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Figures
- Fig.1.
Definitions of kinematic parameters. (A) Coordinate system for the mechanical fly wing. The cartoon shows the three Euler angles that define the wing position at each instant in time. The mean stroke plane is a horizontal slice through the sphere described by the radial coordinates of the wing tip. Instantaneous stroke position, φ(t), is defined as the angular position of the wing in the mean stroke plane, measured from dorsal reversal (start of downstroke) to ventral reversal (start of upstroke). Instantaneous stroke deviation, θ(t), is defined as the angle that the base-to-tip line on the wing makes with the mean stroke plane. A plane that is normal to the base-to-tip line of the wing (shown in blue) cuts through the wing at the wing chord, shown here as a line with a filled circle denoting the leading edge. The instantaneous angle of attack, α(t), is the angle that the wing chord makes with the tangent of the wing’s trajectory. (B) Sample wing kinematics plotted over two complete cycles. Grey and white backgrounds mark downstroke and upstroke, respectively. Stroke position (green) follows a smoothed triangular waveform. Stroke deviation (red) varies as either a half or full sinusoid in each half-stroke. Half-sine variation yields an ‘oval’ tip trajectory (shown in C), whereas full-sine variation yields a ‘figure-of eight’ tip trajectory. Angle of attack (blue) varies as a trapezoidal function. The shape of the function is determined by setting the starting point of the flip, τ0, and flip duration, Δτ. (C) Schematic diagram of the six parameters that were varied in the experiments: total stroke amplitude, Φ, maximum stroke deviation, Θ, mid-stroke angle of attack, α, flip start, τ0, flip duration, Δτ, and the shape of the wing tip trajectory. This cartoon represents a two-dimensional projection of a three-dimensional kinematic pattern, as if viewed within the blue plane in A. The broken blue line shows the mean stroke plane. This representation of the stroke is repeated throughout the paper.
- Fig.2.
Contribution of added mass inertia to total measured aerodynamic force. The traces shown were taken from a representative kinematic pattern (stroke amplitude Φ=180°, maximum stroke deviation Θ=0, angle of attack α=45°, flip start, τ0=0.05, flip duration Δτ=−0.1). The blue trace represents the total normal force measured on the wing. The static gravitational component has been subtracted. The black trace represents the added mass inertia estimated using equation 1. Added mass inertia is zero throughout most of the stroke because the linear velocity of the wing is constant. The red trace represents the total measured force after subtracting added mass inertia. The contribution of added mass inertia to the measured aerodynamic forces is small, as indicated by the similarity of the red and blue traces.
- Fig.3.
Sample instantaneous forces for various combinations of total stroke amplitude Φ and mid-stroke angle of attack α. In each case, the wing rotation parameters were kept constant (flip duration Δτ=0.16, flip start τ0=−0.08, flip timing τf=0). Each panel (A–H) shows a plot of measured drag (solid red line), the quasi-steady estimate of drag (broken red line), measured lift (solid blue line) and the quasi-steady lift (broken blue line). Since the radial forces were zero for all these kinematic patterns, they are not plotted. A two-dimensional diagram of the wing kinematics is plotted above each set of traces using the convention described in Fig.1C. The wing chord is shown in light blue, and the superimposed black vector indicates the magnitude and direction of the instantaneous aerodynamic force. For convenience, the kinematic values of stroke amplitude and angle of attack used in each trial are printed in the upper left of each panel. Values for the measured mean force coefficients (c̅D and c̅L) are printed adjacent to each set of traces. Axis labels given in A apply to all panels. (A,B) Forces generated at a 90° angle of attack with no wing rotation (α=90°) for a short stroke amplitude (A, Φ=60°) and a long stroke amplitude (B, Φ=180°). Note the enormous transients in drag at the start of each stroke due to wake capture. (C,D) Forces generated at a 50° angle of attack for a short stroke amplitude (C, Φ=60°) and a long stroke amplitude (D, Φ=180°). The contribution of rotational circulation is apparent at the end of each stroke. (E,F) Forces generated at a 30° angle of attack for a short stroke amplitude (D, Φ=60°) and a long stroke amplitude (F, Φ=180°). (G,H) Forces generated at a 0° angle of attack for a short stroke amplitude (G, Φ=60°) and a long stroke amplitude (H, Φ=180°).
