Induced power calculation
This protocol is extracted from research article:
Biomechanics of hover performance in Neotropical hummingbirds versus bats
Sci Adv, Sep 26, 2018; DOI: 10.1126/sciadv.aat2980

As birds and bats flap their wings to hover, they accelerate air downward. The induced power required to accelerate this air can be calculated as follows (29)Embedded Image(1)where κ is the induced power factor that accounts for tip losses, nonuniform inflow, and other nonideal effects; ρ is the density of air (1.04 kg m−3); A is the area swept by the rotors or wings; and W is the weight of the animal. Lighthill (39) noted that, for animals, this ideal induced power must be a minimum, as animals do not generate a jet of uniform velocity below. Ellington (40) broke down this κ value into a spatial correction factor to account for downwash profile (σ) and a temporal correction factor for wake periodicity (τ)Embedded Image(2)

While Ellington estimated this spatial correction factor (σ) to be about 0.1 and the temporal correction factor (τ) to be around 0.05 for horizontal stroke planes and 0.5 for inclined stroke planes, we can determine the temporal factor precisely using our directly measured time-resolved force trace. When we decouple the correction factors into a spatial cost factor, κσ (which equals 1 for a uniform wake), and temporal cost factor, κτ (which equals 1 for a constant wake), we can calculate the stroke-averaged induced power asEmbedded Image(3)

By substituting the time-varying vertical aerodynamic force [F(t)] for the constant vertical aerodynamic force equal to weight (W) and integrating over the wingbeat, we can account for power losses due to temporal force fluctuationEmbedded Image(4)where T is the wingbeat period. This allows us to calculate the temporal cost factor based on our directly measured instantaneous vertical force as followsEmbedded Image(5)

Because the time-varying normalized vertical force is raised to the power 1.5, the animal takes an extra penalty when it generates a fluctuating force unequal to its weight. We calculate temporal cost factors of 1.11 and 1.19 for hummingbirds and bats, respectively, as shown in Fig. 4D. The instantaneous induced power [Pind(t)] can be calculated byEmbedded Image(6)where we will use a spatial cost factor of κσ = 1.1 from Ellington (40). To compare across species, the average body mass–specific induced power (Embedded Image) is calculated asEmbedded Image(7)where m is the mass of the animal, and g is the gravitational acceleration. While the asymmetric weight support of bats leads to a higher temporal cost factor, they reduce body mass–specific induced power by lowering the actuator disk loading (W/A), as shown in Fig. 4E.

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