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First published online March 31, 2007
Journal of Experimental Biology 210, 1362-1377 (2007)
Published by The Company of Biologists 2007
doi: 10.1242/jeb.02746

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The aerodynamic benefit of wing–wing interaction depends on stroke trajectory in flapping insect wings

Fritz-Olaf Lehmann^* and Simon Pick

BioFuture Research Group, Institute of Neurobiology, University of Ulm, Albert-Einstein-Allee 11, 89081 Ulm, Germany

View larger version (15K): [in this window] [in a new window]   Fig. 1. Pear-shaped wing tip trajectory (red) during wing flapping of a tethered fruit fly Drosophila melanogaster. Black dot indicates wing insertion point (WH) on the insect body; circled cross shows the position of the centre of body mass (COG) at the transition between thorax and abdomen. Superimposed body posture was reconstructed from free flight experiments. Drosophila wing trajectory is most similar to kinematic pattern D in Fig. 2.

View larger version (36K): [in this window] [in a new window]   Fig. 2. (A–Q) Multiple categories of stroke patterns as used in the present study. The black line shows the tip trajectory of the moving wing with the position of superimposed chordwise wing segments every 0.125 s. Arrows indicate the direction of wing motion during up- and downstroke. The small circle at each wing segment indicates the leading edge. In all experiments the angle of attack of the beating wing with respect to the horizontal was kept constant at 50° while the vertical elevation of the wing trajectory (heaving motion) from the horizontal stroke plane systematically varied within the up- and downstroke. Stroke amplitude (160°), cycling frequency (0.17 Hz), start (–4% stroke cycle) and end (4% stroke cycle) of wing rotation and up- to downstroke ratio (0.5) are the same in all figures. Wing separation angle at ventral excursion is approximately 40°. Grey area indicates the time during dorsal stroke reversal at which left and right wing perform clap-and-fling without physically touching each other.

View larger version (66K): [in this window] [in a new window]   Fig. 3. Rotational axes, angles, and length of relative moment arms for pitching within the stroke cycle. (A) Instantaneous stroke angle , stroke amplitude ; (B) body angle , heaving angle during wing translation and inclination of total force vector . (C) Lift normal to the wing surface (black) is the vector sum between vertical force Fv and a radial force component Fr. See text for more details. (D–E) Location of the animal's centre of mass in the z-plane and length of moment arm for pitching moments in the horizontal dlx and vertical direction dly. (F) Horizontal force of the flapping wing is the vector sum of a horizontal component Fh(x) and a force component Fh(z) in z-direction. COF, centre of force producion. (G) Simplified hypothetical alteration in length of the moment arm for each angular position of the longitudinal wing axis within a horizontal stroke plane and without heaving motion. Total pitching moments (black) are the sum of moments produced by vertical force Fv (red) and horizontal force Fh(x) (blue). Moments were calculated using Eqn 13 and Eqn 14 and plotted for a complete stroke cycle with 180° stroke amplitude. In the example shown body angle is 30°, normalized distance between wing hinge and COG is 0.2 wing length, and wing length R, horizontal Fh(x) and vertical force Fv are 1.0, respectively. (H) Examples of changes in length of moment arm for pitching plotted at various body angles (=0–60°, R=1, Fh(x)=1, Fv=1, =0.2, =0°). (I) Examples of alterations in length of moment arm for pitching at various distances between wing hinge and COG (=0–0.5, R=1, Fh(x)=1, Fv=1, =30°). Positive and negative arm for pitching moments produce pitching down and up moments, respectively. *Point of attack for mean force vector acting on the wing at 0.65 wing length; circled cross, centre of body mass (COG) and filled circle, wing hinge (WH) of the virtual insect.

View larger version (19K): [in this window] [in a new window]   Fig. 4. Vertical force augmentation due to dorsal wing–wing interaction in 17 gross kinematic patterns. (A) Augmentation of vertical force (two-wing minus one-wing performance) plotted as a function of mean vertical force produced throughout the stroke cycle, (B) vertical force augmentation plotted against mean vertical force coefficient based on a conventional quasi-steady approach, (C) vertical force augmentation plotted against the relative contribution of wing rotation (rotational effect) and wake capture to quasi-steady translational vertical force, and (D) change in orientation of total force vector (pictogram) due to dorsal clap-and-fling wing–wing interaction. Letters plotted on or close to data points correspond to the letters of the kinematic patterns shown in Fig. 2.

