SPECTRAL SOLUTION TO A PROBLEM ON THE AXISYMMETRIC NONLINEAR DEFORMATION OF A CYLINDRICAL MEMBRANE SHELL DUE TO PRESSURE AND EDGES CONVERGENCE (р. 6–13)

A geometrically and physically nonlinear model of a membrane cylindrical shell, which has been built and tested, describes the behavior of a airbag made of fabric material. Based on the geometrically accurate relations of “strain-displacement”, it has been shown that the equilibrium equations of the shell, written in terms of Biot stresses, together with boundary conditions acquire a natural physical meaning and are the consequences of the principle of virtual work. The physical properties of the shell were described by Fung’s hyper-elastic biological material because its behavior is similar to that of textiles. For comparison, simpler hyper-elastic non-compressible Varga and Neo-Hookean materials, the zero-, first-, and second-order materials were also considered. The shell was loaded with internal pressure and convergence of edges. The approximate solution was constructed by an spectral method; the exponential convergence and high accuracy of the equilibrium equations inherent in this method have been demonstrated. Since the error does not exceed 1 % when keeping ten terms in the approximations of displacement functions, the solution can be considered almost accurate. Similar calculations were performed using a finite element method implemented in ANSYS WB in order to verify the results. Differences in determining the displacements have been shown to not exceed 0.2 %, stresses – 4 %. The study result has established that the use of Fung, Varga, Neo-Hookean materials, as well as a zero-order material, lead to similar values of displacements and stresses, from which displacements of shells from the materials of the first and second orders significantly differ. This finding makes it possible, instead of the Fung material whose setting requires a significant amount of experimental data, to use simpler ones – a zero-order material and the Varga material.

Solving the problem of continuum mechanics has revealed the defining generalizations using the function argument method. The aim of this study was to devise new approaches to solving problems of continuum mechanics using defining generalizations in the Cartesian coordinate system.
Additional functions, or the argument of the coordinates function of the deformation site, are introduced into consideration. The carriers of the proposed function arguments should be basic dependences that satisfy the boundary or edge conditions, as well as functions that simplify solving the problem in a general form.
However, there are unresolved issues related to how not the solutions themselves should be determined but the conditions for their existence. Such generalized approaches make it possible to predict the result for new applied problems, expand the possibilities of solving them in order to meet a variety of boundary and edge conditions. The proposed approach makes it possible to define a series of function arguments, each of which can be a condition of uniqueness for a specific applied problem. Such generalizations concern determining not the specific functions but the conditions of their existence. From these positions, the flat problem was solved in the most detailed way, was tested, and compared with the studies reported by other authors.
Based on the result obtained, a mathematical model of the flat applied problem of the theory of elasticity with complex boundary conditions was built. Expressions that are presented in coordinateless form are convenient for analysis while providing a computationally convenient context. The influence of the beam shape factor on the distribution of stresses in transition zones with different intensity of their attenuation has been shown.
By bringing the solution to a particular result, the classical solutions have been obtained, which confirms its reliability. The mathematical substantiation of Saint-Venant's principle has been constructed in relation to the bending of a beam under variable asymmetric loading.
Keywords: generalized approaches, function argument, Cartesian coordinates, Laplace equations, Cauchy-Riemann relations. Materials of beams, plates, slabs, strips have been commonly applied in various fields of industry and agriculture as flat elements in the structures for machinery and construction. They are associated with the design of numerous engineering structures and facilities, such as the foundations of various buildings, airfield and road surfaces, floodgates, including underground structures.
This paper reports a study into the interaction of the material (of beams, plates, slabs, strips) with the deformable base as a three-dimensional body and in the exact statement of a three-dimensional problem of mathematical physics under dynamic loads.
The tasks of studying the interaction of a material (beams, plates, slabs, strips) with a deformable base have been set. A material lying on a porous water-saturated viscoelastic base is considered as a viscoelastic layer of the same geometry. It is assumed that the lower surface of the layer is flat while the upper surface, in a general case, is not flat and is given by some equation.
Classical approximate theories of the interaction of a layer with a deformable base, based on the Kirchhoff hypothesis, have been considered. Using the well-known hypothesis by Timoshenko and others, the general three-dimensional problem is reduced to a two-dimensional one relative to the displacement of points of the median plane of the layer, which imposes restrictions on external efforts. In the examined problem, there is no median plane. Therefore, as the desired values, displacements and deformations of the points in the plane have been considered, which, under certain conditions, pass into the median plane of the layer.
It is not possible to find a closed analytical solution for most problems while experimental studies often turn out to be time-consuming and dangerous processes.
Keywords: construction of mathematical models, interaction of material with base, dynamic load, boundary condition, general solution.

