HYDRODYNAMIC KELVIN-VOIGT MODEL TRANSPORTATION SYSTEM

The hydrodynamic Kelvin-Voigt model of production systems with a flow method of organizing production is considered. The main macro parameters of the state of the production line and the relationship between them are determined. The analysis of the main characteristics of models of elastic elements, which are used to analyze the occurrence of the dynamic stresses in a moving conveyor belt, is presented. A boundary value problem for elastic longitudinal vibrations in a conveyor belt with a moving material is formulated. It is assumed that the deformation of the conveyor belt element corresponds to the Kelvin-Voigt model and there is no sliding of the moving material on the belt. When determining the forces of resistance to motion acting on an element of the belt, the recommendations of DIN 22101: 2002-08 were used. The analysis of the Kelvin-Voigt model of the elastic element is carried out and the distinctive features of the model are demonstrated. The justification of the choice of the Kelvin-Voigt model of an elastic element for describing the process of occurrence of the longitudinal vibrations in a conveyor belt is given. The dependence of the non-uniform flow of material and the magnitude of tensions in the belt is estimated. An expression is written for the speed of propagation of disturbances along a moving conveyor belt with the material. The reasons for the acceleration and deceleration of the conveyor belt associated with the uneven supply of material at the entrance of the transport system are determined. The relationship between the speed of a conveyor belt and the mass of material along a section of the conveyor is demonstrated. It is shown that an increase in the power of the electric motor at the start and acceleration of the conveyor belt, as well as a decrease in power during the braking and stopping of the conveyor belt, is the cause of the appearance of dynamic stresses in it. The characteristic phases of the initial movement of the conveyor belt with the material are analyzed. The process of occurrence of dynamic tensions with the constant and variable acceleration of the conveyor belt for the phase of acceleration and deceleration of the conveyor belt is investigated. For the analysis, a dimensionless model of a conveyor line was used. An expression is obtained for static and dynamic tensions in the conveyor belt. The amplitude of oscillations of dynamic stresses and the characteristic time of damping of oscillations in a conveyor belt is estimated. A quadratic dependence of the speed of damping of a wave of dynamic tensions with an increase in the oscillation frequency is demonstrated. An inversely proportional dependence of the characteristic decay time of the generated dynamic tensions on the value of the viscosity coefficient of the composite material of the belt is shown.            n n pn f pn n n K n e W n . Over time, the amplitude of fluctuations in the magnitude of dynamic stresses decreases exponentially, so that after a time of the order of several n n   1 ~ , the oscillations completely damp out. The decay time of oscillations is inversely proportional to the square of the circular frequency of oscillations 2 ~  n n   . CONCLUSION Changing the acceleration mode of the conveyor belt of the transport system is one of the sources of dynamic stresses along the conveyor belt. At the same time, the mechanical properties of composite materials that are used for the manufacture of conveyor belts have occurrence of elastic vibrations and The use corresponding the Kelvin-Voigt model of an of the resulting elastic vibrations. The paper investigates two modes of acceleration of the conveyor and acceleration ramping over time. It has been demonstrated that the mode of movement of a conveyor belt with a linear change in the magnitude of acceleration in time is the cause of the occurrence of dynamic tensions. The damping rate of vibrations

