Inertial and Aerodynamic Torques on the Wings of Diptera in Flight

The torques generated by insects at the wing base to overcome aerodynamic and inertial forces have been quite widely studied in the context of flight energetics (Weis-Fogh, 1973; Ellington, 1984d). The torques that will be set up all the way along the wing as a consequence, however, have largely been ignored: only Pennycuick (1967) seems to have examined the bending and pitching torques set up in a wing during flight, and only at certain points of the wings of his pigeons.

The bending torques produced in flight will vary both temporally and spatially. Inertial forces will predominate at stroke reversal when acceleration is greatest. Aerodynamic forces, however, will predominate in the middle of each halfstroke during normal horizontal stroke plane hovering (Weis-Fogh, 1973), whether circulation is produced by quasi-steady or rotational mechanisms, when the wing velocity is greatest. Both forces vary along the wing. This paper seeks to investigate the relative magnitudes of the bending torques due to inertia, both of the wing itself and of the virtual mass of air bound to the wing (Vogel, 1962; Ellington, 1984a), and those due to aerodynamic forces, and to examine their distribution along the wing.

Since the centre of mass of insect wings is located behind the torsional axis (Norberg, 1972), acceleration of the wings should also tend to set up inertial pitching torques which would tend to twist the wing. At stroke reversal, however, when these torques would be set up, the inertia instead causes the wings to rotate by over 90° at the torsionally compliant wing base (Ennos, 1987) into the correct orientation for the coming halfstroke. Inertial pitching moments are therefore ignored in this paper. I have shown elsewhere (Ennos, 1988b) that these inertial effects are sufficient to cause the rotational velocities of the wing which are observed at stroke reversal.

The chordwise centre of pressure of insect wings is generally behind the torsional axis (Norberg, 1972), so aerodynamic forces will also produce pitching torques along the wings. During the middle of each halfstroke the wing pitch is more or less constant, showing that the pitching torques produced then are equal to the restoring torque produced by the compliant wing base (Ennos, 1987). The magnitude and distribution of the pitching torques set up by aerodynamic forces at this time were investigated.

The distribution of torques will be very important in determining the wing design: insect wings generally withstand the torques that are produced in flight without noticeable bending, but they are generally twisted nose-down along their length during the downstroke (Wootton, 1981; Ellington, 1984b). A wing of optimal design would be as light as possible, to increase the energetic efficiency of flight, while combining adequate bending and torsional stiffness.

Five species of flies were investigated: the hoverfly Eristalis tenax (L.); the March fly Bibio marci (L.); the bluebottle, Calliphora vicina Robineau-Desvoidy; the conopid fly, Conops strigata Wied, in Meig.; and the crane fly Tipula paludosa Meig.

Information that I had obtained on the morphology and kinematics of free flight of these species (Ennos, 1989) was used to calculate the torque distribution along the wings in flight due to both inertial and aerodynamic forces.

To simplify the calculations, several assumptions were made. Measurements of wing acceleration and velocity from kinematic records would be subject to large errors, so it was assumed that the wings oscillated along the stroke plane with simple harmonic motion. The actual kinematics (Ennos, 1989) were very similar to simple harmonic motion, as were those observed in free flight by Ellington (1984 b) and Dudley (1987). Acceleration at the ends of the beat was usually higher than in simple harmonic motion and peak velocity lower. This would mean that my analysis would underestimate inertial torques and overestimate aerodynamic torques.

To calculate aerodynamic torques it was further assumed that the wings produced equal lift on each stroke and did so using the quasi-steady mechanism, in which the lift produced is proportional to the square of the local velocity. The quasi-steady account of insect flight is increasingly under attack (Ellington, 1984b; Dudley, 1987; Ennos, 1989), but remains the only mechanism yielding information about the spatial and temporal distribution of force on the wing. Furthermore, since the average force produced by the wings is known (lift must balance the weight of the insect in flight), the estimates of the peak force and torque that the quasi-steady assumption yields will be of the correct order of magnitude.

Let us examine the moment set up in a wing section a distance r from the wing base (Fig. 1) by a wing strip of mass dm, centred a distance r_i from the wing base, when the wing is subjected to an angular acceleration of d²ϕ/dt², where ϕ is the instantaneous angular position of the wing along the stroke plane (Ennos, 1989). The acceleration, a, of the strip is:

The maximum inertial bending torque, T_Imax, is the sum of that due to mass and that due to virtual mass and was calculated using kinematic data from Ennos (1989) (Table 1).

