Analytical Dependences Research Articles

ГИДРОДИНАМИКА ДВУХФАЗНОГО ПОТОКА В ЦЕНТРОБЕЖНЫХ АППАРАТАХ С ПРОФИЛИРОВАННОЙ КОНТАКТНОЙ ЗОНОЙ

Interest in detailed study and modelling of processes occurring in heat and mass exchange equipment is determined by the variety of designs and types of apparatuses, as well as the number of different ways to intensify the processes occurring in the equipment. The study of hydrodynamics of technological media and interaction of phases (components) on contact devices allows to optimise the geometry of designed equipment in order to increase the efficiency of heat and mass exchange processes. For modelling and analysis of the processes occurring in the technological equipment, the software complexes of computational fluid dynamics based on the methods of finite element analysis are often used, which allow to obtain contours of velocity, concentrations, temperature and pressure distribution only for single-phase media. However, modelling of technological processes implies description of hydrodynamics of multiphase media, which today in conditions of influence of centrifugal field and Coriolis acceleration on the nature of interaction of flows is a rather complicated task requiring significant computational resources, machine time and human resources. In connection with the above mentioned, it is of interest to develop an approach to modelling, combining simplicity of analytical dependencies and providing sufficient accuracy for calculation, determined by satisfactory agreement of the obtained results with literature data and results of numerical studies in software complexes of computational fluid dynamics. In this paper we propose a mathematical description of the solution of the hydrodynamic problem associated with the determination of the flow velocity of a two-phase flow in a profiled nozzle of a centrifugal apparatus. The basic differential equations and boundary conditions describing the flow of interacting phases in the profiled nozzle are given in the text.

ВЫБОР РЕЖИМОВ НАНЕСЕНИЯ ПОКРЫТИЙ ПРИ ГАЛЬВАНОКОНТАКТНОМ МЕТОДЕ

The paper presents studies aimed at studying the movement of the tool when applying dispersion-hardened composite electroplating coatings and its effect on the change in the area of plastic contact. (Research purpose) The research purpose is obtaining analytical dependences that make it possible to perform calculations for frictional contact of friction surfaces when applying dispersion-hardened composite electroplating coatings. (Materials and methods) Conducted research based on the methods of system analysis and the theory of formation of the surface layer. We performed experimental studies using modern standardized techniques on certified equipment. The obtained data were processed using the analytical modeling method, which is aimed at obtaining dependencies linking the desired characteristics with the specified parameters. (Results and discussion) Based on the conducted studies of plastic contact during the application of copper and iron composite coatings, it was concluded that the contact area increases, which requires a shift by a distance equal to 30-40 or more diameters of the contact spots. Consequently, with static friction, the in-crease in the contact area does not exceed 10-15 percent, and it is accompanied by a significant change in the microgeometry of the surfaces. (Conclusions) Theoretically and experimentally proved that the microgeometry of rubbing surfaces is reproduced, the transition from static contact to motion will not cause a significant change in the contact area. These conclusions are confirmed by a change in the convergence of two rough surfaces during sliding. It is possible to perform calculations for friction contact with sufficient accuracy using the formulas obtained for static conditions.

