In most insects, the kinematics of flapping flight consists of
two translational phases during which the wings sweep through
the air with relatively slow changes in the angle of attack,
followed by rapid rotations at the end of each stroke. These
wing flips, termed ‘pronation’ for the upstroke-to-downstroke
transition and ‘supination’ for the downstroke-to-upstroke
transition, allow insects to maintain a positive angle of attack
and thus to generate lift during both forward and reverse
strokes. An understanding of the actual aerodynamic
significance of these wing rotations has long been hindered by
a lack of precise instantaneous force measurements on flapping
airfoils. However, with recent advances in our knowledge of
the instantaneous forces on wings (Dickinson et al., 1999), it
is possible to characterize the role of wing rotation and to
incorporate rotation in existing quasi-steady models of insect
flight.
Aerodynamic theorists have long recognized the importance
of airfoil rotation in the context of fluttering wings. Munk
(1925) predicted that, when two-dimensional airfoils translate
while simultaneously rotating with small amplitudes,
additional circulation is required to maintain the Kutta
condition at the trailing edge. He calculated that the magnitude
of rotational circulation should depend on the axis of rotation
such that, when the axis of rotation crosses a critical point
along the chord, the circulation will reverse sign. Thus, there
exists a critical axis on the wing about which rotation
contributes no net circulation. The relative position of the
rotational axis with respect to this critical axis determines
whether rotational circulation enhances or attenuates the lift
generated via translation.
These ideas were further developed by Glauert (1929) and
Theodorsen (1935) and later by Fung (1969), who proposed a
quasi-steady model for flutter and predicted that the critical
axis resides at a distance of 0.75chordlengths from the leading
edge. Reid (1927), Silverstein and Joyner (1939) and Halfman
(1951) provided experimental support for these models by
demonstrating that oscillating airfoils placed in a steady air
stream generate aerodynamic forces that differ from the steady-
state case in accordance with the theoretical predictions. Most
notably, Farren (1935) investigated how forces varied with
both increasing and decreasing angle of attack on an airfoil
placed in a wind tunnel and showed that, when the angle of
1087
The Journal of Experimental Biology 205, 1087–1096 (2002)
Printed in Great Britain © The Company of Biologists Limited
JEB3926
We used a dynamically scaled model insect to measure
the rotational forces produced by a flapping insect wing. A
steadily translating wing was rotated at a range of
constant angular velocities, and the resulting aerodynamic
forces were measured using a sensor attached to the base
of the wing. These instantaneous forces were compared
with quasi-steady estimates based on translational force
coefficients. Because translational and rotational velocities
were constant, the wing inertia was negligible, and any
difference between measured forces and estimates based
on translational force coefficients could be attributed to
the aerodynamic effects of wing rotation. By factoring
out the geometry and kinematics of the wings from
the rotational forces, we determined rotational force
coefficients for a range of angular velocities and different
axes of rotation. The measured coefficients were
compared with a mathematical model developed for two-
dimensional motions in inviscid fluids, which we adapted
to the three-dimensional case using blade element theory.
As predicted by theory, the rotational coefficient varied
linearly with the position of the rotational axis for all
angular velocities measured. The coefficient also, however,
varied with angular velocity, in contrast to theoretical
predictions. Using the measured rotational coefficients, we
modified a standard quasi-steady model of insect flight to
include rotational forces, translational forces and the
added mass inertia. The revised model predicts the time
course of force generation for several different patterns of
flapping kinematics more accurately than a model based
solely on translational force coefficients. By subtracting
the improved quasi-steady estimates from the measured
forces, we isolated the aerodynamic forces due to wake
capture.
Key words: quasi-steady, model, insect, flight, aerodynamics, wing
rotation, kinematics, flapping, rotational forces.
Summary
Introduction
The aerodynamic effects of wing rotation and a revised quasi-steady model of
flapping flight
Sanjay P. Sane* and Michael H. Dickinson
Department of Integrative Biology, University of California, Berkeley, CA 94720, USA
*e-mail: sane@socrates.berkeley.edu
Accepted 30 January 2002
1088
attack increases, aerodynamic force coefficients are enhanced
compared with corresponding steady-state values. In contrast,
when the angle of attack decreases, the force coefficients are
lower than the steady-state values. Because these experiments
aimed to simulate inviscid conditions, they were performed at
high Reynolds numbers and may not be directly applicable to
the low-to-intermediate Reynolds numbers relevant to insect
flight. In addition, these previous experiments explored a range
of angular velocities of wing rotation that are at least one order
of magnitude lower than those used by insects when pronating
and supinating their wings.
