Justification of design parameters of a disk working body with a changing radius of curvature

In recent years, the improvement of the technological process of soil treatment is based on the use of working bodies with a curved shape of the working surface, which can increase the agro-technical and reduce the energy-intensive parameters of soil treatment. Agricultural units with disk working bodies are one of the most common machines in agriculture. The article presents a design scheme of a disk working organ with a working surface made along the Archimedes ’ spiral. Expressions are obtained for designing the working surface of the disk with the specified initial parameters.


Relevance of research
Most often for post-harvest surface or main soil treatment in all countries of the world, disk tillage units are used, the working bodies of which are steel spherical, conical or straight disks with a cut-out or solid pointed cutting edge. Such a wide distribution of disk units is characterized by their versatility, since one pass is performed pruning and crumbling of soil and weeds, shredding crop residues and mixing them. In addition, at the end of the XX century, disk units usually had a battery arrangement of working bodies, which often led to a decrease in their technological reliability. At the present time, most of the disk units are represented by a row-by-row arrangement of working bodies, which allows them to work without clogging in severe soil and climatic conditions, including with a large number of crop residues and high soil humidity [1].
The expediency of improving any soil treatment technology or a single technological operation is justified not only by economic, but also by agronomic aspects, which generally lead to an increase in the profitability of crop production [2][3][4].

Problem statement
In recent years, the improvement of the technological process of soil treatment is increasingly based on the use of various curved forms of working bodies, which allows to achieve a reduction in the proportion of compression deformations and increase in shear and torsion deformations, as well as to redistribute the energy supplied by the working body to the soil layer, and thereby reduce its dynamic resistance and non-productive energy consumption. From the agronomic point of view, such working bodies allow better crumbling and mixing of the processed layer [5][6][7][8].
One of the most promising surfaces of working bodies are surfaces with a changing radius of curvature, which allow to concentrate the pressure exerted by the working body inside the formation, which will contribute to its crumbling [9,10].
With this in mind, we proposed the design of a disk working body (Fig. 1), the working surface of which is made with a variable radius of curvature, and the curvature of the working surface is made along the Archimedes' spiral with a greater value at the cutting edge [11]. Using the Archimedes' spiral as a forming spiral will allow you to concentrate the stresses arising in the formation when it moves along the working surface of the disk, and also ensures their smooth increase and it reduces the resulting dynamic resistance due to changes in speed. However, the proposed design scheme is not sufficient for the design of the working body, since there will be countless options for constructing the working surface according to the specified initial parameters. Thus, it is necessary to conduct an analytical study of the design parameters of a disk working body with a changing radius of curvature, which will allow the design to obtain the required working surfaces [12].

Presentation of main material of research
The Archimedes' spiral is a flat curve that is formed when a point moves uniformly along the ray with its origin at the center of the coordinates and the ray moves uniformly around its center at the same time. The Archimedes' spiral in the polar coordinate system is described by the expression [13]: where ρ -distance from the center of coordinates to the present point, mm; k -value of displacement of an arbitrary point along the beam when the beam is rotated at an angle of one radian, mm/rad; φ -angle of rotation of the beam, rad; a -step of the spiral, mm.
Let's start the justification of the design parameters of the disk working body with the choice of parameters of the Archimedes' spiral, which are necessary for the formation of the working surface of the disk. To do this, initially take the coordinate system as shown in figure 2. At the same time, we will arrange the plane of the disk blade so that the point of the blade F touches the plane XO, and the axis of rotation of the disk coincides with the axis OY, and the disk blade will have a radius equal to R. To determine the parameters of the spiral, take a section of the disk surface with a plane perpendicular to the plane of the disk blade and passing through the point F (Fig. 2).

