Flight range maximization problem for a simplified aircraft model

封面

如何引用文章

全文:

开放存取 开放存取
受限制的访问 ##reader.subscriptionAccessGranted##
受限制的访问 订阅存取

详细

The flight range maximization problem for a simplified aircraft model is considered, considering the influence of the amount of fuel on the dynamics of the center of mass. It is assumed that the motion occurs in a vertical plane under the influence of homogeneous gravity forces and homogeneous resistance of the medium. In addition, there are an active thrust force and the ability to change the angle of inclination of the trajectory. These parameters are accepted as controls. An area in the space of initial variables is constructed, for which the problem of optimal synthesis is solved. It is shown that in this area the thrust can be of maximum value, zero value or singular value. The number and consequence of the trajectory arcs with the corresponding thrust have been established.

作者简介

E. Malykh

Lomonosov Moscow State University

编辑信件的主要联系方式.
Email: wyvling@gmail.com
俄罗斯联邦, Moscow

O. Cherkasov

Lomonosov Moscow State University; Shenzhen MSU-BIT University

Email: oyuche@yandex.ru
俄罗斯联邦, Moscow; Shenzhen, China

参考

  1. Goldstine H.H. A History of the Calculus of Variations from the 17 Th Through the 19 Th Century, Studies in the History of Mathematics and Physical Sciences. V.5. New York-Heidelberg-Berlin, Springer-Verlag, 1980. P. 410.
  2. Ashby N., Britten W. E., Love W. F., Wyss W. Brachistochrone with Coulomb Friction // Amer. J. Phys. 1975. V. 43. № 10. P. 902–905.
  3. Гершман М.Д., Нагаев Р. Ф. О фрикционной брахистохроне // МТТ 1976. № 4. С. 85–88.
  4. Lipp S.C. Brachistochrone with Coulomb Friction // SIAM J. Control Optim. 1997. V. 35. № 2. P. 562–584.
  5. Van der Heijden A.M.A., Diepstraten J.D. On the Brachistochrone with Dry Friction // Intern. J. Non-Linear Mech. 1975. V. 10. № 2. P. 97–112.
  6. Šalinić S. Contribution to the Brachistochrone Problem with Coulomb Friction // Acta Mech. 2009. V. 208. P. 97–115.
  7. Sumbatov A.S. Brachistochrone with Coulomb friction as the Solution of an Isoperimetrical Variational Problem // Intern. J. Non–Linear Mech. 2017. V. 88. P. 135–141.
  8. Hayen J.C. Brachistochrone with Coulomb Friction // Int. J. Non–Linear Mech. 2005. V. 40. P. 1057–1075.
  9. Голубев Ю.Ф. Брахистохрона с трением // Изв. РАН. ТиСУ. 2010. № 5. С. 41–52. https://doi.org/10.1134/S1064230710050060
  10. Vratanar B., Saje M. On the Analytical Solution of the Brachistochrone Problem in a Non-conservative Field // Intern. J. Non-Linear Mechanics. 1998. V. 33. № 3. P. 489–505.
  11. Зароднюк А.В., Черкасов О.Ю. Качественный анализ оптимальных траекторий движения материальной точки в сопротивляющейся среде и задача о брахистохроне // Изв. РАН. ТиСУ. 2015. № 1. С. 41–49.
  12. Šalinić S., Obradović A., Mitrović Z., Rusov S. Brachistochrone with Limited Reaction of Constraint in an Arbitrary Force Field // Nonlinear Dynamics. 2012. V. 69. P. 211–222.
  13. Lemak S.S., Belousova M.D. The Brachistochrone Problem with Constraints on the Curvature of the Trajectory // IFAC PapersOnLine. Moscow. V. 54. P. 437–442.
  14. Брайсон A., Хо Ю Ши. Прикладная теория оптимального управления. М.: Мир. 1972. С. 544.
  15. Feehery W.F. Dynamic Optimization with Path Constraints (Ph. D. Thesis) Massachusetts Institute of Technology. Cambridge, 1998.
  16. Cherkasov O.Yu., Smirnova N.V. On the Brachistochrone Problem with State Constraints on the Slope Angle // Intern. J. Non-Linear Mech. 2022. V. 139.
  17. Drummond J.E., Downes G.L. The Brachistochrone with Acceleration: A Running Track // J. Optimization Theory and Applications. 1971. V. 7. № 6. P. 444–449.
  18. Вондрухов А.С., Голубев Ю.Ф. Брахистохрона с разгоняющей силой // Изв. РАН. ТиСУ. 2014. № 6. C.42–57.
