## 2 Structure Design and Numerical Method  
### Flapping Wing Rotor Model  
In this investigation, a FWR model of 50 g weight was designed and built in the configuration illustrated in Figure 1. An ultrasonic motor normally provides higher power density (power-to-volume ratio) and quicker actuation response [27] than an ordinary DC motor/servo motor. Therefore, an in-house ultrasonic motor (USR30) is used to drive the FWR model. The motor mass is 28 g, while the maximum continuous power and torque of the motor is 1.8 W, and 0.1 N\(\cdot\)m, respectively. It is capable of changing the flapping frequency in several milliseconds to study the effect of variant flapping frequency on the FWR's motion and lift. The components of the FWR model (in green), shown in Figure 1a, was manufactured by 3D printing, and the rod was made of carbon/epoxy composite. The aerodynamic forces, the rotary angle (\(\psi\)), twisting angle (\(\theta\)), and stroking angle (\(\varphi\)) are indicated using arrows. Among the above motions in three degrees of freedom (3-DoF), only the stroking motion is driven by the ultrasonic motor. The twisting is caused by the deformation of the flexible wing spars and the skin made from the 20 \(\upmu\)m polyimide film. The flapping wing produces lift and thrust from both wings in axial symmetric configuration. The thrust forms a torque to drive the wings rotation around the shaft. Associated with the rotation, the wings also produce drag that equal the thrust at a specific rotational speed. Two optical sensors, shown in Figure 1b, are mounted on the FWR, in order to measure the flapping frequency for feedback and control of the motor speed during up-stroke and down-stroke.  
In order to fulfil the FWR stroking motion, the driving disk was mounted at the centre of the shaft of the ultrasonic motor to provide continuous rotational moment. The assembly of the driving disk, driving bar, and linear rod forms a slider crank mechanism, as shown in Figure 2a. If the small deviation (1 mm), from the mid-bar axis and output shaft axis, is ignored in order to avoid the occurrence of  
Figure 1: (**a**) Working principle; (**b**) driving mechanism of the flapping wing rotor.  
dead-point and the setting of the origin at the top stop point, the stroke position of the mid bar and the stroke angle (\(\varphi\)) could be written as per Equations (1) and (2),  
\[Z_{mid\_bar}=r(1-cos\alpha)+0.25\lambda r[1-\cos(2\alpha)] \tag{1}\]  
\[\varphi=\left[Z_{mid\_bar}-Z_{avg}\right]/B_{1} \tag{2}\]  
where \(\alpha\) is the crank angle defined in Figure 2, \(\lambda\) denotes the crank-link rod ratio (\(r/l\)) and \(Z_{avg}=0.5\cdot(r+0.5r\cdot\lambda+0.5r/\lambda)\) is defined as average value between the maximum \(Z_{mid\_bar}\) and the minimum \(Z_{mid\_bar}\). \(B_{1}\) means the distance between A and B or A' and B', as shown in Figure 2b. The lengths of the two connecting rods (\(r\) and \(l\)) are listed in Table 1.  
The wing of rectangle platform was initially installed at 30\({}^{\circ}\) on the FWR model through the wing front spar beam. Two symmetric rotation axes (\(A\) and \(A^{\prime}\)) are present for the flapping motion as per Figures 2 and 3. As shown in Figure 3, the front spar, chord-wise beam and two reinforced oblique beams of each wing are plotted in black lines. The span-wise and chord-wise length of the wing membrane skin are 205 mm, and 150 mm, respectively.