Wing inertia and whole-body acceleration: an analysis of instantaneous aerodynamic force production in cockatiels (Nymphicus hollandicus) flying across a range of speeds
Concord Field Station, Museum of Comparative Zoology, Harvard University, 100 Old Causeway Road, Bedford, MA 01730, USA
* Author for correspondence (e-mail: thedrick{at}oeb.harvard.edu)
Accepted 12 February 2004
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Summary |
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Key words: cockatiel, Nymphicus hollandicus, flight, inertia, accelerometer
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Introduction |
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Whereas force production during locomotion in terrestrial animals is
confined to stride phases during which the feet are in contact with the
ground, forces in aquatic or aerial locomotion may be produced whenever the
fluid and animal move in relation to one another. Additionally, the
instantaneous magnitude and direction of the forces produced in terrestrial
locomotion may be readily quantified with a force plate or similar device. No
equivalent technology exists for fluid locomotion, although different
experimental and modeling approaches provide some of the same information.
Fluid visualization via digital particle image velocimetry (DPIV)
allows measurement of some of these forces, but the results are typically
interpreted over full- or half-cycle intervals due to the difficulty of
visualizing flows close to the animal
(Stamhuis and Videler, 1995;
Drucker and Lauder, 1999
;
Spedding et al., 2003
).
Physical modeling allows detailed investigation of the time-course and
magnitude of the locomotor forces but has thus far been most effectively
applied to the relatively simple flight surfaces, kinematics and low Reynolds
number regime of insect flight (Ellington
et al., 1996
; Dickinson et
al., 1999
). Computational fluid dynamics (CFD) models offer the
possibility of quantifying fluid flow and force production over the entire
surface of the animal but have thus far been limited to cases where existing
physical models can be used to validate the CFD results
(Liu et al., 1998
;
Ramamurti and Sandberg, 2002
;
Sun and Tang, 2002
).
In the present study, we employed a third approach. We used accelerometers
attached to the dorsal body center of cockatiels (Nymphicus
hollandicus Kerr), together with high-speed 3-D kinematics, to measure
the net instantaneous forces produced during complete wingbeat cycles. We
obtained these measurements as the birds flew in a low-turbulence wind tunnel
(Hedrick et al., 2002) across
a range of steady flight speeds. Our approach followed that of Bilo et al.
(1984
), who examined several
wingbeats of a pigeon (Columba livia) in steady fast flight. The
accelerometers measured the net effect of internal (inertial) and external
(aerodynamic) forces acting on the bird's body. Inertial forces are generated
by oscillation of the bird's wings about its body, whereas aerodynamic forces
result from the interactions between the bird and the surrounding fluid. To
obtain estimates of net aerodynamic forces, we therefore used 3-D kinematics
of wing motion to quantify and remove the inertial forces experienced by the
bird's body.
Measuring the resultant aerodynamic forces produced within a wingbeat cycle
allowed us to test two predictions of wing stroke function during upstroke.
First, at slow flight speeds of approximately 03 m
s1, the cockatiels employ a `tip-reversal' type of upstroke
in which the wing flexed at the wrist early in the upstroke, the proximal
portion of the wing was elevated, and the distal portion of the wing then
swept around and upward with the feathers rotated on their axes
(Fig. 1). This type of upstroke
has been hypothesized to allow the individual feathers to act as airfoils,
producing lift as they sweep back (Brown,
1963; Aldridge,
1986
; Norberg,
1990
; Azuma, 1992
).
However, because our previous kinematic analysis of flight in cockatiels
(Hedrick et al., 2002
) did not
reveal any obvious upward movement of the body during upstroke, we
hypothesized that a tip-reversal upstroke would not produce substantial lift
or thrust in this species.
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In addition to the tip-reversal upstroke, various investigators
(Spedding, 1987;
Rayner, 1993
;
Hedrick et al., 2002
;
Spedding et al., 2003
) have
suggested that the wing may be aerodynamically active during upstroke at
medium to fast flight speeds (511 m s1 in
cockatiels). By positioning the wing at a positive angle of attack during
upstroke and allowing their own forward velocity to drive flow past the
airfoil, birds may be able to generate lift with consequent additional drag
during the upstroke. Because our previous work on cockatiels
(Hedrick et al., 2002
)
provided some support for this, we hypothesized that cockatiels would employ a
lift-producing upstroke at faster flight speeds (511 m
s1). Recent flow visualization (DPIV) analysis of the wake
of a thrush nightingale (Luscinia luscinia L.;
Spedding et al., 2003
) has
also shown that the amount of energy added to the wake during upstroke
gradually increased with flight speed in that species. This suggests that
upstroke lift may also gradually increase with speed. However, our previous
kinematic analysis (Hedrick et al.,
2002
) indicates that upstroke lift in cockatiels may decline again
at the fastest speed (13 m s1) achieved during experiments
in the Harvard-CFS wind tunnel.
