Статьи журнала - Siberian Aerospace Journal
Все статьи: 341
Статья научная
The article contains the results of scientific research on modeling the technological parameters of electron beam welding. The modeling used a material VT-14 with a thickness of 0.16 cm. The purpose of the simulation is to improve the quality of the weld due to the optimal shape and the absence of defects in the form of pores and cracks. A concentrated energy source equivalent to an electron beam is used in the calculations. During the study of the thermal process of heating the material, the authors developed and tested criteria that allow optimizing welding parameters such as welding speed and the position of the focal spot relative to the surface of the heated part. In their calculations, the authors applied an original method of finding the welding speed and the coordinates of the focal spot according to the functionals of the thermal model. The algorithm developed by the authors was successfully tested on AMG-6 material with a thickness of 10 cm. In the process of modeling welding for large thicknesses, results have been obtained that must be taken into account when optimizing the welding parameters of products with large thickness. The relevance of the presented material is confirmed by the demand for the quality of the technology of welding structures with an electron beam. Research by the authors of this direction will significantly expand the possibilities in the application of electron beam technology for rocket and space technology.
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Статья научная
One of the promising areas for improving the methods of manufacturing structural elements of rocket and space technology is the use of selective laser melting technology which represents a unique opportunity to manufacture metal products by melting powder and producing a one-piece solid phase structure. However, pores and other structural defects can appear in the formed element during laser sintering which causes a decrease in the strength characteristics of the parts produced. An important step in the additive technologies introduction is the development of methodology for the preliminary prediction of the strength characteristics of manufactured structural elements under the influence of mechanical loads with the help of mathematical modeling. The methodology for estimating the material strength reduction of a rocket-space technology element obtained using additive technologies by simulating a porous structure and calculating the characteristics of the stress-strain state is presented. The proposed mathematical model and the methodology for calculating the specimen loading on the basis of the distortion energy theory allow calculating the stress-strain state in the process of numerical simulation for different values of the pore diameter. The reduction in yield strength due to the material porosity of the part is estimated using a coefficient equal to the ratio of equivalent stresses arising when a load is applied to a specimen manufactured using traditional and additive technologies. The value of the introduced coefficient characterizes the structure of the grown product and is considered as a function of the random arrangement of pores in the specimen under study. The appearance of pores is the result of a combination of factors: the composition and dispersion of the original metal powder, feed rate, removal distance and laser power during sintering, part orientation and sintering direction, the height of the level of powder deposited on a special base before sintering, etc. The paper evaluates the reduction in strength for the working part of a series of tensile test specimens grown from metal powder of different dispersity. The non-linear nature of the dependence of the yield strength on the particle diameter of the original metal powder is established. The maximum value of the yield strength corresponds to the specimen with the minimum value of the total surface area of the pores.
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Статья научная
The current models of aviation artillery weapons (AAW) are the pulsed heat engines that convert the energy of a powder charge into the energy of highly compressed and heated powder gases (hereinafter referred to as gases), which, when expanding, perform work on communicating kinetic energy to the projectile. In the context of artillery science, aviation artillery weapons and ammunition are structured as a system that interacts with heat sources and the environment, sequentially completing thermodynamic cycles. The main element that is most intensively subjected to thermophysical loads and has a significant impact on the combat qualities and cost of aviation artillery weapons is a small-caliber artillery barrel (hereinafter referred to as the barrel). As a result, the problem of determining the temperature field of the barrel is one of the central problems of designing aviation artillery weapons and optimizing firing modes. The successful solution of this problem largely depends on the accuracy of modeling the processes of heat transfer to the channel and from the outer wall of the barrel during firing. At the same time, an adequate synthesis and calculation of the relations describing the phenomenon of convection accompanying the shot is difficult, which is due to the presence of phase transformations in the state of gases; the simultaneous presence of supersonic and subsonic zones in the solution regions; the existence of laminar, turbulent flows and other non-linear formations. The aim of the work is to develop a relatively simple and acceptable for engineering practice mathematical model of heat transfer inside and around the barrel with near-wall coolant flows (hereinafter referred to as the model). Achieving the goal of the work is carried out by a concentrated choice of criterion equations of the apparatus of thermodynamic similarity, corresponding to the geometric and physical conditions for the uniqueness of the processes of loading the barrel. The introduction of functions that take into account the dependence of the thermophysical properties of gases on temperature made it possible to increase the accuracy of determining the parameters of heat transfer during a shot by 19% in comparison with the known results. The developed model can be used in applied calculations related to determining the thermal state of the barrel. The specialization of the object of study does not exclude the possibility of refining the model for the purpose of mathematical representation of thermal effects in thermally stressed structures of complex shape.