To address the role of wing rotation during insect flight,
Bennett (1970) conducted experiments with a dynamically
scaled flapping model wing and showed that rotations alter
aerodynamic forces at Reynolds numbers in the range 10
2
to approximately 10
3
. In a detailed overview of insect flight
aerodynamics, Ellington (1984c) proposed a scheme to include
wing rotation with translation in quasi-steady models.
However, in the one instance in which wing rotation was
incorporated into a quasi-steady framework and tested against
measurements, instantaneous forces were monitored on the
body of the tethered insect rather than on individual wings
(Wilkin and Williams, 1993). As a result, it was difficult to
separate aerodynamic forces from inertial forces and to
distinguish among the various sources of lift. Recently, direct
measurements of aerodynamic forces on the wings of a
dynamically scaled model fruit fly Drosophila melanogaster
showed that, during the stroke, the wings produce aerodynamic
forces in excess of those predicted by steady-state translation.
The increasing angle of incidence prior to stroke reversal
augmented instantaneous values of lift, whereas the decreasing
angle of incidence after stroke reversal attenuated lift below
quasi-steady translational predictions (Dickinson et al., 1999).
These results confirmed similar findings by Bennett (1970)
based on stroke-averaged values of the lift estimated from the
flow velocity measured using a mechanical model of the
cockshafer Melolontha vulgari.
With recent advances in computational fluid dynamics
(CFD), efforts in mathematical modeling of insect flight have
moved away from quasi-steady approximations to full-scale
Navier–Stokes simulations of fluid dynamics (Liu et al., 1998;
Wang, 2000). However, while CFD models show promise as
an important tool, their application to insect flight is as yet
limited by the complex nature of three-dimensional flows
in intermediate-to-low Reynolds numbers and by the
computational resources required to simulate such conditions.
Although not as rigorous as computational simulations, quasi-
steady models continue to offer a tractable means of
calculating instantaneous forces from measured kinematics,
are readily applicable to the analysis of energy and power
requirements and are more easily incorporated into dynamic
control models of insect flight. As we begin to identify the
various mechanisms by which insect wings generate
aerodynamic forces, it is worthwhile revisiting quasi-steady
models to re-assess their validity after taking all known
aerodynamic phenomena into account.
In this study, we attempt to characterize the effects of wing
rotation on aerodynamic force generation under conditions that
are appropriate for analysis of insect flight. At constant
translational wing velocity, we vary both the angular velocity
and the axis of wing rotation of a dynamically scaled model
wing and measure the corresponding rotational force
coefficients. We compare these values with a theoretical model
based on a two-dimensional rotating flat plate. Although subtle
differences do exist, the theoretical predictions provide a
reasonably close fit to measured values of rotational force
coefficients. By incorporating the rotational effects into a
translational quasi-steady model of flapping flight, the
predictions of instantaneous forces on insect wings are
substantially improved. This revised quasi-steady model may
help researchers to better estimate the time course of the forces
generated by wings flapping with arbitrary kinematics. Further,
because the improved quasi-steady model accurately accounts
for both translational and rotational components, as well as the
added mass inertia, it may be used to selectively isolate
unsteady forces such as those due to wing/wake interactions.
Materials and methods
The design of the mechanical model used in this study and
the procedures for data analyses are identical to those described
previously (Dickinson et al., 1999; Sane and Dickinson, 2001).
We used an isometrically enlarged planform of a Drosophila
melanogaster wing made from a 2.3mm thick acrylic sheet,
with a length of 25cm and mean chord length of 6.7cm
(calculated aspect ratio of the wing pair 7.5) (see Ellington,
1984b). The proximal edge of the wing was equipped with
multiple, equally spaced slots, allowing us to change the axis
of rotation (Fig. 1A). Through a pair of these multiple slots,
the wing was attached to a two-dimensional force transducer
that measured forces normal and parallel to the wing surface.
The wings, force sensor and gearbox were immersed in a
tank of mineral oil with a kinematic viscosity of 120cSt
(1.2×10
–4
m
2
s
–1
) at room temperature (approximately 25°C).
All experiments were conducted at a Reynolds number of
approximately 115, calculated as described in Ellington
(1984d).
In addition to aerodynamic forces, the sensor at the base of
the wing measures the forces due to gravity and the inertia of
the wing and sensor. To remove the effect of gravity, we
measured the weight of the sensor and wing mass and
subtracted it from the measured force traces. The contribution
of the inertial effects of the wing mass and sensor was
examined by replacing the Plexiglas wing with a brass knob of
the same mass. We also ensured that the centers of mass of
both the Plexiglas wing and the brass model were identical.