ОЦЕНКА ВЛИЯНИЯ ТЕХНОЛОГИЧЕСКИХ РЕЖИМОВ ПЛАЗМЕННОГО НАНЕСЕНИЯ ПОКРЫТИЙ НА ТОЛЩИНУ ФОРМИРУЕМОГО СЛОЯ

Parts of technological machines, complexes and equipment used in many industries are subject to various types of mechanical wear, which leads to a significant decrease in their durability and an increase in maintenance and operating costs. Promising directions for improving the technologies of plasma application of wear-resistant coatings have been identified. Special attention was paid to the thickness of the functional layer, since when using plasma spraying methods, it has a significant impact on the quality of coatings. (Research purpose) The research purpose is developing mathematical models that make it possible to study the influence of the technological parameters of the plasma application of wear-resistant coatings on the thickness of the functional layer being formed. (Materials and methods) Samples made of 45 steel were used for experimental studies, on which a nickel-based powder PG-CP4 (PR-NH17CP4) was sprayed by plasma method. The current strength of the plasma torch arc, the spraying distance and the consumption of plasma-forming gas were adopted as the main technological modes of plasma spraying of coatings. The thickness of the formed coatings was measured on standardized equipment. Statistical processing of the obtained coating thickness values was carried out using a complete factor experiment using the Box-Wilson method. (Results and discussion). We have developed an experimental mathematical model of the technological process of plasma coating and demonstrated its operability. The criteria used to evaluate the variance of experiments, the adequacy and significance of the obtained regression equations are described. Using the Statistica program, a graphical interpretation of the studied dependencies was given. (Conclusions) Based on the mathematical planning of the experiment, models were obtained that make it possible to establish the relationship between the coating thickness and the specified technological parameters of the plasma spraying process. The results of the analysis of the presented analytical dependencies showed that the thickness of the coating is most influenced by the current strength of the plasma torch arc.

Modeling heat and mass transfer in grain drying processes under the influence of electromagnetic field

Drying of grain materials by traditional convective methods typically involves significant energy consumption, primarily due to unproductive losses of thermal energy with the exhausted drying agent. Optimizing this process entails reducing the volume of drying agent used and compensating for reduced thermal energy through non-contact heat transfer methods such as microwave (MW) or infrared (IR) irradiation. However, mathematical modeling of these processes requires refinement as they differ from traditional convective heat drying processes. Grain drying is a complex thermophysical process, with heat and moisture transfer occurring within the capillary-porous material in a mutually dependent manner. To better understand these processes, it is necessary to establish mathematical models that account for the influence of the electromagnetic field. This article aims to theoretically investigate the drying process of plant raw materials under the influence of MW and IR electromagnetic fields based on analytical mathematical models. The conducted research indicates the potential of using electromagnetic fields to intensify the drying of plant raw materials, particularly through MW and IR irradiation, which allows for targeted heating of the moist zones of the material. The obtained analytical dependencies enable the calculation of temperature and moisture content fields in materials and determine the degree of influence of electromagnetic field parameters on heating and drying processes. These dependencies also help to more accurately identify heat and mass transfer model coefficients based on experiments. Such an analytical-empirical approach can be utilized to calculate rational processing regimes for grain raw materials under the influence of electromagnetic fields.

АППРОКСИМАЦИЯ ПРОГИБОВ ПЛАСТИНОК, ЛЕЖАЩИХ НА ВИНКЛЕРОВОМ ОСНОВАНИИ

The purpose of this research is to develop the method of shape factor interpolation for calculating the maximum deflection of thin plates on an elastic Winkler base, which are widely used in modeling the operation of elements of building constructions of buildings and structures. The above calculation method allows to obtain solutions based on direct analytical dependences, the argument of which is an integral characteristic of a flat convex one-connected area - the shape factor. This characteristic has applications in a number of problems of mathematical physics and is known from the works of scientists G. Polia and G. Szegö. The shape factor was first applied to the calculation of plates by Professor V.I. Korobko. The method of interpolation by shape factor was developed by Professor A.V. Korobko. When determining the maximum deflection of thin plates on an elastic base, some parameters of the problem are considered as functions of the shape factor of the plate in consideration and are determined by the type of boundary conditions on its contour. The present study is devoted to the construction of approximating functions for continuous sets of plates of characteristic outlines and boundary conditions. The paper presents functions for calculating the maximum deflection of elastic plates in the form of isosceles triangles, rhombuses and rectangles. The plates with various combinations of hinged support and rigid pinch along their individual sides, loaded with a continuous uniformly distributed load, are considered. The established functional dependences are intended for direct use in the calculation of plates of the specified outlines, as well as for obtaining reference solutions during interpolation of the values of maximum deflections of plates of more complex outlines.

