Flexural stiffness in insect wings II. Spatial distribution and dynamic wing bending
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 flexibility, Manduca sexta, Aeshna multicolor, finite element model.
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Introduction |
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Insect wings perform these roles extremely successfully, despite the fact
that they are largely passive structures, with no muscular control past the
wing base (Wootton, 1992).
Although they are strengthened by a network of tubular veins, the wings of
many species deform noticeably during flight, especially during slow flight
and hovering (Willmott and Ellington,
1997a
). These dynamic changes in the three-dimensional shape of
wings could potentially affect many aspects of force production, yet few
models of insect flight have successfully incorporated passive wing
flexibility or examined the effects of flexibility on force production.
In the past two decades, models of insect flight, both mathematical and
physical, have contributed enormously to our understanding of the basic
mechanisms of force production, despite assuming that insect wings are rigid
structures [e.g. mathematical models of Ellington
(1984a), Lighthill
(1973
), Sane and Dickinson
(2002
), Savage et al. (1978),
Smith et al. (1996
) and Wilkin
and Williams (1993
), and
physical models of Bennett
(1966
,
1970
), Dickinson et al.
(1999
), Ellington et al.
(1996
) and Spedding and
Maxworthy (1985)]. Some aspects of wing flexibility have been mimicked in
models by altering the relative positions of wing regions
(Liu et al., 1998
;
Vest and Katz, 1996
) or by
modeling deformations as harmonic waves
(Combes and Daniel, 2001
;
Daniel, 1987
;
Wu, 1971
). These approaches
provide unique insights into the mechanisms of force generation during flight,
but often neglect one or more critical components of wing deflection (e.g.
spanwise bending, chordwise bending or torsion), which can have large effects
on aerodynamic force production (Batchelor,
1967
). Models of insect flight that incorporate passive wing
flexibility (in which shape changes are driven by forces imposed upon the wing
rather than being specified in advance) are exceedingly rare (e.g.
Smith, 1996
).
One difficulty in modeling passive wing flexibility is that forces applied
at the wing base lead to bending and twisting that are influenced not only by
overall stiffness and gross anatomical features (e.g. flexion or fold lines),
but also by the distribution of flexibility throughout the wing
(Wootton, 1999). Extremely
detailed finite element models of insect wings (incorporating vein
configuration, three-dimensional relief, and variations in vein and membrane
characteristics) could potentially reproduce the distribution of flexibility
in real wings by accounting for the precise structural and material properties
of the wing. Unfortunately, precise data about the local properties of insect
wings are often unavailable and finite element models must be constructed
using simplifying assumptions. For example, many finite element models assume
that the material stiffness of the wing (Young's modulus, E) is
equivalent to that of insect chitin and constant throughout the wing
(Smith, 1996
;
Kesel et al., 1998
); however,
recent measurements reveal that E can vary widely within a wing
(Smith et al., 2000
) and that
other proteins, such as resilin, occupy key positions (e.g. wing vein joints)
in insect wings (Gorb, 1999
;
Haas et al.,
2000a
,b
).
Measuring spatial variation in these material properties, as well as the
details of vein and membrane structure, is a time-consuming process that would
need to be repeated for each new species studied.
Although these detailed approaches can provide important information about
functional wing morphology in pivotal, well-studied species (e.g.
Herbert et al., 2000), a more
general approach to wing stiffness measurements and modeling could facilitate
comparative studies and attempts to incorporate passive flexibility into
models of force production. Rather than measuring (and modeling) geometric and
material properties separately, the overall bending response of the wing, or
flexural stiffness (EI, the product of material stiffness E
and second moment of area I) can be determined. Performing
measurements of EI averaged over the whole wing is relatively
straightforward (see Combes and Daniel,
2003a
), but measuring spatial variation in flexural stiffness
throughout a wing is more challenging. Steppan
(2000
) approached this issue
by measuring average flexural stiffness over increasingly larger sections of
dried butterfly wings, and Wootton et al.
