Attraction Basins in the Generalized Kapitsa Problem

Doklady Physics volume 64, pages335–339(2019)Cite this article

Abstract

The stability of the vertical position of an inverted pendulum under the action of support vibration and the attraction basin of this position are considered. In addition to the classical Kapitsa problem for the harmonic vibration of a support, the polyharmonic and random vibration of the support are investigated. The condition of stability of the vertical position is determined, and the attraction basin of this stable position is investigated.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

US$ 39.95

Tax calculation will be finalised during checkout.

Fig. 1.
Fig. 2.

REFERENCES

  1. 1

    A. Stephenson, Phil. Mag. 15, 233 (1908).

    Article  Google Scholar 

  2. 2

    P. L. Kapitsa, Usp. Fiz. Nauk 44 (1), 7 (1951).

    ADS  Article  Google Scholar 

  3. 3

    I. I. Blekhman, Vibration Mechanics (Nauka, Moscow, 1994) [in Russian].

  4. 4

    I. I. Blekhman, Theory of Vibration Processes and Devices (ID Ruda i metally, St. Petersburg, 2013) [in Russian].

  5. 5

    N. F. Morozov, A. K. Belyaev, P. E. Tovstik, and T. P. Tovstik, Dokl. Phys. 63 (9), 380 (2018).

    ADS  Article  Google Scholar 

  6. 6

    A. K. Belyaev, N. F. Morozov, P. E. Tovstik, and T. P. Tovstik, Vestnik SPb Univ. Mathematics. Mechanics. Astronomy 51 (3), 296 (2018).

    Google Scholar 

  7. 7

    G. A. Leonov and M. M. Shumafov, Methods of Stabilization of Linear Controlled Systems (Izd. SPb. un-ta, St. Petersburg, 2005) [in Russian].

  8. 8

    N. N. Bogolyubov and Yu. A. Mitropol’skii, Asymptotic Methods in the Theory of Nonlinear Vibrations (Nauka, Moscow, 1969) [in Russian].

  9. 9

    V. S. Pugachev, Theory of Random Functions (Fizmatlit, Moscow, 1960) [in Russian].

  10. 10

    B. V. Gnedenko, Course of Probability Theory (Fizmatlit, Moscow, 1961) [in Russian].

Download references

Author information

Affiliations

  1. St. Petersburg State University, 199034, St. Petersburg, Russia

    Academician N. F. Morozov, P. E. Tovstik & T. M. Tovstik

  2. Institute for Problems in Mechanical Engineering, Russian Academy of Sciences, 199178, St. Petersburg, Russia

    A. K. Belyaev & T. P. Tovstik

Authors
  1. Academician N. F. Morozov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. K. Belyaev
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. P. E. Tovstik
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. T. M. Tovstik
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. T. P. Tovstik
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Academician N. F. Morozov or T. P. Tovstik.

Additional information

Translated by V. Bukhanov

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Morozov, N.F., Belyaev, A.K., Tovstik, P.E. et al. Attraction Basins in the Generalized Kapitsa Problem. Dokl. Phys. 64, 335–339 (2019). https://doi.org/10.1134/S1028335819080056

Download citation