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Problem. Among the many problems of the solid mechanics, there is a whole class of problems that are related to inverse problems. In turn, among the inverse problems, many problems are ill-posed. Obtaining an exact analytical solution of such problems is related to certain mathematical difficulties and requires using special methods. Goal. The goal of the study is to obtain analytical solutions for inverse problems of the identification of non-stationary load and the control of nonstationary vibrations of a cylindrical shell with asymmetric boundary conditions. Methodology. In this investigation, a refined theory of medium-thickness shells was used. Fourier series expansion, the theory of integral equations and the Laplace transform were used to obtain the solution of the direct problem. Tikhonov’s regularization method was used to solve inverse problems. Results. As a result of the investigation, the solutions of two inverse problems of the solid mechanics were obtained. The first task is to identify a fixed and moving concentrated axisymmetric non-stationary force acting on a cylindrical shell, based on the displacement values at any point of the shell; identification of two fixed concentrated forces. The second task is to control vibrations at any point of the cylindrical shell by introducing an auxiliary concentrated force. Numerical results obtained demonstrate the fulfillment of the control criterion as a result of the action of the given and auxiliary force. Originality. Analytical solutions of the inverse problems of the solid mechanics for a cylindrical shell of medium thickness with asymmetric boundary support conditions are obtained. Practical value. The technique received allows effective identification of an unknown non-stationary load. It’s important for the rational design of reliable cylindrical shell structures. Its use also makes possible to create a theoretical basis to control the deflected mode parameters of cylindrical shell structural elements.

The paper presents a mathematical modeling of dynamic stress-strain state of the rotation of the shell elasto-viscoplastic material. We solve the modified system of S. P. Timoschenko’s partial differential equations by constructing a system of equations on moving surfaces of the gap with the initial conditions in the form of a shock at the end, written in the form of a power series in time, whose coefficients have initial conditions for the differential equations. The solution is presented in the form of the Taylor’s row series up to the fourth order in the shell coordinate. To simulate the waves reflected from the boundaries, the conditions at the boundary of two types (rigidly restrained and stress-free), independent of time, are introduced. A set of programs written in Fortran 90 on the Code::Blocks platform is developed. Two programs for the simulation of dynamic deformation of shell in elastic and elasto-viscoplastic state are implemented. We use the difference representation of the derivatives, the calculation of the integrals by the trapezoid method with a given step of partitioning the segment. The result of the programs is the grid functions of the coefficients of the Taylor rows, which are used to construct the displacement graphs as functions of time and the longitudinal coordinate of the shell.

The mathematical simulation of diffraction of TEM-wave on the periodic 2D structures of graphene micro-ribbons with finite length is performed using the computational algorithm developed by projection method for the normal incident TEM-wave at terahertz frequency range.The results of electrodynamic calculation of the transmission coefficient for the TEM mode through the graphene micro- and nanoribbons with finite length depending on the frequency were obtained for different values of chemical potential (the external electric field) for different sizes of ribbons and the periodicity of the array.

The asymptotic solution to the problem on the hydrodynamic calculation of the porous double - layer r a dial finite length bearing is provided on the eccentricity ratio levels. The solution is based on full nonlinear Navier — Stokes equations, and on Darcy’s equation. The distinction of the suggested design model is that it allows for the perme ability anisotropy of the porous layers, and availability of a greasing source. A case when greasing forcibly gets thro ugh pores of an external porous layer is considered. A velocity and pressure field in the lubricant and porous layers is found. As a result, analytical expressions for the main performance data of the bearing ar e obtained. The effect of the permeability an isotropy of porous layers, nonlinear factors, as well as the influence of a greasing source on the basic performance chara c teristics of the bearing are evaluated.

A method for calculation of electromagnetic parameters of single vertical grounding electrode of finite length has been developed. The results of mathematical simulation of electromagnetic parameters of the stated grounding electrodes are presented in the paper. The paper proposes approximate analytical expressions for calculation of grounding electrode resistances and voltages considering electromagnetic field attenuation in ground and electrode. These expressions are acceptable for application.

