Flexural stiffness in insect wings I. Scaling and the influence of wing venation
Department of Biology, University of Washington, Seattle, WA 98195, USA
* Author for correspondence (e-mail: scombes{at}u.washington.edu)
Accepted 3 June 2003
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Summary |
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Key words: insect flight, flexural stiffness, wing, wing flexibility, wing vein, independent contrast, finite element model
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Introduction |
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Many insect wings undergo significant bending and twisting during flight
(Dalton, 1975;
Wootton, 1990a
), which may
alter the direction and magnitude of aerodynamic force production. Wing
deformations enhance thrust production in some species by creating a force
asymmetry between half-strokes, and can increase lift production by allowing
wings to twist and generate upward force throughout the stroke cycle
(Wootton, 1990a
). The
structure of insect wings thus appears to permit certain beneficial passive
deformations while minimizing detrimental bending that would compromise force
production. Yet our understanding of how insect wing design affects
flexibility and passive wing deformation is limited.
Insect wing veins are the primary supporting structures in wings. The
arrangement of veins and complexity of vein branching varies widely among
insects, and venation pattern is often used to characterize orders and
families. Basal groups of insects (such as odonates) generally possess wings
with a large number of cross-veins (also present in early fossil wings),
whereas more derived groups have wings in which the number of cross-veins is
reduced and the main wing support is shifted anteriorly
(Wootton, 1990a).
Several studies have demonstrated the functional significance of specific
vein arrangements in insect wings. For example, the pleated, grid-like
arrangement of leading edge veins in dragonfly wings helps strengthen the wing
to spanwise bending (Newman and Wootton,
1986; Wootton,
1991
), posteriorly curved veins in flies generate chordwise camber
when a force is applied to the wing
(Ennos, 1988
), and the
fan-like distribution of veins in the locust hindwing causes the wing margin
to bend downward when the wing is extended
(Herbert et al., 2000
;
Wootton, 1995
;
Wootton et al., 2000
). Beyond
these specialized mechanisms, however, the functional significance of the
enormous differences in overall venation pattern in insect wings remains
unclear.
Given the large phylogenetic changes in cross-venation, vein diameter and spatial distribution of veins (Fig. 1), one might expect insect wings to display large mechanical differences that would affect their deformability during flight. On the other hand, differences in venation pattern may reflect alternative designs that provide insect wings with similar overall mechanical and bending properties while allowing veins to be rearranged for other reasons.
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Quantitative measurements of wing stiffness that would allow one to
distinguish between these hypotheses remain limited to a small number of
studies. Wing stiffness has been assessed by applying point forces to isolated
wing sections, in dragonflies (Newman and
Wootton, 1986) and locusts
(Wootton et al., 2000
), or at
the center of pressure to produce torsion, in flies
(Ennos, 1988
) and butterflies
(Wootton, 1993
). Steppan
(2000
) measured bending
stiffness in dried butterfly wings, and Smith et al.
(2000
) measured material
stiffness (Young's modulus, E) of insect wing membrane from locust
hindwings. Although each of these studies provides insight into the functional
wing morphology of the species examined, the measurements are difficult to
compare in a broader phylogenetic context because of variations in
technique.
In this study, we examined the relationship between insect wing flexibility and venation by measuring flexural stiffness (EI) and quantifying venation pattern in 16 insect species from six orders. Flexural stiffness is a composite measure of the overall bending stiffness of a wing; it is the product of the material stiffness (E, which describes the stiffness of the wing material itself) and the second moment of area (I, which describes the stiffness generated by the cross-sectional geometry of the wing). Because insect wings bend spanwise (from wing base to wing tip) and chordwise (from leading to trailing edge) during flight, we measured flexural stiffness in both of these directions.
Correlations between venation pattern and wing flexural stiffness may arise
either from a functional relationship between these traits or simply as a
result of the shared phylogenetic history of the species examined. To remove
the effects of phylogeny from this study, we calculated standardized
independent contrasts (Felsenstein,
1985; Garland et al.,
1999
) of venation and stiffness measurements, and examined the
correlations between these contrasts to assess the relationship between wing
venation pattern and flexural stiffness.
Finally, we created a simplified finite element model of an insect wing, in which we can alter the stiffness of specific wing veins (or remove veins entirely) and perform numerical experiments to assess the resulting flexural stiffness of the whole wing. This modeling approach allows us to examine the functional significance of various wing veins in generating the overall patterns of flexural stiffness measured in real wings.
