Wing-Beat Frequency

The main function of wing bones is to transmit force to the external environment during flight. This force cannot, however, be too high because of the risk for bone or muscle failure. These limitations, along with the amount of power available from flight muscles, settle the upper and lower limits of wing-beat frequency for flapping animals [59]-[61]. Based on the insight into the wing-beat or flapping frequency, it is possible to estimate the power output from a bird’s flight muscles and achieve an esti­mation of the power required for flying. According to Pennycuick [62], this means we can estimate the maximum wing-beat frequency /w, max for geometrically similar ani­mals, as shown in the following discussion. Because the force Fm exerted by a muscle is assumed to be proportional to the cross-sectional area of its attachment, we get

Fn « S — l2. (1-8)

Pennycuick [62] assumes that the stresses in muscles and bones are constant and that the torque acting about the center of rotation of the proximal end of the limb can be expressed as

Jt = Fml.

The mass of the limb is denoted by mlimb and it is assumed that the limb has a uniform density. The muscle in action has an angular acceleration, which can be determined as

* = і — l5 – o-n)

From Eq. (1-11) it is easy to determine the stroke time scale T, and with the frequency f — T-1, we get

A relation between the body mass m and the maximum wing-beat frequency fw max can also be derived:

fw, max ^ T-1 ~ I-1 ~ m-1/3. (1-13)

With the assumption of geometric similarity, this is the upper limit of the flapping frequency. For the lower flapping-frequency limit, which is the case for most birds in slow forward flight or hovering, the induced velocity w,, – the airflow speed in the wake right beneath the animal – dominates. Still, the weight W of the flyer must be balanced by the lift L so that, referring to Eq. (1-1), we obtain the following relation for the induced velocity wi:

W = L = 2pwSCl * w = Щ. a-14)

The angular velocity of the wings can be dimensionally expressed as

ы ~ w,/I.

With Eqs. (1-3), (1-4), and (1-15), we obtain the final expression for the lower flapping limit as

Because of these two physical limits, animal flight has an upper and lower bound for the flapping frequency.