A COMPUTATIONAL MODEL FOR ESTIMATING THE MECHANICS OF HORIZONTAL FLAPPING FLIGHT IN BATS : MODEL DESCRIPTION AND VALIDATION
1
Department of Ecology and Evolutionary Biology, Brown University,
Providence, RI 02912, USA
2
Division of Engineering, Brown University, Providence, RI 02912,
USA
*
Author for correspondence (e-mail:
sharon_swartz{at}brown.edu
)
Accepted June 5, 2001
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Summary |
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Key words: flight, wing mechanics, aerodynamics, computational modelling, Chiroptera, bat, Pteropus poliocephalus
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Introduction |
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Both classic and recent results demonstrate that the bat wing is unique
among mammalian limbs in anatomical design and mechanical function and suggest
that the specialized features of the bat musculoskeletal system are linked
directly to flight capabilities (Findley et al.,
1972; Hermanson and
Alternbach, 1983
; Holbrook and
Odland, 1978
; Norberg,
1970a
; Norberg,
1972a
; Norberg,
1972b
; Papadimitriou et al.,
1996
; Strickler,
1978
; Swartz,
1997
; Swartz,
1998
; Swartz et al.,
1992
; Swartz et al.,
1996
; Vaughan,
1959
; Vaughan,
1970b
). Similarly, significant
contributions have been made to our understanding of the kinematics of bat
flight (e.g. Aldridge, 1986
;
Aldridge, 1987
; Baggøe,
1987
; Brandon,
1979
; Norberg,
1970b
; Norberg,
1976a
; Norberg
1976b
; Norberg,
1990
; Rayner,
1987
; Rayner et al.,
1986
; Vaughan,
1970a
). Some components of
these kinematic and morphological studies have also contributed directly to
our understanding of flight mechanics, including the interrelationship between
wing membrane tension and aerodynamic force (Norberg,
1972a
; Pennycuick,
1973
), the velocity-dependence
of wing kinematics (Aldridge,
1986
) and the relationship
between wing inertia, energetics and maneuverability (Norberg,
1976a
; Thollesson and Norberg,
1991
).
However, despite decades of study, we understand much less about the
mechanics and energetics of flying than we do of walking, running or swimming
in vertebrates; this, in turn, limits our knowledge of how flight capacity and
the intricate array of physiological and morphological specializations
associated with it have evolved. In part, this is because our insight is
constrained by our limited understanding of the nature of the complex and
dynamically changing forces experienced by the wing during flapping flight and
the technical difficulties of approaching this subject empirically. To meet
these challenges and thereby gain new insight into the mechanics and
energetics of the bat wing, we have developed a three-dimensional computer
simulation of bat flight. Computer modeling is a powerful approach to gaining
insight into the complexities of animal flight and has been adopted with
success in diverse ways by students of bat, bird and insect flight (e.g.
Bennett, 1977; DeLaurier,
1993a
; Ellington,
1975
; Ellington,
1978
; Ellington,
1984
; Ennos,
1988
; Norberg,
1975
; Norberg,
1970b
; Norberg,
1976a
; Pennycuick,
1968
; Pennycuick,
1975
; Rayner,
1986
; Rees,
1975
; Spedding,
1992
; Spedding and DeLaurier,
1995
; Withers,
1981
).
Our model computes wing bone stresses, joint forces and moments and other mechanical and energetic parameters from wing kinematics and structural geometry placed in a context of a well-founded, realistic and detailed aerodynamic model. The wing anatomy of bats is particularly well suited to this kind of engineering analysis: the wing comprises a jointed network of virtually rigid structural supports interconnected by an essentially two-dimensional elastic membrane; bats operate at Reynolds numbers high enough for appropriate application of inviscid aerodynamic theory; and bat flight occurs at Strouhal numbers at which complex unsteady forces are far less important than for insect flight. In this context, our model is structured to estimate accurately the forces experienced by the wing, to analyze in detail one of these forces, the internal forces developed in the wing, and, ultimately, to detail further the contributions to a critical element of the internal forces, the wing membrane forces.
In addition to its detail and accuracy, this model is noteworthy in that we
are able to validate key aspects by direct comparison of the model's estimates
of wing bone stresses with empirically measured values (Swartz et al.,
1992). A well-validated model
of bat flight should be able to reproduce, in order of increasing
sophistication, appropriate skeletal loading (tension versus
compression), the general pattern of change in skeletal stress in relation to
the wingbeat cycle, stresses of realistic magnitude and details of changes in
stresses during the wingbeat. We are also able to compare the predictions of
the model concerning the vertical movements of the animal's center of mass
with measurements made directly from wind-tunnel flights (Bartholomew and
Carpenter, 1973
).
Here, we describe the structure of the model, evaluate its ability to reproduce important aspects of bat flight mechanics realistically and examine the sensitivity of the model to various inputs and assumptions. Once validated and evaluated for parametric sensitivity, we intend to apply the model to a broader comparative analysis of the flight mechanics of bats that differ in wing morphology and/or flight behavior. We believe this model can be employed to gain insight into diverse problems in the mechanics and evolution of bat flight in future studies.
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Materials and methods |
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We employed Newton's second law of motion to balance the inertial force of
a wing strip with the external forces as well as the internal force within the
wing segment, and then used the segmental balance of forces at each instant in
time to solve for the internal force carried by the wing structures. We
calculated the stresses developed at the midshaft of the humerus and radius by
summing the internal forces over all strips distal to the bone site of
interest and computing the moments of these forces about the bone's midshaft.
We modified these maximum possible internal forces and moments by subtracting
the forces and moments transmitted directly by the plagiopatagium (armwing) to
the body, and then employed standard engineering beam theory to convert the
remaining three orthogonal forces and moments at the midshaft into normal and
shear stresses at the bone surface, and compared stresses calculated for the
humeral and radial midshafts with stresses inferred from strains measured
empirically in flying bats (Swartz et al.,
1992). The close
correspondence between the model estimates and the empirical data provides
robust evidence that the model approximates the mechanics of bat flight in
biologically meaningful ways. As further validation, we also compared the
vertical displacement of the animal's center of mass during the wingbeat
cycle, as predicted by the model, with the empirically observed vertical
motion.
The computer program that solves the equations and algorithms presented below was written in Fortran 77 (Pro Fortran, version 5.0, Absoft) and run on a Power Macintosh G3. The program provides outputs (bone stresses, joint moments, etc.) at 40 time steps evenly spaced over the course of a single wingbeat cycle. Hence, the time t corresponding to the integer value of the time step q is given by:
Choice of species
We selected the species Pteropus poliocephalus, the greyheaded
flying fox, for the first application of our bat flight model for several
reasons. First, this species occurs at relatively high densities in the
proximity of Brisbane, Australia, and the University of Queensland and has
therefore been a subject of previous research (e.g. Carpenter,
1985; Swartz,
1998
; Swartz et al.,
1992
; Swartz et al.,
1993
). Second, individuals are
large (adult body mass typically 550-950 g), facilitating mechanical
assessment of functionally important wing structures, including in
vivo measurements of wing bone stresses. Therefore, we possess detailed
information concerning the in vivo loading of the humerus, radius,
metacarpals III and V and proximal phalanges III and V for this species.
