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Classical Kapitsa’s problem of stability of an inverted pendulum and some generalizations

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Abstract

The classical Kapitsa’s problem of stability of an inverted pendulum subjected to vertical vibration of the support and some generalizations are investigated. The asymptotic method of two-scale expansions allows one to determine the level of vibrations that stabilizes the vertical position of the rod. The cases of inextensible and extensible rod are studied, and a benchmark of results is carried out. An attraction basin of the stable vertical position of the rod is found. Both harmonic and random stationary vibrations of the support are considered, too. Stability and attraction basin of the vertical position for a flexible rod are studied in detail. A single-mode approximation is used for the approximate solution to the nonlinear problem of stability of a flexible rod.

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Acknowledgements

The work is supported by Russian Foundation of Basic Research, Grants 19-01-00280-a, 20-51-S52001 MHT-a. The authors thank Prof. I.I. Blekhman who paid attention to the studied problem, and Prof. D.A. Indeitsev who suggested to construct an attraction basin for a flexible rod.

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Authors and Affiliations

  1. Institute for Problems in Mechanical Engineering of RAS, St. Petersburg, Russian Federation

    A. K. Belyaev & T. P. Tovstik

  2. St. Petersburg State University, St. Petersburg, Russian Federation

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

Authors
  1. A. K. Belyaev
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  2. N. F. Morozov
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  3. P. E. Tovstik
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  4. T. M. Tovstik
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  5. T. P. Tovstik
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Correspondence to P. E. Tovstik.

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Belyaev, A.K., Morozov, N.F., Tovstik, P.E. et al. Classical Kapitsa’s problem of stability of an inverted pendulum and some generalizations. Acta Mech 232, 1743–1759 (2021). https://doi.org/10.1007/s00707-020-02907-0

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  • DOI: https://doi.org/10.1007/s00707-020-02907-0

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