Research of spatial motion of an uncontrolled spacecraft with the triharmonic moment characteristics around its center of mass during its reentry into atmosphere


Spatial motion of an uncontrolled spacecraft around its center of mass is considered, the restoring aerodynamic moment of the spacecraft being described by an odd Fourier series in the angle of attack with the three first harmonics. Analytical formulas are deduced for the action integral along seperatrix, expressed, in terms of elliptic integrals of the first, second and third kind and of elementary functions. The moments of transition between various phase planes are determined. Formulas for determining the probability of capture into any region are deduced. The moments of additional equilibrium state appearing on the phase plane are determined. The analytical condition that determines the initial angular velocity ensuring the absence of additional equilibrium states ou the angle of attack in the given time interval is specified.

About the authors

Ye. V. Barinova

Samara State Aerospace University

Author for correspondence.

Post-graduate Student

Russian Federation

I. A. Timbay

Samara State Aerospace University


Doctor of Technical Science, Professor

Head of the Department of Higher Mathematics

Russian Federation


  1. Kuzmak, G. E. Dynamics of Uncontrolled Motion of Spacecraft at Their Reentry into the Atmosphere [Text] / G. E. Kuzmak. – Moscow: Nauka, 1970.
  2. Yaroshevskii, V. A. Motion of an Uncontrolled Body in the Atmosphere [Text] / V. A. Yaroshevskii. – Moscow: Mashinostroenie, 1978.
  3. Aslanov, V. S. Transient Modes of Spacecraft Angular Motion on the Upper Section of the Reentry Trajectory [Text] / V. S. Aslanov, I. A. Timbai // Kosm. Issled., 1997, vol. 35, no. 3, p. 279 – 286.
  4. Barinova, E. V. Research of Planar Motion of an Uncontrolled Spacecraft with the Treegarmonic Moment Characteristics around its Center of Mass at its Reentry into Atmosphere, [Text] / E. V. Barinova, I. A. Timbai // Vestnik SSAU, 2010, no. 1 (21), p. 9 – 19.
  5. Aslanov, V. S. Space Motions of a Reentry Spacecraft in the Atmosphere [Text] / V. S. Aslanov. – Moscow: FIZMATHLIT, 2004.
  6. Neishtadt, A. I., Variation of Adiabatic Invariant Crossing the Separatrix [Text] / A. I. Neishtadt // Fiz. Plazmy, 1986. vol.12. no. 8. p. 992 – 1001.
  7. Lifshits, I. M. Features of Charged Particle Motion in Changing and Inhomogeneous Electromagnetic Fields [Text] / I. M. Lifshits, A. A. Slutskin, V. M. Nabutovskii // Zh. Eksp. Teor. Fiz., 1961, vol. 41, no. 3, p. 939 – 948.



Abstract: 3052

PDF (Russian): 1592




  • There are currently no refbacks.

Copyright (c) 2015 VESTNIK of the Samara State Aerospace University

This website uses cookies

You consent to our cookies if you continue to use our website.

About Cookies