14

Wingbeat Frequency Theories—A Mathematical Approach

195

Wing span loading is an important aerodynamic and bio-aerodynamic flight

parameter in comparative studies.

8.

DEAKIN’S THEORY (2010)

Deakin’s theory [8] is also based on dimensional analysis. Deakin derived formula

for wing beat frequency similar to Newton’s formula. But he adopted dimensional

method similar to that of Pennycuick. It can be seen that the constant (317) evaluated

by Deakin is slightly higher than that of Newton. He worked on several species of

insects having typical mass, wing area and wing beat frequencies. He used large

samples of insects for conformation. Deakin developed a formula for wing beat

frequency of insects considering the “Buckingham Pi Theorem”, f (π2, π3) = 0 and

writing the frequency as

νh ∝g^1/2 A^−1/4 f (π3) ∝g^1/2 A^−1/4 f (ρ A^1/2/m)

Function f (π3) maybe expanded by means of Frobenius series α.

f (π3) = kπ^−α

2 ^{(1 + α1π1 + α2π2}

2 ^{+ · · · ),}

where k and α are dimensionless constants, i.e. pure numbers.

Power is assigned with negative sign for convenience and also as is small,

neglecting terms beyond 1, the frequency can be written as

νh = kg^1/2 A^−1/4 f



ρ^−α A^−(3/2)α/m^−α

νh = kg^1/2ρ^−αm^α A^{−(3/2)α−(1/4)}

(14.2)

νh = Km^α A^−β

(14.3)

where K = k g^1/2ρ^−α is considered as another constant.

By solving we get α = 1/2, β = 1 and substituting these values in (14.2)

νh = k(g/ρ)^1/2(m^1/2/A)

(14.4)

Or

νh = ^K√m

A

where K = 317

m = mass of the flier

A = wing area.

This theory has been applied for calculating the wingbeat frequency of insects.