The effect of advance ratio on the aerodynamics of revolving wings
California Institute of Technology, Mail Code 138-78, Pasadena, CA 91125, USA
* Author for correspondence (e-mail: wbd{at}caltech.edu)
Accepted 27 August 2004
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Summary |
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Key words: flapping flight, quasi-steady force, unsteady aerodynamics, insect flight, Reynolds number, insect aerodynamics
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Introduction |
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Robotic models have proved a powerful tool in the investigation of
aerodynamic mechanisms during flapping flight
(Bennett, 1970;
Maxworthy, 1979
;
Dickinson and Gotz, 1993
;
Dickinson et al., 1999
;
Ellington et al., 1996
;
Sane and Dickinson, 2001
).
Such models have allowed investigators to examine the effects of wing rotation
as well as wingwake and wingwing interactions. A complication
encountered when studying flapping flight using robotic models is that of
isolating and quantifying the effect of a particular variable, such as wing
rotation or forward velocity, upon force production. One technique commonly
used by researchers to circumvent such complications is to employ extremely
simplified sets of wing kinematics in order to elucidate and characterize a
particular feature of force production. An example of the effective use of
such a simplified set of kinematics is the study of `revolving' wings
(Usherwood and Ellington,
2002a
,b
)
in which a propeller arrangement is used to isolate the force generation
mechanisms of the downstroke and upstroke from the complicating effects of
pronation and supination.
The effect of advance ratio on revolving wings has been considered
previously in the context of helicopter aerodynamics
(Isaacs, 1946;
van der Wall and Leishman,
1994
). It is difficult, however, to apply these directly to insect
flight because helicopters use high aspect ratio wings operating at relatively
low angles of attack, conditions atypical of insect flight. As a result, they
are more amenable to a blade element model in which sectional force
coefficients derived from two-dimensional (2D) studies are used to predict
total aerodynamic forces. In contrast, insect wings have a low aspect ratio,
approximately 210 (Dudley,
2000
), and typically operate at high angles of attack, often
greater than 40°. Low aspect ratio wings revolving at high angles of
attack are known to form a stable leading-edge vortex that is responsible for
elevated force coefficients (Ellington et
al., 1996
; Dickinson et al.,
1999
; Birch and Dickinson,
2001
). For this reason, previous models of insect flight have used
mean sectional force coefficients derived from three-dimensional (3D) studies
employing a revolving wing. These differences between the aerodynamics of
helicopter rotors and insect wings highlight the need for a rigorous study of
the effect of advance ratio on the forces produced by revolving wings of a
shape, speed and angle of attack typical of insects.
In this study, we characterize the effect of advance ratio on aerodynamic force generation during forward flight using a dynamically scaled mechanical model of Drosophila melanogaster. Forces are measured over a range of advance ratios spanning the transition from hovering to fast forward flight. The kinematic pattern we used consists of a wing revolving in a horizontal stroke plane with constant angular velocity at a fixed angle of attack. From the instantaneous force records we estimate the contribution due to added mass and compare it with theoretical predictions. The added mass component is then subtracted from the force traces and mean sectional lift and drag coefficients are calculated. The mean sectional lift and drag coefficients are found to depend upon the angle of attack and the velocity profile experienced by the wing. We show that this dependence upon angle attack follows the same trigonometric relationships as that of hovering flight. However, the variation of the force coefficients with velocity profile is new and implies that modifications to the quasi-steady model are required in order to accurately predict forces during forward flight. We show that the variation of the force coefficients with velocity profile can be effectively characterized in terms of the tip velocity ratio of the wing. A modified version of the quasi-steady model is presented that incorporates this variation.
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Materials and methods |
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Kinematics
In a manner similar to that described previously
(Sane and Dickinson, 2001),
the kinematics of the wings are specified by the time course of three angles:
stroke position
, the angle of attack
and stroke deviation
(Fig. 1B). The
relatively simple kinematic patterns used in this study were chosen to isolate
the effects of advance ratio, stroke position, and angle of attack upon
aerodynamic force generation without the additional complications of
rotational forces or wingwake interactions.
