6

Theories on Hovering Flight of Insects

81

ϑh = ^M

L²

1

Bef f

K

′

(6.10)

∴ϑh = f



W SL, Bef f



(6.11)

Mechanical Oscillator Theory

Greenewalt [1] considered the oscillating wing of a biological flier as a driven

mechanical harmonic oscillator and suggested the following equation as

ϑ.lⁿ = K (constant)

(6.12)

with a value of 3540 for the constant K and l is being the wing length in mm.

Accordingly,

ϑ.l^{(1 to 1.25)} = 3540

(6.13)

The value of n has to be varied between 1 and 1.25 for tallying with the natural

frequency of the flier by the trial-and-error method. The value of the wing length

(l) has to be expressed in millimetres. Greenewalt confined his experiments to small

hummingbirds. This is a very cumbersome method that involves many trial-and-error

calculations by changing the power values of the index of l. Therefore, when we apply

this formula for the calculation of frequency in biological fliers, it is a long drawn

process. Hence, this theory is not generally used for calculating the insect wingbeat

frequency.

Crawford’s Theory

Crawford [2] proposed modified mass flow theory and obtained a relationship for

the wingbeat frequency of small fliers and the equation is as follows:

ϑh =

 g

4πρ

1/2

x

M f

Sw

(6.14)

where

M f

Mass of the flier.

Sw

Wing swept area = Stroke angle (radians) × (wing length, mm)².

ρ

Density of air 0.001225 gm/ cm³ (for standard air at mean sea level).

g

Acceleration due to gravity = 981 cm/ s².