and Computational Methods

#### 621.464.3 Mathematical modeling hydraulic systems for synchronous movement of actuators based on a throttle flow divider

**Bushuev A. Y. (Bauman Moscow State Technical University), Danilov N. A. (Bauman Moscow State Technical University)**

doi: 10.18698/2309-3684-2022-2-313

A mathematical model of functioning of synchronization systems of actuators based on a throttle flow divider was developed to solve design problem. The solution of the optimization problem of a mismatch time of movement of actuators operating under conditions of external alternating-sign force effects is given, performed using the genetic algorithm and refined using the Nelder-Mead algorithm

Бушуев А.Ю., Данилов Н.А. Математическое моделирование гидравлической системы синхронизации исполнительных органов на основе дроссельного делителя потока. Математическое моделирование и численные методы, 2022, № 2, с. 3–15

#### 539.26 Analysis of empirical models of deformation curves of elastoplastic materials (review). Part 2

**Golovina N. Y. (Industrial University of Tyumen), Belov P. A. (Institute of Applied Mechanics of RAS)**

doi: 10.18698/2309-3684-2022-2-1427

This article is a continuation of the review of works devoted to the study of the properties of elastic-plastic materials. In the first part, universal laws of deformation containing less than four formal parameters considered. As result of the review, requirements for the formulation of empirical laws of deformation of elastic-plastic materials formulated.In particular, it concluded that the deformation law must be at least four-parameter. In the second part of this paper, empirical laws of deformation containing four or more parameters considered and analyzed. Comparison of the considered empirical curves with a sample of experimental points carried out according to the standard procedure of minimization of the total quadratic deviation and using the method of gradient descent to determine the minimum of a function of many variables. A representative sample of 158 experimental points of the deformation curve of the Russian titanium alloy VT6 used to evaluate the predictive ability of the models for experimental agreement. Universal empirical strain laws containing four formal parameters allow describing the strain curve with specified stresses and tangential moduli at the ends of the curve. This fact allows us to state that the elastic-plastic properties of materials can expressed through the geometric parameters of the strain curve. In turn, the relationship between the elastic-plastic properties of the material and the geometry of the strain curve can interpreted as the principle of "geometrization" of the elastic-plastic properties of materials.

Головина Н.Я., Белов П.А. Анализ эмпирических моделей кривых деформирования упругопластических материалов (обзор). Часть 2. Математическое моделирование и численные методы, 2022, № 2, с. 16–29

#### 539.36 Asymptotic theory of multilayer thinelastic plates with layer slip

**Dimitrienko Y. I. (Bauman Moscow State Technical University), Gubareva E. A. (Bauman Moscow State Technical University)**

doi: 10.18698/2309-3684-2022-2-2862

The problem of development a theory for calculating the stress-strain state of thin multilayer elastic plates, for which linearized slip conditions are specified at the interface between the layers, is considered. The solution of this problem is constructed using an asymptotic analysis of the general equations of the 3-dimensional theory of elasticity with the conditions of non-ideal contact of the layers. The asymptotic analysis is carried out with respect to a small geometric parameter representing the ratio of the plate thickness to its characteristic length. Recurrent formulations of local quasi-one-dimensional problems of the theory of elasticity with slip are obtained. Explicit analytical solutions are obtained for these problems. The derivation of the averaged equations of elastic equilibrium of multilayer plates is presented, taking into account the slippage of the layers. It is shown that due to the effect of slippage of layers, the system of averaged equations of the theory of multilayer plates has an increased - 5th order of derivatives, in contrast to the classical 4th order, which takes place in the theory of Kirchhoff-Love plates. It is shown that the asymptotic theory makes it possible to obtain an explicit analytical expression for all 6 components of the stress tensor in the layers of the plate. As a special case, the problem of calculating the stress-strain state of a 4-layer plate under uniform pressure bending with one slip coefficient is considered. A complete analytical solution of this problem is obtained, including explicit expressions for all non-zero components of the stress tensor. A numerical analysis of the solution of the averaged problem for a composite plate is carried out, in which the layers are unidirectional reinforced fibrous materials oriented at different angles. A comparative analysis of the influence of the fiber reinforcement angles and the slip coefficient of the layers on the displacement of the plate and the distribution of stresses in the layers was carried out. It is shown that the problem of bending a plate with slip admits the existence of a spectrum of critical values of the slip coefficient, when passing through which the displacements and stresses in the layers of the plate change significantly, and these critical values depend on the angle of reinforcement of the composite layers.

