## 3 Component Design
### Motor Selection
Based on these calculation results and in conjunction with the standardized specifications of the motor and gearbox product series, a certain brand of motor and gearbox combination has been preliminarily selected to drive the small gear. The DC motor is equipped with an encoder that provides displacement, velocity, and acceleration control functions. The parameters are detailed in Table 3.  
The switching time is verified based on the actual parameters of the selected motor. The adjacent switching time is 0.098 s, the shape of the motor displacement curve is similar to Figure 7b. The maximum switching time is 0.34 s, and the shape of the motor displacement curve is similar to Figure 7a; this is because the actual acceleration (416.67 r/s  
Figure 7: Motor acceleration curve: (**a**) longer displacement; (**b**) shorter displacement. \begin{table}
\begin{tabular}{l c c c c c} \hline \hline  & **Reduction** & **Torque** & **Maximum** & **Rated** & **Maximum** & **Maximum** \\  & **Ratio** & **Stalling Torque** & **Speed** & **Speed** & **Acceleration** \\ \hline Motor & & 0.0025 N-m & 0.005 N-m & 166.67 r/s & 233.33 r/s & 1666.67 r/s\({}^{2}\) \\ \cline{3-6} Gearbox & 4:1 & 0.01 N-m & 0.02 N-m & 41.67 r/s & 58.33 r/s & 416.67 r/s\({}^{2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameters of motor and gearbox. than needed (160 r/s\({}^{2}\)), so the motor reaches its rated speed faster, while also resulting in a shorter time than expected. Next, the torque of gear is verified. The purpose of torque verification method is to ensure that the motor can drive the filter wheel to achieve maximum acceleration. The verification process refers to Formula (4). \[ \begin{split} T_{D}-T_{1}-T_{2}-T_{3}=J_{z}\cdot\alpha\\ J_{Z}=J_{M}+\frac{J_{1}}{i_{1}{}^{2}}+\frac{J_{2}}{i_{2}{}^{2}}+ \frac{J_{3}}{i_{3}{}^{2}}\\ T_{1}=T_{up}\frac{1}{i_{2}\eta}\\ T_{2}=T_{low}\frac{1}{i_{2}\eta}\\ T_{3}=T_{small}\frac{1}{i_{3}}\\ T_{up}=T_{low}=T_{small}=n\cdot f_{s}\cdot N_{load}\cdot\frac{(D+d)} {4}\end{split}  \tag{4}\]  
where \(T_{D}\) is the required output torque of the gearbox, \(\alpha\) is the maximum acceleration of gearbox, which is 2616.67 rad/s; \(T_{up}\), \(T_{low}\), \(T_{small}\) are the static loads of the upper gear, lower gear, and small gear; \(n\) is the number of bearings that match each gear, which is 2; \(N_{load}\) is the load of bearings--to improve the reliability coefficient, \(N_{load}\) is set to the weight of all gears, which is 0.2 N; \(D\) is the outer diameter of bearings, which is 8 mm; \(d\) is the inner diameter of bearings, which is 5 mm; \(f_{s}\) is the coefficient of friction, which is 0.3; \(i_{1}\), \(i_{2}\), \(i_{3}\) are the transmission ratio of the gearbox shaft relative to the upper gear, lower gear, and small gear, which are 5, 4, and 1, respectively; \(\eta\) is the transmission efficiency of the gear, which is 0.9; \(T_{1}\), \(T_{2}\), and \(T_{3}\) are the static loads converted from the upper gear, lower gear, and small gear to the gearbox shaft, which are 8.7 \(\times\) 10\({}^{-5}\) N\(\cdot\)m, 1.08 \(\times\) 10\({}^{-4}\) N\(\cdot\)m, 3.9 \(\times\) 10\({}^{-4}\) Nm, respectively; \(J_{M}\) is the moment of inertia of the motor and gearbox, which is 2.4 \(\times\) 10\({}^{-7}\) kg\(\cdot\)m\({}^{2}\); \(J_{1}\), \(J_{2}\), and \(J_{3}\) are the moment of inertia of the upper gear, lower gear, and small gear relative to their own rotation axis, which are 5.1 \(\times\) 10\({}^{-6}\) kg\(\cdot\)m\({}^{2}\), 4.3 \(\times\) 10\({}^{-6}\) kg\(\cdot\)m\({}^{2}\), 3 \(\times\) 10\({}^{-8}\) kg\(\cdot\)m\({}^{2}\), respectively; \(J_{Z}\) is the total moment of inertia converted to the gearbox shaft, which is 7.4 \(\times\) 10\({}^{-7}\) kg\(\cdot\)m\({}^{2}\). After the calculation, \(T_{D}\) is 0.0025 N\(\cdot\)m, which is far less than the torque of the gearbox 0.01 N\(\cdot\)m. Here, the motor selection is completed.