Forces involved in flight
Flight involves competing forces: lift versus the animal’s weight, and thrust versus drag. When thrust is not provided, gliding occurs; the loss in a creature’s altitude compensates for energy lost to drag.
Thrust is relatively minor in magnitude compared to lift. To be more specific, when birds glide, their glide ratios vary by species, such as 11:1 in vultures or 22:1 in albatrosses – either 11 or 22 meters of distance covered for each meter of altitude lost (Doane, 2011; Parrott, 1970), meaning only 9% to 5% of the magnitude of lift force is needed for thrust while cruising.
On the other hand, while energy input is required for thrust, lift is a passive activity. Therefore, in an ideal system, the only necessary energy input is for thrust.
Pectoralis muscle forces
The pectoralis muscles are the major flight muscles of birds. Located on the torso, they pull each wing downwards, exerting force against the air. However, bird wings are like “bad” levers, where lift occurs far from the fulcrum (the shoulder), and the pectoralis muscles are much closer to the shoulder. Due to this lever effect, pectoralis muscles need to exert much larger force when contracting as compared to the ultimate force of lift. In addition, whereas the effect of gravity is constant, muscles apply varying force during the wingbeat.
We will use a study of ducks in flight for estimation of muscle force needs (Williamson et al., J Exp Biol, 2001). The ducks measured in the study had average masses of 0.995 kg, implying a lift requirement of 9.75 N in level flight. Since pectoralis muscles are working on a relatively short lever arm (compared to the size of the wingspan), pectoralis muscles need to exert more peak force than the force of lift. Each unilateral muscle exerts peak force of approximately 107.5 N in level flight. (Only slightly more force is exerted during takeoff and in ascending flight, so we will focus on level flight for ease of comparison). Bilaterally, 215 N of force is exerted, suggesting that the relation of pectoralis force to lift force can be estimated as 215 / 9.75 = 22 fold.
While the scales are different from the needs for an anthro, the basic relations between forces would be similar, as long as the relative positioning of the pectoralis along the wingspan is the same. Therefore, using the relation between total pectoralis force and lift force of 22 fold, we can estimate that a creature weighing 80 kg (necessitating lift of 80 kg * 9.8 m/(s^2) = 784 N) would need a bilateral pectoralis force specification of 17,248 N in level flight. (Note that this force specification is lower than the maximum force capacity of isotonic muscle. Therefore, when considering muscle cross-sectional area, it is important to distinguish whether forces are routine or maximum/isotonic per unit of cross-sectional area).
As well, ducks inform how a muscle’s cross-sectional area relates to this measure of pectoralis force. Measurements for ducks in Williamson et al 2001’s study give a unilateral myofibrillar area of around 5.47 cm^2. Myofibrillar area is approximately 60% of total muscle cross-sectional area (Dial and Biewener, J Exp Biol, 1993; Williamson et al., J Exp Biol, 2001), implying ducks have a total unilateral muscle cross-sectional area of 9.12 cm^2 per pectoralis (or 18.24 cm^2 for bilateral CSA). Therefore, in ducks, the relation is 215 N of force per 18.24 cm^2, or 11.78 N per cm^2 in level flight. Again, this is the force capacity of muscles in practice, in evolutionarily optimized animals. The isotonic force capacity of these muscles is higher (and would yield a higher N per cm^2 number), but not observed in living ducks.
Therefore, for equally optimized pectoralis muscles in anthro flyers, we would obtain 17,248 N / (11.78 N/(cm^2)) = 1,464 cm^2 of bilateral pectoralis muscle cross-sectional area (or 732 cm^2 per unilateral muscle). Simplistically, if the muscles are 60 cm (24 inches) along the torso (along the rostral-caudal axis), each muscle would be around 12 cm thick (4.7 inches). On the surface, this sounds reasonable.
Moreover, it is possible that muscle pennation would reduce the apparent cross-sectional area, but the literature of muscle anatomy requires close reading to make sure that we are not unfairly double-counting ‘discounts’ that would let us reduce apparent muscle size (existing literature tends to not give fully traceable inputs to published calculations). Therefore, to avoid the risk of double-counting pennation if it is already included in our muscle cross-sectional area calculations, we will not include a pennation correction factor here.
Pectoralis muscle length
We still need to estimate appropriate pectoralis muscle length. We will assume that muscle length will scale linearly with wingspan, since the muscle ultimately needs to span from places equivalent to a bird’s keel to where it inserts on the wing humerus. This is a fair assumption because muscle length needs to match the range of motion during a contraction, not the force being exerted. In other words, muscle length will not directly scale with body mass, nor with muscle cross-sectional area.
In the Williamson et al 2001 paper, ducks’ fascicle length, which we take as a proxy of overall muscle length (length from the muscle’s origin on the keel to its insertion on the wing humerus), is around 6.97 cm for animals with bilateral wingspans of 86.2 cm. This suggests that each pectoralis should have fascicle length of 0.081 cm per cm wingspan.
Similarly, in Silver King pigeons (a very large breed of pigeons), the relation is 5.5 cm of fascicle length for animals with wingspans of around 73.6 cm (calculated from “body mass” and “disk loading” in Table 1 of Biewener et al, J Exp Biol, 1998), yielding a ratio of 5.5 cm per 73.6 cm -> 0.074 cm per cm of wingspan, consistent with the above 0.081 cm per cm wingspan estimate.
If we take the slightly more conservative estimate of 0.81 cm/cm, and assume that the 4 m wingspan discussed in the “Wingspan and airspeed” section is reasonable, this would suggest that pectoralis fascicle length should be 0.081 (cm/cm) * 400 cm = 32.4 cm.
