An integral equation of the three-dimensional contact problem for an orthotropic half-space (9 independent elastic parameters in Hooke’s law) is obtained where its kernel does not include integrals, but it depends on the solution of a characteristic binary cubic. The interaction between two identical symmetrically embedded punches is considered for the case of the elliptic paraboloids. Galanov’s method of nonlinear boundary integral equations is used for solving the problem with an unknown contact domain that makes it possible to determine simultaneously the contact domain and the contact pressure. The exact solution to one elliptical punch is used for debugging the computer program. Contact pressures, contact zones and pressing forces are calculated for various orthotropic materials at the specified settlement, base forms of the punches, and relative distances between the punches. The orthotropic body model is applicable for describing lots of materials which are in-demand in the machinery and industry: sulfur, Rochelle salt, wolframite, barite, and various wood species.

New mathematical models including an oscillation generator and semi-bounded non-uniform in depth foundation possessing porosity, fluid saturation, and viscoelasticity, are considered. The foundation is represented by a poroelastic layer saturated with gas-liquid mixture, a heterogeneous layer with a viscoelastic coating, and a heterogeneous layer with a subsurface liquid sheet. The foundation of the pack of layers is hard. The operation of the surface oscillator is represented as Fourier series, and the problem of steady-state oscillatory conditions is solved. Applying the Fourier integral transform to the equations that describe continuous media under satisfying boundary conditions allows the construction of integral formulas describing the stress-strain condition in the layer package. A numerical algorithm to study the dependence of the ground-wave propagation on the mechanical and geometrical characteristics of the problem is proposed. The models described are widely used in Geophysics, seismic exploration, construction, railway design, and new material designing.

The work objective is to evaluate how the geometric transformations of the trailing edge of some airfoils affect the aerodynamic characteristics. Such transformations as the trailing edge thickening and rounding are considered. The performance of Wortmann FX 63-137, NACA 23024 и Clark Y is calculated. The effect of the trailing edge changes on their aerodynamic performance at Reynolds numbers from 200,000 to 1,000,000 is assessed. It is shown that the thickening of the trailing edge increases the drag coefficient up to some Reynolds number dependent on the airfoil, and thereafter it reduces. At that the increase of the lift coefficient is observed. The rounding of the trailing edge reduces both the lift and drag forces, and it can lead to the improvement of the airfoil lift-to-drag ratio for certain angles of attack. The data obtained can be used in the manufacture of blades with the profiles discussed.

The work objective is to analyze the attracting manifolds generated under the endmilling. The formation of the periodic stationary trajectories of the tool deformation displacements in relation to the workpiece and their sustainability is previously considered. In this case, the movements form the attracting manifold defined by the trajectories of the periodic changes in the thickness and width of the cut-off layer by each cutter tooth taking into account the deformation displacements. As opposed to the previously considered cases, the paper focuses on the attracting manifolds generated at the buckling failure of the stationary trajectory. It is shown that depending on the system parameters and the process conditions in the dynamic milling system, the attracting manifolds of limit cycles, invariant tori, and strange (chaotic) attractors can be formed. In this context, two cases are analyzed. The first relates to the system parameters (primarily the cutting speed) which allow neglecting the coefficient variations in the differential equations within the impulsive reaction of the system. In the second case, the system parameters vary within the impulsive reaction of the system, and an additional source of the parametric self-excitation is formed in it. Considerable attention is paid to the analysis of the attracting manifold bifurcations in a parameter space: an overview and examples are provided. The attracting sets are analyzed from the perspective of their impact on the quality parameters of the parts production.

