Nonlinear Dynamics

, Volume 62, Issue 3, pp 499–506 | Cite as

Stabilization of Kapitza oscillator by symmetric periodical forces

Original Paper


With the application of Kapitza method of averaging for an arbitrary periodic force, the oscillator is stabilized by minimizing its effective potential energy function. The aim is to lower the frequency and amplitude of fast oscillation as compared to external harmonic/periodic kicking pulses, which is achieved by introducing special symmetric periodical kicking pulses.


Kapitza oscillator Stable points 


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  1. 1.
    Sontag, E.D.: Mathematical Control Theory, Deterministic Finite Dimensional Systems, 2nd edn. Texts in Applied Mathematics, Vol. 6. Springer, New York (1998) MATHGoogle Scholar
  2. 2.
    Blekhman, I.I.: Vibrational Mechanics—Nonlinear Dynamic Effects, General Approach, Applications. World Scientific, Singapore (2000) CrossRefGoogle Scholar
  3. 3.
    Damgov, V.: “Quantized” oscillations and irregular behavior of a class of kick-excited self-adaptive dynamical system. Prog. Theor. Phys., Suppl. 139, 344–352 (2000) CrossRefGoogle Scholar
  4. 4.
    Astakhov, V., Shabunin, A., Anishchenko, V.: Synchronization of self-oscillations by parametric excitation. Int. J. Bifurc. Chaos 8, 1605–1612 (1998) MATHCrossRefGoogle Scholar
  5. 5.
    Blekhman, I.I.: Selected Topics in Vibrational Mechanics. World Scientific, Singapore (2004) MATHCrossRefGoogle Scholar
  6. 6.
    Brown, L.S.: Quantum motion in a Paul trap. Phys. Rev. Lett. 66, 527–529 (1991) MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Paul, W.: Electromagnetic traps for charged and neutral particles. Rev. Mod. Phys. 62, 531–540 (1990) CrossRefGoogle Scholar
  8. 8.
    Gilary, I., Moiseyev, N., Rahav, S., Fishman, S.: Trapping of particles by lasers: the quantum Kapitza pendulum. J. Phys. A, Math. Gen. 36, L409–L415 (2003) MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Bullo, F.: Averaging and vibrational control of mechanical systems. SIAM J. Control Optim. 41, 542–562 (2002) MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Nakamura, Y., Suzuki, T., Koimura, M.: Nonlinear behavior and control of a nonholonomic free-joint manipulator. IEEE Trans. Robot. Autom. 13, 853–862 (1997) CrossRefGoogle Scholar
  11. 11.
    Hslyst, J.A., Wojciechowski, W.: The effect of Kapitza pendulum and price equilibrium. Phys. A 324, 388–395 (2003) CrossRefMathSciNetGoogle Scholar
  12. 12.
    Stephenson, A.: On a new type of dynamic stability. Mem. Proc. Manch. Lit. Philos. Soc. 52, 1–10 (1908) Google Scholar
  13. 13.
    Stephenson, A.: On induced stability. Philos. Mag. 15, 233–236 (1908) Google Scholar
  14. 14.
    Stephenson, A.: On induced stability. Philos. Mag. 17, 765–766 (1909) Google Scholar
  15. 15.
    Kapitza, P.L.: Dynamic stability of a pendulum with an oscillating point of suspension. J. Exp. Theor. Phys. 21, 588–597 (1951) Google Scholar
  16. 16.
    Broer, H.W., Hoveijn, I., van Noort, M.: The inverted pendulum: a singularity theory approach. J. Differ. Equ. 157, 120–149 (1999) MATHCrossRefGoogle Scholar
  17. 17.
    Levi, M.: Stability of the inverted pendulum—a topological explanation. SIAM Rev. 30, 639–644 (1988) MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Levi, M., Weckesser, W.: Stabilization of the inverted pendulum by high frequency vibrations. SIAM J. Math. Anal. 35, 844–867 (2003) CrossRefMathSciNetGoogle Scholar
  19. 19.
    Bardin, B.S., Markeyev, A.P.: The stability of the equilibrium of a pendulum for vertical oscillations of the point of suspension. J. Appl. Math. Mech. 59, 879–886 (1995) MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Landau, L.D., Lifshitz, E.M.: Mecanics, 3rd edn. Pergamon, Oxford (2005) (Butterworth 15) Google Scholar
  21. 21.
    Ahmad, B., Borisenok, S.: Control of effective potential minima for Kapitza oscillator by periodical kicking pulses. Phys. Lett. A 373, 701–707 (2009) CrossRefGoogle Scholar
  22. 22.
    Hsu, H.P.: Analysis of periodic waveforms. Appl. Fourier Anal. (1930) Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.Mirpur University of Science and TechnologyMirpurPakistan

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