*4.2. Design of Pulling Roller*

In order to analyze the movement of the cabbage on the pulling roller, the cabbage and the pulling roller are formed into a rigid body system for force analysis. In order to simplify the model, the position of the mass center is not considered in the analysis process. See Figure 8.

$$
\sum m v\_x = \sum F\_x^{(\varepsilon)}, \\
 m\_j a\_j - m(a\_\varepsilon - a\_r \cos \delta) = F\_N \cos \delta = \mu F\_j \cos \delta \tag{7}
$$

$$
\sum mv\_z = \sum F\_z^{(\varepsilon)} \, \_\prime F\_m - (m\_\flat + m)g - F\_\flat \sin \delta = ma\_r \sin \delta \tag{8}
$$

**Figure 7.** Trajectory diagram simulation of reel movement: (**a**) motion trajectory integration; (**b**) rotation speed = 30 r/min; (**c**) rotation speed = 50 r/min; (**d**) rotation speed = 70 r/min.

**Figure 8.** Dynamic analysis of pulling operation: (**a**) 1. Reel. 2. Pulling roller; (**b**) pulling roller and cabbage movement direction; (**c**) rotation speed = 50 r/min.

According to the centroid motion law of rigid body motion, the force balance equations in the *x*-direction and *z*-direction are listed:

$$
\mathfrak{m}(a\_{\bar{l}}\cos\delta - a\_r) = \mathfrak{m}\mathfrak{g}\sin\delta\tag{9}
$$

$$m\lg\cos\delta = F\_{\bar{j}}\tag{10}$$

From the above formula, it can be deduced that:

$$a\_{\circ} = \frac{mg(\mu \cos^2 \delta + \sin \delta \cos \delta)}{m\_{\circ} - m + m \cos^2 \delta} \tag{11}$$

$$a\_{\hat{\jmath}} = \frac{g(\mu \cos^2 \delta + \sin \delta \cos \delta)}{K\_M - 1 + \cos^2 \delta} \tag{12}$$

The conditions that must be met to make the cabbages move upwards without falling are: *α<sup>r</sup>* > 0. It can be derived from Formulas (8) and (12):

$$\frac{m g (\mu \cos^2 \delta + \sin \delta \cos \delta)}{m\_{\circ} - m + m \cos^2 \delta} > g \frac{\sin \delta}{\sin \delta} \tag{13}$$

$$\mu > \frac{\tan \delta (K\_M - 1 + \cos^2 \delta)}{\cos^2 \delta} - \tan \delta \tag{14}$$

where *μ* is the friction coefficient of cabbage; *Fm* is the overall force of the pulling roller on the cabbage, N; *α<sup>r</sup>* is the relative acceleration of the motion of the cabbage, in m/s2; *α<sup>e</sup>* is the acceleration of the conveying system, in m/s2; *δ* is the angle between the cone on the pulling roller and the horizontal line, (◦); *Fj* is the pull-out force of the pull-out roller on cabbage, N; *α<sup>j</sup>* is the absolute acceleration of the conveying system, in m/s2; *mj* is the weight of a single pulling roller, in kg; *Fn* is the friction force between cabbage and the pulling roller, N; and *Km* is the mass ratio of the pulling roller with cabbage.

To ensure that the cabbages do not fall during harvest, it is necessary to meet Formula (14). Therefore, the material properties should be considered when designing the pulling roller. It can be seen from Formula (12) that the relative motion between the pulling roller and the cabbage has a great influence on transportation, so the rotation speed of the pulling roller plays a vital role in the qualified rate of harvesting.

As shown in Figure 9, the force analysis of cabbage in the pulling process is carried out.

$$f\_1 \cos \alpha = F\_{N1} \sin \alpha \tag{15}$$

$$f\_1 \sin \alpha + F\_{N1} \cos \alpha = mg \tag{16}$$

$$f\_1 = \mu F\_{\text{N1}} \tag{17}$$

where *f* <sup>1</sup> is the resistance and friction of the cabbage in the pulling process, N; *FN*<sup>1</sup> is the supporting force of the working surface of the pulling roller on the cabbage, N; and *α* is the pulling angle of the pulling roller, (◦).