In the first set of kinematic patterns the wing was towed through the oil at constant forward velocity while at the same time revolving through a 500° arc at a constant angular velocity of ±72 deg. s1. During each trial we maintained the angle of attack at a fixed value. Four forward velocities were used in these experiments: 0, 0.04, 0.08, 0.12 and 0.16 m s1. For each forward velocity the angle of attack was systematically varied in 10° increments, from 110° to 10° for a total of 96 runs. For all of these trials, stroke deviation angle was fixed at zero. Angle of attack is defined as the angle between the wing's chord and the tangent of the wing's trajectory.
In the second set of kinematic patterns the wing was towed through the oil at constant forward velocity of 0.16 m s1 at a fixed stroke position angle of 0°. During each trial we maintained the angle of attack at fixed value, which was varied from 10° to 110° in 10° increments.
Dynamic scaling
Two non-dimensional parameters are required in order to achieve an accurate
dynamic scaling of the forces obtained via the robotic model: the
Reynolds number (Re), and the advance ratio
(Spedding, 1993). The Reynolds
number is given by:
![]() | (1) |
![]() | (2) |
A third dimensionless parameter that will prove useful in our analysis is
the tip velocity ratio µ:
![]() | (3) |
Data acquisition and analysis
Force data from the 2D strain gauges were sampled at 1500 Hz using a
Measurement Computing PCI-DAS-1000 Multifunction Analog digital I/O board
(Middleboro, MA, USA) and filtered offline using a zero phase delay low-pass
4-pole digital Butterworth filter, with a cut-off frequency of 3 Hz. The
positions of the four servo-motors were acquired simultaneously using the
multifunction card and custom electronics for decoding the quadrature encoders
of the servo-motors. In this manner it was possible to determine the
instantaneous position of the motors, and thus the wing.
Because the stroke amplitude of most insects is less than 180° the
condition when is between 90° and 90° is of particular
interest. With this in mind, the stroke length used in this study was selected
to meet two criteria. First, the strokes needed to be long enough so that
there was sufficient time for the force transients resulting from the
acceleration of the wing to disappear before
was within the region
90° to 90°. Second, the strokes needed to be short enough so as
not to incur any wingwake interactions in this region. Accordingly, we
chose a pattern in which the wing revolved from 250° to 250° or
from 250° to 250°.
The force measured by the strain gauges at the base of the wing can be
decomposed into gravitational, inertial and fluid dynamic components. The
gravitational component of the measured forces is due to the mass of the wing
and the mass of the sensor, and may be calculated and subtracted from the
measured forces. In practice the subtraction was determined by moving the wing
through sample kinematic patterns at very low velocity, for which the
aerodynamic and inertial forces are negligible, and fitting the functions for
the parallel and normal measured forces:
![]() | (4) |
![]() | (5) |
The inertial component of the measured forces consists of two components:
the action of the acceleration forces on the mass of the wing and sensor, and
the added mass of the fluid around the wing (see
equation 21). The contribution
of the acceleration forces of the wing and sensor masses to the total measured
forces for the robotic apparatus are negligibly small
(Sane and Dickinson, 2001).
The added mass component experienced by the wing was estimated from the data
obtained when
was between 90° and 90° in the following
manner. The force produced by the wing consists of the sum of the
translational and added mass force components. The translational force
component is typically proportional to the square of the flow velocity.
Because the flow velocity is a symmetric function of stroke position, the
translational force component Ft should also be a symmetric
function, and thus Ft(
) should be equal to
Ft(
) for equal angles of attack. The added mass
force is proportional to the acceleration of the flow in the direction normal
to the surface of the wing. As the acceleration of the flow is an
antisymmetric function of stroke position, the added mass force component
Fa should be an antisymmetric function of stroke position,
and thus Fa(
) should be equal to
Fa(
) for equal angles of attack. This
observation shows that the difference between the force measurements at stroke
positions
and
can be attributed solely to added mass
component of the forces because the translational force components cancel.