Optimization of open optical communication with multiple UAV over non homogenous turbulent paths

The article is devoted to optimizing communication through open atmospheric channels with many UAV by using inhomogeneous turbulent paths. A probabilistic assessment of the total signal loss in the communication system between the central communication node and a group of UAV located at the same distance from this node is analyzed. The case is considered when the scintillation indices in the subzones of optical communication with the UAV are different, which leads to unequal signal losses in the corresponding optical channels. The target functional is compiled by using a certain integral of the lognormal probability density function, which includes the newly introduced function of the losses dependence from the scintillation index. An analytical dependence of these indicators, characterizing the worst case when the total losses in the network reach a maximum, is presented. Keywords optical communications, turbulence, UAV, optimization, atmosphere

Оценка характеристик фрагмента контроля и устранения дефектов (ошибок), допущенных при реализации технологического процесса

This article considers a fragment of the technological process related to multi-stage monitoring of the production results (check, diagnostics, and elimination of the identified product defects). A general control algorithm is presented. This algorithm includes preliminary control at the workplace and diagnostic controls at thesite and workshop level, with subsequent elimination of the defects identified. An atypical elementary operation with three outputs was identified. All possible paths were analyzed followed by obtaining of analytical dependencies for probabilistic and time estimates, as well as for estimates of the number of operations, stereotypicality, and logical complexity. A method for reducing the algorithm to an enlarged work operation, whose characteristics adequately reflect the operational peculiarities of the algorithm (technological process – control and elimination of defects), is proposed. The obtained estimates can be used to solve a number of specific problems related to improving the algorithm (reducing the likelihood of defects in the resulting product, changing the structure (structural optimization) of the algorithm, etc.). The implementation of an approach based on identifying paths in the algorithm, assessing their characteristics, and calculating the characteristics of the entire algorithm confirmed the validity of the proposed approach. The results obtained can be used as a methodological and mathematical basis for creating various decision support tools in the design and development of human-machine systems for various purposes.

Виброактивность электромеханических устройств вследствие наличия технологических погрешностей

This article discusses a theoretical justification for the presence of vibration in operating modes of electromechanical devices due to technological errors occurring during the machining of parts and assembly of products. The reason for the presence of a dense spectrum of vibration caused by technological errors in the structural elements of ball bearings (outer and inner rings, balls, and cage) is also denoted. The analytical dependence of the vibration amplitude on the magnitude of technological errors is shown. It follows from the results of the analysis that for the most effective minimization of the vibration activity of an electromechanical device, it is necessary to use broadband damping, which is liquid–viscous in most cases. In addition to the above, it is advisable to apply damping in the cascade version. When solving the system of vibration equations for an electromechanical device, replacing ball bearings with sliding bearings eliminates the first four terms, which characterize a wide and dense vibration spectrum of the ball bearings in the operating mode of the electromechanical device.

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It is noted that combat readiness characterizes the ability of a military unit to perform a combat mission and is being a defining element of its combat readiness. The factors on which combat readiness depends are given. A formalized mathematical model for assessing the combat readiness of military units for performing the assigned tasks has been developed, which is being a system of analytical dependencies. A control algorithm that implements a formalized model for assessing the combat readiness of military units for performing the assigned tasks has been proposed.

Моделирование гидродинамики, теплообмена и процесса усреднения гранулированных сред в пневматическом циркуляционном аппарате

The mathematical modeling of hydrodynamics, heat transfer, and averaging of a highly concentrated granular medium in the working area of a vertical pneumatic circulation apparatus is performed. The hydrodynamics of a dense layer of the granular medium is described by a non-Newtonian model with partial slip conditions on solid walls. Such a representation of the granular medium dynamics allows one to approve theoretical calculations with available experimental data for a steady flow in a flat channel. The numerical applicability of the partial slip condition for a non-Newtonian fluid is confirmed by the analytical dependences obtained by the authors for a steady flow in a circular tube and a flat channel, which are transformed into the known analytical formulas under no-slip conditions for a non-Newtonian medium. On the basis of the proposed model, the heat exchange intensity is analyzed when setting the constant heat flux density or the constant temperature on the annular shelves in the apparatus. The intensity of the granular mixture averaging in the bunker with the introduction of additional annular shelves is studied, and the efficiency of granular mixture mixing during rotation is considered. A method for analyzing the mixing of a granular medium in dynamics in each local area of the apparatus is proposed, which is based on the determination of the inhomogeneity coefficient. The results of this study can be used when developing powder technology devices for drying, mixing, dosing, and transport.