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Materials and methods |
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Insects were collected locally or obtained from laboratory colonies and placed in a humidified container at 4°C until experiments were performed. Within 1 week of capture, we cold-anaesthetized an insect, recorded its mass, removed one forewing, and placed the insect back in the humidified container. We photographed the wing and measured its flexural stiffness in the spanwise direction within 1 h of removing the wing from the insect (during which time the stiffness of the wing does not change appreciably, according to trials). We then repeated this process for the other forewing, measuring flexural stiffness in the chordwise direction.
Flexural stiffness measurements
We measured flexural stiffness of wings by applying a point force to bend
the wing in either the spanwise or chordwise direction, and using the measured
force and wing displacement to calculate overall flexural stiffness
EI with a beam equation (see below). We glued wings to glass slides
using cyanoacrylate glue cured with baking soda, at either the wing base (for
spanwise measurements) or the leading edge (for chordwise measurements;
Fig. 2A). We then fixed the
slide to the left side of the apparatus shown in
Fig. 2B and adjusted the right
arm so that the pin contacted the wing on its dorsal surface. We applied the
point force at 70% of wing span or chord length because the pin slipped from
the wing if it was placed too close to the edge. We performed four
measurements with point loads of varying magnitude, lowering the right side of
the apparatus with a micrometer, measuring force and displacement and
returning to the zero (unloaded) position between each measurement. We then
flipped the slide over and repeated the measurements, loading the wing from
the ventral side.
|
We removed the slide and measured the distance from the point of wing
attachment to the point of force application to determine the effective beam
length (L). We calculated EI over this distance as in Gordon
(1978):
![]() | (1) |
We plotted spanwise and chordwise flexural stiffness against several measures of wing and body size: wing span and maximum chord length (measured in NIH Image), wing area (measured in Matlab; see below) and body mass. We also combined these size measurements using principal components analysis into a measure of overall size (including all four variables) and a measure of wing size (including wing span, chord and area).
To verify our technique for measuring EI, we measured force and
displacement of a thin, rectangular glass coverslip (a homogeneous beam of
linearly elastic material) at three different lengths (with 3, 4 and 4.85 cm
of the coverslip extending past the point of attachment). We then calculated
EI for each beam length, measured the thickness and width of the
coverslip, and calculated I, the second moment of area as in Gordon
(1978):
![]() | (2) |
Wing shape and venation measurements
From photographs of each wing, we created a black silhouette in Photoshop
and used NIH Image to measure wing span and chord. For one wing of each
species, we hand-digitized the wing veins in Photoshop so that both the
position and precise diameter of the veins were represented
(Fig. 1).
From the digitized vein images, we counted the number of vein intersections
in each wing as a measure of the complexity of vein branching. We imported
images of the wing silhouette and veins into Matlab and measured the planform
area of the whole wing and of the wing veins. From these measurements, we
calculated the vein density (proportion of planform wing area occupied by
veins) as the planform vein area divided by total wing area. We also
determined average vein thickness by finding the total length of veins in the
wing (see Combes, 2002) and
dividing planform vein area by total vein length. We divided this value by
wing span to scale for overall wing size.
Finally, we measured two venation characteristics related to the leading
edge of the wing: the proportion of veins in the leading edge and the density
of veins in the leading edge. We defined the leading edge visually as a
cohesive unit of veins running spanwise along the anterior edge of each wing
(Combes, 2002). We imported
digitized images of these leading edge veins and a silhouette of the leading
edge area (leading edge veins plus the area they surround) into Matlab. We
then calculated the proportion of veins in the leading edge as the leading
edge vein area divided by total planform vein area, and calculated leading
edge vein density as the leading edge vein area divided by the area of the
leading edge silhouette.
Phylogenetic analysis of correlated characters
Because the species tested share a phylogenetic history, some characters
(e.g. wing venation and flexural stiffness) may be correlated in closely
related groups simply because these species share a common ancestor that
possessed these characters, and not because the characters are related in any
functional way. As a result, the species used in this study cannot be treated
as independent data points for statistical analysis. However, any differences
in traits between two adjacent (closely related) groups can be assumed to have
occurred independently, after the two groups diverged. Therefore, we can
calculate the independent contrasts (or differences) between values of a trait
in adjacent groups, and plot the standardized contrasts of one trait against
those of another to see if a relationship exists between the traits when the
effect of phylogeny has been removed
(Felsenstein, 1985;
Garland et al., 1999
).
