title: Forward flight of birds revisited. Part 2: short-term dynamic stability and trim.
abstract: 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.
authors.affiliation: Faculty of Aerospace Engineering, Technion, Haifa 32000, Israel.
authors.forename: G
authors.initials: G
authors.lastname: Iosilevskii
keywords: Hill's equation; dynamic stability; flapping flight; parametric resonance