## 2 Structure Design and Numerical Method
### Kinematics of the Flapping Wing Rotor  
The kinematics of the FWR motion can be decomposed into twisting \(\theta\), flapping \(\varphi\), and rotary angle \(\psi\). The function of the flapping motion is determined by the geometric parameters of the flapping wing rotor mechanism, and could be deduced by using Equations (1) and (2), given the flapping frequency. The rotary motion can be assumed as a rotation with constant speed, as in previous studies [22; 23], and  
\begin{table}
\begin{tabular}{l c c c c} \hline \hline
**Variables** & \(r\) & \(I\) & \(OA\) (\(OA\)’) & \(AB\) (\(A\)’ B’) \\ \hline Dimensions (mm) & 6 & 35 & 9 & 8 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Geometric parameters for the flapping wing rotor’s mechanism.  
Figure 3: Dimensions of the rectangle wings for the flapping wing rotor.  
Figure 2: Schematic diagram of (**a**) the slider crank mechanism; and (**b**) the U-shape mechanism.  
the flexibility of the wings is not taken into consideration in the CFD model for numerical calculation. Above all, the kinematic parameters, required for the numerical calculation, are the rotational speed \(\dot{\psi}\), the flapping frequency during down-stroke (\(f_{d}\)), and up-stroke (\(f_{u}\)). As shown in Figure 4, three reflective stickers (denoted as white and green dots) are attached on the flapping wing rotor. A laser velocimeter (DM6230) is applied to measure both, the rotational speed and the flapping frequency. To generate the variant flapping frequencies, such as \(f_{d}\) and \(f_{u}\), it is needed to trigger the upper sensor and disable the circuits to trigger the lower sensor before making the motor rotate at \(f_{d}\), and vice versus. After both \(f_{d}\) and \(f_{u}\) are set, variant flapping frequencies between the up-stroke and the down-stroke can be achieved if the circuits to trigger lower sensor and upper sensor work.  
Figure 5 illustrates a general set of variation curves of flapping angles, twisting angles, and rotary angles in three successive periods for numerical calculation of the flapping wing rotor's lift, where \(\hat{t}\) stands for non-dimensional time normalized by flapping period (\(0.5/f_{u}+0.5/f_{d}\)). The definition of \(\hat{t}_{d}\) and \(\hat{t}_{u}\). could be given as below:  
\[\hat{t}_{d}=\frac{\frac{0.5}{f_{d}}}{\left(\frac{0.5}{f_{u}}+\frac{0.5}{f_{d}} \right)}=\frac{1}{f_{d}\Big{(}\frac{1}{f_{u}}+\frac{1}{f_{d}}\Big{)}},\,\hat{ t}_{u}=\frac{\frac{0.5}{f_{u}}}{\left(\frac{0.5}{f_{u}}+\frac{0.5}{f_{d}} \right)}=\frac{1}{f_{u}\Big{(}\frac{1}{f_{u}}+\frac{1}{f_{d}}\Big{)}}. \tag{3}\]  
It is noted that the flapping frequency during the down-stroke (\(f_{d}\)) is larger than that during the up-stroke (\(f_{u}\)) in Figure 5. The curve of flapping angles follows Equations (1) and (2) during either, the single up-stroke or down-stroke. Twisting angles remain unchanged throughout the flapping process, and the slope of the curves for rotary angles over 360 represents the revolutions of rotary motion during every flapping cycle.  
Finally, based on the above assumption, the trace line of the point P at flapping wing rotor's wing tip in one rotary cycle could finally be plotted in Figure 6. It only consists of both flapping motion and rotary motion, while the flexible twisting motion is neglected in the following numerical calculation. The wings are mounted on the FWR in an installation angle (\(\theta_{0}\)) and installation angle is defined as the  
Figure 4: Measurement of flapping frequency and rotational speed.  
Figure 5: Time-course curves of stroking, twisting, and rotary angles in three flapping periods.  
angle between the wing chord and the horizontal plane when the flapping angle is zero during mid down-stroke, as illustrated in Figure 6.