Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Sontag, E.D.: Mathematical Control Theory, Deterministic Finite Dimensional Systems, 2nd edn. Texts in Applied Mathematics, Vol. 6. Springer, New York (1998)
Blekhman, I.I.: Vibrational Mechanics—Nonlinear Dynamic Effects, General Approach, Applications. World Scientific, Singapore (2000)
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)
Astakhov, V., Shabunin, A., Anishchenko, V.: Synchronization of self-oscillations by parametric excitation. Int. J. Bifurc. Chaos 8, 1605–1612 (1998)
Blekhman, I.I.: Selected Topics in Vibrational Mechanics. World Scientific, Singapore (2004)
Brown, L.S.: Quantum motion in a Paul trap. Phys. Rev. Lett. 66, 527–529 (1991)
Paul, W.: Electromagnetic traps for charged and neutral particles. Rev. Mod. Phys. 62, 531–540 (1990)
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)
Bullo, F.: Averaging and vibrational control of mechanical systems. SIAM J. Control Optim. 41, 542–562 (2002)
Nakamura, Y., Suzuki, T., Koimura, M.: Nonlinear behavior and control of a nonholonomic free-joint manipulator. IEEE Trans. Robot. Autom. 13, 853–862 (1997)
Hslyst, J.A., Wojciechowski, W.: The effect of Kapitza pendulum and price equilibrium. Phys. A 324, 388–395 (2003)
Stephenson, A.: On a new type of dynamic stability. Mem. Proc. Manch. Lit. Philos. Soc. 52, 1–10 (1908)
Stephenson, A.: On induced stability. Philos. Mag. 15, 233–236 (1908)
Stephenson, A.: On induced stability. Philos. Mag. 17, 765–766 (1909)
Kapitza, P.L.: Dynamic stability of a pendulum with an oscillating point of suspension. J. Exp. Theor. Phys. 21, 588–597 (1951)
Broer, H.W., Hoveijn, I., van Noort, M.: The inverted pendulum: a singularity theory approach. J. Differ. Equ. 157, 120–149 (1999)
Levi, M.: Stability of the inverted pendulum—a topological explanation. SIAM Rev. 30, 639–644 (1988)
Levi, M., Weckesser, W.: Stabilization of the inverted pendulum by high frequency vibrations. SIAM J. Math. Anal. 35, 844–867 (2003)
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)
Landau, L.D., Lifshitz, E.M.: Mecanics, 3rd edn. Pergamon, Oxford (2005) (Butterworth 15)
Ahmad, B., Borisenok, S.: Control of effective potential minima for Kapitza oscillator by periodical kicking pulses. Phys. Lett. A 373, 701–707 (2009)
Hsu, H.P.: Analysis of periodic waveforms. Appl. Fourier Anal. (1930)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ahmad, B. Stabilization of Kapitza oscillator by symmetric periodical forces. Nonlinear Dyn 62, 499–506 (2010). https://doi.org/10.1007/s11071-010-9736-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-010-9736-0