Study of the movement of semi-finished products between working shafts

. The article examines the movement of semi-finished products in the field of covering with rotating working shafts. As a result of the research, it was determined that in order to improve the quality of processed ore semi-finished products, it is necessary to increase the coverage angle and reduce the radius of the working shafts accordingly. In order to maintain the natural quality of the semi-finished ore product being processed, it is known and justified that if the value of the external pressure force is increased, the radii of the working shafts should also be increased accordingly.


Introduction
The contact curves of the lower and upper shafts are modelled taking into account the deformation properties of the leather.They have been found to represent all the special interactions of leather with a pair of shafts of shaft compression machines [1].
Using the lower (upper) shaft contact curve and friction stress models, a mathematical model of distribution of contact stresses along the lower (upper) shaft curve was obtained.These models have been found to represent the contact stress profiles of all specific interaction states of the shaft compression machines with a pair of shafts [2].
In [3], the mechanical behaviour of three types of hard wood impregnated with various swelling fluids, compressed at a high strain rate, was studied.Variations in modulus of elasticity, proportional limit, and maximum stress were studied depending on the type of treatment.It was found that saturated samples could be as hard as dry ones.This result was explained by the behaviour of the fluid present in the large cavities of the wood, i.e. in the pores; the results differ from the ones observed at low strain rates.At high strain rates, fluid cannot flow out of the pores and must behave like a solid, so the structure of the material is strengthened.
The study in [4] is devoted to the research of the mechanical properties of leather.It is noted that at ambient temperature and humidity, most types of leather exhibit mainly elasticity, although delayed elasticity can impart plasticity to leather.The dependence of the stress relaxation time under constant linear strain showed that the stress decreases linearly with time.The stress-strain ratio for the leather specimens stretching was non-linear and fixed even at low strains.
In [5], composite materials made from natural rubber with industrial leather waste were studied.The microstructure, mechanical properties, and rheological properties of these composites were studied.The results showed that leather waste increased tensile strength and hardness with good reproducibility and uniform distribution of residues as fillers.The authors concluded that the investigated composite material could be used as raw material for the manufacture of shoes and other products.
One of the factors influencing the process of fluid squeezing from a semi-finished product is its feeding.The feeding is conducted by a belt conveyor; a significant drawback in the designs of squeezing machines is the draining of the squeezed fluid to the workplace of attendants.In this regard, the authors of [6][7][8][9][10][11][12][13][14][15] proposed devices and performed studies, the results of which make it possible to eliminate these drawbacks.
In [16], experiments were conducted, where a fibrous layer covering the primary cavity was observed during the invasion of a fluid of higher density into a less dense medium and a fluid of lower density into a heavier one.The thickness of the fibrous shell of the primary cavity increases with the growth of the drop rate.
The method for modelling the destruction of river dams was developed in [17].The authors considered two sets of Lagrangian material points that represent soil and pore water layers.
In [18], a method was obtained for modelling flows of a non-viscous fluid with a large deformation of the free surface.To simulate turbulent flows with a free surface, the particle displacement method was introduced into this method.This method was used to simulate extremely large deformation and free surface re-connection.
In [19], the fluid flow in industrial pumps was studied.The results of the study showed that with an increase in the concentration of solids in the pump, its casing was subjected to the greatest wear.
An experimental device was developed in [20] for testing a two-phase flow of solid and liquid phases in a centrifugal pump.The test results showed that the device allows measuring the flows of solid and liquid phases in a centrifugal pump with high reliability.
In [21], a sludge pump of a mining-and-processing plant was studied.The Euler method was used to analyze the slurry flow for various particle concentrations and low-volume flow rates.
The study in [22] describes the wear mechanisms related to various types of materials that affect them.Theoretical and empirical models were given to predict wear.
In reference [23], the properties of leather processed in water were investigated.The tanning process was carried out with natural tannin at different duration of the process.Leather samples were processed by traditional methods in liquids and treated.The results showed that the 2-hour and 4-hour tanning was insufficient in terms of the expected physical properties, however, the 6-hour and 8-hour tanning resulted in leather of good quality.
The authors of article [24] investigated the process of tanning leather without chromium.An environmental impact analysis showed that the developed process resulted in a significant reduction (compared to chrome tanning) in total solids content and improved biodegradability of organic compounds present in wastewater.
The flow along a flat plate was experimentally studied in [25] using different fluids.The results showed that the decrease in surface tension has a stabilizing effect on the flow, damping the capillary waves that would otherwise have to occur.As the slope increases, the fluid film thickness decreases and as the fluid flow rate increases, it increases too.
In [26], the characteristics of regeneration of a liquid desiccant on multi-row vertical cylindrical surfaces were experimentally studied.The method of mechanical surface modification was researched to improve the performance level of the cylinder surface.It was determined that the regeneration rate of the surface of a conventional cylinder exceeds the regeneration rate of the surface of a conventional plate by 50.5%.The authors proposed an empirical correlation to evaluate the efficiency of regenerators with falling films.
An analysis of the studies discussed above showed that in the formulation of the problems of these studies, the flow of fluid against the motion of the plane was not considered.In this paper, for the first time, the problem of fluid flow under its own weight along a moving plane inclined relative to the vertical and horizontal lines is considered.
Based on the works reviewed, we have developed, improved and studied designs and a process for mechanical processing of leather on roller-type machines with vertical transportation [27][28][29][30][31][32][33].