We calculated independent contrasts of flexural stiffness and wing venation
data using the Phenotypic Diversity Analysis Programs (PDAP), version 6.0,
developed by Garland, Midford, Jones, Dickerman and Diaz-Uriarte [originally
from Garland et al. (1993),
with modifications in Garland et al.
(1999
) and Garland and Ives
(2000
)]. This program allows
one to construct a phylogenetic tree and enter tip values for traits; it then
calculates independent contrasts and performs various diagnostics on the
output.
Some aspects of the phylogenetic relationships of the pterygotes (winged
insects) are under debate, and the divergence times of groups and families are
far from certain. However, we constructed a phylogeny of the species used in
this study by combining the available information, and in some cases averaging
or choosing midpoints between estimates of branching times in the phylogeny
(Fig. 1). To determine how
sensitive the analysis is to phylogenetic branch length, we also calculated
independent contrasts using arbitrary branch lengths
(Pagel, 1992), in which all
internode branches are equal to one, but the tips of the tree are
contemporaneous. We tested for correlations between standardized contrasts in
JMP 4 on a Macintosh computer.
Finite element modeling
To explore how wing veins contribute to the flexural stiffness of a wing,
we used MSC Marc/Mentat to create a simplified finite element model of an
insect wing based on the Manduca wing shown in
Fig. 1. Our goal was not to
reproduce the behavior of a real Manduca wing, but rather to create a
general model of a wing to explore how simply adding or strengthening veins in
certain regions of the wing affects the overall flexural stiffness of the
structure. Therefore, we did not attempt to recreate the precise
three-dimensional shape of a Manduca wing (including changes in
membrane thickness, vein cross-sectional shape and vein/membrane attachments),
but instead modeled the wing as a flat plate of uniform thickness, composed of
thin shell elements arranged to mimic the planform shape and vein
configuration of a Manduca wing
(Fig. 3A).
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This generalized model allowed us to increase the stiffness of various groups of veins beyond that of the surrounding membrane to determine how these veins contribute to overall wing flexural stiffness. The flexural stiffness of individual veins in real insect wings is determined by both their geometric properties (generally those of hollow tubes) and the material properties of their walls. A hollow tube has a second moment of area I that can be several orders of magnitude higher than that of a flat plate of the same width (and equivalent wall thickness). Because our model is composed solely of flat elements (with a lower second moment of area), we adjusted the flexural stiffness of the veins by increasing the material stiffness (E, Young's modulus) of these elements beyond that of the surrounding membrane elements.
We chose an element density of 1200 kg m-3 (as measured in
insect wings; Wainwright et al.,
1982) and an element thickness of 45 µm. We used a Poisson's
ratio of 0.49, as measured in some biological materials
(Wainwright et al., 1982
);
because the Poisson's ratio of insect wings is unknown, we tested the effects
of using a Poisson's ratio of 0.3 and found that the difference in model
behavior was negligible. To determine the minimum number of elements needed,
we performed a sensitivity analysis with models composed of 200, 350, 865 and
2300 total elements, and found that 865 elements are sufficient to ensure
asymptotic performance of the model.
We subjected the model wing to virtual static bending tests that mimic the tests performed on actual wings, fixing the base with zero displacement or rotation and applying a point force to the wing tip (Fig. 3A, blue dot). The model calculates the tip displacement due to this point force (and given the material properties of the membrane and veins). We then used the applied force, displacement, and wing span to calculate overall spanwise EI for the model wing as above (Equation 1). Similarly, we fixed the model wing at the leading edge and applied a point force at the trailing edge (Fig. 3A, orange dot) to calculate chordwise flexural stiffness.
In all simulations, we used a material stiffness of 1x109
Nm-2 for the membrane elements (as measured in locust wings;
Smith et al., 2000). In the
first set of simulations, we used a material stiffness of
1x109 Nm-2 for the vein elements as well (thus the
wing was essentially veinless in these simulations). We then increased the
material stiffness of the vein elements by orders of magnitude up to
1x1015 Nm-2, while membrane stiffness remained at
1x109 Nm-2. This increased vein material stiffness
represents not only potential differences in the material properties of wing
veins and membrane, but also differences in the second moment of area caused
by the three-dimensional shape of veins. To test the effect of leading edge
veins alone, we increased the material stiffness of the leading edge veins (in
pink, Fig. 3A) by orders of
magnitude to 1x1015 Nm-2, while fixing the
material stiffness of the remaining vein and membrane elements at
1x109 Nm-2.
