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An extension of the Levi-Weckesser method to the stabilization of the inverted pendulum under gravity

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Abstract

Sufficient conditions are given for the stability of the upper equilibrium of the mathematical pendulum (inverted pendulum) when the suspension point is vibrating vertically with high frequency. The equation of the motion is of the form

$$ \ddot{\theta}-\frac{1}{l}\bigl(g+a(t)\bigr) \theta=0, $$

where l,g are constants and a is a periodic step function. M. Levi and W. Weckesser gave a simple geometrical explanation for the stability effect provided that the frequency is so high that the gravity g can be neglected. They also obtained a lower estimate for the stabilizing frequency. This method is improved and extended to the arbitrary inverted pendulum not assuming even symmetricity between the upward and downward phases in the vibration of the suspension point.

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Acknowledgements

The authors would like to thank the referees for their valuable comments and suggestions.

The authors were supported by the European Union and co-funded by the European Social Fund under the project “Telemedicine-focused research activities on the field of Matematics, Informatics and Medical sciences” of project number “TAMOP-4.2.2.A-11/1/KONV-2012-0073”. The second author was supported also by the Hungarian National Foundation for Scientific Research (OTKA K75517) and Analysis and Stochastics Research Group of the Hungarian Academy of Sciences.

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  1. Bolyai Institute, University of Szeged, Aradi vértanúk tere 1., 6720, Szeged, Hungary

    L. Csizmadia & L. Hatvani

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  1. L. Csizmadia
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  2. L. Hatvani
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Correspondence to L. Hatvani.

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Csizmadia, L., Hatvani, L. An extension of the Levi-Weckesser method to the stabilization of the inverted pendulum under gravity. Meccanica 49, 1091–1100 (2014). https://doi.org/10.1007/s11012-013-9855-z

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  • DOI: https://doi.org/10.1007/s11012-013-9855-z

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