# Optimal Design and Experiment of Corn-Overlapped Strip Fertilizer Spreader
## 2 Materials and Methods
### Analysis of Fertilizer Spreading Motion
#### 2.2.2 Analysis of the Vertical Throwing Motion of Fertilizer Particles  
The upper end of this fertilizer spreading device is connected to the fertilizer discharger. The fertilizer blending device based on EDEM software by [PERSON] [22; 23] et al. is similar to this investigation in that its fertilizer allocation device is connected to the fertilizer discharger, and during its analysis of the discharge velocity of the discharger outlet, the velocity at the discharge port of the discharger is derived in the theoretical output velocity range of 0.44 m/s-0.57 m/s. As shown in Figure 4, the fertilizer enters the fertilizer spreading tube at speed \(V_{\text{in}}\), with the height \(H_{1}\) of the fertilizer particles from the inlet to the outlet of the spreading device and energy loss \(E_{1}\) generated during the impact with the inner wall surface of the fertilizer spreading tube, while the process increases the conversion of gravitational potential energy into kinetic energy, and finally, at the outlet of fattening small plates, it is thrown out at speed \(V_{\text{out}}\). The kinetic energy of its exit is calculated as: \(\frac{1}{2}mV_{out}^{2}=\frac{1}{2}mV_{in}^{2}+2 mgH_{1}-E_{1}\), and the outlet velocity \(V_{\text{out}}\) is obtained as \(V_{out}=\sqrt{V_{in}^{2}+2 gH_{1}-2E_{1}}\). Ignoring other motions, such as the rotation and collision of fertilizer particles in the fertilizer spreading device, and simplifying the vertical spreading movement of fertilizer, the vertical spreading motion is shown in Figure 4. The direction of fertilizer particles is set to falling in the \(Z\) direction, and the direction of movement of the corn along the side of the monopoly is the horizontal \(X\) direction. \(V_{\text{x}}\) and \(V_{\text{y}}\) denote the velocity of fertilizer particles moving in the \(X\) and \(Y\) directions, respectively; \(F\) denotes the sum of air resistance to fertilizer particles; and \(F_{\text{x}}\) and \(F_{\text{z}}\) signify air resistance to fertilizer particles in the \(X\) and \(Z\) directions, respectively. Figure 4: Fertilizer vertical spreading movement. Fertilizer pellets are thrown from the fattening small plates by gravity and air resistance. The air resistance applied is:  
\[F=\frac{1}{2}\rho scv^{2} \tag{4}\]  
where \(F\) is the air resistance of fertilizer particles, \(N\); \(\rho\) is the air density, kg/m\({}^{3}\); \(S\) is the windward area of fertilizer particles, m\({}^{2}\); \(V\) is the velocity of fertilizer particles, m/s; \(C\) is the air resistance coefficient, m\({}^{2}\); and \(H\) is the throwing height of fertilizer particles, mm. Combining [PERSON]'s second law and related studies [24; 25], the equation of motion of a fertilizer particle in the vertical plane is:  
\[\left\{ \begin{array}{l}F_{z}=\frac{\rho sc}{2}(v_{2}\sin\alpha)^{2}-mg\\ \frac{d^{2}H_{Z}}{dt^{2}}=\frac{F_{z}}{m}-g\end{array} \right. \tag{5}\]  
where \(F_{z}\) is the air resistance of fertilizer particles in the vertical plane, \(N\); \(t\) is the time required to throw to the ground, s; and \(H_{z}\) is the displacement of fertilizer particles in the vertical plane, m.  
It can be obtained as:  
\[t=\frac{2 mH_{z}}{F_{Z}+mg}=\frac{4 mH_{z}}{\rho scv_{2}\sin\alpha^{2}+2 mg} \tag{6}\]  
The three fattening small plates are evenly distributed in a 120\({}^{\circ}\) homogeneous direction; the throwing direction of the two side fattening small plates and the rear fattening small plates is different from the forward direction of the machine. Therefore, the motion of the fertilizer particles in the horizontal direction needs to be solved separately. When the throwing direction of the fattening small plates is opposite to the forward direction of the machine, the equation of motion in the horizontal direction is:  
\[\left\{ \begin{array}{l}F_{L_{2}}=\frac{\rho sc}{2}(v_{1}+v_{2}\cos\alpha)^{2 }\\ \frac{d^{2}L_{2}}{dt^{2}}=-\frac{F_{L_{2}}}{m}\end{array} \right.