Material and method of research
To derive the equation of motion, we calculate the kinetic energy of the semi-finished product and the shafts.The semi-finished product moves forward, and its kinetic energy is equal to: here m1 -the mass of a piece of the semi-finished product.vc -he velocity of the center of mass of the semi-finished product piece.From figure 1, we can see that the upper working shaft moves parallel to the plane, and their kinetic energy is equal to the following expression: here, m2 -the mass of the upper working shaft, Joz -moment of inertia of the working shaft.ω -working shaft angular velocity.Since the lower working shaft rotates, the kinetic energy is determined as follows: We have the following expression for the coverage angle in the considered coordinate system: here, R -the radius of the distance from the center of the working shaft to the contact area; h0 -initial thickness of the semi-finished product; δ0 -the initial thickness of the coatings of the working shafts.Deriving from the expression (4), it is equal to: Taking into account the above expressions ( 1), ( 2), ( 3) and ( 4), the total kinetic energy was equal to the following expression: Taking into account that equation ( 6) represents the complete kinetic energy, we determine the specific derivatives of the generalized velocity and generalized coordinates:  (10) Using the principle of possible displacements, the generalized force for the system under consideration is determined as follows: here, Mrot -torque, Mres -resistance torque generated during the rotation of the working shafts, Q -compressive force acting on the upper shaft, Psh -the gravity of the upper shaft.
δx and δy displacements in the expression (4) are equal to: Substituting the expression ( 12) into (11), we get the expression for finding the generalized force.(13) For a mechanical system with an ideal, holonomic and non-releasing coupling, we write Lagrange's 2-type equation: Substituting expressions (10) and (12) into equation ( 14), the equation of motion of the half-product and shaft pairs is formed: We write the moment of inertia for a cylindrical shaft: here, R1 -outer radius of cylindrical shaft lining, rinner radius of the cylindrical shaft, m3 -the mass of the elastic coating.We determine the moment of resistance that appears on a pair of working shafts: Given that the thickness of the half-product is small enough, the corresponding angle of coverage is also small, sinα=α and cosα=α can be replaced by the values.
Thus, we have determined the conditions for uniform feeding and capture of leather between the roller pair.

Results
The resulting differential equation ( 15) is calculated in numerical methods and allows to select the kinematic and geometric parameters of the device that affect the process of mining semi-finished products.
In the considered problem, ten and a half products move at a constant speed when they are transferred from the conveyor.Then we create the graphical solution of the resulting motion equation ( 18)     and 0   for the case (Figure 2).It can be seen from the graphic solution that as a result of the study of the movement of the semi-finished product in the area of coverage of the working shafts, the radius of the working shafts decreases accordingly as the angle of coverage increases in order to improve the quality of the processed semi-finished product.

Conclusion
If the value of the external pressure force increases in order to maintain the quality of the semi-finished product being processed, it allows to select the radii of the working shafts in large values.

Fig. 1 .
Fig. 1.The scheme of movement of the semi-finished product between the shaft pairs.