Thus, for fixed angle of attack the added mass force can be estimated by:
![]() | (6) |
The translational component of the forces was isolated by subtracting the
estimates of the forces due to added mass from the measured forces. The
instantaneous mean force coefficients for lift and drag were then calculated
using:
![]() | (7) |
![]() | (8) |
Equations 7 and
8 were derived from blade element
theory and take into account the changing instantaneous velocity profile
experienced by the wing. When µ is equal to zero the usual mean force
coefficients used for a stationary revolving wing
(Osborne, 1951;
Sane and Dickinson, 2001
;
Usherwood and Ellington,
2002a
) are obtained:
![]() | (9) |
![]() | (10) |
![]() | (11) |
![]() | (12) |
The force coefficients given in equations
7 and
8 can be viewed as functions of
two parameters: the angle of attack and the tip velocity ratio µ.
The variation of the lift and drag coefficients with angle of attack for
hovering flight is known to be well approximated by trigonometric expressions
(Dickinson et al., 1999
). In
order to determine if these relationships are still approximately true,
normalized lift and drag coefficients were derived for each angle of attack
. The normalized lift coefficient is defined by:
![]() | (13) |
![]() | (14) |
In order to examine behavior of the lift and drag coefficients as a
function of tip velocity ratio, µ, the measured lift and drag coefficients
were fit via least squares, for each µ, to the following
equations:
![]() | (15) |
![]() | (16) |
Quasi-steady model
In this section we extend the quasi-steady model for hovering flight (Sane
and Dickinson, 2001,
2002
) to the special case of
forward flight consisting of a revolving wing translating at constant forward
velocity. For simplicity we assume that the angle of attack
, the
angular velocity of the wing, and the forward velocity Vf,
of the wing are all constant. Further, we set the deviation angle
to
zero so that the stroke plane is horizontal. In our model the instantaneous
force generated by the wing is represented by the vector sum of two
components:
![]() | (17) |
For a wing revolving at instantaneous angular velocity and moving forward
at velocity Vf, the magnitude of the sectional flow
velocity is given by:
![]() | (18) |
![]() | (19) |
![]() | (20) |
![]() | (21) |
![]() | (22) |
|
Under the quasi-steady assumption, the translational force term
Ft depends solely upon the instantaneous angle of attack and
velocity profile experienced by the wing. Ft can therefore
be expressed in terms of the mean sectional force coefficients in the
following manner:
![]() | (23) |
![]() | (24) |
![]() | (25) |
![]() | (26) |
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Results |
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The forces shown in Figs
3AD,
4AD vary with time as
the wing sweeps through the background flow. Because the angular velocity of
the wing is constant, the stroke position of the wing is a linear function of
time. Thus, the forces in the figures may alternatively be viewed as varying
with stroke position. Such a view explicitly ignores any time dependence in
the flows and forces. This simplification is justified, however, because the
effect of the initial stroke position did not measurably influence the
-dependence of the forces. Thus, while exhibiting a dependence upon
, the forces showed no intrinsic time dependence once the transients due
to the starting accelerations decayed. Because the flow velocity at each wing
section is a function of the stroke position,
, the aerodynamic forces
experienced by the wing also depend upon
. During the downstroke, when
the wing sweeps against the net flow, the sectional flow velocities increase
from r| | to r|
|+Vf as
goes from 90° to 0°,
and then decrease to r| | again as
goes from
0° to 90°. The lift and drag forces, which depend on the square of the
flow velocity, reflect these changing velocities reaching a maximum near
=0°. During the upstroke, when the wing sweeps with the background
flow, the flow velocities decrease from r| | to
r| |Vf as
goes from
90° to 0°, and increase to r| | as
goes
from 0° to 90°. Again, the effects of the changing sectional
flow velocities are reflected in the force traces. As expected, the effect of
stroke position on force production is greater as advance ratio increases.
Added mass
Because the flow velocity experienced by each wing section varies with time
it will experience an added mass force. This acceleration is the same for each
wing section and is given by equation
19. In Fig. 5, we
plot the added mass force estimated using
equation 6 as a function of the
absolute value of the acceleration. The theoretical estimate for the added
mass force (equation 21) is
shown for comparison. The magnitude of the added mass force is quite small
compared to the aerodynamic forces and approaches the noise limit of our
measurements for low accelerations. However, the trend is quite clear and the
match between the theoretical estimate and the measured values is reasonable.