- Fig.4.
Parameter maps of net aerodynamic force and net force coefficient as functions of stroke amplitude and mid-stroke angle of attack. For fixed values of wing rotation (flip duration Δτ=0.16; flip start τ0=−0.08, flip timing τf=0), stroke amplitude was varied from 60 to 180° and angle of attack was varied from 0 to 90°. In each diagram, the small open circles indicate the positions of actual measurements. Values between these measured points have been interpolated using a cubic spline. Values are encoded in pseudocolor according to the scales shown beneath each plot. This same format is used in Fig.5, Fig.7 and Fig.10. (A) Net aerodynamic force, the vector sum of lift and drag, increases monotonically with increasing angle of attack and stroke amplitude. (B) Net aerodynamic force coefficient increases with angle of attack, but decreases with stroke amplitude.
- Fig.5.
Parameter maps of stroke-averaged lift coefficient, drag coefficient and lift-to-drag ratio as functions of angle of attack and stroke amplitude. The data are plotted as in Fig.4. Axis labels given in A apply to all panels. Pseudocolor scales apply to both figures in a given column. The top panels (A,C,E) show maps of values measured from the mechanical model, and the bottom panels (B,D,F) show the corresponding values calculated for a translational quasi-steady model using empirically measured force coefficients. (A,B) Measured and quasi-steady values for mean lift coefficient. (C,D) Measured and quasi-steady values for mean drag coefficient. The quasi-steady predictions grossly underestimate the drag coefficient. (E,F) Measured and quasi-steady values for mean lift-to-drag ratio.
- Fig.6.
Sample instantaneous forces for various combinations of flip start τ0 and flip duration Δτ. In all kinematic patterns, stroke amplitude was 180° and angle of attack was 45°. The format for each panel is that described for Fig.3. As in Fig.3, the radial forces for all these kinematics are zero and have not been plotted. (A) Forces generated with a slow flip (Δτ=0.5), symmetrical with respect to stroke reversal (τ0=−0.25, flip timing τf=0). Under these conditions, the quasi-steady model (broken lines) accurately predicts measured lift, but not drag. (B) Forces generated with moderate flip duration (Δτ=0.25), advanced with respect to stroke reversal (τ0=−0.25, τf=−0.125). With these kinematics, the augmentation of lift by rotational circulation and wake capture is evident. (C) Forces generated with a long, advanced flip (Δτ=0.5; τ0=−0.5, τf=−0.25). This pattern of kinematics produced elevated drag due to wake capture at the start of each stroke. (D) Same kinematics as in C, but with a delayed flip (Δτ=0.5; τ0=0, τf=+0.25). The delay in flip timing causes a small decrease in mean drag, but an enormous decrease in lift. (E–H) The influence of rotational timing on a short-duration flip. (E) Forces generated by a short flip advanced by almost a full half-cycle with respect to stroke reversal (Δτ=0.1; τ0=−0.5, τf=−0.45). Note that the angle of attack is negative during most of translation because the wing flips much too soon. As a consequence, the pattern generates negative lift. (F) Forces generated by a slightly advanced short flip (Δτ=0.1; τ0=−0.1, τf=−0.05). This near-optimal pattern augments lift by both rotational mechanisms. (G) Forces generated by a short symmetrical flip (Δτ=0.1; τ0=−0.05, τf=0). (H) Forces generated by a slightly delayed short flip (Δτ=0.1; τ0=0, τf=0.05). The small delay of 0.05 decreases the mean lift coefficient by 20% compared with the symmetrical case shown in G. C̅D̅, mean drag coefficient; C̅L̅, mean lift coefficient.
- Fig.7.
Parameter maps of force coefficients as a function of flip start and flip duration. Each map was generated from a 9×11 array of kinematic patterns. For all experiments, the stroke amplitude Φ and angle of attack α were held constant (Φ=180°, α=45°), while flip duration and flip start were systematically varied. The data and interpolated values are plotted as in Fig.4 and Fig.5. Isolines of flip timing (given by equation 2) are indicated by the diagonal lines in each panel and correspond to the red labels on the right axis. The top panels (A,C,E) show the measured force coefficients, and the bottom panels (B,D,F) show values for the quasi-steady-state estimated coefficients. Axis labels given in A apply to all panels. (A,B) Measured and quasi-steady values for stroke-averaged mean lift coefficient. The pseudocolor scale for both panels is shown below the parameter map in B. (C,D) Measured and quasi-steady values for stroke-averaged mean drag coefficient. Note the large discrepancy between the estimated and measured values. (E,F) Measured and quasi-steady values for stroke-averaged mean lift-to-drag ratio. (G). Measured values for stroke-averaged mean net force coefficient. Note the strong similarity to the drag coefficient map.
