Forward flight of birds revisited. Part 2: short-term dynamic stability and trim
Thrust generation by flapping is accompanied by alternating pitching moment. On the down-stroke, it pitches the bird down when the wings are above its centre of gravity and up when they are below; on the up-stroke, the directions reverse. Because the thrust depends not only on the flapping characteristics but also on the angle of attack of the bird's body, interaction between the flapping and body motions may incite a resonance that is similar to the one that causes the swinging of a swing. In fact, it is shown that the equation governing the motion of the bird's body in flapping flight resembles the equation governing the motion of a pendulum with periodically changing length. Large flapping amplitude, low flapping frequency, and excessive tilt of the flapping plane may incite the resonance; coordinated fore–aft motion, that uses the lift to cancel out the moment generated by the thrust, suppresses it. It is probably incited by the tumbler pigeon in its remarkable display of aerobatics. The fore–aft motion that cancels the pitching moment makes the wing tip draw a figure of eight relative to the bird's body when the wings are un-swept, and a ring when the wings are swept back and fold during the upstroke.
2. Introduction
At a first glance, the dynamic stability in flapping flight should not be any different from the dynamic stability in non-flapping flight. After all, the periodic lift, thrust and pitching moment generated by the wings can be viewed as periodic perturbations to the nominal (non-flapping) state. If the latter is stable, the bird is stable, flapping or not. Of course, there can be an interaction between flapping and rigid-body natural modes, but if the flapping frequency is large as compared with the rigid-body natural frequencies, the bird should hardly be affected by flapping at all. Essentially, this was the conclusion of Taylor & Thomas [1].
At a second glance, however, things are much more complicated. Thrust generation by flapping is indeed accompanied by periodic pitching moment: on the down-stroke, it pitches the bird down when the wings are above its centre of gravity, and up, when they are below; on the up-stroke, drag replaces thrust, and hence these directions reverse. The problem is that both the thrust and the drag depend on the angle of attack, and hence an interaction between the motion of the bird's body and flapping may incite a (parametric) resonance, similar to the one reported by Taylor & Zbikowski [2] for desert locust (Schistocerca gregaria). Analysis of this resonance is the first objective of this study.
If flapping can incite a resonance, the bird needs an active control that operates on the time scale of the flapping period. In principle, active control can be furnished by moving the tail up and down, or by twisting, cambering and sweeping the wings. In forward flight, the tail is closed, and hence cannot be used for control. Periodic wing twist, proportional to the flapping rate, was shown in Part 1 [3] to be the key element in the effective production of thrust. It is therefore unlikely that the periodic twist is also used to control the pitch. We did not find any account on appreciable variations of camber on birds' wings during the flapping cycle—bats are not addressed in this study. It leaves the fore–aft sweeping motion of the wing to fill the function of the primary active control.
Sweeping motion is manifested in the intricate trajectories drawn by a wing tip during the flapping cycle [4]. These trajectories change among species and change with flight conditions. In some cases, they look like an oblique figure of eight; in other cases, they look like a deformed ring, with up-stroke trajectory passing aft of the down-stroke one. A posteriori, our obtaining similar trajectories by simply enslaving the sweeping motion to keep the pitch attitude or the angle of attack makes the conclusion that the sweeping motion is used as the primary control in flight plausible. Trim analysis in the flapping flight is the second objective of this study.
Generic equations of motion can be found in any textbook on flight mechanics [5]. In order to formulate them in explicit form, one needs a model relating the aerodynamic forces generated by the wings with parameters characterizing the flapping and body motions. For this study, we have used the model developed in Part 1 [3]. It has the advantage of furnishing the aerodynamic forces in closed analytical form, and it was shown in Part 1 to be sufficiently accurate to capture the correct behaviour of these forces during the flapping cycle. Since this model is central for this study, we implicitly adopt all the notations and the assumptions of Part 1. For completeness of this presentation, they are briefly recapitulated in the next section.
The rest of the manuscript is organized as follows: equations of motion are derived in §4; the aerodynamic derivatives underlying them are explicated in appendix A. The problem of the short-term dynamic stability is addressed in §5; mathematical details underlying the analysis are found in appendix B. Sweeping motion of the wings is studied in §§6–8. Section 9 concludes the paper. Data of all numerical examples shown in this study are concentrated in appendix C. Morphological data of birds that were used to estimate the range of coefficients in the equations of motion is concentrated in appendix D.
3. Preliminaries
As already mentioned, we adopt the notation, the assumptions and the aerodynamic model of Part 1 [3] practically ‘as is’. A bird to be considered here has a pair of identical wings; the length of a single wing (semi-span) is s; its area is S; the aspect ratio (of the two wings together) is A=2s2/S. The average flight velocity is v; the density of the air in which the bird flies is ρ; the acceleration of gravity is g. s, v, s/v, v/s, ρSs, ρSs3, ρsv2, ρSv2 and ρSv2s will serve as convenient units of length, velocity, time, frequency, mass, moment of inertia, force per unit span, force and moment, respectively. Note that although S is half the quantity commonly used as the wing area, the unit of force is standard. Use of dimensionless quantities will be implicitly understood hereafter. Should a dimensional quantity be required (other than ρ, v, g, s and S), it will be marked by an asterisk. A list of nomenclature can be found in table 1.
The reduced mass of the bird is m (this is the single significant deviation from the notation of Part 1, where m was dimensional, rather than reduced, mass). For the sake of simplicity, the mass of the wings is neglected. The centre of gravity is located at the distance xcg aft of the quarter-chord at the root, and at the same height. The reduced moment of inertia about the centre of gravity is
where ry is the respective (reduced) radius of gyration.
The bird's body is allowed to pitch and heave; relative to the body, the wings are allowed to flap, sweep fore and aft, and twist. It is assumed that the wing twists in such a way that its sections do not deform and remain parallel to each other; moreover, the twist axis crosses all sections at their respective quarter-chord points and remains straight at all times. The sweep angle of the twist axis is λ (positive aft), flapping angle is ϕ (positive down); the twist angle is αg (positive for leading edge up); pitch angle is τ (positive for nose up); vertical translation of the bird's body (and hence of the twist axis) is h. Following equations (3.1) and (5.10) of Part 1, it is assumed that the twist varies linearly along the span; moreover, it follows the flapping rate