(2000
) measured average
flexural stiffness in three isolated sections of locust hindwing. However, no
studies to date have demonstrated a method of measuring the spatial variation
in local flexural stiffness of intact insect wings.
In this study, we developed a method of approximating spatial variation in
flexural stiffness along two axes of the wing (in the spanwise and chordwise
direction). We measured flexural stiffness variation in two insects, the
hawkmoth Manduca sexta and the dragonfly Aeshna multicolor.
These insects have wings of similar size and are both agile fliers, capable of
hovering as well as fast, forward flight. However, they are very distantly
related and display large differences in wing shape and venation pattern (see
Combes and Daniel, 2003a) that
may underlie differences in the distribution of flexural stiffness in their
wings.
To determine the spatial pattern of flexural stiffness in wings, we first
developed a method to measure displacement (due to a point force) along the
wing in the spanwise and chordwise directions. Next, we proposed various
alternatives for how flexural stiffness might vary along the wing (represented
by simple mathematical functions) and predicted the patterns of displacement
that a loaded wing with these theoretical stiffness distributions would
display. We then found the flexural stiffness distribution that produced a
pattern of displacement most similar to the pattern measured in a real wing.
We used this method to estimate the flexural stiffness distribution of the
wing in response to forces applied from both the dorsal and the ventral side,
as dorsal/ventral asymmetry has been noted in previous studies
(Ennos, 1988;
Steppan, 2000
;
Wootton, 1993
;
Wootton et al., 2000
).
We also created two different finite element models of an insect wing, based on the planform geometry of a Manduca sexta forewing. Rather than realistically reproducing the structure and behavior of Manduca wings, these generalized models provide a method of assessing the consequences of flexural stiffness distributions to wing bending. We attempted to capture major elements of wing structure in the models (e.g. vein position and flexural stiffness distribution) while avoiding detailed structural features, such as variation in wing thickness and three-dimensional relief. We used these models to examine how different spatial patterns of EI determine bending patterns in wings subjected to both static and dynamic loads.
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Materials and methods |
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We measured displacement in response to loads of varying magnitude, returning to the unloaded position to photograph the wing before applying each new load. We checked the repeatability of displacement measurements on the wings of a hawkmoth (in both the spanwise and chordwise direction) by applying five different loads to the dorsal surface of the wing and repeating each load three times (applying loads in random order).
For all other hawkmoth and dragonfly wings, we measured displacement on the
dorsal surface of the wing at four different loads, gently removed the glued
pin from the edge of the wing (without tearing the wing membrane), flipped the
slide over, and measured displacement on the ventral side of the wing at four
different loads. Dorsal stiffness in this study describes the stiffness of the
wing in response to loading on the dorsal surface (resulting in a dorsally
convex surface; see Steppan,
2000), which is equivalent to ventral flexion in other studies
(Ennos and Wootton, 1989
;
Wootton, 1981
,
1992
).
After completing the flexural stiffness measurements, we photographed the
wings on a white background from above and measured wing area, span and
maximum chord length in NIH Image. For Manduca sexta (whose wing
morphology may depend on gender; Willmott
and Ellington, 1997b), we also calculated aspect ratio
(span2/area) of the forewings and wing loading (body mass divided
by the area of both forewings).
In each hawkmoth, we measured spanwise flexural stiffness on one forewing and chordwise flexural stiffness on the other. We did not measure the flexural stiffness of the hindwings, which are much smaller and overlap with the forewings to varying degrees during flight. In dragonflies (which have independent fore- and hindwings of similar size), we measured spanwise flexural stiffness on one forewing and one hindwing, and chordwise flexural stiffness on the other forewing and hindwing.
We analyzed images of loaded and unloaded wings with a custom Matlab
program (developed by A. Trimble) that finds the center of the laser line
running from the wing base to the tip or from the leading to the trailing edge
(Fig. 1A,B). We used a
2nd order Butterworth filter to remove noise from the line position
data, and splined the unloaded and loaded data to an equal number of points
for comparison. We found the difference in line position between the two data
sets [y(x); Fig.