Димитриенко Ю.И., Губарева Е.А. Асимптотическая теория многослойных тонких упругих пластин с проскальзыванием слоев. Математическое моделирование и численные методы, 2022, № 2, с. 30–64

#### 004.9:621.7 Modeling of the temperature field in the productionof metal products on a casting and forging modulewith one-sided action of the side strikerand fixed plate

**Dmitriyev E. A. (Komsomolsk-na-Amure State University), Potianikhin D. A. (Komsomolsk-na-Amure State University), Odinokov V. I. (Komsomolsk-na-Amure State University), Evstigneev A. I. (Komsomolsk-na-Amure State University), Kvashnin A. E. (Komsomolsk-na-Amure State University)**

doi: 10.18698/2309-3684-2022-2-6377

The paper presents a mathematical statement and presents the results of calculations in the problem of determining the temperature field during deformation of a strip made of aluminum alloy AD0 on a vertical-type casting and forging module of a new modification. The design of the casting and forging module assumes that one of the four walls of the mold is stationary, the second performs a rotational movement on eccentric shafts, the other two move in a vertical plane, ensuring that the deformed billet is fed down. When solving the problem, a proven numerical method is used. For a moving medium, the heat equation is written in finite difference form in a curvilinear orthogonal coordinate system. The solution of the problem is carried out by the iterative method. When calculating the initial temperature field and its further change, the heat removal on the contact surfaces of the metal with the deformation tools is taken into account. The result of the solution is the temperature field in the spatial domain for discrete times corresponding to the steps of the numerical calculation. At each step, the boundary of the liquid and solidified metal is determined

Дмитриев Э.А., Потянихин Д.А., Одиноков В.И., Евстигнеев А.И., Квашин А.Е. Моделирование поля температур при получении металлоизделий на литейно-ковочном модуле с односторонним воздействием бокового бойка и неподвижной плитой. Математическое моделирование и численные методы, 2022, № 2, с. 65–79

#### 533.6.011.5 A new dependence for the enthalpy profile in the boundary layer model

**Kotenev V. P. (Bauman Moscow State Technical University), Tonkih E. G. (Bauman Moscow State Technical University)**

doi: 10.18698/2309-3684-2022-2-7887

A new dependence for the enthalpy profile in the boundary layer model. A new analytical dependence is proposed for the enthalpy profile in the boundary layer, the use of which makes it possible to quickly and efficiently obtain the distribution of heat flux over the surface of blunted bodies

Котенев В.П., Тонких Е.Г. Новая зависимость профиля энтальпии в модели пограничного слоя. Математическое моделирование и численные методы, 2022, № 2, с. 80–89

#### 519.6 Simulation and optimization of spacecraft flight control from Earth orbit to Venus orbit using ion thrusters

**Mozzhorina T. Y. (Bauman Moscow State Technical University), Zakurazhnaya A. A. (Bauman Moscow State Technical University)**

doi: 10.18698/2309-3684-2022-2-88101

In this paper, optimization of the flight of a low-mass satellite from Earth orbit to the orbit of Venus using ion engines is considered. The first flight to the planet took place in 1961 by the Soviet automatic interplanetary station "Venus-1", which passed 100,000 kilometers from Venus. In addition, in 1962, the American station "Mariner-2" was flown. The most recent spacecraft launched to the planet was the European Space Agency's Venus Express in 2005, which flew to Venus in 153 days. When solving the current problem, the following assumptions were made: an interorbital flight is considered without taking into account the attraction of the planets, and the orbits of the planets are considered circular and lying in the same plane. The angle between the tangential velocity of the spacecraft and the thrust direction was chosen as the control. Optimization of satellite control was carried out using the Pontryagin maximum principle. The resulting boundary value problem for a system of ordinary differential equations was solved by a numerical method — the targeting method. Newton's method was used to solve systems of nonlinear algebraic equations. The calculation program was written using the C++ programming language. As a result of the work, it was possible to minimize the flight time between orbits, thus the operability of the shooting method for solving optimization problems was shown.

Мозжорина Т.Ю., Закуражная Д.А. Моделирование и оптимизация управления полетом космического аппарата с орбиты Земли на орбиту Венеры с помощью ионных двигателей. Математическое моделирование и численные методы, 2022, № 2, с. 90–103

#### 519.8 A "mixed" model of the confrontation of numerous groups with linear dependencies on the time of the effective speed of strikes by units of the parties

**Chuev V. U. (Bauman Moscow State Technical University), Dubogray I. V. (Bauman Moscow State Technical University)**

doi: 10.18698/2309-3684-2022-2-102113

Using the method of dynamics of averages, a "mixed" model of numerous groupings confrontation has been developed with linear dependencies on the time of effective speeds of striking by units of the parties. An algorithm is constructed that allows to investigate the course of the process and calculate its main indicators. It is established that the use of confrontation models with constant effective speed of strikes in many cases leads to significant errors in the calculation of the main indicators of the process. The influence of a preemptive strike by one of the opposing sides on the course of the progress and the final outcome are investigated.

Чуев В.Ю., Дубограй И.В. «Смешанная» модель противоборства многочисленных группировок при линейных зависимостях от времени эффективных скоростей воздействий единицами сторон. Математическое моделирование и численные методы, 2022, № 2, с. 104–115