In the paper, based on the solutions of an infinite system of non-regular linear algebraic equations, an approach is found that allows to reduce essentially the number of shortened equations through which solutions the stresses and displacements are calculated as well as to determine the behavior of the stresses when approaching to the angular point. [ABSTRACT FROM AUTHOR]

The electric field strength is one of the main factors influencing sensitive microprocessor equipment and personnel on power stations and substations, power lines. Determining its level is an important applied task for ensuring the safe operation of electrical installations. The aim is to develop calculation relationships for determining the electric field strength created by the busbar of high-voltage substations in the working areas of personnel. The solution of the problem was based on the use of the method of equivalent charges to determine the strength of the electric field created by the complex busbar of high-voltage substations. Methodology. The development was based on solving the problem of the potential of the electric field of a point charge located in a dielectric half-space for a cylindrical coordinate system. By representing the electrode in the form of a set of point charges and subsequent integration, an expression for calculating the potential is obtained, created by a busbar of arbitrary orientation of finite length in an analytical form. Using the principle of superposition of fields and the definition of the derivative, expressions were obtained for calculating the vertical component of the electric field strength at given heights. Results. Based on the obtained expressions, using Visual Basic, the simulation of the distribution of the electric field strength under a three-phase power line with a voltage of 150 kV was performed. Comparison with the known calculation results obtained on the basis of analytical expressions for infinitely long conductors showed that the obtained expressions have an error of no more than 7%. The scientific novelty lies in the fact that for the first time expressions were obtained for determining the electric field strength created by a system of electrodes of finite length, based on the analytical method for solving differential equations. Practical significance. The proposed technique is implemented as a test module of the LiGro specialized software package, which allows modeling complex busbar systems typical for power stations and substations and power lines. A test calculation was carried out for an operating substation of regional electric networks with a voltage class of 110 kV. By comparing the duration of the calculation of switchgears with a diagonal of about 500 m, it was found that the calculation time in the LiGro complex based on the analytical method is several tens of times less than the calculation based on the finite element method. In addition, a more powerful computer was used for the end element simulation.

Copyright of National Academy of Sciences of Armenia Reports / Doklady Nacionalnaâ Akademiâ Nauk Armenii is the property of National Academy of Sciences of the Republic of Armenia and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Journal of Guangxi Normal University - Natural Science Edition is the property of Gai Kan Bian Wei Hui and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of National Academy of Sciences of Armenia Reports / Doklady Nacionalnaâ Akademiâ Nauk Armenii is the property of National Academy of Sciences of the Republic of Armenia and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of National Academy of Sciences of Armenia Reports / Doklady Nacionalnaâ Akademiâ Nauk Armenii is the property of National Academy of Sciences of the Republic of Armenia and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Proceedings of the National Academy of Sciences of Armenia. Mechanics is the property of National Academy of Sciences of the Republic of Armenia and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

The simplest transformation model of the dynamic deformation along the length of a rod-strip consisting of two segments was constructed. The model is based on the classical geometrically linear Kirchhoff–Love model for the unfixed segment, while the fixed segment of finite length is assumed to be connected to a rigid and immovable support element through elastic interlayers. On the fixed segment, the deflections of the rod and interlayers were considered zero. For axial displacements within the thicknesses of the rod and interlayers, approximations were adopted according to S.P. Timoshenko’s shear model, subject to the conditions of continuity at the points where they connect to each other and immobility at the points where the interlayers connect to the support element. The conditions for kinematic coupling of the unfixed and fixed segments of the rod were formulated. Taking them into account and using the D’Alembert–Lagrange variational principle, the equations of motion and boundary conditions for the considered segments were derived, and the conditions for force coupling of the segments were obtained. With the help of the derived equations, exact analytical solutions of the problems of free and forced harmonic vibrations of the rod of the studied type were found. These solutions were employed in the numerical experiments to determine the natural modes and frequencies of bending vibrations, as well as the dynamic response during the resonant vibrations of the rod-strip made of a unidirectional fibrous composite based on ELUR-P carbon tape and XT-118 binder. The findings show a significant transformation of transverse shear stresses when passing through the boundary from the unfixed segment of the rod to the fixed one, as well as their pronounced localization in the region of the fixed segment near this boundary.

