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Stability of a Vertical Rod on a Vibrating Support

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Abstract

This work contains a generalization of Kapitsa’s classical problem. The stability of the vertical position of a flexible rod with a lower support point under gravity and vibrations is considered. It has been shown that an unstable position can become stable in the presence of vertical harmonic vibrations of the base. Both rigid and hinge fixing of the lower rod end are considered. In the linear approximation, the problem is reduced to transverse oscillations of the rod under the action of periodic axial compression. The solution is obtained in two formulations—taking into account the propagation of longitudinal waves in the rod and without regard for it. It turns out that longitudinal waves significantly reduce the base vibration level necessary for the stability.

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ACKNOWLEDGMENTS

The authors are grateful to I.I. Blekhman, who drew attention to the problems under consideration.

This work was supported by the Russian Foundation for Basic Research, project nos. 16.51.52025 MNT-a and 16.01.00580-a.

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Authors and Affiliations

  1. St. Petersburg State University, Staryi Peterhof, 198504, St. Petersburg, Russia

    N. F. Morozov & P. E. Tovstik

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

    A. K. Belyaev & T. P. Tovstik

Authors
  1. N. F. Morozov
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  2. A. K. Belyaev
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  3. P. E. Tovstik
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  4. T. P. Tovstik
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Corresponding author

Correspondence to P. E. Tovstik.

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Translated by A. Nikol’skii

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Morozov, N.F., Belyaev, A.K., Tovstik, P.E. et al. Stability of a Vertical Rod on a Vibrating Support. Dokl. Phys. 63, 380–384 (2018). https://doi.org/10.1134/S1028335818090069

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  • DOI: https://doi.org/10.1134/S1028335818090069

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