## 4 Finite Element Analysis
### Dynamic Analysis of Switching Between Adjacent Combinations  
Figure 8 shows the dynamic analysis model, with parameters set according to the details provided in Section 3.2. The static loads on all gear are 3.9 \(\times\) 10\({}^{-4}\) N\(\cdot\)m. Each gear is configured to rotate specifically around its individual axis. According to the content in Section 3.2, the input conditions applied to the small gear set are shown in Figure 9a below. The rotation angle is 360\({}^{\circ}\), and the maximum  
Figure 8: Dynamic analysis model. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Order** & **Vibration Mode** & **Frequency (Hz)** & **Vibration Form** \\ \hline
1 & & 470.81 & PCB board rotates around the X-axis \\
2 & & 479 & Reinforcing vibrates along the Y-axis \\
3 & & 479.23 & Reinforcing vibrates along the X-axis \\ \hline \hline \end{tabular}
\end{table} 
Table 5: The results of the first three modal analyses. angular velocity is 128.217 rad/s, with an acceleration of 2616.67 rad/s\({}^{2}\). The speed unit is converted from r/s to rad/s. The dynamics analysis results are shown in Figure 9b,c. It can be seen that the angular displacement and velocity of the upper and lower gears are inversely proportional to the transmission ratio with respect to the small gear. While the small gear rotates 360\({}^{\circ}\), the upper gear rotates 72\({}^{\circ}\) and the lower gear rotates 90\({}^{\circ}\). The angular acceleration exhibits certain fluctuation characteristics, which is due to the nonlinear interaction between gear meshing and elastic deformation [18]. The average angular acceleration is also proportional to the transmission ratio with the small gear. Figure 9d shows the torque of the small gear. The average torque during acceleration is about 0.0015 Nm, while it is slightly smaller during deceleration. This is because the frictional resistance acts opposite to the driving force during acceleration and in the same direction during deceleration. The torque during both acceleration and deceleration is much smaller than the rated torque of the gearbox (0.01 Nm), and the torque margin coefficient is larger than 6. The dynamic analysis results demonstrate that the component has the ability to complete the switching of adjacent combinations within the set time. The maximum stress of mechanism during switching process is calculated by dynamic analysis, as shown in Figure 10. Figure 11 shows the stress cloud maps at three different times. The maximum stress is about 0.88 Mpa, occurring at the contact position between two gears, much less than the strength of the PEK material 105 MPa [19], and the strength margin coefficient is larger than 117. Figure 10: The maximum stress results of the gear for a switch to an adjacent combination. Figure 9: The transient dynamics analysis results for the switch to an adjacent combination: (**a**) the dynamics parameters of the small gears; (**b**) the kinematic results of the upper gear; (**c**) the kinematic results of the lower gear; (**d**) the torque of the small gear. ### Dynamic Analysis of Switching Between Adjacent Combinations  
According to the content in Section 3.2, the input conditions applied to the small gear set are shown in Figure 12a below. The rotation angle is 3600\({}^{\circ}\), while the maximum angular velocity is 261.67 rad/s, the two inflection points of the speed trapezoidal curve are at 0.1 s and 0.24 s, and the acceleration is 2616.67 rad/s\({}^{2}\). The dynamics analysis results are shown in Figure 12b,c. Similar to the analysis in Section 4.2, the average of angular acceleration, the angular displacement, and velocity of the upper and lower gears is inversely proportional to the transmission ratio with the small gear. While the small gear rotates 3600\({}^{\circ}\), the upper gear rotates 720\({}^{\circ}\), and the lower gear rotates 900\({}^{\circ}\). Figure 12d shows the torque of the small gear; the average value during acceleration is also about 0.0015 Nm, with a torque margin coefficient larger than 6.