## 4 Data Analysis of Mechanism Design
flapping trajectory, which can be seen that the flapping amplitude increased linearly with the increase in \(k_{r}\), which overall followed a straight line trend. To approximate it, we obtained a linear function relationship between the flapping amplitude \(\theta_{f\_max}\) of the string drive mechanism and the ratio \(k_{r}\) of the installation position of the bearing group to the radius of the roller group, which was given by the Equation (13). \[\theta_{f\_max}=K\cdot k_{r}+B=112.5k_{r}+0.218, \tag{13}\]  
where \(K\) and \(B\) represent the slope and the intercept of the fitting line, respectively. The values for this equation were obtained when the radius of the roller group was 1.285 mm, the radius of the bearing group was 1.785 mm, and the installation position of the roller group was 10.15 mm. To verify this, we attempted to change these values and compared the impact of the different values on \(K\) and \(B\), thereby forming a parameter comparison, as can be seen in Table 3. It can be seen that changing the radius of the bearing group and the installation position of the roller group has a very weak effect on the slope of the fitting line. The difference between the value of the intercept of the fitting line and the slope was two orders of magnitude, which had a minimal impact. Therefore, the linear function relationship between the flapping amplitude \(\theta_{f\_max}\) of the string drive mechanism and the ratio \(k_{r}\) of the installation position of the bearing group to the radius of the roller group can be expressed as Equation (14). From the simulation analysis of the angle change, when the radius of the left and right rollers where under ideal tight conditions, it can be seen that the peak value was taken when the phase of the center gear was about 171\({}^{\circ}\) and 351\({}^{\circ}\), which was not the ideal values of 180\({}^{\circ}\)and 360\({}^{\circ}\). This difference was due to the distance of the center gears between the rollers, which tends to be zero when the distance of the center of circles increases toward infinity. \[\theta_{f\_max}=112.5\frac{r_{0}}{r_{\_}}+0.3 \tag{14}\]  
Figure 9: **(a)** The influence of the installation position of the double-layer bearings on the flapping amplitude. **(b)** The influence of the installation position of the double-layer bearings on the flapping trajectory. Figure 11: (**a**) The influence of the ratio of \(r_{0}\) and \(r_{1}\) on the flapping amplitude. (**b**) The influence of the ratio of \(r_{0}\) and \(r_{1}\) on the flapping trajectory. \begin{table}
\begin{tabular}{c c c c c} \hline \(r_{1}\) (mm) & \(r_{3}\) (mm) & \(d\) (mm) & \(K\) & \(B\) \\ \hline
1.285 & 1.785 & 10.15 & 112.5 & 0.218 \\
1 & 1.785 & 10.15 & 112.6 & 0.1258 \\
1.3 & 1.785 & 10.15 & 112.5 & 0.2241 \\
1.6 & 1.785 & 10.15 & 112.3 & 0.3595 \\
1.285 & 1.4 & 10.15 & 113.3 & 0.1323 \\
1.285 & 1.8 & 10.15 & 112.5 & 0.2218 \\
1.285 & 2.2 & 10.15 & 111.5 & 0.3375 \\
1.285 & 1.785 & 8 & 111 & 0.6161 \\
1.285 & 1.785 & 10 & 112.5 & 0.233 \\
1.285 & 1.785 & 12 & 113.2 & 0.1065 \\ \hline \end{tabular}
\end{table} 
Table 3: Characteristics of the design parameters. Figure 10: (**a**) The influence of the radius of the rollers on the flapping amplitude. (**b**) The influence of the radius of the rollers on the flapping trajectory. The simulation above is based on the assumption that both strings are tight. However, the actual situation was that, due to the soft nature of the strings, the phase keeper strings had to be used to keep the strings tight. Based on the connection method and parameters shown in Table 2, when the phase of the center gear was from \(-9^{\circ}\) to \(171^{\circ}\), the left roller was pulled directly by the center gear and right roller was pulled by the phase keeper strings, which meant that the change in value in the left rollers was the same as in the right rollers.