The presented study discusses the structures used for restoring the bearing capacity of the main pipeline sections in case of the corrosion damage. It is shown that a welded crimping socket the most frequently used in repair can efficiently reduce stress in the pipe walls only in close joints. To increase the repair structure efficiency, it is proposed to place an elastic layer between a pipe and a socket which can fill the space and transfer a part of load from the pipe to the socket. The elastic layer material must be, first, soft enough to completely fill a gap between the pipe and the socket considering the pipe surface irregularity. Secondly, the material must be characterized by a sufficiently large elastic modulus (in the range of 1000 MPa

The work objective is the optimization of the part cutting process to ensure the specified surface layer quality when using the dynamic methods of the surface plastic deformation (SPD). The research subject is the process of forming the surface layer parameters that ensure the rational SPD processing modes. The adjusted dependences for determining the hardened layer depth, the degree of hardening and roughness of the surface layer are presented. The analytical prediction technique of the parts surface layer quality parameters is developed on the basis of the abovementioned relations. These relations are experimentally tested and can be used for designing high-performance, cost-effective technological processes based on the application of the dynamic SPD methods. The importance of the process optimization based on the criterion of the workpieces durability is noted. A control-flow chart of the dynamic SPD machining optimization is given.

The work objective is to study the wear resistance of the experimental hard alloys (EHA) and complex alloyed high-speed steels (HSS) under cutting, as well as to establish the interconnection between their tribological characteristics and the entropy values (thermal EMF). The experimental studies on the EHA wear are carried out under the longitudinal turning, and HSS - under the friction conditions and drilling operations. Molybdenum, iron, titanium and copper mono carbide alloys are inserted into the cobalt bonds. It is shown that in this case, both entropy and wear-resistance increase (up to 2.5-3 times) as compared to the VK8 base alloy. At the same time the optimum machining spectrum extends. The functional relation between entropy and thermal EMF of the HSS is established, and it is shown that the minimum values of the thermal EMF correspond to the high entropy values. The HSS wear rate under friction is lower and the guaranteed life of drills is higher in steels with a high entropy value. The results obtained should be applied to the forming operations for the cutting-tool materials (CTM) of the maximum wear-resistance, and when developing new CTM.

The work objective is to search for new non-conventional structural low-activated materials. The dispersion-strengthened microalloyed chrome-based alloys that show high mechanical-and-physical properties under the initial condition are described. The results of the comparative studies of the chromium structural behavior obtained by the galvanic and hot spraying methods applied to the piston rings are quoted. This allows identify and assess the advantages of these coatings depending on the performance. Chromium rings wear - engine operating time experimental dependences are shown. The results of the quick and full-scale testing of various engines to determine the effect of the chromium coating composition on the size and behavior of the friction factor depending on the load are described. The feasibility of applying the diesel piston ring coatings under study is assessed.

The paper objective is to solve the issues related to the increase of reliability, profitability, and safety of modern washing machines on the basis of decrease in dynamic loads under whizzing. The principles of forming operation factors in the liquid self-balancing device (SBD) with the displaced fluid (on which design a number of patents are received) when passing a resonance zone under whizzing are considered. A comparative analysis of the SBD function implementation for the conditions of forming an adequate force response to the force input for two earlier established most rational conditions: for the force generated by resonance, and if reached the maximum unbalanced force, is carried out. The dependences of the SBD internal capacity traverse on the design factors of the SBD and the cleaning unit suspension system, and also on the operation parameters of the extraction process, are obtained. The research results are submitted in diagrams. The results analysis allows drawing the conclusions on the obtained data application perspectiveness to ensure maximum efficiency of the process of balancing irregular masses of textiles in the drum during the whizzing cycle.

It is shown that the magnetodynamic treatment implemented under the conditions of a rotating electromagnetic field is a kind of the dynamic methods of hardening free-moving indenters by the convection flows. It is proposed to use analytic dependences obtained by the approximation of the test data of effect of the surface plastic deformation (SPD) dynamic methods on the component hardened layer quality to evaluate quality parameters of the surface layer formed under the magnetodynamic processing. The resulting expression describing the experimentally established fact that the main contribution to the formation of the quality parameters of the hardened layer is made by the initial continuous coating of the treated surface with plastic prints is obtained. An analytical functional model establishing the surface layer quality parameters behavior depending on the multiplicity of covering the surface by the shock-pulse impact of the free-moving indenters is proposed. A comparative analysis of the experimental data obtained in the process of the magneto-dynamic processing with the calculated ones shows the applicability of the proposed models for predicting quality parameters of the hardened surface layer.