The article deals with the issue of determining the necessary dependencies between the characteristic elastic and geometric parameters that ensure the boundedness of the contact pressures (stresses) at the boundaries of the contact areas in the classical contact problems of the theory of elasticity, namely, in plane and axisymmetric problems, as well as in the problem of contact interaction of a stringer of finite length with an elastic half-plane. To determine the size of contact areas or other parameters when contact stresses are restricted, along with the widely used principle of continuity of contact stresses on smooth contact surfaces, the Castigliano energy principle is also used for elastic bodies, for which Hooke's law is valid. Based on the identity of the results obtained by both principles, their equivalence is shown. In addition, new results have been obtained for these two classical contact problems, supplementing those previously obtained. In the problem of the contact of a stringer with an elastic halfplane, by the principle of stress continuity, the conditions for the boundedness of contact stresses at the ends of a stringer are obtained only in two special cases. In addition, new results have been obtained for these two classical contact problems, supplementing those previously obtained. In the problem of the contact of a stringer with an elastic half-plane, by the principle of stress continuity, the conditions for the boundedness of contact stresses at the ends of a stringer are obtained only in two particular cases. [ABSTRACT FROM AUTHOR]

Copyright of National Academy of Sciences of Armenia Reports / Doklady Nacionalnaâ Akademiâ Nauk Armenii is the property of National Academy of Sciences of the Republic of Armenia and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Background Today, one of the problems of modern society in the health are the high rates of mortality from cardiovascular disease.Aim of study Study of physico-mechanical properties of the vascular implants and aortic wall to support an adequate choice of the plastic material when performing reconstructive surgeries in patients of various age groups.Material and methods As an object of study, sections of the anterior wall of the thoracic and abdominal aorta were taken from 30 corpses. Study groups: Group 1 - from 33 to 60 years, Group 2 - from 60 to 92 years. We also evaluated the physical and mechanical properties (finite length, plastic strain ratio, tensileuniaxial load (longitudinal)) of the three vascular implants made of polyethylene terephthalate and polytetrafluoroethylene. Kruskal–Wallis H test was used to determine the significance of differences.Results The finite length of the aortic wall varies slightly in the study groups, while plastic strain rate of the abdominal aorta in Group 1 was 1.3 times higher than that in the chest. When assessing physical and mechanical properties of thoracic and abdominal aorta, values of “finite length” (1.4 times) and “plastic strain ratio” (1.5 times) were higher in Group 1, and the value of the tensile load indicator is higher in Group 2 compared to Group 1 (1.47 times). The tensile load of the thoracic aorta in Group 1 is 14 H and 27 H lower than the samples of Groups 1 and 3, respectively, but 1.6 times higher than the sample of Group 2. The tensile load of the abdominal aorta of Group 1 is 13 H higher compared to the sample of Group 1 and twice higher compared to the sample of Group 2. The study Group 2 values of tensile load for thoracic aorta are 1.3 times, 2.4 times and 1.16 times higher than respective implant values (sample 1, 2 and 3, respectively).Conclusion We used samples of Group 3 vascular implants for replacement of abdominal aorta and Group 1 implants for thoracic aorta. In study Group 2 vascular implants of Group 3 may be used for abdominal and thoracic aorta replacement due to the proximity of the values of the evaluated characteristics.

The stress state of a piecewise-homogeneous elastic strip is considered, composed of two strips from different materials, one of which contains a crack of finite length, perpendicular terminating at the interface of materials. It is assumed that the composite strip on the side of the strip, in which there is a crack, is rigidly restrained, and the other side is free from stress. The strip is deformed under the influence of a normal load symmetrically applied to the crack edges. The stress intensity factors at the crack tips, contact stresses at the material interface, as well as the shape of the crack opening were determined. The study of the dependence of the indicated values on the elastic constants of the strip materials and the relative widths of the strips has been carried out. [ABSTRACT FROM AUTHOR]

