## 2 Structure Design and Numerical Method  
### Structure of the FWR Model  
As shown in Figure 1, a flapping-wing rotor (FWR) model with variable rigid twisting angle was designed and manufactured. Most 3D-printed components of the FWR model (in black) as shown in Figure 1a were made of high-performance nylon and the black rods were made of carbon/epoxy composite. At the same time, the wings' membrane is made from 12.5 \(\upmu\)m polyimide film. The customized coreless motor shares 7.4 V rated voltage, 15 W rated power, and 30000 rpm rated speed respectively. Through the double-reduction gear unit, the motor provides force for pushing the pushrod while the reciprocating motion of the pushrod would generate the stroke motion. Due to the reverse Karman vortex, the stroke motion would also produce axial-symmetric lift and thrust on both wings as shown in Figure 1. Then, compared with a typical flapping-wing system such as Delfly [20], the two thrusts of the FWR in opposite direction form a moment that leads to a unique rotary motion (\(\psi\)). Although the FWR uses an ordinary crank-slider mechanism to realize its stroke motion, it is noted that the angle limiter and the stop blocks are specially designed to realize the variation of the rigid twisting angle (\(\theta\)) ranging from 10\({}^{\circ}\) to 50\({}^{\circ}\) during the flapping process of the FWR without extra active control. Then, the total twisting angle can be divided into two parts, that is, the rigid twisting motion part caused by the aerodynamic forces and inertial forces acting on the wings' surface, as well as the twisting motion part due to the flexibility. According to the Table 1, the total weight of the FWR is 18.7 g wherein the motor accounts for 5.5 g of the weight. Apart from the motor, the frame, the wing assembly, and U-shape mechanism weighs 2.8 g, 3.4 g, and 2.95 g, accounting for 15%, 18.2%, and 15.8% of the FWR's total weight. The weight of other parts including the bearings, pushrod, box, gear set, and connections are relatively small. As depicted before, although the rotational speed of the coreless motor is as high as 30,000 rpm, the output torque is only about 4.78 mN\(\cdot\)m. Thus, in order to provide enough thrust to push the pushrod, three gears are used to form the double-reduction gear unit as shown in Figure 2, wherein the addendum circle diameter of these three gears are D\({}_{d}\) = 29 mm, D\({}_{b}\) = 14 mm, and D\({}_{c}\) = D\({}_{d}\) = 5 mm respectively. The depth of the teeth (D\({}_{t}\)) is 1 mm. By assembling these gears properly, a double-reduction gear unit is designed with its gear ratio equal to 22.75:1. Moreover, the distance between the two holes on the connections is 5 mm which limits the feathering amplitude of the pushrod to \(\pm\)5 mm. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Component Name** & **Weight (g)** & **Quantity** \\ \hline Frame & 2.8 & 1 \\ Box & 0.5 & 1 \\ Motor & 5.5 & 1 \\ Gear set & 1.6 & 1 \\ Bearings & 0.25 & 5 \\ Pushrod & 0.2 & 1 \\ Connections & 0.25 & 2 \\ U-shape, echanism & 2.95 & 1 \\ Wing assembly & 1.7 & 2 \\ \hline Total weight & 18.7 & 15 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The weight of the FWR’s components. Figure 1: (a) Degree of freedom; (b) actuator mechanism of the FWR. The wing assembly composed of stop blocks, angle limiter, leading-edge beam, subsidiary beams, and wing membrane is shown in Figure 3a, wherein both the stop blocks and angle limiter are made from nylon plastic. The leading-edge beam is made from 1.5 mm carbon/epoxy rod (0.5 g) while the other subsidiary beams are made from 1 mm carbon/epoxy rod (0.4 g). Moreover, the weight of the wing membrane is 0.4 g. The length of the spanwise beam and chordwise beam is 145 mm and 90 mm, respectively. In fact, with the wing membrane, the force obtained by the load cell should be the total force composed of inertial forces and aerodynamic forces [16].