Pitch, Roll, and Yaw Damping of a Flapping Wing

a = lift slope C = damping coefficient Cd = drag coefficient c = chord length of the wing c = mean chord length F = tension of the thread f = flapping frequency g = gravitational acceleration h = distance between the tip path plane and the center of gravity I = moment of inertia of the total system (i.e., the flapping-wing aircraft, suspension bar, and counterbalancing weights) K = a constant in Eq. (2) L = rolling moment Lp = roll damping of the flapping-wing aircraft l = distance between the two threads used to suspend a bar attached to the X-flapping-wing aircraft ls = length of each of two threads used to suspend a bar attached to the flapping-wing aircraft M = pitching moment Mq = pitch damping of the flapping-wing aircraft m = total mass of the flapping-wing aircraft, suspension bar, and counterbalancing weight ma = mass of the flapping-wing aircraft mw = mass of one wing N = yawing moment N r = yaw damping of the flapping-wing aircraft P = cycle period of rotation p = roll rate q = pitch rate R = wing length r = spanwise position r = yaw rate Sw = area of one wing t = time T = thrust vi = induced velocity = flapping angle 0 = mean flapping angle = amplitude of flapping motion = rotational angle 0 = initial rotational angle = amplitude of the rotational motion = feathering angle = amplitude of feathering motion = density of air = angle of the thread measured from the vertical direction of the gravitational force