Biomedical Engineering Reference
In-Depth Information
using retroreflective markers in conjunction
with motion-tracking algorithms, was input to
the computational model. The computational
model showed good agreement with measured
forces at low flapping frequencies and was less
accurate at higher flapping frequencies.
Two-dimensional strip theory is the approach
most often adopted by researchers, in which the
wings are discretized into chordwise strips dis-
tributed along the span. The flows over the strips
are assumed to behave independently of each
other, and so each strip is treated as a two-dimen-
sional airfoil section. The forces and moments on
each strip are calculated based on local flow
velocities, angles, and airfoil characteristics, and
the contributions of all the strips are summed to
find the total forces on the flyer. A simplified
unsteady aerodynamic analysis based on modi-
fied strip theory was developed by DeLaurier
[82, 83]
to model the flight performance of a har-
monically flapping wing. In this analysis, the
wing was assumed to be spanwise rigid in bend-
ing but flexible in torsion. A harmonic variation
of pitching and flapping motion was assumed. A
modified Theodorsen function was used to incor-
porate the unsteadiness of the flow as well as the
finite aspect ratio of the wing. Post-stall charac-
teristics were incorporated in the analysis in addi-
tion to a leading-edge suction force that account
for the majority of forward-thrust production.
Figure 5.22
shows a schematic of a wing dis-
cretized into chordwise sections along its span.
The wings flap about their axis of symmetry, at
the mid-span location (only one wing is shown
in the figure). The incident velocities, angles,
and forces on the two-dimensional airfoil sec-
tion are similar to that shown in
Figure 5.16
.
This analysis was used to find the performance
of the flapping wings on an 18 ft span pterosaur
model, which included a spanwise variation in
airfoil chord as well as sweep, similar to that inves-
tigated by DeLaurier
[82]
. An empirical model was
used to calculate the appropriate flapping fre-
quency based on the total mass of the pterosaur
(around 40 lbs). The flapping frequency
f
for any
natural flyer was given by Pennycuick
[84]
as
f
=
m
3/8
g
1/2
b
−
23/24
S
−
1/3
ρ
3/8
.
(5.7)
Here,
m
is the mass of the bird,
g
is the accelera-
tion due to gravity,
b
is the wingspan,
S
is the
wing area, and
ρ
is the density of air. Using this
relation, the flapping frequency was found to be
1.2 Hz. This is low in comparison to typical birds
of today but can be attributed to the large wing
span and low wing loading of pterosaurs. The
calculations were performed at a flight speed
of 44 ft/s, a flapping amplitude of 20°, and an
angle of incidence of the flapping axis of 7.5°.
The average lift produced over one cycle as a
function of the dynamic twist angle amplitude
β
0
is shown in
Figure 5.23
a. It is seen that for
values of
β
0
>
2.25°/ft, the lift produced is about
42 lb, which is more than the weight of the ptero-
saur and hence sufficient to sustain flight.
The average thrust produced as a function of
dynamic twist angle is plotted (
Figure 5.23
b). It
is seen that the thrust produced peaks at a
dynamic twist of around
β
0
=
2.25°/ft. Beyond
this value of
β
0
, the thrust rapidly decreases to
around zero. This trend in thrust can be explained
by the fact that upon increasing the dynamic
twist to a larger value, the outboard sections of
the wing become prone to stall, causing them to
lose thrust.
A similar trend is seen in the propulsive effi-
ciency curve (
Figure 5.24
), where a distinct max-
imum of 42% is reached at
β
0
=
2.25°/ft. The
propulsive efficiency is low compared to the
z
dD
dL
y
β
o
x
V
∞
FIGURE 5.22
Schematic of flapping wing modeled
using two-dimensional strip theory
[82]
.
