Vol 23, No 2 (2023)

The article considers the anisotropic  Lorentz – Karamata space of periodic functions of several variables and the Nikol'skii – Besov class in this space. The order-sharp estimates are established for the best $M$-term trigonometric approximations of functions from the Nikol'skii-Besov class in the norm of another Lorentz – Zygmund space.

The article discusses a method for constructing a spline function to obtain estimates that are exact in order to approximate bounded functions by trigonometric polynomials in the Hausdorff metric. The introduction provides a brief history of approximation of continuous and bounded functions in the uniform metric and the Hausdorff metric. Section 1 contains the main definitions, necessary facts, and formulates the main result. An estimate for the indicated approximations is obtained from Jackson's inequality for uniform approximations. In section 2 auxiliary statements are proved. So, for an arbitrary $2\pi$-periodic bounded function, a spline function is constructed. Then, estimates are obtained for the best approximation, variation, and modulus of continuity of a given spline function. Section 3  contains evidence of the main results and final comments.

In this paper, the authors present the results of calculations of the stress field of a cylindrical shell weakened by a circular hole and under the influence of various loads: uniaxial tension, internal pressure and torsion. Six simplified equations of the theory of cylindrical shells with a high variability index (coinciding with the equations of the theory of shallow shells) are reduced to an equation of mathematical physics with respect to the stress potential, which is solved by the Fourier method. The main obstacle to obtaining an answer is the need to search for coefficients in the decomposition of the solution into the sum of the basis functions for which this solution satisfies the boundary conditions. Also, this equation depends on the parameter $\beta$, which is responsible for the relationship between the geometric characteristics of the shell and the hole. From a mechanical point of view, for small and medium holes, this parameter has limitations of $\beta\leq 4$, because for large values, the hole is considered large, and the general equations of the theory of cylindrical shells are used to describe the stress-strain state. At the same time, a detailed study of classical works has led to the understanding that none of the previously proposed methods for finding coefficients can be considered definitively successful, and the results obtained by these methods vary. Among the variety of works by Soviet and Western scientists of the 1960-70s years of the twentieth century, the numerical results of engineer Van Dyke, which he obtained by collocation, stand out. Unlike his contemporaries, who lay out the solution in a row for a small parameter $\beta$ and therefore get results only close to the flat case, Van Dyke first published results for the entire working range of the parameter $\beta$ in the framework of considering small and medium holes. The authors proposed a new approach based on the decomposition of basic functions into a Fourier series. For the first time, it was possible to compose an infinite system of linearly independent equations for finding unknown coefficients. It is essential that the proposed method, unlike the well-known small parameter method, has no mathematical limitations and can be used not only for the values of the parameter $\beta$ close to zero, but for any values. Restrictions up to $\beta=4$ are imposed by the mechanical model. In this paper, systems for finding unknown coefficients for basic functions for three types of loads are compiled, and the results obtained by the authors are compared with the results obtained by the numerical method. At the same time, if in most sources only the results of calculating the circumferential stresses at the boundary of the hole are given, in the proposed work the stress field for the entire cylindrical shell is found, arising due to the presence of the hole, depending on the polar coordinates $(r,\theta)$.

A method for solving the problem of relaxing residual stresses after bilateral surface hardening of a hollow cylinder under creep conditions with rigid constraints on the initially specified axial deformation and twist angle is presented. The solution is developed for complex loading regimes including pure thermal exposure, axial loading, torque, internal pressure, and their combinations. A numerical simulation was conducted on a thin-walled cylindrical specimen comprised of X18N10T steel, subjected to a temperature of $T\!=\!600\,^\circ$C, with the inner and outer surfaces subjected to ultrasonic peening. The reconstruction of the initial fields of residual stresses and plastic deformations was carried out based on the known experimental information on the distribution of axial and circumferential stress components in thin surface-hardened areas on the inner and outer surfaces. A phenomenological model of creep of steel alloy X18N10T at $T\!=\!600\,^\circ$C is constructed. The rheological deformation problem within the first two stages of creep was numerically solved using time and radius discretization. The calculations established that the presence of rigid constraints on angular and linear axial displacements resulted in a decrease in the rate of relaxation of residual stresses compared to the case where these constraints are absent. Graphs illustrating the kinetics of residual stresses with respect to the sequence of temperature and loading forces at different timestamps are presented.

When developing electronic shooting simulators for manual automatic weapons that do not use ammunition, it is necessary to achieve the maximum realistic modeling of the bullet flight path for each shot taking into account a set of factors. Traditionally, a system of differential equations of outer ballistics is used in modeling the trajectory. The use of a constant value of the ballistic coefficient in such a mathematical model does not allow to achieve high accuracy of modeling the trajectory for such important for solving the “task of the meeting” parameters as complete flight time and excess of the trajectory for all targeted range of small arms. The initial values in the mathematical model based on the system of differential equations of the outer ballistic are the casting angle (depends on the settings of the sight), the initial speed and the ballistic coefficient of the bullet, and such parameters as the current excess, range, time, speed and direction are calculated. Estimates of the errors of the calculation of the coordinates of the ballistic trajectory at various approaches to the use of the value of the ballistic coefficient are given. It is concluded that at the moment when modeling the flight trajectory of the bullet, simplification based on the use of a constant value of the ballistic coefficient is quite justified but with the relevant requirements of the tactical and technical task the study of ways to increase the accuracy of the trajectory modeling will become relevant. One of these paths is  using the value of the ballistic coefficient, depending on the casting angle proposed in this article.