Third, high-speed dorsal, lateral and oblique films of individuals of this
species flying in a wind tunnel provide detailed information concerning wing
kinematics (Bartholomew and Carpenter,
1973
). Fourth, level flight is
probably an ecologically relevant flight mode for this species given that
these fruit-eating bats often fly 30-50 km during foraging bouts on a single
night and may migrate hundreds of kilometers in a year (Eby,
1991
).
Validation by in vivo strain recordings
Measurements of bone surface principal strain magnitudes and orientations
were made from eight individual P. poliocephalus in previous studies
(Swartz et al., 1992; Swartz
et al., 1993
). In these
studies, animals were wild-caught and trained to fly the length of a 30 m
flight cage without stopping. Within 2 weeks of capture, rosette strain gauges
were surgically implanted on the subperiosteal surfaces of the midshafts of
wing bones, and the animals recovered fully from the effects of surgery. We
then collected data from up to nine strain gauge elements simultaneously
via a lightweight cable (100 Hz) and synchronized the data with video
recordings of the animals' wing movements. Data from individual rosette
elements were analyzed to obtain maximum and minimum principal strain
magnitudes and orientations, and strain values were converted to stresses
assuming that the compact cortical bone of the long bones of the wings of
large bats is similar in its mechanical properties to that of other mammals
and birds (Carter, 1978
; Beer
and Johnston, 1981
; Biewener,
1983
; Currey,
1987
). Model predictions of
bone stresses were then calculated for specific anatomical sites from which
strain data had been collected, and the experimentally determined stress
profiles for a given recording site were normalized to a standardized wingbeat
cycle, synchronizing the mid-downstroke, downstrokeupstroke transition
and upstrokedownstroke transition.
Morphological parameters and general flight characteristics
Mass, wing size and shape and the dimensions of individual wing bones were
assessed by direct weight measurement, tracings of wing outlines and
measurements of high-resolution radiographs of a member of the study
population used in previous work (Swartz et al.,
1992)
(Table 1). Although the model
data were measured from a single individual killed following completion of
bone strain recording, these parameters vary relatively little among
individuals of a given body mass, and the small intraspecific variation would
have no significant effect on model results. Video recordings of flying foxes
were used to estimate the flight speed of the bats (Swartz et al.,
1992
). A typical flight speed
U of approximately 6 m s-1 was estimated from the time
needed to fly approximately 25 m; this represents a moderate speed for this
species. We measured the wingbeat period and detailed kinematics (see also
section on wing kinematics) from a high-speed film of a flying fox in a wind
tunnel. Wing length was measured as the distance from the proximal shoulder to
the proximal carpus plus the distance from the carpus to the wingtip for a
fully extended handwing. Body width is taken as the shoulder-to-shoulder
distance. We calculated the mean area of a single wing from plan-view
photographs of a deeply anesthetized study subject with its wings fully
extended.
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Wing segmentation
In the computer model, we conceptually divided the wing at mid-downstroke
into a series of rectangular segments of variable chord and width, determined
anatomically (see below) (DeLaurier,
1993b; Norberg,
1976a
; Thollesson and Norberg,
1991
). We subdivided the wing
into 14 rectangular segments: seven segments of equal width between the
shoulder and the carpus, three equal-width segments between the carpus and the
third metacarpophalangeal (MCP) joint and four equal-width segments between
the third MCP joint and the wingtip (Fig.
1). Because the wing changes its three-dimensional conformation
during the wingbeat, we calculated the instantaneous segment widths and
segment motions by linear interpolation of the distances between the carpus,
third MCP joint and wingtip positions, respectively. Because all segments
between two proximodistally adjacent landmarks are defined to have equal
widths, elbow flexion causes the width of the seven armwing segments to
decrease, although we assumed that the chord of each rectangular segment
remains constant. We write the width of a wing segment as
wpq, where the subscript p denotes the wing
segment and increases from proximal to distal, and the subscript q
denotes the time step during the wingbeat cycle. Since the model does not
employ lift coefficients, the angle of attack of each wing segment is not
required as input to the model. As a result, we consider each wing segment to
be parallel to the cranialcaudal axis at all times. However, wing
segment pitching angle and angle of attack variations are modeled
intrinsically through the calculation of the spatial orientation of
aerodynamic forces over time.
We sectioned the wing and, for each strip, we measured mass,
mp, chord along the strip's midline,
cp, leading edge position (perpendicular distance from the
center of the leading edge to a reference line connecting the left and right
glenohumeral joints), ep, and center of mass position with
respect to the shoulder-to-shoulder reference line, dp
(e.g. Thollesson and Norberg,
1991)
(Fig. 1, Table 2). Values of
ep are negative, indicating a position cranial to the
reference line. We determined the position of the center of mass of each
segment by attaching it to a rigid cardboard rectangle of known mass and
locating the center of mass of the cardboardwing ensemble. We shaped
the wing to match the plan view of an individual wing in the middle of the
downstroke; at this point in the wingbeat cycle, the wing is nearly horizontal
and coplanar. For the wing kinematics employed in the model (see below), the
middle of the downstroke corresponds to the dimensionless time
t/T=0.38. During the middle of the downstroke, we determined the
midline leading edge positions of all wing segments relative to straight lines
adjoining wing landmarks in order to position wing segments in the
cranialcaudal direction. We calculated the position and motion of
coplanar wing segments by linear interpolation between adjacent wing
landmarks. We kept the distances from segmental leading edges to the straight
lines connecting adjacent wing landmarks constant over the entire wingbeat
cycle, in effect conserving the shape of the flapping wing. Wing segment
position determined where the forces acting on that wing segment were
applied.
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Wing kinematics
In this section, we outline our method of describing in three dimensions
the motion of the carpus, the third MCP joint and the wingtip. The
three-dimensional positions of these major wing landmarks over one wingbeat
cycle suffice to capture the large-scale features of wing motion; for the
purposes of our model, we define the beginning of downstroke as the onset of
downward motion of the carpus.
Origin and axes for movement description
The body of a flying bat accelerates and decelerates vertically during
`level' flight and cannot therefore be used as an inertial frame of reference
from which to measure the accelerations of the animal's wing. Here, instead,
we describe the motion of the wing relative to the mean position of the
glenohumeral joint averaged over the entire wingbeat cycle, which constitutes
the origin of an inertial coordinate system moving with the mean forward
flight speed of the bat. We fixed the origin of a second coordinate system to
the right glenohumeral joint and related the motion of this non-inertial
coordinate system to the inertial coordinate system
(Fig. 2). We defined our axes
such that the x axis points horizontally to the right of the
direction of flight for the right wing (distally along the wing at
mid-downstroke), the y axis points vertically upwards (in the dorsal
direction) and the z axis points in the opposite direction to flight
(in the caudal direction along the cranialcaudal axis). We denote the
absolute acceleration of the glenohumeral joint (or shoulder) as the global
acceleration a with respect to the non-inertial origin. The global
acceleration of the shoulder is not known a priori, but is instead
computed internally by the model through the force balance on the body. The
unit vectors giving the direction of each coordinate axis are i for the
x axis, j for the y axis and k for the
z axis. We model the left wing implicitly through symmetry across the
yz, or midsagittal, plane. This symmetry is a reasonable assumption
for forward flight and gliding, but would not be appropriate when modeling
more complex flight maneuvers.
