Into thin air: contributions of aerodynamic and inertial-elastic forces to wing bending in the hawkmoth Manduca sexta
Department of Biology, University of Washington, Seattle, WA 98195, USA
* Author for correspondence (e-mail: scombes{at}u.washington.edu)
Accepted 18 May 2003
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Summary |
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Key words: insect flight, wing flexibility, wing bending, aerodynamic forces, inertial forces, finite element model, Manduca sexta
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Introduction |
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In some insect species, such as Drosophila, wing bending is
limited, and physical or mathematical models that assume the wings are rigid
can provide significant insights into mechanisms of unsteady force production
(e.g. Dickinson et al., 1999;
Ramamurti and Sandberg, 2002
;
Sane and Dickinson, 2002
;
Sun and Tang, 2002a
). However,
the wings of many species, such as Manduca, bend and twist
dramatically during flight (Dalton,
1975
; Wootton,
1990
), particularly during slow flight and hovering
(Willmott and Ellington,
1997
). Most computational models of flight in Manduca
have accounted for wing bending by incorporating simplified shape changes that
are specified in advance (Liu et al.,
1998
; Liu and Kawachi,
1998
); these approaches have contributed substantially to our
understanding of fluid dynamic force generation in specific situations, such
as during hovering flight. However, models of insect flight incorporating
passive wing deformations could be used to address further questions of
functional wing morphology and evolution, as well as to explore the effects of
alternative kinematic patterns on dynamic wing shape and insect flight
performance.
Unfortunately, the development of these integrative models has been
hindered by uncertainty about the relative importance of fluid-dynamic and
inertial-elastic forces in determining dynamic wing shape. Some estimates of
overall wing inertia (averaged spatially and/or temporally) suggest that
inertial forces are generally higher than aerodynamic forces
(Ellington, 1984b;
Lehmann and Dickinson, 1997
;
Wilkin and Williams, 1993
;
Zanker and Gotz, 1990
),
whereas other studies conclude the opposite
(Sun and Tang, 2002b
;
Wakeling and Ellington, 1997
).
A limited number of theoretical studies addressing local bending moments in
flapping wings suggest that inertial-elastic forces may play a larger role
than aerodynamic forces in determining instantaneous wing shape. For example,
Ennos (1989
) estimated that
spanwise bending moments due to the inertia of flapping wings are at least
twice as large as those due to aerodynamic forces, and showed that wing
inertia alone could cause the tip-to-base torsional wave seen in many insect
wings during supination (Ennos,
1988
). Daniel and Combes
(2002
) showed that chordwise
bending moments generated by elastic wave propagation in flapping insect wings
(inertial-elastic effects) are significantly larger than the moments exerted
on wings by the surrounding fluid.
In this study, we used an experimental approach to examine the relative contributions of inertial-elastic and fluid-dynamic forces to passive wing bending. We attached fresh Manduca sexta wings to a motor and flapped them around the dorsal-ventral axis of the wing hinge at a realistic wing-beat frequency and stroke amplitude, mimicking the large-amplitude motions of freely flying moths. We used high-speed video recording to compare instantaneous wing deformations of wings flapped in normal air versus helium (approx. 15% air density). The lower density of helium substantially reduces the contribution of fluid-dynamic forces to the observed wing deformations, allowing us to determine the relative importance of these forces in passive wing bending.
At the same time, however, this lower fluid density also reduces external damping of the wing's motions. We used a simplified finite element model based on a Manduca wing to explore how damping alone (in the absence of fluid-dynamic forces) affects wing motions. Because the finite element analysis does not include fluid-dynamic forces, the motions of the model wing depend solely on structural features and inertial-elastic effects. We subjected the model to the same motions as real wings, and compared bending patterns in the undamped model wing to those of the model with damping added.
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Materials and methods |
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The motor was attached to a platform inside a 30 cm Plexiglass box (with 1
cm thick walls) and its motion was controlled with a function generator. Wings
were flapped sinusoidally around the dorsal-ventral axis of the wing hinge at
room temperature. Total amplitude varied between 107° and 110°,
corresponding to intermediate stroke amplitudes in free-flying hawkmoths
(Willmott and Ellington,
1997). Wing motions were recorded by two high-speed video cameras
(Redlake Inc., San Diego, CA, USA) at 1000 frames s-1, one viewing
the wing from its leading edge and the other from its tip
(Fig. 1A).