The hydrodynamic Kelvin-Voigt model of production systems with a flow method of organizing production is considered. The main macro parameters of the state of the production line and the relationship between them are determined. The analysis of the main characteristics of models of elastic elements, which are used to analyze the occurrence of the dynamic stresses in a moving conveyor belt, is presented. A boundary value problem for elastic longitudinal vibrations in a conveyor belt with a moving material is formulated. It is assumed that the deformation of the conveyor belt element corresponds to the Kelvin-Voigt model and there is no sliding of the moving material on the belt. When determining the forces of resistance to motion acting on an element of the belt, the recommendations of DIN 22101: 2002-08 were used. The analysis of the Kelvin-Voigt model of the elastic element is carried out and the distinctive features of the model are demonstrated. The justification of the choice of the Kelvin-Voigt model of an elastic element for describing the process of occurrence of the longitudinal vibrations in a conveyor belt is given. The dependence of the non-uniform flow of material and the magnitude of tensions in the belt is estimated. An expression is written for the speed of propagation of disturbances along a moving conveyor belt with the material. The reasons for the acceleration and deceleration of the conveyor belt associated with the uneven supply of material at the entrance of the transport system are determined. The relationship between the speed of a conveyor belt and the mass of material along a section of the conveyor is demonstrated. It is shown that an increase in the power of the electric motor at the start and acceleration of the conveyor belt, as well as a decrease in power during the braking and stopping of the conveyor belt, is the cause of the appearance of dynamic stresses in it. The characteristic phases of the initial movement of the conveyor belt with the material are analyzed. The process of occurrence of dynamic tensions with the constant and variable acceleration of the conveyor belt for the phase of acceleration and deceleration of the conveyor belt is investigated. For the analysis, a dimensionless model of a conveyor line was used. An expression is obtained for static and dynamic tensions in the conveyor belt. The amplitude of oscillations of dynamic stresses and the characteristic time of damping of oscillations in a conveyor belt is estimated. A quadratic dependence of the speed of damping of a wave of dynamic tensions with an increase in the oscillation frequency is demonstrated. An inversely proportional dependence of the characteristic decay time of the generated dynamic tensions on the value of the viscosity coefficient of the composite material of the conveyor belt is shown. КEY WORDS: hydrodynamic model of a transport system, two-moment description, Hooke model, balance equations, PDE production model In article [1], the hydrodynamic Hooke's model of a transport system is considered, which was used to analyze the mechanism of the occurrence of longitudinal vibrations in a conveyor belt when material moves along a transportation route. To construct a model of the transport system, the equations of two-moment description (2) were used in the form [1,2]: 1  are respectively, the density of the material and the flow of material at the moment in time t at the point of the transport route, which is determined by the coordinate S ,  is the intensity of material receipt at the entrance of the transport system at the point is the force that acts on the material per unit mass of the material and the belt [3] is Dirac delta function.
We will assume that the specific density of the conveyor belt is constant and equal   С 0  , the conveyor section is located horizontally, and the material does not crumble from the conveyor belt.
The force acting on the section dS of the density conveyor belt   is located can be calculated as follows [1, Fig. 1]: Descriptions of each component of the secondary resistance (5) are discussed in detail in [1]. Detailed information on the methods for calculating secondary resistances is given in [4].
Primary resistances H F are related to the frictional resistance along the conveyor belt, with the exception of specific resistances. The primary resistances H F , assuming a linear relationship between the resistances and the transported load, are determined by the expression The force St F , characterizing the gradient resistance of the conveyor belt and transported material [4]  The calculation of the force S F , associated with special resistances in the transport system is determined by the design features of the transport system. For most conveyor-type transport systems, it is assumed substituting the result into (2), we obtain an equation that, together with equation (1), determines the propagation of stress disturbances along the conveyor belt, taking into account the distribution of material along the transport route To solve the system of equations (1), (7), the relationship between the tensions and the relative deformation of the section of the conveyor belt must be known This dependence is determined by the properties of the composite material from which the conveyor belt is made and is a model of an elastic element.