Estimating pectoralis muscle mass
We now have a cross-sectional area, as well as length, of pectoralis muscles. However, we are not yet ready to determine an overall mass of pectoralis, because pectoralis muscles have a shape very divergent from a rectangular prism volume with 90 degree angles. We will perform calculations that attempt to preserve the biological relations we have already established.
The pectoralis muscle in ducks has a unilateral mass of 0.067 kg (Williamson et al., J Exp Biol, 2001). Again, they also have a unilateral muscle cross-sectional area of 9.12 cm^2. Therefore, in ducks, pectoralis mass is 0.00735 kg per cm^2 for muscles with a fascicle length of 6.97 cm, or 0.00105 kg per cm^2 for 1 cm-long fascicles. Applying this to anthros, with muscles with a bilateral CSA of 1,464 cm^2 and an unrealistic fascicle length of 1 cm, we obtain muscles that are 0.00105 kg * 1,464 cm^2 = 1.54 kg in bilateral mass.
Finally, we need to lengthen the muscles from 1 cm to 32.4 cm, and we will simplistically use a linear multiplier of 32.4 cm/cm. Therefore, 1.54 kg * 32.4 = 50 kg of bilateral pectoralis mass. Ouch!
Possible adjustments to pectoralis mass while preserving nature-inspired anatomical design
50 kg of pectoralis mass is not practical to specify for anthro flyers, suggesting that it is not realistic to anticipate the use of biologically-inspired, bird-like pectoralis wing anatomy, while expecting similar levels of performance and comfort over long, sustained flight sessions.
If we reduce the performance demands, such as by allowing shorter-duration flight, it will be possible to reduce muscle mass below 50 kg.
For example, specifying that muscles should produce maximum forces of around 32.1 N/(cm^2) might be permissible, considering measurements in humans – though the measurements were for maximum strength, not over a longer span of time (de Monsabert et al, Med Biol Eng Comput, 2017). So, if we modify the force per CSA specification for flight muscles from 11.78 N/(cm^2) to 31.2 N/(cm^2), we could reduce bilateral muscle mass from 50 kg to perhaps 18.9 kg. These would still be very large muscles, and could be expected to be inconvenient for flyers when they are on the ground.
Still, for the sake of argument, let us propose the following parameters:
- 17,248 N / (31.2 N/(cm^2)) = 553 cm^2 of bilateral pectoralis muscle cross-sectional area
- Pectoralis muscles each 32.4 cm long
- Bilateral mass of 18.9 kg
How about the lever arm? Above, we specified that the pectoralis would be acting at a proportionally identical region along the wing in anthros as in extant ducks. Per Williamson et al., 2001, the ratio between the lever arm of the pectoralis, versus the lever arm of air pressure acting upon the wing, is approximated by r_pectoralis/r_wing = 0.100. However, if we specify that the pectoralis inserts onto the humerus relatively more laterally than in ducks (e.g., by moving its joint with the wing-scapula more medially), we might obtain a ratio between the lever arms of 0.2, giving the pectoralis relatively higher mechanical advantage, though presumably incurring a need for a longer muscle to accomplish the same work.
Let us modify the specification, again for the sake of argument:
- Bilateral pectoralis cross-sectional area = 553 cm^2 * 0.100 / 0.2 = 277 cm^2
- Assuming fascicles 1 cm long: 0.00105 kg per cm^2 for 1 cm-long fascicles
- Pectoralis muscle length = 32.4 cm * 0.2 / 0.100 = 64.8 cm
- Muscle mass of 0.00105 kg/(cm^2_CSA * cm_fascicle_length) * 277 cm^2 * 64.8 cm = 18.8 kg.
In other words, the lever arm (of where the pectoralis meets the humerus) does not appear to impact the overall mass of the muscle.
Rationally designed structures, divergent from nature, appear crucial to satisfactory performance
Overall, these calculations suggest that relatively large pectoralis muscles would be needed if we were to pursue biologically inspired design. Pectoralis muscles likely still need to be large even if performance demands are reduced. As well, even if the mechanical advantage for pectoralis muscles is increased, this likely necessitates a longer muscle length to accomplish the same resulting movements of the wing. The use of biological muscles and biologically inspired flight anatomy appears difficult.
Still, it is important to remember that even though lift is the major force in flight, lift is a passive, rather than active, outcome of the mechanical system. Thrust is the force that overcomes drag (gliding does not require thrust input). Therefore, a ligament or tendon can substitute for much of the pectoralis muscle, in order to provide the cyclical force throughout wingbeats needed to hold wings downward against the force of air pressure. In fact, this is a strategy that albatrosses use: they lock their wings in place during gliding without muscle input.
Other energy input is of course required to overcome drag, and permit climbing rather than just level flight. But, this energy can be applied through muscles that are not already doing double-duty with holding the wings in position. Still, there is a caveat: we are not aware of any biological examples of tendons or ligaments that are used in parallel with muscles, rather than in series with them.
Conclusions about forces in flight
We come away with the following conclusions about forces involved in flight:
- Lift and thrust (and weight and drag) are the major forces in flight. Lift/weight is larger than thrust/drag by between 11:1 and 22:1.
- The vast majority of force experienced by flight muscles during wing beats goes towards providing lift, rather than thrust.
- If using muscles in a similar biological design to natural birds, pectoralis muscles would need to be inconveniently large, especially for comfortably sustained flight.
- The use of elastic structures to help support lift might dramatically reduce force demands on pectoralis and other flight muscles, allowing muscles to focus on providing energy input rather than both energy input and load. However, more detailed study would be needed.