To validate our finite element modeling approach, we created a model of the
glass coverslip used as an experimental control. We attached the coverslip at
its base, with 4.85 cm of the coverslip extending past the point of
attachment, and used a Young's modulus for glass of 7x1010
Nm-2 (Gordon,
1978). We applied a series of point forces to the tip of the
coverslip (equivalent to those applied in the experiment) and compared the tip
displacements predicted by the model to those measured in the experiment.
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Results |
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The measurements of overall wing flexural stiffness in this study did not
reveal a significant dorsal-ventral difference in the spanwise or chordwise
direction in any species tested; we therefore averaged dorsal and ventral
meaurements of EI in each direction. These flexural stiffness
measurements were significantly correlated with all size variables tested (see
Combes, 2002). However,
spanwise stiffness was more strongly correlated with wing span
(r2=0.95; Fig.
4A) than with any other size variable, or with the principal
components of wing or body size. Similarly, chordwise stiffness was most
strongly correlated with chord length (r2=0.91;
Fig. 4B). The measurements of
flexural stiffness also revealed a large anisotropy in all species tested,
with spanwise flexural stiffness approximately 1-2 orders of magnitude larger
than chordwise flexural stiffness (Fig.
4).
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Phylogenetic analysis of correlated characters
To verify that the correlations between flexural stiffness and size remain
significant when the effect of phylogeny is removed, we calculated
standardized independent contrasts of log-transformed spanwise EI,
wing span, chordwise EI and chord length. The relationships between
standardized independent contrasts of these wing size and flexural stiffness
traits were significant, and the slopes were nearly the same as in the
original data (span: y=3.04x, r2=0.96; chord:
y=2.02x, r2=0.96). The residuals from these
relationships can be used as an estimate of flexural stiffness with the effect
of size (and phylogeny) removed (for examples of similar uses in correcting
for body size, see Garland and Janis,
1993; Rezende et al.,
2002
).
These size-corrected estimates of flexural stiffness were not significantly
correlated with standardized independent contrasts of any of the five wing
venation characters measured. However, the residual of spanwise flexural
stiffness was positively correlated with the residual of chordwise flexural
stiffness (r2=0.37, P=0.012), and several
contrasts of vein characters were correlated with each other
(Combes, 2002). When Pagel's
arbitrary phylogenetic branch lengths
(Pagel, 1992
) were used, the
residuals of flexural stiffness remained uncorrelated with wing venation
characters, and the residuals of spanwise and chordwise flexural stiffness
remained positively correlated (Combes,
2002
).
Finite element modeling
The model of the glass coverslip provided estimates of tip displacement
that were within 5% of those measured in the experiments, validating the
finite element method. Virtual static bending tests of the model wing showed
that chordwise flexural stiffness was higher than spanwise flexural stiffness
when no veins were present (when membrane and vein material stiffness were the
same; Fig. 3B). However, when
veins were added and their material stiffness was increased, spanwise flexural
stiffness increased beyond chordwise flexural stiffness. When only leading
edge veins were added, spanwise flexural stiffness increased as above, while
chordwise flexural stiffness did not change significantly from the veinless
model (Fig. 3B).
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Discussion |
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This strong relationship is apparent despite several assumptions inherent in the use of a homogeneous beam equation (Equation 1). Because wings resemble plates more closely than beams, we used simple finite element models to assess how robust the beam equation is to variations in the shape of the beam. We created models of rectangular plates spanning the size and flexural stiffness range measured in real wings, applied point forces at the free end, and recorded the displacement of the model plate. We then calculated EI using Equation 1 and compared this to the known EI of the plate (based on its cross-sectional dimensions and material stiffness). We found that Equation 1 slightly overestimates flexural stiffness (by up to 12%) when the plate is longer than it is wide (i.e. for wings measured in the spanwise direction), and underestimates flexural stiffness (by up to 80%) when plates are wider than they are long. However, there is no overall trend in wing shape with increasing span or chord length, and these changes in EI are relatively small compared to the large range of flexural stiffness measured both within and between species (Fig. 4).