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Three-dimensional coordinates of wing landmarks were taken from high-speed
movies of wind-tunnel flight (Bartholomew and Carpenter,
1973). We selected these films
for analysis of wing kinematics because of their far greater resolution than
the video recordings made during previous in vivo strain measurement
experiments (Swartz et al.,
1992
; Swartz et al.,
1993
). The films were
converted to video and analyzed with the Peak Performance Motion Analysis
System (Peak Performance Technologies, Englewood, CO, USA) to obtain detailed
information regarding the dynamically changing positions of the carpus, third
MCP joint and wingtip relative to the right shoulder. The spatial resolution
of the digitizing process corresponds to approximately ±2 cm, or
approximately ±5 % of the mean wing length and ±12 % of the mean
wing chord. From both head-on (xy plane) and lateral (yz
plane) views, approximately 170 sets of two-dimensional coordinates were
obtained for one complete wingbeat cycle. The right shoulder coordinates were
subtracted from each wing landmark position. We combined the two views into a
composite of wing motion by overlaying the x coordinates of wing
motion at the carpus, third MCP joint and wingtip. We illustrate
representative motion of the carpus and wingtip projected onto the xy
plane with respect to the shoulder (Fig.
3).
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The wingbeat cycle of flying foxes, like that of some birds, can be partitioned into approximately 65 % downstroke and 35 % upstroke. Sinusoidal motion, necessarily 50 % downstroke and 50 % upstroke, is therefore not appropriate to describe these wing motions, and one would have to resort to Fourier series expansions to describe wing kinematics; we have instead selected cyclic polynomials to describe wing motion realistically. We curve-fitted the time change in the x, y and z positions of the carpus, third MCP joint and wingtip relative to the right shoulder with eighth-order polynomials using a least-squares algorithm (KaleidaGraph, version 3.0.5, Abelbeck Software) (Table 3). For example, we wrote the position of the carpus along the x axis as:
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Wing landmark positions near t/T=1 were not precisely periodic
because of rounding errors in the curve-fit coefficients
(Table 3). Moreover, double
differentiation of position curves with continuous acceleration guarantees
continuity of the acceleration at end points but not continuity in the slope
of the acceleration. These are subtle effects that become very pronounced when
taking derivatives of the position data. We enforced periodicity in all
position curves by using the value at t/T=0 in place of the value
obtained from t/T=1. Continuity and smoothness in velocity and
acceleration profiles was obtained by applying SavitzkyGolay smoothing
filters to the position, velocity and acceleration curves (Press et al.,
1989). The amount of smoothing
was carefully tested to have no perceptible effect on landmark positions and
yet still eliminate the growth of spurious discontinuities near end points. We
calculated wing landmark velocities from second-order finite differences of
the smoothed position curves and accelerations from second-order finite
differences of the smoothed velocity curves. Finally, we calculated the
position, velocity and acceleration of each wing segment from linear
interpolation between coplanar wing segments, as described in the previous
section.
Dynamics
From the curve fits of position data, we calculated the magnitude,
orientation and location of the gravitational, inertial, added mass and
aerodynamic forces acting on each wing segment over the course of one wingbeat
cycle. We then solved for the internal force carried by wing structures within
a segment by invoking Newton's second law of motion at discrete instants of
time. We summed the internal forces within all wing segments to calculate the
instantaneous global acceleration of the bat shoulder. We corrected the
magnitudes of the segmental aerodynamic forces in order to enforce the
level-flight criterion (no net vertical acceleration of the body over an
entire wingbeat cycle). Finally, we calculated reaction forces and moments at
the humeral and radial midshafts by assuming that each bone carries the entire
internal load of all more distal wing structures minus the load carried
directly to the body by the plagiopatagium (see below, section on forces,
moments and stresses on the wing skeleton).
Gravitational force
The action of gravity on the wing of a large bat is typically an order of
magnitude smaller than the aerodynamic force encountered in level flight
because the former acts only on the wing mass while the latter supports the
entire body mass (e.g. Thollesson and Norberg,
1991). Nevertheless, the
segmental gravitational force is given by:
![]() | (6) |
Near the wingtip, the trapezoidal and triangular shape of the last few wing segments would cause the center of mass to be located more proximal to the shoulder than the midline of the wing segment. We did not include this effect because segment widths and masses near the wingtip are sufficiently small that the effect of any correction is negligible.
Inertial force
Inertial forces on the wing vary with wing motion and thus can range in
magnitude from zero during episodes of gliding to values comparable with the
large aerodynamic forces exerted on the wing during flapping flight. By
definition, the inertial force also represents the net sum of all external
forces as well as the force carried by the wing structures for a given wing
segment. On the basis of segmental accelerations, the inertial force on wing
segment p at time q is given by:
![]() | (7) |
Added mass force
The added mass force resists the acceleration of the wing and is sometimes
considered part of the inertial force since it is proportional to the
acceleration perpendicular to the plane of a wing segment (DeLaurier,
1993b). The magnitude of this
acceleration is a.n and it is always aligned with the unit normal
vector n. We define the normal vector npq at any time
q to point away from the dorsal face of wing segment p. The
component nz.pq of the normal vector npq
is always zero given that we assume, as a first approximation, that wing
segments remain parallel to the cranialcaudal axis of the body at all
times. We write the added mass force for wing segment p at time
q as:
Aerodynamic force
The aerodynamic force consists of lift and thrust components that arise
from the rates at which air is drawn up ahead of the wing and pushed down
behind the wing. In our study species, the aerodynamic forces are the largest
and most significant forces experienced by the wing, although segmental
inertial forces during a flapping cycle can momentarily achieve the magnitude
of segmental aerodynamic forces (see Results). Together, the aerodynamic and
inertial forces account for most of the external load exerted on wing
structures. Skin friction along the surface of the wings, form drag on the
body and induced drag from vortices in the wake provide the total drag
experienced during flight. We combined the total drag and the animal's weight
to find the total aerodynamic force and distributed this force over all the
wing segments. In the absence of detailed wing profiles in flight, we
prescribed the aerodynamic force generated by each wing segment through an
elliptical, spanwise distribution of circulation (Norberg,
1990) at the middle of the
downstroke as a plausible approximation for a bat wing in fast forward flight
(Rayner et al., 1986
; Rayner,
1986
; Spedding,
1987
) (see also section on
aerodynamic force distribution and constant-circulation flight). Variation in
wing span is a fundamental method of generating thrust under the constraint of
constant circulation (Spedding and DeLaurier,
1995
); bats, including P.
poliocephalus, experience significant variations in tip-to-tip wing span
during steady flight which makes constant-circulation flight plausible (see
also Fig. 3). We also took care
to ensure that the directions of the segmental aerodynamic forces over one
entire wingbeat cycle were such that mean lift equaled weight and mean thrust
equaled total drag. We assume that the small radius of curvature of the
leading edge of the wing produces low leading-edge suction efficiency
(DeLaurier, 1993b
); the
consequence of this assumption is that the aerodynamic force must act almost
normal to the wing instead of in a vertical direction.
