Authors:M. Olshevskyi Pages: 11 - 22 Abstract: In this paper we consider a specific case of the diameter search problem for finite groups, thecase where the system of generators is fixed. This problem is well-known and can be formulated in the following way: find the diameter of a group over its system of generators. The diameter of the corresponding Cayley graph is the diameter of a group over its specific system of generators. The main object of the research is the alternating group with the system of generators consisting of cycles having length three and the form (1,2,k). This system of generators is a classical irreducible system of generators of the alternating group. It is introduced the property of even permutations to be balanced. We consider the set of balanced permutations and permutations close enough to balanced and find minimum decompositions of them over defined system of generators. The main result of the paper is the lower bound of the diameter of Alternating group over con-sidered system of generators. The estimation is achieved using minimal decompositions of balanced permutations. Pages of the article in the issue: 11 - 22 Language of the article: English PubDate: 2021-12-21 DOI: 10.17721/1812-5409.2021/4.1

Authors:V. A. Voskoboinick, O. A. Voskoboinyk, L. M. Tereshchenko, A. V. Voskobiinyk Pages: 25 - 28 Abstract: Group constructions of bluff bodies are widely used in bridge construction practice. The junction flows of such structures are characterized by considerable complexity, nonstationarity and instability. In the vicinity of bluff bodies, systems of horseshoe vortex structures, shear layers, separated regions, jet flows, wake vortices and vortex Karman’s streets are formed. The study of the features of the generation and evolution of vortex and jet flows, the mechanisms of interaction of these flows with streamlined surfaces requires considerable effort during numerical and physical modeling. The purpose of the work is to determine the features of vortex and jet flow in the region of junction of three-row pile grillage with a rigid flat surface. Experimental studies were carried out in laboratory conditions in a hydrodynamic channel, where the three-row group of cylinders was installed on the hydraulically smooth rigid surface. Visual investigations and measurements of the velocity field were carried out inside and around the three-row grillage. The features of the formation and evolution of vortex and jet flows inside and near the cylindrical group were established. Integral and spectral characteristics of the velocity fluctuation field were obtained. Pages of the article in the issue: 25 - 28 Language of the article: Ukrainian PubDate: 2021-12-21 DOI: 10.17721/1812-5409.2021/4.2

Authors:V. P. Golub Pages: 29 - 34 Abstract: The problems of stress relaxation analysis in thin-walled cylindrical shells made of linear viscoelastic materials under uniaxial and biaxial loading have been solved. The analysis is based on a there-dimensional model of viscoelasticity starting from the hypothesis of the deviators proportionality. The viscoelastic properties of a material are given with relationships that establish the dependence between stress and strain intensities as well as between the mean stress and volumetric strain by the Bolzmann-Volterra type equation. The kernels of relaxation intensity and volumetric relaxation are given with the Rabotnov exponential-fractional functions. The parameters of relaxation kernels are determined from creep test result using the relationships between creep kernels under the complex stress state and creep kernels under the one- dimensional stress state. The problems of the analysis of normal and tangential stresses relaxation in thin-walled cylindrical shells made of high density polyethylene “ПЭВП” under uniaxial tension, pure torsion and combined tension with torsion loading have been solved and experimentally approved. Pages of the article in the issue: 29 - 34 Language of the article: Ukrainian PubDate: 2021-12-21 DOI: 10.17721/1812-5409.2021/4.3

Authors:G. M. Zrazhevsky, V. F. Zrazhevska Pages: 35 - 38 Abstract: The paper considers the problem of natural harmonic oscillations of an elastic rod with stress-free ends in the presence of one or a set of defects. Defects are modeled by the inhomogeneity of the Young's modulus. The location of the defects, their geometric size, which is considered small, and the change in elastic properties are the parameters of the defects. The analysis of natural frequency shifts caused by the defect of the rod is the subject of the study. The aim of the work is a mathematical substantiation for the construction of fast and stable algorithms for determining the defect parameters of elastic bodies by analyzing free oscillations. The paper uses and compares fundamentally different research methods. The first methods are classical mathematical methods of mechanics, applied to the analysis of deterministic systems and based on analytical studies combined with numerical implementation. In contrast, a composite machine learning meta-algorithm used in standard statistical classification and regression - Bootstrap-aggregated Regression Trees (BART) - is used to solve the inverse problem. When comparing the constructed algorithms, the statistical method Sampling was used, which allowed to quantify the accuracy and stability of the algorithms. Pages of the article in the issue: 35 - 38 Language of the article: Ukrainian PubDate: 2021-12-21 DOI: 10.17721/1812-5409.2021/4.4

