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Fig. 3. Validation of the revised quasi-steady model for representative kinematic
patterns. (A) Advanced rotation. (B) Symmetrical rotation. (C) Delayed
rotation. (Ai,Bi,Ci) Two-dimensional cartoons of the three-dimensional
kinematics and the translational (broken line) and rotational (solid line)
velocities as a function of non-dimensional time within the stroke cycle. The
red arrows signify the net force measured on the surface of the flapping wing.
(Aii,Bii,Cii) Drag; (Aiii,Biii,Ciii) Lift. The red lines indicate the measured
forces, the green lines indicate the quasi-steady translational component
based on empirical measurements, the purple lines indicate the quasi-steady
rotational component and the blue lines indicate calculated estimates of added
mass. In Aiv,v, Biv,v and Civ,v, the summed quasi-steady predictions for drag
(Aiv,Biv,Civ) and lift (Av,Bv,Cv) are shown in black and compared with the
measured values (red lines) for the corresponding kinematics.
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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 Ut and rotates about a fixed
axis of rotation with varying angular velocity, 
. 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 1.5 rad
s-1 (D) and 0.667 rad s-1 (E). In both cases, the
translational velocity is 0.272 m s-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.
) are presented in red under the
corresponding dimensional value. The vertical dashed lines signify the section
through this plot at values of 
, non-dimensional axis
of rotation. (B) Rotational coefficient versus axis of rotation. Two
representative plots of rotational force coefficients at two values of