The theoretical estimate of the constant of proportionality that relates
acceleration to force is 0.96 kg, whereas a linear regression to the data
collected in all 96 trials yields an estimated constant of proportionality of
0.98 kg, which is statistically indistinguishable from the theoretical value.
This result suggests that added mass forces account for the slight asymmetry
in the lift and drag forces about =0, which is evident in Figs
3 and
4.
|
Angle of attack
Using equations 7 and
8, lift and drag coefficients
were constructed from the force traces, after subtracting the added mass
component. Previous studies of hovering flight (Dickinson,
1996,
1994
;
Ellington and Usherwood,
2001
), observed that, aside from a small contribution due to skin
friction, the translational component of the force experienced by the wing is
approximately normal to the surface of the wing.
Fig. 6 shows a plot of force
angle, the angle between the total force vector and the wing's surface,
versus
for all 96 trials. At angles of attack above about
15° the force is approximately normal to the surface of the wing. This
suggests that for high angles of attack differences in pressure normal to the
surface of the wing dominate force production. For small angles of attack less
than 15°, the force angle is less than 90°, an effect that can be
attributed to skin friction.
|
Prior studies of revolving or flapping model wings
(Dickinson et al., 1999;
Usherwood and Ellington,
2002a
,b
)
have shown that the mean sectional lift coefficient is proportional to
sin(
)cos(
), whereas the mean sectional drag coefficient, minus
skin friction, is proportional to sin2(
). The quasi-steady
model presented earlier in equations
25 and
26, suggests that for a fixed
tip velocity ratio µ, these functional relationships will still hold.
However, the constants of proportionality in the relationships are, in
addition, functions of the tip velocity ratio in the case of forward flight.
To test whether or not this approximation is valid, we calculated the
normalized lift and drag coefficients using equations
13 and
14
(Fig. 7A,B). Plots of the
functions 2sin(
)cos(
) and sin2(
) are shown for
comparison. Agreement between the normalized coefficients and the
trigonometric functions is quite close. This suggests that the mean sectional
lift and drag coefficients during forward flight behave in a manner analogous
to that during hovering with respect to angle of attack, provided the effects
of tip velocity ratio are properly taken into account.
|
Tip velocity ratio
Given that the lift and drag coefficients obey the trigonometric functional
relationships given by equations
15 and
16 with respect to angle of
attack, the task of determining the effect of tip velocity ratio µ is
reduced to characterizing the amplitude and offset functions
K1(µ), K2(µ) and
K3(µ). In Fig.
8 we plot the drag coefficient versus lift coefficient
for several tip velocity ratios. A fit of equations
15 and
16 for each tip velocity ratio
is shown for comparison. For angles of attack greater than approximately
30°, both the lift and drag coefficients decrease with increasing tip
velocity ratio. For the drag coefficients at small angles of attack, this
trend is reversed. Also shown in Fig.
8 is a fit of equations
15 and
16 to hovering data from Birch
et al. (2004). The values of
the lift and drag coefficients from the hovering data coincide with the lift
and drag coefficients from the zero tip velocity ratio case. In general equal
tip velocity ratios, regardless of the advance ratio, result in equivalent
force coefficients. However, at higher advance ratios a greater range of tip
velocity ratios is achieved during each stroke.
|
The quasi-steady model, equations
25 and
26, suggests that an appropriate
functional form for the amplitude and offset functions is that of a rational
function whose numerator and denominator are second order polynomial functions
of µ. The values of K0(µ),
K1(µ) and K2(µ) estimated from
the data are shown in Fig. 9.