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Models and algorithms for automatic grouping of objects based on the k-means model
Статья научная
The paper is devoted to the study and development of new algorithms for automatic grouping of objects. The algorithms can improve the accuracy and stability of the result of solving practical problems, such as the problems of identifying homogeneous batches of industrial products. The paper examines the application of the k-means algorithm with the Euclidean, Manhattan, Mahalanobis distance measures for the problem of automatic grouping of objects with a large number of parameters. A new model is presented for solving problems of automatic grouping of industrial products based on the k-means model with the Mahalanobis distance measure. The model uses a training procedure by calculating the averaged estimate of the covariance matrix for the training sample (sample with pre-labeled data). A new algorithm for automatic grouping of objects based on an optimization model of k-means with the Mahalanobis distance measure and a weighted average covariance matrix calculated from a training sample is proposed. The algorithm allows reducing the proportion of errors (increasing the Rand index) when identifying homogeneous production batches of products based on the results of tests. A new approach to the development of genetic algorithms for the k-means problem with the use of a single greedy agglomerative heuristic procedure as the crossover operator and the mutation operator is presented. The computational experiment has shown that the new mutation procedure is fast and efficient in comparison with the original mutation of the genetic algorithm. The high rate of convergence of the objective function is shown. The use of this algorithm allows a statistically significant increase both in the accuracy of the result (improving the achieved value of the objective function within the framework of the chosen mathematical model for solving the problem of automatic grouping), and in its stability, in a fixed time, in comparison with the known algorithms of automatic grouping. The results show that the idea of including a new mutation operator in the genetic algorithm significantly improves the results of the simplest genetic algorithm for the k-means problem.
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Статья научная
The article discusses formalization of the problem of heterogeneous distributed information processing systems (HDIPS) software and hardware configuration management. A formal description of possible optimality criteria for the HDIPS software and hardware configuration is given. The HDIPS model in terms of queuing theory is proposed. The problem of allocating the HDIPS computational resources is formulated as a transport problem according to time criterion with atomic needs. The algorithm for solving this problem is proposed and the boundaries of its applicability to the HDIPS are determined. To meet the selected optimality criterion, the analysis of the HDIPS software and hardware configuration applying its formal model, using the queuing theory methods is presented. HDIPS is presented as a queuing network, where each computing node and route control unit is a mass service system. The problem of computing resource allocation in HDIPS is presented as a transport problem according to the time criterion with atomic needs. The least time algorithm for indivisible needs takes into account the indivisibility condition.
