Figure 7a shows a plot of the time history of the flapping angle of the leading edge

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4.1. Wing kinematics

The wing commenced a flapping cycle from the beginning to the end of the downstroke in the period (t/T) from 0 to approximately 0.50. Following this, the wing began the upstroke in the period from approximately 0.50 to 1.00. The measured peak-to-peak values of the flapping angle ranged from approximately 97.2° to ?95°, and thus the measured flapping amplitude was approximately 192.2°. In order to precisely track the time history of the measured flapping angle, 8-term sine and cosine functions were used as a fitting function for the CFD model inputs. The fitted values of the flapping angle at the end of each stroke were approximately 95.6° and ?93.5°, and the flapping amplitude was 189.1°. Hence, the fitted amplitude was approximately 3.1° smaller than the measured angle, and this is an acceptable error. Figure 7b shows the fitted wing rotation angles at seven wing sections using 8-term sine and cosine functions. The variation of the rotation angle indicated that the wing was twisted from the wing root to the wing tip during the translational phase (0.10 ? t/T ? 0.45 and 0.60 ? t/T ? 0.95). This feature was similar to the rotation angle of a beetle’s hind wing (see fig. 4(c) in Le et al. ). The wing was not only twisted in the spanwise direction but was also cambered in the chordwise direction. The variation of the wing camber (which was fitted by 8-term sine and cosine functions) at each wing section in a flapping cycle is shown in figure 7c. The camber is defined as the ratio of the height of the mid-chord, denoted by h in figure 7c, and the chord length, denoted by c, at each wing section. The cambers at the seven wing sections from the wing root to the wing tip were less than 20% of the wing chord during both the downstroke and the upstroke, in a manner similar to the chordwise camber in the beetle’s hind wing (see https://hookupdate.net/nl/passion-com-overzicht/ fig. 4(d) in Le et al. ).

Figure 7. Time histories of (a) flapping angle, (b) wing rotation angle and (c) camber deformation at different wing sections measured along the wingspan during the flapping motion.

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4.2. Forces produced by flapping wings

The time histories of the measured vertical force (F_z) and horizontal force (F_y) generated by the flapping-wing system with and without implementing the clap-and-fling effect are plotted along with those obtained by the CFD simulation in figure 8a and b, respectively. The inertial force was not considered in the computational simulation. A study by Truong et al. suggested that the inertial force did not affect the average force values but contributed to the change in the time history of forces during the flapping motion. In the current flapping-wing system, the vertical force direction (z-direction in figure 4b) was perpendicular to the flapping stroke plane (xy-plane). Therefore, the inertial force did not significantly affect the time history of the vertical force. As shown in figure 8a, the measured time histories of the vertical forces showed similar tendencies to those of the simulated ones even though there were some differences. However, the time histories of the horizontal forces shown in figure 8b were strongly affected by the inertial force. As seen in table 1, the average vertical forces (F_z) obtained by the numerical simulation are about 3.2% and 7.5% larger than the measured vertical forces for the cases with and without the clap-and-fling effect, respectively. This proved that the numerical simulation could be used to properly estimate the average forces generated by the FW-MAV.