1B] and converted this difference to actual wing displacement
along the wing (x) with a factor derived from analysis of the
calibration object.
With this displacement data and the applied force, we were able to estimate
local flexural stiffness by using a continuous beam equation to approximate
EI variation along the wing (see Appendix). This equation defines
local flexural stiffness as a function of the local curvature (second spatial
derivative of displacement) and the local moment (applied force times distance
from the point of force application). To avoid errors caused by
differentiating displacement data, we solved this as an inverse problem; we
posed several simple mathematical functions that might approximate flexural
stiffness variation along the wing (constant, linear, exponential, or
2nd degree polynomial) and calculated the expected pattern of
displacement due to the applied force and these possible EI
distributions. We then used a simplex minimization program to find the
EI distribution that provided the best fit to the measured
displacement along the wing. Finally, we calculated average flexural stiffness
() by integrating the equation describing flexural stiffness
variation along the wing's length (see Appendix).
To verify the method of determining flexural stiffness distribution, we
measured (x) and calculated local flexural stiffness of a
rectangular glass coverslip, which is made of homogeneous material and has a
constant material stiffness (E) and second moment of area
(I).
Comparisons within and between individuals
Because wing displacement in hawkmoth and dragonfly wings does not increase
linearly with applied force (see Combes,
2002), our estimates of flexural stiffness varied with the
relative displacement of the wing (
T/L, tip
displacement divided by wing span or trailing edge displacement divided by
chord length). To facilitate comparisons between individuals and species, we
standardized measurements at a relative displacement of 0.05 for spanwise
measurements and 0.08 for chordwise measurements (values that are within the
range measured on real wings). For each individual, we used the slope of the
relationship between EI variables (
and coefficients
describing the distribution of flexural stiffness) and relative displacement
to estimate the value of these variables at a given
T/L.
In Manduca sexta, we collected spanwise data from nine males and four females, and chordwise data from ten males and nine females. In Aeshna multicolor, we used only males, but tested both fore and hindwings (which have different morphologies), so we examined stiffness separately in the two sets of wings. We collected spanwise and chordwise data on both fore and hindwings from eight individuals.
Within each group, we tested for differences between the dorsal and ventral side of the wing with a Wilcoxon signed rank test. We also tested for differences between male and female Manduca sexta with a Mann-Whitney U-test.
Finite element modeling
To investigate how flexural stiffness variation affects wing bending, we
created two simplified finite element models of an insect wing (based on the
forewing of a male Manduca sexta) with the same geometry, but
dramatically different patterns of spatial variation in flexural stiffness
(Fig. 2). We used MSC
Marc/Mentat 2001 to construct model wings with an accurate planform
configuration of veins and membrane, but a simplified three-dimensional
geometry (with no camber or three-dimensional relief; see Discussion). We
increased the material stiffness of vein elements beyond that of the
surrounding membrane to mimic the higher second moment of area of tubular
veins and produce the measured spanwise-chordwise anisotropy in flexural
stiffness (see Combes and Daniel,
2003a).
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The models were composed of thin shell elements with a density of 1200 kg
m-3 (as measured in insect wings;
Wainwright et al., 1982) and a
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. The uniform mass distribution of
the models may cause dynamic bending in the distal regions to be
overestimated, but qualitative differences in wing bending between the two
models should not be affected. To determine the minimum number of elements
necessary to capture the bending behavior of wings, we performed a sensitivity
analysis with models composed of 200, 350, 865 and 2300 total elements, and
found that 865 elements were sufficient to ensure asymptotic performance of
the model.
