Influence of viscous friction on stability balance of the body with a vibrating suspension

Applied Mathematics, Mechanics and Physics


Safonov A. I.

InfoSistem-35, 16, 3-Mytischinskaya str., bld. 37, Moscow, 129626, Russia



Influence of high-frequency vibrations on stability of stationary motions of mechanisms used in aviation and cosmic techniques is a very actual problem. Vibrations can change the nature of stability and cause new dynamic effects. Furthermore, friction should be taken into account when studying real systems.
We consider here a model problem of a motion of a rigid body, one point of which (a suspension point) accomplishes vertical harmonic high-frequency oscillations of small amplitude. A force of viscous friction is assumed to act on the body, with a torque proportional to its angular velocity. Stability of two relative equilibriums for which the center of mass and the suspension point lie on the same vertical is studied. The investigation is carried out in the framework of the approximate autonomous system of differential equations of motion obtained in the paper [11]. Conditions for stability of the vertical relative equilibriums in the case of absence of friction have been also obtained in [11].
Three special cases of the body mass geometry are considered here when the bodys center of mass lie on one of the principal axes of inertia or in the principal plane of inertia, or the body is dynamically symmetric.
A characteristic equation of linearized equations of a perturbed motion has a zero root. The other roots have negative real parts if friction is small and the vibration frequency exceeds the special value (for the upper equilibrium) and has any value (for the lower one). The nonlinear parts of the whole perturbed equations are proved to have a special structure. This is so-called special case of one zero root in terminology of [1]. Investigation of stability is carried out by using algorithm of this monograph. It is shown that the equilibriums under consideration are asymptotically stable in five variables and stable (not asymptotically) in the sixth (critical) variable.


stability of rigid body, vibration of the suspension point, the equilibrium position, the viscous friction


  1. Malkin I.G. Teoriya ustoichivosti dvizheniya (The theory ofstability ofmotion), Moscow, Nauka, 1966, 531p.
  2. Stephenson A. Ona New Type of Dinamical Stability, Memoirs and Proceedings of the Manchester Literary and Philosophical Society, 1908, vol.52, no2, pp. 1— 10.
  3. Kapitsa P.L. Uspekhi fizicheskikh nauk, 1951, vol.44, no1, pp. 7-20.
  4. Kapitsa P.L. Zhurnal eksperimentalnoi iteoreticheskoi fiziki, 1951, vol.21, no5, pp. 588-597.
  5. Bardin B.S., Markeev A.P. Prikladnaya matematika imekhanika, 1995, vol.59, no6, pp. 922-929.
  6. Markeev A.P. Prikladnaya matematika imekhanika, 1999, vol.63, no2, pp. 213-219.
  7. Kholostova O.V. Teoreticheskaya mekhanika. Sbornik statei, Moscow, MGU, 2003, vol.24, pp. 157-167.
  8. Kholostova O.V. Izvestiya Rossiiskoi Akademii Nauk. Mekhanika tverdogo tela, 2009, no2, pp. 25-40.
  9. Strizhak T.G. Metody issledovaniya dinamicheskikh sistem tipa «mayatnik» (Methods ofresearch ofdynamic systems such as«pendulum»), Alma-Ata, Nauka, 1981, 254p.
  10. Yudovich V.I. Uspekhi mekhaniki, 2006, vol.4, no3, pp. 26-158.
  11. Markeev A.P. Prikladnaya matematika imekhanika, 2011, vol.75, no.2, pp. 193-203.
  12. Markeev A.P. Doklady Akademii nauk, 2009, vol.427, no6, pp. 771-775.
  13. Kholostova O.V. Vestnik Rossiiskogo universiteta druzhby narodov. Matematika. Informatika. Fizika, 2011, no2, pp. 111-122.
  14. Markeev A.P. Izvestiya Rossiiskoi akademii nauk. Mekhanika tverdogo tela, 2012, no4, pp. 3-10. — informational site of MAI

Copyright © 1994-2024 by MAI