2.3.4. Cotton Stalk Pulling Process: Establishment of Mechanical Model

Generally, the forces exerted on the cotton stalk root by the soil act to resist the stalk-pulling process; this force system is quite complex. Therefore, for the purpose of simplification, the soil force system was modeled to have a resultant force, which is referred to as the cotton stalk soil resistance and denoted as *F*b. *F*<sup>b</sup> was determined to occur along the length of the cotton stalk.

As the cotton stalk is being pulled, its initial state of static equilibrium is disrupted by sudden movement, which involves an acceleration component. This acceleration was set as a1, the force producing the acceleration, which is equivalent to the resultant force *P*, was set as *F*a, the frictional force was expressed as Ft, the extrusion force was set as *F*s, and the soil resistance was set as *F*b. Additionally, the angle between the force vector *F*<sup>a</sup>

and the surface of the toothed plate was set as *γ*. A schematic illustrating the force system associated with this state of transition is presented in Figure 11.

**Figure 11.** Schematic of forces acting on a cotton stalk transitioning from its initial static equilibrium to a dynamic state.

From the stress analysis of the cotton stalk in Figure 11, Formulas (11) and (12) can be obtained according to the dynamic knowledge such as D'Alembert's principle while taking the acceleration into consideration:

$$F\_\mathbf{a} \sin \gamma + F\_\mathbf{b} = F\_\mathbf{s} \cos \alpha + F\_\mathbf{t} \tag{11}$$

$$F\_\mathbf{a} = a\_1 m = \frac{v\_2 - v\_1}{t\_1} m = \frac{v\_2}{t\_1} m \tag{12}$$

By (10) and (11), (13) can be obtained:

$$
tau\_1 \sin \gamma + F\_\mathbf{b} = F\_\mathbf{s} \cos \alpha + 2F\_\mathbf{h1} + \frac{P}{\sin \frac{\theta}{2}} f \tag{13}
$$

By (8), (12)–(14) can be obtained:

$$m\frac{v\_2}{t\_1}\sin\gamma + F\_\mathbf{b} = F\_\mathbf{s}\cos\mathfrak{a} + 2F\_\mathbf{h1}f + \frac{6\Delta yfelf}{ab(a^2 + b^2 - l^2)\sin\frac{\theta}{2}}\tag{14}$$

In the formula,

*a*1—the acceleration (m/s2) (in actual conditions, the acceleration of cotton stalk a1 is a variable, which is related to the position and angle of the cotton stalk clamping. It is simplified for analysis here);

*m*—weight of cotton stalk (kg);

*v*2—speed of the cotton stalk when being pulled out (m/s);

*v*1—speed of cotton stalk at the static state (m/s, set as 0 m/s here);

*t*1—time for the pull-out of the cotton stalk (s);

*F*a—force-producing acceleration, equal to the combined force of the tooth plate to the cotton stalk and the soil to the cotton stalk (N).

Formula (14) was used to model and analyze the extraction, missed extraction, and fracture states of the cotton stalk pulling process. The following conclusions were made:

*F*h1 generation is mainly dependent on the collision strength between the toothed plate and cotton stalk; more specifically, the elastic recovery of the cotton stalk is dependent on the speed of the collision. Within a certain range, a higher speed corresponds to larger deformation of the cotton stalk. Additionally, the forces associated with deformation recovery are strong and serve to facilitate the cotton stalk clamping process. Upon making contact, when the effects of the collision between the toothed plate and cotton stalk cannot be endured by the cotton stalk, phloem rupture, a missed extraction, or even cotton stalk fracture can occur, as shown in Figure 12c. Under these conditions, the cotton stalk is broken, and the root remains in the soil. Thus, the rotational speed of the V-shaped roller should not be too low or too high.

**Figure 12.** Conditions of the cotton stalk epidermis following a stalk extraction attempt. (**a**) Example of the epidermis following successful cotton stalk extraction. (**b**) Example of the epidermis of a cotton stalk that failed to be extracted. (**c**) Example of the epidermis of broken cotton stalks.

*F*s is the upward force of the toothed plate that acts against the cotton stalk. The effects of *F*<sup>s</sup> are mainly concentrated on the phloem of the cotton stalk. Thus, the magnitude of *F*<sup>s</sup> is primarily dependent on the characteristics of the cotton stalk phloem; this may be a significant reason for considerable differences in the stalk-pulling effects between autumn and spring.

Conclusions <sup>1</sup> and <sup>2</sup> above could be used to explain the phenomenon depicted in Figure 10. Figure 12a shows an example of the state of the cotton stalk epidermis at the V-shaped toothed plate clamping position after a cotton stalk was successfully extracted in autumn. The epidermis was obviously ruptured; however, in terms of length, the rupture was not remarkably substantial. (Most ruptures were less than 25 mm in length according to the results of rough statistics.) Figure 12b shows the state of the epidermis of a cotton stalk that was unsuccessfully extracted. It can be seen that a large area of the epidermis was destroyed, and some side branches were broken. This state is referred to as the missed extraction state.

In Formula (14), *a* represents the ground clearance of the contact point between the toothed plate and cotton stalk. Theoretically, a lower value of *a* should correspond to a better cotton stalk-pulling outcome. *θ* is the angle between the toothed plates. When *P* is constant, a lower value of *θ* corresponds to better gripping force.

Following the analysis of these results, successful cotton stalk extraction has been determined to be associated with a relative displacement between the V-shaped toothed plate and cotton stalk that does not exceed 25 mm. This was determined based on whether the extrusion force *F*s exceeded the endurable limit of the cotton stalk. Thus, the extrusion force *F*<sup>s</sup> should be maintained at an appropriate value during the stalk-pulling process. Furthermore, according to Formula (13), under the condition that the soil resistance *F*<sup>b</sup> is constant, the cotton stalk pulling outcome can be improved by reducing the speed *v*2, the stalk-pulling height a, or the cogging angle *θ*.

$$m\frac{\upsilon\_2\downarrow}{t\_1}\sin\gamma + F\_{\rm b} = F\_{\rm s}\cos\mathfrak{a}\downarrow + 2F\_{\rm h1}f\uparrow + \frac{6\Delta yfelf}{ab(a^2\downarrow + b^2 - l^2)\sin\frac{\theta\downarrow}{2}}\uparrow\tag{15}$$

The left side of Formula (15) describes the forces of the soil acting on the cotton stalk root, as well as the force associated with cotton stalk acceleration, which has been set as a passive load. The right side of the equation describes the force exerted by the toothed plate on the cotton stalk; it has been set as the active force. Among all the parameters influencing the active force, the bilateral forces *F*h1 and *F*h2 applied to the clamping tooth by the cotton stalk during the deformation recovery stage of the phloem, the ground clearance *α*, and the cogging angle *θ* are controllable. With the exception of the speed of the toothed roller *n* that influences *F*h1 and *F*h2 that is adjustable, the remaining parameters, such as the friction coefficient *f*, elastic modulus *E*, cotton stalk height *l*, and extrusion force *F*s cannot be controlled. Thus, in this study, the speed of the toothed roller *n*, ground clearance *α*, and cogging angle *θ* were determined to be test factors.
