*2.2. Jujube Picking Mechanism*

Figure 2 shows a three-dimensional model of the jujube picking equipment. The equipment is mainly composed of five parts: plane and convey mechanisms, a pair of roller picking mechanisms, crush mechanisms, screen mechanisms and walk mechanisms. The tractor was used as the traction power to realize the field picking process, with the jujube picking equipment connecter to the traction frame. In the process of forward movement, the jujube was gathered in front of the shovelling teeth by the rolling brush, and the shovelling teeth were used to transport the jujube branches to the upper and lower gear components to complete the levelling and transportation of the jujube. The jujube could be collected under the action of upper and lower gears. The collected jujube was crushed by the crushing hammer of the crushing device, and the crushing process of the jujube was realized. Through the sieve plates, the crushed straw residue was evenly sprinkled back onto the field. The research emphasis of the equipment is to improve the picking rate, which can not only improve the clogging problem of the collecting system, but also reduce the power consumption; therefore, the research on picking rate is of great significance to the equipment used for picking up branches.

**Figure 2.** Three-dimensional model of the jujube picking equipment. 1. Rolling brush 2. Traction frame 3. Device shell 4. Crushing mechanism 5. Walking wheels 6. Sieve plates 7. Lower gear assembly 8. Upper gear assembly 9. Collect shovelling teeth.

To obtain the reliable data of the picking rate of the equipment, the numerical simulation model was simplified according to the structure of the test bench, as shown in Figure 3. The rotation direction of the lower gear shaft was opposite to the forward direction, so that the picking mechanism produced a sufficient picking process, the lifting distance and time of the material were extended, and finally, the jujube branches were fully separated. Jujube picking can be regarded as a kind of supernormal particle movement, in which the study of jujube branch size can be regarded as the study of particle size, and the study of total feeding amount can be regarded as the study of the effects of particle number on supernormal particle flow. What is more, the gear will collide with the jujube branch during the rotation, which will affect the speed and direction of the jujube branch. Therefore, the research on the speed of the gear can be regarded as the study of the influence of external force on supernormal particle flow. Due to the frequent occurrence of missing branches during the picking process, the picking rate is an important indicator of jujube picking. The picking rate of jujube branches is shown in Formula (1).

$$Q\_1 = \frac{C\_1}{S\_1} \times 100\% \tag{1}$$

where *Q*<sup>1</sup> is picking rate, *C*<sup>1</sup> is the number of jujube branches successfully picked up into the collection box by the jujube picking agency, and *S*<sup>1</sup> is the total number of jujube branches fed in the picking test.

**Figure 3.** Simplified numerical model of jujube branch picking mechanism. 1. Upper gear shaft 2. Upper gear 3. Device shell 4. Conveyor 5. Movement direction of conveyer belt 6. Lower gear 7. Lower gear shaft 8. Counterclockwise rotation 9. Clockwise rotation.

### *2.3. Numerical Methods*

For the first time, the jujube branch is used as a supernormal particle model, and the meshless Galerkin method was first used to simulate the motion of supernormal particles. This method is to eliminate the influence of the shape of the jujube branch on the flow. Compared with the traditional CFD method, the meshless Galerkin method takes the advantage of not considering the force on the interface between the particles and the flow field, and the effect of particle movement on the mesh deformation of the fluid [25–29]. The study found that the meshless Galerkin method mainly constructs an approximate weight function for the field function according to the node, which is a basic feature of the method. The meshless method includes two key steps: (1) constructing an approximate weight function for the field function; and (2) solving the meshless discretization of partial differential equations.

The jujube branches were transported, entangled, collided, escaped and collected during the mechanical picking process. Among them, the particle size determines the particle weight, the particle force, particle movement speed, and displacement value in the field function. The number of particles affects the relative position between particles, which in turn affects the force between particles and particle movement speed. The gear rotation speed directly affects the force between particles and walls, resulting in changes in particle movement speed and particle displacement. Therefore, the field functions to be calculated are the force, the velocity and the displacement values; the force of the particles is the volume force and the surface force, including the collision force between the particles, the collision contact force, and the rolling force of the particles. The particle movement speed and displacement values were calculated using the current particle position, while speed and time data were used to calculate the relevant data of the particle in the next time step. The specific field function equations are shown in Formulas (2)–(4).

$$m\_i \frac{d\upsilon\_i}{dt} = \sum F\_{net} = \sum F\_{body} + \sum F\_{surface} = \sum F\_{body} + \sum F\_{contact} + \sum F\_{rolling} \tag{2}$$

$$v\_{ncv} = v\_{old} + \int\_{t}^{t + \Delta t} \frac{\sum F\_{nct}}{m} dt \tag{3}$$

$$\chi\_{new} = \chi\_{old} + \int\_{t}^{t + \Delta t} v\_{new} dt \tag{4}$$

where *mi* is the particle mass at the node *xi*, kg·m−3, *<sup>F</sup>* is the particle force, N, *vi* is the particle velocity at the node *xi*, m·s<sup>−</sup>1, *<sup>x</sup>* is the particle displacement at the node *xi*, m.

