On the problem of free deceleration of a rigid body in a resisting medium



A mathematical model of the influence of a medium on a rigid body with some part of its external surface being flat is considered with due allowance for an additional dependence of the moment of the medium action force on the angular velocity of the body. A full system of equations of motion is given under quasi-steady conditions; the dynamic part of this system forms an independent third-order system, and an independent second-order subsystem is split from the full system. A new family of phase portraits on a phase cylinder of quasi-velocities is obtained. It is demonstrated that the results obtained allow one to design hollow circular cylinders (“shell cases”), which can ensure necessary stability in conducting additional full-scale experiments.


rigid body resisting medium equations of motion phase portrait 


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  1. 1.
    N. E. Joukowski, “Incidence of Light Extended Bodies Rotating around their Longitudinal Axis,” in Total Collected Papers (Fizmatgiz, Moscow, 1937), Vol. 5, pp. 72–80 and 100–115) [in Russian].Google Scholar
  2. 2.
    S. A. Chaplygin, Selected Papers (Nauka, Moscow, 1976) [in Russian].Google Scholar
  3. 3.
    S. A. Chaplygin, “On Motion of Heavy Bodies in an Incompressible Fluid,” in Total Collected Papers (Izd. Akad. Nauk SSSR, Leningrad, 1933), Vol. 1, pp. 133–135 [in Russian].Google Scholar
  4. 4.
    M. V. Shamolin, Methods of Analysis of Variable Dissipation Dynamic Systems in Rigid Body Dynamics (Ekzamen, Moscow, 2007) [in Russian].MATHGoogle Scholar
  5. 5.
    M. V. Shamolin, “Dynamic Systems with Variable Dissipation: Approaches, Methods, and Applications,” Fundam. Prikl. Mat. 14 (3), 3–237 (2008).Google Scholar
  6. 6.
    H. Lamb, Hydrodynamics (Cambridge Univ. Press, Cambridge, 1932).MATHGoogle Scholar
  7. 7.
    M. V. Shamolin, “Variety of Integrable Cases in Dynamics of Low- and Multi-Dimensional Rigid Bodies in Nonconservative Force Fields,” Itogi Nauki Tekh., Ser. Sovr. Mat. Pril. Temat. Obzory 125, 3–251 (2013).Google Scholar
  8. 8.
    B. Ya. Lokshin, V. A. Privalov, and V. A. Samsonov, Introduction into the Problem of Motion of a Body in a Resisting Medium (Izd. Mosk. Gos. Univ., Moscow, 1986) [in Russian].Google Scholar
  9. 9.
    V. A. Eroshin, G. A. Konstantinov, N. I. Romanenkov, and Yu. L. Yakimov, “Experimental Determination of Pressure on a Disk Submerged into a Compressible Fluid at an Angle to the Free Surface,” Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza, No. 2, 21–25 (1988).Google Scholar
  10. 10.
    V. A. Eroshin, G. A. Konstantinov, N. I. Romanenkov, and Yu. L. Yakimov, “Experimental Determination of the Moment of Hydrodynamic Forces in the Case of Asymmetric Penetration of a Disk into a Compressible Fluid,” Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza, No. 5, 88–94 (1990).ADSGoogle Scholar
  11. 11.
    M. I. Gurevich, Theory of Jets of an Ideal Fluid (Nauka, Moscow, 1979) [in Russian].Google Scholar
  12. 12.
    V. G. Tabachnikov, Steady-State Characteristics of Wings at Small Velocities in the Entire Range of the Angles of Attack,” Tr. TsAGI, No. 1621, 18–24 (1974).Google Scholar
  13. 13.
    L. Prandtl and A. Betz, Ergebmisse der Aerodinamischen Versuchsastalt zu Gottingen (Aerodinam. Versuchsastalt zu Gottingen, München–Berlin, 1932).Google Scholar
  14. 14.
    G. S. Bushgens and R. V. Studnev, Aircraft Dynamics. Spatial Motion (Mashinostroenie, Moscow, 1988) [in Russian].Google Scholar
  15. 15.
    V. A. Eroshin, V. A. Samsonov, and M. V. Shamolin, “Model Problem of Body Deceleration in a Resisting Medium in a Jet Flow,” Izv. Ross. Akad. Nauk, Mekh. Zhidk. Gaza, No. 3, 23–27 (1995).Google Scholar
  16. 16.
    V. A. Samsonov, M. V. Shamolin, V. A. Eroshin, and V. M. Makarshin, “Mathematical Modeling in the Problem of Body Deceleration in a Resisting Medium in a Jet Flow,” Report No. 4396 (Inst. Mechanics, Moscow State University, Moscow, 1995).Google Scholar
  17. 17.
    V. A. Eroshin, “Experimental Study of High-Velocity Penetration of an Elastic Cylinder into Water,” Izv. Ross. Akad. Nauk, Mekh. Zhidk. Gaza, No. 5, 20–30 (1992).Google Scholar
  18. 18.
    Yu. K. Bivin, V. V. Viktorov, and L. P. Stepanov, “Study of the motion of a Rigid Body in a Clay Medium,” Izv. Akad. Nauk SSSR, Mekh. Tverd. Tela, No. 2, 159–165 (1978).Google Scholar

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© Pleiades Publishing, Ltd. 2016

Authors and Affiliations

  1. 1.Institute of Mechanics at the Lomonosov Moscow State UniversityMoscowRussia

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