6

Theories on Hovering Flight of Insects

79

where,

R

is reacting force of the flier.

dm

dt

is the rate of mass flow of air passing over the wings.

M f

is the mass of the flier and

g

is the acceleration due to gravity.

The rate of mass flow of air during flight is considered as a function of disc area

(Sd), effective wing breadth (Beff), the density of air ρ and wingbeat frequency of

flier (ϑh).

dm

dt ^{α Sd Bef f ρ ϑh}

2

(6.3)

The wingbeat cycle consists of an upstroke followed by a downstroke. In a flap-

ping cycle, as the downstroke of the wing is considered to be more powerful and

contributes to maximum lift than upstroke (recovery stroke), half of the value of wing-

beat frequency

 ϑh

2



is included in the above equation as it is derived on an empirical

basis. The frequency arrived at the above equation agrees well with experimental

observations.

R α ^dm

dt

R α Sd Bef f ρ ^ϑh

2

(6.4)

or

R = kSd Bef f ρ ^ϑh

2

(6.5)

From Eqs. 6.2 and 6.5, we get

M f g = kSd Bef f ρ ^ϑh

2

(6.6)

where

Sd = πr² = ^{π L2}

4

,

and L = wingspan

M f g = k ^{π L2}

4

Bef f ρ ^ϑh

2

Rewriting this equation, we get