Scaling of bird wings and feathers for efficient flight

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Science Advances  16 Jan 2019:
Vol. 5, no. 1, eaat4269
DOI: 10.1126/sciadv.aat4269
  • Fig. 1 A replot of Tennekes’ “Great Flight Diagram” focusing exclusively on birds.

    Weight (W), cruising speed (v), and wing loading (S) of various birds follow notable correlations over almost four orders of magnitude in weight (25). The solid line describes predicted values based on isometric scaling of the wingspan and weight.

  • Fig. 2 Scaling the humerus bone with weight.

    (A) Humerus, ulna, and manus bones modeled as a bending beam. The humerus is considered to be a hollow cylinder with length LH, diameter 2c, and thickness t. The total distributed force, w = W/(2L) multiplied by the wing length L is equal to the weight of the bird (W) divided by 2. Although the bone extremities articulate in the plane of the wing, they can be considered as a single beam resisting the lift forces perpendicular to the wing plane. (B) Bone strength limits the length of the humerus bone. Experimental data demonstrate that the humerus length LH scales allometrically with the weight W of the bird with an exponent equal to 0.45. The data conform closely to the prediction of Eq. 8, which is based on the assumption that bone strength is limited and that the humerus dimensions change isometrically. Deviations are seen for heavier birds. Isometric scaling would require LHW0.33 (lower curve) and is not followed in nature.

  • Fig. 3 Cruising speed v plotted against wing area A on a log-log scale for a variety of birds.

    The weight of the bird is indicated by the color coding. Diagonal lines represent calculated values of constant lift, which are nearly equal to the weight of the bird. The color of these lines corresponds to the color map used to plot the weight of birds.

  • Fig. 4 Humerus dimensions and cruising speed of birds.

    (A) Cruising speed v plotted against wing area A. The percent wingspan composed of humerus (humerus length/half of wingspan) is color coded. (B) Wing area versus humerus length. Birds 1 and 2 have the same mass and humerus length. Bird 1, however, has a much smaller wing area and therefore has higher wing loading. To compensate for this, a larger percentage of bird 1’s wingspan is composed of humerus bone. (C) Percentage of the wingspan LH/L composed of the humerus plotted against wing loading W/A on a log-log scale. The cruising velocity is color-mapped. (D) Humerus diameter DH and length LH plotted as in (A), with the cruising speed color-mapped. The cruising speed does not appear to correlate with the humerus dimensions.

  • Fig. 5 Various dimensions of the flight feather scaled with mass.

    (A) The total feather shaft length scales with bird mass following the trend y = 2.3x0.34 with an R2 value of 0.95 (measurement uncertainty is ±0.05 cm). (B) Width of the feather shaft at its midpoint scales with bird mass exponentially following the trend y = 0.19x0.35 with an R2 value of 0.95 (measurement uncertainty is ±0.02 mm). (C) The barb length of the trailing and leading feather vane follows y = 4.29x0.27 (R2 = 0.91) and y = 2.58x0.25 (R2 = 0.83), respectively (SDs range from 0.02 to 0.2mm) (30). The trends shown in (A) to (C) scale closely to the trend expected through isometric scaling with bird mass. The spacing between trailing hooked barbules (D) does not follow this trend and ranges between 8 and 16 μm across all bird masses (30). Images were taken from (30).

  • Fig. 6 Barbules as connecting elements between feathers.

    Their spacing is measured as the distance between barbules, as shown in (A). An additively manufactured bioinspired model (B) demonstrates the function of the barbule membrane flaps. This model is shown with air blown dorsally [as in the wing upstroke (C)] and ventrally [as in the downstroke (D)] at the vane. Blue circles represent the location of airflow. Micrographs of the feather vane of Anna’s hummingbird (Calypte anna) (left) and the Andean condor (Vultur gryphus) (right) demonstrate dimensional similarities on the microscale (E), while macroscale differences are shown in (F). A single barbule is highlighted in yellow in each image shown in (E). Images were taken from (30).

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