Violation of the Uniqueness of Solutions for the Mathematical Model of the Motion of an Aerodynamic Pendulum in the Flow of a Quasi-Static Medium
The article presents the construction and study of a mathematical model of small oscillations of an aerodynamic pendulum in a flow of a moving medium. The model of the impact of the medium on the body is taken as the model of the quasi-static flow around the plate by the medium. According to this hy...
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| Format: | Article |
| Language: | Russian |
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The Fund for Promotion of Internet media, IT education, human development «League Internet Media»
2022-10-01
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| Series: | Современные информационные технологии и IT-образование |
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| Online Access: | http://sitito.cs.msu.ru/index.php/SITITO/article/view/900 |
| _version_ | 1811162399413108736 |
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| author | Dmitry Belyakov |
| author_facet | Dmitry Belyakov |
| author_sort | Dmitry Belyakov |
| collection | DOAJ |
| description | The article presents the construction and study of a mathematical model of small oscillations of an aerodynamic pendulum in a flow of a moving medium. The model of the impact of the medium on the body is taken as the model of the quasi-static flow around the plate by the medium. According to this hypothesis, the aerodynamic forces acting on the body are applied at the center of pressure. In our task, the pressure centre is movable relative to the plate. The equations of motion for the considered body are obtained. A transition to new dimensionless variables is carried out. Uniqueness is shown to be impaired when determining the angle of attack at points where the air velocity of the pressure center is near zero. Some areas of ambiguity are constructed using multiple solutions of algebraic nonlinear equations obtained from kinematic relations. This method of creating areas of ambiguity is rather laborious and does not allow it to be done within a wide range of phase variables. An easier way of dealing with ambiguity areas is to define the coordinates of return points and draw the boundaries of ambiguity areas as two envelopes. The algorithm for obtaining these envelopes is described in detail. In the mathematical package MATLAB 18, a program is written that builds the upper and lower boundaries of the ambiguity regions. With its help, a parametric analysis of folds was carried out for various values of phase variables. It is shown that the sizes of the obtained areas of ambiguity are small. It is specified that the change of the angle of the attack during integration must be continuous, otherwise there is a failure in the search for a numerical solution. A set of programs has also been created, which makes it possible to build areas of ambiguity by repeatedly solving a nonlinear equation. Thus, a mathematical model of plate vibrations has been developed and a geometric analysis of surfaces and folds has been carried out using a set of programs based on a specialized computer mathematics system MATLAB 18. In this way, a very interesting problem in which there is a multi-valued angle of attack is considered, which is more understandable if we consider a mechanical interpretation, but non-trivial from the point of view of the general theory of solving differential equations. |
| first_indexed | 2024-04-10T06:29:01Z |
| format | Article |
| id | doaj.art-de49cf48c49144a5a4f4201454738e25 |
| institution | Directory Open Access Journal |
| issn | 2411-1473 |
| language | Russian |
| last_indexed | 2024-04-10T06:29:01Z |
| publishDate | 2022-10-01 |
| publisher | The Fund for Promotion of Internet media, IT education, human development «League Internet Media» |
| record_format | Article |
| series | Современные информационные технологии и IT-образование |
| spelling | doaj.art-de49cf48c49144a5a4f4201454738e252023-03-01T14:17:06ZrusThe Fund for Promotion of Internet media, IT education, human development «League Internet Media»Современные информационные технологии и IT-образование2411-14732022-10-0118361762410.25559/SITITO.18.202203.617-624Violation of the Uniqueness of Solutions for the Mathematical Model of the Motion of an Aerodynamic Pendulum in the Flow of a Quasi-Static MediumDmitry Belyakov0https://orcid.org/0000-0001-5093-2963Moscow Aviation Institute (National Research University), Moscow, RussiaThe article presents the construction and study of a mathematical model of small oscillations of an aerodynamic pendulum in a flow of a moving medium. The model of the impact of the medium on the body is taken as the model of the quasi-static flow around the plate by the medium. According to this hypothesis, the aerodynamic forces acting on the body are applied at the center of pressure. In our task, the pressure centre is movable relative to the plate. The equations of motion for the considered body are obtained. A transition to new dimensionless variables is carried out. Uniqueness is shown to be impaired when determining the angle of attack at points where the air velocity of the pressure center is near zero. Some areas of ambiguity are constructed using multiple solutions of algebraic nonlinear equations obtained from kinematic relations. This method of creating areas of ambiguity is rather laborious and does not allow it to be done within a wide range of phase variables. An easier way of dealing with ambiguity areas is to define the coordinates of return points and draw the boundaries of ambiguity areas as two envelopes. The algorithm for obtaining these envelopes is described in detail. In the mathematical package MATLAB 18, a program is written that builds the upper and lower boundaries of the ambiguity regions. With its help, a parametric analysis of folds was carried out for various values of phase variables. It is shown that the sizes of the obtained areas of ambiguity are small. It is specified that the change of the angle of the attack during integration must be continuous, otherwise there is a failure in the search for a numerical solution. A set of programs has also been created, which makes it possible to build areas of ambiguity by repeatedly solving a nonlinear equation. Thus, a mathematical model of plate vibrations has been developed and a geometric analysis of surfaces and folds has been carried out using a set of programs based on a specialized computer mathematics system MATLAB 18. In this way, a very interesting problem in which there is a multi-valued angle of attack is considered, which is more understandable if we consider a mechanical interpretation, but non-trivial from the point of view of the general theory of solving differential equations.http://sitito.cs.msu.ru/index.php/SITITO/article/view/900bodyarea of ambiguitysingular points |
| spellingShingle | Dmitry Belyakov Violation of the Uniqueness of Solutions for the Mathematical Model of the Motion of an Aerodynamic Pendulum in the Flow of a Quasi-Static Medium Современные информационные технологии и IT-образование body area of ambiguity singular points |
| title | Violation of the Uniqueness of Solutions for the Mathematical Model of the Motion of an Aerodynamic Pendulum in the Flow of a Quasi-Static Medium |
| title_full | Violation of the Uniqueness of Solutions for the Mathematical Model of the Motion of an Aerodynamic Pendulum in the Flow of a Quasi-Static Medium |
| title_fullStr | Violation of the Uniqueness of Solutions for the Mathematical Model of the Motion of an Aerodynamic Pendulum in the Flow of a Quasi-Static Medium |
| title_full_unstemmed | Violation of the Uniqueness of Solutions for the Mathematical Model of the Motion of an Aerodynamic Pendulum in the Flow of a Quasi-Static Medium |
| title_short | Violation of the Uniqueness of Solutions for the Mathematical Model of the Motion of an Aerodynamic Pendulum in the Flow of a Quasi-Static Medium |
| title_sort | violation of the uniqueness of solutions for the mathematical model of the motion of an aerodynamic pendulum in the flow of a quasi static medium |
| topic | body area of ambiguity singular points |
| url | http://sitito.cs.msu.ru/index.php/SITITO/article/view/900 |
| work_keys_str_mv | AT dmitrybelyakov violationoftheuniquenessofsolutionsforthemathematicalmodelofthemotionofanaerodynamicpendulumintheflowofaquasistaticmedium |