Computer analysis of the differential equation of the model of the synchronous electric motor that does not include electric currents
DOI:
https://doi.org/10.21638/spbu01.2023.305Abstract
Phase portraits of the well-known equation of the no-current model of the synchronous electric motor are obtained with use of computer, and some properties of its solutions are noted, which are imperceptible at the phase portraits, obtained analytically. With use of computer, a graph is built for the curve, representing the critical value versus the principal stationary value of the angular variable. Linear and sinusoidal approximations are proposed for this curve, maximum absolute and relative errors of these approximations are calculated.Keywords:
synchronous electric motor, phase portrait, critical value, global stability, reduction method
Downloads
References
Литература
1. Леонов Г. А. Фазовая синхронизация. Теория и приложения. Автоматика и телемеханика 10, 47-85 (2006).
2. Леонов Г. А. Второйметод Ляпунова в теории фазовойсинхронизации. Прикладная математика и механика. 40 (2), 238-244 (1976).
3. Гелиг А. Х., Леонов Г. А., Якубович В. А. Устойчивость нелинейных систем с неединственным состоянием равновесия. Москва, Наука (1978).
4. Коносевич Б. И., Коносевич Ю. Б. Достаточное условие глобальнойустойчивости модели синхронного электромотора при нелинейном моменте нагрузки. Вестник Санкт-Петербургского университета. Математика. Механика. Астрономия 5 (63), вып. 1, 74-85 (2018). https://doi.org/10.21638/11701/spbu01.2018.109
5. Tricomi F. Integrazione di unequazione differenziale presentasi in electrotechnica. Annali della Roma Schuola Normale Superiore de Pisa 2 (2), 1-20 (1933).
6. Барбашин Е. А., Табуева В. А. Динамические системы с цилиндрическим фазовым пространством. Москва, Наука (1969).
7. Климов Д. М., Харламов С. А. Динамика гироскопа в кардановом подвесе. Москва, Наука (1978).
8. Коносевич Б. И., Коносевич Ю. Б. Об устойчивости стационарных движений гироскопа в кардановом подвесе, снабженного электродвигателем. Известия РАН. Механика твердого тела 3, 57-73 (2013).
9. Коносевич Б. И., Коносевич Ю. Б. Критерийустойчивости стационарных решенийуравнениймноготоковоймодели синхронного гироскопа в кардановом подвесе. I. Известия РАН. Механика твердого тела 2, 124-141 (2020).
10. Коносевич Б. И., Коносевич Ю. Б. Критерийустойчивости стационарных решенийуравнениймноготоковоймодели синхронного гироскопа в кардановом подвесе. II. Известия РАН. Механика твердого тела 1, 50-68 (2021). https://doi.org/10.31857/S0572329920020075
11. Леонов Г. А., ЗарецкийА. М. Глобальная устойчивость и колебания динамических систем, описывающих синхронные электрические машины. Вестник Санкт-Петербургского университета. Математика. Механика. Астрономия 4, 18-27 (2012).
12. Коносевич Б. И., Коносевич Ю. Б. Аппроксимация критического значения параметра демпфирования для синхронного электромотора. Труды Института прикладной математики и механики 29, 121-126 (2014).
13. Карманов В. Г. Математическое программирование. 2-е изд. Москва, Наука (1980).
14. Amerio L. Determinazione delle condizioni di stabilit`a per gli integrali di un’equazione interessante l’electrotecnica. Ann. Mat. pura ed appl. 2 (2), 75-90 (1949
References
1. Leonov G. A. Phase synchronization. Theory and applications. Automatics and Telemechanics 10, 47-85 (2006). (In Russian)
2. Leonov G. A. Lyapunov’s second method in the theory of phase synchronization. Applied Mathematics and Mechanics 40 (2), 238-244 (1976). (In Russian)
3. Gelig A. Kh., Leonov G. A., Yakubovich V. A. Stability of nonlinear systems with a nonunique equilibrium state. Moscow, Nauka Publ. (1978). (In Russian)
4. Konosevich B. I., Konosevich Yu. B. Sufficient global stability condition for a model of the synchronous electric motor under nonlinear load moment. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy 5 (63), iss. 1, 74-85. (2018). https://doi.org/10.21638/11701 /spbu01.2018.109 (In Russian) [Engl. trans.: Vestnik St. Petersburg University. Mathematics 51, iss. 1, 57-65 (2018). https://doi.org/10.3103/S1063454118010053].
5. Tricomi F. Integrazione di unequazione differenziale presentasi in electrotechnica. Annali della Roma Schuola Normale Superiore de Pisa 2 (2), 1-20 (1933).
6. Barbashin E. A., Tabueva V. A. Dynamical systems with the cylindrical phase space. Moscow, Nauka Publ. (1969). (In Russian)
7. Klimov D. M., Kharlamov S. A. Dynamics of a gimbals mounted gyroscope. Moscow, Nauka Publ. (1978). (In Russian)
8. Konosevich B. I., Konosevich Yu. B. On stability of steady-state motions of a gimbals mounted gyroscope supplied with the electric motor. Izvestiia RAN. Mekhanika tverdogo tela 3, 57-73 (2013). (In Russian) [Engl. trans.: Mechanics of Solids 48 (3), 285-297 (2013). https://doi.org/10.3103/S0025654413030059].
9. Konosevich B. I., Konosevich Yu. B. Stability criterion for stationary solutions of multi-current model equations for a synchronous gimbal-mounted gyroscope. I. Izvestiia RAN. Mekhanika tverdogo tela 2, 124-141 (2020) (In Russian) [Engl. trans.: Mechanics of Solids 55 (2), 258-272 (2020). https://doi.org/10.3103/S0025654420020119].
10. Konosevich B. I., Konosevich Yu. B. Stability criterion for stationary solutions of multi-current model equations for a synchronous gimbal-mounted gyroscope. II. Izvestiia RAN. Mekhanika tverdogo tela 1, 50-68 (2021). https://doi.org/10.31857/S0572329920020075 (In Russian) [Engl. trans.: Mechanics of Solids 56 (1), 40-54 (2021). https://doi.org/10.3103/S0025654421010088].
11. Leonov G. A., Zaretskiy A. M. Global stability and oscillations of dynamical systems describing synchronous electrical machines. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy 4, 18-27 (2012). (In Russian) [Engl. trans.: Vestnik St. Petersburg University. Mathematics 45 (4), 157-163 (2012)].
12. Konosevich B. I., Konosevich Yu. B. Approximation of the critical value of the damping parameter for the synchronous electric motor. Proceedings of the Institute of applied mathematics and mechanics 29, 121-126 (2014). (In Russian)
13. Karmanov V. G. Mathematical programming. 2nd ed. Moscow, Nauka Publ., 1980. (In Russian)
14. Amerio L. Determinazione delle condizioni di stabilit`a per gli integrali di un’equazione interessante l’electrotecnica. Ann. Mat. pura ed appl. 2 (2), 75-90 (1949).
Downloads
Published
2023-09-23
How to Cite
Konosevich, B. I., & Konosevich, Y. B. (2023). Computer analysis of the differential equation of the model of the synchronous electric motor that does not include electric currents. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 10(3), 499–515. https://doi.org/10.21638/spbu01.2023.305
Issue
Section
Mathematics
License
Articles of "Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy" are open access distributed under the terms of the License Agreement with Saint Petersburg State University, which permits to the authors unrestricted distribution and self-archiving free of charge.