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Materials and methods |
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Flight kinematics
Flight trials were recorded using three synchronized, high-speed digital
video cameras [one Photron Fastcam-X 1280 PCI (Photron USA Inc., San Diego,
CA, USA) and two Redlake PCI 500 (Redlake Inc., San Diego, CA, USA] operating
at 250 frames s1 with a shutter speed of 1/1000th of a
second. The Photron camera was placed above and behind the wind tunnel flight
chamber; the two Redlake cameras were positioned on the side of the tunnel
opposite the operator with one lateral to the bird and the other
postero-lateral (Fig. 2A). The
camera data were synchronized with the accelerometer signals by recording the
cameras' digital stop trigger together with the accelerometer outputs
via an A/D converter (Axoscope Digidata 1200; Axon Instruments Inc.,
CA, USA). The cameras were calibrated using the modified direct linear
transformation (DLT) technique with a 70 point calibration frame (measuring
0.457 mx0.967 mx0.900 m in xyz coordinate space) that was
recorded at the end of each set of trials
(Hatze, 1988). Trials were
recorded at flight speeds of 113 m s1 in 2 m
s1 intervals. Flight speed sequence was not restricted to a
particular order, and the birds were allowed to rest between trials as
necessary to maintain satisfactory performance (typically 25 min of
sustained flight).
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Three points (dorsal and ventral surfaces of the shoulder, wrist and tip of the ninth primary) were marked on the right wing of each bird using 5 mm-diameter dabs of white correction fluid. The dried correction fluid was marked with a small black central dot. In addition, three points defining two orthogonal axes were attached to the accelerometers using two short lengths (2.5 cm) of wire, with a known orientation to the accelerometer sensitive axes (Fig. 2B). These axes were used to orient the accelerometers in the global reference frame later in analysis (see below).
Flight sequences consisting of a minimum of four successive wingbeats with minimal lateral and vertical movement within the flight chamber (velocity relative to the flight chamber of <0.5 m s1) were selected from the video data and the points noted above digitized using custom software written in MATLAB v. 6.5 (The Mathworks Inc., Natick, MA, USA). In the few cases (3 of 28) where sufficient sequential wingbeats with minimal change in wind tunnel position were not available, we selected additional wingbeats from the recorded flight sequence, digitizing at least four wingbeats for each individual at each speed.
The raw coordinate data obtained from the digitized trials were resolved
into a single 3-D space using the DLT coefficients derived from the
calibration frame (Hatze,
1988). In addition to resolving the dorsal and lateral 2-D camera
views into a single 3-D space, the modified DLT method also corrects for
parallax and other linear and lens distortions. Individual points having a DLT
root mean square error (RMSE) two standard deviations greater than the median
RMSE for that point (approximately 4% of the points) were considered outliers
and removed prior to analysis. Median RMSE ranged from 1.19 mm for the
orthogonal axis markers to 1.49 mm for the ninth primary tip. Occasionally, a
point was not in the view of at least two of the three cameras (approximately
7% of all points digitized), resulting in a gap in the reconstructed point
sequence. After the digitized coordinate data were filtered, missing or
dropped points were interpolated with a quintic spline fit to known RMSE using
the `Generalized Cross Validatory/Spline' (GCVSPL) program (Woltring, 1986).
This method uses the RMSE from the DLT reconstruction to filter the positional
data and then fills any gaps with a quintic spline interpolation. The results
from this technique were similar to those obtained by smoothing the positional
data using a 37 Hz digital Butterworth low-pass filter. However, the quintic
spline method also allows direct calculation of velocity and acceleration
derivatives from the spline curves, providing the most accurate method for
obtaining higher order derivatives from positional data
(Walker, 1998
).
