Upper and Lower Limits

Can scaling arguments provide any information about limits on the size of flapping flyers capable of sustained flight? As mentioned earlier, large pterosaurs once flew long ago; some of these species were much larger than the birds of today. There are discussions about whether they were able to flap or only to soar [4]. There are many parameters to consider when flapping flight is studied, but limitations to this kind of flight mainly depend on the power available and structural limits.

These limitations are intimately connected, because flapping frequency affects both power and structural limits. To generate the power required for flight, most birds and other flapping animals have well-developed flight muscles. For birds these muscles are the pectoral muscles, which power the downstroke of the wings, and the supracoracoideus muscles powering the upstroke. Much effort has been made to determine power output and frequency levels and to compare the masses of these muscles with the mass of the whole specimen. According to Rayner [56], relations between body mass m and the mass of the pectoral and supracoracoideus muscles, mp and ms, respectively, can be expressed as

mp = 0.15 m099, (1-26)

ms = 0.016 m101. (1-27)

This means that the flight muscles constitute approximately 17 percent of the total weight. In comparison, the muscle of human arms accounts for about 5 percent of total body weight, according to Collins and Graham [76]. The power output from bird muscles and “fast” human muscles is about the same, 150 W/kg. Because the wings are often flexed during the upstroke and therefore not exposed to the same aerodynamic force or moment of inertia as during the downstroke, the weight of the supracoracoideus muscle is generally low compared with the weight of the pectoral muscle. Hummingbirds are different, having an aerodynamically active upstroke (producing lift). In their case, the weight of the supracoracoideus is higher; according to Norberg [4], this muscle group can constitute up to 12 percent of the body weight. The smallest supracoracoideus muscles are found among species with large wingspan, where the muscle mass is about 6 percent of the total mass. This value is comparable to that of the human body, and hence, these species have difficulties taking off without a headwind, running start, or a slope-start from a height. However, species with a long aspect ratio are usually able to soar, so the duration of the flapping flight mode can therefore be decreased.

Pennycuick [62] [77] [78] defined the power margin as the ratio of the power available from the flight muscles to that required for horizontal flight at the minimum power speed. As already mentioned, the power available depends on the flapping frequency, which determines the upper and lower limits of the size of flying animals. Pennycuick [78] concluded that the upper limit for flapping flight, based on actual sizes of the largest birds with powered flight, is a body mass of about 12 to 15 kg. Larger birds do not have the possibility of beating their wings fast enough to generate lift to sustain horizontal flight. Smaller birds have the advantage of being able to use different flapping frequencies, but for animals with a weight of about 1 gram, there is another upper limitation. Their muscles need time to reset the contractile mechanism after each contraction [4]. For insects with wing-beat frequencies up to 400 Hz, this problem is solved with special fibrillar muscles capable of contracting and resetting at very high frequencies. This limitation results in a minimum mass for birds of

1.5 grams and, for bats, 1.9 grams.

The upper and lower wing-beat frequencies are also restricted because of the structural limits. Bones, tendons, and muscles are not capable of performing wing motions above a certain wing-beat frequency. Wing bones that have to transmit forces to the external environment during flight must be strong enough to not fail under the imposed loads. This means that the bones have to be stiff and strong and at the same time not too heavy. Kirkpatrick [61] investigated the scaling rela­tionships between body size and several morphological variables of bird and bat wings in order to estimate the stress levels in their wings. He also estimated the bending, shearing, and breaking stresses in the wing bones during flight. He sug­gested that the breaking stress for a bat humerus bone is around 75 MPa and for birds 125 MPa. This structural limit helps explain why no bat weighs more than 1.5 kg. Kirkpatrick [61] found that no relationship exists between either bending or shearing stresses and wingspan during gliding flight and during the downstroke in hovering flight. In general, the safety factors are greater for birds than for bats. Hence birds are more capable of withstanding higher wing load­ing. A final conclusion by Kirkpatrick [61] is that the stresses examined are scale independent.