3

Aerodynamic Considerations

31

a wingbeat frequency of 90 Hz and is having a Reynolds number of 1000 (Chari,

N—Personal Communication).

In an insect, the wing will have three velocity scales which lead to two more

dimension less parameters (U0/u) and (c/u) along with the Reynolds number.

Here, u is flapping wing velocity with respect to insect body,U0 is forward velocity

of the body and c is pitching velocity. U0/u is also known as the advance ratio,

which is also related to reduced frequency f c/U0 [24].

By considering a relatively rigid non-flexible insect wing (viz., Drosophila), the

motion of the wing is relative to the fixed body, which can be explained by three

variables [24].

(a)

Position of the tip in spherical coordinates (t) and (t) and

(b)

Pitching angle (t), which is above the axis connecting root and tip.

The role of AOA, α, is vital to calculate the aerodynamic forces, which ranges

from 25° to 45° for hovering insects. The role of the Angle of Attack (AOA) is vital

to calculate the aerodynamic forces. The AOA ranges from 25° to 45° for hovering

insects with a high wing span. Most of the insects hover to stay at one spot in the air by

flapping their wings rapidly. Though this process of hovering is complex, the insects

need to stabilize and need to overcome the body weight due to gravitational force

for required lift. The lift forces on the wings during upstroke are small and during

downstroke, the forces on the wings will be high, and due to which, the insect moves

up and down oscillating to keep itself in the same position. The wingbeat frequency of

an insect to maintain stability and amplitude can be calculated by making reasonable

assumptions as suggested by [25]:

• Lift force to be a constant when the wings are moving down; which means that

the force during upstroke is very less.

• The insect drops by a distance h due to gravity during upstroke for a time interval

t.

• And h will be equal to h = ^{g( t)2}

2

= 0.1 mm (typical).

• The maximum allowable time ( t) can be calculated for the free fall of the insect

as

t =

2h

g



=



2 × 10⁻²

980

≈4.5 × 10⁻³ s.

• By considering Stroke Period (T) for a cycle of insect flight including both up

and down movements, we get T = 2. t = 9 × 10⁻³ s or 9 ms.

• The wingbeat frequency per second can be calculated by ϑ = f = ¹

T ^{≈110 Hz.}

The wingbeat frequency of insects ranges from 4 to 1000 Hz. Davidovits [25]

explained the important relation between upward force and insect body weight to

restore the insect’s original position. However, it has to be mentioned that the average

upward force on the flying insect is equal to its weight (L = W).