demonstrated alternatively an increase, no change or a decrease in wingbeat frequency as forward airspeed increases. Sotovalta (1947) reported for a hoverfly (Eristalis arbustorum) that wingbeat frequency in free flight increased approximately from 260 Hz to 275 Hz over a flight speed range of hovering to 5 ms−1, while for Calliphora the increase was from 120Hz to 210Hz over a range of 0·5 to 5 m s−1. In free-flying Locusta migratoria, there is a positive correlation between flight speed and wingbeat frequency, albeit with considerable scatter in the data (Baker et al. 1981). This relationship has also been found in tethered migratory locusts (Gewecke, 1975). In the long-eared bat Plecotus auritus wingbeat frequencies showed a small decrease over the speed range 2·3ms−1 (Norberg, 1976), while for the bat Myotis lucifugus wingbeat frequency decreased from approximately 16 Hz at a flight speed of 2 m s−1 to a value of 10 Hz at 7 m s−1 (Schnitzler, 1971). Carpenter (1985) found that in the flying fox Pteropus poliocephalus the wingbeat frequency decreased by a factor of 2 over the range 3–7 ms−1, and then increased at higher airspeeds. Aldridge (1986) showed for the greater horseshoe bat Rhinolophus ferrumequinum that wingbeat frequency decreased with increased airspeed. Tucker (1966) found for a free-flying budgerigar Melopsittacus undulatus that wingbeat frequency varied by less than 6 % from the mean value over a speed range 5–9 ms−1, while Schnell (1974) found for the magnificent frigatebird Fregata magnificens no significant correlation between flight speed and wingbeat frequency over the range 8–14 ms−1. In a study of flight mechanics of Columba livia, Pennycuick (1968) found a decrease in wingbeat frequency as flight speed increased from 7 to 13 m s−1, followed by a slight increase at higher airspeeds. A similar pattern was found for the stroke amplitude, although for both kinematic parameters scatter in the data was considerable. McGahan (1973) found no significant correlation between forward airspeed and wingbeat frequency in the Andean condor Vultur gryphus, but it is not clear whether these measurements were made on the same individuals at different airspeeds. Butler & Woakes (1980) found no relationship between wingbeat frequency and airspeed for barnacle geese (Branta leucopsis)’. wingbeat frequency was approximately constant over a speed range of 16–24ms−1. The variable consequences of changes in airspeed for the wingbeat frequency of flying animals suggests, not surprisingly, that considerable diversity exists in the aerodynamic mechanisms used by animals of widely varying shapes and sizes to effect forward flight.
As with wingbeat frequency, the stroke amplitude of buff-tailed bumblebees in free flight showed no significant change with airspeed. A wide diversity of changes in stroke amplitude with airspeed has been observed in flying animals. Magnan (1934) noted in free-flying insects a general decrease in stroke amplitude with increased airspeed. Gewecke (1967, 1970, 1974) has documented for tethered Calliphora, Locusta and the dragonfly Orthetrum a significant decrease in stroke amplitude with increased airspeed. Hollick (1940) noticed in tethered individuals of the fly Muscina stabulans a decrease in stroke amplitude with increased airspeed, as well as with decreased body inclination relative to horizontal. Urquhart (1960), apparently on the basis of a visual estimate in the field, reported that in cruising flight of the monarch butterfly Danaus plexippus the stroke amplitude was only 30°, but that the amplitude while in ‘speeding flight’ increased to 120°. Sotovalta (1947) suggested that the stroke amplitude in slow forward flight of the sphingid Sphinx ligustri increased from the value in hovering flight, without a significant change in wingbeat frequency. In the long-eared bat Plecotus auritus the stroke amplitude while hovering was 120°, which decreased to 91° at a flight speed of approximately 2·4ms−1 (Norberg, 1976). Reviewing these results, it is clear that no consistent relationship between stroke amplitude and airspeed has been observed for flying animals, and perhaps none should be expected, given again the morphological diversity of the animals being considered.