Accelerometers
We measured instantaneous accelerations of the cockatiels via a
block of three accelerometers (1 EGA2-10 dual axis accelerometer and 1 EGA-10
single axis accelerometer; Entran Devices Inc., Fairfield, NJ, USA) mounted at
orthogonal axes and attached to the dorsal midline of the cockatiels just
above the estimated center of mass. The center of mass position was estimated
by first locating it in a frozen, wingless cockatiel cadaver by hanging the
specimen at various angles, then relating this to the position on the
experimental animal estimated via visual and tactile landmarks. The
accelerometers were anchored to the dorsal midline by suturing the
accelerometer base plate to the intervertebral ligaments with two loops of 3-0
silk suture while the bird was under light anesthesia (isoflurane;
Fig. 2B). Accelerometer signals
were collected through a lightweight (5.6 g) multi-lead cable that ran a
distance of 1 m from the accelerometers on the animal to a small (0.75 cm
diameter) opening at the top of the wind tunnel's working section. This
lightweight cable connected to a heavier, shielded cable outside the flight
chamber that ran to the recording amplifiers (Micromeasurements 2120 bridge
amplifiers; Vishay Intertechnology Inc., Malvern, PA, USA). The amplifier
outputs were sampled by the A/D converter at 5 kHz and stored on a computer
for subsequent analysis.
The mass of the accelerometers and the portion of the data cable supported by the bird was 11.4 g, approximately 13% of the total body mass of the bird. Measurements of the drag produced by the cable and attached accelerometer ranged from 0.05 N at 1 m s1 to 0.23 N at 13 m s1. These measurements were made without an associated cockatiel; drag from the accelerometer and cable may therefore differ somewhat when associated with a bird's body. However, the effect of this drag, when related to the inertial and aerodynamic forces produced over the entire downstroke and upstroke phases, can be expected to be small and of negligible significance to how the patterns of inertial and aerodynamic force relate to each other. The cockatiels typically ignored the accelerometers and data cable while flying in the wind tunnel. The presence of the accelerometers did result in a reduction in the maximum flight speed we were able to record in the wind tunnel (from 15 m s1 to 13 m s1), probably due to the additional drag from the cable and accelerometers. Flight duration and the position within the tunnel that each bird selected, however, were unaffected.
After each recording session, we recorded accelerometer calibration
voltages by positioning each accelerometer's sensitive axis at 0°,
45°, 90° and 180° with respect to gravity. An accelerometer
calibration equation was calculated from least squares regression of the
recorded voltages and the expected accelerations of g, 0.707
g, 0 and g. In all cases, the
r2 for the calibration regression was 0.99. After
calibration, we used the position information obtained by digitizing the three
markers attached to the accelerometer block to rotate the accelerometer
outputs from their native `bird-fixed' orientation on the dorsal surface of
the animal to the standard global coordinate space defined by the camera
calibration frame. Rotations were performed via a series of Euler
angle transformations:
![]() | (1) |
Although the inertial forces produced by wing motion cannot accelerate the
bird's center of mass (CT), they can produce accelerations
at the center of the body (CB;
Fig. 2B), above which the
accelerometers were attached and that will be included in the accelerometer
recordings. Following Bilo et al.
(1984), we accounted for these
accelerations of CB due to inertial forces (subsequently
referred to as inertial accelerations) by reconstructing them from the 3-D
wing kinematics (see below) and subtracting them from the accelerometer
recordings.
Reconstruction of the inertial accelerations requires a mass distribution
for the wing as well as the wing's kinematics. We created a standard cockatiel
wing mass distribution by sectioning and weighing wings from three cockatiels.
The resulting standard cockatiel wing was composed of 18 slices, each of which
was 1.3 cm wide, and included both the actual section mass and an estimated
virtual mass predicted from the volume of air accelerated with the wing
(Fig. 3). The virtual mass
contributed 12.6% to the total wing mass
(Fig. 3B) and 25.8% to the
moment of inertia (Fig. 3A) for
a fully extended wing. The total moment of inertia for the outstretched
standardized wing shown here was 4.02x105 kg
m2; the standard deviation between the three individual
wings was 3.12x106 kg m2. Virtual
mass for each section was computed using the following equation from Norberg
(1990):
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Our model treats each wing section as a point mass. This is a reasonable
assumption given the concentration of mass in each strip at the leading edge
in the bone and muscle rather than in the feathers extending posterior and the
large number of wing slices we employed
(Van den Berg and Rayner,
1995). We merged the mass information from the standard wing with
the 3-D kinematics by computing the position of each wing strip in each video
frame, then distributing the appropriate number of strips between the
shoulder, wrist and wingtip. We then derived the acceleration of each wing
section in the global frame of reference by taking the 2nd derivative of a
quintic spline fit between the successive positions of each wing strip. The
resulting X-, Y- and Z-axis section accelerations
were used to reconstruct the inertial accelerations with equations
3,
4,
5,
6,
7(below). Lastly, we subtracted
the predicted inertial accelerations from the accelerometer recordings leaving
only the accelerations due to aerodynamic forces. We assumed that the two
wings operate symmetrically and that the Y-axis inertial
accelerations cancel each other.