  19. Зароднюк А.В., Черкасов О.Ю. О максимизации горизонтальной дальности и брахистохроне с разгоняющей силой и вязким трением // Изв. РАН. ТиСУ. 2017. № 4. С. 3–10.
  20. Smirnova N.V, Cherkasov O.Yu. Range Maximization Problem with a Penalty on Fuel Consumption in the Modified Brachistochrone Problem // Applied Mathematical Modelling. 2021. V. 91. P. 581–589. https://doi.org/10.1016/j.apm.2020.10.001
  21. Руссаловская А.В., Иванов Г.И., Иванов А.И. О брахистохроне точки переменной массы с трением и экспоненциальным законом истечения массы // Докл. АН УССР. Сер. А. 1973. C. 1024–1026.
  22. Jeremić O., Šalinić S., Obradović A., Mitrović Z. On the Brachistochrone of a Variable Mass Particle in General Force Fields // Mathematical and Computer Modelling. 2011. V. 54. P. 2900–2912.
  23. Menon P.K.A., Kelley H.J., Cliff E.M. Optimal Symmetric Flight with an Intermediate Vehicle Model // J. GUIDANCE. 1984. V. 8. № 3. P. 312–319.
  24. Indig N., Ben-Asher J.Z., Sigal E. Singular Control for Two-Dimensional Goddard Problems Under Various Trajectory Bending Laws // J. Guidance, Control and Dynamics 2018. V. 42. № 3. P. 1–15. https://doi.org/10.2514/1.G003670
  25. Indig N., Ben-Asher J.Z., Sigal E. Optimal Guidance with Additional Thrust Control for Various Flight Tasks // AIAA Guidance, Navigation and Control. Conf. Texas AIAA, 2017. P. 1737. https://doi.org/10.2514/6.2017-1737
  26. Goddard R.H. A Method of Reaching Extreme Altitudes. Washington, Smithsonian Institute Miscellaneous Collections, 1919. V. 7. P. 71. (Reprinted by American Rocket Society. 1946.)
  27. Охоцимский Д.Е. К теории движения ракет // ПММ. 1946, Т. 10. № 2. С. 251–272.
  28. Tsien H.S., Evans R.C. Optimum Thrust Programming for a Sounding Rocket // J. American Rocket Society. 1951. V. 21. № 5. P. 99–107.
  29. Leitmann G.A. Calculus of Variations Solution of Goddard’s Problem // Astronautica Acta. 1956. V.2. № 2. P. 55–62.
  30. Seywald H., Cliff E.M. Goddard Problem in Presence of a Dynamic Pressure Limit // J. Guid. Control Dyn. 1993. V. 6. № 4. P. 776–781. https://doi.org/10.2514/3.21080
  31. Graichen K., Kugi A., Petit N., Chaplais F. Handling Constraints in Optimal Control with Saturation Functions and System Extension // Systems & Control Letters. 2010. V. 59. № 11. P. 671–679. https://doi.org/10.1016/j.sysconle.2010.08.003
  32. Bonnans F., Martinon P., Trélat E. Singular Arcs in the Generalized Goddard’s Problem // J. Optim Theory Appl. 2008. V. 139. P. 439–461. https://doi.org/10.1007/s10957-008-9387-1
  33. Miele A. Extremization of Linear Integrals by Green’s Theorem // Mathematics in Science and Engineering. 1962. V. 5. P. 69–98 https://doi.org/10.1016/S0076-5392(08)62091-3
  34. Tsiotras P., Kelley H.J. Goddard Problem with Constrained Time of Flight // J. Guidance, Control and Dynamics. 1992. V. 15. № 2. P. 289–296. https://doi.org/10.23919/ACC.1988.4789942
  35. Охоцимский Д.Е., Энеев Т.М. Некоторые вариационные задачи, связанные с запуском искусственного спутника Земли // УФН. 1957. № 1а. С. 5–32.
  36. Голубев Ю.Ф. Метод Охоцимского-Понтрягина в теории управления и аналитической механике. Ч. 1. Метод Охоцимского-Понтрягина в теории управления // Вестн. МГУ. Сер.1. Математика, механика. 2008. № 6. С. 49–55.
  37. Cherkasov O.Y., Malykh E.V., Smirnova N.V. Brachistochrone Problem and Two-dimensional Goddard Problem // Nonlinear Dyn. 2023. V.111. P. 243–254. https://doi.org/10.1007/s11071-022-07857-x
  38. Понтрягин Л.С., Болтянский В.Г., Гамкрелидзе Р.В., Мищенко Е.Ф. Математическая теория оптимальных процессов. М.: Наука, 1983. 393с.
  39. Габасов Р., Кириллова Ф.М. Особые оптимальные управления. М.: Наука, 1973. 256с.

补充文件

附件文件
动作
1. JATS XML

版权所有 © Russian Academy of Sciences, 2024