FORMULATION OF THE PROBLEM
The cost of transporting material from the place of extraction to the place of material processing reaches 20% of the total cost of mining material [5]. These costs can be significantly increased in the case of underloaded transport systems. This is especially important for long-distance transport systems [6,7]. To reduce unit costs, systems are used to control the speed of the conveyor belt μ and the value of the incoming flow (t)  to the input of a separate section from the accumulating bunker. The control of the parameters of the transport system changes the linear density of the along the transportation route. Control algorithms assume the operation of the conveyor section in the mode of acceleration or deceleration of the conveyor belt [8,9]. This leads to the generation and propagation of tension disturbances along the conveyor belt [10]. If the maximum permissible tension value of the conveyor belt is CONVEYOR TYPE PRODUCTION LINE MODEL In this paper, the Kelvin-Voigt element model will be used to analyze the arising stresses in the conveyor belt ( Fig. 1): where E is the elastic modulus of the element;  is element viscosity.  (7) is used to calculate the normal tensions of an element. If the tension is constant At 0 t t  , the limiting value for the element deformation is obtained: , the limit transition to the model of an elastic element follows, in which the stress and strain can be calculated in accordance with Hooke's law:

EEJP. 4 (2020)
Oleh M. Pihnastyi, Valery D. Khodusov The analysis of the arising deformations in accordance with Hooke's law was investigated in [1]. Substituting expression (7), which determines the relationship between stress and strain, into equation (6), we obtain Let us introduce the absolute elongation of the conveyor belt ) , at the moment of time t for the technological position S . The ratio of the elongation ) , ( S t dW of the element by the conveyor belt to the length of the segment dS is the relative deformation of the element The speed of the conveyor belt μ , on which the material is located, consists of the speed of the belt in equilibrium   and the oscillatory part of the belt speed Since the relative deformation of the element ) , For the relative deformation, represented in the form of a plane wave where  is the oscillation frequency; k is wave vector; k  is disturbance wavelength; ph μ is the phase velocity of propagation of the disturbance wave, that is, the velocity of movement of a point with a constant phase of oscillatory motion in space, along a given direction. A negative value of the wave vector corresponds to the case of propagation of a backward wave. In this paper, we will consider perturbations whose propagation satisfies the condition (8). Assuming that for the case when the functions ) , ( S t W , μ have a large gradient, the destruction of the conveyor belt occurs.
Taking this into account, we write down the expression for the change in the speed of the belt, neglecting the values of higher orders of smallness where the order of smallness is given below: When the length of a segment dS of the transport route changes, the density changes   . Let the length of the segment dS change and become equal   . In this case, the linear density will change and become equal to . For a given segment of the transport route, we have

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Hydrodynamic Kelvin-Voigt Model Transportation System EEJP. 4 (2020) Neglecting a quantity of the second order of smallness The function ) , determines the speed of propagation of disturbances along the conveyor belt [8]. The local change in density as a result of stretching the belt does not significantly affect the propagation of stress disturbances along the conveyor belt. Thus, we represent equation (8) in the form where is the acceleration of the conveyor belt for the steady state The solution of the equation for the case [1]. In this paper, we consider the case for which the relation Let's assume that at the initial moment of time the linear density of the material is distributed along the transport route according to the law Let us supplement the system of equations (10) with boundary conditions for the equation describing oscillatory processes in the transport system. The are determined by the tension forces of the conveyor belt 1 T and 4 T (Fig. 2).

Figure 2. Conveyor belt tension diagram
Let us 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:

EEJP. 4 (2020)
Oleh M. Pihnastyi, Valery D. Khodusov are forces associated with the acceleration or deceleration of the conveyor belt. We believe that the effects associated with a change in the angular speed of rotation of the drum are small due to the insignificant value of the moment of inertia of the rollers. The solution of the system of equations (11) makes it possible to determine the tractive effort for the transport conveyor for a steady motion: where b k is the coefficient of adhesion between the drum and the belt; s k is drum loss factor "A";  is total belt wrap angle of drive drums. For steel drum without moisture is [17] and In accordance with (9), the expression for the tension in the conveyor belt takes the form , the conveyor belt is engaged with the drive and driven shafts of the conveyor section. In this regard, it can be assumed that Taking into account the values of the acting forces 1 T , 4 T , let write down the boundary conditions Let us supplement the system of equations with initial conditions. Consider the mode of occurrence of oscillations, assuming that at the initial moment there were no oscillations The tension of the conveyor belt at the initial moment of time is determined by the initial distribution of material along the technological route Let us estimate the value of the acceleration of the conveyor belt ) (t f  . Asynchronous motors with phase rotor are usually installed on powerful belt conveyor sections. The acceleration and deceleration of the conveyor belt of a separate section occurs with the help of a rheostat, which sequentially changes the resistance in the rotor circuit. Simulation of starting up of the drive member of a mechanical system is given into [18]. The qualitative characteristic connecting the torque of the electric motor eng M and the engine speed eng n is shown in Fig.3 [19], where The change in force 1 T occurs due to uneven material receipt 0 ) (