Fig. 2. To determination of the spiral's parameters
As can be seen from figure 2, the disk forming area is bounded by the maximum and minimum radius of curvature. We should set the angle ψ4 between the OX axis and the segment AF to fix the minimum radius of curvature on the axis of the disk's rotation. Then the Archimedes's spiral equation in general form in the specified coordinate system will be: cos cos sin sin , cos sin sin cos .
x k k a y k k b ϕ ϕ γ ϕ ϕ γ ϕ ϕ γ ϕ ϕ γ where x and y -present values of coordinates, mm; γ -angle of the spiral's rotation to touch the specified section of the spiral at points A and F, rad; a and b -accordingly the replacement of the starting point of the spiral along the OX and OY axes, or the coordinates of the starting point of the spiral, mm To determine the structural parameters of disk working body we need to determine the replacement of the starting point of a spiral a along the axis OX and b on OY axis, the angles ψ1 and ψ2 between the OX axis and the maximum ρmax and the minimum ρmin radii of curvature, and the value k of the displacement of an arbitrary point on the beam while it is rotated at an angle in one radian.
Parametric expression (2.2) of the Archimedes's spiral of in the specified coordinate system should be used to address specific types of research tasks and we can only define the replacement value of the beginning of helix a and b and the replacement value k at this stage. In addition, since the maximum and minimum radii of curvature when forming the working surface of the disk working body will rotate at the same angle γ, its definition can also be ignored at this stage. Figure 2 shows that: where ρmax and ρmin -maximum and minimum radii of curvature, mm, respectively; ψ1 and ψ2 -value of the sharp angle between the OX axis and the maximum and minimum radius of curvature, respectively.
The angles ψ1 and ψ2 will be equal: where φmax and φmin -angles of the turn of the beam forming the spiral to form maximum and minimum radii of curvature, respectively, rad; n -number of turns of the helix to form the required radius of the curvature at a specific value of the displacement k of an arbitrary point along the ray when it is rotated; ψ3 -angle between the maximum and minimum radii of curvature, rad. The angle ψ3 between the maximum and minimum radii of curvature forms the radius of the disk working body and depends on the angle ψ4 between the OX axis and the segment AF. The value of the angle ψ3 is determined from the expression: At the same time 4 , cos where R -radius of the disk working body, mm. We can get the conversion after substituting the expression 7 with the expression 6: The expression 10 allows us to determine the necessary value of displacement k of an arbitrary point along the beam at its rotation to form the working surface of a disk working body with radius R and a fixed location of its maximum and minimum radius of curvature. Understanding the maximum ρmax and minimum ρmin curvature radii and the value of displacement k of an arbitrary point along the ray when it is rotated, using the expression 1, it is easy to find the required angles of rotation of the ray φmax, φmin and the number of turns of the spiral, which must be rounded to a larger integer. The graph of the replacement value k dependence is an arbitrary point on the beam while it is rotated from the range of variation of the radius of curvature Δρ, the angle ψ4 between the horizontal and a line connecting the minimum and maximum radii of curvature and the radius of the disk are presented in figure 3. Figure 3 shows that an increase in the range of variation of the radius of curvature Δρ leads to a linear increase in the value of the displacement k of an arbitrary point along the ray when it is rotated, while a smaller value of the range corresponds to a smaller value k. The character of the change in the value of the displacement k of an arbitrary point along the ray when the angle ψ4 changes between the horizontal and the segment connecting the minimum and maximum radii of curvature as well as the radius R of the disk is curved. In this case the increase of the radius R of the disk and the angle ψ4 the replacement value k of an arbitrary point on the beam decreases, however the intensity of decrease of the replacement k of an arbitrary point on the beam by increasing the radius R of the disk is much higher than the increase in the angle ψ4.
where ψ5 -angle between the segment AF and the maximum radius of curvature pmax . Then after substituting expression 7 into expression 11 after converting the conversion, we get: Taking into account the expression 12, the angle ψ1 is equal to the sum of the angles ψ4 and ψ5 as lying crosswise will be the following: To find the angle of rotation of the spiral γ for its connection at points A and F we should use the expression: The value of the segment BC will be equal to the replacement value of the spiral along the OX axis, and the segment CD will be equal to:

Conclusions
The obtained expressions 3, 4, 5, 10, and 13 allow us to design the working surface of a disk working body with a varying radius of curvature at the known maximum ρmax and minimum ρmin of curvature radii, the radius of the disk R, and the angle ψ4. Expression 16 allows you to determine the angle of rotation of the Archimedean spiral relative to its start with coordinates [a;b] to combine ρmax with the blade and ρmin with the center of rotation of the disk working body.