Because the compact brass knob generates negligible
aerodynamic forces compared with the Plexiglas wing, the
measured forces for the brass model may be ascribed entirely
to gravity and inertia. Compared with the gravitational
contribution, the inertia of the wing was very small and was
therefore ignored.
S. P. Sane and M. H. Dickinson
1089Quasi-steady model of flapping flight
The force data were filtered on line using an active four-pole
Bessel filter with a frequency cut-off at 10Hz. They were
further processed off line using a low-pass digital Butterworth
filter with a zero phase delay and a cut-off at 3Hz, which was
17.6 times the wing stroke frequency. Increasing the cut-off
frequency of the digital filter amplified the influence of jitter
resulting from the high-frequency motor steps but did not alter
the time course of the recordings.
Stroke kinematics
We performed 171 separate experiments. In all these
experiments, the wing began moving from rest at zero angle of
attack and accelerated to a constant translational velocity
within 0.05s. Each stroke was completed in 2.94s. After
attaining a constant translational velocity, the wing rotated
with constant angular velocity (Fig. 1B–E). To avoid the
influence of wake vorticity from previous strokes on force
production (Birch and Dickinson, 2001), only forces measured
during the initial forward stroke were used to calculate
rotational coefficients (Fig. 1F,G). The absolute angular
velocity was systematically varied in separate trials from 0
to 1.5rads
–1
in increments of 0.085rads
–1
. The absolute
Time (s)
0
2.94
0
1
2
–2
0
2
Time (s)
0
1
2
0
2.94
Net force (N)
–0.4
0
0.4
ω
U
t
ω
U
t
B
D
F
Translational
velocity (m s
–1
)
A
0
0.167
0.083
0.25
0.33
0.417
0.5
0.583
0.66
Leading edge
Trailing edge
x
0
^
C
E
G
–0.4
0
0.4
–2
0
2
Angular velocity (rad s
–1
)
Fig. 1. Wing design and the experimental method. (A) The wing planform used for all experiments. The wing was scaled directly from a
Drosophila melanogaster wing and equipped with multiple slots at the base to allow the axis of rotation to be changed. The leading edge
corresponds to a non-dimensional rotational axis (xˆ
0
) value of 0, whereas the trailing edge corresponds to a value of 1. (B,C) Two-dimensional
cartoons showing the kinematics of rotation and translation. The wing translates from left to right at a velocity U
t
and rotates about a fixed axis of
rotation with varying angular velocity, ω. The leading edge of the wing is indicated by a filled circle. (D,E) Kinematic variables as a function of
time. Translational velocity is shown in blue, rotational velocity in red. Data are shown for two representative rotational velocities of 1.5rads
–1
(D)
and 0.667rads
–1
(E). In both cases, the translational velocity is 0.272ms
–1
. (F,G) Net aerodynamic forces as a function of time. The continuous red
line indicates the measured forces and the dotted red line indicates the quasi-steady translational estimates. The difference between these traces
(double-headed arrow) is used to calculate rotational force coefficients over the shaded region. The early peaks in both force traces are due to
inertial transients caused by rapid acceleration of the wing at the start of each trial. Similar inertial effects also occur as a result of rapid rotational
acceleration, as is evident in the force traces. Although detectable, these effects are small in comparison with the circulatory forces.
1090
translational velocity of the wing in these experiments was
0.272ms
–1
. To compare our results with those in the literature
(Walker, 1931; Kramer, 1932; Fung, 1969), we also express
angular velocity in non-dimensional terms (ωˆ) (after Ellington,
1984c):
ωˆ= ωc¯/U
t
, (1)
where ω is the absolute angular velocity, c¯ is the mean chord
length and U
t
is the wing tip velocity. The range of ωˆ values
explored in these experiments was 0–0.374.
By attaching the wing to the sensor in different positions,
we conducted a series of angular velocity trials over a range of
non-dimensional axes of rotation (xˆ
0
) from 0 to 0.66 in
increments of 0.083, where 0 indicates the leading edge and 1
indicates the trailing edge. We could not examine values of xˆ
0
greater than 0.66 because, at these locations, the large moments
around the rotational axis threatened to damage the wing
sensor at high values of angular velocity.
Components of the quasi-steady model
In the absence of skin friction, the instantaneous forces
generated by a thin, flapping wing may be represented as the
sum of four force components, each acting normal to the wing
surface:
F
inst
= F
a
+ F
trans
+ F
rot
+ F
wc
, (2)
where F
inst
is the instantaneous aerodynamic force on the wing,
F
a
is the force due to the inertia of the added mass of the fluid,
F
trans
is the instantaneous translational force, F
rot
is the
rotational force and F
wc
is the force due to wake capture. Note
that each term of this quasi-steady model is implicitly, but not
explicitly, dependent on time. Thus, any time-dependence of
these force components arises only from the time-dependence
of the kinematics.