- Fig.8.
Sample instantaneous forces for various combinations of stroke deviation and tip trajectory. All other kinematic variables were constant (amplitude Φ=180°, angle of attack α=45°, flip duration Δτ=0.5, flip timing τf=−0.05). The format for each panel is that described for Fig.3 and Fig.6, except for an additional panel showing the instantaneous radial forces (green). (A) Forces generated with a large oval deviation in which the downstroke starts with upward motion and the upstroke starts with downward motion Ddev=+25°, Udev=−25°, where Ddev is equal to the maximum angle of downstroke deviation and Udev indicates maximum angle of upstroke deviation. Both lift and drag transients are higher at the start of the upstroke when the wing travels downwards than at the start of the downstroke. The absolute average radial forces are also correspondingly large. (B) Reversed condition compared with A, the downstroke starts with downward motion and the upstroke starts with upward motion (Ddev=−25°, Udev=+25°). Lift and drag transients are now much larger at the start of the downstroke. (C) Forces generated with a figure-of-eight deviation in which both strokes start with upward motion (Udev=Ddev=+25°). Both lift and drag are low at the start of each stroke, but reach elevated values at midstroke. (D) Reversed condition compared with C, both strokes start with downward motion (Udev=Ddev=−25°). Both strokes now start with large transients in both lift and drag. (E) Forces generated by comparable kinematic pattern to those in A–D but with no stroke deviation.
- Fig.9.
Effects of stroke deviation on mean force coefficients. (A) Cartoon illustrating the kinematic patterns used to study stroke deviation. Throughout the figure, blue data points refer to oval kinematic patterns, and red data points refer to figure-of-eight patterns. Measured and quasi-steady predicted values are given by filled and open circles, respectively. (B) The magnitude of the mean net aerodynamic force coefficient is only mildly influenced by stroke deviation. (C) Measured mean lift coefficient decreases with stroke deviation. For the figure-of-eight patterns, the drop in performance with increasing deviation is greater for strokes that begin with upward motion. (D) Drag coefficient decreases with stroke deviation. Note the large discrepancy between measured values of mean drag and quasi-steady predictions. (E) Ratio of lift-to-drag and radial-to-drag forces as a function of stroke deviation. Radial-to-drag forces for oval kinematic patterns are represented by filled black circles and for figure-of-eight patterns by open black circles. Because deviation affects lift and drag almost equally, the influence on their ratio is quite small. The radial forces are dependent on the sine of stroke deviation angle. Because of the linearity of sine functions for small angles, the values of radial-to-drag force increase linearly with increasing absolute deviation.
- Fig.10.
The effects of wing kinematics on profile power P̅p̅r̅o̅ and the lift-to-power ratio L̅ / P̅p̅r̅o̅. Procedures for plotting data are as described in Fig.4, Fig.7 and Fig.9. The pseudocolor scale for A and B is shown below B, and the scale for D and E is shown below E. (A) Profile power as a function of stroke amplitude and angle of attack. (B) Profile power as a function of flip start and flip duration. Flip timing is shown on the right axis. Profile power varies by a factor of 2 within the parameter space, indicating that flip kinematics are important determinants of flight cost. (C) Profile power as a function of stroke deviation for oval (blue) and figure-of-eight (red) patterns. Stroke deviation has only a minor effect on profile power. (D) The ratio of mean lift to mean profile power, L̅/P̅p̅r̅o̅, as a function of stroke amplitude and angle of attack. Like the mean lift coefficient (Fig.5A), there is a single angle of attack that maximizes L̅/P̅p̅r̅o̅, for each value of stroke amplitude. (E) The ratio of mean lift to mean profile power as a function of flip start, flip duration and flip timing. (F) The ratio of mean lift to mean profile power as a function of stroke deviation for oval (blue) and figure-of-eight (red) deviation.
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