Included in the figure for comparison are least-squares fits of the functions:
![]() | (27) |
|
Fig. 10 shows a plot of the
lift coefficients versus drag coefficients as a function of angle of
attack for a non-revolving wing, with a constant forward velocity of 0.16 m
s1 and a fixed stroke position angle of 0°. For this set
of kinematics both the advance ratio J and the tip velocity ratio
µ are essentially infinite. The quasi-steady model with coefficients
determined by the fit to the first set of kinematic patterns, with µ
between 0.5 and 0.5, can be extrapolated to predict the lift and drag
coefficients for the non-revolving wing by taking the limit of equations
25 and
26 as µ approaches infinity:
![]() | (28) |
![]() | (29) |
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Discussion |
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Added mass forces
The added mass forces estimated from experimental data closely agree with
the theoretical predictions made using
equation 21. Both the measured
and predicted forces were quite small in magnitude and represent less than 10%
of the total force generated by the wing. Also, over the course of an actual
stroke cycle they would average to zero so that net their effect on average
forces is insignificant. Nevertheless, it is possible that they remain large
enough to play a role in the delicate force and moment balance that takes
place during aerial maneuvers.
For the kinematics considered in this study, the theoretical predictions of
the added mass force, based on an approximation given in Sedov
(1965), match the estimates
from experimental data quite well. It has been shown, however, that for some
types of wing kinematics this is not the case. Birch and Dickinson
(2003
) considered the forces
produced by a back-and-forth flapping pattern in which the time course of
stroke position is a filtered triangle wave. They observed that the time
course of the forces generated at the start of a stroke were not well matched
by the same added mass model considered here. This discrepancy held even for
impulsive starts, when wingwake interactions are not present. A
significant difference between the two cases is that magnitude of the peak
acceleration in the back-and-forth pattern was approximately 10 times greater
than those of the revolving and translating wing in this study. Thus, it
appears the Sedov model is reasonably accurate for the more gentle
accelerations but underestimates forces during higher accelerations.
Translational forces
The quasi-steady model of the translational force coefficients, equations
25 and
26, is based on a blade element
derivation. In this treatment, the sectional force coefficients vary with
spanwise location, a dependence embodied by the functions
kj() in equations
A1 and
A2. In contrast, previous work
on revolving wings under hovering conditions have employed mean sectional
force coefficients that are assumed constant with respect to spanwise location
(Sane and Dickinson, 2001
;
Usherwood and Ellington,
2002a
). In the zero advance ratio limit such an assumption is not
detrimental, because for a given angle of attack the mean sectional force
coefficients are not sensitive to variations in velocity provided that the
dependence of the sectional force coefficients on span, regardless of form,
does not vary over the range of velocities considered. However, the
simplification does not hold at finite advance ratio. With the addition of
forward velocity, the mean force coefficients may become sensitive to
variations in the flow velocity profile experienced by the wing. Even assuming
that the functional dependence of the sectional force coefficients upon span
remains the same, the mean force coefficients may depend upon the
instantaneous velocity profile experienced by the wing. Only in the special
case where the sectional coefficients are constant with respect to span does
the dependence of the mean force coefficients upon the velocity profile
disappear. This effect complicates the analysis of forward flight and was the
reason we adopted a more general approach here. Theoretical considerations
that take into account the effect of tip vortices
(Katz and Plotkin, 2001
) as
well as recent experimental results (Birch
and Dickinson, 2003
) suggest that for each angle of attack the
sectional force coefficients do indeed depend upon span. The exact form of
this dependence, and whether for each angle of attack the sectional force
coefficients are dependent or independent of the velocity profile, is not yet
known.
Mean sectional force coefficients determined from zero advance ratio data
as a function of angle of attack are available for various wing planforms and
at various Reynolds numbers (Sane and
Dickinson, 2001; Usherwood and Ellington,
2002a
,b
;
Birch et al., 2004
). For this
reason it is interesting to compare the total force coefficients estimated
from the forward flight data using equations
25 and
26, with those from hovering
data. The force coefficients from hovering data agree with the coefficients
from forward flight data when the tip velocity ratio µ=0. For angles of
attack typical of insect flight (3090°) at tip velocity ratios
<0, the lift and drag coefficients are greater than those during hovering
flight, and at tip velocity ratios >0 the lift and drag coefficients are
less than those during hovering flight. For low advance ratios (<0.1), this
discrepancy can probably be ignored without incurring too much error. However,
as advance ratio increases modifications are required in order to predict
forces accurately.
