*2.3. Cotton Stalk Pulling Process: Analysis of Toothed Roller Motion Trajectory*

During operation, the V-shaped toothed roller, i.e., the main functioning component, makes forward and circular motions. As shown in Figure 4, the process of stalk pulling can be divided into the following four stages according to the motion trajectory: the clamping stage, pulling stage, delivering stage, and detaching stage. The analysis of tooth roller trajectory will provide a theoretical reference of the parameters design in the subsequent stalk removal mechanism. The motion trajectory formula for a certain point on the toothed plate is as follows:

$$\begin{cases} \begin{array}{c} \begin{array}{c} x = v\_q t + \frac{R}{1000} \sin\left(\frac{\pi nt}{30}\right) \\ y = \frac{R+h}{1000} + \frac{R}{1000} \cos\left(\frac{\pi nt}{30}\right) \end{array} \end{cases} \end{cases} \tag{6}$$

In the formula,

*v*q—speed of forward motion (m/s);

*R*—radius of gyration of V-shaped tooth roller (mm);

*h*—height of central axis of V-shaped tooth bar above the ground (mm);

*n*—rotation speed of V-shaped tooth roller (rad/min);

*t*—time (s).

**Figure 4.** Motion trajectory sketch.

The simulation curves at different speeds and comparison chart of stalk pulling trajectory are exhibited in Figures 5 and 6. It can be clearly seen that the stalk-pulling trajectory is different at different speeds. Assuming that the cotton stalk is contacted and clamped at M, the machine advanced a distance of S and reaches N. It can be obtained from Figure 6 that for the cotton stalk with different clamping heights, as the speed increases, the stalk pulling distance can be increased faster.

**Figure 5.** Simulation curves at different speeds.

**Figure 6.** Comparison chart of stalk pulling trajectory.

2.3.1. Cotton Stalk Pulling Process: Collision Analysis

The mechanical changes that occur as the V-shaped toothed roller extracts stalks are very complex; these complex changes occur because of the interaction between the cotton stalk and the toothed plate and between the cotton stalk and soil. Thus, in this study, a simplified mechanical model was established by performing static analysis using data corresponding to a certain moment during the pulling process [21].

The type of collision between the rotating toothed plate and each cotton stalk was assumed to be elastic under ideal conditions [25], meaning that the mechanical properties of each cotton stalk that were temporarily altered as a result of deformation can be recovered without heating, sound, or kinetic energy loss. As such, the deformation of the cotton stalk was set to have a deformation stage and recovery stage. As shown in Figure 7, the force applied by the toothed plate was converted into deformation energy in the deformation stage; in the recovery stage, this deformation energy was defined as the bilateral forces applied to the cotton stalk by the toothed plate, i.e., *F*h1 and *F*h2. Under the action of *F*h1 and *F*h2, the cotton stalk and toothed plate moved relative to each other, generating frictional forces, *F*<sup>1</sup> and *F*<sup>2</sup> , between them along the direction of the axis, as well as the corresponding resultant force, *F*12.

**Figure 7.** Sketch of the forces applied to a cotton stalk by the toothed plate.

2.3.2. Cotton Stalk Pulling Process: Mechanical Analysis of Cotton Stalk Push and Pull Forces

The toothed plate was designed to clamp around the cotton stalk after the initial collision to enable extraction by applying push and pull forces that can overcome the soil resistance. This stage included the following two processes: the clamping process and the push and pull process. The forces considered in the clamping process stress analysis for a single cotton stalk are illustrated in Figure 8. In general, the toothed plate initially makes contact with the phloem of the cotton stalk before extruding it. Then, the phloem applies bilateral forces, i.e., *F*h1 and *F*h2, on the toothed plate during the deformation recovery stage. As the machine moves forward, the toothed plate pushes the cotton stalk forward. During this time, the cotton stalk xylem begins undergoing flexible deformation and the push and pull process is initiated. The bilateral forces applied to the toothed roller by the xylem were set as *F*h3 and *F*h4.

**Figure 8.** Sketches of cross section and longitudinal plane of cotton stalk clamped by the toothed plate: (**a**) schematic diagram of cross section under clamping state and (**b**) schematic diagram of longitudinal section in a clamped state.

During the push and pull process, the toothed plate continuously exerts force on the cotton stalk, consequently significantly deforming the phloem and creating the expanded zone A. At this moment, the cotton stalk is in an extrusion state, as shown in Figure 9. Thus, the pulling force applied to a single cotton stalk is the resultant force of (1) the upward force *F*<sup>s</sup> exerted on the cotton stalk by the toothed plate, which is perpendicular to the surface of the toothed plate; and (2) the bilateral frictional forces *F*<sup>12</sup> and *F*<sup>34</sup> applied to the cotton stalk by the clamping tooth, which are directed upward along the length of the cotton stalk.