Mykhailo Keremet
Volodymyr Dahl East Ukrainian National University, Severodonetsk, Ukraine ORCID: https://orcid.org/0000-0003-4058-8083 The dynamic loading and strength of the frame of the "East-West" type covered wagon were determined. To increase the efficiency of operation of covered wagons in international traffic, it is proposed to improve their frames. This improvement consists in using a sectional partition in the body in order to divide it into two separate sections. This allows the transportation of different goods in one wagon, and therefore decreasing empty mileage.
The longitudinal loading of the covered wagon frame was determined. The case of shunting impact was considered. The studies were carried out in a flat coordinate system. The loading mode of the frame of the covered wagon in the empty and loaded states was considered. The acceleration acting on the covered wagon frame in the loaded state was 0.37g, empty -0.42g, which does not exceed the standard values. The wagon motion is rated "excellent".
The main strength indicators of the covered wagon frame were determined. The calculation was made by the finite element method. It was found that the maximum equivalent stresses are concentrated in the area of interaction of the center sill with the bolster beam and amounted to 340 MPa, which is lower than the yield stress of the material. Maximum displacements occur in the middle of the frame beams and are about 12 mm. The natural vibration frequencies of the covered wagon frame were calculated.
The research will help to increase the efficiency of using covered wagons in international traffic. Also, the research results can be useful developments in the creation of innovative rolling stock structures.
Keywords: transport mechanics, covered wagon, frame, dynamic loading, stress state, East-West wagon. This paper reports determining the energy efficiency of a vibratory machine consisting of a viscoelastically fixed platform that can move vertically, and a vibration exciter whose operation is based on the Sommerfeld effect. The body of the vibration exciter rotates at a steady angular speed while there are the same loads in the form of a ball, a roller, or a pendulum inside it. The load, being moved relative to the body, is exposed to the forces of viscous resistance, which are internal within the system.
It was established that under the steady oscillatory modes of a vibratory machine's movement, the loads are tightly pressed to each DOI: 10 A method for determining effective elastic constants of a composite unidirectionally reinforced with two types of transtropic hollow fibers is developed. Determining these characteristics is an integral step in the design of composite structures. The approach is based on analytical formulas for determining the elastic characteristics of a two-component composite with a transtropic matrix and hollow fiber. Hexagonal fiber lay-up with periodic reinforcement structure is considered. Double homogenization is used. The composite is conventionally divided into hexagonal regions of two types. The first is a hollow fiber of one material and the surrounding matrix. Similarly, the second one -with a hollow fiber of another material. In the first homogenization, elastic constants of the transtropic material of each of the two regions are determined. In the repeated homogenization, the region of the first type is taken as a "conditional" fiber, the region of the second type is taken as a "conditional" matrix. Effective elastic constants for a composite reinforced with two types of isotropic hollow fibers are calculated. The proposed method gives a good convergence of the results with calculations by known formulas. The maximum relative calculation error for the longitudinal elastic characteristics compared to known formulas does not exceed 0.05 %.
other, thereby forming a combined load. Energy is productively spent on platform oscillations and unproductively dissipated due to the movement of the combined load relative to the body.
With an increase in the speed of the body rotation, the increasing internal forces of viscous resistance bring the speed of rotation of the combined load closer to the resonance speed, and the amplitude of platform oscillations increases. However, the combined load, in this case, increasingly lags behind the body, which increases unproductive energy loss and decreases the efficiency of the vibratory machine.
A purely resonant motion mode of the vibratory machine produces the maximum amplitude of platform oscillations, the dynamic factor, the total power of viscous resistance forces. In this case, the efficiency reaches its minimum value.
To obtain vigorous oscillations of the platform with a simultaneous increase in the efficiency of the vibratory machine, it is necessary to reduce the forces of viscous resistance in supports with a simultaneous increase in the internal forces of viscous resistance.
An algorithm for calculating the basic dynamic characteristics of the vibratory machine's oscillatory motion has been built, based on solving the problem parametrically. The accepted parameter is the angular speed at which a combined load gets stuck. The effectiveness of the algorithm has been illustrated using a specific example.