The predicted inertial accelerations of CB were
calculated from equations 1015a of Bilo et al.
(1984), restated here for
convenience as equations 3,
4,
5,
6,
7:
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![]() | (6) |
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Inertial power
We calculated the inertial power requirements for downstroke and upstroke
by taking the change in wing Ek from the start of each
half-stroke to its point of maximum Ek and dividing these
energies by the duration of the entire wingbeat cycle. Wing
Ek was computed from the mass distribution and kinetic
analysis described above. We consider only the work of wing acceleration in
both upstroke and downstroke because deceleration is unlikely to require
substantial metabolic input in any circumstance due to the high metabolic
efficiency of vertebrate muscle when actively generating force to absorb
energy (Abbott et al., 1952).
By contrast, we calculate power from the entire wingbeat duration because the
muscle contractions involved in powering the movements of the wing occur over
an entire wingbeat cycle.
Wing kinetic energy recovery
Current models of forward flight in birds assume that the inertial power
requirements for downstroke do not incur any metabolic cost because the wing
Ek is recovered as aerodynamic work done to support or
propel the bird (Pennycuick,
1996; Askew et al.,
2001
; Tobalske et al.,
2003
). This is presumed to occur in the latter half of downstroke
as the wing loses kinetic energy while doing aerodynamic work to produce
forces (lift and thrust) that support and propel the bird. Following Askew et
al. (2001
), we assume that if
the aerodynamic work required to produce the observed whole-body accelerations
exceeds the wing Ek then all energy is transferred. Only
in cases where the wing Ek exceeds aerodynamic work does
the inertial power incur a metabolic cost. This analysis ignores any
additional muscle work done during the wing deceleration phase but provides a
simple benchmark to evaluate the importance of inertial power requirements in
downstroke.
To carry out this analysis, it was necessary to estimate the aerodynamic
work required to produce the observed aerodynamic forces. We employed the
aerodynamic model described in equations
2,
3,
4,
5,
6,
7 of Hedrick et al.
(2003) to estimate the
instantaneous aerodynamic power output based on the bird's wing kinematics and
whole-body accelerations. For the purposes of comparison with wing
Ek, we computed the aerodynamic work performed during the
period of wing deceleration in downstroke. This can be summarized as:
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Results |
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Despite continuous variation in acceleration throughout a wingbeat, certain features of the acceleration profile of the wingbeat cycle remained consistent across flight speeds. We found that maximum horizontal acceleration occurred at a cycle phase of 0.65 (zero being defined as the start of upstroke, and 1.0 the end of downstroke) and maximum vertical acceleration occurred at a phase of 0.74. Both of these occurred near the kinematic mid-downstroke (phase of 0.73), with the wings fully outstretched and horizontal to the bird. These phases did not vary significantly with flight speed (F=0.38 and F=1.78 for vertical and horizontal acceleration phase, respectively; P>0.05, repeated-measures ANOVA) but were significantly different from one another (P<0.05, t-test of individual means).
We also observed that the initial two-thirds of downstroke generally produced positive (forward) thrust, while the latter third resulted in negative (rearward) thrust. The negative thrust was associated with a large angle of incidence adopted by the wing late in downstroke as the wing supinated prior to upstroke. Positive (upward) lift was produced over the entire downstroke at all speeds. By contrast, lift and thrust production during upstroke generally varied more, especially at slow and fast flight speeds. During the upstroke, lift and drag tended to vary together. For example, at 7 m s1 (Fig. 5B), the cockatiels reduced both lift and drag whereas at 13 m s1 (Fig. 5C) lift production was coupled with increased drag. Finally, net accelerations due to aerodynamic forces were near zero at the end of downstroke at all speeds, despite the rapid wing rotations that occurred at this time during slower flight speeds (13 m s1; Fig. 5).