We adjusted the material stiffness E of vein and membrane elements
in the two models to produce different spatial patterns of flexural stiffness,
but the same overall bending performance (so that tip and trailing edge
displacement in response to a point load were the same as displacements
measured in real wings). In the first model, we assigned all vein elements a
single (homogeneous) Young's modulus of E=1.5x108 N
m-2 (similar to values measured in locust hindwing;
Smith et al., 2000), and all
membrane elements a Young's modulus of E=2.1x1012 N
m-2 (representing both increased material stiffness and the
increased second moment of area of tubular veins;
Fig. 2A). In the second model,
we applied declining values of material stiffness to the model wing in 12
strips oriented diagonally (Fig.
2B); these strips are perpendicular to most of the wing veins,
which decrease in diameter towards the wing edge and thus are likely to
decrease in stiffness along this axis. We adjusted the values of material
stiffness in these strips to approximate patterns of overall wing flexural
stiffness measured in real Manduca wings (E in the model
varies from 4.7x107 N m-2 to
4.5x109 N m-2 in membrane elements, and from
1.9x1011 N m-2 to 1.8x1013 N
m-2 in vein elements).
To determine the resulting pattern of flexural stiffness variation in the spanwise direction of the model wings, we fixed each wing at its base (with no displacement or rotation) and applied a point force at the tip. For chordwise measurements, we fixed the model at its leading edge (from the base to 2/3 span) and applied a point force at the trailing edge. We recorded the displacement at 22 nodes (junctions between elements) aligned between the point of attachment and the point of force application, and used this information to estimate spanwise and chordwise flexural stiffness distribution with the Matlab simplex minimization program, as in real wings.
We compared static bending performance of the two models by fixing each
wing at its base and applying either a point load of 0.003 N at the tip
(within the range of loads applied to real wings) or a pressure load of -14.43
Pa to the wing surface (equivalent to the average lift force that the wing
would experience during steady flight). We compared dynamic bending by
flapping each model wing at a realistic wing beat frequency and stroke
amplitude. We applied boundary conditions to the nodes at the wing base so
that they could not translate in any direction and could rotate only in the
dorsal-ventral direction (around the y-axis, see
Fig. 2). We began the
simulation with initial conditions of zero displacement and zero velocity at
all nodes, and gradually increased the rotation at the wing hinge to a
sinusoidal motion with the following function:
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Because the finite element program does not calculate aerodynamic forces
acting on the wing, a damping factor had to be applied to stabilize the
models. We compared the motions of the model wing with declining material
stiffness to those of a real Manduca wing attached to a motor and
rotated in the same way (Combes and Daniel,
2003b; Daniel and Combes,
2002
). We found that a mass damping factor of 10 reduced high
frequency vibrations and provided the closest match to motions of the real
wing, and applied this damping factor to both model wings.
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Results |
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Our control measurements validated the procedure for estimating local
flexural stiffness and revealed that the method is repeatable. Bending tests
on a glass coverslip showed that a constant distribution of flexural stiffness
provided the best approximation of measured displacement (as expected for a
homogeneous beam). In addition, measurements of Manduca flexural
stiffness in which each load was repeated several times provided consistent
estimates of the EI distribution each time that a given load was
applied (see Combes, 2002).
Morphological measurements showed that female Manduca sexta are
significantly heavier and have larger wings than males (body mass,
P=0.004; wing area, P=0.003; wing span, P=0.002,
chord length, P=0.004; wing mass, P=0.012), but aspect ratio
and wing loading were not significantly different in the individuals sampled
(aspect ratio, P=0.946; wing loading, P=0.262). Although
overall flexural stiffness scales with wing size across a broad range of
species (see Combes and Daniel,
2003a), spanwise
was not significantly different in male
and female Manduca (dorsal, P=0.165; ventral,
P=0.308; Fig. 3A,B).
In the chordwise direction,
was higher in females than in males on
the dorsal side of the wing (P=0.021), but not on the ventral side
(P=0.477). The exponents of the stiffness distribution in the
chordwise direction were significantly higher in females than in males
(dorsal, P=0.006; ventral, P=0.008), but exponents in the
spanwise direction were the same in males and females (dorsal,
P=0.165; ventral, P=0.089).
