14

Wingbeat Frequency Theories—A Mathematical Approach

189

Norberg [4] showed Regression Equations for wing morphology and aerody-

namic characteristics for various fliers based on available empirical data. Her general

formula has also shown relation between body mass and wing span, body mass and

wing loading, body mass and aspect ratio which contribute to a better understanding.

For studying the wingbeat frequency she has made use of wing span, wing area, wing

loading, and aspect ratio. She has calculated the wing beat frequency and expressed it

in terms of mass values. Norberg has considered mass of the flier alone for the calcu-

lation of wingbeat frequency and assumed that other morphological flight parameters

are related to the mass. The formula is applicable to all birds other than Humming

birds. Frequency in Norberg’s theory is inversely proportional to mass raised to the

fractional power (Table 14.1).

Using linear least squares regression lines she fitted power function for the

wingbeat frequency with mass of the body alone as

νh = 3.98 M^−0.27.

The abbreviations used are as below:

M = mass of the flier (in kg) and



M^−0.27

is variable.

However, for Humming birds, the formula is νh = 1.32 M^−0.60.

Wing dimensions and flight parameters against body mass of some birds, bats

and pterosaurs have been calculated and enlisted by Norberg [4] in the book entitled

“Vertebrate flight”. The modified table has been enclosed in the present studies (Table

14.2).

4.

PENNYQUICK’S THEORY (1996)

In determining the wing beat frequencies of fliers, Pennyquick [5] adopted a hybrid

method, i.e. use of

(a)

Multiple regression analysis and

(b)

Dimensional analysis.

By multiple regression analysis he will determine how the frequency depends on

variables such as body mass, wing span and wing area. Pennyquick in his dimensional

analysis method identified constraints to which a physically valid solution must

confirm.

The list of the variables which are likely to influence the frequency are as follows:

Body mass—m

Wing span—b

Wing area—S

Wing moment of inertia—I

Acceleration due to gravity—g

Air density—ρ.