Subject and Purpose. Narrow-band filters are among the basic components of modern communication systems, instruments for spectroscopy, high-sensitivity sensors, etc. Photonic crystal structures open up broad possibilities for creating compact-sized, narrow-band filters in the optical and terahertz ranges. Tuning of spectral characteristics of photonic crystal filters is usually carried out through introduction of certain elements into their structure that are sensitive to external electric and magnetic fields. This work has been aimed at investigating electrodynamic characteristics of one-dimensional magnetophotonic crystals with structural layers characterized by "hyperbolic" dispersion, and suggesting a multichannel, narrow-band filter on their base. Methods and Methodology. The dispersion equation for excitations in an infinite magnetophotonic crystal has been obtained within the framework of the Floquet-Bloch theory, with the use of fundamental solutions of Hill’s equation. The transfer matrix approach has been used to obtain an analytical expression for the transmission coefficient. Results. The band diagram of the one-dimensional magnetophotonic crystal has been analyzed for the case where one of the layers on the structure’s spatial period is characterized by a hyperbolic dispersion law. The areas of existence of surface wave regimes have been found for such layers for the case of normal incidence of the wave upon the finite-seized magnetophotonic crystal. Frequency dependences of the transmission coefficient are characterized by a set of high-Q resonant peaks relating to Fabry-Perot resonances in a periodic structure of finite length. Conclusions. Application of a finite-seized, one-dimensional magnetophotonic crystal is considered as of a means forachieving multichannel optical filtering and formation of a frequency comb. Expressions for the dispersion equation and transmission coefficient have been obtained within the framework of the Floquet-Bloch theory and with the use of the transfer matrix. The feasibility of surface mode excitation has been shown for gyrotropic layers of the periodic structure characterized by a hyperbolic dispersion law, for the case of normal incidence upon the magnetophotonic crystal. The spectral response of the filter contains narrow-band peaks with a high transmission efficiency. By increasing the number of the structure’s periods it is possible to form a frequency comb, which effect can be useful for applications in metrology and modern optical communication systems.

Copyright of Proceedings of the National Academy of Sciences of Armenia. Mechanics is the property of National Academy of Sciences of the Republic of Armenia and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Proceedings of the National Academy of Sciences of Armenia. Mechanics is the property of National Academy of Sciences of the Republic of Armenia and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Subject of Research.The paper deals with questions of operation safety for continuous welded rail track under dynamic loads. We study the gap value formed in the rail bar in view of its possible break under the conditions of dynamic loads caused by the moving train and varying temperatures. Occurring strains are linked to a variable stiffness of the rail track base (ballast). Methods. We studied longitudinal oscillations in a semi-infinite rod under dynamic loads and in the process of thermal expansion of the finite length rod on the elastic-damped foundation. Oscillatory processes in a semi-infinite rod on an elastic foundation was modeled using the extended equation of longitudinal vibrations, taking into account the reaction of the foundation, performing the role of a damper. To determine the state of the model after the damping of oscillations we used the time limiting transition. In solving the problem of thermal expansion of the finite length rod on the elastic foundation we used equilibrium equation for small fragment, which takes into account the thermal load and the elastic force of both the rod and the foundation. Main Results. We have calculated the maximum value of the dynamic displacement of the rod end at different values of parameters. We have obtained the distributions of stresses and displacements along the rod length for different stiffness distribution in the elastic foundation. In the event of accidental break in the rail, the gap value after the damping of oscillations is set at the level of 5-7 cm. We have shown that the amplitude of these oscillations can reach 10-12 cm. The availability of limited stiffness change zones in the rail base causes no significant deformation of rail bars, but can lead to the growth of the gap in the rail bar break. Practical Relevance. The proposed model can be used in the calculation and design of continuous welded rail tracks.

The technique of calculating the radial sliding bearings of infinite and finite length running on a double-layer stratified lubricant is presented. The technique is based on the use of the self-simulated variable that allows obtaining the exact self-similar problem solution both in polar and in cylindrical coordinates, as well as the parameter value characterizing the interface of the stratified layers. These tasks are complicated by the simultaneous consideration of the lubricant viscosity-pressure relation, the presence of the friction-adapted supporting profile of the bearing bush, and the effect of the axial lubricant feed in the finite-length bearing. As a result, computational models are obtained for two-layer stratified lubricants the numerical analysis of which allows establishing the impact of variables on the basic bearings performance - components of the bearing capacity vector, friction forces, as well as optimal values of the reference profile parameter, the lubricant supply, and its viscous ratio in the stratified layers.