The work objective is to describe the numerical solution method for the stationary Schrödinger equation based on the application of the integral equation identical to the Schrödinger equation. The structure of this integral equation is close to the structure of the Fredholm equation of second kind and allows obtaining the problem numerical solution. The method under study allows finding the energy eigenvalues and eigensolutions to the quantum-mechanical problems of various dimensions. The test results of the solving problems method for one-dimensional and two-dimensional quantum oscillators are obtained. The found numerical values of eigenenergy and eigenfunctions of the oscillator are compared to the known analytical solutions, and then the error of result is evaluated. The highest accuracy of the solution is obtained for the first energy levels. The numerical solution error increases with the number of the energy eigenvalue. For the subsequent energy level, the error increases almost by an order of magnitude. The solution error for the fourth energy level is less than 2% if the integration domain contains 500 elements. If the energy level is degenerate, it is possible to obtain all eigenfunctions corresponding to the given level. For this purpose, various auxiliary functions the symmetry of which is coherent with the eigenfunction symmetry are used.

A method of problem solution of the eigenvalues and eigenfunctions for the Helmholtz equation in the domains with arbitrary configuration is worked out. In developing the approach of the numerical solution of problems, the point-source method (PSM) is used. The proposed method is based on the analysis of the condition number of the PSM system or error of the problem numerical solution. The concept of “eigenvalues criterion” is introduced. The research result is a developed effective method - an algorithm for solving problems of eigenvalues and eigenfunctions for the Helmholtz equation. It is shown that at the approach of the Helmholtz parameter to the problem eigenvalue, the condition number of the PSM system and the error of the numerical solution rise sharply. Therefore, the dependence of the condition number of the PSM system or the error of the problem numerical solution can be calculated from the Helmholtz parameter. Then, according to the position of the maximum of the obtained dependences, the eigenvalues of the Helmholtz equation in a given domain are found. It allows searching the eigenvalues. After finding the eigenvalues, it is possible to proceed to the determination of the eigenfunctions. At that, if the eigenvalue appears degenerate, that is some eigenfunctions correspond to it, then it is possible to find all the eigenfunctions taking into account the symmetry of the solution domain. The two-dimensional and three-dimensional test problems are solved. Upon the results obtained, the conclusion about the efficiency of the proposed method is made.

The work objective is to build and investigate the modified adaptive method of minimum amendments (MAMMA) which is destined for the numerical simulation of the two-phase compressible fluid filtration in porous media. This approach allows overcoming the known use limitations of other methods of the finite-difference equations solution, such as: crucial differential pressures acting on the oil-and-water bearing formation; and the compressibility of the medium at the considerable gas content in the oil phase. An approximation method - an explicit one for defining the function of water saturation, and an implicit one for the pressure function computation - is selected as the research basis. When setting the initial boundary value problem and its sampling, the process of the two-phase compressible fluid filtration in the space-dimensional domain with the lateral area bounded below by the subface of stratum, and above - by the bed top, is considered. A two-layer iterative method of the variational type - a modified method of minimal amendments adapted for solving finite-difference equations of the two-phase compressible fluid with a non-selfadjoint operator under the most general assumptions on the properties of the grid-problem operator is built. It is shown that a MAMMA has the asymptotic convergence rate characteristic of the “classical” alternate triangular method that does not use the Chebyshev acceleration technique and can be applied to the problems with a self-adjoint operator. Numerical experiments have confirmed the high efficiency of MAMMA. It is established that to achieve the specified accuracy, the number of iterations at the MAMMA reduces to 3-20 times as compared to the method of Seidel and the overrelaxation method.