When the cabbage is pulled out, it is subjected to resistance, friction, and support. After overcoming gravity, the pull-out force of cabbage is:

$$F = f\_1 \sin \alpha + F\_{N1} \cos \alpha - mg \tag{18}$$

It can be derived from Formulas (15)–(17):

$$F = \frac{mg}{\tan \alpha \sin \alpha} (\mu \sin \alpha + \cos \alpha) \tag{19}$$

The weight of cabbage is determined by Table 1 at 1.91 kg. According to Formula (19), the pulling force *F* of the pulling roller on cabbage depends on the pulling angle α. Therefore, a better pulling effect can be obtained by selecting the appropriate pulling angle. The material of the pulling roller designed in this paper is stainless steel, the friction coefficient is 0.3, the distance between the inside of the pulling roller is 50–80 mm, the diameter of the end is 120 mm, the total length of the conical pulling roller is 450 mm, and the angle between the pulling roller (taper) and the horizontal angle is 13.9◦. At this time, the pulling force of the two pull-out rollers on cabbage is:

$$2F = \frac{1.91 \times 9.8}{\tan 13.9^\circ \sin 13.9^\circ} \times (0.3 \sin 13.9^\circ + \cos 13.9^\circ) = 677 \tag{20}$$

At this time, the theoretical pulling force of the pulling roller designed in this paper is greater than the pulling force measured in Table 2, which meets the design requirements.

**Figure 9.** Pulling force analysis.

### *4.3. Design of Clamping Conveying Device*

As shown in Figure 9, the low-loss harvesting test platform for cabbage designed in this paper adopts a new mechanism and new method called "vertical clamping + flexible conveying". By using the flexible feeding and flexible clamping methods, we can improve the adaptability of different ball diameters of cabbage and realize low-loss transportation [22,23].

The analysis of the movement process of the cabbage in the clamping and conveying device is shown in Figure 10, where 1 is the cabbage feeding link, 2 is the clamping and conveying link, and 3 is the harvesting of the finished product link.

**Figure 10.** Clamping and conveying device: (**a**) 1: driving motor; 2: CR flexible sponge conveyor belt; 3: tensioner pulley; 4: tightening spring; 5: conveyor belt drive wheel. (**b**) Clamping conveying device physical diagram n. (**c**) High-density CR flexible sponge.

The clamping conveying process of cabbage is shown in Figure 11.

**Figure 11.** Clamping conveying process of cabbage.

The motion analysis and force analysis of the feeding link of the cabbage are shown in Figure 12a,b.

**Figure 12.** Clamping conveying process of cabbage. (**a**) Force analysis of the feeding link. (**b**) Motion analysis of the feeding link.

If the cabbages are not blocked in the feeding link and smoothly enter the clamping and conveying link, the following formula should be satisfied:

$$
\tan \alpha \le \frac{F\_T}{F\_N} = \mu \tag{21}
$$

$$V\_1 \sin \beta > V\_0 \tag{22}$$

According to the above formula, it can be calculated by the following formula:

$$\mu \ge \sqrt{\frac{(D+d)^2 - (D-l)^2}{\left(D+l\right)^2}}\tag{23}$$

where *α* is the angle between pressure and the horizontal line of cabbage; *V*<sup>1</sup> is the linear velocity of the conveyor belt, m/s; *β* is the lifting angle of the conveyor belt, (◦); *V*<sup>0</sup> is the operating speed of the conveying system, m/s; *FT* is the friction force on cabbage, N; *FN* is the pressure of the conveyor belt on cabbages, N; *μ* is the friction coefficient between the conveyor belt and cabbages; *D* is the diameter of the conveyor belt drive wheel, mm; *d* is the diameter of cabbage, mm; and *l* is the distance between the two conveyor belt drive wheels, mm.

