• #### 539.3 Modeling of stresses in a composite nonlinear elastic panel under cylindrical bending

##### Dimitrienko Y. I. (Bauman Moscow State Technical University), Gubareva E. A. (Bauman Moscow State Technical University), Karimov S. B. (Bauman Moscow State Technical University), Kolzhanova D. Y. (Bauman Moscow State Technical University)

doi: 10.18698/2309-3684-2021-1-330

The problem of calculating the stress–strain state of a composite laminated panel during cylindrical bending under conditions of finite deformations is considered. To solve the problem, the method of asymptotic averaging of periodic nonlinear elastic structures with finite deformations was applied, which was developed in detail earlier in the previous works of the authors. A feature of this problem is the use of universal models of constitutive relations for isotropic components of the composite, as well as for the composite as a whole, which is a transversely isotropic nonlinear elastic medium. Universal models make it possible to obtain solution of problems within the framework of a single solution algorithm simultaneously for several classes of models of nonlinear elastic media corresponding to different conjugate pairs of stress tensors–deformation. An analytical solution is obtained for the problem of cylindrical bending of a composite panel. A numerical analysis of the solution is carried out using the example of a composite, the periodicity cell of which consists of two layers: polyurethane and rubber. It is shown that for thin panels the stresses, both averaged and true, practically do not depend on the class of the model of the constitutive relations. At the same time, for thicker panels, the stresses differ significantly for different classes of models of composite layers.

Димитриенко Ю.И., Губарева Е.А., Каримов С.Б., Кольжанова Д.Ю. Моделирование напряжений в композитной нелинейно упругой панели при цилиндрическом изгибе. Математическое моделирование и численные методы, 2021, № 1, с. 3–30.

• #### 517:519.6 On the construction of adequate and economical mathematical models in the tasks of continuous monitoring of especially dangerous and technically complex objects

##### Trutaev S. Y. (АО "Иркутский научно-исследовательский и конструкторский институт химического и нефтяного машиностроения")

doi: 10.18698/2309-3684-2021-1-3148

The possibility of constructing adequate and economical mathematical models of especially dangerous and technically complex objects when solving problems of monitoring their dynamic stress-strain state using the finite element method is discussed. The desired result is achieved on the basis of the application of the developed volumetric finite elements with a mixed degree of interpolation, both geometry and displacements for all edges. At the same time, it is possible to combine both subparametric and isoparametric approaches by varying the degree of interpolating polynomials from one to three within each element. The results of evaluating the effectiveness of the proposed approach are presented on the example of solving a model problem on free oscillations of a pipeline. The proposed family of volumetric finite elements is implemented in the COMPASS finite element package, as well as in the specialized software package MStruct, designed for continuous monitoring of industrial equipment, as well as buildings and structures in real time. An example of the practical application of the developed software for monitoring the dynamic stress-strain state of equipment at a hazardous production facility is presented.

Трутаев С.Ю. О построении адекватных и экономичных математических моделей в задачах постоянного мониторинга особо опасных и технически сложных объектов. Математическое моделирование и численные методы, 2021, № 1, с. 31–48.

• #### 004.942 Automation algorithm for time-frequency analysis of transients obtained by UAV motion modeling

##### Tochilova O. L., Kolgotin A. V.

doi: 10.18698/2309-3684-2021-1-4965

A wavelet based data processing algorithm providing automation of the process of time–frequency analysis of transients obtained by statistical UAV motion modeling for a large number of randomly realized sets of tolerances has been developed. Statistical UAV motion modeling is the main tool for analyzing and checking out the functioning of the developed control algorithms according to possible scatter of UAV characteristics and environmental parameters. Based on the quality of the transients obtained in the statistical modeling, acceptability of the selected control algorithm parameters in terms of ensuring a stable UAV flight in a given range of permissible trajectories according to the tolerances is determined. The greatest difficulty in this case is automation of the time-frequency analysis of the obtained transients, since standard diagnoses do not allow identifying critical variants of the tolerance combinations that have spectral components in the transients relevant to the studied parts of the trajectory exceeding the specified values of frequency, amplitude and duration. To solve this problem, the developed algorithm uses wavelet and wavelet–packet transforms of one-dimensional signals, which are precisely the type of time– frequency transforms of the signals, in order to obtain and then analyze the time–frequency representations of the transients with specified parameters. An example of using the developed algorithm to estimate the parameters of spectral components, such as duration, maximum and average amplitude value, frequency in the neighborhood of the point with the maximum amplitude value, required to determine the quality of the transients obtained by the modeling is given.

Точилова О.Л., Колготин А.В. Алгоритм автоматизации частотно–временного анализа переходных процессов, полученных при моделировании движения БПЛА. Математическое моделирование и численные методы, 2021, № 1, с. 49–65.