Authors:N. M. Kizilova, N. L. Rychak Pages: 39 - 45 Abstract: Stormy events in recent years have shown that the destructive effects of wind on urban technical structures and plants pose a special threat. The paper provides an overview of mathematical models and approaches to experimental and theoretical studies of the problems associated with the effects of wind gusts and tornadoes on urban areas. Computer simulations of wind action on standard multistorey buildings in Ukraine are given. The coefficients of normal and shear components of forces and moments of forces acting on the surface of buildings, as well as vortex tracks over the residential complex at different wind speeds from moderate to severe have been computed. The calculations were performed by the finite element method using the model of turbulent air flow in the package AnSys2020. It is shown how with the help of a slight change in shape (roofs, additional passages, shields) the destructive effects of wind on the buildings and plants, as well as the threat to human life can be reduced. Pages of the article in the issue: 39 - 45 Language of the article: Ukrainian PubDate: 2021-12-21 DOI: 10.17721/1812-5409.2021/4.5

Authors:Yu. M. Kobzar Pages: 46 - 49 Abstract: A fatigue model based on a decrease in the carrier mass of a substance in the first quarter of a cycle. Also a fatigue model based on an increase in its density in the second quarter of a counterclockwise rotation cycle. As well as this model based a decrease in a carrier mass in a third quarter cycle and an increase in its density in a fourth quarter of a clockwise rotation cycle. The tangential stress and shear angle are related by the Hooke linear relationship. Depending on the initial physical and mechanical properties of the rod, its structural changes are controlled, which quantitatively reflect the changes in mass, density, stresses, shear modulus, which are calculated on each cycle. It is accepted that the brittle fracture of the rod occurs in a cycle in which the inequality of the initial fracture energies and the potential elastic energy pumped on this cycle is not fulfilled. The criterion for achieving the limit of fatigue is not to fulfill the inequality outside the accepted test base. The model algorithm is implemented in the software environment of computer algebra. Pages of the article in the issue: 46 - 49 Language of the article: Ukrainian PubDate: 2021-12-21 DOI: 10.17721/1812-5409.2021/4.6

Authors:V. V. Mozharovsky, D. S. Kuzmenkov, M. V. Moskaleva, S. V. Kirhintsava Pages: 50 - 55 Abstract: Based on the solutions of contact problems for fibrous materials and coatings of composite materials, the article considers the implementation of the method of calculating the pliability of gears of composite materials, determining the contact parameters of coatings of fibrous materials. To calculate the contact deformations of gears made of metals and composites, a program was developed in the Delphi environment, which makes it possible to calculate the coefficients of contact deformation of the gear tooth, as well as the calculation of contact deformations for a tooth with a fibrous coating. An integral equation is presented, which gives a solution of the contact problem of pressing a stamp into an orthotropic coating. To study the influence of the properties of the material and the thickness of the coating on the contact parameters, a computer program was compiled, which was used to calculate for different thicknesses. Pages of the article in the issue: 50 - 55 Language of the article: Ukrainian PubDate: 2021-12-21 DOI: 10.17721/1812-5409.2021/4.7

