capacity **He flies** They evolved independently in different groups of animals. To reduce the energy needed, biologists hoped for a pattern in… **repetition** Which has different types **Flapping their wings**But finding a mathematical description common to all of them was difficult.

Now researchers from Roskilde University (Denmark) have succeeded in achieving this. In an open access journal *One plus* Provide a global equation that describes **Wingbeat frequency** In birds, insects, bats, and even in swimming animals such as whales, although their size, evolutionary history, and shape of their wings or swimming skeletons vary.

One of the authors, **Tina Heckscher**, explains to SINC the essence of this formula, derived from physical principles: “The flapping frequency of the wing (f) is proportional to the square root of the mass (m) of the animal divided by the area of (a) the wing or fin.” Other parameters such as gravitational field strength and air density are also included in the full version of the equation.

A simple version of the global equation (left), where the wing beat frequency (f) is proportional to the square root of the animal’s mass divided by the area of the wing or fin. On the right is the full formula, which also includes the gravitational field strength (g), a dimensionless value called C (which describes the shape, motion, and other properties of the wings) and the air density (ρair). /Jensen et al./PLOS ONE

“We have it all figured out **Flying animals follow this proportionality** And almost the same factor – and he confirms -. Furthermore, as this formula is extended to swimming animals, it is shown that it follows the same global relationship (when buoyancy and the difference between the density of air and water are corrected for). “There is no simpler, physically meaningful formula that can describe the frequency of ‘flutter’ in flying or swimming animals.”

There is no simpler, physically meaningful formula that can describe the frequency of flapping of wings or flippers in flying or swimming animals.

Tina Heckscher (Roskilde University)

Authors **They made sure their equation was correct** With published data on the wingbeat frequencies of bees, moths, dragonflies, beetles, mosquitoes, bats, and birds of various sizes, from hummingbirds to swans. They also confirmed this with information on flipper movement frequencies in penguins and several species of whales, such as humpback whales and northern bottlenose whales.

The results for 414 animals, from these large cetaceans to small mosquitoes, which differ by a factor of about 10,000 in the frequency of flapping of wings and fins, lie on the same line as the graph they represent. “As physicists, we were surprised by how well our simple prediction of wingbeat formula worked for such a diverse group of animals,” the authors say.

## Wing movement in the largest flying animal

Finally, they calculated that A **Pterodactyl is an extinct animal** extinct (*Quetzalcoatlus northropi*) – the largest flying animal ever known – was able to move its 10-square-meter wingspan at a frequency of 0.7 Hz, or once per second.

“the **equation**“Together with experimental data, it provides a recipe for calculating wingbeat frequency given a given mass and wing surface,” Heckscher highlights, and although it is not intended for animals that do not fly or swim, it gives an example of how this can be applied. In our species: “If a human weighs 70 kg and builds wings with a total area of 2 m^{2}You’ll need to beat them about 4 times per second to stay in the air. This is very difficult, even without the wings! From our own experience when moving our arms, we can conclude that humans were not created to fly, although the equation does not rule this out.

The study shows that flying animals are different from butterflies and bats, for example **And it developed towards this relationship** Relatively constant between body mass, wing surface, and frequency of wingbeats or ‘flapping’ (although the latter term applies to wing movement without flight, so in this context the former is more correct).

Regarding swimming animals, the researchers admit that they did not find studies that contained all the necessary information, so they combined data from several to make comparisons and estimates.

This “recipe” also works for diving animals that, like whales and penguins, must constantly move water upward to stay submerged.

However, with some modifications, this “recipe” is also valid for positively buoyant diving animals which, like whales and penguins, must constantly move water upward with their flippers to remain submerged. **Fish is excluded**Because they adjust their buoyancy using their swim bladder.

## Equation limits and nanorobotics

According to the authors, it is possible **Very small animals **They wouldn’t fit into the equation either. “We expect that there is a limit set by fluid dynamics. For animals smaller than any insect measured so far, we derive another equation, which is that the frequency of wing flapping is proportional to the mass divided by the area, i.e. without a square root. It would be interesting to investigate the matter by New studies,” says Heckscher, which may have implications for future evolution **Flying nanobots.**

rights: **Creative Commons.**

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