Study on modeling of contact interaction in roll modules

The main problem of the mechanics of contact interaction in roll modules is to determine the stress-strain state in the technological zone. The article presents mathematical models of the laws of distribution of contact stresses and analytical dependencies of their main indices, which make it possible to calculate the power parameters of roll modules of technological machines. When developing mathematical models, empirical formulas for the laws of distribution of normal stresses along roll contact curves and a friction stress model were used, considering the effect of external forces, roll drive, and compliance of the roll coating on the distribution of shear stresses. It is shown that normal stresses in the compression zone increase from zero to the maximum value by the harmonic law, and in the recovery zone, they decrease from the maximum value to zero by the parabolic law. It was revealed that in the driven roll, the neutral point is located on the left side of the point of maximum normal stress, and the points that determine the maximum normal stress and compression of the interacting bodies coincide and are shifted from the line of centers towards the supply of the processed material into the contact zone.

ПРОБЛЕМА РАСКРЫТИЯ ТРЕЩИН В ЖЕЛЕЗОБЕТОНЕ

The article discusses various aspects of the problem of assessing crack opening in reinforced concrete based on experimental data obtained in recent studies. The author proposed a classification of crack types, introduced a number of new hypotheses, established experimentally the effects of deformation of reinforced concrete in the crack zone. The principles formulated on their basis include schemes for the distribution of force flows between cracks, the concept of progressing main cracks and the deformation effect in reinforced concrete - a special two-console element in the local region near the crack banks, new generalized hypotheses, theorems and functionals about linear and angular deformations of the compressed and tensile zones of reinforced concrete element sections at all levels of elastic-plastic deformation. The proposed model of composite rods in the form of single strips for determining the stiffness of a section of reinforced concrete with intersecting cracks has made it possible to reduce the differential equations of the theory of composite rods by an order of magnitude when solving such problems. The connection is established and analytical dependences for displacements in the crack with opening and shear of the crack banks are given. The main vector of displacements and the angle of equidirectional forces in the reinforcement crossing the crack are determined. Calculated dependences for determining the level distances between cracks and crack opening widths are constructed. Within the framework of the general methodology of the considered problem of crack opening in reinforced concrete, using the formulated principles, a general combined numerical-analytical model of the structural mechanics of reinforced concrete is constructed, which takes into account the deformation effect in the crack modeled by a double cantilever element, the types of cracks, the spatial surface of strain distribution in the cross section with the crack and other experimentally determined features of the mechanics of reinforced concrete, cracks types, spatial surface of strain distribution in the cross-section with a crack and other experimentally determined features of reinforced concrete mechanics.

Control of Dynamic Biped Walking Robot

The objective of this article is to present control algorithms of biped dynamic walking robot. A mathematical model of the walking robot is compiled. The construction of the robot is based on minimalist design principles - absolutely solid body with two legs. Analytical dependencies between six coordinates of the body, coordinates of the left and right legs and absolute coordinates of the feet are defined. The block diagram of control system is presented. Differential equations of walking are compiled. The dynamic stability of biped walking robot near the required trajectory is provided by discrete control of the parameters: «when and where to put foot». To define these parameters, it is proposed to use an inverted mathematical pendulum as an ideal mechanism. Its linearized differential equations allow present the feet coordinates according to the measured initial and final parameters (at the begging and at the end of the step). The article paid particular attention to results of such control method with various walking modes. The coordinates of the robot body, its velocities and accelerations, angles of rotation were investigated. Calculations confirmed the small deviations of the proposed model of walking from the ideal one. The developed methodology allows improve the construction and control system of biped dynamic walking robots at the design stage. Some of the tasks, it can be used: design machines simulating human walking, with high speed and maneuverability, with point feet and synchronized with natural uncontrolled movements.