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Multi-grid finite elements in calculations of multilayer oval cylindrical shells
Статья научная
The method of finite elements (FEM) is actively used in calculations of composite shell constructions (rotation shells, circle and oval cylindrical shells), which are widely used in space-rocket and aviation equipment. To calculate multi-layer oval cylindrical shells three-dimensional curvilinear Lagrange multi-grid finite elements (MGFE) are suggested. When building a k-grid finite element (FE), k nested grids are used. The fine grid is generated by the basic split of MGFE that takes into account its complex heterogeneous structure and shape. On k-1 large grids the move functions used for decreasing MGFE dimension are determined. The stress-strain state in MGFE is described by the elasticity theory three-dimensional task equations (without introduction of additional hypotheses) in local Cartesian coordinates systems. The procedure of building shell-type Lagrange MGFE with the use of Lagrange polynomials presented in curvilinear coordinate systems is demonstrated. With the size reduction of discrete models MGFE have constant thickness equal to the thickness of the shell. The Lagrange polynomials nodes coincide in thickness with the MGFE large grid nodes and are located on the shared borders of different module layers. The use of such MGFE generates approximate solutions sequences that uniformly and quickly converge to precise solutions. The main advantages of MGFE are as follows: they form discrete models with the dimension 102–106 times smaller than the basic models dimension and they generate small error solutions. Examples of calculations are given for four- and three-layer oval shells of various thickness and shape under both uniform and local loading with the use of 3-grid FE. Comparative analysis of the obtained solutions with the solutions built with the help of the software package ANSYS shows high efficiency of the suggested MGFE in calculations of multi-grid oval shells.
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Multiversion model of software control systems for space vehicles with range of decision-making
Статья научная
The paper presents a multi-version model with ranking of alternatives in order of preference, taking into account the dependence of the attributes in the design of software for spacecraft control systems of various classes. The applied software with a set of algorithms, based on the general scheme of the method of branches and borders allow determining the exact solution of the optimization problem. To achieve the highest reliability of the software component of spacecraft control systems built with the use of multi-version programming methodology, a large number of versions of software modules are combined into a single structure. While software complexes even without introduction of redundant elements are characterized as complex systems, there is no need to speak about wide use of enumerative methods for their formation. Using the proposed modified method of ordered preference through similarity to an ideal solution, will allow to solve the problem of choosing the best computing system from a number of available systems. This approach is becoming increasingly possible because of the tremendous progress in computing design and manufacturing technology. Even the so-called personal computers provide computational capabilities that some time ago seemed impossible even for computers representing a much more powerful class of computing equipment - supercomputers.
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N-version programming for nanosatellite telemetry processing
Статья научная
Software is a key element that ensures the functioning of any modern complex technical system. One such system is the constellation of spacecraft and associated ground control complexes that provide reception, transmission, and processing of collected telemetry. The process of data acquisition and its subsequent processing is critical to the flight control of the spacecraft and its onboard scientific equipment. Furthermore, telemetry data processed by ground control systems involves large volumes of raw data, the processing of which is a complicated and time-consuming task. In order to solve this problem, various methods of automatic data processing are used. Improving them is a key factor in ensuring the fault tolerance of onboard software and hardware, improving its reliability. Of all the existing widely-used methods of data processing, we shall focus on N-version programming (NVP) approach. N-version programming has firmly established itself as an effective method for increasing software reliability and designing fault-tolerant systems. Since its inception in the 1970s, this approach has been deeply connected with the development of aerospace software systems, including, among others, satellite ground control stations. In light of the aforementioned, this paper discusses the application of NVP for processing telemetry data gathered from nanosatellites (CubeSats). Due to the fact that there exists a skeptical view on the NVP approach in terms of its efficiency, the author covers this issue in existing literature in terms of the approach’s applicability for processing satellite telemetry.