The dynamic response of a finite length thermo-piezoelectric rod with variable material properties is investigated in the context of the fractional order theory of thermoelasticity. The rod is subjected to a moving heat source and fixed at both ends. The governing equations are formulated and then solved by means of Laplace transform together with its numerical inversion. The results of the nondimensional temperature, displacement and stress in the rod are obtained and illustrated graphically. Meanwhile, the effects of the fractional order parameter, the velocity of heat source and the variable material properties on the variations of the considered variables are presented, and the results show that they significantly influence the variations of the considered variables. [ABSTRACT FROM AUTHOR]

The problem of axisymmetric vibrations of a functionally graded finite hollow cylinder is considered. The ends of the cylinder are thermally insulated and are in a sliding fit. Zero temperature is maintained on the inner surface of the cylinder, free from stress, and a combined thermal and power load acts on the outer surface. The direct problem after applying the Laplace transform is solved based on the method of separation of variables. A set of canonical linear systems of differential equations of the 1st order is obtained, the solution of each of which is obtained numerically using the shooting method. The coefficient inverse problem of finding the thermomechanical characteristics of a finite length cylinder using additional information in Laplace transforms, given on the outer surface of the cylinder, is posed. The dimensionless thermomechanical characteristics of the cylinder were restored in two stages. At the first stage, the initial approximation was determined in the class of positive bounded functions. At the second stage, based on the solution of the corresponding Fredholm integral equations of the 1st kind, corrections of the reconstructed functions were found, and an iterative process of their refinement was constructed. In the course of computational experiments, it was found that monotonic characteristics are restored with good accuracy; the reconstruction procedure is resistant to input information noise.

The method of spatial average of Maxwell equations for VHF electromagnetic waves propagation through a wooland, as heterogeneous dielectric medium with losses, retrieves analytical expressions depicting damping of monochromatic upright - of polarized radio waves. The wood massif is submitted by the way arbitrary of distributed vertical finite-length cylinders. [ABSTRACT FROM AUTHOR]

The aim of research is to compare two calculation methods using the example of solving the axisymmetric thermoelasticity problem. Methods. The calculation of a thick-walled cylindrical shell on the temperature effect was carried out by the numerical-analytical method and the finite element method, implemented in the LIRA-CAD software package. The shell consists of three layers: two layers of heat-resistant concrete and an outer steel layer. In the calculation, a piecewise linear inhomogeneity of the shell due to its three-layer structure and continuous inhomogeneity caused by the influence of a stationary temperature field is taken into account. The numerical-analytical method of calculation involves the derivation of a resolving differential equation, which is solved by the sweep method, it is possible to take into account the nonlinear nature of the deformation of the material using the method of successive approximations. To solve this problem by the finite element method, a similar computational model of the shell was constructed in the LIRA-CAD software package. The solution of the problem of thermoelasticity for an infinite cylinder (under conditions of a plane deformed state) and for a cylinder of finite length with free ends is given. Results . Comparison of the calculation results is carried out according to the obtained values of ring stresses σθ.

In the article we consider methods for calculating the entropy of two-dimensional words of finite length over in finite alphabet, with reference to the problem of self-organization of rod-like particles on a torus. Monitoring the change in entropy allows a better understanding of the processes occurring during self-organization. To calculate the entropy, we use the sliding window method, which consists in finding unique configurations of windows in the matrix and counting the number of these configurations. Then, the entropy is calculated from the obtained counter values. The basic time complexity is required by the algorithms for counting the number of windows. Value of entropy is calculated quickly using known counters. Therefore, we purpose variants of asymptotically time efficient algorithms for counting the number of identical windows using the methods of conversion into an index, a search tree and a prefix tree. In this case, entropy is always calculated in the same way. The results of an experimental study of the software operating time implementations of these algorithms are presented: the minimum, maximum, and average operating time on a number of experiments for certain window sizes. The conclusions about the effectiveness of various algorithms, depending on the size of the window were made. We show the non-representativeness of the entropy calculated by the direct path is due to the disproportionately large number of possible existing unique configurations, even with small dimensions of the alphabet and the window. Finally, we purpose the method of quickly generalized entropy of two-dimensional words calculating instead of direct entropy.

