# Influencing Factors Analysis and Simulation Calibration of Restitution Coefficient of Rice Grain
## 2 Materials and Methods
### Test Principle  
The restitution coefficient is an index to measure the rebound ability of granular materials. Based on the principle of kinematics, the composite value of the velocity components of rice grain before and after collision along three coordinate axes is measured, and the coefficient of restitution is calculated [22]. \[e=\frac{v_{t}}{v_{0}} \tag{1}\]  
where \(e\) is the restitution coefficient between rice grain and tested materials; \(v_{t}\) is the instantaneous rebound velocity of rice grain after collision, m/s; and \(v_{0}\) is the instantaneous falling velocity of rice grain before collision, m/s. According to the collision state between rice grain and contact materials, the falling collision process of rice grain is analyzed. It is assumed that the rice grain fall from the seed sticking plate under impact at a certain initial velocity and collide with the horizontal material plane (the collision platform is horizontal, i.e., the collision angle is 0\({}^{\circ}\)). The horizontal space rectangular coordinate system \(xyz\) was established, as shown in Figure 4a. Before the collision, the rice grain accelerates uniformly in the vertical direction, and the coordinate value of the spatial reference wall before the \(m\) frame is \((y_{m},z_{m})\); the coordinate value in the mirror is \((x_{m},z_{m})\). The coordinate values of the space reference wall at the moment of collision are \((y_{0},z_{0})\). In the mirror, the coordinates are \((x_{0},z_{0})\). In this case, the values of \(x_{0}\) and \(x_{m},y_{0}\), and \(y_{m}\) are equal. Figure 4: Kinematics analysis of rice grain in collision process: (**a**) simplified model of collision with horizontal collision platform; (**b**) simplified model of collision with inclined collision platform. In the vertical direction, the rice grain is uniformly accelerated, and the instantaneous velocity in the middle is equal to the average velocity in this period. \[\overline{v}_{m}=\frac{z_{m}-z_{0}}{t_{m}} \tag{2}\]  
The instantaneous velocity before the collision between the rice grain and the collision platform is as follows:  
\[v_{0}=\overline{v}_{m}+\frac{1}{2}gt_{m} \tag{3}\]  
The combined Formulas (2) and (3) are as follows:  
\[v_{0}=\frac{z_{m}-z_{0}}{t_{m}}+\frac{1}{2}gt_{m} \tag{4}\]  
By decomposing the separation velocity of rice grains after collision, the separation velocity along the triaxial direction in the three-dimensional space is obtained as follows:  
\[v_{t}=\sqrt{v_{x}^{2}+v_{y}^{2}+v_{z}^{2}} \tag{5}\]  
where \(v_{x}\) is the partial velocity of rice grain in \(x\) direction after collision, m/s; \(v_{y}\) is the partial velocity of rice grain in \(y\) direction after collision, m/s; and \(v_{z}\) is the partial velocity of rice grain in \(z\) direction after collision, m/s. Equations (4) and (5) are substituted into Equation (1), and the restitution coefficient between rice grain and horizontal test material is obtained. \[e=\frac{v_{t}}{v_{0}}=\frac{2t_{m}\sqrt{v_{x}^{2}+v_{y}^{2}+v_{z}^{2}}}{2(z_{ m}-z_{0})+gt_{m}^{2}} \tag{6}\]  
On this basis, the bouncing motion of rice grains in the process of collision with the inclined collision platform was analyzed, as shown in Figure 4b. The horizontal space coordinate system \(xyz\) is transformed into the inclined space coordinate system \(x^{\prime}y^{\prime}z^{\prime}\) (the collision platform was inclined at a certain angle; the collision angle is \(\alpha_{i}\)). The displacement relation between the coordinate systems is obtained. \[\left\{ \begin{array}{l}x^{\prime}=x\\ y^{\prime}=z\cos\alpha_{i}+y\sin\alpha_{i}\\ z^{\prime}=z\sin\alpha_{i}-y\cos\alpha_{i}\end{array} \right. \tag{7}\]  
where \(v_{x^{\prime}}\) is the partial velocity of rice grain in \(x^{\prime}\) direction after collision, m/s; \(v_{y^{\prime}}\) is the partial velocity of rice grain in \(y^{\prime}\) direction after collision, m/s; and \(v_{z^{\prime}}\) is the partial velocity along \(z^{\prime}\) direction after the collision of rice grains, m/s.