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Статья научная
In the presented work, an assessment of the noise immunity of the parallel algorithm of the broadband signals with frequency manipulation with minimal shift time search is given. The threshold value of the Signal-to-Noise ratio per one L signal is used as a criterion, which ensures the required accuracy of code synchronization with a given error probability. In this case, the task of temporary search is formed as a task of L signals recognition which modulating code sequences { } kl d differ by a time cyclic shift τk (k 1) τэ (k 1,L,L the length of the modulating code sequence, э the duration of the signal element or the element of the modulating code sequence) equal to energy. The L -dimensional probability density of the modules 1,..., LVV ( i V the correlation module of the analyzed and comparison signals at the output of the quadrature correlator) is preliminarily determined. The probability of error is determined by L -multiple integration of the L -dimensional probability density of modules 1,..., L VV. The problem of determining the energy losses in the analyzed signal under the influence of structural interference is solved. In this case, a quadrature recognition circuit of one of two discrete signals is used and the Signal-to- Noise ratio at the output of the recognition circuit is determined, taking into account the effect of structural interference, and a comparison is made with the Signal-to-Noise ratio for the analyzed signal acting at the input of the quadrature recognition circuit. An assessment of the structural interference impact on the temporary search is given, taking into account the normalized periodic autocorrelation function (NPACF) of the analyzed signal and structural interference (at L 16383). It is shown that it is most rational to work with NPACF in its ascending section (cycle interval [ 361τ , 540τ э э]), while the reciprocal correlation coefficient will not exceed 4 · 10–3 and the equivalent energy losses of the analyzed signal will not exceed 3 dB. At an intensity of 34 dB of structural interference, the use of an ascending section of the NPACF allows minimizing the energy losses of the analyzed signal and temporary searching for a time of 0.35 s (at τэ = 2.5 microseconds).
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Non-parametric identification and control algorithms for T-processes
Статья научная
In this paper, nonparametric identification and control methods are considered for multidimensional discrete-continuous processes with a delay inherent in many real productions. Such systems are typical for practice, including in the rocket and space industry, as well as in the technological processes of space technology production. Considering multidimensional processes, it is necessary to take into account the relationships between input and output variables, as well as their relationships with each other. Moreover, these connections are not always known to the researcher. When taking into account unknown connections of input variables, the researcher will deal with tubular processes or H-models, and when taking into account unknown connections of output variables, the model for one or another channel of the object will be analogs of implicit functions. In general, the model of a multidimensional object will be represented as a system of nonlinear implicit equations. In this case, the solution of the identification problem will be reduced to finding the prediction of the vector of output variables from the known values of the vector of input variables and can be obtained only as a result of solving the corresponding system of equations, which were called T-models. The solution of a system of nonlinear implicit equations by parametric identification methods will not lead to the desired result due to the lack of sufficient a priori information, and here there is a need for the use of nonparametric identification methods, as well as the use of system analysis methods. A priori information in the tasks of nonparametric statistics is insufficient, which conventional identification methods cannot cope with. When controlling multidimensional processes, it is necessary to take into account the dependencies of output variables, in connection with which another important feature arises, namely: random values from the domain of determining output variables cannot be used as setting influences; they must be selected from their common intersection.
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Статья научная
The paper discusses new classes of models of multidimensional inertia-free systems with a delay in the condi-tions of a lack of a priori information. The subject is multidimensional discrete-continuous processes, the com-ponents of the vector of output variables of which are stochastically dependent in an unknown way. There are also processes, through some channels of which aprior information corresponds simultaneously to both the par-ametric and nonparametric type of source data about the studied process. The mathematical description of such processes leads to a system of implicit nonlinear equations, some of which will be unknown, while others will be known with accuracy to the parameter vector. The main purpose of a model of an object having stochastic de-pendencies of output variables is to find a forecast of output variables with known input variables. To find the predicted values of the output variables from known inputs, it is necessary to solve a system of im-plicit nonlinear equations. The problem is to solve a system that is actually unknown, when only equations for some channels of a multidimensional system are known. Thus, a rather nontrivial situation arises when solving a system of implicit nonlinear equations under conditions when, in one channel of a multidimensional system, the equations themselves are not in the usual sense, and in others they are known accurate to parameters. Therefore, an object model cannot be constructed using the methods of the existing identification theory because of a lack of aprior information. The purpose of this work is the solution of the identification problem in the presence of a partially-parameterized discrete-continuous process, and despite the fact that the parameterization stage cannot be overcome without additional priori information about the process under study. The control algorithm for multidimensional processes with dependencies of output variables is a sequential multi-step algorithmic chain that allows finding the control action and bring the object to the desired state. Computational experiments to study the proposed models and to control multidimensional discrete-continuous processes have shown quite satisfactory results. The article presents the results of computational experiments illustrating the effectiveness of the proposed technology for predicting the values of output varia-bles from known input variables, as well as for managing these processes.
