Stability of Various Stationary Points with Small Oscillations of the Aerodynamic Pendulum in the Flow of a Quasi-Static Medium
Abstract
In the article, a mathematical model of small oscillations of an aerodynamic pendulum in the flow of a moving medium is constructed and investigated. As a model of the effect of the medium on the body, the model of quasi-static flow around the plate by the medium is adopted. According to this hypothesis, the aerodynamic forces acting on the body are applied at the center of pressure. In our problem, the pressure center is movable relative to the plate. The equations of motion for the body under consideration are obtained. The transition to new dimensionless variables has been carried out. The violation of uniqueness in determining the angle of attack is shown. The parametric analysis of the ambiguity areas is carried out. All stationary points that are solutions of the equilibrium equations are found. It is shown that there is no ambiguity in the most characteristic equilibrium position corresponding to the state of rest. A study of the stability of various non-trivial equilibrium positions in which the Hurwitz criterion is implemented, and the stability regions are depicted is carried out. It is shown that the forces of aerodynamic action for bodies with some shapes can contribute to the development of self-oscillations, and for others to attenuation. The mathematical package MATLAB 18 contains a set of programs that allows you to find stationary points, build stability regions for each of them and perform numerical integration of equations describing body vibrations in order to confirm the adequacy of the constructed mathematical model.
References
2. Belyakov D.V. The Problem of Studying the Self-Oscillations of an Aerodynamic Pendulum in the Flow of a Medium. Sovremennye informacionnye tehnologii i IT-obrazovanie = Modern Information Technologies and IT-Education. 2020; 16(2):449-459. (In Russ., abstract in Eng.) doi: https://doi.org/10.25559/SITITO.16.202002.449-459
3. Belyakov D.V. Matematicheskoe modelirovanie dvizhenija rotirujushhego spuskajushhegosja v vozduhe ob'ekta [Mathematical modeling of the motion of a rotating object descending in the air]. Proceedings of the Fifth international Aerospace Congress IAC-06. Dedicated to the 20th anniversary of the launch of the MIR Space Station. Moscow; 2006. p. 62-63. (In Russ.)
4. Belyakov D.V., Samsonov V.A., Filippov V.V. Motion Investigation of Asymmetric Solid in Resistant Environment. Vestnik Moskovskogo jenergeticheskogo instituta = Vestnik MEI. Bulletin of Moscow Power Engineering Institute. 2006; (4):5-10. Available at: https://elibrary.ru/item.asp?id=9455853 (accessed 23.09.2021). (In Russ., abstract in Eng.)
5. Belyakov D.V. Development and Features of Mathematical Model of Movement Asymmetrical Autorotating Bodies in Quasi-static to Environment. Mehatronika, Avtomatizacija, Upravlenie = Мechatronics, Automation, Control. 2007; (11):20-24. Available at: https://elibrary.ru/item.asp?id=9609383 (accessed 23.09.2021). (In Russ., abstract in Eng.)
6. Samsonov V.A., Dosaev M.Z., Selyutskiy Yu.D. Methods of Qualitative Analysis in the Problem of Rigid Body Motion in Medium. International Journal of Bifurcation and Chaos. 2011; 21(10):2955-2961. (In Eng.) doi: https://doi.org/10.1142/S021812741103026X
7. Strickland J.H., Webster B.T., Nguyen T. A Vortex Model of the Darrieus Turbine: An Analytical and Experimental Study. Journal of Fluids Engineering. 1979; 101(4):500-505. (In Eng.) doi: http://doi.org/10.1115/1.3449018
8. Lyatkher V.M. High Jet Power Station with Orthogonal Power Units. Alternativnaya Energetika i Ekologiya = Alternative Energy and Ecology. 2014; (7):21-38. Available at: https://elibrary.ru/item.asp?id=21497653 (accessed 23.09.2021). (In Russ., abstract in Eng.)
9. Moskatov G.K., Chepelev A.A. Reliability and safety of feedback flight control systems. Scientific Bulletin of the military-industrial complex of Russia. 2013; (2):41-63. Available at: https://elibrary.ru/item.asp?id=24276464 (accessed 23.09.2021). (In Russ., abstract in Eng.)