Mean aerodynamic forces in upstroke and downstroke
As expected, the mean magnitude of whole-body acceleration resulting from
aerodynamic forces in both the vertical and horizontal directions varied
significantly between upstroke and downstroke (P<0.001 for both
vertical and horizontal, paired t-test). The mean vertical
acceleration during upstroke and downstroke also varied significantly with
speed (P<0.05, F=3.81 and F=4.52 for downstroke
and upstroke, respectively; repeated-measures ANOVA). Differences in mean
vertical acceleration between upstroke and downstroke were minimized at
intermediate flight speeds and maximized at both faster and slower speeds
(Fig. 6). The same trends also
characterize the mean horizontal accelerations. Furthermore, the specialized
tip-reversal upstroke employed by cockatiels at slow flight speeds (1 m
s1 and 3 m s1) was not associated with
large net accelerations in any direction.
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Wing kinetic energy and the inertial power requirements of flight
The inertial work required to accelerate the wing varied significantly with
flight speed downstroke (P<0.01, F=4.18;
repeated-measures ANOVA) but not upstroke (P>0.05;
Fig. 7A). However, because of
speed-related variation in the duration of upstroke relative to downstroke,
wing inertial power varied significantly with speed for both upstroke and
downstroke (P<0.05, F=3.81 and P<0.01,
F=6.49, respectively; repeated-measures ANOVA;
Fig. 7B). Not surprisingly,
wing inertial work and power during the downstroke exceeded that during the
upstroke. This results from the wings' outstretched configuration and
increased moment of inertia during the downstroke compared with their flexed
configuration during upstroke. However, we did find that in slow-speed flight
(13 m s1) the exceptionally brief duration of wing
acceleration in upstroke elevated the mean power output to a level nearly
equal to that in downstroke.
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These inertial work requirements translate into substantial mass-specific
inertial power requirements when measured over a complete wingbeat cycle.
Downstroke pectoralis mass-specific inertial power requirements averaged 24.0
W kg1 across the entire speed range
(Fig. 8). This represents 21.9%
of the cockatiels' pectoralis power output, based on the results of Tobalske
et al. (2003). Nevertheless,
the maximum wing Ek developed during downstroke in
excess of the aerodynamic work performed during wing deceleration
(Ek,rd) was negative at speeds less than 7 m
s1 and only slightly positive at faster speeds, averaging
7.25 W kg1 or 7.0% of pectoralis power output for speeds of
>7 m s1.
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Muscle mass-specific inertial power requirements for upstroke were greater than those for downstroke due to the large mass difference between the upstroke and downstroke musculature (Table 1). Assuming that the wings' acceleration and Ek during upstroke are achieved via contraction of the supracoracoideus and deltoideus major muscles, upstroke mass-specific inertial power reached a maximum of 122 W kg1 at a flight speed of 1 m s1 (Fig. 8) and averaged 89 W kg1 over all flight speeds. At slower flight speeds (15 m s1), upstroke mass-specific inertial power was similar to the pectoralis mass-specific power output required for aerodynamic force production (Fig. 8). At faster speeds, upstroke inertial power declined somewhat relative to pectoralis power output.
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Discussion |
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Given these assumptions, we can compare the following results from the two
studies: (1) the magnitude of inertial accelerations, (2) the magnitude of the
aerodynamic accelerations and (3) the timing of peak accelerations during the
wingbeat cycle. The pigeon was reported to produce inertial accelerations that
were approximately 2533% of the magnitude of the aerodynamic
accelerations. In cockatiels, the inertial accelerations were nearly
equivalent to their aerodynamic equivalents in the vertical direction and were
approximately 50% in the horizontal direction at a flight speed of 7 m
s1. This discrepancy is probably explained by three factors:
(1) the inclusion of added mass in the determination of cockatiel wing moment
of inertia, (2) differences in camera recording frequency and (3) the scaling
of wing size with body size. The added mass component elevates the cockatiel
moment of inertia and peak inertial acceleration by 26%. Removing this
component would decrease the cockatiel horizontal acceleration into the same
range as that of the pigeon, for which added mass was not included
(Bilo et al., 1984), but would
not fully account for the differences in vertical inertial acceleration.
Higher imaging frequencies will more accurately estimate the peak
accelerations and therefore result in greater inertial forces. We recorded at
250 Hz whereas Bilo et al.
(1984
) recorded at 80 Hz. This
difference probably explains at least part of the remaining discrepancy in
inertial accelerations. Any remaining differences are likely to be the result
of differences in wing morphology and moment of inertia between the two
species. Cockatiels have relatively long wings with a high moment of inertia
for their body mass (4.02x105 kg m2
in the present study). This is well above that of other bird species of a
similar body mass (Van den Berg and
Rayner, 1995
), although less than the value of
1.65x104 kg m2 for the much larger
pigeon studied by Bilo et al.