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The wings of all Manduca sexta displayed dorsal/ventral asymmetry; the average flexural stiffness of their wings was higher in response to forces applied to the dorsal side of the wing, in both the spanwise and chordwise direction (spanwise, P=0.002; chordwise, P<0.001; Fig. 3A,B). In contrast, Aeshna multicolor displayed no dorsal/ventral difference in its hindwings or in the chordwise direction of its forewings (hindwing spanwise, P=0.327; hindwing chordwise, P=0.735; forewing chordwise, P=0.069; Fig. 3C,D). In the spanwise direction, the forewings were stiffer on the dorsal side than on the ventral side (P=0.018), but the difference between dorsal and ventral flexural stiffness was far smaller than that seen in Manduca.
Finite element modeling
The finite element model with diagonal strips of declining material
stiffness accurately reproduced the sharp decline in flexural stiffness
measured in male Manduca wings
(Fig. 3A, red lines), while the
model with homogeneous vein and membrane regions displayed a vastly different
pattern of flexural stiffness (Fig.
3A, blue lines).
The effects of these differences in EI distribution were apparent in static tests on the model wings. Although an applied point force resulted in the same tip displacement, most bending in the exponential wing was confined to the outer third of the wing, while the homogeneous wing bent gradually along much of its span (Fig. 4A). When subjected to a pressure load, the homogeneous wing again bent along a large portion of its length, and its maximum displacement was nearly twice as large as the displacement of the exponential wing (Fig. 4B). In the exponential wing, displacement was localized to the tip and trailing edge of the wing, and bending was apparent in both the spanwise and chordwise directions.
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The effects of an exponential decline in flexural stiffness were further illustrated by dynamic tests on the models. The homogeneous wing showed little chordwise bending and spanwise bending was most pronounced near the base, while the distal portion of the wing remained relatively rigid (Fig. 5, blue wings). In contrast, the exponential wing bent considerably in the chordwise direction, and bending in the spanwise direction was confined mainly to the outer portion of the wing (Fig. 5, red wings). For a movie of the model wings in motion, see http://faculty.washington.edu/danielt/movies.
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Discussion |
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Our measurements of spatial variation in flexural stiffness agree with these qualitative observations; flexural stiffness declines sharply from wing base to tip and from the leading to the trailing edge in the wings of both Manduca sexta and Aeshna multicolor. Displacement in the spanwise and chordwise directions of these wings can be predicted fairly well when the flexural stiffness distribution is approximated by an exponential equation. The observed similarities in flexural stiffness distribution in these two species (despite large differences in wing shape and venation pattern) suggest that a sharp decline in wing flexural stiffness towards the tip and trailing edge may be a common feature of many insect wings.
Although detailed inter-specific comparisons are not feasible with only two
species, this study demonstrates several interesting intra-specific
differences, particularly in Manduca sexta. Male and female
Manduca sexta appear to differ in the spatial patterns of flexural
stiffness in their wings; chordwise flexural stiffness declines far more
sharply in male moths than in female moths
(Fig. 3A,B), and as a result,
the wings of female moths are significantly stiffer in the chordwise direction
(dorsal is higher). The functional significance of these differences
is unclear, but other aspects of Manduca wing morphology (such as the
physical coupling of fore- and hindwings) also display sexual dimorphism
(Eaton, 1988
), which may be
related to higher wing loading in females carrying eggs
(Willmott and Ellington,
1997b
). The traits measured in female hawkmoth wings could in some
way compensate for this intermittent higher wing loading, for example by
influencing the pattern or extent of chordwise wing deflections during
flight.
The wings of both male and female Manduca sexta display a large
dorsal/ventral difference in average flexural stiffness, and in the exponents
of flexural stiffness distribution. Average flexural stiffness is greater in
response to forces applied on the dorsal side than on the ventral side (in
both the spanwise and chordwise direction). In the spanwise direction, this
difference is greatest near the base (Fig.
