## 2 Mechanism Introduction and Modeling Analysis
### Seedling Dropping Process Segmentation
#### 2.3.2 Analysis of Dropping Seedling Duration  
The dropping process of pot seedlings starts from the rotation of the seedling plate to the end of the pot seedling dropping into the seedling planting groove. The length of time from the start of the seedling plate to the start of the potted plant is called the start time, which is independent of the position of the potted plant. According to the previous analysis, the dropping trajectories of pot seedlings at the No. 1, No. 2, No. 4, and No. 5 positions are similar. Simplifying the dropping process of the potted seedlings at the No. 3 position as a free dropping motion, the dropping duration \(t_{i}\) of the pot seedlings at each position can be divided into  
\[\left\{\begin{array}{ll}t_{i}=t_{Q}+t_{iAB}+t_{iBC}+t_{iCD}+t_{iDE}&(i=1,2,4, 5)\\ t_{i}=t_{Q}+t_{iAE}&(i=3)\end{array}\right. \tag{14}\]  
where \(t_{Q}\) is the time from the onset of the rotation of the seedling plate to the start of the dropping of the pot seedling, s; \(t_{iAB}\) is the dropping duration from \(B_{i}\) to \(C_{i}\), s; \(t_{iBC}\) is the dropping duration of the pot seedlings from \(B_{i}\) to \(C_{i}\), s; \(t_{iCD}\) is the dropping duration of the pot seedlings from \(C_{i}\) to \(D_{i}\), s; \(t_{iDE}\) is the dropping duration of the pot seedlings from \(D_{i}\) to \(E_{i}\), s; and \(t_{iAE}\) is the dropping duration of the pot seedlings from \(A_{i}\) to \(E_{i}\), s;  
The drop duration was analyzed using the potted seedlings at the No. 1 position as an example. After the seedling carrying plate is activated, the pot seedlings begin to fall after a time \(t_{Q}\). The time from point A to point B is  
\[t_{1 AB}=\sqrt{\frac{2L_{AB}\cos\alpha}{g}} \tag{15}\]When analyzing the force of the pot seedlings in the BC section in Figure 9b, the motion in the BC section can be simplified to a uniform variable motion with an initial velocity of \(v_{2}\) and a final velocity of \(v_{3}\), and the time from point B to point C is  
\[t_{1\text{BC}}=\frac{2L_{\text{BC}}}{v_{2}+v_{3}} \tag{16}\]  
The duration of the oblique projectile motion from point C to point D is  
\[t_{1\text{CD}}=\frac{-v_{3}\cos\alpha+\sqrt{\left(v_{3}\cos\alpha\right)^{2}-2 gL_{\text{CD}}\cos\alpha}}{g} \tag{17}\]  
The approximate free-dropping motion duration from point D to point E is  
\[t_{1\text{DE}}=\sqrt{\frac{2L_{DE}}{g}} \tag{18}\]  
The dropping duration of pot seedlings in each time period can be added to obtain the total dropping time of pot seedlings at the No. 1 position as follows:  
\[t_{1}=t_{Q}+\sqrt{\frac{2L_{AB}\cos\alpha}{g}}+\frac{2L_{\text{BC}}}{v_{2}+v_{3 }}+\frac{-v_{3}\cos\alpha+\sqrt{\left(v_{3}\cos\alpha\right)^{2}-2 gL_{\text{CD }}\cos\alpha}}{g}+\sqrt{\frac{2L_{\text{DE}}}{g}} \tag{19}\]  
Since the No. 2 position is only different from the No. 1 position in the collision position, the total dropping duration of the pot seedlings at the No. 2 position can be similarly obtained as follows:  
\[t_{1}=t_{Q}+\sqrt{\frac{2L_{AB}{}^{\prime}\cos\alpha}{g}}+\frac{2L_{\text{BC}} {}^{\prime}}{v_{2}{}^{\prime}+v_{3}{}^{\prime}}+\frac{-v_{3}{}^{\prime}\cos \alpha+\sqrt{\left(v_{3}{}^{\prime}\cos\alpha\right)^{2}-2 gL_{\text{CD}}{}^{ \prime}\cos\alpha}}{g}+\sqrt{\frac{2L_{\text{DE}}{}^{\prime}}{g}} \tag{20}\]  
The pot seedlings at the No. 3 position directly fall vertically from the cavity of the seedling separating cylinder, and the total dropping seedling duration is  
\[t_{3}=t_{Q}+\sqrt{\frac{2L_{AE}}{g}} \tag{21}\]  
As the left and right seedling separating cylinders are symmetrically arranged, the total time for the pot seedlings to fall at each position is \(t_{1}=t_{5}\), \(t_{2}=t_{4}\), and \(t_{1}=t_{5}>t_{2}=t_{4}>t_{3}\).