For two-dimensional motion in an inviscid fluid, the first
term, F
a
, is calculated for each blade element and integrated
along the span of the wing to estimate the force on a three-
dimensional airfoil (Sane and Dickinson, 2001) (see Sedov,
1965):
where ρ is the fluid density, R is the wing length, rˆ is the non-
dimensional radial position along the wing, cˆ(rˆ) is the non-
dimensional chord length (Ellington, 1984b), φ is the angular
position of the wing and α is the morphological angle of attack.
Note that, for a wing of infinitesimal thickness, the force due
to added mass inertia acts normal to the wing surface.
The quasi-steady translational estimate for the net force was
obtained through vector addition of the mutually orthogonal
lift and drag estimates:
where S is the projected surface area of the wing and rˆ
2
2
(S) is
the non-dimensional second moment of wing area (Ellington,
1984b). Note that, in these equations, α and U
t
are the only
terms that vary throughout the stroke. Values for the lift and
drag coefficients, C
Lt
(α) and C
Dt
(α), have been previously
measured for the model wing used in this study and are
accurately fitted by the following equations (Dickinson et al.,
1999):
C
Lt
(α) = 0.225 + 1.58sin(2.13α – 7.2) (5)
and
C
Dt
(α) = 1.92 – 1.55cos(2.04α – 9.82), (6)
where α is in degrees. These equations allow us to determine
the forces that arise from the translational mechanism of
dynamic stall.
To measure the contribution of rotational forces, we
programmed the stroke kinematic patterns as described above.
In the absence of added mass inertia or wake capture effects,
we can rewrite equation 2 as:
F
rot
= F
inst
− F
trans
. (7)
Thus, we isolate rotational force by subtracting a quasi-steady,
translational estimate from the forces measured at a time when
inertia and wake capture are insubstantial (Fig. 1F,G). This
force difference may be used to derive the experimental
values for rotational coefficients. In subsequent sections,
experimentally determined values for rotational forces will be
denoted by F
rot,exp
, whereas theoretically determined values for
rotational forces will be denoted by F
rot,theo
.
Theoretical estimation of rotational forces
A quasi-steady treatment of the aerodynamic force due to
wing rotation was derived by Fung (1969) (see also Theodorsen,
1935; Sedov, 1965; Ellington, 1984c) for small-amplitude flutter
on thin, rigid wings. For a two-dimensional wing of chord length
c, flapping in an inviscid fluid and rotating with an angular
velocity ω (where ω=α˙) around an axis at xˆ
0
, the quasi-steady
assumption requires two boundary conditions to be satisfied at
every instant. First, the direction of fluid velocity near the surface
must be equal to the slope of the airfoil surface, which is
equivalent to requiring that no fluid travels across the surface of
the wing. Second, the vorticity generated by the trailing edge
chord element must be zero (Kutta condition). The resultant
theoretical value of rotational circulation, Γ
rot,theo
, that satisfies
these boundary conditions is given by:
Γ
rot,theo
=C
rot,theo
ωc
2
, (8)
where the theoretical value of rotational coefficient, C
rot,theo
, is
given by:
C
rot,theo
= π(0.75 − xˆ
0
). (9)
As xˆ
0
varies from 0 to 1, C
rot,theo
changes sign atxˆ
0
=0.75. Thus,
a xˆ
0
value of 0.75 represents the critical axis around which the
wing generates no force as it rotates.
In standard Kutta–Jukowski theory, the approximation of
a small angle of attack applies, and there is no net drag.
Consequently, the lift per unit span on the airfoil equals the
ρSU
t
2
rˆ
2
2
(S)
2
F
trans
=
(4)
[C
Lt
2
(α) + C
Dt
2
(α)]
1/2
,
⌠
⌡
1
0
rˆcˆ
2
(rˆ)drˆ
F
a
= ρ
(3)
R
2
c¯
2
(φ
¨
sinα + φ
˙
α˙cosα)
π
4
−α¨ρ
c¯
3
R
π
16
⌠
⌡
1
0
cˆ
2
(rˆ)drˆ,
S. P. Sane and M. H. Dickinson
1091Quasi-steady model of flapping flight
total aerodynamic force per unit span. Assuming that the
theory holds true for large angles of attack, we can use the
Kutta–Jukowski equation to relate instantaneous rotational
circulation to the rotational component of the total
aerodynamic force:
F′
rot,theo
(t) = ρU
∞
Γ
rot,theo
(t) , (10)
where F′
rot,theo
(t) is the theoretically estimated net force per
unit span due to rotation, 6t is time, ρ is the density of the fluid
medium, U
∞
is the free-stream velocity and Γ
rot,theo
(t) is the
theoretical value for rotational circulation around the wing.