We also used the finite element model of an insect wing to explore the
effects of inherent wing camber on flexural stiffness estimates. The maximum
camber measured in Manduca wings with a laser ranging technique (see
Combes and Daniel, 2003a) was
5% in the chordwise direction and 4% in the spanwise direction. Applying these
levels of camber to the model wing had almost no effect on displacement when
the wing was cambered parallel to the wing attachment (i.e. when the model
wing was cambered in the chordwise direction and chordwise flexural stiffness
was measured). When the wing was cambered in the direction perpendicular to
the attachment (i.e. when the wing was cambered in the spanwise direction and
chordwise flexural stiffness was measured), displacement varied up to a
maximum of 40% from the value measured in a flat plate, indicating a
relatively minor effect on flexural stiffness.
Finally, the assumption in Equation 1 that flexural stiffness is
homogeneous across a wing may lead to a systematic error in the reported
values, which could potentially underlie the observed scaling relationships.
If flexural stiffness varies along the wing span or chord, the reported values
of overall flexural stiffness represent some weighted integration of
EI along the length measured. If this integral varies systematically
as beam length increases, it could account for part of the size scaling in the
data. We explored this hypothesis numerically by integrating various simple
functions (that represent how stiffness might vary in the wing) over
increasing beam length. We found that the integral of EI over the
wing may vary slightly with length depending on the function used, but this
variation is far smaller than the range of values measured in real wings, and
is therefore unlikely to cause the observed scaling relationships
(Combes, 2002).
Scaling of flexural stiffness
The strong correlations between wing size and flexural stiffness suggest
that size scaling is the dominant factor determining overall flexural
stiffness in insect wings. Because EI is a composite measure that
incorporates the second moment of area as well as the material stiffness of a
wing, it is not surprising that spanwise and chordwise flexural stiffness
increase with wing size; wings with larger spans generally also have larger
average chord lengths (and thus I, the second moment of area, is
higher). I is proportional to a beam's width times its thickness
cubed (Equation 2). For measurements of spanwise flexural stiffness, the width
of the beam is the average chord length, and in the species tested average
chord length is directly proportional to wing span
(y=0.2546x-0.0004, r2=0.8574). The
average thickness of the wings tested, however, is unknown. Average thickness
may be proportional to span (if wings grow isometrically), or could be
independent of span (since all cuticle consists of a single cell layer with
extracellular deposits). If we assume that E, the material stiffness
of wing cuticle, does not vary with wing size, we would predict that flexural
stiffness should scale with length (if thickness is independent of span) or
with length to the fourth power (if thickness is directly proportional to
span). The results of this study do not agree with either of these
predictions; spanwise EI scales with the cube of chord length,
whereas chordwise EI scales with the square of span. Thus, increased
second moment of area alone cannot account for the observed scaling of
spanwise and chordwise flexural stiffness.
Scaling of flexural stiffness has been examined previously in dried
butterfly wings (Steppan,
2000) and in the primary flight feathers of birds
(Worcester, 1996
), both of
which show a positive correlation between flexural stiffness and size. The
authors compared the observed scaling relationships with the theory of
geometric similarity (see Alexander et al.,
1979
), where structures maintain a similar shape regardless of
size, as well as with the theory of elastic similarity (see
McMahon, 1973
), in which
loaded structures maintain a similar angular deflection regardless of size.
Neither study (Steppan, 2000
;
Worcester, 1996
) found scaling
patterns that could be explained by geometric similarity, and the results of
our work appear to agree with these conclusions. The exponents of the
relationship between insect wing flexural stiffness and body mass (0.91
spanwise and 0.61 chordwise; Combes,
2002
) are far from the expected value of 1.67 proposed by
Worcester (1996
) for geometric
similarity, and the exponents of the relationship between flexural stiffness
and wing area (1.50 spanwise and 0.99 chordwise;
Combes, 2002
) are also far from
the expected value of 2.0 proposed by Steppan
(2000
).