(1984
).
While we expect the pigeon to produce larger absolute aerodynamic forces due to its greater mass, whole-body accelerations of the cockatiels and pigeon should be comparable in magnitude. However, the peak horizontal and vertical accelerations of the pigeon reached nearly 40 m s2 compared with 22.6 m s2 peak vertical and 7.4 m s2 peak horizontal accelerations in the cockatiels. Because vertical and horizontal accelerations over the entire wingbeat cycle must average near 9.81 m s2 and 0 m s2, respectively, for both species, the lower peak values for the cockatiels indicate a smoother wingbeat cycle with less variation in acceleration at this flight speed. We believe this may reflect the cockatiels' use of an aerodynamically active upstroke, which we discuss below. At faster and slower speeds, peak accelerations in the cockatiels approached 40 m s2, comparable with those in the pigeon.
The timing of peak accelerations was similar between the two studies. Bilo
et al. (1984) reported that
peak aerodynamic accelerations occurred near mid-downstroke as the wings
passed through the horizontal plane, with peak vertical accelerations slightly
preceding peak horizontal accelerations. In the cockatiels, we also found peak
accelerations near mid-downstroke, although peak horizontal accelerations
slightly preceded peak vertical accelerations.
Lift production during upstroke
By obtaining 3-D kinematic and whole-body acceleration data, we confirmed
our hypothesis that the tip-reversal upstroke employed by cockatiels in slow
flight (13 m s1) is not an important source of lift
or thrust, contrary to some previous hypotheses
(Brown, 1963;
Aldridge, 1986
;
Norberg, 1990
;
Azuma, 1992
). The tip-reversal
upstroke did result in a slight upward acceleration, but this came at the cost
of a larger rearward acceleration (Figs
5,
6). The magnitude of both these
accelerations was much less than those produced during the downstroke.
Although the tip-reversal does not make an important contribution to weight
support, it was surprisingly effective at minimizing the inertial work
required to accelerate the wing in upstroke
(Fig. 7A). Despite the reduced
duration and increased amplitude of upstroke at slow flight speeds, the wings'
peak kinetic energy was not significantly greater than that at other speeds
(P>0.05; repeated-measures ANOVA). In a tip-reversal upstroke, the
proximal portion of the wing is accelerated in early upstroke while the distal
portion is allowed to travel freely. Later in upstroke, the proximal wing is
decelerated while the distal wing is accelerated. By accelerating different
portions of the wing at different times in upstroke the tip-reversal motion
effectively reduces the wing's peak kinetic energy and the work required to
accelerate the wing. This probably permits a more rapid upstroke than would
otherwise be possible, given the limited size of the cockatiel upstroke
musculature. Thus, the tip-reversal upstroke appears to be an effective means
for long-winged birds to rapidly elevate their wings without a substantial
increase in inertial work and negative (downward or rearward) aerodynamic
forces that might otherwise be produced by upward wing motion.
We also confirmed our hypothesis that upstroke lift at intermediate flight
speeds provides more substantial weight support than at slower and very high
speeds. At flight speeds from 5 m s1 to 11 m
s1, aerodynamic forces in upstroke produced upward
accelerations that exceeded 2 m s2 but were still less than
the 9.81 m s2 required to counter gravity
(Fig. 6). Upstroke lift
production was greatest at 5 m s1 and 7 m
s1, resulting in vertical accelerations as high as 6 m
s2. At these speeds, mean vertical acceleration during
downstroke was correspondingly reduced. As a result, upstroke lift was 35% of
that produced in downstroke. These results confirm expectations based on our
earlier work (Hedrick et al.,
2002) and are also consistent with the gradual increase in
upstroke wake energy with flight speed recently found in the thrush
nightingale (Spedding et al.,
2003
).
The energy source used to power aerodynamic force production during upstroke is most likely the bird's own kinetic and potential energy. This is because the upstroke musculature is of small size and is not well positioned to produce upward or forward aerodynamic forces. To estimate the changes in whole-body energy during upstroke, we integrated the instantaneous accelerations using initial velocities taken from the kinematics. This analysis showed generally similar decrements in whole-body kinetic and potential energies, consistent with their role in providing upstroke aerodynamic force. However, when examined more closely, we found that the cockatiels favored a slightly greater kinetic energy loss at flight speeds below 9 m s1 and greater potential energy loss at flight speeds above 9 m s1.
