HYDRODYNAMIC MODEL OF TRANSPORT SYSTEM

A hydrodynamic model of production systems with a flow method of organizing production is considered. The basic macroparameters of the state of the production flow line and the relationship between them are determined. The choice of a lot of moment approximation for modelling the production line is justified. It is shown that the conveyor-type flow line is a complex dynamic system with distributed parameters. The boundary value problem is formulated for the longitudinal vibrations of the conveyor belt when the material moves along the transportation route. It is assumed that there is no sliding of material along the conveyor belt, and the deformation that occurs in the conveyor belt is proportional to the applied force (Hooke's elastic deformation model). The significant effect of the uneven distribution of the material along the transportation route on the propagation velocity of dynamic stresses in the conveyor belt is shown. When constructing the boundary and initial conditions, the recommendations of DIN 22101: 2002-08 were used. The mechanism of the occurrence of longitudinal vibrations of the conveyor belt when the material moves along the transportation route is investigated. The main parameters of the model that cause dynamic stresses are determined. It is shown that dynamic stresses are formed as a result of a superposition of stresses in the direct and reflected waves. Analytical expressions are written that make it possible to calculate the magnitude of dynamic stresses in a conveyor belt and determine the conditions for the occurrence of destruction of the conveyor belt. The characteristic phases of the initial movement of the material along the technological route are considered. The process of the emergence of dynamic stresses with the constant and variable acceleration of the conveyor belt is investigated. The dynamics of stress distribution along the transportation route is presented. It is shown that the value of dynamic stresses can exceed the maximum permissible value, which leads to the destruction of the conveyor belt or structural elements. The transition period is estimated, which is required to ensure a trouble-free mode of transport operation during acceleration or braking of the conveyor belt. The use of dimensionless parameters allows us to formulate criteria for the similarity of conveyor systems. КEY WORDS: hydrodynamic model of a transport system, two-moment description, Hooke model, balance equations, PDE production model

The methods of statistical physics are one of the tools for modelling production systems with the flow method of organizing production [1], [2]. The developed models of production systems in the hydrodynamic approximation are widely used in the design of highly efficient flow production line control systems at leading world enterprises [3]. The values of the macro parameters of the state of the production system are determined through the values of the state parameters of a large number of products that are in different stages of processing in technological operations along the technological route [4]. The main macro-parameters of the state used to describe production systems with the in-line method of organizing production are the density of products in inter operational backlogs [ ] ( ) along the technological route are determined by the laws established by the technological process of manufacturing the finished product and are reflected in the route maps of the enterprise. As a result of technological processing, the product passes from one state to another as a result of exposure to technological equipment and the interaction between individual products [1,4]. The set of points that specify a continuous change in the state of the product determines the technological path of the product in the phase state space. The change in the condition of the product as a result of the influence of technological equipment occurs as a result of the transfer of technological resources to products. The balance equations for the macroscopic parameters of the production system with the flow type of organization of production are determined, to a large extent, by the technological laws of the interaction of products with each other and technological equipment.

DISTRIBUTED MODEL OF THE PRODUCTION LINE
In a multi-moment approximation, the system of balance equations for the macro parameters of the production flow line has the form [1][2][3]:  are used. When constructing models in the one-moment description, the closure condition is often applied: The disadvantage of the one-moment description is that such a description does not provide an opportunity to study the fluctuations of the flow parameters of the production line. To describe production lines for which the presence of fluctuations in flow parameters is of practical importance, a two-moment description can be used: Such systems may include conveyor-type production lines, for which fluctuations in flow parameters exceeding the limit level can lead to break down of technological equipment. As an alternative approach for studying fluctuations in the flow parameters of production lines, kinetic models of production lines can be called [5]. However, kinetic models are currently not widely used for designing highly efficient production line control systems. In this regard, the main focus of this article will be on the study of fluctuations in the flow parameters of the production line using a twomoment model in the form (2).

