MODELING OF VIBRATIONAL PROCESSES IN DISCRETE MATRIX STRUC-TURES APPROACH
In the article are presented general principles of modeling vibrations in discrete structures formed in the form of special matrix forms of the Latin square (Sudoku type) are presented. The signs of structural and functional self-similarity for the matrix structures of standard Sudoku grids are form...
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| Format: | Article |
| Language: | English |
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National Aerospace University «Kharkiv Aviation Institute»
2020-04-01
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| Series: | Радіоелектронні і комп'ютерні системи |
| Subjects: | |
| Online Access: | http://nti.khai.edu/ojs/index.php/reks/article/view/980 |
| _version_ | 1797707939357655040 |
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| author | Yuriy Penkin Georgi Hara Alina Fedoseeva |
| author_facet | Yuriy Penkin Georgi Hara Alina Fedoseeva |
| author_sort | Yuriy Penkin |
| collection | DOAJ |
| description | In the article are presented general principles of modeling vibrations in discrete structures formed in the form of special matrix forms of the Latin square (Sudoku type) are presented. The signs of structural and functional self-similarity for the matrix structures of standard Sudoku grids are formulated. It is shown that the structural principle can be interpreted as the implementation of the second iteration in the scale scaling algorithm characteristic of fractal objects. The signs of functional self-similarity of structures include the property of additive conservation of grid shapes to the requirements of Sudoku, which is formulated as a theorem. It is proved that the matrix sums of Sudoku constants and grids of arbitrary sizes, obtained taking into account the introduced cyclic ranking rule, will satisfy the three required Sudoku requirements. It is determined that by performing a given sequence of group shift operators, it is possible to establish a specific scenario for dynamically changing the state of a structure on a discrete time scale. It has been established that the evolution operators of linear-type group translations lead to matrix transformations of Sudoku grids from the set of equivalent ones (concerning the original ones), and the vortex-type group shifts operators to matrix transformations from many essentially different networks. The modes of harmonic, chaotic, and hybrid vibrations for structures of arbitrary size are considered. The requirements for transformations of the operators of the evolution of structures that provide the implementation of the considered modes are formulated. The results of modeling chaotic oscillatory processes by cycles of states of a discrete system that form similarities of attractor paths are analyzed. The principle of synchronization of chaotic states of matrix structures is established. The possibility of simulating the modes of beatings of oscillations in discrete cellular structures organized in the form of two-level matrix forms is substantiated. Specific examples show the results of simulating beatings of oscillations in cycles of changing states of a discrete system for two types of beats: similar to the result of a superposition of harmonic vibrations at multiple frequencies in the theory of radio signals, as well as noise-like beats. |
| first_indexed | 2024-03-12T06:13:56Z |
| format | Article |
| id | doaj.art-a883c0b8e86542eb81e25ff3db0a3ffd |
| institution | Directory Open Access Journal |
| issn | 1814-4225 2663-2012 |
| language | English |
| last_indexed | 2024-03-12T06:13:56Z |
| publishDate | 2020-04-01 |
| publisher | National Aerospace University «Kharkiv Aviation Institute» |
| record_format | Article |
| series | Радіоелектронні і комп'ютерні системи |
| spelling | doaj.art-a883c0b8e86542eb81e25ff3db0a3ffd2023-09-03T02:47:01ZengNational Aerospace University «Kharkiv Aviation Institute»Радіоелектронні і комп'ютерні системи1814-42252663-20122020-04-0102677910.32620/reks.2020.2.061025MODELING OF VIBRATIONAL PROCESSES IN DISCRETE MATRIX STRUC-TURES APPROACHYuriy Penkin0Georgi Hara1Alina Fedoseeva2National Pharmaceutical University, KharkovNational Pharmaceutical University, KharkovKharkiv radiotechnical College, National Aerospace University “Kharkiv Aviation Institute”In the article are presented general principles of modeling vibrations in discrete structures formed in the form of special matrix forms of the Latin square (Sudoku type) are presented. The signs of structural and functional self-similarity for the matrix structures of standard Sudoku grids are formulated. It is shown that the structural principle can be interpreted as the implementation of the second iteration in the scale scaling algorithm characteristic of fractal objects. The signs of functional self-similarity of structures include the property of additive conservation of grid shapes to the requirements of Sudoku, which is formulated as a theorem. It is proved that the matrix sums of Sudoku constants and grids of arbitrary sizes, obtained taking into account the introduced cyclic ranking rule, will satisfy the three required Sudoku requirements. It is determined that by performing a given sequence of group shift operators, it is possible to establish a specific scenario for dynamically changing the state of a structure on a discrete time scale. It has been established that the evolution operators of linear-type group translations lead to matrix transformations of Sudoku grids from the set of equivalent ones (concerning the original ones), and the vortex-type group shifts operators to matrix transformations from many essentially different networks. The modes of harmonic, chaotic, and hybrid vibrations for structures of arbitrary size are considered. The requirements for transformations of the operators of the evolution of structures that provide the implementation of the considered modes are formulated. The results of modeling chaotic oscillatory processes by cycles of states of a discrete system that form similarities of attractor paths are analyzed. The principle of synchronization of chaotic states of matrix structures is established. The possibility of simulating the modes of beatings of oscillations in discrete cellular structures organized in the form of two-level matrix forms is substantiated. Specific examples show the results of simulating beatings of oscillations in cycles of changing states of a discrete system for two types of beats: similar to the result of a superposition of harmonic vibrations at multiple frequencies in the theory of radio signals, as well as noise-like beats.http://nti.khai.edu/ojs/index.php/reks/article/view/980data discrete structuresudoku gridsigns of self-similarityoperators of matrix transformationsoscillations modesdeterministic chaosnoise-like beats |
| spellingShingle | Yuriy Penkin Georgi Hara Alina Fedoseeva MODELING OF VIBRATIONAL PROCESSES IN DISCRETE MATRIX STRUC-TURES APPROACH Радіоелектронні і комп'ютерні системи data discrete structure sudoku grid signs of self-similarity operators of matrix transformations oscillations modes deterministic chaos noise-like beats |
| title | MODELING OF VIBRATIONAL PROCESSES IN DISCRETE MATRIX STRUC-TURES APPROACH |
| title_full | MODELING OF VIBRATIONAL PROCESSES IN DISCRETE MATRIX STRUC-TURES APPROACH |
| title_fullStr | MODELING OF VIBRATIONAL PROCESSES IN DISCRETE MATRIX STRUC-TURES APPROACH |
| title_full_unstemmed | MODELING OF VIBRATIONAL PROCESSES IN DISCRETE MATRIX STRUC-TURES APPROACH |
| title_short | MODELING OF VIBRATIONAL PROCESSES IN DISCRETE MATRIX STRUC-TURES APPROACH |
| title_sort | modeling of vibrational processes in discrete matrix struc tures approach |
| topic | data discrete structure sudoku grid signs of self-similarity operators of matrix transformations oscillations modes deterministic chaos noise-like beats |
| url | http://nti.khai.edu/ojs/index.php/reks/article/view/980 |
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