It can be seen that the moment of inertia of the wing virtual mass is typically only slightly less than that of the wing mass, even though the wing virtual mass is generally less than half the wing mass. This is because the wing mass is concentrated nearer the base than the virtual mass, and (m) is consequently lower than (v). Fig. 2 is an example of the raw data from which the moments were calculated, showing the masses and virtual masses of wing strips for the hoverfly, Eristalis tenax Et.l. The concentration of wing mass at the base can clearly be seen.

The spanwise distribution of inertial bending torque was very similar in all the. species studied, so the results for the hoverfly sequence Et. la alone are shown (Fig. 3A). Bending torque falls extremely rapidly away from the wing base.

During each halfstroke the wing must also withstand the bending torques caused by the aerodynamic forces it produces: lift and drag. Using quasi-steady aerodynamics, and assuming that equal lift is produced during symmetrical hovering on the two halfstrokes, and that the wings are flapped with simple harmonic motion, these torques can be estimated.

The peak velocity in simple harmonic motion is √2 times the average velocity. Since lift is proportional to the square of velocity, the peak lift force, produced at midstroke, will be about twice the average lift. To hover, therefore, each wing will produce a peak lift force equal to the weight of the insect, mg. As aerofoils move through the air at relative angles of attack of less than 20°, this lift will act nearly at right angles to the wing chord, whereas the drag produced, generally a much smaller force, will act nearly parallel to the chord. Drag may therefore be Conveniently ignored when calculating the peak force perpendicular to the wing chord, which will be approximately equal to the peak lift.

The spanwise torque distribution will be similar to that produced by inertial forces. More force will be produced by the faster-moving distal parts of the wing, just as they possess more inertia about the wing base. The moment will decrease rapidly away from the wing base for the same reasons: the smaller fraction of the wing outboard of the section, and the shorter moment arm of each force to the section.

The approximate torque distribution is shown in Fig. 3A for the sequence Et.la of the hovering flight of the drone fly Eristalis tenax (Ennos, 1989).

The aerodynamic forces produced by a wing section at the Reynolds numbers characteristic of insect flight are centred approximately one-quarter of the chord, c/4, back from the leading edge (Norberg, 1972; Ellington, 1984c). The torsional axis of the wing, along which forces will not cause pitch changes in the wing (Norberg, 1972), however, is usually closer to the leading edge. In the flies that I examined (Ennos, 1988b), it was typically only one-eighth of the wing chord, c/8, behind the leading edge. Aerodynamic forces will therefore act behind the torsional axis and tend to produce a nose-down pitching moment on it.

The spanwise distribution of this torque will be quite different from that of the bending torques, since the moment arm, c/8, of each section remains high until very near the tip, and does not decrease away from the base as does the moment arm for bending. Aerodynamic force production is greatest at the fast-moving wing tip during hovering, and falls to zero at the base, so most of the aerodynamic pitching torque about the torsional axis is set up near the wing tip. The torque will therefore decrease only gradually away from the base (Fig. 3B).

Bending torques on insect wings decrease rapidly away from the base, values halfway along the wing being typically only about one-quarter of those at the base, and falling to zero at the tip. It is not surprising, therefore, that the wings of insects are thicker at the base than at the tip. Mass is concentrated near the base to resist the higher bending moments, and outer regions are as light as possible, to reduce the stress caused at the wing base and to reduce the energy expenditure involved in accelerating the wing, although much of the wing’s energy is recovered in the flight muscles and thoracic box (Machin & Pringle, 1959; Ellington, 1984d). Torsional moments stay at a high value until near the wing tip, but are far smaller than bending moments.

The surprising aspect of the behaviour of the wings is that they twist but do not noticeably bend in flight, despite the fact that the bending torques on them are far larger than the pitching torques which tend to twist them by around 10° during each wingbeat. This is due to their open, corrugated structure (Ennos, 1988a), which combines high bending stiffness with low torsional stiffness.

The pitch of the wings also changes greatly between strokes. The very high torsional compliance of the wing base (Ennos, 1987) allows the inertia of the wing to cause rotation of over 90° at stroke reversal. The aerodynamic pitching torque can then maintain it at the correct pitch attitude during the next halfstroke.

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