CONVEYOR TYPE PRODUCTION LINE MODEL
Among the models of conveyor systems, two large groups should be distinguished. The first group includes models for calculating the flow parameters of the conveyor line. When building models of this group are used: the finite element method (FEM) [6]; finite difference method (FDM) [7]; Lagrange equations [8]; aggregated equation of state [9,10,11]; equations for neural network layers [12]; system dynamics equations [13] and multiple regression equations [14][15][16]. The models of the first group are used in the tasks of operational planning of production activities of the enterprise. For a given quality criterion, the algorithms of optimal control of the flow parameters of the transport system are built on the foundation of these models. The second group includes models for the force calculation of structural elements of the transport system [16][17][18][19][20][21][22]. Of particular practical interest are the models that determine the conditions for the destruction of the conveyor belt [6,8,[23][24][25][26][27][28][29]. This allows you to determine the design parameters of the conveyor belt and the dynamic load modes, which ensure the stable operation of the transport system. The energy consumption required for the transport system (belt conveyor type 2LU120V) can be represented by the expression [30][31][32]: where xx N is the power of the conveyor idling; 1 n is increment of power consumption with an increase in the mass of cargo on the conveyor by 1 ton. According to the experiment for a belt conveyor type 2LU120V xx N =160(kW) , where xx n is unit costs of electricity for moving a running meter of conveyor belt. Expression (4)  when applying algorithms to control the speed of conveyor belts [25,32,33], which makes it possible to save up to 30% in the specific energy spent on moving the material [25,30]. As a result of using conveyor belt speed control systems, a 30% reduction in specific energy costs is achieved by increasing the load on conveyor systems, for which the traditional filling level of conveyor belt material is 60-100%. However, as a result of controlling the speed of the belt, there is a constant acceleration and braking of the conveyor belt loaded with material. This causes additional stresses on the material of the conveyor belt and, as a result, its damage. One of the ways to avoid damage to the conveyor belt is to increase the thickness of the conveyor belt, and therefore the mass of a running meter of moving parts [ ] С 0 χ . An increase in the mass of a running meter of the conveyor belt leads to an increase in the specific energy costs required to move the extracted material of a single mass. In this regard, in the present work, the main attention is paid to the interconnection of the stream parameters of the conveyor and the design parameters of the conveyor belt.
To build a model of the transport system, we use the two-moment description equations (2) in the following form [35]: The system of equations (5) corresponds to the case when the material does not crumble during transportation along the technological route. We will also assume that the material does not spread along the technological route: We introduce the notation for the speed of the belt in the equilibrium ψ μ and nonequilibrium state μ [35]: Taking into account (6), we multiply the first equation (5) (5), we obtain a system of equations for a two-moment description of the transport system: For a conveyor line, the force moving the material acts on the element of the conveyor belt and can be represented in the form ( Fig. 1): where dm is total mass acting on the belt; B is width of the conveyor belt;; h -the thickness of the conveyor belt; W F is the sum of the total resistance to movement of the belt [21, p.12]: Primary resistances H F are associated with the friction of resistance along the conveyor belt, with the exception of special resistances. Primary resistance H F , assuming a linear relationship between the resistances and the transported load, are determined by the expression [21, p.13]: For belt conveyor systems with filling factors φ in the range from 0.7 to 1.
The calculation of the force S F , associated with special resistances in the transport system, is presented in [21, p.18]. The value of S F is determined by the design features of the transport system. For most conveyor-type transport systems, it is assumed For the model under consideration, it is believed that stress and strain are related by a linear relationship where E is elastic modulus; ε is relative deformation of the conveyor belt element. If we introduce the absolute elongation of the conveyor belt ) , ( S t ω at a point in time t for the technological position S , then the ratio of the elongation ) , of the element of the conveyor belt to the length of the segment dS is the relative deformation of the element ( 1 1 ) If we substitute (9) -(11) in (7) then obtain the equations of oscillation of the conveyor belt The speed μ of the conveyor belt, on which the material is located consists of the speed of the belt in the Since the relative deformation of the element ) , ( 1 3 ) We substitute the expression for the speed of movement of the conveyor belt in the oscillation equation (12), obtain ( 1 4 ) Belt speed for steady motion is a known quantity that is set by program control of the conveyor line One of the main features of conveyor-type transport systems is that within the conveyor section for a steady-state mode of operation, the material at each point of the transport route moves at the same speed, which is equal to the speed of the conveyor belt .
( 1 5 ) We introduce the notation and taking into account expression (15), we represent equation (14) with small perturbations The speed of the conveyor belt μ can have a large gradient, for example as a result of sudden acceleration or deceleration of the conveyor belt. In this paper, we will pay attention to the steady mode for which condition (10) is satisfied, i.e., there are no large gradients for variable speed. Using the passage to the limit using the limit ratio