The method of determining the parameters of the soil prism formed in front of the dump during digging

Problem. The studied dynamic process of digging, taking into account the variable total moving mass of wheeled ZTM equipped with a dump, is not sufficient for the development of recommendations for selecting rational parameters for the performance of such a technological operation. The issue of determining and analyzing the parameters of the variable mass and volume of the material prism moved by the dumper has not been sufficiently investigated. There in no engineering technique that allows you to determine the parameters of the material prism that is gradually forming before dumping. Methodology. The measures are aimed at determining the parameters of the soil prism before the dump in the digging process using analytical dependencies. Results. For the processes of development and analysis of dynamic models of machines with a shovel working body for the stages of digging and transportation of material, it is necessary to take into account its mass in the form of mass attached to the mass of the machine itself. Originality. The obtained dependencies allow to take into account the variable mass of the attached material when performing the analysis of the bulldozer movement in the process of digging the environment being developed. Practical value. The obtained results can be recommended when studying the disciplines of industrial mechanical engineering

DETERMINATION OF THE THERMODYNAMIC PARAMETERS OF α-IRON IN BINARY SYSTEMS Fe-Cr AND Fe-Ti

This article presents the results of calculations of the thermodynamic parameters of binary Fe-Cr and Fe-Ti systems based on their state diagrams using the Bjerrum-Guggenheim osmotic coefficient, for the region of iron crystallization from the iron activity of an ideal α-γ solution for the above systems. When calculating the lines of phase equilibrium in the Fe-Cr and Fe-Ti systems, it was established that the Bjerrum-Guggenheim coefficient can serve as an assessment parameter characterizing the role of exchange forces between atoms, and by its dimension (Фi><1) one can qualitatively judge the nature of the intermolecular interactions in the system. The results of this work made it possible to correctly solve the direct Gibbs problem, namely, to obtain an analytical dependence of phase compositions on temperature at phase equilibria based on the laws of phase formation. It is shown that with the help of the Bjerrumm-Guggenheim osmotic coefficient (Fi), the real equilibrium in the system under study can be determined in detail and their correct analytical expression can be obtained. When calculating phase equilibrium lines in systems with different types of interaction between components, it was established that the Bjerrum-Guggenheim coefficient can serve as an assessment parameter characterizing the role of exchange forces between atoms. Keywords: thermodynamics; Bjerrum-Guggenheim coefficient; state diagram; osmotic coefficient; binary systems.

DETERMINATION OF ELASTIC CHARACTERISTICS OF BUILDING INDUSTRY VIBRATION SYSTEMS

The design of the difficult dynamic systems is examined with removable descriptions ofresilient elements and analytical dependences are got for realization of multimode resonances and the card ofstability of vibration of shock machines is offered.

Investigation of the influence of the heterogeneity of the load node on its comparability with the power of the energy source in an autonomous energy system

The article considers the investigation of the influence heterogeneity of the load node on the required power of the power source and on their comparability. By checking fulfillment of loading conditions of the electric generator unit, the obtained analytical dependencies in relation to determining parameters of an autonomous power supply source for a multi-motor load are solved. The criterion of comparability was proposed for matching the power of the source with the installed power of consumers of the autonomous energy system. A dependency for determining the comparability coefficient was given for the system the electric generator unit and the load node. The analysis of the permissible solution the comparability coefficient has been carried out for ensuring rational loading of the electric generator unit in the continuous operating. The necessity of using devices for reducing or compensating the reactive power of individual electric loads was assessed to reduce the power of the electric generator unit.