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Nonparametric identification of dynamic systems under normal operation
Статья научная
The research gives nonparametric identification algorithms under the conditions of incomplete a priory information. The identification case differs from the previously known ones due to the fact that, besides the control action, an uncontrollable variable, but a measurable one, impacts on the object input. In contrast to parametric identification, the research considers the situation when the equations describing dynamic objects are not given with accuracy to the parameters. In this case, there are some features to study while getting the recovery characteristics of various object channels. The main characteristic is that the transition response of a channel is taken when the other channel is in a stable position. Moreover, the identification problem is analyzed under normal object operation, opposite to the previously known nonparametric approach based on Heaviside function input to the object and further Duhamel integral application. An arbitrary signal is input to the object during normal operation as a result we have a corresponding response of the object output. It should be noted that the measurements of the input and output variables are carried out with random noise. As a result, we have a sample of input-output variables. As linear dynamical system can be described by the Duhamel integral, with known input and output object variables, corresponding values of the weight function can be found. This is achieved by discrete representation of the latter. Having such realization, nonparametric estimate of the weight function in the form of the nonparametric Nadaraya-Watson estimate is used later. Substituting this with the Duhamel integral, we obtain a nonparametric model of a linear dynamical system of unknown order. The article also describes the case of constructing nonparametric model when a delta-shaped function is input to the object. It is interesting to find out how delta-shaped function might differ from the delta function. The weight function is determined in the class of nonparametric Nadaraya-Watson estimates. Previously proposed nonparametric algorithms consider the case when Heaviside function is applied to the object; this narrows the scope of nonparametric identification practical use. It is important to construct nonparametric model of the dynamic object under conditions of normal operation.
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Статья научная
Testing the hypothesis of independence of random variables is one of the main stages of sys-tem analysis of statistical data. Based on its results, the synthesis of effective decision-making algorithms is carried out. The traditional method of testing the hypothesis of independence of random variables is based on the use of the Pearson criterion, which contains a difficult to formalize stage of dividing the range of the values of random variables into multidimensional intervals. A method for testing the hypothesis of independence of random variables is proposed, which uses a nonparametric pattern recognition algorithm corresponding to the maximum likelihood criterion. Its application makes it possible to circumvent the problem of decomposing the range of the values of random variables into intervals. The idea of the approach is to form a training sample based on the initial statistical data to solve a two-alternative pattern recognition problem. Each class is defined under the assumption of independence or dependence of random variables, which is manifested in the difference in their distribution laws in the classes. Under these conditions, it becomes possible to replace the initial hypothesis with the task of checking the reliability of the difference in the probabilities of pattern recognition errors in classes. Using the apparatus of graph theory, the proposed method is developed in the formation of sets of independent random variables. The obtained results are generalized when testing the hypothesis of independence of random variables for large volumes of statistical data based on compression of the original information. This allows increasing the computational efficiency of the problem being solved. The article substantiates the method for testing the hypothesis of independence of random variables, based on the use of a nonparametric pattern recognition algorithm in conditions of large volumes of statistical data. The results of comparing the technique with the generally recognized Pearson consensus criterion in the study of ambiguous dependences between random variables of varying complexity are presented. The effectiveness of the proposed method is confirmed by the results of its application in processing remote sensing information from anthropogenic territories in the vicinity of the city of Krasnoyarsk.