View larger version (69K): [in this window] [in a new window]   Fig. 5. Forces generated by three different kinematic patterns that yield approximately the same mean vertical force coefficients (1.51) during wing flapping. Stroke pattern in the left, middle and right column is A, J and M, respectively (Fig. 2). (A–C) Vertical force Fv and (D–F) horizontal force Fh acting on the wing for one wing flapping (black) and while flapping a mirror wing in close distance on the other side of the robotic flapper (red). (G–I) Translational angular wing motion (black), the wing's angle of attack (blue) and heaving motion (green) of two stroke cycles for the three kinematic patterns A, J, M.

View larger version (31K): [in this window] [in a new window]   Fig. 6. (A–C) Temporal distribution of vertical force augmentation for three kinematic patterns and (D) mean vertical force augmentation during clap-and-fling wing beat. Data are plotted as the difference (2–1) between the performance of one (1) and two (2) flapping wings. (A–C) Kinematic patterns (pictograms) produced the same mean vertical force coefficient during wing motion (1.51). (D) Mean values (red) ± s.d. (grey) of all 17 kinematic patterns. Force peaks are labelled according to Lehmann et al. (Lehmann et al., 2005).

View larger version (51K): [in this window] [in a new window]   Fig. 7. (A–I) Kinematics and moments around the imaginary pitch axis of the electromechanical flapper produced at three experimental conditions. (A–C) Pictograms above each column indicate the kinematic patterns used for comparison (mean vertical force coefficient in all patterns is 1.51). (G–I) Translational angular wing motion (black), the wing's angle of attack (blue) and heaving motion (green) of a single stroke cycle for the three kinematic patterns A,J,M. Black=single wing performance; red=performance when flapping two wings; body angle =30° and distance =0.2 wing length.

View larger version (40K): [in this window] [in a new window]   Fig. 8. Single wing and clap-and-fling induced pitching moments at various flapping conditions. (A) Mean pitching moment produced by a single wing, assuming a body angle of 30° and a distance between wing hinge and COG of 0.2 wing length. (B) Mean temporal distribution of pitching moment augmentation (red, left scale) ± s.d. (grey area) due to clap-and-fling wing beat of all 17 kinematic patterns used in the present study. Black line (right scale) indicates variation in the data set during two complete stroke cycles. (C,D) Augmentation of pitching moment due to clap-and-fling plotted against mean vertical force in C and mean vertical force augmentation in D. (E,F) Alteration in pitching moment for the three kinematic examples in Figs 5, 6, 7, shown as a function of body angle at =0.2 wing length in E and as a function of distance between wing hinge and COG at a body angle of 30° in F.

View larger version (43K): [in this window] [in a new window]   Fig. 9. Schematic reconstruction of wake pattern during wing–wake interaction in fruit fly model wings and effect of heaving motion during clap-and-fling. (A,B) The graphs show chordwise wing segments during clap-and-fling at the end of the upstroke, during the clap, and during the fling phase before the two wings separate for the downstroke. The low-pressure region evolving between the wings during the fling pulls fluid around the leading and the trailing wing edge into the opening cleft. Inflow of fluid during fling potentially increases during heaving down motion, as shown in A, and decreases when the wings move upwards at the beginning of the downstroke, as shown in B (cf. length of black straight arrows). (C) Simplified hypothetical analytical simulation modelling the inflow velocity between both leading wing edges during fling. Angular velocity during dorsal wing rotation and wing size are taken from a tethered flying fruit fly. The model predicts a reduction in flow velocities into the opening cleft during upward heaving motion whereas flow velocity increases during downward heaving compared to a wing beat without heaving motion (blue). (D) Heaving rate and direction plotted against vertical force augmentation during clap-and-fling wing beat. Pictogram illustrates the change in stroke angle (, black), the wing's angle of attack (m, blue) and heaving angle (, green). Heaving rate was derived from the angular change within a time window of 0.1 stroke cycle after the wing has started the downstroke. (E) Effect of heaving up and down motion during fling on vertical force coefficient of a single wing (left), absolute vertical force augmentation when flapping both wings (middle) and relative vertical force augmentation due to clap-and-fling (right). NS, not significant; ***P<0.001 significance level.