EEJP. 1 (2020)
Oleh M. Pihnastyi, Valery D. Khodusov and taking into account (13), it follows ( 1 9 ) When you change the length of a segment of the transport route, the changes. Let the length of the segment dS changes and becomes equal ( ) . In this case, the linear density will change and becomes . For this segment of the transport route we Neglecting the value of the second-order of smallness This allows us to represent the square of the function ) , as an expansion in the vicinity of the unperturbed The function ) , determines the propagation speed of disturbances along the conveyor belt [36]. Thus, using assumption (10) on the linear dependence of stress and relative deformation, the system of equations for determining the vibrations of the conveyor belt takes the form: We assume that at the initial moment of time the linear density of the material is distributed along the transport route according to the law [ ] We supplement the system of equations (20) with boundary conditions for an equation that describes oscillatory processes in a transport system. Stresses T ( fig.2). We write the system of equations for the forces i T , that determine the movement of the belt at the characteristic points of the horizontal conveyor section: are forces associated with the acceleration or deceleration of the conveyor belt. We believe that the effects associated with a change in the angular velocity of rotation of the drum are small due to the insignificant magnitude of the moment of inertia of the rollers. Traction moment rotates drum "B". Drum "A" rotates under the action of the frictional force of the belt, resisting movement. Then from the equality is determined the tension of the tape at characteristic points The stress of the conveyor belt at the initial time is determined by the initial distribution of the material along the technological route ) (S Ψ and the acceleration of the conveyor belt ) (t f ψ . Then Combining the system of equations (20), (21) with the boundary (24) and initial (25), (26) conditions, we obtain a two-moment model of the transport system that allows us to study dynamic stresses in the conveyor belt depending on changes in flow parameters [ ] ) , , ( 1 S t χ . Equations (7) and (9) can be used to study the scattering of the material during its transportation.
equation (20) with boundary conditions (22) takes a dimensionless form: The solution of equation (28) allows you to determine the state of the linear density of the material ) ( 0 ξ τ θ , along the transport route at an arbitrary point in time at an arbitrary point ξ The linear density of the material along the transport route ) ( An analysis of the solution (29), (30) is given in [33]. Using the notation equation (21), which describes oscillatory processes in a transport system, takes a dimensionless form and initial conditions (25), (26)

SOLUTION ANALYSIS FOR SMALL LOADED CONVEYOR LINES
Consider the solution of the system of equations (30)- (35) for the case of the initial movement of the conveyor, when the conveyor line is underloaded. The specific weight of the material along such conveyor lines is small compared to the specific weight of the conveyor belt Based on the above assumption, the system of equations (30) -(35) takes the form with boundary conditions (32), (33) and initial conditions (34), (35) During the initial movement of the conveyor belt, three characteristic phases should be distinguished: a) the period of time of the initial start-up, when the conveyor belt goes from rest to moving along the entire route; b) the phase of formation of the static force along the conveyor belt; c) the phase of acceleration of the conveyor belt to the rated speed.  , where 2 1 ν , 2 2 ν , 1 α , 12 α are constant coefficient, is the stresses, due to vibration of the traction drum as a result of providing the required traction moment.
The solution to equation (9)  Then the coefficients we obtain as a result of solving the system of equations The constant 01 C is determined from the condition of the minimum allowable stress min ω , ensuring the adhesion of the tape with the drum to create traction in the transport system with a given limit on the allowable amount of sagging of the tape. This allows us to write down the inequality We determine the unknown coefficients n G from the initial condition (48) Integrating the equalities, we obtain ( ) ( ) ( 5 6 ) We define the point of the transport route at which the maximum dynamic stresses of the conveyor belt occur. We introduce the variable n τ so that We calculate the value of dynamic stresses (53) arising in the conveyor belt The expression for the function ( )