The quasi-steady model, with coefficients derived from finite advance ratio data, was found to extrapolate fairly well to steadily translating wings (Fig. 10). The model as currently posed attributes the difference in force coefficients entirely to the effect of the instantaneous velocity profile on the constant spanwise distribution of sectional force coefficients. In particular, it is assumed that the the spanwise distribution of sectional forces coefficients for a given angle of attack does not itself depend on the velocity profile. This is probably not entirely true. However, it appears to be a reasonable approximation for tip velocity ratio between 0.5 and 0.5. It also captures the trend correctly at high advance ratios. Validation of this assumption will require measurements of the spanwise loading of a wing at various tip velocity ratios.
Interest in the possible role of unsteady effects in insect flight was
stimulated in large part by the comprehensive analysis of Ellington
(1984a), in which he tested
the feasibility of quasi-steady models using a `proof by contradiction'. He
compared available experimental measures of the maximum steady-state lift
coefficients in the literature with the values required to support hovering
flight based on body morphology and simplified wing kinematics. His conclusion
was that experimental values were typically too low to account for the forces
required to sustain flight, thus justifying a search for unsteady effects that
might account for the elevated performance of insect wings under flapping
conditions. However, the conclusions of Ellington's thorough analysis are in
conflict with recent studies demonstrating that revolving wings create
constant force in the Reynolds number range used by insects
(Dickinson et al., 1999
;
Usherwood and Ellington,
2002a
). More specifically, although revolving wings separate flow
and create a leading edge vortex, this flow structure is stable over many
chord lengths. Given these recent results it is perplexing why Ellington's
metanalysis demonstrated an insufficiency of quasi-steady models based on
previous measures of force coefficients on real and model wings in steady
translating flow. The results of our analysis offer a possible explanation for
this discrepancy. Namely, that the maximum steady-state lift coefficient
depends upon the velocity profile experienced by the wing, and use of lift
coefficients from steadily translating wings, with essentially infinite tip
velocity ratio, leads to an underestimate of the possible lift for a flapping
or revolving wing. From these results it is clear that unsteady mechanisms may
not be required in order to explain the force balance for a hovering insect,
but only that the appropriate force coefficients be used.
Implications of kinematics
During steady forward flight it is likely that an insect must adopt
appropriate wing kinematics to balance lift, thrust and body moments at each
forward velocity. Several studies (David,
1978; Willmott and Ellington,
1997
) that have examined the relationship between forward flight
speed and body angle found an inverse correlation, such that the angle between
the insects body and the horizontal plane decreases with increasing flight
speed. Further, in a study of Manduca sexta, Willmott and Ellington
(1997
) demonstrated that there
is a positive correlation between stroke plane angle and forward speed. During
forward flight the angle of attack, and thus the instantaneous forces
produced, depend strongly upon the stroke plane angle. From these studies it
is clear that wing kinematics, at least via changes in stroke plane
angle, do indeed vary in a systematic manner with forward velocity. Without a
comprehensive understanding of force production for arbitrary wing kinematics
over a suitable range of advance ratios it is difficult to interpret how the
observed changes in wing motion effect the appropriate force and moment
balance.
In order to keep things as simple as possible the kinematics employed in this study all had a stroke plane angle of zero, which we know to be unrealistic. It is not yet known how changes in stroke plane angle will further modify the measured lift and drag coefficients. Further studies are required to determine the combined effect of forward velocity and nonzero stroke plane angles.
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Appendix |
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![]() | (A1) |
![]() | (A2) |
![]() | (A3) |
![]() | (A4) |
Integrating the sectional lift and drag forces along the span of the wing
and substituting equations A1
and A2 for the sectional lift
and drag coefficients yields the following expressions for the magnitudes of
the total lift and drag forces experienced by the wing:
![]() | (A5) |
![]() | (A6) |
![]() | (A7) |
Equating the expressions for lift and drag given by equations
23 and
24 and by equations
A5 and
A6, respectively, and then
solving for the lift and drag coefficients, yields the desired expressions for
the mean sectional lift and drag coefficients:
![]() | (A8) |
![]() | (A9) |
List of symbols
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Acknowledgments |
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References |
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