## 2 Materials and Methods
### Structure and Working Principle of Organic Fertilizer Trenching Device
#### 2.1.2 Kinematics Analysis of Ditching Blades  
Based on the requirements on ditching operation, the travelling of cutting wheel during ditching is divided into three stages: soil cutting stage, soil removal stage and soil throwing stage. In the ditching operation, the ditching blade cuts the soil from the bottom of the trench to the surface, the direction of rotation of the ditching blade is opposite to the forward direction of the tractor wheel, and the corresponding horizontal force is also opposite to it, the trenching disc rotates and cuts the soil while moving forward together with the tractor, so its absolute motion is the synthesis of rotation and movement, and the trajectory is a cosine motion [17]. As shown in Figure 3,With the center of rotation of the cutter as the origin, X axis for the unit forward direction, the Y-axis is perpendicular to the X-axis in the positive direction to establish a right-angle coordinate system, let the forward speed of the unit be \(V_{m}\), cutter rotation angular velocity is \(\omega\), the beginning of trenching, the blade end point is located at point A, in time \(t\), the center of rotation of the cutter moves from point O to point P, at the same time, the blade end point from point A to point B, then the trajectory of point B equation is:  
\[\left\{\begin{array}{c}x=V_{m}t+Rcos\omega t\\ y=Rsin\omega t\end{array}\right. \tag{1}\]  
where \(x\), \(y\) is the trajectory coordinates of the ditching blade at any moment, m; \(R\) is the blade rotation radius, m; \(V_{m}\) is the forward speed of the trencher, km/h; \(\omega\) is the angular speed of the blade rotation, rad/s; \(t\) is the moving time, s; The absolute velocity of the end point of the ditching blade can be obtained by deriving the trajectory equation as:  
\[\left\{\begin{array}{c}V_{x}=\frac{dx}{dt}=V_{m}-R\omega sin\omega t=V_{m}-R \omega sin\theta\\ V_{y}=\frac{dy}{dt}=R\omega cos\omega t=R\omega cos\theta\end{array}\right. \tag{2}\]  
where \(\theta\) is the angle of blade rotation in time T.  
Figure 2: Trenching tool.  
In order to study the soil throwing performance, The variable \"\(\mu\)\" is introduced here, and \"\(\mu\)\" is the ratio between the blade speed \(V_{k}\) and the unit forward speed \(V_{m}\), so that \(\mu=\frac{\omega\mathbb{R}}{V_{m}}\), then the absolute speed of the blade end point can be obtained as follows:  
\[V=\sqrt{V_{x}^{2}+V_{y}^{2}}=\sqrt{R^{2}\omega^{2}+{V_{k}}^{2}+2V_{k}R\omega cos \omega t} \tag{3}\]  
Let the instantaneous angle between the velocity and the X-axis be \(\varphi\).  
\[\varphi=\arctan\frac{v_{x}}{v_{y}}=\frac{v_{m}-R\omega sin\theta}{R\omega cos\theta} \tag{4}\]  
Further deformation of the horizontal fractional velocity yields  
\[V_{x}=V_{m}(1-\mu\sin\theta) \tag{5}\]  
From the above equation, we can see that when \(\mu<1\), that is, \(V_{k}<V_{m}\), then no matter what position the ditching blade rotates to, there is always \(V_{x}>0\), the soil particles on the blade have to move in the direction of the advance of the unit, a large amount of soil will be thrown into the ditch and cannot be thrown well out of the ditch. When \(\mu>1\), \(V_{k}>V_{m}\), The blade end point trajectory is a residual cycloid, and the horizontal velocity of the end point is opposite to the forward direction of the unit, so the soil will have the tendency to be thrown backward and reduce the soil return flow.  
Similarly, the acceleration equation of point B is obtained by deriving  
\[\left\{\begin{array}{l}a_{x}=\frac{dv_{x}}{dt}=-R\omega^{2}cos\theta\\ a_{y}=\frac{dv_{y}}{dt}=-R\omega^{2}sin\theta\end{array}\right. \tag{6}\]  
where the direction of acceleration is always directed to the center of the cutter.