3

Aerodynamic Considerations

33

A general expression for Moment of Inertia is given by

I =

n

i=1

mir²

i ^,

(3.14)

where

mi

is the mass of i strip of the wing in grams.

ri

is the distance between the fulcrum and centre of i strip.

n

is the strip number.

Themaximumangularvelocity(wmax)iscalculatedfrommaximumlinearvelocity

(vmax) at the centre of the wing.

wmax = ^vmax

l/2 ^.

(3.15)

Average linear velocity can be calculated at the centre of the wing for a distance d

traversed at the centre of the wing divided by duration t during each wing stroke.

For the above example with d = 0.57 and t = 4.5 × 10⁻³s, the average linear

velocity is calculated as

vavg = ^d

t ⁼

0.57

4.5 × 10^{−3 = 127 cm/s.}

At the beginning and end of the wing strokes, the velocity of the wings will be

zero and by assuming a sinusoidal variation of velocity along the wing path, the

maximum velocity of the insect will be twice and it will be as high as the average

velocity. Hence, the maximum angular velocity (from Eq. 2.15) is given by

wmax = ²⁵⁴

l/2 ^.

And the kinetic energy from Eqs. 2.12, 2.13 and 2.15 is given by

K E = ¹

2 ^Iw2

max ^{= 1}

2

10−3l2

3

254

l/2

2

= 43 erg.

The KE for both (up and down) strokes of a cycle of wing movement is 2 ×

43 = 86ergs. It is possible to apply these formulae for smaller pentatomids such as

Chrysocoris purpureus.