## 2 Materials and Methods
### Mechanical Analysis
#### 2.2.2 Cutting Stage  
When the castor stalks are cut with the disc cutter, the edge and surface of the disc cutter are the key parts that cut the stalk and resist the cutting force. Therefore, this study analyzed the mechanical model of cutting resistance in terms of differentials. As shown in Figure 4, the cutting resistance of the disc cutter at the moment of cutting mainly includes the reaction force \(F_{0}\) exerted by the stalk against the motion of the disc cutter at the cutting edge, the extrusion force \(F_{1x}\), \(F_{1y}\) and friction \(T_{2}\) of the stalk pressed on the external edge surface, and the extrusion force \(F\) and friction \(T_{1}\) of the stalk pressed on the internal edge surface. Figure 4: Cutting force of the stalk. The stalk is analyzed as a whole unit with certain physical characteristics. The external edge surface is divided using a differential method. The extrusion force \(dF_{1x}\) on the external edge differential element plane in the \(X\) direction (that is the direction \(v_{m}\) of the disc cutter motion velocity) is obtained as follows:  
\[dF_{1x}=\sigma ldy=\sigma ldx\cdot\tan\alpha \tag{16}\]  
where \(F_{1x}\) is the extrusion force on the external edge surface of the disc cutter in the \(X\) direction N; \(\sigma\) is the crushing stress in the \(X\) direction, MPa; \(l\) is the cutting width, m; and \(\alpha\) is the edge angle, \({}^{\circ}\). The stress and strain of the stalk during the cutting process was determined using Hooke's law, and the stress-strain relationship of the stalk cutting is as follows:  
\[\varepsilon=\frac{X_{e}}{x}=\frac{\sigma}{E_{x}} \tag{17}\]  
where \(\varepsilon\) is the shear strain; \(X_{e}\) is the total stalk thickness, m; \(x\) is the stalk thickness when pressed by the cutter disc, m; and \(E_{x}\) is the elastic modulus of the stalk in the \(X\) direction, MPa. Next, we substitute Equation (17) into Equation (16) and integrate \(dF_{1x}\). The extrusion force \(F_{1x}\) is as follows:  
\[F_{1x}=\int_{0}^{x}\sigma l\tan\alpha dx=\int_{0}^{x}\frac{x}{X_{e}}E_{x}l\tan \alpha dx=\frac{x^{2}l}{2X_{e}}E_{x}l\tan\alpha \tag{18}\]  
The extrusion force on the external edge differential element plane in \(Y\) direction \(dF_{1y}\) is as follows:  
\[dF_{1y}=\varepsilon_{y}E_{y}ldx=\varepsilon_{x}\mu E_{y}ldx \tag{19}\]  
where \(F_{1y}\) is the extrusion force on the external edge surface of disc cutter in the \(Y\) direction, N; \(\varepsilon_{y}\) is the cutting strain of the castor stalk in the \(Y\) direction; and \(E_{y}\) is the elastic modulus of the stalk in the \(Y\) direction, MPa. The extrusion force \(F_{1y}\) of the external edge surface in the \(Y\) direction is as follows:  
\[F_{1y}=\int_{0}^{x}\varepsilon_{x}\mu E_{y}ldx=\int_{0}^{x}\frac{x}{X_{e}}\mu E _{y}ldx=\frac{x^{2}l}{2X_{e}}\mu E_{y}l \tag{20}\]  
The friction force \(T_{1}\) on the external edge is as follows:  
\[T_{1}=\mu\big{(}F_{1x}\cdot\sin\alpha+F_{1y}\cdot\cos\alpha\big{)}=\frac{x^{2 }l}{2X_{e}\cos\varepsilon}\Big{(}E_{x}\sin^{2}\alpha+\mu_{\omega}E_{y}\ \cos^{2}\alpha\Big{)} \tag{21}\]  
where \(T_{1}\) is the friction force on the external edge, N; and \(\mu\) is the coefficient of dynamic friction between the blade surface and stalk. \[F_{0}=\delta l\sigma_{c} \tag{22}\]  
Here, \(F_{0}\) is the acting force of the stalk on the disc cutter, N; \(\delta\) is the disc cutter thickness, m; and \(\sigma_{c}\) is the yield strength, MPa. \[T_{2}=\mu_{i}F \tag{23}\]  
Here, \(T_{2}\) is the friction force on the internal edge, N; \(\mu_{i}\) is the coefficient of stalk internal friction; and \(F\) is the pressure of the disc cutter against the stalk, N.  
At the cutting moment, the total cutting resistance of the disc cutter is as follows:  
\[\left\{ \begin{array}{l}F_{X}=F_{0}+F_{1x}+T_{1}+T_{2}\cdot\cos\alpha\\ F_{Y}=F+T_{2}\cdot\sin\alpha\end{array} \right.