Drag
We modeled three distinct drag components that do not arise from
circulation about a wing segment: skin friction over the surface of the wing,
form drag of the body and induced drag from wingtip vortices that exist
because of the finite wing span. The first two drag components are
proportional to the square of the flight speed, whereas the induced drag is
inversely proportional to the square of the flight speed; hence, the induced
drag is an order of magnitude larger than skin friction or form drag on
account of the relatively low flight speed. The maximum segmental velocity can
reach several times the forward flight velocity, and this argues against using
the forward flight velocity in the model; however, we found that the skin
friction drag is negligible for this bat species at the moderate flight speeds
employed in this model and that a more precise treatment is therefore
unwarranted here. As suggested by Norberg (Norberg,
1972a), we assume fully
turbulent boundary layer flow over the entire bat wing because of the small
leading edge radius and the presence of hair near the leading edge of the
wing. The turbulent drag coefficient CD,f of a flat, rough
plate is approximately:
We approximated body shape as a sphere and assumed that air flow around the
body has a turbulent boundary layer and a typical drag coefficient
CD,b0.5 (White,
1991
). In contrast to the
bodies of birds, those of bats are not streamlined, and can, indeed, resemble
a sphere to some extent because of the proportions of the ribcage. Moreover,
form drag is smaller than the induced drag, which mitigates against searching
for a precise numerical value of the drag coefficient. Our results for this
species (see below) demonstrate that there is not much evolutionary pressure
to drive bat body streamlining from a biomechanical perspective. It follows
that the form drag Db is:
Induced drag represents the continuous conversion of freestream momentum
into the wasted angular momentum of wingtip vortices. We assumed that a
constant and uniform induced drag exists across the entire wing span,
including behind the body. However, since we do not consider segmental angles
of attack in our model, we added induced drag explicitly to each wing segment,
rather than implicitly through a downwash angle that alters the orientation of
the aerodynamic force (DeLaurier,
1993b). We calculated the
induced drag on the basis of the mean wing span and an assumed elliptical
distribution of lift along the wing span (Norberg,
1990
). The induced drag
Di becomes:
The total drag Dt on the bat in forward flight is the sum
of skin friction drag, body form drag and induced drag:
![]() | (13) |
We assumed for all drag components that the forward flight velocity
U is constant over one wingbeat cycle. To evaluate this
approximation, we considered the deceleration caused by the total drag over
one-quarter of a wingbeat cycle. If the deceleration is uniform and
approximately equal to
aszDt/M over a duration
T/4, then we can write the fractional change in flight velocity as
approximately:
Thrust
Bat wings must generate thrust from wing segment circulation to counteract
the drag, as described above. Thrust can be achieved during part of a wingbeat
cycle by temporarily orienting a component of the aerodynamic force in the
direction of flight and always arises from an asymmetry in the wingbeat cycle
(Spedding and DeLaurier,
1995). Sustained horizontal
flight requires that the mean total thrust over one wingbeat cycle equals the
total drag. In the absence of more specific information on thrust generation,
we modeled the variation in net thrust over one wingbeat cycle by making the
following plausible assumptions: the body does not generate thrust, and the
skin friction of a wing segment is overcome by thrust from the same segment.
Therefore, the mean thrust Tf.p required over one wingbeat
cycle to overcome skin friction on both sides of a wing segment p
is:
![]() | (16) |
![]() | (17) |
We also assumed that form drag and induced drag are each counteracted by thrust generated uniformly over the mean length of the two wings, 2B. We write the mean thrust needed to overcome form drag and induced drag on a wing segment p as:
![]() | (20) |
Passive aeroelastic wings supported by leading edge spars tend to produce
maximum thrust during the middle of the downstroke as a result of wing
twisting that rotates the aerodynamic force towards the direction of flight
(DeLaurier, 1993c). Therefore,
we assumed sinusoidal variation in thrust over the wingbeat cycle, with the
maximum thrust occurring during the middle of the downstroke. We calculated
the magnitude of the thrust explicitly to find the orientation of the
aerodynamic force. In the absence of specific thrust observations, we verified
a posteriori that sinusoidal thrust variation does not strongly
affect our simulation results. Ideally, a more precise thrusting function is
desirable, particularly one that captures the 65% downstroke versus
35% upstroke cycle, but we did not feel that such an effort was germane to
this work at present. We write the magnitude of the thrust at wing segment
p and time q as:
![]() | (21) |
![]() | (22) |
Aerodynamic force distribution and constant-circulation flight
The magnitude of the total aerodynamic force Fa needed
to sustain forward flight is a combination of weight and total drag components
with a mean value over one wingbeat cycle of:
![]() | (23) |
There are three constraints for all plausible spanwise distributions of the
total aerodynamic force. First, the increase in circulation proximal to the
wingtip is dictated by the mathematical singularity of the wingtip vortex
itself: all plausible spanwise distributions of the aerodynamic force are
similar near the wingtip, with spanwise circulation becoming zero. Second,
flow visualization around a gliding model bat indicates that the body and
uropatagium are capable of producing lift (P. Watts, unpublished data). As a
result, we postulate the spanwise distribution of aerodynamic force generated
by the armwings and the body to be a relatively constant plateau. Third, the
constraint that mean lift over one wingbeat cycle equals body weight bounds
the possible numerical values of a constant aerodynamic force plateau. An
elliptical distribution of the total aerodynamic force over the wing span is a
common and convenient choice for engineers designing subsonic fixed-wing
aircraft and even ornithopters (DeLaurier,
1993b). It produces minimum
induced drag as well as a uniform downwash across the wing span; thus, an
elliptical distribution of aerodynamic force over the wing span would be
energetically favorable for flying mammals. Given the three constraints
mentioned above, an elliptical distribution of aerodynamic force is an
appropriate and plausible way of connecting an aerodynamic force plateau to
wingtip vortices. To balance weight, any other plausible distribution that is
less than an elliptical distribution somewhere along the span must also be
greater than the elliptical distribution elsewhere along the span. We expect
errors incurred by actual deviations in the spanwise distribution of
aerodynamic force from the chosen elliptical form to be small in magnitude
compared with other approximations made in the model.
