## 2 Mechanism Introduction and Modeling Analysis
### Seedling Dropping Process Segmentation
#### 2.3.1 Kinematics Analysis of Seedling Separation Process
According to the kinetic energy theorem, the following equation can be obtained  
\[mgL_{BC}\cos\alpha-\int_{L_{BC}}f(x,y)\mathrm{d}s=\frac{1}{2}mv_{3}{}^{2}- \frac{1}{2}mv_{2}{}^{2} \tag{7}\]  
where \(L_{BC}\) is the path length from point B to point C, mm; \(f(x,y)\) is the functional relation of the friction between the potted seedling and the inner wall of the seedling dropping cylinder, N; and \(v_{3}\) is the relative velocity of the pot seedling sliding to point C, m/s. According to Equation (7), the following equation can be obtained  
\[v_{2}=\sqrt{v_{3}{}^{2}-2 gL_{BC}\cos\alpha+\frac{2\int_{L_{BC}}f(x,y)\mathrm{ d}s}{m}} \tag{8}\]  
As shown in Figure 9c, after the pot seedling reaches point C, there is an oblique launching angle with an initial velocity of \(v_{3}\) and an oblique launching angle of \(\alpha\). In this process, it is only affected by its own gravity until it collides with point D on the opposite side of the seedling cylinder wall, and the unilateral deformation caused by the collision is \(\varepsilon_{2}\). According to the oblique launching motion formula, the following equation can be obtained  
\[\left\{ \begin{array}{l}v_{3}\cos\alpha\Delta t_{2}+\frac{1}{2}g(\Delta t_{2 })^{2}=H_{6}\\ v_{3}\sin\alpha\Delta t_{2}=L_{6}\end{array} \right. \tag{9}\]  
where \(\Delta t_{2}\) is the projectile motion time, s.  
According to the kinetic energy theorem, the following equation can be obtained  
\[\frac{1}{2}mv_{3}{}^{2}+mgL_{CD}\cos\alpha=\frac{1}{2}mv_{4}{}^{2} \tag{10}\]where \(v_{4}\) is the relative velocity of pot seedlings before the collision of D point, m/s, and \(L_{\text{CD}}\) is the path length from point C to point D, mm. According to Equation (9), the relative velocity of the pot seedling sliding to point C can be obtained as follows:  
\[v_{3}=\sqrt{\frac{gL_{6}{}^{2}}{2(H_{6}-L_{6}\tan\alpha)\sin^{2}\alpha}} \tag{11}\]  
Substituting Equation (11) into Equation (10), the velocity before the collision between the pot seedling and point D can be obtained  
\[v_{4}=\sqrt{(\frac{gL_{6}{}^{2}}{2(H_{6}-L_{6}\tan\alpha)\sin^{2}\alpha})-2 gL_ {CD}\cos\alpha} \tag{12}\]  
The relative velocity of the pot seedling after collision can be obtained by substituting Equation (11) into Equation (8)  
\[v_{2}=\sqrt{(\frac{gL_{6}{}^{2}}{2(H_{6}-L_{6}\tan\alpha)\sin^{2}\alpha})-2 gL_ {BC}\cos\alpha+\frac{2\int_{L_{BC}}f(x,y)\text{d}s}{m}} \tag{13}\]  
As shown in Figure 9d, the pot seedling continues to fall after colliding with point D, during which the pot seedling blade is subjected to the supporting force of friction at the dropping mouth: \(f_{2}(x,y)\) and \(N_{2}(x,y)\). Under the action of this supporting force, the dropping posture of the pot seedling gradually changes downward and vertically until it falls to the position of the planting groove E. As the cans collide at point D, the angle between the velocity direction and the collision surface is small, the energy loss in the collision process is neglected, and the velocity direction of the potted seedlings remains constant. The process was thus simplified to a free dropping motion with initial velocity \(v_{4}\).