The possibilities of using the fuzzy clustering method in the automated system of evaluation working efficiency of the experts’ validation status on the assessment of the state final certification in the form of the Unified State Exam (USE) are considered. Various parameters of experts’ evaluation and their calculation formula are presented. The parameters are divided into the following groups: basic statistical parameters, advanced statistical parameters, normalized parameters, and quality parameters. The application of the fuzzy clustering method using a weighted Euclidean metric is illustrated by the case of the assessment of the experts’ work in the subject commission on mathematics (profession-oriented level). The experiments are performed with and without noises with the subsequent conclusions of the impact of noises on the results. Besides, the experimental results are presented in tables and graphs with partitioning of experts into clusters.

The work objective is to use the capacities of the science-intensive and universal mechanical CAD TP by adapting them to solve complex specialized problems of the footwear process design that will allow industry enterprises transfer to a whole new level of solving design-engineering problems. The theoretical and methodological basis of the study is the theory of complex systems simulation; the operations research theory; the method of system- structural analysis of objects; the classification and coding technique; the methodology of data systems development, databases and knowledgebases creation; the mathematical logic; the method of aprior ranking of factors. On the basis of the standard shoemaking techniques, and similar production techniques used in the modern shoe factories, an aggregate engineering process of the shoes assembly of various construction methods consisting of 310 production steps, each of which is assigned a code, is developed. A list of structuring units characterizing the design-processing footwear specifications is worked out to formalize data on the shoe bottom parts. And the clusters that affect a particular set and operational sequence are allocated within each block. On the basis of the data obtained from the system- structural analysis, structural-logical models of designing the shoes assembly of various construction methods are developed. A schematic form clearly reflects the interrelation logic between the clusters and operational steps. The database “ObuvPro” that allows solving problems on the design process in the shoe factory is developed to ensure the operational, regulatory, and information support for the territorial-production footwear companies.

The weighed and unweighted minimal set-cover problem, its applicability for the solution of the major optimization practical tasks, such as arrangement of service points, assignment of crews in transport, as well as the integrated-circuit and conveyer lines designing is considered. The paper objective is to describe methods of improving efficiency of this task solution. The principle of operation of the genetic algorithm and the applicability of its modification as a method of the solution to the set-cover problem are defined. Greedy strategy of Chvatal for the set-cover problem solution is considered. An exhaustive algorithm as an exact algorithm for the solution of small-size tasks is developed. The modified genetic algorithm developed by Nguyen M. H. is described. A software tool for comparing the performance of these algorithms is created. It is concluded that the solution of the set-cover problem by our genetic algorithm modification is more effective than by the genetic algorithm of Nguyen M. H. or the greedy strategy. And the results obtained in small-size tasks are noted for insignificant error.

The personnel safeguards issues and the evacuation algorithm on the urban passenger ropeways are considered. The paper objective is to formulate the requirements for personnel to ensure safety during the evacuation of passengers with the use of the redundant systems. The innovative constant redundant system with parallel redundancy is described. The implementation of the required function by the object is provided by the remaining elements switchless in case of any element breakdown in the reserve group. Technical, technological and organizational risks for various cases of critical failures occurrence are considered. In the result, extra operations, professional knowledge and skills are formulated; besides, risks of the personal injury are assessed by expertise. Recommendations on the decision-making to reduce the injury risk for the personnel taking into account the risk parameters are offered.

The study aims at finding solutions to ensure the safe operation of welders for hard-to-reach objects in the limited enclosed space with a rapid rise in the content of harmful substances and ambient air temperature. The research subject is the systems of lung protection for welders working under the constrained conditions with high gaseousness and lack or insufficient natural circulation of the ambient air. The research method is a numerical simulation of heat and mass transfer of the harmful substances, welding aerosols, and excess heat under the transient conditions in the three-dimensional formulation. The work methodology is the construction of mathematical models of heat and mass transfer of the contaminants on the basis of the classical gas flue models with their adaptation to the specific boundary conditions, the characteristics of different types of welding in hard-to-reach places including deepwater welding, and taking into account various types of heat transfer. A classification of hard-to-reach objects and types of welding is worked out. It is shown that the resulting mathematical models allow specifying in detail the physical fields of dangerous and harmful factors impact on the welders.