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Object tracking with deep learning
Статья научная
Tracking objects is a key task of video analytics and computer vision, which has many applications in various fields. A lot of tracking systems include two stages: detecting objects and tracking changes in the position of objects. At the first stage, objects of interest are detected in each frame of the video sequence, and at the second, the correspondence of the detected objects in neighboring frames is assessed. Nevertheless, in difficult conditions of video surveillance, this task has a number of difficulties associated with changing the illumination of the frame, changing the shape of objects, for example, when a person is walking, and the task is also complicated in the case of camera movement. The aim of the work is to develop a method for tracking objects on the basis of deep learning, which allows to track several objects in the frame, including those in the rough conditions of video surveillance. The paper provides an overview of modern methods for solving objects tracking tasks, among which the most promising one is deep learning neural networks application. The main approach used in this paper is neural networks for detecting regions (R-CNN), which has proven to be an effective method for solving problems of detection and recognition of objects in images. The proposed algorithm uses an ensemble containing two deep neural networks to detect objects and to refine the results of classification and highlight the boundaries of the object. The article evaluates the effectiveness of the developed system using the classical in the field MOT(Multi-Object tracking) metric for objects tracking based on the known databases available in open sources. The effectiveness of the proposed system is compared to other well-known works.
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On an alternative method for testing the dynamic strength of a small spacecraft structure
Статья научная
This article presents an analysis of the possibility of applying an alternative approach to testing the mechanical effects of the design of a small spacecraft for remote sensing of the Earth, which has an analog product that has passed a full cycle of ground experimental testing. However, despite the similar power scheme and the maximum borrowing of onboard equipment with minimal modifications, the spacecraft planned for testing has a number of significant differences. The application of the main alternative strategies in foreign and domestic practice in the ground-based experimental development of space technology is considered, their advantages and disadvantages are described. Some criteria for decision-making on the rejection of the use of traditional methods of ground-based experimental testing of space technology for mechanical effects are given. The analysis of the normative and technical documentation adopted in the domestic industry in terms of clarifying the list of development tests of spacecraft, the assumptions of applying the computational and experimental method to the development of dynamic (vibration) strength and the analysis of the design of the spacecraft planned for testing in comparison with an analog product showed that the most preferred method of testing dynamic (vibration) strength is the strategy “protocol qualifications”. In accordance with the chosen strategy, a list of tasks was defined that will clarify the nomenclature of the development tests of the research object.
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On incorrect representation of the shock process on shock polars in a viscous heat-conducting gas
Статья научная
The shock gas-dynamic processes that have found wide application in rocket and space technology in the design and optimization of devices and power plants are considered. The analysis of known exact and asymptotic relations/conditions on a shock wave, in particular, generalized differential relations (GDRs) on a curvilinear oblique shock wave (SW) (COSW) for a model of a viscous heat-conducting gas at high Reynolds numbers is made. The advantages of using the discrete-analytical approach are shown, for exam-ple: 1) the ability to make maximal use of the smoothness of the shock gas-dynamic formation (shock wave) in the tangential direction; 2) to build efficient computational algorithms devoid of the negative effect of approximation/artificial viscosity on the schematized discontinuity. In parallel, a very common graphical method for mapping the results of gas-dynamic calculations on the plane of shock polars, proposed by Busemann, and a volumetric (3D) polaroid, proposed by V. N. Uskov, are considered. The mathematical apparatus of shock polars itself is based on exact relations of the Rankine-Hugoniot type and has proven itself well even in modeling the flows of a viscous heat-conducting gas. However, in numerous literature sources there are results (shock solutions) of both physical and computational experiments, which are not mapped strictly on shock polars. In this paper, we show that in rare cases this very common way of such mapping may be incorrect. It has been proven that the main causes of such a defect are the combined ac-tion of three main factors: the nonuniform flow in front of the shock formation, the edge/boudary effect be-hind it, the action of the external factor of viscosity and the heat conduction mechanism.
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On remote sensing of the Earth by spacecraft
Статья научная
Remote sensing is a process which implies collecting information about an object. Due to their properties, satellite images are widely used in both practical and scientific fields. Satellite imagery is used in research aimed at the comprehensive study of natural resources, the dynamics of natural phenomena, and in the tasks of environmental protection. Special attention is paid to the use of space information for daily operational monitoring of the state of the environment in the implementation of geo-ecological monitoring of regions. In particular, this poses the problem to find the regions of the earth's surface with the characteristics determined by the considered parameters using the values of established parameters at certain points of the earth's surface. In this paper, we consider the special case of this problem when the given four points of the earth's surface determine the regions of the earth's surface (the so-called kernels of generalized squares) that have a specified configuration (square).