FOUR-COMPONENT ZOOM-AFOCAL SYSTEMS ZOOM-TRANSFOCACATORS OF THREAD RANGE METERS

The problem addressed in this research is the parametric synthesis of zoom-afocal optical systems for transfocusors used in measurement technology, specifically thread-type rangefinders. The research aims to develop a universal analytical algorithm that can be computerized for calculating the optical and dimensional parameters of four-component zoom-afocal lens systems for trans-focusors.The implementation methodology involves representing the zoom-afocal system as a sequence of thin components.Using a system of equations where the optical powers of the components and the distances between them are unknown, the values of these parameters are determined to achieve the desired range of angular magnification and zero optical power of the afocal system. The optical powers of the components and their ratios must ensure positive values for the distances between the components. The calculation of the light diameters of the components takes into account the permissible vignetting of the ray bundle from the field of view edge by determining the angles and heights of the zero rays on the system's components.The research results include analytical dependencies that enable the determination of component optical powers and their mutual arrangements throughout the entire range of angular magnification based on the input data. The research shows that the first component has the largest light diameter, which reaches its maximum value as the angular magnification approaches the maximum within the specified range.The obtained results allow for the parametric synthesis of four-component zoom-afocal systems in an analytical form, considering the dimensional requirements, range of angular magnification, and permissible vignetting of ray bundles from the field of view.

Bitumen emulsion effect on properties of asphalt-concrete mix for non-rigid road pavements (Tambov) .

During major repairs of non-rigid road surfaces and reconstruction of automobile roads, it is necessary to remove old coating with road cutters. During cold milling of the old coating, the material acquires the form of granules with the surface covered with a bitumen film. Over time, the bitumen film loses its properties, and to reuse the material, it is necessary to restore the binder properties. The introduction of the bitumen emulsion in the composition of asphalt-concrete mix, allows to restore the binder properties affecting not only strength properties of the asphalt-concrete mix, but also deformability and stiffness of the mixture layer under the compressive load.Purpose: Evaluation of the influence of the binder content on deformability and stiffness of the asphalt-concrete mix.Methodology: Asphalt granulate fractions are used in proportions 8 to 16 and 18 to 36 mm with a residual bitumen content of 2%. The bitumen emulsion EBK-2 is used as a binder. The ultimate strength of the asphalt layer is measured according to GOST 12801. Deformation of the mixture layer is measured on a TP-1-1500 hydraulic press.Research findings: The bitumen emulsion effect is shown for deformation and stiffness of the asphalt-concrete mix for the pavement construction.Value: It is shown that the bitumen emulsion in regeneration of asphalt granulate affects deformability and stiffness of the layer subjected to compaction load. The development of deformation and stiffness of the mixture layer under the load with different binder content, particles size, and layer thickness are established. Analytical dependences are suggested for deformation and stiffness of the loaded layer, its thickness during paving, and percentage of bitumen emulsion in the mixture.

Calculation of horizontal collective drainage pipelines of reclamation systems in the presence of a ground water level slope

The purpose of this work is to develop a methodology for calculating the parameters of pressure horizontal collective drainage pipelines of reclamation systems that operate in the presence of a ground water level slope.
 The presented materials are based on the results of personal analytical studies of the main mathematical dependencies, which describe the pressure liquid flow in horizontal collective perforated pipelines, which operate in the presence of the ground water level slope.
 System of two differential equations, which describe the liquid motion in a drainage pipeline with variable flow rate and the conditions of liquid entry from the surrounding soil through the walls of the pipe in filtration mode, is considered. The explored pipeline is laid in the presence of the ground water level slope. By introducing new variables, the original system and its solution is presented in a dimensionless form.
 In this case the solution of the system depends on the value of three main factors: the resistance factor of the collecting drainage pipeline "ζl"; the generalized parameter "A", which comprehensively takes into account the structural and filtering characteristics of the stream; the ground water level slope «І». The analysis used the concept of an infinitely long drainage pipeline, which operates in the presence of the ground water level slope, or, what is the same, an inclined pipeline with an infinite filtering capacity of the side surface walls. Relatively simple and easy-to-use analytical dependencies were obtained on the basis of the conducted analysis. They allow to calculate the nature of the flow rate variations and pressure drop along the length.
 Corresponding graphic dependencies were constructed to simplify the calculations. It is shown that the value of the geometric slope of the ground water level, along with the resistance factor and the generalized parameter, significantly affects the calculated parameters of such pipelines.