3A,B). Steppan
(2000) found a similar
dorsal/ventral asymmetry near the base of butterfly wings, and higher dorsal
flexural stiffness has also been measured in the leading edge of locust
hindwings (Wootton et al.,
2000
).
Although measurements of such bending asymmetry are rare, theoretical
studies have suggested that insect wings might display dorsal/ventral
stiffness asymmetry due to the camber inherent in most wings
(Ennos, 1995,
1997
;
Wootton, 1993
). We explored
the effects of camber on bending asymmetry in the finite element model of a
Manduca wing, adding 4% spanwise and 5% chordwise camber (the maximum
values measured in real wings). Adding this degree of camber to the model
wings had a relatively small effect on tip and trailing edge displacement (see
Combes and Daniel, 2003a
), and
failed to produce any dorsal/ventral bending asymmetry (tip displacement was
identical whether the force was applied to the convex or the concave side; see
Combes, 2002
).
Camber may in fact influence the maximum load each side of the wing can resist before buckling (the buckling load), and the finite element model does not simulate elastically stable buckling. However, our measurements on real wings do not display the strong non-linear dependence on loading that would be expected if elastically stable buckling were occurring during these static tests. In a more dynamic situation (such as flapping flight), where continuous shape changes can alter flexural stiffness, camber may in fact lead to dorsal/ventral differences in wing deformation. However, wing camber does not appear to explain the large differences measured in this study on wings subjected to static point loading.
The structural source of this bending asymmetry remains unclear. Although
Manduca wings have none of the gross anatomical features often
associated with bending asymmetry (e.g. a ventral flexion line), their veins
may contain one-way hinges or other micro-structural features that facilitate
asymmetric bending in other insect wings
(Wootton, 1981;
Wootton, 1992
). In addition,
the stress-stiffening effect explored by Kesel et al.
(1998
) in models of dipteran
wings could potentially apply to Manduca. When unloaded, the
membranes of Manduca wings lie rather loosely between the veins
extending to the trailing edge. A point force applied to the dorsal (convex)
side of the trailing edge appears to pull the trailing edge veins further
apart, which would first remove any slack in the membrane, and then possibly
increase the stiffness of the wing as force is applied to the already-taut
membrane (the stress-stiffening effect). In contrast, when a point force is
applied to the ventral side of the trailing edge, the veins are pushed
together and the membranes between them become looser.
Although the finite element models do not reproduce this dorsal/ventral asymmetry, the strikingly different bending patterns of the two models demonstrate that flexural stiffness distribution is critical in determining how insect wings bend. In the model wing with exponentially declining flexural stiffness, both static and dynamic bending are limited to the tip and trailing edge, whereas bending in the homogeneous model occurs along the entire span.
These results suggest that the sharply declining flexural stiffness
measured in real wings helps maintain rigidity near the wing base (despite
larger bending moments), while localizing bending to the tip and trailing
edge, which are regions of particular importance in controlling aerodynamic
force production. The trailing edge is a critical control surface in airplane
wings (affecting both the magnitude of lift and the lift-to-drag ratio;
Anderson, 1991), and passive
deformations in this region are likely to play a similarly important role in
controlling flight forces in insects. The integration of passive flexibility
and spatial patterns of flexural stiffness into models of insect flight will
be an important step in determining the functional significance of wing
structure and dynamic bending to insect flight performance.
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Appendix |
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![]() | (A1) |
![]() | (A2) |
If tip displacement is extremely large relative to beam length, the beam is
bent down so far that its coordinates in the x-dimension while loaded
are significantly different from its coordinates while unloaded, invalidating
Equation A2. However, Equation A2 is valid for our measurements for the
following reasons: (i) all measurements were standardized to a relative
displacement (T/L) of 0.05 in the spanwise
direction and 0.08 in the chordwise direction, (ii) original measurements were
rarely performed at relative displacements over 0.08 in the spanwise direction
and 0.15 in the chordwise direction, and (iii) even in the most extreme cases
(where relative displacement exceeded 0.15), the error in coordinates
introduced by length changes in the x-dimension was less than 4%.
