Feather measurements
The mechanical forces involved in diving, plunging, and alighting are
not accessible to direct measurement by current technologies in any
reliable or representative way. Any such data would not be meaningfully
correlated to the resulting yield or flexure of barbs and vanes during
forceful interaction with water. However, the bending and flexing of
materials of different shapes and sizes have been well described in
engineering physics and it is from these considerations that a number of
conclusions in relation to our hypothesis can be drawn.
When a force F is applied over the length of a single barb, the
barb will bend in the direction of the applied force with its tip
flexing over a distance S . This relates to the barb lengthl and barb radius r as
S = F . l3/2π .r4 . E (1)
where E stands for the Young’s elastic modulus of the feather
keratin (Bonser and Purslow 1995, Greenwold et al. 2014). For the
purpose of modeling, barbs are here assumed to be cylindrically shaped.
When the force is applied to the vane, the flexural displacement of the
tips of the vane per repeating unit 2(r + d ) can be
written as
Sv = Fv .l3 . 2(r + d)/2π .r4 . E (2)
where the subscript v refers to the repeating unit of the vane.
Rearrangement of Eqn 2 then yields
\(\pi\).E.Sv/Fv =
(l/r)3 . (r + d)/r (2a)
Apart from π and the elastic modulus E , the left-hand side of Eqn
2a represents the extent of flexing of the tips of barbs per unit of
force applied over the lengths of the barbs and measured over a distance
2(r + d ). For the bending of the entire vane,Fv needs to be considered for the number of
repeating units per vane. Note that the right-hand side of the equation
is made up of the feather variables l , r and d ,
which, unlike Sv and Fv ,
are easily and directly accessible to measurement. These considerations
allow us to predict semi-quantitatively the bending of the vane under an
applied force from the dimensions and spacing of the barbs alone.
The role of the barbules in resisting bending of the vane is to be
considered in the light of their primary function, i.e., keeping the
barbs from separating under an applied force and doing so by their hooks
sliding in the flanges of the barbule next more distal. For this reason,
but mostly for their small size, barbules are assumed to make only a
minimal, if any, contribution to the over-all resistance to bending.
According to Eqn 2a, the bending of the vane of the contour feather
under the impact of forces associated with diving or alighting - here
referred to as the deflection parameter - consists of two factors: (1)
the ratio of the length to the thickness of the barbs expressed asl/r and (2) the wettability parameter (r + d)/r . The first
factor indicates that short and thick barbs make the vane stiff
resisting bending, whereas long and thin barbs favor flexibility that
promotes bending. The appearance of the wettability parameter in the
deflection parameter shows that feathers resistant to water penetration
also help prevent their bending, whereas highly water repellent feathers
do not. Note that l/r enters the equation in the form of a third
power which markedly enhances its contribution to the deflection
parameter and dwarfs that of the other factor: over its range of 2.5 to
7 or higher, (r + d)/r increases by only a factor of 3 or 4,
whereas (l/r )3 does so by about three orders of
magnitude.