Spedding (Spedding, 1987)
used stereoscopy of small helium bubbles in the wake of a kestrel, Falco
tinnunculus, to reveal that flapping flight need not be accompanied by
the shedding of spanwise vortices from the trailing edge of a wing. The
absence of vortex shedding implies a constant circulation about each spanwise
segment of the wing in fast forward flight. Consequently, constant circulation
implies that each spanwise wing segment maintains an aerodynamic force that
changes orientation but not absolute magnitude during the wingbeat cycle. It
has been demonstrated that bats can use a constant-circulation gait during
fast forward flight and that bat wings possess sufficient passive and active
control to achieve constant-circulation flight over an entire wingbeat cycle
in fast forward flight (Rayner et al.,
1986
). The approximately 6 m
s-1 flights from which our data were collected may not fall
strictly within the fast flight category; Norberg and Rayner (Norberg and
Rayner, 1987
) estimate that
the maximum range and minimum power flight speeds for flying foxes of this
mass are approximately 10 and 7.5 m s-1, respectively, on the basis
of the general relationship they estimated between body mass and flight
velocity. In any case, neither the constant-circulation theory of flapping
flight nor the flow visualization on which it is based is capable of providing
the spanwise distribution of the aerodynamic force. Therefore, an elliptical
aerodynamic force distribution combined with the constant-circulation
hypothesis appears to be a reasonable starting point for modeling
purposes.
In our model, we defined the magnitude of the aerodynamic force on a wing
segment p as:
![]() | (24) |
We approximated the smooth elliptical distribution traversing each wing
segment with the value of the circulation at the segment midline, resulting in
a spanwise distribution of circulation that resembles a descending staircase
over the entire wing length from root to tip
(Fig. 4). The circulation
pq is calculated according to the x axis position of
the midline of each wing segment relative to the center of the body at the
instant of the middle of the downstroke. Because the maximum circulation
exists midway between the shoulders, the circulation about a wing segment
p is given by:
|
Net drag and lift
We defined net drag by adding the drag and thrust acting on each wing
segment. The net drag on wing segment p at time q is given
by:
![]() | (28) |
Internal force
The internal force is the force carried by the wing structures that enables
the wing to undergo the observed accelerations and to resist the external
forces applied to the wing. We invoke the common form of Newton's second law
of motion for wing segment p at time q so that:
![]() | (30) |
Global acceleration
To calculate the body's global acceleration, we employed the estimate of
total internal force transferred from the wings to the body. We assumed that
the position of the shoulder joint relative to the center of mass is fixed
then, working from the inertial reference frame of the global axes,
approximated the force transfer from the wings to the body by summing the
segmental internal forces from the wing root to the wingtip at each instant of
time. The components of the total internal force transferred from both wings
are:
![]() | (34) |
![]() | (35) |
Level flight criterion
As mentioned above, we ensured that the mean lift was equal to body weight
over the wingbeat cycle by adjusting the aerodynamic constant C.
Without explicit knowledge of the correct value of C, we adopted the
following approach: we made two initial estimates near unity, observed the
resulting trend in the mean vertical component of the global acceleration and
used these results to guide more accurate estimates of the aerodynamic
constant C. To do this, we averaged the equation for the vertical
component of the global acceleration over one wingbeat cycle to obtain:
Plagiopatagium tension
Lift acting on the armwing causes the plagiopatagium to billow and thereby
increases the tension of the skin. Because of this coupling between lift and
skin tension, we replaced total armwing forces with skin internal forces; skin
force represents the sum of skin tension integrated along the chordwise
direction and does not include the frictional drag, which is accounted for
separately through the net drag. We therefore developed a simple model of
passive plagiopatagium deflection due to quasi-steady lift to determine the
internal forces of the skin that are transferred directly to, and indirectly
through, the humerus and radius.
To calculate the magnitude and location of this force, we assumed that the skin of the plagiopatagium is linearly elastic and orthotropic, that skin thickness remains constant during the wingbeat cycle, that shear stresses in the skin may be neglected and that the intrinsic musculature of the membrane, particularly the mm. plagiopatagiales, does not affect skin tension near the fifth digit during the wingbeat. This final assumption comes from our hypothesis that the muscles will only have a localized effect on membrane tension, perhaps by influencing local camber for aerodynamic purposes or by damping oscillations in the wing membrane via modulating skin stiffness. These localized effects will probably have little influence on skin tension measured elsewhere in the wing. The nature of elastic problems is such that a local perturbation of the global solution is rapidly smoothed out of existence. This local versus global dichotomy is a consequence of the elastic partial differential equation being elliptical: the stress at every point on the wing membrane is in some way an average of the stresses surrounding that point. Therefore, elastic materials quickly erase any evidence of local perturbations.
We characterized the plagiopatagium as a rectangular sheet coplanar with the plane connecting the shoulder, elbow and carpus if the wing membrane was not deformed by lift (Fig. 5). The long axis of this rectangular membrane is parallel to the line connecting the shoulder to the carpus. The rectangle's chordwise dimension is the mean armwing chord ca and its spanwise dimension is the instantaneous shoulder-to-carpus distance ba. For this analysis, we used a coordinate system with its origin at the carpus, its x axis parallel with the long edge of the rectangle and positive proximally, and its z axis parallel with the fifth digit and positive caudally (Fig. 5). The location and orientation of the total skin internal force Fs justifies the approximation of the plagiopatagium as a rectangle since the line of force action passes distal of the armwing bones (see below). Newton's third law requires that the plagiopatagium induce a force normal to the armwing that is equal to the lift that it is replacing. Part of the lift is accommodated by the angle tan-1s that the force Fs makes with the plane of the fifth digit and the body, and we distributed the remainder of the lift evenly along the armwing bones acting normal to the armwing.
|
We estimated the spanwise (Exx) and chordwise
(Ezz) elastic moduli of P. poliocephalus
plagiopatagia from published mean values of moduli of plagiopatagia of other
bats, yielding Exx =3 MPa and Exzz=37
MPa; these values correspond closely to the elastic moduli found for the
plagiopatagia of A. jamaicensis, the only large fruit-eating bat from
which these data have been collected (Swartz et al.,
1996). However, because this
species is an order of magnitude smaller in body mass than P.
poliocephalus, we also explored the effects of lower moduli on force
transfer. We further assumed the Poisson's ratio
zx=1 as a
typical value for skin composed of crossed fiber sheaths (Frolich et al.,
1994
). Symmetry of the skin
stress tensor associated with mechanical equilibrium requires that
xz=(Exx
zx)/Ezz=0.8,
where
xz is Poisson's ratio of the plagiopatagium due to
loading along the x axis.
