The Inertial Cause of Wing Rotation in Diptera

The kinematics of flight in Diptera has been extensively studied (Nachtigall, 1966,1981; Vogel, 1966; Weis-Fogh, 1973; Ellington, 1984; Dudley, 1987), and the basic pattern of wing movements is well known. During the two half-strokes the wings are moved through the air along a stroke plane at a more or less constant angle of pitch: about 30-45° from the stroke plane. At the ends of each half-stroke the wings rotate quickly by around 90° to the orientation for the coming stroke. The trailing edge sweeps under the leading edge, until the wing has rotated a few degrees past the pitch angle for the coming stroke, after which it recoils to this position (Ellington, 1984). The wing does not rotate as a flat plate. Anterior and distal areas of the wing rotate first and then a tip to base ‘torsion wave’ runs along the wing until rotation is complete (Fig. 1).

It has generally been assumed that torque is actively applied by wing base sclerites (Boettiger & Furshpan, 1952; Pfau, 1985; Miyan & Ewing, 1985) which rotate the leading edge of the wing. The torsion is then transmitted quickly along the stiff leading edge to the wing tip, and from there the torsion travels more slowly through the more compliant rear wing region to the base (Wootton, 1981; Ellington, 1984).

I have found, however, that in flies the base of the leading edge is very compliant to torsion (Ennos, 1987, 1988). Torsion applied at the base would not be transmitted immediately to the outer wing. It therefore seems unlikely that control of the wing can be effected in this manner.

In his study on the function of the pterostigma in dragonflies, Norberg (1972) demonstrated the importance of inertial effects on wing rotation. He showed that wing flutter could be caused in gliding flight if the centre of mass of the wing was too far behind its torsional axis, and suggested that wing inertia could control pitch angles at stroke reversals during flapping flight, though neuromuscular control would also operate in many insects.

In this paper I hope to show that no active twisting need be invoked in the case of the Diptera and that the rotational velocity of the wing at stroke reversal and its associated pattern of deformation can both be explained by inertial effects alone.

In a wing flapping with a sinusoidal motion, the torques which cause angular acceleration will be constantly changing. These torques may have several causes. (1) Inertia. The centre of mass of an insect wing is often behind its torsional axis (Norberg, 1972). If such a wing is accelerated the inertia of the wing will cause torque to be set up around this axis, and this should rise to a maximum at stroke reversal, when acceleration is high. (2) Aerodynamic. The centre of pressure of aerodynamic forces tends to act behind the torsional axis (Norberg, 1972, 1973; Wootton, 1981). Aerodynamic force will therefore also tend to produce torque, which should rise to a maximum at the middle of each wingbeat, when the wing velocity is greatest. (3) Elastic recoil. Elastic recoil of the wing will produce a torque which tends to return it to its rest position. (4) Active torsion. The pattern of active torsion is unknown.

A complex differential equation could be built up of the angular wing motion, with estimates of the magnitudes of inertial, aerodynamic and elastic torques and the motion of the wing. The expected rotational velocities could then conceivably be computed numerically. Such an approach, however, as well as being lengthy and possibly inaccurate, does not give an intuitive feel of the problem.

Instead, I have developed a simplified analysis which is based on the principle of conservation of angular momentum. In the middle of a half-stroke there is no rotation in the pitching plane, but the wing has linear momentum which is centred off the torsional axis (Fig. 2A). It thus possesses angular momentum about the axis. When the wing is decelerated at the end of the beat this angular momentum will remain and cause rotation in the pitching plane. The velocity of rotation is calculated when the torsional axis has been brought to rest and hence in the middle of stroke reversal, halfway through wing rotation. As the wing accelerates into the coming stroke the rotational velocity should become even greater, but further rotation is prevented by mechanical stops on the wing base (Ennos, 1987) and the pitch angle becomes set for the coming half-stroke.

Given knowledge of the mass distribution of the wing and the position of the torsional axis, the resulting angular velocity can be estimated and compared with values from high-speed films.

The virtual mass of the wing is also likely to affect the rotation of the wing. Wing virtual mass, however, is only around a third of the wing mass. If represented by a cylinder of air around the wing, the centre of virtual mass will also be behind the torsional axis of the wing, at the mid-chord. It should therefore produce an effect similar to the wing mass on wing rotation. For these reasons, and also because the actual distribution of the virtual mass is unknown, it is ignored in these calculations.

Specimens of the hoverfly, Eristalis tenax, and the bluebottle, Calliphora vicina were killed with ethyl acetate vapour and the torsional axis of a wing was found by pushing it with the point of a mounted needle. The centre of torsion was identified as the line along which a force applied to the wing did not result in either pronation or supination (Norberg, 1972).

The wing was then removed at the base and weighed intact on a Mettler UM3 microbalance, accurate to within 0·1 μg. It was then placed between two microscope slides and inserted into a photographic enlarger which gave an image magnified by 10 times, and an outline drawing of the wing was made on graph paper. The wing was then cut into rectangular elements 2 mm long and 1 mm broad from the base of the leading edge and each element was weighed. The rest of the wing was kept between the microscope slides to reduce desiccation and consequent weight loss. Typically weight losses were of the order of 10% of wing mass. Element masses were multiplied by the ratio of wing mass to total strip mass to give corrected figures.

The centre of mass of each element was then estimated from the wing outline, and the wing moment of inertia, y2dm and the value of xydm about the torsional axis were calculated. Wing length and area were found using the magnified image and hence the aspect ratio was calculated.

Error in identifying the exact position of the torsional axis was not critical. If the torsional axis was actually further forward than was assumed, the moment arm to the centre of the wing strips would be underestimated, but so would the moment of inertia of the wing. As a result there would be little error in the calculated angular velocity.

The results of the measurements on individual wings are given in Table 1 and the mass distribution and torsional axis of the wing of the hoverfly Et.2 are shown as an example of the raw data in Fig. 3. In each case the torsional axis of the wing seemed to be along the radial vein, 0·5mm, or about 15%, of the wing chord behind the leading edge, whereas the centre of wing mass was about 1 mm, or 30% of the wing chord, behind the leading edge. The calculated values for the relative magnitudes of rotational to flapping velocity are also given in the table. Values range from 0·67 to 0·76 in Eristalis tenax and from 0·66 to 0·86 in Calliphora vicina. Values observed in films range from 0·70 to 0·90 in Ellington’s (1984) films of Eristalis and 0·58 to 0·79 in my own; values I obtained for Calliphora ranged from 0·61 to 1·02. The calculated values fall neatly into the observed range of velocity ratios. Clearly the inertia of the wing is adequate to cause the observed velocities of wing rotation. No active twisting of the wing base need be invoked.

Though wing inertia can explain pitch alteration at stroke reversal it is clear that the insects must exert some control, since the timing of wing rotation can be altered in flight (Ellington, 1984).

I thank Dr R. J. Wootton for his help and criticism during the work and in the production of the manuscript. The work was carried out while under tenure of an SERC research studentship.