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On the application of the Bayesian search method for detecting underwater objects
Статья научная
Underwater archaeology and, in particular, the study of shipwrecked vessels, is one of the most advanced fields in the discipline, covering a range of theoretical and applied problems. In the practice of searching for sunken ships, it is quite rare to find a ship on the basis of previously (a priori) known data. In this regard, as well as in connection with the use of definitions from the field of probability theory and mathematical statistics, Bayesian statistics, namely, a search method that has found application in a number of well-known international search projects, has become the natural direction for the development of search systems. The Bayesian search method for determining the location of sunken ships, as well as their identification, has been almost never used in domestic underwater archeology practice. However, there is a need for its use, as it was demonstrated by the 2024 expedition to search for the transport ship Tbilisi, sunk during the Great Patriotic War in the Yenisei Gulf: despite the relatively small area of the search zone, determining the ship’s location became a very labor-intensive process. At the same time, the use of Bayesian search could significantly simplify this task. In this regard, the paper examines the methodology of using Bayesian search to detect sunken ships (an example of constructing a probability distribution in the search zone of the ship Tbilisi is given). In addition, the author considered the issue of using the Bayesian method for identifying objects (a database model is proposed with the inclusion of various search parameters). As a real example, the results of the 2024 expedition are presented with a description of the historical object, search conditions, and problems that arose during this work.
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On the choice of orbits for spacecrafts
Статья научная
The problem of distribution of a given number of spacecrafts over a certain structured set of orbits consisting of k np orbits is considered. The solution to this problem is given under the condition that the set of possible orbits for spacecraft coincides with the number of spacecrafts. In addition, it is assumed that the given set is divided into disjoint subsets of orbits, and the number of orbits in the indicated subset is the same. In the situation under consideration, it is equal to some prime number p. Currently, several orbits are used to place satellites on them, depending on the tasks they solve. Geostationary orbit is used for live TV broadcasting. Low satellite orbits are used for communication between satellite phones. Their own orbits exist for satellites of navigation systems GPS, Navstar, GLONASS, military satellites, satellites for various scientific research. Naturally, under these conditions, the problem of structuring a set of orbits with some restrictions on the location of the spacecraft in given orbits, depending on the purpose of the spacecraft arises. The problem of the complexity of calculating the number of orbits under these constraints is considered.
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On the construction of stress discontinuity lines for a two-dimensional plastic region
Статья научная
The paper considers the plasticity equations in case of two dimensions and the construction of stress discontinuity lines. The construction of stress discontinuity lines is based on the fact that they are located at the intersection point of lines of the same family (characteristics) and are directed along the bisector of the angle formed by these characteristics. Therefore, to find these lines, we constructed characteristics. Such a problem is easier to solve in case of plastic torsion, then there is only one characteristic, it is directed along the normal to the outer contour, and finding the sliding lines and their intersection points is quite simple. Consequently, most of the works devoted to the construction of stress discontinuity lines solve the problem of plastic torsion for isotropic and anisotropic media. For problems of plane strain of plastic ma-terial, this method is not sufficiently developed. This is due to the complexity of constructing sliding lines for such problems and the presence of two families of sliding lines. In this paper, we construct a homotopy of two known exact solutions: that of Prandtl and of Nadai, that is, a continuous transformation of one solution into another. In this case, one can observe the evolution of characteristics that depend on the group parameter a: for a = 1, the characteristics of the Prandtl solution are obtained; at a = 0, the characteristics of the Nadai solution, at a = 0.5, the characteristics of one fami-ly begin to intersect and lines of stress discontinuity appear. These lines are constructed in this paper.
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