Although inertial forces exerted on the skin can probably approach the
magnitude of aerodynamic forces near the carpus during periodic rapid wing
acceleration, we modeled only the skin deformation induced by the
instantaneous lift acting on the plagiopatagium. We divided total armwing lift
by the instantaneous plagiopatagium area
baca to find mean pressure differences
P (proportional to the wing loading), assumed to act uniformly
at each instant of time over the plagiopatagium as the source of membrane
deflection. We adopted a quasisteady model of membrane deflection out of the
plane by neglecting the retarding effect of membrane inertia on changes in
plagiopatagium deflection over time. We computed the linear solution of the
partial differential equation governing membrane deflection Y using
the method of separation of variables by assuming uniform values of the skin
tensions Tx and Tz in the spanwise and
chordwise directions respectively. To capture the most important contribution
to deflection (and hence skin tension) and yet maintain a tractable analytical
solution, we approximated the complete linear solution of the partial
differential equation by the first eigenmode solution corresponding to the
boundary conditions that there is no membrane deflection along the distal,
cranial and proximal edges, and that the slope of the membrane deflection be
zero along the caudal edge of the rectangle. The linearized partial
differential equation applies whenever lateral membrane deflections are small
enough to be considered remaining in the undeflected plane, which is an
assumption that must be verified a posteriori. We find the
approximate solution for membrane deflection out of the plane:
![]() | (38) |
![]() | (39) |
We calculated the skin tension Tx at 20 discrete
locations along the distal edge of the plagiopatagium using a
NewtonRaphson convergence scheme at each location to solve iteratively
the balance between deflection and tension. We found uniform skin tensions
along the fifth digit except for a brief period during the upstroke at which
time the skin tension approached zero and also became non-uniform. To
calculate a particular Tx, we transformed the integral for
zz into a complete elliptical integral of the second kind,
E(m,
/2):
Forces, moments and stresses on the wing skeleton
To analyze the effect of flight-related forces on the skeleton, we
designated the point at which an inertial or external or skin membrane force
vector Ff acts on a given wing segment as
(xf, yf, zf) and
the point on the skeleton about which we seek the moment M0
as (x0, y0, z0).
By varying the location of the point for which the analysis is carried out, we
may then analyze the skeletal loading and bending at any location on the wing
skeleton of the armwing that is biologically significant and through which a
known fraction (usually taken as 100%) of the remaining internal force is
transmitted. Although the analysis applies generally to any location along the
armwing, we focused here on the moments about the humeral and the radial
midshafts, to carry out comparisons of model estimates with empirically
measured values (Swartz et al.,
1992). We defined three moment
arms:
![]() | (42) |
![]() | (43) |
![]() | (44) |
![]() | (45) |
![]() | (46) |
![]() | (47) |
From the total force and moment vectors at a bone midshaft, we calculated
stresses at dorsal and ventral sites on a bone's circumference treating the
bone as a beam of circular cross section (Popov,
1978). We considered the
center of the bone cross section as the origin of a local coordinate system
parallel to the global coordinate system
(Fig. 2,
Fig. 6). We converted total
force and moment components written in the local x, y, z coordinate
system into a primed x', y', z' coordinate system
oriented to the bone long axes. This primed system shares the origin of the
local coordinate system, but possess an x' axis oriented
parallel to the bones' long axes (positive distally), a z' axis
in the plane of the wing (positive caudally) and a y' axis
normal to the wing plane (positive dorsally)
(Fig. 6). We defined the angle
as the angle between the armwing and the horizontal x axis.
The angle
defines a new y' axis normal to the armwing.
We also defined
as the angle by which we rotate the y'
axis to align the x' axis with the long axis of the humerus or
radius. The coordinate transformations from the local axes to the primed axes
are:
![]() | (48) |
![]() | (49) |
![]() | (50) |
|
We assumed that the humerus and radius are straight circular cylinders of
inner radius Ri, outer radius Ro and
length L, and that the medullary cavity has no capacity to resist
bending or torsion. We also assumed that the neutral axis intersects the
centroid of the cylinder's cross section. To calculate stresses, we balanced
the total force and moment vectors at midshaft with longitudinal normal
stresses L and shear stresses
1,2
distributed within the bone at the surface of a perpendicular midshaft cut.
When integrated over the proximal surface of the cut, the normal and shear
stress distributions represent three force and three moment components that
are equal in magnitude but opposite in sign to the total force and moment
components. We considered tensile normal stresses and left-handed (or
clockwise) shear stresses as positive (Fig.
7). We also assumed that the resulting stresses can be
superimposed linearly given that superposition of stresses is quite accurate
up to approximately 1% strain; we note that the peak humeral and radial
strains in P. poliocephalus rarely exceed 0.3% (3000 µ
)
(Swartz et al., 1992
).
|
Assuming that a bending moment induces a linear increase in normal stress
L with increasing distance from the neutral axis, we
calculated the magnitude of the normal stress induced by
My' as:
The normal stress induced by the force Fx was calculated by assuming that the force is distributed evenly over the cross section, giving rise to a normal stress:
Swartz et al. (Swartz et al.,
1992) assumed that compact
cortical bone is a transversely orthotropic material and used published values
for compact bone moduli and Poisson's ratio (Carter,
1978
) to convert empirical
measured strains into estimated stresses. They used a value of
E11
15 Gpa, where E11 is the
compressional longitudinal elastic modulus of bone, a typical value for
mammalian compact cortical bone (Currey,
1984
). Since there are no
obvious sources of transverse stress
T applied to the
humerus or radius, we assumed that applied
T=0. Hence, the
longitudinal strain is simply
1=
1/E11 and transverse
strains
2=-0.46
1 exist due to Poisson's
ratio effects alone, even in the absence of externally applied transverse
forces.
![]() |
Results |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
During the portions of the wingbeat cycle in which the wing is extended, the wing membrane is placed under significant tension. This tension exerts forces on the humerus and radius that tend to bend them in the plane of the wing in a cranially convex manner, placing the cranial bone surfaces in tension and the caudal surfaces in compression. Values of model predictions for stresses on the cranial and caudal surfaces of the bones are therefore strongly affected by input values of skin moduli (Fig. 8A; see also sensitivity analysis, below). Because this bending is largely restricted to the plane of the membrane, skin modulus input has little effect on estimates of shear and longitudinal dorsal and ventral normal stresses (Fig. 8B,C).
|
To date, no studies of the mechanical properties of bat wing membranes have
included large megachiropterans (Studier,
1972; Swartz et al.,
1996
). Therefore, we began our
analyses using values for mean spanwise and chordwise skin moduli measured
from bats an order of magnitude smaller in body mass than our model species.
On the basis of a physical inspection of the wing membranes of many bat
species, we expected these values to be considerably higher than the true
moduli of P. poliocephalus wing membrane skin, and carried out
analyses using lower values for moduli as well, maintaining the empirically
measured ratio of spanwise to chordwise modulus
(Fig. 8). When the model
employs a spanwise modulus of 0.5 MPa, one-sixth of the mean spanwise
stiffness of the wing membranes of smaller bats (Swartz et al.,
1996
), the magnitudes of
estimated bone stresses on the cranial and caudal bone surfaces are most
similar to those on the dorsal and ventral surfaces, so we conducted all
further analyses with this lower value for the skin moduli; this is reflected
in the results for the cranial and caudal surfaces of the humerus.