Authors:V. V. Novytskyy, V. V. Novytskyy (Jr) Pages: 56 - 61 Abstract: Practical medicine requires new research to better understand the processes of blood flow through the vascular system. In particular, the processes of blood movement in capillaries, when their diameter is smaller than the diameter of erythrocytes, are of interest. It is believed that the center of mass of the erythrocyte lies on the midline of the capillary. While in the arterioles, the erythrocyte releases nutrients, so its mass decreases. When moving in the venule, the mass of the erythrocyte increases because it receives spent substances from the tissue space. The vascular wall of the capillary and its midline are modeled using the equation of the parabola, which makes it possible to calculate within the specified limits the length of the wall and the midline. The movement of an erythrocyte is described by the Meshchersky equation for bodies with variable mass. The proposed article is devoted to the construction of static models of capillaries in the norm and a dynamic model of movement in the capillary of an erythrocyte with variable mass. Pages of the article in the issue: 56 - 61 Language of the article: Ukrainian PubDate: 2021-12-21 DOI: 10.17721/1812-5409.2021/4.8

Authors:Y. V. Pavlyuk Pages: 62 - 65 Abstract: The problem of calculating the deformations of the cyclic creep of nonlinear viscoelastic materials is considered, which is given in the form of cyclic alternations of loads and unloadings of equal amplitude over rectangular cycles, where the duration of loading and unloading half-cyclescoincide. The program of loading is realized in the form of sequence of elementary loadings set by means of unit functions of Heaviside. A nonlinear creep model with a time-independent nonlinearity of Yu. Rabotnov's model is used to describe the deformation process. The fractional-exponential function is used as the nucleus of heredity. The paper develops a nonlinear viscosity model with time-independent nonlinearity due to the use of instantaneous deformation diagrams as isochronous for zero time and smoothing cubic splines, as approximations of nonlinear instantaneous deformation diagrams that define the nonlinearity of the model. The concept of a single isochronous deformation diagram for the studied material is experimentally substantiated. A system of solution equations of nonlinear creep under cyclic loading is formulated. The problem of calculating the deformations of stationary and cyclic creep for nylon fibers FM 10001 is solved and experimentally tested. Pages of the article in the issue: 62 - 65 Language of the article: Ukrainian PubDate: 2021-12-21 DOI: 10.17721/1812-5409.2021/4.9

Authors:O. V. Romanov Pages: 66 - 69 Abstract: The question of the influence of the type of stress state (TSS) on the process of scleronomic and rheonomic deformation of structural materials under simple and close to simple loading conditions is considered. The parameters characterizing the TSS are indicated. This is a parameter (indicator) of the rigidity of the stress state, which reflects the influence of the first invariant of the stress tensor and is, with an accuracy of a constant factor, the ratio of the hydrostatic component of the stress tensor to its deviator component. The influence of the third invariant of stress tensors is taken into account by a number of parameters – the Nadai-Lode parameter, as well as the TSS angle in its various interpretations. A variant of taking this influence into account by introducing the governing equation of state of the function with the parameter of the TSS angle is shown. This function is determined based on analysis of basic tensile and torsion experiments. It is shown that taking into account VNS increases the accuracy of the calculation and improves the agreement between the calculation results and experiment. Pages of the article in the issue: 66 - 69 Language of the article: Ukrainian PubDate: 2021-12-21 DOI: 10.17721/1812-5409.2021/4.10

Authors:I. K. Senchenkov, M. V. Jurzhenko, O. P. Chervinko, O. V. Dolia Pages: 70 - 75 Abstract: The paper is devoted to the assessment of the thermo-stress-strain state of elements of structures obtained by 3D printing using FDM (Fusing Deposition Modeling) technology. Three stages of solving this problem are considered: (1) - mathematical formulation of the problem, that includes universal balance relations, constitutive equations of mechanical behavior of the material and is based on the model of growing bodies; (2) the technique of finite-element solution with increasing mesh due to the addition of new elements; (3) - study of a specific problem of growing plate-like element by polymer PLA (polylactide) with temperature-dependent physical properties. Options of deposed layers of different thickness are considered. The residual stress-strain state of the body has a two-scale structure, which reflects the change of state characteristics in the scale of body size, as well as the thickness of the layers. As the thickness of the layers increases, the average values of the residual stresses decrease, but the amplitude of the stress fluctuations along the thickness of the layer slightly increase. Pages of the article in the issue: 70 - 75 Language of the article: Ukrainian PubDate: 2021-12-21 DOI: 10.17721/1812-5409.2021/4.11