If the scaling of wing flexural stiffness provides functional, rather than
geometric similarity across a range of body sizes, wing angular deflection
should remain constant, as in elastic similarity
(McMahon, 1973). If we take
tip displacement divided by wing span (or trailing edge displacement divided
by chord length) as a rough measure of strain or curvature in the wing, we can
rearrange Equation 1 as:
![]() | (3) |
To assess whether /L remains constant in flying insects
over a range of sizes, we need to know how the forces on flapping insect wings
scale with size. If we assume that the primary forces on an insect's wings are
aerodynamic, then force is proportional to body mass. However, several studies
have suggested that the inertial forces generated by flapping wings may be
considerably larger than the aerodynamic forces
(Combes and Daniel, 2003b
;
Daniel and Combes, 2002
;
Ellington, 1984
;
Ennos, 1989
;
Lehmann and Dickinson, 1997
;
Zanker and Gotz, 1990
), and
therefore inertial forces may be more important in determining wing
deformations. A generalized scaling argument for inertial force in insect
wings is difficult to derive because wingbeat frequency does not scale
strongly with size in the insects studied here. However, small insects often
have significantly higher wingbeat frequencies, so the ratio of inertial to
aerodynamic forces acting on their wings may be as high or higher than in
large insects with heavier (but slower) wings
(Combes and Daniel, 2003b
;
Daniel and Combes, 2002
).
Effects of wing venation on flexural stiffness
Although both spanwise and chordwise flexural stiffness scale with wing
length, the magnitude of flexural stiffness in these directions differs
greatly; spanwise EI is approximately 1-2 orders of magnitude higher
than chordwise EI in all species tested
(Fig. 4). Because spanwise
flexural stiffness increases as L3 and chordwise flexural
stiffness only as L2, this anisotropy is generally bigger
in larger-winged insects.
The finite element analysis of an insect wing shows that this structural anisotropy is due to a common venation feature of insect wings: leading edge veins. The model without any strengthening veins demonstrates that the basic planform shape of the wing would lead to similar spanwise and chordwise flexural stiffness if no veins were present (Fig. 3B). Adding leading edge veins to the model increases spanwise flexural stiffness dramatically, generating spanwise-chordwise anisotropy (Fig. 3B).
Clustered or thickened veins in the leading edge of the wing are found in
nearly all insects, even insects that have lost all other wing veins (such as
some hymenopterans and small dipterans). Thus, spanwise-chordwise anisotropy
may be a universal trait among insects. This anisotropy would serve to
strengthen the wing from bending in the spanwise direction while allowing
chordwise bending to generate camber. It could also facilitate spanwise
torsion, which is seen in many species during supination
(Ennos, 1988;
Wootton, 1981
).
Although leading edge veins appear to play a crucial role in determining the relative magnitudes of spanwise and chordwise flexural stiffness, the details of venation pattern measured in this study do not appear to affect the overall flexural stiffness of the wing. We did find, however, that the residuals of spanwise flexural stiffness are correlated with the residuals of chordwise flexural stiffness. This indicates that some insects have wings that are generally stiffer (in both directions) than expected for their size, while others have wings that are more flexible than expected. The residuals from the original data show that dragonflies, hawkmoths, flies (except for craneflies) and bumblebees all have wings that are stiffer than expected for their size. Damselflies, craneflies and lacewings have more flexible wings than expected, while butterflies and wasps are intermediate.
The functional significance of phylogenetic changes in wing venation (such
as loss of cross veins and increased vein thickness) remains unclear. Perhaps
more derived groups of insects have simply evolved a venation pattern that
allows them to maintain the essential scaling of wing stiffness in a more
economical way (e.g. using less vein material), or perhaps the venation
patterns are related to something entirely different, such as the distribution
of sensory receptors on the wing (Kammer,
1985).
Alternatively, venation pattern may in fact affect wing stiffness, but in
ways that could not be detected in this study. For example, venation pattern
may not affect overall stiffness, but could influence how stiffness varies
throughout the wing (see Combes and Daniel,
2003a). In addition, the stiffness measurements in this study
exclude the outer 30% of the wing, which is likely to be the most flexible
region. Differences in wing stiffness between insects with veins that extend
to and delineate the trailing edge (such as odonates; see
Fig. 1) and insects with
primarily unsupported membrane in the trailing edge (such as hymenopterans)
would most likely be found in this region. How the spatial distribution of
stiffness contributes to the instantaneous shape of a dynamically moving wing
is a subject of further study, and will be crucial to understanding the
implications of mechanical design of wings to insect flight performance.
