# Design and Evaluation of a Shaping and Pruning Machine for Dwarf and Densely Planted Jujube Trees
## 2 Materials and Methods
### Design of Pruning Device
#### 2.3.2 Analysis of Cutter Rotation Speed  
When the pruning machine is working, the pruning cutters perform both rotation and translation, and the cutter movement trajectory is cycloid (Figure 4). Through theoretical analysis of the cutting process of the cutters, the trajectory equations of blade \(a\) and \(b\) are shown in Equations (1) and (2), respectively. Trajectory equations of blade \(a\):  
\[\left\{ \begin{array}{c}x_{a}=vt+R\sin\omega t\\ y_{a}=R\cos\omega t\end{array} \right. \tag{1}\]  
where \(R\) is the cutter radius, in mm; \(v\) is the forward speed of the pruning machine, in m/s; \(\omega\) is the angular velocity of the cutter, in rad/s. Trajectory equations of blade \(b\):  
\[\left\{ \begin{array}{c}x_{b}=vt+R\sin(\omega t+\beta)\\ y_{b}=R\cos(\omega t+\beta)\end{array} \right. \tag{2}\]  
where \(\beta\) is the angle between the two adjacent blades of the knife disc, in rad. During the cutting, missed cuts may appear if the two neighboring blades \(a\) and \(b\) turn at the same angle and the horizontal coordinate space between the two edges is larger than  
Figure 4: Path of cutter motion. Figure 3: Schematic diagrams of pruning device structure. (1) Supporting rod. (2) Circular saw blades. (3) Guard plate for pruning. (4) Hydraulic motor. (5) Synchronous belt drive mechanism. (6) Cutter shaft. (**a**) Side pruning assembly. (**b**) Top pruning assembly. the height of the cutting edge \(h\). To avoid such missed cuts, the circular saw blade shall satisfy the following condition:  
\[x_{a1}-x_{b1}\leq h \tag{3}\]  
where \(x_{a1}\) is the horizontal coordinate of the cutting edge \(a\), in mm; \(x_{b1}\) is the horizontal coordinate of the cutting edge \(b\), in mm; \(h\) is the height of the cutting edge, in mm. When the blade \(a\) is at position \(\omega t=\pi/2\) or turns \(\pi/2\), it reaches the horizontal axis and prunes the jujube branch at the axis x; substituting these coordinates into Equation (1) yields:  
\[x_{a1}=\frac{\pi v}{2\omega}+R \tag{4}\]  
When the blade b is at position \(\omega t=\pi/2-\beta\) or turns \(\pi/2-\beta\), it prunes the jujube branch at the axis x; substitute this value into equation (2) to obtain:  
\[x_{b1}=\frac{v}{\omega}\Big{(}\frac{\pi}{2}-\beta\Big{)}+R \tag{5}\]  
Equations (3)-(5) are solved simultaneously; meanwhile, \(\omega=2\pi u\), the rotation speed of the cutter \(n\), is:  
\[n\geq\frac{v\beta}{2\pi h} \tag{6}\]  
where \(n\) is the rotation speed of cutter, in r/s. According to Equation (6), when missed cuts are avoided during pruning jujube trees, there is a critical rotation speed of the cutter. If the rotation speed of the cutter is lower than this critical speed, the cutter will be unable to cut off the jujube branches, causing branch skewing or lacerations. As specified in Tungsten Carbide Tipped Circular Saw Blade for Woodworking (GB/T14388-2010), the number of saw teeth for tungsten carbide tipped circular saw blades is usually 20-128, i.e., \(\beta\) is between \(\pi/64\)-\(\pi/10\), the forward speed is usually 8-30 m/min, and the actual effective length of the cutter edge is 1.5-4.0 mm; if these parameters are substituted into Equation (6), it can be found that missed cuts appear when the rotation speed of the circular saw blade cutter is lower than 1020 r/min. The pruning device designed in this study has the following advantages for pruning dwarf and densely planted jujube trees in Xinjiang. Firstly, the top and side branches can be pruned at the same time. Secondly, the device can complete the pruning operation for the whole line of jujube trees, so as to ensure the uniformity of the pruned tree shapes. It is conducive to the fully mechanized management of Xinjiang jujube gardens. Thirdly, the distance between the two sets of side pruning assemblies can be adjusted to enable the pruning of jujube canopies of different sizes. Fourth, while hard jujube branches in Xinjiang have characteristics that demand laborious pruning, this device uses a disc cutter to effectively cut hard jujube branches under the high speed of the cutter.