Authors:Ya. P. Trotsenko Pages: 76 - 81 Abstract: The flow of a viscous incompressible liquid in a cylindrical duct with two serial diaphragms of a rectangular profile is studied by the numerical solution of the unsteady Navier-Stokes equations. The discretization procedure is based on the finite volume method using the TVD scheme for the discretization of the convective terms and second order accurate in both space and time difference schemes. The resulting system of non-linear algebraic equations is solved by the PISO algorithm. It is shown that the fluid flow in the region between the diaphragms is non-stationary and is characterized by the presence of an unstable shear layer under the certain parameters. A series of ring vortices is formed in the shear layer that causes quasi-periodic self-sustained oscillations of the velocity field in the vicinity of the orifice of the second diaphragm. In comparison with the case of rounded diaphragms, an increase in the maximum jet velocity is observed, which in turn leads to an increase in the frequency of self-sustained oscillations and a decrease in the Reynolds numbers at which quasi-periodic oscillations are excited. Pages of the article in the issue: 76 - 81 Language of the article: Ukrainian PubDate: 2021-12-21 DOI: 10.17721/1812-5409.2021/4.12

Authors:I. A. Ulitko, O. B. Kurylko, M. B. Zathei Pages: 82 - 87 Abstract: The use of a tuning fork resonator as sensitive element of a gyroscopic sensor has some advantages in comparison with other types of the resonators. For instance, it allows to compensate lateral accelerations in the direction perpendicular to the axis of rotation. At the same time, the task of accurate determination of the carrying frequency of the primary mode of a non-moving tuning fork is of great importance. Thus, in [1] the analysis of vibrations of a gyroscope is built on the evaluation of the first frequency of flexure vibrations of Timoshenko's beam with one rigidly fixed end [2]. As a result, the sensing frequencies of the Bryan's splitting pair [3] of the fork lie below the frequency of Timoshenko's beam, and the resonant frequency of the non-moving tuning fork remained uncertain. The purpose of a present paper is to establish this frequency. In the statement of a problem, concerning real geometric dimensions of the tuning fork elements, we assume that the length of the tuning fork rods l is much more then the radius of the base r: r/l << 1. Then, frequencies of the flexure vibrations of the half-ring lie much higher than the frequencies of the bending vibrations of the rods. It allows us to give a solution for the base in a quasi-static approximation, and to take into account the dynamics of the tuning fork in the solution for bimorph piezoceramic rods. Conditions of coupling between the rods and the half-ring are reduced to the conditions of elastic fixing of the rods, which take into account the geometric parameters r and l. Pages of the article in the issue: 82 - 87 Language of the article: Ukrainian PubDate: 2021-12-21 DOI: 10.17721/1812-5409.2021/4.13

Authors:K. G. Khoroshev, S. V. Kykot Pages: 88 - 93 Abstract: The classical approach in the investigation of natural oscillations of discrete mechanical oscillatingsystems is the solution of the secular equation for finding the eigenfrequencies and the system of algebraic equations for determining the amplitude coefficients (eigenforms). However, the analytical solution of the secular equation is possible only for a limited class of discrete systems, especially with a finite degree of freedom. This class includes regular chain oscillating systems in which the same oscillators are connected in series. Regular systems are divided into systems with rigidly fixed ends, with one or both free ends, which significantly affects the search for eigenfrequencies and eigenforms. This paper shows how, having a solution for the secular equation of a regular system with rigidly fixed ends, it is possible to determine the eigenfrequencies and eigenforms of regular systems with one or both free ends. Pages of the article in the issue: 88 - 93 Language of the article: Ukrainian PubDate: 2021-12-21 DOI: 10.17721/1812-5409.2021/4.14