Because the net force acts perpendicular to the airfoil surface
rather than normal to the wing motion, lift and drag emerge as
orthogonal components of the net force.
By combining equations 8 and 10, and replacing U
∞
with U
t
as required for a non-dimensional form of blade element
analysis, the net estimated rotational force on a flapping,
rotating wing of finite span is:
Experimental determination of rotational force coefficients
To measure the rotational force coefficient, we may
substitute for F
rot,theo
and C
rot,theo
in equation 11 with the
corresponding symbols F
rot,exp
and C
rot,exp
for experimental
values:
This equation allows us to evaluate coefficients directly from
measured rotational forces and compare them with the
theoretical estimates given by equation 9. Note that, although
the theoretical estimates in equation 9 depend on all the standard
assumptions of Kutta–Jukowski theory, equation 12 is
independent of any such assumptions. As a result, the effects of
a finite wing span and other aspects of three-dimensional flow
are contained in measured values of instantaneous C
rot,exp
. To
attain a single value of C
rot,exp
in each trial, we averaged
measurements over a temporal window, as shown in Fig. 1D–G.
The sign of the rotational circulation, as well as the
rotational forces, is either the same as, or opposite to, that of
the circulation due to translation, depending on whether the
angle of attack increases or decreases over time. In the
experimental situation described above, positive rotation is
simply the mirror image about the horizontal axis of the
negative rotation case. As a result, separate experiments were
not required for measuring coefficients during positive and
negative rotation.
Results
Rotational coefficients versus angular velocity
Fig. 2A illustrates the dependence of the measured
rotational force coefficient (C
rot,exp
) on the non-dimensional
angular velocity of wing rotation (ωˆ) for a series of non-
dimensional axes of rotation (xˆ
0
). For values of ωˆ less than
0.123 (or ω<0.5rads
–1
), values of C
rot,exp
tend towards infinity
(see equation 12) and should be ignored. For values of ωˆ above
0.123, C
rot,exp
rises linearly with ωˆ. This trend contradicts the
theoretical prediction (equation 9) that C
rot,exp
should remain
constant with respect to ωˆ. However, for axes of rotation nearer
the trailing edge, the dependence of C
rot,exp
on ωˆ is less steep
and the data more closely resemble theoretical expectations.
Among insects, values of xˆ
0
are thought to lie between 0.25
and 0.5 (Ellington, 1984d), although very few studies have
attempted to measure this parameter precisely. Within this
range of rotational axes, the dependence of C
rot,exp
on ωˆis
substantial. Nevertheless, as a first approximation, it seems
reasonable to model the rotational coefficients as a constant.
The validity of this assumption will be tested by the accuracy
with which a quasi-steady rotational model based on a constant
force coefficient can predict measured forces.
⌠
⌡
1
0
rˆcˆ
2
(rˆ)drˆ .
C
rot,exp
= F
rot,exp
/ρU
t
ωc¯
2
R
(12)
⌠
⌡
1
0
rˆcˆ
2
(rˆ)drˆ .
F
rot,theo
= C
rot,theo
ρU
t
ωc¯
2
R
(11)
Table 1. Parameters from C
rot,exp
versus xˆ
0
regressions for various values of
ω
Angular velocity, (ω) Non-dimensional
(rads
–1
) angular velocity, (ωˆ) Slope xˆ
0
-intercept C
rot,exp
-intercept r
2
0.677 0.166 –1.21*** 0.57*** 0.69 0.80
0.762 0.187 –1.35** 0.56** 0.75 0.71
0.847 0.208 –1.38** 0.65** 0.90 0.67
0.931 0.229 –1.84** 0.62** 1.14 0.87
1.016 0.249 –2.24** 0.62** 1.39 0.94
1.101 0.270 –2.49* 0.64* 1.59 0.93
1.185 0.291 –2.75 0.64* 1.75 0.95
1.270 0.312 –2.85 0.68 1.93 0.95
1.354 0.333 –3.13 0.68 2.13 0.97
1.439 0.353 –3.26 0.70 2.27 0.97
1.523 0.374 –3.51 0.69 2.43 0.97
Values for the slope and x-intercept that are significantly different from the theoretical predictions based on equation 9 (slope=−π, xˆ
0
-
intercept=0.75) are marked with asterisks (*P<0.05, **P<0.01, ***P<0.001).
xˆ
0
, axis of rotation; C
rot,exp
, experimental rotational force coefficient.