Downstroke inertial power
Our results for the inertial power requirements of downstroke generally
support the currently accepted view that the energy required to accelerate the
wing in downstroke is wholly subsumed within the aerodynamic power
requirements of avian flight (Pennycuick
et al., 2000; Askew et al.,
2001
; Hedrick et al.,
2003
). Although our calculations show that there is kinetic energy
in excess of aerodynamic work at speeds of >7 m s1, this
excess kinetic energy is negligible in comparison to the overall power
requirements for flight at these higher speeds
(Fig. 8). Our simple test of
the importance of wing kinetic energy
(equation 8) assumes that the
pectoralis muscle does no work during the latter half of downstroke as the
wing decelerates. However, prior in vivo measurements of pectoralis
force and length change obtained from cockatiels and other species show that
this is not the case the avian pectoralis continues to shorten and
produce force throughout the downstroke
(Dial et al., 1997
;
Biewener et al., 1998
;
Hedrick et al., 2003
).
Re-examination of our previous results for cockatiel pectoralis power
output (Hedrick et al., 2003)
shows that 24.2±4.6% (mean ± S.D.) of the work done
by the pectoralis is performed as the wing decelerates during the latter half
of the downstroke. Also, this fraction does not vary systematically with
flight speed. Thus, equation 8
should contain an additional term adding the work done by the pectoralis
muscle to the aerodynamic work and wing kinetic energy. At slow flight speeds
(15 m s1), the work done by the pectoralis during
wing deceleration does not affect the conclusion that inertial power
requirements are unimportant because the aerodynamic work greatly exceeded
wing kinetic energy (Fig. 8).
However, at flight speeds greater than 5 m s1, the wing
kinetic energy and aerodynamic work done during wing deceleration were similar
in magnitude. Consequently, accounting for work performed by the pectoralis
during downstroke wing deceleration increases the likelihood that not all of
the wing kinetic energy can be usefully transferred to the surrounding air and
that a significant fraction must be absorbed or stored through other
mechanisms at moderate to fast flight speeds.
An attractive possibility for elastic energy storage is the long robust
tendon of the supracoracoideus muscle. Electromyographic recordings of the
supracoracoideus of pigeons (Dial,
1992) show that it is activated during the terminal phase of the
downstroke. This suggests that, in addition to developing force to decelerate
and elevate the wing, its tendon may also store and recover excess kinetic
energy of the wing. Future study of the cockatiel supracoracoideus will be
needed to examine this possibility. Otherwise, although reduced at faster
speeds, any excess inertial kinetic energy during downstroke should be added
to the overall power requirements of flight for this species.
Upstroke inertial power
Although the inertial power required to accelerate the wing in upstroke
averaged only 14% of the cockatiels' total aerodynamic power requirements,
this inertial power requirement is probably incremental to the bird's
aerodynamic power requirements. Unlike the downstroke, no aerodynamic transfer
mechanism has been proposed. It is difficult to envision how the upwardly
moving wing of a bird can produce lift or thrust without the use of muscles
and while losing velocity during the second half of the upstroke, when
inertial kinetic energy would need to be recovered. Elastic strain energy
stored in the supracoracoideus tendon could be used to power the initial
acceleratory phase of the upstroke, reducing the amount of additional energy
input via the upstroke muscles. Although a reasonable candidate for
this, until now energy storage in the supracoracoideus tendon has not been
demonstrated and should not therefore be considered to account for the
inertial power required for upstroke.
In addition to the upstroke musculature, aerodynamic forces might also be used to elevate the wing at the beginning of upstroke. However, this would not eliminate the inertial power required for upstroke because aerodynamic forces require their own energy source. As we discussed above, this is most likely derived from losses in the bird's own potential or kinetic energy. Even so, this energy ultimately must be produced by the bird's downstroke musculature (pectoralis) during the subsequent wing beat cycle. Furthermore, our comparison of the wings' kinetic energy in upstroke relative to concurrent losses in whole-body kinetic and potential energy suggests that all whole-body energy losses are applied to weight support rather than wing elevation.
Elastic energy storage offers another possible mechanism for minimizing the
additional energy required for upstroke. As in downstroke, it is possible that
the kinetic energy is stored elastically as the wing decelerates, most likely
in the tendinous attachment of the pectoralis to the humerus and within the
pectoralis muscle itself. However, recordings of muscle force and length
change in the pectoralis of cockatiels
(Tobalske et al., 2003)
indicate that the muscle produces little force as the wing decelerates.