Introduction. The investigation of the contact problems for cylindrical bodies is urgent due to the engineering contact strength analysis on shafts, cores and pipe-lines. In the present paper, a new contact problem of elastostatics on the interaction between a rigid band and an infinite two-layered cylinder, which consists of an internal continuous cylinder and an outer hollow one, with a frictionless contact between the cylinders, is studied. The outer cylindrical band of finite length is press fitted. By using a Fourier integral transformation, the problem is reduced to an integral equation with respect to the unknown contact pressure.Materials and Methods. Different combinations of linearly elastic materials of the composite cylinder are considered. Asymptotics of the symbol function of the integral equation kernel at zero and infinity is analyzed. This plays an important role for the application of the analytical solution methods. A key dimensionless geometric parameter is introduced, and a singular asymptotic technique is employed to solve the integral equation.Research Results. On the basis of the symbol function properties, a special easily factorable approximation being applicable in a wide variation range of the problem parameters is suggested. The Monte-Carlo method is used to determine the approximation parameters. The asymptotic formulas are derived both for the contact pressure, and for its integral characteristic. Calculations are made for different materials and for various relative thickness of the cylindrical layer including thin-walled layers.Discussion and Conclusions. The asymptotic solutions are effective for relatively wide bands when the contact zone length is bigger than the diameter of the composite cylinder. It is significant that the method is applicable also for those cases when the outer cylindrical layer is treated as a cylindrical shell. The asymptotic solutions can be recommended to engineers for the contact strength analysis of the elastic barrels with a flexible coating of another material.

Closedness equation for biorthogonal system of functions has been derived. Closed-loop function systems characteristics are defined. They allow to solve contact problems of elasticity theory for finite bodies – a rectangle, a circular plate, a cylinder of finite length, etc.

The linear system of partial differential equations is considered. It is assumed that there is the irreversible linear operator in the main part of the system, which enjoy the skeletal decomposition. The differential operators is such system are assumed to have a sufficiently smooth coefficients. In the concrete situations the domains of such differential operators are linear manifolds of smooth enough functions with ranges in Banach space. Such functions are assumed to satisfy an additional boundary conditions. The concept of a skeleton chain of linear operator is introduced. It is assumed that the operator generates a skeleton chain of the finite length. In this case, the problem of solution of given system is reduced to a regular split system of equations. The system is resolved with respect to the highest differential expressions taking into account the certain initial and boundary conditions. The possible generalization of the approach and the application to the formulation of boundary value problems in the nonlinear case. Presented results develop the theory of degenerate differential equations in the monographs N. A. Sidorov [General regularization questions in problems of branching theory. (1982; MR 87a:58036)]; N. A. Sidorov, B. V. Loginov, A. V. Sinitsyn and M. V. Falaleev [Lyapunov–Schmidt methods in nonlinear analysis and applications (Math. Appl. 550, Kluwer Acad. Publ., Dordrecht) (2002; Zbl 1027.47001)].

Introduction. Prof A.V. Belokon’s (1941-2013; former Rector of Rostov State University and President of Southern Federal University) PhD thesis was devoted to asymptotical methods in contact problems of the elasticity theory for bodies of cylindrical shape (1969). In the present paper, a contact problem of the elasticity theory on torsion of an infinite hollow cylinder by a rigid insert is investigated. The outer cylinder surface is rigidly fixed. The insert of a finite length is inside the cylinder. In 1971, this problem was formulated and analyzed by A.V. Belokon. He reduced it to an integral equation with respect to the unknown contact stress by using the Fourier integral transformation. A.V. Belokon derived a complete solution to this problem for the case of thick-walled cylinders when the kernel symbol of the integral equation can be approximated by the function corresponding to the torsion of the space with a cylindrical cavity. In the present paper, the case of thin-walled cylinders being complementary to Belokon’s case is mainly considered. Materials and Methods . The cylinder material is supposed to be linearly elastic. The method of integral transformations is used to solve the problem. The singular asymptotic method is applied to solve the integral equation. Research Results . On the basis of studying the properties of the integral equation kernel symbol function, a new special easily factorable approximation applicable for any cylinder thickness is suggested. The Monte-Carlo method is used to determine optimal approximation parameters. Calculations are mainly made for thin-walled cylinders. As a result, an analytic asymptotical solution to the integral equation is obtained. Discussion and Conclusions . The new solution can be effective for relatively long rigid inserts whose length is bigger than the internal diameter of the cylinder. The method based on new approximation remains applicable also for the cases when a cylinder can be regarded as a cylindrical shell. The asymptotical solution can be recommended to engineers for the strength analysis of elastic machine parts of the cylindrical form twisted by a rigid insert.