Each wing was filmed while flapping at 0.5 Hz, as a control for the shape
and position of the wing when no dynamic bending occurs, and at 26 Hz, a
typical wing-beat frequency for Manduca sexta
(Willmott and Ellington,
1997). The density of fluid inside the chamber was then reduced by
repeatedly removing air through an opening near the bottom of the box and
adding helium through an opening near the top
(Fig. 1A). The wing was filmed
flapping at 26 Hz in a mixture of no less than 95% helium, which has a density
of 0.164 g l-1 (approximately 14% of normal air density;
CRC, 2001
). Finally, the box
was opened to release the helium and the wing was again filmed at 26 Hz in
normal air, to check for potential wing damage. All filming was completed
within 1 h, during which time the flexural stiffness of the wing does not
change appreciably (Combes,
2002
). The procedure was repeated on four different wings from
three individuals.
Wing bending analysis
We analyzed frames from three complete flaps in the middle of each filming
sequence, to avoid bending artefacts at the onset of motion. A custom Matlab
program (developed by M. S. Tu) identified the coordinates of the wing tip,
leading edge and trailing edge in each frame. These three-dimensional
Cartesian coordinates were converted to spherical coordinates, using the wing
base as the origin and measuring the position of the wing (viewed from the
leading edge) in degrees, with 0° at the center of rotation
(Fig. 1B). Flapping frequency
was found by dividing the number of complete flaps by the total number of
frames and multiplying by 1000. Amplitude applied by the motor was measured at
the leading edge in the control sequence (0.5 Hz) to avoid wing bending, using
the maximum excursion of the leading edge marker to define the sides of a
right triangle. To determine if amplitude applied at the base changes
significantly with flapping frequency, a brass rod of the same length and mass
as a Manduca wing was attached to the motor and filmed at 0.5 Hz and
26 Hz.
To examine temporal patterns of bending at each wing location, we compared the trajectory of a wing flapping at 26 Hz and at 0.5 Hz (where no dynamic bending occurs), adjusting the time base of the control sequence to match that of the experimental sequence and splining data to equal time intervals in Matlab. We then calculated the difference in position at each time point and performed a Fourier analysis on this wing bending data to determine the dominant frequencies of wing motion and the amplitude coefficient at each frequency.
Finite element modeling
As a wing is flapped through the air and deformed by inertial and/or
aerodynamic forces, its motions are damped by some combination of internal
(e.g. elastic or structural) and external (fluid) mechanisms. When the
surrounding fluid is less dense (as is the case with the Manduca
wings flapped in helium) the wing experiences less external damping. We
explored how damping affects wing bending by constructing a simplified finite
element model (FEM) based on a male Manduca forewing, and comparing
bending in the undamped wing to bending in the model wing with damping added.
The model was created in MSC Marc/Mentat and is composed of thin shell
elements of uniform thickness, recreating the planform configuration of veins
and membranes in a real wing (but omitting details of three-dimensional wing
structure; see Combes and Daniel,
2003b). We applied declining values of material stiffness to the
model wing in 12 strips, oriented diagonally
(Fig. 2); 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.
This configuration results in an exponential decline in flexural stiffness
EI (the product of Young's modulus E and the second moment
of area I) in both the spanwise and chordwise directions of the wing,
approximating patterns of flexural stiffness measured in real wings
(Combes, 2002
;
Combes and Daniel, 2003b
).
Within each strip, vein elements have a higher material stiffness than
membrane elements, mimicking the increased flexural stiffness of tubular
veins. We used an element density of 1200 kg m-3 (as measured in
insect wings; Wainwright et al.,
1982
), a thickness of 45 µm, and a Poisson's ratio of 0.49
(consistent with measured values of biological materials;
Wainwright et al., 1982
). 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 are
sufficient to ensure asymptotic performance of the model.