We solved Equation A2 in reverse, posing a functional distribution for
EI(x), calculating the expected displacement
(x), and using simplex minimization to find the parameters
that minimize the difference between predicted and measured wing displacement.
We proposed that stiffness in the spanwise or chordwise direction might be
constant, or that its spatial pattern might be approximated by a linear,
exponential, or 2nd-degree polynomial equation:
![]() | (A3) |
![]() | (A4) |
![]() | (A5) |
![]() | (A6) |
Each of the proposed EI distributions (Equations A3-A6) was
inserted into the following equation to find the expected displacement along
the span or chord given an applied moment [F(L-x)]:
![]() | (A7) |
After estimating the spatial distribution of flexural stiffness, we
calculated average flexural stiffness of the wing by integrating the
continuous distribution EI(x):
![]() | (A8) |
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List of symbols |
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A. Trimble was critical in devising the method of measuring displacement by a laser ranging technique and in developing Matlab code to analyze loaded and unloaded images. M. Tu and E. Goldman generously provided useful comments and Matlab wisdom. J. Henry provided advice on collecting dragonflies and donated specimens to the study. J. Dierberger at MSC Software provided crucial troubleshooting of the FEM models, without which the dynamic simulations would not have been possible. This work was supported by NSF grant F094801 to T. Daniel, the John D. and Catherine T. MacArthur Foundation, an NSF graduate fellowship to S. Combes, and an ARCS fellowship to S. Combes.
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References |
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Anderson, J. D., Jr (1991). Fundamentals of Aerodynamics. New York: McGraw-Hill, Inc.
Batchelor, G. K. (1967). An Introduction to Fluid Dynamics. Cambridge, UK: Cambridge University Press.
Bennett, L. (1966). Insect aerodynamics: vertical sustaining force in near-hovering flight. Science 152,1263 -1266.
Bennett, L. (1970). Insect flight: lift and rate of change of incidence. Science 167,177 -179.
Combes, S. A. (2002). Wing flexibility and design for animal flight. PhD thesis, University of Washington.
Combes, S. A. and Daniel, T. L. (2001). Shape, flapping and flexion: Wing and fin design for forward flight. J. Exp. Biol. 204,2073 -2085.[Medline]
Combes and Daniel (2003a). Flexural stiffness
in insect wings. I. Scaling and the influence of wing venation. J.
Exp. Biol. 206,2979
-2987.
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.
Daniel, T. L. (1987). Forward flapping flight from flexible fins. Can J. Zool. 66,630 -638.
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.
Dickinson, M. H., Lehman, F.-O. and Sane, S. P.
(1999). Wing rotation and the aerodynamic basis of insect flight.
Science 284,1954
-1960.
Eaton, J. L. (1988). Lepidopteran Anatomy. New York: John Wiley & Sons.
Ellington, C. P. (1984a). The aerodynamics of hovering insect flight. I. The quasi steady analysis. Phil. Trans. R. Soc. Lond. B 305,1 -15.
Ellington, C. P. (1984b). The aerodynamics of hovering insect flight. II. Morphological parameters. Phil. Trans. R. Soc. Lond. B 305,17 -40.
Ellington, C. P., Van den Berg, C., Willmott, A. P. and Thomas, A. L. R. (1996). Leading-edge vortices in insect flight. Nature 384,626 -630.[CrossRef]
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.
Ennos, A. R. (1995). Mechanical behavior in torsion of insect wings, blades of grass and other cambered structures. Proc. R. Soc. Lond. B 259, 15-18.
Ennos, A. R. (1997). Flexible structures in biology. Comments Theor. Biol. 4, 133-149.
Ennos, A. R. and Wootton, R. J. (1989). Functional wing morphology and aerodynamics of Panorpa germanica (Insecta: Mecoptera). J. Exp. Biol. 143,267 -284.
