Birds of different feathers fly to the same beat

Sub: Science and tech

Sec: Msc

Universal Formula for Flapping Frequencies in Animals:

Almost all animals that fly through the air and many that swim through water have evolved to flap their wings (or fins) at a frequency given by a simple formula, three scientists at Roskilde University in Denmark have found.
Discovery: A simple formula relating the flapping frequency of wings or fins to the mass and wing area of the animal.

f here is the wingbeat frequency, m the mass of the animal, ⍴air the atmosphere’s density, and A the size of the wing. Et voila.

Formula Verification:

Applicability: Validated across a wide range of species including insects, birds, bats, penguins, whales, and a robotic bird (ornithopter).
Method: Plotting calculated values against actual frequencies showed a nearly straight line, confirming the formula’s accuracy.

Derivation Process:

Starting Point: Newton’s second law F=ma applied to airborne animals.
Considerations:
- Force (F) is generated by flapping wings to counteract gravity.
- Momentum of air pushed by wing strokes.
- Airflow velocity and atmospheric density.
Result: A theoretical equation was derived and empirically validated.

Proportionality Constant (C):

Importance: Represents dimensionless quantities related to wing shapes and flapping angles.
Implications: Can provide deeper insights into the efficiency of flight and potential limitations for heavier birds.

Application to Swimming Animals:

Extension: The formula also applies to swimming animals needing continuous motion to stay submerged.
Modification: Replace air density with water density and adjust for buoyancy.

Empirical Testing:

Data Used:
- 176 insect data points (bees, moths, dragonflies, beetles, mosquitoes).
- 212 bird data points (hummingbirds to swans).
- 25 bat data points.
Limitations: Formula assumes high Reynolds number (Re) conditions where fluid density is more critical than viscosity.

Future Work:

Low Reynolds Number: Adjustments needed for animals operating in low Re conditions where viscosity is more significant.
Density Variations: Simplification of the formula is constrained unless density varies significantly by an order of magnitude.

Source: TH