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We applied boundary conditions to the nodes at the wing hinge so that they
could not translate in any direction and could rotate only along the
dorsal-ventral axis, as in experiments on real wings
(Fig. 2, red arrows). 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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We performed one simulation with no damping added to the computational analysis, and another in which we added mass damping, adjusting the level of damping to best represent the observed changes in motion between real wings flapped in normal air versus helium. We ran each simulation for 19200 time steps, recreating 12.5 flaps in 0.48 s, and measured displacement of nodes at the wing tip, leading edge and trailing edge (in the same locations as on real wings; Fig. 2), through six flaps after the wing motion had reached full amplitude. To provide a control with no dynamic bending (analogous to the slow rotation in the experiments), we created a stiff wing by changing the Young's modulus of all elements to 1x1016 Nm-2, and subjected this wing to the same motions as the flexible wings. We quantified temporal patterns of wing bending at each of the wing locations by finding the difference in position between the stiff wing and the flexible wing (with and without damping), and performed a Fourier analysis on the resulting data.
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Results |
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In control sequences, the angular positions of the wing tip, leading edge and trailing edge were nearly identical (black lines, Fig. 3) and total amplitude was equivalent to the amplitude applied at the base, demonstrating that no dynamic bending occurs at these low frequencies. When flapping frequency was increased during experimental sequences, peak amplitudes at the wing tip and trailing edge increased, indicating that wings bent considerably at the end of each stroke (Fig. 3Ai-Ci), while amplitude at the leading edge changed only slightly (Fig. 3Bi).
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Wings flapped in helium displayed slightly higher peak amplitudes than those flapped in normal air, but patterns of bending were similar (Fig. 3Ai-Ci; see also http://faculty.washington.edu/danielt/movies for movies of flapping wings). Fourier analysis reveals that the dominant frequencies of wing bending were the same in both helium and normal air, and that only the amplitude of some higher harmonics differed (Fig. 3Aii-Cii; Table 1). Amplitude coefficients were similar in normal air and helium at the driving frequency (26 Hz), but were often larger in helium at higher harmonics, particularly at the second harmonic (78 Hz; Table 1).
|
Finite element modeling
In the stiff FEM wing, the angular positions of the wing tip, leading edge
and trailing edge were identical (Fig.
4, black lines) and equivalent to amplitude at the base; thus, as
in control sequences on real wings, the stiff FEM wing displayed no dynamic
bending. In the flexible FEM wings, maximum tip and trailing edge amplitudes
were higher than the amplitude applied at the base, while the leading edge
amplitude changed only slightly (Fig.
4Ai-Ci). Although bending amplitude at the trailing edge of the
FEM wings was lower than in real Manduca wings, temporal patterns of
wing bending were similar (Figs
3Ai-Ci,
4Ai-Ci).
|
In addition, the differences between the FEM wing with no damping and the wing with mass damping were similar to those seen between real wings flapped in helium and in normal air. The undamped model wing showed slightly higher peaks in wing bending (Fig. 4Ai-Ci), but the same overall bending pattern as the damped wing. Fourier analysis revealed that the dominant frequencies of bending were the same in the two simulations, and that amplitude coefficients were similar at the driving frequency and larger in the undamped model at higher frequencies (Fig. 4Aii-Cii).
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Discussion |
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Although overall patterns of bending were remarkably similar, high-frequency components of bending motion were more pronounced in wings flapped in helium (Fig. 3Aii-Cii); this was manifested visually as rapid oscillations in the more flexible regions of the wing, particularly as the wing slowed and began to move in the opposite direction. Simulations of wing bending in the finite element model suggest that reduced damping may explain this difference. Adding damping to the finite element analysis reduced higher-frequency components of motion in the model wing, just as increasing the density of the fluid (by using normal air as opposed to helium) reduced higher-frequency components of motion in real wings (Figs 3Aii-Cii, 4Aii-Cii).
These results suggest that a damped finite element model (with realistic, three-dimensional forces applied at the base) could be successful in predicting the overall pattern and magnitude of Manduca wing deformations during flight, independent of aerodynamic calculations. The finite element model used in this study contains several simplifications in three-dimensional geometry that may limit its ability to predict wing motions precisely. In addition, we did not incorporate an accurate distribution of wing mass, which declines sharply towards the tip and trailing edges (although preliminary simulations suggest that mass distribution affects primarily the magnitude, not the pattern of wing bending). Yet even this simplified model was able to simulate temporal and regional wing bending patterns relatively well, suggesting that a slightly more detailed finite element model could provide very accurate results.