Assuming for steady motion
, the equation takes the following form Taking into account that where d T is the time it takes for the material to pass the transportation route. The inequality allows us to formulate an important result: the uneven flow of material at the entrance of the transport system does not accelerate or decelerate of the conveyor belt. Uneven material flow leads to the destruction of the conveyor belt as a result of shock loads [20].

DIMENSIONLESS MODEL OF CONVEYOR LINE
Let's introduce dimensionless designations [1,21,22]:  determines the ratio of the resistance force to the maximum permissible tension force, which ensures the belt break

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Hydrodynamic Kelvin-Voigt Model Transportation System EEJP. 4 (2020) Then equation (10), which describes the oscillatory processes in the transport system, takes the dimensionless form with boundary conditions: and initial conditions:

ANALYSIS OF THE SOLUTION FOR LOW-LOADED CONVEYOR LINES
Let us consider a solution for the case of initial conveyor movement when the conveyor line is low loaded. The specific density of the material along such conveyor lines is low compared to the specific gravity of the conveyor belt Taking into account this assumption, equation (13) takes the form with boundary conditions: and initial conditions:

EEJP. 4 (2020)
Oleh M. Pihnastyi, Valery D. Khodusov 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 from a state of rest turns into movement along the entire route; b) the phase of the formation of a static force, along the conveyor belt; c) the phase of acceleration of the conveyor belt to the rated speed. Let us dwell on the analysis of the last phase of the start of the conveyor belt. The phase of acceleration of the conveyor belt to the nominal speed is characterized by a quasi constant value of the traction torque (Fig.3), [19] and a quasi constant value of the acceleration of the elements of the conveyor belt. In this regard, the function ) (  f will be assumed to be slowly varying during the characteristic time of the acceleration process a t .
Let us choose the solution to the equation in the form Let us represent a function ) , ( where 01 C is an unknown constant. We define the coefficients in such a way as to ensure the presence of boundary conditions for the function ) , ( This implies the conditions for determining the coefficients ) From the solution of the system of equations it follows 2 ) ( ) ( with boundary conditions:

a) the acceleration of the conveyor belt is absent or constant
  If the acceleration value is constant or absent, then by definition ) ( 12

CONCLUSION
Changing the acceleration mode of the conveyor belt of the transport system is one of the sources of dynamic stresses along the conveyor belt. At the same time, the mechanical properties of composite materials that are used for the manufacture of conveyor belts have a significant effect on the occurrence of elastic vibrations and their propagation. The use of composite materials with mechanical properties corresponding to the Kelvin-Voigt model of an elastic element ensures damping of the resulting elastic vibrations. The paper investigates two modes of acceleration of the conveyor belt: constant acceleration mode and acceleration ramping over time. It has been demonstrated that the mode of movement of a conveyor belt with a linear change in the magnitude of acceleration in time is the cause of the occurrence of dynamic tensions. The damping rate of vibrations is proportional to the viscosity of the elastic element and the square of the vibration frequency in the transport system. The magnitude of the generated dynamic tensions is determined by the amplitudes of the oscillations of the first harmonics. of variation of the oscillation amplitude and, therefore, the oscillation energy. The use of composite materials with a high value of the viscosity coefficient ensures attenuation of the arising dynamic stresses, which increases the reliability and durability of the functioning of transport systems.