Given that neither wingbeat frequency nor stroke amplitude vary with airspeed for flying bumblebees, the mechanism by which changes in body angle and stroke plane angle are effected are not clear. An increase in the mean positional angle with airspeed will move the center of lift progressively more dorsally, and could therefore provide the nose-down pitching moment necessary to tilt the body, and hence the resultant force vector, to the angles characteristic of forward flight (Ellington, 1984b). Nose-down pitching moments required from the wings to produce the appropriate body angles are at all airspeeds greater than zero (Dudley & Ellington, 1989). No such increase in mean positional angle with airspeed was apparent in this study. It should be noted that pitching moments about the wing base axis, as predicted by a quasi-steady aerodynamic analysis, are not simply proportional to the mean positional angle but are also inversely proportional to the stroke plane angle (Dudley & Ellington, 1990).
The high-speed cinematographic studies described herein of bumblebees in free flight have revealed no systematic variation with airspeed of wingbeat frequency, stroke amplitude or minimum, mean or maximum positional angle over the range of airspeeds from hovering to 4·5 ms−1. Instantaneous values of these kinematic parameters at any given airspeed were, however, found for bumblebees to deviate considerably from mean values for flights of several minutes. If such temporal variation in kinematics is typical of insects in free flight, the kinematic parameters obtained from a single high-speed film covering a very short time period may not necessarily be representative of mean values. Multiple high-speed films of the same insect flying at a constant airspeed would address this issue.
The remaining kinematic parameter to be discussed is the angle of attack of the wing. While drastic changes in wing profile cannot be used to create an asymmetry between the downstroke and upstroke, changing the angle of attack of the wing relative to the stroke plane (Figs 11–13) can be used to control the production of thrust and, thereby, to regulate forward airspeed. Such changes in the angle of attack are easily made by altering the extent of wing rotation at the ends of halfstrokes (Fig. 14), which may be controlled by the activity of only one muscle or muscle group involved in rotation. Because wing orientation changes gradually with airspeed, the increase in asymmetry between the downstroke and upstroke should also be gradual, as will be shown using a quasi-steady aerodynamic analysis for bumblebees in free forward flight (Dudley & Ellington, 1990).
The hypothesis that asymmetry between the downstroke and upstroke, and hence variable production of thrust, results from changes in wing orientation is supported by considering the direction of forces produced on the wing, as determined by the effective angles of incidence. The effective angle of incidence is measured from the relative velocity vector, which is the sum of the forward, flapping and induced velocities. Calculations of the effective angles of incidence for bumblebees in forward flight will be made in the companion paper (Dudley & Ellington, 1990), where it will be shown that effective angles of incidence are independent of airspeed, and do not change significantly from the downstroke to the upstroke. Because the stroke plane angle varies with airspeed, therefore, the direction of forces produced by the downstroke and upstroke must also change as a function of airspeed. In hovering flight with a horizontal stroke plane, the upstroke and downstroke contribute equally to the production of vertical forces, but as airspeed increases the downstroke contributes progressively more and the upstroke less to sustaining the body weight. At higher airspeeds the upstroke functions primarily to generate thrust, which arises from lift forces on the wing.
In conclusion, bumblebees in forward flight use changes in the angle of attack of the wing relative to the stroke plane to generate a force asymmetry between the downstroke and upstroke, and hence to regulate forward airspeed. Effective angles of incidence remain constant during the downstroke and upstroke; because of wing rotation, the sense of wing circulation is reversed. The consequences for force production are a progressive change in the direction of the lift and drag, with the vertical forces increasing in magnitude during the downstroke and the horizontal forces increasing during the upstroke. The gradual change in wing orientation during the downstroke and upstroke corresponds to the gradual change in the asymmetry between half-strokes required to offset the additional body drag at higher airspeeds.
We are indebted to the late Dr Ken Machin for his irreplaceable advice and discussion during the course of this study. Mr Neal Maskell and Mr Dennis Unwin provided invaluable technical expertise. We thank Dr James Mayberry for helpful discussions. The financial assistance of the Marshall Commission, the Smithsonian Tropical Research Institute and the Science and Engineering Research Council is gratefully acknowledged.