10. Paraschivoiu I., Delclaux F. Double multiple streamtube model with recent improvements (for predicting aerodynamic loads and performance of Darrieus vertical axis wind turbines). Journal of Energy. 1983; 7(3):250. (In Eng.) doi: http://doi.org/10.2514/3.48077
11. Alqurashi F., Mohamed M.H. Aerodynamic Forces Affecting the H-Rotor Darrieus Wind Turbine. Modelling and Simulation in Engineering. 2020; 2020:1368369. (In Eng.) doi: http://doi.org/10.1155/2020/1368369
12. Parashivoiu I. Aerodynamic loads and rotor performance for the Darrieus wind turbines. Journal of Energy. 1982; 6:406-412. (In Eng.) doi: http://doi.org/10.2514/6.1981-2582
13. Dosaev M.Z., Samsonov V.A., Seliutski Yu.D. On the Dynamics of a Small-Scale Wind Power Generator. Doklady Physics. 2007; 52(9):493-495. (In Eng.) doi: http://doi.org/10.1134/S1028335807090091
14. Samsonov V.A., Selyutskii Yu.D. Comparison of Different Notation for Equations of Motion of a Body in a Medium Flow. Mechanics of Solids. 2008; 43(1):146-152. (In Eng.) doi: http://doi.org/10.1007/s11964-008-1015-x
15. Samsonov V.A., Selyutskii Yu.D. About Vibrations of a Plate in a Flow of a Resisting Medium. Mechanics of Solids. 2004; (4):24. Available at: https://elibrary.ru/item.asp?id=17636289 (accessed 23.09.2021). (In Russ., abstract in Eng.)
16. Privalov V.A., Samsonov V.A. Ob ustojchivosti dvizhenija tela, avtorotirujushhego v potoke sredy [On the Stability of Motion of a Body Autorotating in the Flow of a Medium]. Izv. USSR Acad. Sci. MTT. 1990; (2):32-38. (In Russ.)
17. Zhang J.Z., Liu Y., Sun X., Chen J.H., Wang L. Applications and Developments of Aeroelasticity of Flexible Structure in Flow Controls. Advances in Mechanics. 2018; 48(1):299-319. (In Eng.) doi: http://doi.org/10.6052/1000-0992-16-034
18. Klimina L.A. Rotational modes of motion for an aerodynamic pendulum with a vertical rotation axis. Moscow University Mechanics Bulletin. 2009; 64(5):126-129. (In Eng.) doi: https://doi.org/10.3103/S0027133009050069
19. Klimina L.A., Lokshin B.Ya. On a constructive method of search for rotary and oscillatory modes in autonomous dynamical systems. Russian Journal of Nonlinear Dynamics. 2017; 13(1):25-40. (In Russ., abstract in Eng.) doi: http://doi.org/10.20537/nd1701003
20. Klimina L.A. Method for Generating Asynchronous Self-Sustained Oscillations of a Mechanical System with Two Degrees of Freedom. Mechanics of Solids. 2021; 56(7):1167-1180. (In Eng.) doi: https://doi.org/10.3103/S0025654421070141
21. Yao J., Yeo K.S. Free hovering of hummingbird hawkmoth and effects of wing mass and wing elevation. Computers & Fluids. 2019; 186; 99-127. (In Eng.) doi: https://doi.org/10.1016/j.compfluid.2019.04.007
22. Hesamian G., Akbari M.G. A fuzzy additive regression model with exact predictors and fuzzy responses. Applied Soft Computing. 2020; 95:106507. (In Eng.) doi: https://doi.org/10.1016/j.asoc.2020.106507
23. Radionov A.A., Gasiyarov V.R. Proceedings of the 7th International Conference on Industrial Engineering (ICIE 2021). Lecture Notes in Mechanical Engineering. Vol. II. Springer Cham; 2022. 849 p. (In Eng.) doi: https://doi.org/10.1007/978-3-030-85230-6
24. Li Q., Hou P. Three-dimensional quasi-static general solution for isotropic thermoelastic medium with applications. Case Studies in Thermal Engineering. 2021; 25:100897. (In Eng.) doi: https://doi.org/10.1016/j.csite.2021.100897
25. Liu X., Xu J., Liu Y. Trajectory tracking and point stability of three-axis aero-dynamic pendulum with MPC strategy in disturbance environment. Assembly Automation. 2021; 41(3):358-368. (In Eng.) doi: https://doi.org/10.1108/AA-11-2020-0181
This work is licensed under a Creative Commons Attribution 4.0 International License.
Publication policy of the journal is based on traditional ethical principles of the Russian scientific periodicals and is built in terms of ethical norms of editors and publishers work stated in Code of Conduct and Best Practice Guidelines for Journal Editors and Code of Conduct for Journal Publishers, developed by the Committee on Publication Ethics (COPE). In the course of publishing editorial board of the journal is led by international rules for copyright protection, statutory regulations of the Russian Federation as well as international standards of publishing.
Authors publishing articles in this journal agree to the following: They retain copyright and grant the journal right of first publication of the work, which is automatically licensed under the Creative Commons Attribution License (CC BY license). Users can use, reuse and build upon the material published in this journal provided that such uses are fully attributed.