R. Sugg and J. Edwards graciously assisted in identifying the insects used in this study. D. O'Carroll, T. Morse and J. Kingsolver provided specimens, and D. Combes assisted in collecting and transporting the spiderwasps. T. Garland provided the program for calculating independent contrasts, as well as helpful advice on its use. D. Grunbaum and R. Huey contributed useful comments on both the project and drafts of the paper. This work was supported by NSF grant F094801 to T.D., the John D. and Catherine T. MacArthur Foundation, an NSF graduate fellowship to S.C. and an ARCS fellowship to S.C.
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References |
---|
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---|
Alexander, R. McN., Jayes, A. S., Maloiy, G. M. O. and Wathuta, E. M. (1979). Allometry of the limb bones of mammals from shrews (Sorex) to elephant (Loxodonta). J. Zool., Lond. 189,305 -314.
Benton, M. J. (ed.) (1993). The Fossil Record 2, 845pp. London: Chapman & Hall. http://palaeo.gly.bris.ac.uk/frwhole/FR2.html
Borror, D. J., Triplehorn, C. A. and Johnson, N. F. (1989). An Introduction to the Study of Insects, 6th edition. Fort Worth, TX: Harcourt Brace College Publishers.
Combes, S. A. (2002). Wing flexibility and design for animal flight. PhD thesis, University of Washington, USA.
Combes, S. A. and Daniel, T. L. (2003a).
Flexural stiffness in insect wings. II. Spatial distribution and dynamic wing
bending. J. Exp. Biol.
206,2989
-2997.
Combes, S. A. and Daniel, T. L. (2003b). Into
thin air: Contributions of aerodynamic and inertial-elastic forces to wing
bending in the hawkmoth Manduca sexta. J. Exp.
Biol. 206,2999
-3006.
Dalton, S. (1975). Borne On The Wind: The Extraordinary World of Insects in Flight. New York: Reader's Digest Press.
Daniel, T. L. and Combes, S. A. (2002). Flexing wings and fins: bending by inertial or fluid-dynamic forces? Int. Comp. Biol. 42,1044 -1049.
Ellington, C. P. (1984). The aerodynamics of hovering insect flight. Part VI: Lift and power requirements. Phil. Trans. R. Soc. Lond. B 305,145 -181.
Ennos, A. R. (1988). The importance of torsion in the design of insect wings. J. Exp. Biol. 140,137 -160.
Ennos, A. R. (1989). Inertial and aerodynamic torques on the wings of Diptera in flight. J. Exp. Biol. 142,87 -95.
Felsenstein, J. (1985). Phylogenies and the comparative method. Am. Nat. 125, 1-15.[CrossRef]
Garland, T., Jr, Dickerman, A. W., Janis, C. M. and Jones, J. A. (1993). Phylogenetic analysis of covariance by computer simulation. Syst. Biol. 42,265 -292.
Garland, T., Jr, Midford, P. E. and Ives, A. R. (1999). An introduction to phylogenetically based statistical methods, with a new method for confidence intervals on ancestral states. Am. Zool. 39,374 -388.
Garland, T., Jr and Ives, A. R. (2000). Using the past to predict the present: Confidence intervals for regression equations in phylogenetic comparative methods. Am. Nat. 155,346 -364.[CrossRef][Medline]
Garland, T., Jr and Janis, C. M. (1993). Does metatarsal/femur ratio predict maximal running speed in cursorial mammals? J. Zool., Lond. 229,133 -151.
Gordon, J. E. (1978). Structures: or Why Things Don't Fall Down. New York: Penguin Books.
Herbert, R. C., Young, P. G., Smith, C. W., Wootton, R. J. and
Evans, K. E. (2000). The hind wing of the desert
locust (Schistocerca gregaria Forskål). III. A finite element
analysis of a deployable structure. J. Exp. Biol.
203,2945
-2955.
Kammer, A. E. (1985). Flying. In Comprehensive Insect Physiology, Biochemistry and Pharmacology. Vol. 5, Nervous system: structure and motor function (ed. G. A. Kerkut and L. I. Gilbert), pp.491 -552. Oxford: Pergamon Press.
Kent, G. C. (1992). Comparative Anatomy of the Vertebrates. St Louis, Missouri: Mosby-Year Book, Inc.
Kristensen, N. P. (1991). Phylogeny of extant hexapods. In The Insects of Australia: A textbook for students and research workers, 2nd edition, Vol.1 , pp. 125-140. Ithaca, NY: Cornell University Press.