Authors:N. D. Yakovenko, O. P. Chervinko, S. M. Yakymenko Pages: 94 - 97 Abstract: In the present work we solve the axially symmetric problem of a half-space under thermal loading. The statement of the problem includes: Cauchy relations, equations of motion, heat conduction equation, initial conditions, thermal and mechanical boundary conditions. The thermomechanical behavior of an isotropic material is described by the Bodner–Partom unified model of flow generalized in the case of microstructure influence on inelastic characteristics of steel. To determine the parameters of the model corresponded to yield stress and yield strength the mixture rule is utilized. The problem is solved with using the finite element technique. The numerical realization of our problem is performed with the help of step-by-step time integration. Equations of the evolution for the inelastic flow model are integrated by the second-order Euler implicit method. The equations of motion are integrated by the Newmark method, whereas the heat-conduction equation is integrated by the first-order implicit method. We use quadrangular isoparametric elements. The parameters of a fine grid are chosen with the help of the criterion of practical convergence of the solutions. The stress state taking into account linear and nonlinear rules of mixtures is described. Pages of the article in the issue: 94 - 97 Language of the article: Ukrainian PubDate: 2021-12-21 DOI: 10.17721/1812-5409.2021/4.15

Authors:A. G. Rudnitskii, M. A. Rudnytska, L. V. Tkachenko Pages: 98 - 107 Abstract: Optoacoustic tomography is effective in applying to the visualization of objects that have a high coefficient of light absorption relative to the environment. Such tasks include, for example, defectoscopy, vascular imaging, detection and monitoring of tumors, diagnosis of porosity of composite materials, high-precision local measurement of the modulus of elasticity. However, the quality of optoacoustic images largely depends on factors such as noise (equipment or environmental noise) and distortion due to the characteristics of the model and the calculation algorithm. The article proposes an iterative algorithm for improving the quality of optoacoustic images, based on the observation that artifacts increase in magnitude with each iteration, while the nature and location of the distortions remain unchanged. Numerical simulations of the propagation of ultrasonic waves in environments close to soft biological tissues have been performed. In terms of eliminating distortion and artefacts inherent to the method of image reconstruction, an iterative filter was found to be highly effective The effectiveness of the approach is manifested in the use of a small number of iterations. Pages of the article in the issue: 98 - 107 Language of the article: Ukrainian PubDate: 2021-12-21 DOI: 10.17721/1812-5409.2021/4.16

Authors:O. A. Kapustian, Y. V. Petlenko Pages: 110 - 115 Abstract: The paper aims to investigate the model of a complicated educational system, built on the basis of general principles of mathematical modeling of complex hierarchical systems. The model consists of three components, the interaction between them is provided by certain conditions. This interaction is the basis for modeling the main parameter of the study - the level of funding required for higher education. An important feature of the modeled system is its controllability, because in a real educational system the controlling influence can be performed as an external management of the system to ensure an average level of funding. For the initial problem of optimal control, which consists of a parabolic equation with a multivalued reaction function of the system, the approximate synthesis problem for a certain class of target functionals with control restrictions is solved. The results of modeling can be used in the development of various models of complex systems, as well as for decision-making in public administration and higher education management. Pages of the article in the issue: 110 - 115 Language of the article: Ukrainian PubDate: 2021-12-21 DOI: 10.17721/1812-5409.2021/4.17

Authors:Valentyn V. Sobchuk, Iryna V. Zamrii, Oleg V. Barabash , Andrii P. Musienko Pages: 116 - 127 Abstract: The article analyzes the concept of process management in an industrial enterprise, which provides functional stability of technological processes using information systems of various types and allows to increase productivity of all production centers while reducing the number of people employed in production and the impact of various destabilizing factors. The research of existing mathematical models of functionally stable complex technical systems and formalization of the main types of procedures of the process of functioning of the intellectual information system of the enterprise is carried out. The method of ensuring the properties of functional stability of the intelligent information system of the enterprise is improved by presenting the functioning of the system in the form of a formalized process, in which the main types of procedures are accumulation of tests, analysis of test links, diagnosing a failed module and restoring the system. This technique allows to ensure the property of functional stability of the information system of the enterprise. Pages of the article in the issue: 116 - 127 Language of the article: Ukrainian PubDate: 2021-12-21 DOI: 10.17721/1812-5409.2021/4.18