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Table 1. _z), horizontal forces in the y-direction (F_y) and horizontal drag (F_?) in the ?-direction in a complete flapping cycle. NC&F, non-clap-and-fling; C&F, clap-and-fling.

Then, the wings approached each other at the end of clap (second quarter) and contributed to an increase of 1

Table 1 also shows the average horizontal forces in the y-direction (F_y) obtained by simulation and measurement. The horizontal forces were less than 3 mN and less than 3% of the vertical forces. The simulated horizontal forces are approximately 17.2% and 7.7% smaller than the measured horizontal forces for the cases with and without the clap-and-fling effect, respectively. This was reasonable given that the minor asymmetric flapping motion generated a small amount of the horizontal force. The results from the CFD simulation indicated that the average horizontal forces in a complete cycle generated by the flapping-wing mechanisms with and without the clap-and-fling effect were the same. However, for the measurement shown in table 1, the difference between the average horizontal forces in the two cases was 11.5%. In reality, this increment was a result of the asymmetric contribution to the average horizontal force of two clap-and-fling mechanisms at the dorsal and ventral stroke reversals, which produced forces in the opposite direction, and was not a result of the clap-and-fling effect. This difference may not significantly affect the horizontal force in the y-direction as the average horizontal force in the y-direction was small and close to the resolution of the load cell.

The horizontal force in the y-direction should not be a drag force in the horizontal plane because of the three-dimensional motion of the flapping wings. Instead, the horizontal drag force should be the force in the ?-direction (F_?), which is tangential to the wing motion direction as shown in figure 4b. However, the measurement was not able to directly capture this horizontal drag force (F_?). Therefore, the time histories and average values of F_? in a flapping cycle were obtained by the CFD simulation and shown in figure 8c and table 1, respectively. The average horizontal drag force (F_?) generated by the flapping-wing system with the clap-and-fling effect (5.1 mN) was 41.7% higher than that of the system with a minimized clap-and-fling effect (3.6 mN). This enhancement was not caused by the clap-and-fling effect because of the opposite motions of the wings during the downstroke and upstroke, and the presence of the clap-and-fling mechanisms at both stroke reversals. Instead, in a manner similar to the CFD situation for the horizontal force in the y-direction, the difference was caused by the asymmetric contribution of the clap-and-fling effects at the dorsal and ventral stroke reversals.

4.3. Contribution of the clap-and-fling effect to force generation

The average simulated and measured vertical forces in table 1 show that the clap and flings at dorsal and ventral stroke reversals contributed to increases of 11.5% and 16.2%, respectively, when compared with those in the non-clap-and-fling case. Although the average measured forces over cycles are well matched with the estimated forces, the fluctuations in their time histories are not well repeated due to vibratory forces created by flapping wings and mechanism. Therefore, the time histories of the estimated forces are used to investigate contribution of the clap-and-fling effect to the force enhancement. The vertical force enhancement due to the effect of the clap and fling in a flapping cycle from the CFD simulation is plotted in figure 9a. As shown in figure 9a, the flapping cycles (t/T) included four force peaks of approximately 0.04, 0.46, 0.57 and 0.98. These peaks exhibited flings at the beginnings of the downstroke and upstroke (t/T = 0.04 and 0.57, respectively), and claps at the ends of the downstroke and upstroke (t/T = 0.46 and 0.98, respectively). Thus, the clap as well as the fling contributed to the vertical force enhancement. Table 2 shows the average vertical forces at each quarter of the cycle including the independent clap or fling phase to examine the contribution of each phase to the enhanced vertical force. During the downstroke, the wings flung apart to the translation stage (first quarter) and resulted in an increase of 3.7% in the vertical force. 9% in the vertical force. Hence, the fling and clap at the beginning and the end of downstroke contributed 32.2% and 16.5%, respectively, to the enhanced vertical force in this half flapping cycle. During the upstroke, the fling increased the vertical force by 3.5% (third quarter), while the clap enhanced the vertical force by 2.4% (last quarter). In this half stroke, the contributions of the fling and the clap phases to the enhanced vertical force were 30.4% and 20.9%, respectively. Therefore, in a complete cycle, the clap-and-fling effect improved the vertical force by 11.5%, and the fling phases with a total of 62.6% (from the individual contributions of 32.2% and 30.4%) played a more dominant role than the claps with a total of 37.4% (from the individual contributions of 16.5% and 20.9%) in augmenting the vertical force.