14

Wingbeat Frequency Theories—A Mathematical Approach

191

Table 14.1 (continued)

S. No.

Name of theory

Formula

Proportionality

constant (K)

Mass (M)

Wing dimensions

Remarks

5

Based on Newton’s

laws

vh = Kn

√m

A

where,kn =

g

2πρ = 252

Kn =

 g

4πρ =

252

M1/2

A

Appling 2nd law of

motion and taking

amplitude of vibrating

wing Newton derived

the formula for

frequency

6

Ellington Theory

(1999)

vh =



1

0.387 .



m Ar

φ2 R4CL or

vh = Ke



m Ar

φ2 R4CL

Ke =



1

0.387 =

1.607

M1/2

√Ar &



1

φ2 R4CL

Related wing beat

frequency to the mass.

Frequency formula and

is extracted to usual

form. This involves

wing length (R). Aspect

ratio, coefficient of lift

etc.

7

Mass Flow Theory

(1977, 2015)

vh = K ′

m f

L2 Be f f

where K ′ = 2086

K ′ =

8g

Knπρ =

2086

M f

L2, Be f f

Equating weight of flier

to the lift in hovering

state, wingbeat

frequency is calculated

in terms of mf

8

Deakin’s Theory

(2010)

vh = Kd

√m

A

where Kd = 317

Kd = k

 g

ρ = 317

M1/2

A

Deakin considering

square root mass wing

area of the flier

calculated frequency by

dimensional analysis

methods