The Mathematical Model of Seed Movement on a Concave Profile Rib

The differential equations of movement along the concave profile of the grate, consisting of three broken lines, are integrated on Maple 9.5 under initial conditions, using separate functions, and graphs of the dependence of movement and speed over time are presented. The graphs show the patterns of change in displacement and speed at different angles, friction coefficient of seeds along grate with a broken line of a concave profile.


Introduction
Currently, the quality of fiber obtained at cotton processing enterprises depends on the efficient operation of machines operating in the direct technological process. Each process is to some extent important in obtaining high-quality fiber. The main technological process in the production of fiber at the enterprise is the process of ginning (separation of fiber from seed). The cotton, peeled from small and large litter in the cleaning workshops, is fed to the main machine of the genie shop with saw gin. The cotton enters the working chamber of the gin and is hooked with the teeth of the saws rotating next to the seed comb, leading to the grate. In the working chamber, the cotton flies hooked with the teeth saw, hook the other cotton flies and create a raw roller. The seed roll rotates in the opposite direction of the saw cylinder, and provides a continuous supply of cotton fiber to the teeth of the saws. Gin problems were previously studied by many researchers [1]- [7].
There are also known attempts by the authors of mathematical modeling of various stages of the technology of primary processing of cotton, which are published both in local and foreign scientific journals [8] [9] [10] [11] [12].
The authors of the article conducted a series of studies to improve the working elements of gin. The aim of the study is to create the possibility of timely exit of bare seeds from the working chamber of the saw gin by creating grooves in the grates, creating a device that performs this process, determining the technological dimensions that ensure efficient operation, as well as introducing the device into production.
The selection of the optimal structural and technological parameters of the new grate is a crucial stage of research. The use of mathematical methods in research planning, in contrast to traditional methods of research calculations, makes it possible to determine the influence of each factor together influencing several parameters on optimization parameters. As a result of this, a mathematical model of the studied object will be obtained with a relatively small multiplicity of tests. This model is also used for decision making.
Investigation of the movement of single and systemic seeds along grates with a concave profile.
1) The movement of a single seed.
To describe the movement of the seed, it is necessary to determine the concavity of the grate groove. Typically, this concavity profile consists of a curve, the shape of which should provide the minimum resistance force and the process of maximum separation of the fiber from the seed when the seeds move. Suppose the equation of the concavity profile parameter, and the shape is expressed by a curved line in the plane (Figure 1).
In this case, its curvilinear motion will be expressed by the following equations: Consider the equation in the polar coordinate system. We consider the equation of the curve of the line in the coordinate of the pole systems. The law of motion of the seed is written as follows: ψ -the angle formed by the tangent to the curve of the line with the axis ox, this angle is expressed as follows: For the radius of curvature we obtain the following expression ( ) In Equation (2), the letter indicates the sum of the tangential forces affecting the seed. In this direction, the projection of gravity and the Coulomb friction force in the direction, i.e.
In this equation using equality 2 We can write the differential equation of motion ( ) S S t = of the path covered by the seed for each section as follows:   In Equation (12), taking into account the equation 1 1 S l = , we obtain the equation with respect to 1 t Here we define As a result, we obtain the expression of speed 1 v Similarly, we obtain the following solutions for the second and third sections ( ) ( ) Points 11 x , 11 y lie on an arc, therefore ( ) ( ) In addition, the slope of the line AB ( ) ( ) If the angle coefficient 1 K is given, the coordinates 11 x and 11 y are also determined in Equations (18) and (20).
The coordinates 12 x and 12 y ( ) ( ) We write the equation of motion of the seed for each plot (   The analysis showed the feasibility of using ribs with a concave profile, which helps to accelerate the release of seeds from the working chamber of the fiber separator, which significantly increases the productivity of the cotton processing.

Conclusions
1) An increase in the coefficient of friction leads to a decrease in the movement and speed of the seed in time.
2) You can see that in this section the trajectory and speed of the seeds depend on the friction coefficient, that is, with an increase in the friction coefficient, the