14

Wingbeat Frequency Theories—A Mathematical Approach

193

where dimensionless multiplier 1.08 came from multiple regression.

Pennyquick’stheoryispurelybasedondimensionalanalysis.Hisformulainvolves

different physical parameters raised to different powers and is given by

νh = 1.08 ×



m

3

8 g

1

2 b

−23

24 s

−1

3 ρ

−3

8



,

where f = νh

The abbreviations used are as below:

m = mass of the flier (gm), g = acceleration due to gravity (cm/sec²)

b = semi-wing span (cm), s = wing area (cm²) and

ρ = air density (gm/cm³).

The frequency of the flier is more influenced by the powers of the parameters

(m, g, b, s, ρ) evaluated. The values are in good agreement and are close to natural

frequency of the fliers. He studied marine birds. The value of k is 252.

5.

NEWTON THEORY

In this formula, the frequency of a flier depends directly on square root of mass and

indirectly on wing area. Newton’s theory is based on mass flow concept. However,

larger samples of fliers have to be used for its confirmation. He considered vibration

of wings of a flier similar to an equation of motion of second order differential

equation, as per Newton’s second law of motion of a body or a particle. Newton also

solved the equation assuming sine function.

The formula isνh = ^K√m

A

,

where K = 252 and m = mass of the flier, A = wing area.

6.

ELLINGTON THEORY (1999)

He studied the kinematic changes with the flight speed of bumblebee [6]. Other

insects like ‘fruit fly’ and ‘hawk moth’ were also studied. He derived the mass ‘m’

that can be supported during hovering as

m = ^{0.387φ2n2R4C L}

Ar

,

(This can also be expressed in terms of νh)

νh =



mAr

0.387φ² R⁴C L

1/2

,

where φ is wingbeat amplitude (peak to peak in radians) and

n = νh is strokes/seconds of wingbeat (Hz)