Gorb, S. N. (1999). Serial elastic elements in the damselfly wing: mobile vein joints contain resilin. Naturwissenschaften 86,552 -555.[CrossRef][Medline]
Haas, F., Gorb, S. and Blickhan, R. (2000a). The function of resilin in beetle wings. Proc. R. Soc. Lond. B 267,1375 -1381.[CrossRef][Medline]
Haas, F., Gorb, S. and Wootton, R. J. (2000b). Elastic joints in dermapteran hind wings: materials and wing folding. Arthropod. Struct. Dev. 29,137 -146.[CrossRef]
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.
Kesel, A. B., Philippi, U. and Nachtigall, W. (1998). Biomechanical aspects of the insect wing: an analysis using the finite element method. Comp. Biol. Med. 28,423 -437.[CrossRef][Medline]
Lighthill, M. J. (1973). On the Weis-Fogh mechanism of lift generation. J. Fluid Mech. 60, 1-17.
Liu, H., Ellington, C. P., Kawachi, K., Van den Berg, C. and
Willmott, A. P. (1998). A computational fluid dynamic study
of hawkmoth hovering. J. Exp. Biol.
201,461
-477.
Sane, S. P. and Dickinson, M. H. (2002). The
aerodynamic effects of wing rotation and a revised quasi-steady model of
flapping flight. J. Exp. Biol.
205,1087
-1096.
Savage, S. B., Newman, B. G. and Wong, D. T.-M. (1979). The role of vortices and unsteady effects during the hovering flight of dragonflies. J. Exp. Biol. 83, 59-77.
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.
Smith, M. J. C. (1996). Simulating moth wing aerodynamics: towards the development of flapping-wing technology. AIAA J. 34,1348 -1355.
Smith, M. J. C., Wilkin, P. J., and Williams, M. H.
(1996). The advantages of an unsteady panel method in modelling
the aerodynamic forces on rigid flapping wings. J. Exp.
Biol. 199,1073
-1083.
Spedding, G. R. and Maxworthy, T. (1986). The generation of circulation and lift in a rigid two-dimensional fling. J. Fluid. Mech. 165,247 -272.
Steppan, S. J. (2000). Flexural stiffness patterns of butterfly wings (Papilionoidea). J. Res. Lepid. 35,61 -77.
Vest, M. S. and Katz, J. (1996). Unsteady aerodynamic model of flapping wings. AIAA J. 34,1435 -1440.
Wainwright, S. A., Biggs, W. D., Currey, J. D. and Gosline, J. M. (1982). Mechanical Design in Organisms. Princeton, New Jersey: Princeton University Press.
Wilkin, P. J. and Williams, M. H. (1993). Comparison of the aerodynamic forces on a flying sphingid moth with those predicted by quasi-steady theory. Physiol. Zool. 66,1015 -1044.
Willmott, A. P. and Ellington, C. P. (1997a).
The mechanics of flight in the hawkmoth Manduca sexta. I. Kinematics
of hovering and forward flight. J. Exp. Biol.
200,2705
-2722.
Willmott, A. P. and Ellington, C. P. (1997b).
The mechanics of flight in the hawkmoth Manduca sexta. II.
Aerodynamic consequences of kinematic and morphological variation.
J. Exp. Biol. 200,2723
-2745.
Wootton, R. J. (1981). Support and deformability in insect wings. J. Zool., Lond. 193,447 -468.
Wootton, R. J. (1990). The mechanical design of insect wings. Sci. Am. November,114 -120.
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. (1999). Invertebrate paraxial
locomotory appendages: design, deformation and control. J. Exp.
Biol. 202,3333
-3345.
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.
Wu, T. Y. (1971). Hydromechanics of swimming propulsion. Part 1. Swimming of a two dimensional flexible plate at variable forward speeds in an inviscid fluid. J. Fluid Mech. 46,337 -355.