**Figure 9.** Sketch of longitudinal plane of cotton stalk extruded by the toothed plate.

2.3.3. Cotton Stalk Pulling Process: Analysis of the Bilateral Forces Acting on the Clamping Tooth

Dry friction can be defined as the force that acts to resist the relative motion of two solid objects. In the proposed system, there are two points at which friction occurs between the clamping teeth and the cotton stalk. One is the point at which the bilateral forces *F*h1 and *F*h2 are applied to the clamping tooth by the cotton stalk during the deformation recovery stage. The other is the point at which the cotton stalk undergoing flexible deformation applies bilateral forces to the clamping tooth because of being pushed and pulled. These bilateral forces have been defined as *F*h3 and *F*h4. The respective interactions between the cotton stalk and soil and the toothed plate and reel could abstract the cotton stalk into a simple beam, wherein the force-exerting points correspond to the contact points between the cotton stalk and the toothed plate. Thus, assuming that the contact points between the cotton stalk and soil, reel wheel and cotton stalk, and toothed plate and cotton stalk are A, B, and C, respectively, the force exerted by the toothed plate on the cotton stalk is P. A schematic showing the application of the simple beam theory to illustrate these forces and corresponding contact points is presented in Figure 10. It can be seen that within a certain range, the greater the impact intensity, the greater the deformation of the cotton stalk, and the greater the Fh1 and Fh2 produced by the elastic recovery force of the cotton stalk.

**Figure 10.** Schematic illustrating simple beam theory application.

According to the stress analysis in Figure 8 and the theory of simple beam, the flexible deformation Δ*y* of cotton stalk was calculated as follows:

$$
\Delta y = \frac{Pab}{6lE}(a^2 + b^2 - l^2) \tag{7}
$$

Through the transformation of the above formula, the force *P* of the tooth plate exerted on cotton stalk was obtained as:

$$P = \frac{6\Delta ylEI}{ab(a^2 + b^2 - l^2)}\tag{8}$$

In the formula,

Δ*y*—the flexible deformation (mm);

*E*—elastic modulus (MPa);

*<sup>J</sup>*—moment of inertia (kg·m2);

*l*—contact point height of reel wheel (mm);

*a*—ground clearance of the contact point between tooth plate and cotton stalk (mm);

*b*—space of contact points among reel wheel, tooth plate, and cotton stalk (mm);

*P*—the resultant force on the cotton stalk exerted by the tooth plate (N).

According to the parallelogram rule of force, the forces *F*h3 and *F*h4 of the resultant force *P* on the sides of the tooth plate can be deduced as shown in Formula (9):

$$F\_{\rm h3} = F\_{\rm h4} = \frac{P \sin(\frac{\pi}{2} - \frac{\theta}{2}) \sin \theta}{\sin \theta} = \frac{p \cos \frac{\theta}{2}}{2 \sin \frac{\theta}{2} \cos \frac{\theta}{2}} = \frac{p}{2 \sin \frac{\theta}{2}}\tag{9}$$

Under the action of *P*, the maximum friction exerted by the tooth plate on the cotton stalk was 2*F*h3, which was along the axis of the cotton stalk. To sum up, the friction *F*<sup>t</sup> between the bilateral sides of the cogging and the cotton stalk was shown in Formula (10):

$$F\_{\rm t} = (F\_{\rm 12} + F\_{\rm 34})f = 2(F\_{\rm h1} + F\_{\rm h3})f = 2F\_{\rm h1}f + \frac{P}{\sin\frac{\theta}{2}}f \tag{10}$$

In the formula,

*F*t—friction between the bilateral sides of the cogging and the cotton stalk, N;

*θ*—angel of cogging, ◦;

*f*—static friction coefficient between the cotton stalk and tooth plate;

*F*h3—force exerted by cotton stalk under deformation on bilateral sides of cogging, N;

*F*h4—force exerted by cotton stalk under deformation on bilateral sides of cogging, N.

From Formulas (7)–(10), it can be concluded that under a certain value of Δ*y*, the smaller the value of *a* is, the larger the value of *P* is. In the case of the same deformation of the cotton stalk, the lower the height of the clamped cotton stalk, the greater the force between the tooth plate and the cotton stalk, and the greater the friction that can be provided.