To recreate Manduca wing motions during flight precisely, the
boundary conditions at the base of the model wing would also need to be
altered. The experimental work and dynamic modeling in this study were based
on a relatively simple kinematic pattern, in which the wing was rotated around
only the dorsal-ventral axis of the wing hinge. In most insects, muscular
forces transmitted to the wing base not only propel the wing with large
amplitude motions such as these, but also rotate the wing around its leading
edge, controlling the angle of attack of the wing and, in some cases, causing
significant spanwise twisting. The rapid wing rotations evident in some
species during stroke reversal (e.g.
Dickinson et al., 1999) may
involve increased aerodynamic forces, as well as rapid accelerations and
decelerations that could augment inertial-elastic forces. The extent to which
more detailed kinematics might alter our findings about the relative
contributions of aerodynamic and inertial-elastic forces to wing bending
remains a subject of future study.
It is also important to note that the relative contributions of aerodynamic
and inertial-elastic forces to wing bending are likely to vary along a
continuum, from hovering, where inertial-elastic forces appear to dominate in
Manduca, to the extreme case of steady, forward flight with no
flapping, where inertial forces are negligible and any wing bending would be
due solely to aerodynamic forces. In many insects, however, the most
pronounced wing bending and twisting occurs during slow flight or hovering
(e.g. in Manduca; Willmott and
Ellington, 1997), so passive deformations may in fact decrease as
aerodynamic forces begin to dominate.
Insect size and wing design
Because the Manduca wings used in this study are relatively large
and heavy, it is possible that inertial-elastic effects are more important in
determining wing bending in this species than in other species with smaller,
lighter wings. A simple analysis of average bending moments can be used to
assess the relative magnitudes of inertial-elastic and aerodynamic moments on
the flapping wings of different species
(Daniel and Combes, 2002):
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In addition to large variations in size, insect wings display tremendous
variability in design features, such as planform wing shape and the
arrangement of supporting veins, which could affect how their wings respond to
aerodynamic and inertial-elastic forces. Interestingly, despite dramatic
visual differences in wing design, overall wing stiffness appears to scale
strongly with wing size in a broad range of species
(Combes and Daniel, 2003a).
Patterns of regional stiffness variation in insect wings may in fact be
affected by wing shape and venation; however, we have measured very similar
patterns of stiffness variation in the wings of hawkmoths and aeshnid
dragonflies, insects with strikingly different wing designs
(Combes and Daniel, 2003b
).
These results suggest that large differences in insect wing design do not
necessarily lead to equivalent differences in wing stiffness and bending
behavior.
Concluding remarks
Because aeroelastic effects appear to be relatively unimportant in
determining dynamic wing shape, an integrative model of insect flight that
incorporates passive wing flexibility may be easier to develop than previously
thought. Although the experimental work presented in this study addresses the
relative contributions of aerodynamic and inertial-elastic forces to wing
bending in only one species using a particular kinematic pattern, these
results verify recent theoretical studies
(Daniel and Combes, 2002)
suggesting that fluid-dynamic forces have only a minor effect on passive
bending when flexible structures are flapped in air (versus water,
where fluid forces dominate). In addition, while detailed numerical methods
(e.g. Liu et al., 1998
;
Ramamurti and Sandberg, 2002
;
Sun and Tang, 2002a
;
Wang, 2000
) will undoubtedly
continue to contribute to our understanding of three-dimensional fluid flow
around flapping wings, recent work suggests that far simpler analytical
methods are able to predict temporal patterns of unsteady force production
remarkably well (Sane and Dickinson,
2002
). Inserting the dynamic shape of wing sections (as determined
by finite element analysis or other inertia-based methods) into quasi-steady
models of flight that account for unsteady effects may yield a tractable
modeling tool that could be used to explore the effects of wing flexibility on
unsteady force production. Models of this type could be used to determine when
and how wing flexibility affects aerodynamic force generation, and ultimately
contribute to an integrative model of insect flight linking sensory feedback
and patterns of muscle force production to dynamic wing motions, force
generation and insect flight performance.
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Acknowledgments |
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