Regular and Chaotic Dynamics

, Volume 17, Issue 6, pp 547–558 | Cite as

Self-propulsion of a body with rigid surface and variable coefficient of lift in a perfect fluid

  • Sergey M. RamodanovEmail author
  • Valentin A. Tenenev
  • Dmitry V. Treschev


We study the system of a 2D rigid body moving in an unbounded volume of incompressible, vortex-free perfect fluid which is at rest at infinity. The body is equipped with a gyrostat and a so-called Flettner rotor. Due to the latter the body is subject to a lifting force (Magnus effect). The rotational velocities of the gyrostat and the rotor are assumed to be known functions of time (control inputs). The equations of motion are presented in the form of the Kirchhoff equations. The integrals of motion are given in the case of piecewise continuous control. Using these integrals we obtain a (reduced) system of first-order differential equations on the configuration space. Then an optimal control problem for several types of the inputs is solved using genetic algorithms.


perfect fluid self-propulsion Flettner rotor 

MSC2010 numbers

70Hxx 70G65 


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  1. 1.
    Borisov, A.V. and Mamaev, I. S., Rigid Body Dynamics: Hamiltonian Methods, Integrability, Chaos, Moscow-Izhevsk: R&C Dynamics, 2005 (Russian).zbMATHGoogle Scholar
  2. 2.
    Vetchanin, E. V. and Tenenev, V.A., Motion Control Simulating in a Viscous Liquid of a Body with Variable Geometry of Weights, Computer Research and Modeling, 2011, vol. 3, no. 4, pp. 371–381 (Russian).Google Scholar
  3. 3.
    Voinov, O.V., Inertial Motion of a Body in an Ideal Fluid from the State at Rest, Prikl. Mekh. Tekhn. Fiz., 2008, vol. 49, no. 4, pp. 214–219 [J. Appl. Mech. Tech. Phys., 2008, vol. 49, no. 4, pp. 699–703].Google Scholar
  4. 4.
    Kirchhoff, G. R., Vorlesungen über mathematische Physik: Bd. 1: Mechanik, Leipzig: Teubner, 1876.Google Scholar
  5. 5.
    Kozlov, V.V., General Theory of Vortices, Izhevsk: Izdatel’skij Dom “Udmurtskij Universitet”, 1998 [Encyclopaedia Math. Sci., vol. 67, Berlin: Springer, 2003].Google Scholar
  6. 6.
    Kozlov, V.V. and Ramodanov, S. M., The Motion of a Variable Body in an Ideal Fluid, Prikl. Mat. Mekh., 2001, vol. 65, no. 4, pp. 592–601 [J. Appl. Math. Mech., 2001, vol. 65, no. 4, pp. 579–587].MathSciNetzbMATHGoogle Scholar
  7. 7.
    Kochin, N.E., Kibel, I. A., and Roze, N. V., Theoretical Hydrodynamics, New York: Wiley, 1964.Google Scholar
  8. 8.
    Lavrentyev, M.A. and Lavrentyev, M.M., On One Principle of Creating the Thrust Force in Motion, Prikl. Mekh. Tekhn. Fiz., 1962, no. 4, pp. 3–9 [J. Appl. Mech. Tech. Phys., 1962, no. 4, pp. 6–9].Google Scholar
  9. 9.
    Lavrentiev, M.A. and Shabat, B. D., Hydrodynamics Problems and Mathematical Models, Moscow: Nauka, 1973 (Russian).Google Scholar
  10. 10.
    Benjamin, T.B. and Ellis, A.T., The Collapse of Cavitation Bubbles and the Pressure Thereby Produced Against Solid Boundaries, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 1966, vol. 260, no. 1110, pp. 221–240.CrossRefGoogle Scholar
  11. 11.
    Galper, A. and Miloh, T., Self-Propulsion of General Deformable Shapes in a Perfect Fluid, Proc. Roy. Soc. A, 1993, vol. 442, pp. 273–299.CrossRefzbMATHGoogle Scholar
  12. 12.
    Galper, A. and Miloh, T., Dynamical Equations for the Motion of a Rigid or Deformable Body in an Arbitrary Potential Nonuniform Flow Field, J. Fluid Mech., 1995, vol. 295, pp. 91–120.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Galper, A.R. and Miloh, T., Hydrodynamics and Stability of a Deformable Body Moving in the Proximity of Interfaces, Phys. Fluids, 1999, vol. 11, no. 4, pp. 795–806.MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kelly, S.D. and Hukkeri, R. B., Mechanics, Dynamics, and Control of a Single-Input Aquatic Vehicle with Variable Coefficient of Lift, IEEE Transactions on Robotics, 2006, vol. 22, no. 6, pp. 1254–1264.CrossRefGoogle Scholar
  15. 15.
    Kuznetsov, V. M., Lugovtsov, B.A., and Sher, Y. N., On the Motive Mechanism of Snakes and Fish, Arch. Ration. Mech. Anal., 1967, vol. 25, no. 5, pp. 367–387.MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Lamb, H., Hydrodynamics, 6th ed., New York: Dover, 1945.Google Scholar
  17. 17.
    Landweber, L. and Miloh, T., The Lagally Theorem for Unsteady Multipoles and Deformable Bodies, J. Fluid Mech., 1980, vol. 96, pp. 33–46.CrossRefzbMATHGoogle Scholar
  18. 18.
    Lighthill, J. M., Note on Swimming of Slender Fish, J. Fluid Mech., 1960, vol. 9, pp. 305–317.MathSciNetCrossRefGoogle Scholar
  19. 19.
    Milne-Thomson, L.M., Theoretical Hydrodynamics, 4th ed., London: MacMillan, 1962.Google Scholar
  20. 20.
    Magnus H.G. Über die Abweichung der Geschosse, Poggendorff’s Annalen der Physik u.Chemie, 1853, vol. 88, pp. 1–14.Google Scholar
  21. 21.
    Saffman, P.G., The Self-Propulsion of a Deformable Body in a Perfect Fluid, J. Fluid Mech., 1967, vol. 28, pp. 385–389.MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Taylor, G. I., Analysis of the Swimming of Microscopic Organisms, Proc. Roy. Soc. A, 1951, vol. 209, pp. 447–461.CrossRefzbMATHGoogle Scholar
  23. 23.
    Taylor, G. I., The Action of Waving Cylindrical Tails in Propelling Microscopic Organisms, Proc. Roy. Soc. A, 1952, vol. 211, pp. 225–239.CrossRefzbMATHGoogle Scholar
  24. 24.
    Taylor, G. I., Analysis of the Swimming of Long and Narrow Animals, Proc. Roy. Soc. A, 1952, vol. 214, pp. 158–183.CrossRefzbMATHGoogle Scholar
  25. 25.
    Wu, T.Y., Swimming of a Waving Plate, J. Fluid Mech., 1961, vol. 10, pp. 321–344.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2012

Authors and Affiliations

  • Sergey M. Ramodanov
    • 1
    Email author
  • Valentin A. Tenenev
    • 2
  • Dmitry V. Treschev
    • 3
    • 4
  1. 1.Institute of Computer ResearchUdmurt State UniversityIzhevskRussia
  2. 2.Izhevsk State Technical UniversityIzhevskRussia
  3. 3.Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia
  4. 4.M. V. Lomonosov Moscow State UniversityVorob’evy gory, MoscowRussia

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