## 2 Materials and Methods
### Mechanical Analysis
#### 2.2.1 Contact Stage  
The maximum stress at the contact stage occurs at the moment when the stalk changes from a static state, gains kinetic energy and withstands the impact of the disc cutter. The contact and collision process between castor stalk and disc cutter was analyzed. The force and velocity analysis of the stalk are shown in Figure 3. The disc is in contact with the stalk at a relative velocity \(V_{z}\). When the velocity of the stalk increases to the same level as that of the disc cutter, their relative velocity is zero. The relative displacement \(\delta_{z}\) between the stalk and the disc cutter does not decrease. In this case, the maximum contact force is \(P_{max}\). Figure 2: Double-disc cutting system consisting of wave-type disc cutters. According to inelastic collision theory, the relative displacement \(\delta_{z}\) at a given moment is as follows:  
\[a=\left(\frac{3 PR}{4E^{*}}\right)^{\frac{1}{3}} \tag{1}\]  
\[\delta_{z}=\frac{a^{2}}{R}=\left(\frac{9P^{2}}{16 RE^{*2}}\right)^{\frac{1}{3}} \tag{2}\]  
\[\frac{1}{E^{*}}=\frac{\left(1-v_{1}^{2}\right)}{E_{1}}+\frac{\left(1-v_{2}^{2} \right)}{E_{2}} \tag{3}\]  
\[\frac{1}{R}=\frac{1}{R_{1}}+\frac{1}{R_{2}} \tag{4}\]  
\[\frac{1}{m}=\frac{1}{m_{1}}+\frac{1}{m_{2}} \tag{5}\]  
where \(a\) is the effective dimension of contact segment m; \(P\) is the loading force, N; \(R\) is the relative radius of curvature, m; \(E^{*}\) is the elasticity modulus, MPa; \(\delta_{z}\) is the displacement of contact point, m; \(v_{1}\) is the contact velocity of the stalk, m/s; \(E_{1}\) is the elasticity modulus of the stalk, MPa; \(v_{2}\) is the contact velocity of the disc cutter, m/s; \(E_{2}\) is the elasticity modulus of the disc cutter, MPa; \(R_{1}\) is the effective radius of the stalk, m; \(R_{2}\) is the effective radius of the disc cutter, m; \(m\) is the equivalent mass, kg; \(m_{1}\) is the mass of the stalk, kg; and \(m_{2}\) is the mass of the disc cutter, kg. During the collision process, because of elastic deformation, the centers of the stalk and disc cutter approach each other by a displacement \(\delta_{z}\), and their relative velocity is as follows:  
\[v_{2}-v_{1}=\frac{d\delta_{z}}{dt} \tag{6}\]  
At any instant, the interaction force is as follows:  
\[P(t)=m_{1}\frac{dv_{1}}{dt}=-m_{2}\frac{dv_{2}}{dt} \tag{7}\]  
This can be obtained using the following Equations (6) and (7):  
\[-\frac{m_{1}+m_{2}}{m_{1}m_{2}}P=\frac{d(v_{2}-v_{1})}{dt}=\frac{d^{2}\delta_ {z}}{dt^{2}} \tag{8}\]  
According to Equation (2), the following can be obtained:  
Figure 2:0  
Figure 2:1 Schematic of stalk contact force and velocity. Substituting Equations (5), (8) and (9), the following can be obtained:  
Figure 2:2  
The following can be obtained by integrating \(\delta_{z}\):  
Figure 2:3  
Figure 2:4  
where \(V_{Z}\) is the velocity of the stalk and the disc cutter approaching each other, m/s. At the maximum compression displacement, \(d\hat{\varrho}_{z}/dt=0\), using Equation (11) we can obtain the following:  
Figure 2:5  
According to the [PERSON] criterion, at the time of initial yield, the relationship between maximum contact pressure \(P_{max}\) and loading force is as follows:  
Figure 2:6  
By substituting Equations (9), (13), and (14), the maximum force in the contact process is obtained as follows:  
Figure 2:7  
During the contact process, when the maximum contact stress \(P_{max}\) exceeds the elastic limit of the stalk, the stalk undergoes plastic deformation and enters the cutting stage. According to Equation (15), the maximum contact force of the disc cutter is positively correlated with the contact relative velocity of the stalk and the equivalent mass of the stalk and disc cutter, and negatively correlated with the equivalent radius of curvature.