• #### 519.8 Probabilistic models of bilateral fighting. Comparison of results

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

doi: 10.18698/2309-3684-2021-1-6676

It is made a comparison of the results of calculating of the main indicators of the battle with the use of stochastic models developed by the authors with different dependences of the effective rate of fire of the combat units of the parties on the time of the battle. The influence of various factors on the differences in the calculations of these indicators when using these models is shown. It is established that for the solution of the majority of military -technical and military-tactical tasks it is possible to use any of the models of bilateral military operations developed by authors.

Чуев В.Ю., Дубограй И.В. Вероятностные модели двухсторонних боевых действий. Сравнение результатов. Математическое моделирование и численные методы, 2021, № 1, с. 66–76.

• #### 519.2 Mathematical modeling of assessing the quality of intruder detection by a robotic system in a wooded area

##### Pushkarev Y. A. (Филиал ВА РВСН), Sviridov V. V. (Филиал ВА РВСН)

doi: 10.18698/2309-3684-2021-1-7790

When solving the tasks of protecting critical objects, the problem of detecting an intruder in a non-deterministic dynamic external environment with the use of standard technical means arises. The article considers a methodological approach to assessing the quality of probabilistic characteristics of intruder detection using optoelectronic means of a robotic complex. Based on the obtained parameters of the statistics of received signals from optoelectronic devices, changes in the values of the probability of detecting an intruder, the values of the detection threshold and the signal-to-noise ratio at different values of the false alarm probability are analyzed.

Пушкарёв Ю.А., Свиридов В.В. Модель оценки качества характеристик робототехнического комплекса для обнаружения нарушителя в лесистой местности. Математическое моделирование и численные методы, 2021, № 1, с. 77–90.

• #### 519.6 Методы численного решения дифференциального уравнения смешанного типа в неограниченной области

##### Galanin M. P. (Bauman Moscow State Technical University/Keldysh Institute of Applied Mathematics of the Russian Academy of Scienсes), Sorokin F. D. (Bauman Moscow State Technical University/Keldysh Institute of Applied Mathematics of the Russian Academy of Scienсes), Ukhova A. R. (Bauman Moscow State Technical University)

doi: 10.18698/2309-3684-2021-1-91109

Разработаны методы численного решения задачи для уравнения смешанного типа в неограниченной области в случае, когда решение удовлетворяет уравнению теплопроводности в ограниченной области и уравнению Лапласа в оставшейся части пространства. Предложен способ задания искусственных граничных условий, позволяющий проводить расчёты в ограниченной области. Построен итерационный алгоритм нахождения численного решения в ограниченной области, такой что численное решение сходится к проекции точного решения на ограниченную область. Исследована скорость сходимости итерационного алгоритма. Задача решена в одномерном плоском, в цилиндрически и сферически симметричных случаях. Приведены примеры решений.

Галанин М.П., Сорокин Д.Л., Ухова А.Р. Методы численного решения дифференциального уравнения смешанного типа в неограниченной области. Математическое моделирование и численные методы, 2021, № 1, с. 91–109.

• #### 004.942 Non-autonomous system as a model of the production process of technical innovation

##### Belov V. F. (МГУ им. Н.П. Огарева/АУ «Технопарк–Мордовия»), Gavryushin S. S. (Bauman Moscow State Technical University), Markova Y. N. (АУ «Технопарк–Мордовия»)

doi: 10.18698/2309-3684-2021-1-110131

Development of an analytical method for economic characteristics variation in an innovation process is a challenging task, starting from invention or entrepreneur idea and finishing with market implementation. The practical purpose here is both to minimize the risks and shorten design and implementation time. Theoretical and experimental results presented in the paper show the possibility to solve the mentioned task via applying a non-autonomous differential equation system along with the Lyapunov's First Method for stability analysis. A mathematical model of flow of funds related to the manufacturing and market implementation of the engineering innovation has been studied. It is presented in the form of the system of differential balance equations with unit impulse function in the right-hand side. An algorithm has been developed to analyze the stability of the equilibrium states of the manufacturing process, considering the influence of the external environment “from the right” (preparation of manufacturing) as well as “from the left” (market state). The requirements are determined regarding both the discrete model of manufacturing preparation to formulate initial conditions for the non-autonomous system and the discrete model of the market to calculate the market state dependent coefficients for the system of differential equations. The results of analysis of the economic characteristics variation in the stage of product manufacturing have been presented as 3D phase images.

Белов В.Ф., Гаврюшин С.С., Маркова Ю.Н. Неавтономная система как модель процесса производства технической инновации. Математическое моделирование и численные методы, 2021, № 1, с. 110–131.