Predicted stresses at all bone sites except the cranial humerus share an overall pattern of change with respect to the wingbeat cycle: stresses reach a peak just after the initiation of downstroke and another peak of similar, slightly greater or slightly lower magnitude at the middle of downstroke; stresses then decrease to their lowest values shortly after the initiation of upstroke (Fig. 9). On the cranial aspect of the humerus, the longitudinal stress peaks just after the beginning of downstroke, exhibits a plateau throughout most of the remainder of the downstroke and then decreases steeply at the initiation of upstroke in concert with the stress minima of the other sites. Longitudinal stresses at all sites are higher than shear stresses throughout the greater part of the wingbeat cycle with the exception of the downstrokeupstroke transition, at which time longitudinal stress values approach zero.
|
The underlying basis of the similarity in the stress profiles at all anatomical locations modeled becomes clear when total stress is partitioned into components due to gravitational, inertial, added mass, net drag, handwing lift and armwing membrane (plagiopatagium) tension stresses (Fig. 10, Fig. 11). The inertial stresses display the pattern of timing observed for net stresses at all sites other than the cranial humerus, displaying peaks just after the initiation of downstroke and at mid-downstroke, and minima at the downstrokeupstroke transition. Although the general shape of the inertial stress curve for the cranial humerus is similar in overall form to that of the dorsal and ventral bone surfaces, the magnitude of inertial stress (+5 to -10 MPa) is far smaller here than stress due to tension in the plagiopatagium, so plagiopatagium stresses dominate. Skeletal stresses due to gravity, added mass and drag are virtually always very low, typically below 4 MPa and often close to 1 MPa. The only exception to this pattern is in the longitudinal stress at the cranial mid-humerus; in this case, the thrust engendered at mid-downstroke is large enough to generate significant bending in the plane of the wing, decreasing tension on the cranial surface by up to approximately 7 MPa or 20 % of total stress at this location.
|
|
Stress due to plagiopatagium tension is also relatively small, generally around 2-5 MPa but reaching contributions as high as 10 MPa to the total longitudinal stress on the dorsal and ventral surfaces of the humerus. The contribution of plagiopatagium stress to the total stress at the cranial edge of the humerus is much greater, as high as 40 MPa, for the geometric reason noted above. If the model used a higher value for spanwise membrane modulus, this contribution would be even higher. Stresses in the humerus and radius due to the lift generated by the handwing are significant (5-20 MPa) and show a consistent plateau with little variation in magnitude throughout the downstroke.
Global body motion
The model predicts that the bat's center of mass has a positive (upward)
vertical acceleration through the greater part of the downstroke, downward
acceleration through the end of the downstroke and beginning of the upstroke
and upward acceleration once more during the second half of the upstroke
(Fig. 12). Downward
acceleration reaches a maximum of nearly 13 N; given that this animal's mass
is approximately 800 g, this acceleration is close to 2 g.
This large acceleration occurs because the wing is folded such that some
aerodynamic force acts downwards near the top of the upstroke. In addition,
the upward wing acceleration near the end of the downstroke causes further
downward acceleration of the body.
|
During the middle half of the downstroke, the body is accelerating forward, but it loses its forward acceleration and gradually decelerates during the end of the downstroke and first half of the upstroke (Fig. 12). Near the upstrokedownstroke transition, there is a brief forward acceleration followed by a brief deceleration before the acceleratory phase of the downstroke begins. The magnitude of the foreaft component of the acceleration of the center of mass (±2 N) is small compared with the vertical acceleration. As a consequence, forward flight speed varies little during single wingbeats in level flapping flight.
Model validation: comparisons with empirical data
Humeral and radial stresses
To determine the degree to which our model accurately estimates bone
stresses, we compared midshaft longitudinal and shear stresses calculated from
the model with values computed from directly measured in vivo wing
bone strains of bats from the same study population. In this earlier study
(Swartz et al., 1992), maximum
and minimum principal strain magnitudes and orientations throughout the
wingbeat cycle were computed from data obtained from rosette strain gauges
surgically implanted around the midshaft circumference of the humerus and
radius; post-operative recordings, synchronized with video recordings, were
made as bats flew the length of a 30 m flight cage at moderate speed.
We found that the model can predict effectively not only the general shape of the stress curve, but also the timing of both major and secondary stress peaks in relation to the wingbeat cycle (Fig. 13). Moreover, stress magnitude estimates correspond quite closely to empirically measured values, with predicted values deviating from recorded values by approximately 5-30 % for longitudinal and shear stress peaks.
|
Deviations between the predicted and recorded values are sometimes greater during the lower-stress portions of the loading cycle and occasionally differed by as much as a factor of 2-3. These deviations arise partly from an `offset' that appears in some of our results: predicted and measured stress plots are very closely matched in shape but offset by 20-25 MPa. This phenomenon is particularly striking at the cranial surface of the humerus (Fig. 13).
Empirically recorded bone strains indicated relatively large transverse
stresses at the humeral and radial midshafts (Swartz et al.,
1992). The constitutive model
for bone presented by Carter (Carter,
1978
) is almost
volume-conserving and predicts
2=0.46
1 from
the Poisson's ratio alone in the absence of externally applied transverse
stress. It is therefore possible that transverse stresses observed in bat
bones are largely due to Poisson's ratio effects. We computed transverse
stresses from the model as
T=0.46
L, where
T is transverse and
L is longitudinal
midshaft bone stress, and then compared these with transverse stress values
calculated from the rosette strain gauge data
(Fig. 14). Correspondence
between the predictions and measured values is excellent and detailed in some
cases, but only moderate in others.
|
Global body motion
Because the model predicts the net vertical force exerted on the bat's
center of mass during the wingbeat, a comparison of the vertical changes in
position of the bat's center of mass calculated from the model with those
measured directly from film provides a strong test of model accuracy. Body
oscillations are slightly greater in the wind-tunnel flight than in the
simulation but, overall, oscillations computed from the simulation and
measured directly from film show a high degree of correspondence
(Fig. 15).
|
Sensitivity analysis
The model's output values for bone stresses and global body motion
descriptors are sensitive to various inputs and model assumptions to greatly
varying degrees. To assess the importance of these effects, we carried out a
sensitivity analysis in which we varied each of 10 model inputs by ±5 %
and computed the resulting changes in 24 model outputs
(Table 4).
|
As noted above, the elastic modulus of the wing membrane skin in the spanwise direction has a significant effect on estimates of skeletal stress; a 10 % change in modulus can produce a nearly equal change in bone stress (Table 4; see minimum longitudinal stress, cranial and caudal radius). The sensitivity analysis indicates that this effect arises from the influence of skin modulus on skin tension. Bone shear stresses are also rather sensitive to aerodynamic parameters, e.g. the timing of peak thrust within the downstroke, the timing of mid-downstroke within the entire downstroke, the location of the center of aerodynamic pressure and the added mass coefficient. The humerus and radius are not, however, affected equally; humeral shear stress is more than an order of magnitude more sensitive than radial shear stress. Longitudinal stresses are influenced by the same parameters, but generally to a lower degree. The cranial and caudal maximum longitudinal stresses, however, are moderately sensitive to the location of the center of pressure (Table 4).
In contrast, the model results are quite robust to variation in many model input parameters. Whole-body outputs (accelerations, velocities and displacements of the animal's center of mass) are virtually unaffected by changes in model inputs (Table 4). Longitudinal bone stresses are not influenced by changes in values of skin Poisson's ratio, skin friction or form drag and are only minimally affected by variation in the added mass coefficient.