Consequently, although some storage may occur late in upstroke it does not
appear to be large enough to account for the loss in wing kinetic energy.
Aerodynamic force production and wing rotation
Our recordings of whole-body acceleration also allow us to assess
indirectly whether cockatiels obtain useful lift from wing rotation when
flying at slow speeds. Unsteady aerodynamic force production via wing
rotation has been described in insect flight
(Dickinson et al., 1999) and,
for flies, accounts for a substantial fraction of the lift needed for weight
support. Given the rapid supination and pronation of the wing that occurs in
cockatiels, and other birds, during the end of the downstroke and upstroke at
slower flight speeds, it seems possible that birds may also obtain some
benefit from wing rotation. However, we found that net aerodynamic forces
typically approached zero during stroke reversal at the lower flight speeds
(13 m s1; Fig.
5), when wing rotation is most pronounced. This suggests that
aerodynamic mechanisms associated with wing translation, rather than wing
rotation, predominate in the generation of flight power of cockatiels, as well
as other birds.
Future work
The use of accelerometer techniques first developed by Bilo et al.
(1984) and employed here to
analyze the inertial and aerodynamic power requirements of the steady flapping
flight of cockatiels over a range of speeds provides considerable insight into
the mechanisms for aerodynamic power production and kinetic energy exchange.
Such an approach might also be used to analyze patterns of aerodynamic force
production during unsteady, maneuvering flight to better understand the
aerodynamic mechanisms by which birds maneuver, particularly at slow speeds.
In combination with recordings of muscle activation, force production and
length change, measurements of whole-body acceleration may allow a more
detailed investigation of how birds produce and control the forces required
for maneuvering flight. As we have seen in our analysis of cockatiels during
steady flight across a range of speeds, interactions between wing inertia,
elastic energy recovery and muscle work, and the aerodynamic power
requirements of flight remain uncertain. Experiments that artificially vary
wing inertia while recording pectoralis length change and force production may
also provide better insight into the importance of wing inertia in flight and
how the behavior of a power-producing muscletendon system operates
under an inertial load. Inertial loads have recently been shown to accentuate
the peak power output of the plantaris muscle and tendon in jumping bullfrogs
(Roberts and Marsh, 2003
). A
similar mechanism could operate in the pectoralis muscle of birds during
flapping flight if the pectoralis tendon/aponeurosis can store and release
adequate elastic strain energy and this energy can be effectively transferred
to the air while producing useful aerodynamic force. Finally, the resolution
and recording frequency limitations of current high-speed video technologies
make derivation of whole-body accelerations from video unreliable in most
circumstances, encouraging the use of accelerometers. However, video
technologies are improving rapidly and may soon allow sufficiently accurate
acceleration measurements without requiring the use of accelerometers.
Summary
The combination of high-speed 3-D kinematics and three-axis accelerometer
data allowed us to explore the timing and magnitude of net aerodynamic force
production throughout the wingbeat cycle of cockatiels flying across a range
of steady speeds. Our results reveal that the proposed mechanisms for
aerodynamic force production during upstroke in slow flight result in little
net force. However, useful aerodynamic force production during upstroke at
intermediate speeds was observed, consistent with our earlier estimates of
circulation and lift production (Hedrick
et al., 2002) and with the latest flow visualization analysis of
steady avian flight (Spedding et al.,
2003
). At speeds from 7 m s1 to 11 m
s1, net aerodynamic forces during upstroke acted in an
upward and rearward direction, as expected. Detailed examination of the
inertial power requirements for flapping flight in cockatiels generally
supported the commonly held assumption that the inertial kinetic energy of the
wing during the downstroke is converted into useful aerodynamic work and need
not represent an incremental cost to the power required for flight. This was
best supported at slower flight speeds, where inertial power requirements were
greatest but aerodynamic power requirements were also large. At faster flight
speeds, excess downstroke kinetic energy was observed. We believe that this
excess kinetic energy may be recovered by elastic storage within the tendon of
the supracoracoideus muscle, but this remains to be confirmed. Our analysis of
the kinetic energy imparted to the wing during upstroke indicates that the
majority is probably dissipated and should be added to the total power
requirement for flapping avian flight. However, some fraction of this energy
may also be stored and recovered within elastic structures of the
pectoralis.
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