The feeding inlet clamping position of the conveyor belt should be at the waist of the cabbage. In this paper, the single weight of "Chun xi" cabbage was 1.2–1.5 kg, the bulb was 180–200 mm, the feeding inlet spacing of the clamping conveying mechanism, which can be adjusted by spring and the minimum spacing, was 120 mm, and the diameter of the conveyor belt drive wheel was 110 mm. Therefore, the maximum value of the friction coefficient between the conveyor belt and the cabbage can be calculated as follows:

$$
\mu \ge \sqrt{\frac{\left(110 + 200\right)^2 - \left(110 + 120\right)^2}{\left(110 + 120\right)^2}} = 0.9\tag{24}
$$

The maximum extrusion force of cabbage is:

$$F\_{Nmax} = \frac{G}{\mu} \times 2 = 130.67\tag{25}$$

According to the test of the mechanical harvesting characteristics of the cabbage, the extrusion force is far less than the maximum extrusion crushing force of 1198.4 N. Based on the compressional force test results, the clamping and conveying mechanical structure designed in this paper can avoid the results that the cabbage cannot be clamped due to too small extrusion pressure, and if the extrusion pressure is continuously increased, the cabbage may be blocked, so the structural parameters of the clamping and conveying device are designed reasonably.

By analyzing the force on the base surface of the conveying interval and the deformation of the cabbage [24], the deformation analysis in the extrusion deformation link is shown in Figure 13.

**Figure 13.** Deformation analysis of cabbage.

It can be seen from Figure 12 that the extrusion force *F* = *F*<sup>1</sup> at point B and the forces *FA* and *FC* at points *A* and *C* are:

$$F\_A = F\_\mathbb{C} = \frac{FL\_1}{L} \tag{26}$$

The reaction force of point *B* can be obtained from the equilibrium equation:

$$F\_B = \frac{F L\_2}{L} \tag{27}$$

The deformation bending moment of cabbage is:

$$M\_{\mathbf{x}} = \frac{FL\_2}{L}\mathbf{x} - F(\mathbf{x} - L\_1) \quad \text{ ( $L\_1 \le x \le L$ )}\tag{28}$$

The deformation-bending moment is the same on both sides, and the integral on one side can be obtained:

$$Ef\frac{d^2y}{dx^2} = \frac{FL\_2}{L}x\tag{29}$$

The deflection curve equation is:

$$y = \frac{F\_4 L\_2 \text{x}}{6LE} = \text{x}^2 - L\_2^2 + L^2 \tag{30}$$

Because *x* = *L*<sup>1</sup> = *L*2, the clamping deformation Δ*y* of cabbage is:

$$
\Delta y = \frac{F\_2 L\_2 L\_1 L}{6EJ} \tag{31}
$$

where *E* is the modulus of elasticity; *J* is the momentum of inertia; *L* is the length of *AC*, mm; *L*<sup>1</sup> is the length of *AB*, mm; and *L*<sup>2</sup> is the length of *BC*, mm.

According to Formula (15), when the conveyor belt speed is too low, the conveying efficiency will be reduced, which makes it easy to cause conveying blockage, resulting in unsmooth subsequent operations and incomplete root cutting. Through the preliminary test and observation, the main forms of damage in the transportation of cabbages are friction, extrusion, collision, and other forms of damage. The reason is that the deformation of cabbage is too large under the action of rigid parts in the conveying process. If the deformation deflection Δ*y* of cabbage is too large, it is easy to squeeze and break. In Formula (25), the deformation deflection of the cabbage is determined by the elastic modulus. In order to reduce the damage as much as possible, the high-density CR flexible sponge belt is selected

to wrap the cabbage for clamping and conveying. While reducing the relative friction, the tensioning mechanism transfers part of the deformation of the cabbage to the flexible clamping conveyor belt. It has a certain anti-deformation effect on the cabbages and can ensure that the cabbages do not slide when they are clamped by the conveyor belt and will not cause ineffective root cutting of the cabbage due to sliding.

In order to adapt to different kinds of cabbage and improve the adaptability of harvesting equipment. The maximum center spacing of the conveyor belt designed in this paper can be adjusted to 280 mm, and the feeding inlet spacing can be adjusted in the range of 200–250 mm. Ensure that different varieties of cabbage can be successfully clamped and transported, even if the ball diameter is different.