![]() |
Discussion |
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![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
The greatest strength of computer modeling approaches their ability to simplify intractably complicated problems and to allow investigators to vary parametrically single constituent elements that cannot be teased apart experimentally is counterbalanced, however, by their greatest weakness. If the phenomena we model are those whose complexity defies ready empirical study, how are we to assess the validity of our models? Here, we present a model that facilitates analysis of some of the tremendous complexity of bat flight dynamics while retaining a strong link to empirical studies. This model is sufficiently detailed to predict both the net motion of a flying animal's center of mass and the stresses developed in the bones of the wing throughout the wingbeat cycle. As a consequence, we are able to assess the validity of our model with a degree of rigor unusual for computer models of animal flight.
We find that the model described here is able to make excellent estimates of aspects of wing and whole-body mechanics from kinematics and structurally important aspects of wing form and mass distribution. Indeed, a comparison of model results with empirically measured strains and of model-predicted vertical oscillations of the center of mass with those measured indicates that the model is surprisingly realistic (Fig. 13, Fig. 15). The model is able to predict the overall pattern of bone stress change during the wingbeat, the proper modes of stress (tension, compression, shear), some of the small-scale changes in stress during a wingbeat and, with varying precision, the magnitude of the wing bone stresses. Hence, it is reasonable to conclude that the model captures many of the features of horizontal constant-velocity flight that are important to flight performance. The unverified assumptions built into the model are therefore either likely to be confirmed by future direct measurements or are of relatively little importance to the mechanical behavior of the wing in flight.
In addition to longitudinal stresses, the model is also able to estimate
the stresses oriented perpendicular to the long axes of the humerus and radius
from a reasonable estimate of the Poisson's ratio of compact cortical bone
(Fig. 14). The relatively
large transverse stresses observed in the armwing bones of P.
poliocephalus are, as a consequence, best understood as arising from this
aspect of bone's mechanical properties rather than from an off-axis pull by
the wing musculature or tension within the wing membrane, as previously
suggested (Swartz et al.,
1992). Indeed, decomposition
of total force exerted on the wing into its constituents demonstrated that the
longitudinal stresses in the humerus and radius due to plagiopatagium tension
are quite low, typically less than 5 MPa, and transverse forces should have
roughly similar values. At some points in the wingbeat cycle for some bones,
however, model-based predictions of transverse stresses are considerably lower
than recorded values. These discrepancies may well indicate instances where
local muscle pull strongly influences recorded strains, and they may be
employed in the future to develop specific hypotheses concerning the activity
of particular wing muscles.
Similarly, the model predictions of vertical movements of the bat's center of mass are closely, although not perfectly, matched by empirical measurements (Fig. 15). The vertical oscillations measured from the film records also display a significant amount of variation. This variation and the lack of precise matching of the two curves is probably due in part to the derivation of the empirical values from the mean of a number of wingbeats whose amplitude and frequency were not identical. Although, to date, there have been no studies of variation in kinematics and/or body motion in bats in relation to wing kinematics, flight velocity, etc. at the level of detail needed to assess such features as wingbeat amplitudes on this scale, we believe that in the future it may be possible to relate kinematics to variation in aspects of flight performance such as the lift generation reflected in the motion of the center of mass.
Sensitivity analysis
A notable feature of the sensitivity analyses presented here is the
dependence of stresses on the cranial and caudal aspects of the bones and the
skin tension on the values for skin moduli entered into the model
(Fig. 8). Although relatively
little is known to date about the mechanical properties of wing membrane skin
(Studier, 1972; Swartz et al.,
1996
), our analyses indicate
that they may play an important role in flight mechanics and aerodynamics.
Although some previous work has characterized wing membrane mechanical
properties across taxa and among wing regions (Swartz et al.,
1996
), these comparisons were
based solely on the linear region of a complex, J-shaped stressstrain
curve. The analyses presented here suggest that wing membranes may not reach
these relatively high stiffness values during level flight at moderate speed
and may operate primarily in the toe region of the curve. Further work will be
needed to characterize more accurately this non-linear region of the wing
membrane stressstrain curve and to determine the stiffness of the wing
membrane during natural flight behaviors.
Beyond this effect of skin modulus, the overall model results are quite
robust to variation in a number of input parameters (timing of peak thrust and
mid-downstroke, etc.) and extremely robust to a number of inputs (skin
Poisson's ratio, coefficient of friction of the skin, form drag)
(Table 4). Within this overall
pattern, it is striking that the humerus is approximately an order of
magnitude more sensitive than the radius to the timing of the mid-downstroke
within the wingbeat and the shear stresses due to the location of the center
of mass. It is likely that this is a function of a combination of the
cross-sectional geometry of the bone and the position of the bone within the
wing which, in turn, influence the typical mode of loading of each of the
bones. The aerodynamic forces applied to the wing induce both torsion and
bending in the skeleton, but the torsion is greater for the humerus because it
is oriented such that forces at the center of pressure have a large torsional
moment about the humerus (Swartz et al.,
1992). Alteration in the model
input parameters can act effectively to shift the position of this center of
pressure relative to the humerus, producing the observed sensitivity. The
midshaft of the radius, in contrast, is positioned closer to the center of
pressure, and small changes in its location have a smaller effect on computed
radial stresses.
Extensions and limitations of the model
There are a number of important limitations to the model in its present
form. It applies strictly only to level, constant-velocity flight of moderate
speed and to a single species, Pteropus poliocephalus. With
modifications, this framework can be extended to a greater range of flight
behaviors and to a variety of taxa. However, this will require additional data
and analysis, particularly the detailed documentation of kinematic variation
in relation to activity, body size, wing form, etc., and the detailed analysis
of structural design of the wings of other species. Moreover, as the model is
extended and employed in novel ways, it will not be possible to validate it
via the bone strain analysis of level flight in P. poliocephalus.
In vivo bone strain measurement is not feasible for very small bones and
so is unlikely to serve as an effective tool to assess flight mechanics in bat
species much smaller than grey-headed flying foxes, i.e. most of the nearly
1000 species of bat. It is conceivable that in vivo strain gauge
measurement could be used, however, to obtain data that could serve to
validate models of higher speed or of accelerating or turning flight. It will
be critical, though, to seek additional means of external validation. With
rigorous validation of model accuracy for simple flight behaviors, however, it
is possible to retain a degree of confidence in the model even for behaviors
or taxa for which such detailed validation is impossible.
Finally, we note that this model is not restricted to predicting bone stresses and motions of the animal's center of mass. In future studies, we will use the model to estimate joint forces and moments and flight energetics, as well as extending the model to behaviors including vertical and horizontal acceleration and turning flight. Because the required model inputs, detailed flight kinematics and specific measurements of wing structure and mass distribution are simple in comparison with direct measurement of skeletal stresses, membrane and joint forces, etc., we will be able to extend the model to other taxa. We believe that this kind of model can be employed to provide new insights into flight mechanics that will improve our understanding of wing morphology, kinematics and dynamics. It is a powerful tool for the generation of new hypotheses concerning the mechanics and aerodynamics of bat flight and, by identifying aspects of structural design and flight mechanics that could constrain behavior and influence organismal performance, it may help define and focus future field and morphological studies.
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Acknowledgments |
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References |
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