The general expression of the equivalent integral form of the differential equation system based on the meshless Galerkin method to solve each node in the domain was calculated using Formula (5).

$$\int\_{\bar{\Omega}\_j} wA \left[ \sum\_{i=1}^n N\_i(\boldsymbol{x}) \boldsymbol{u}\_i \right] d\Omega\_{\bar{\boldsymbol{\beta}}} + \int\_{\bar{\Gamma}\_j} \overline{w}B \left[ \sum\_{i=1}^n N\_i(\boldsymbol{x}) \boldsymbol{u}\_i \right] d\Gamma\_{\bar{\boldsymbol{\beta}}} = 0 \tag{5}$$

where Ω represents the solution domain of the problem, Γ represents the boundary of the solution domain (including displacement boundary and force boundary), A and B are the differential operators of independent variables (such as spatial coordinates, time coordinates, etc.). Functions *w* and *w* are test functions (or weight functions); *Ni*(*x*) is the shape function of the approximate function *u*(*x*); *ui* is the value of the field function *u*(*x*) to be solved at node *xi*.

The approximate function *u*(*x*) is constructed by solving a set of discrete points *xi* = (*i* = 1, 2, ···, *N*) in the domain by solving a set of discrete *ui* known in the function *u*(*x*), and the expression of the global approximation function *uh*(*x*) at the solution point *xi* is:

$$u^h(\mathbf{x}, \overline{\mathbf{x}}) = \sum\_{i=1}^m p\_i(\overline{\mathbf{x}}) a\_i(\mathbf{x}) = p^T(\overline{\mathbf{x}}) a(\mathbf{x}) \tag{6}$$

where *x* is the coordinate of each node in the solution domain app, *pi*(*x*) is the basis function, *m* is the number of basis function, and *ai*(*x*) is the undetermined coefficient.

The primary and secondary basis functions in a two-dimensional space can be expressed as:

*pT*(*x*) = [1, *x*, *y*], *m* = 3 (7)

$$p^T(\overline{\mathbf{x}}) = \left[1, \mathbf{x}, y, \mathbf{x}^2, \mathbf{x}y, y^2\right], \mathbf{w} = \mathbf{6} \tag{8}$$

The sum of the global approximation function *uh*(*x*) and weight squared error at the solution point is *xi* is:

$$J = \sum\_{j}^{N} w\_{j} \left[ \sum\_{i=1}^{m} p\_{i}(\overline{\mathbf{x}}) a\_{i}(\mathbf{x}) - u(\mathbf{x}\_{i}) \right]^{2} \tag{9}$$

Using the principle of least squares method to solve the undetermined coefficient *ai*(*x*) when *J* =0, we can obtain:

$$\sum\_{j}^{m} \left[ \sum\_{i=1}^{N} w\_i p\_i(\mathbf{x}\_i) p\_j(\mathbf{x}\_j) \right] a\_i(\mathbf{x}) = \left[ \sum\_{i=1}^{N} w\_i p\_i(\mathbf{x}\_i) \right] u(\mathbf{x}\_i) \tag{10}$$

The undetermined coefficient *ai*(*x*) can be obtained from the above formula

$$a(\mathbf{x}) = A^{-1}Bu\tag{11}$$

It can be obtained by integrating Equations (6) and (11)

$$\mu^h(\mathbf{x}, \overline{\mathbf{x}}) = p^T(\overline{\mathbf{x}}) A^{-1} B \mu = \mathcal{N}(\mathbf{x}, \overline{\mathbf{x}}) \mu \tag{12}$$

The shape function can be obtained as follows:

$$N(\mathbf{x}, \overline{\mathbf{x}}) = p^T(\overline{\mathbf{x}}) A^{-1} B \tag{13}$$

The weight function type obtained by the Rocky software is the Gaussian exponential weight function, and its expression is:

$$w\_i = \begin{cases} \frac{e^{-(d\_i/\epsilon\_i)^{2k}} - e^{-(r\_i/\epsilon\_i)^{2k}}}{1 - e^{-(r\_i/\epsilon\_i)^{2k}}} & , 0 \le d\_i \le r\_i\\ 0 & , d\_i \le r\_i \end{cases} \tag{14}$$

where *di* is the distance from the node *xi* to the field point *x*, m, *ci* is the constant of the control function shape, and *ri* is the solution domain of the weight function *wi*, which is the radius of the node *xi*, m.

Under the simulation control of the above equations, jujube branches' movement phenomena, such as transportation, entanglement, collision, escape and collection, occur. Particle size determines the particle mass, which in turn affects particle force, particle movement speed, and the displacement value in the field function. The number of particles affects the position between particles, as well as the force between particles and the speed of particle movement. The rotation speed of the gear directly affects the force between the particles and wall surface, and then affects the particle movement speed and particle displacement.