Kukalova-Peck, J. (1991). Fossil history and the evolution of hexapod structures. In The Insects of Australia: A textbook for students and research workers, 2nd edition, vol.1 , pp. 141-179. Ithaca, NY: Cornell University Press.
Lehmann, F.-O. and Dickinson, M. H. (1997). The
changes in power requirements and muscle efficiency during elevated force
production in the fruit fly Drosophila melanogaster. J.
Exp. Biol. 200,1133
-1143.
Maddison, D. R. (1995a). Hymenoptera. In The Tree of Life Web Project, (ed. D. R. Maddison and K.-S. Schultz). http://tolweb.org/tree/phylogeny.html
Maddison, D. R. (1995b). Lepidoptera. In The Tree of Life Web Project, (ed. D. R. Maddison and K.-S. Schultz). http://tolweb.org/tree/phylogeny.html
McMahon, T. A. (1973). Size and shape in biology. Science 179,1201 -1204.[Medline]
Newman, D. J. S. and Wootton, R. J. (1986). An approach to the mechanics of pleating in dragonfly wings. J. Exp. Biol. 125,361 -372.
Pagel, M. D. (1992). A method for the analysis of comparative data. J. Theor. Biol. 164,194 -205.
Rezende, E. L., Swanson, D. L., Novoa, F. F. and Bozinovic,
F. (2002). Passerines versus nonpasserines: so far,
no statistical differences in the scaling of avian energetics. J.
Exp. Biol. 205,101
-107.
Smith, C. W., Herbert, R., Wootton, R. J. and Evans, K. E.
(2000). The hind wing of the desert locust (Schistocerca
gregaria Forskål). II. Mechanical properties and functioning of the
membrane. J. Exp. Biol.
203,2933
-2943.
Steppan, S. J. (2000). Flexural stiffness patterns of butterfly wings (Papilionoidea). J. Res. Lepid. 35,61 -77.
Trueman, J. W. H. and Rowe, R. J. (2001). Odonata. In The Tree of Life Web Project (ed. D. R. Maddison and K.-S. Schultz). http://tolweb.org/tree/phylogeny.html
Wainwright, S. A., Biggs, W. D., Currey, J. D. and Gosline, J. M. (1982). Mechanical Design in Organisms. Princeton, New Jersey: Princeton University Press.
Whiting, M. F., Carpenter, J. C., Wheeler, Q. D. and Wheeler, W. C. (1997). The Strepsiptera problem: Phylogeny of the holometabolous insect orders inferred from 18s and 28s ribosomal DNA sequences and morphology. Syst. Biol. 46, 1-68.[Medline]
Wiegmann, B. M. and Yeates, D. K. (1996). Diptera. In The Tree of Life Web Project (ed. D. R. Maddison and K.-S. Schultz). http://tolweb.org/tree/phylogeny.html
Wootton, R. J. (1981). Support and deformability in insect wings. J. Zool., Lond. 193,447 -468.
Wootton, R. J. (1990a). The mechanical design of insect wings. Sci. Am. November,114 -120.
Wootton, R. J. (1990b). Major insect radiations. In Major Evolutionary Radiations, Systematics Association Special Volume No. 42 (ed. P. D. Taylor and G. P. Larwood), pp. 187-208. Oxford: Clarendon Press.
Wootton, R. J. (1991). The functional morphology of the wings of Odonata. Adv. Odonatol. 5, 153-169.
Wootton, R. J. (1992). Functional morphology of insect wings. Annu. Rev. Entomol. 37,113 -140.[CrossRef]
Wootton, R. J. (1993). Leading edge section and
asymmetric twisting in the wings of flying butterflies (Insecta,
Papilionoidea). J. Exp. Biol.
180,105
-117.
Wootton, R. J. (1995). Geometry and mechanics of insect hindwing fans: a modelling approach. Proc. R. Soc. Lond. B 262,181 -187.
Wootton, R. J., Evans, K. E., Herbert, R. and Smith, C. W.
(2000). The hind wing of the desert locust (Schistocerca
gregaria Forskål). I. Functional morphology and mode of operation.
J. Exp. Biol. 203,2921
-2931.
Worcester, S. E. (1996). The scaling of the size and stiffness of primary flight feathers. J. Zool., Lond. 239,609 -624.
Zanker, J. M. and Gotz, K. G. (1990). The wing beat of Drosophila melanogaster. II. Dynamics. Phil. Trans. R. Soc. Lond. B 327,19 -44.