1. On the assumption that steady-state aerodynamics applies, simple analytical expressions are derived for the average lift coefficient, Reynolds number, the aerodynamic power, the moment of inertia of the wing mass and the dynamic efficiency in animals which perform normal hovering with horizontally beating wings.

2. The majority of hovering animals, including large lamellicorn beetles and sphingid moths, depend mainly on normal aerofoil action. However, in some groups with wing loading less than 10 N m-2 (1 kgf m-2), non-steady aerodynamics must play a major role, namely in very small insects at low Reynolds number, in true hover-flies (Syrphinae), in large dragonflies (Odonata) and in many butterflies (Lepidoptera Rhopalocera).

3. The specific aerodynamic power ranges between 1.3 and 4.7 WN-1 (11-40 cal h-1 gf-1) but power output does not vary systematically with size, inter alia because the lift/drag ratio deteriorates at low Reynolds number.

4. Comparisons between metabolic rate, aerodynamic power and dynamic efficiency show that the majority of insects require and depend upon an effective elastic system in the thorax which counteracts the bending moments caused by wing inertia.

5. The free flight of a very small chalcid wasp Encarsia formosa has been analysed by means of slow-motion films. At this low Reynolds number (10-20), the high lift co-efficient of 2 or 3 is not possible with steady-state aerodynamics and the wasp must depend almost entirely on non-steady flow patterns.

6. The wings of Encarsia are moved almost horizontally during hovering, the body being vertical, and there are three unusual phases in the wing stroke: the clap, the fling and the flip. In the clap the wings are brought together at the top of the morphological upstroke. In the fling, which is a pronation at the beginning of the morphological downstroke, the opposed wings are flung open like a book, hinging about their posterior margins. In the flip, which is a supination at the beginning of the morphological upstroke, the wings are rapidly twisted through about 180°.

7. The fling is a hitherto undescribed mechanism for creating lift and for setting up the appropriate circulation over the wing in anticipation of the downstroke. In the case of Encarsia the calculated and observed wing velocities at which lift equals body weight are in agreement, and lift is produced almost instantaneously from the beginning of the downstroke and without any Wagner effect. The fling mechanism seems to be involved in the normal flight of butterflies and possibly of Drosophila and other small insects. Dimensional and other considerations show that it could be a useful mechanism in birds and bats during take-off and in emergencies.

8. The flip is also believed to be a means of setting up an appropriate circulation around the wing, which has hitherto escaped attention; but its operation is less well understood. It is not confined to Encarsia but operates in other insects, not only at the beginning of the upstroke (supination) but also at the beginning of the downstroke where a flip (pronation) replaces the clap and fling of Encarsia. A study of freely flying hover-flies strongly indicates that the Syrphinae (and Odonata) depend almost entirely upon the flip mechanism when hovering. In the case of these insects a transient circulation is presumed to be set up before the translation of the wing through the air, by the rapid pronation (or supination) which affects the stiff anterior margin before the soft posterior portions of the wing. In the flip mechanism vortices of opposite sense must be shed, and a Wagner effect must be present.

9. In some hovering insects the wing twistings occur so rapidly that the speed of propagation of the elastic torsional wave from base to tip plays a significant role and appears to introduce beneficial effects.

10. Non-steady periods, particularly flip effects, are present in all flapping animals and they will modify and become superimposed upon the steady-state pattern as described by the mathematical model presented here. However, the accumulated evidence indicates that the majority of hovering animals conform reasonably well with that model.

11. Many new types of analysis are indicated in the text and are now open for future theoretical and experimental research.

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