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The Method of Averaging for the Kapitza – Whitney Pendulum

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Abstract

A generalization of the classical Kapitza pendulum is considered: an inverted planar mathematical pendulum with a vertically vibrating pivot point in a time-periodic horizontal force field. We study the existence of forced oscillations in the system. It is shown that there always exists a periodic solution along which the rod of the pendulum never becomes horizontal, i. e., the pendulum never falls, provided the period of vibration and the period of horizontal force are commensurable. We also present a sufficient condition for the existence of at least two different periodic solutions without falling. We show numerically that there exist stable periodic solutions without falling.

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Funding

This work has been supported by the Grant of the President of the Russian Federation (Project MK-1826.2020.1).

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Authors and Affiliations

  1. Steklov Mathematical Institute, Russian Academy of Sciences, ul. Gubkina 8, 119991, Moscow, Russia

    Ivan Yu. Polekhin

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  1. Ivan Yu. Polekhin
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Correspondence to Ivan Yu. Polekhin.

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MSC2010

34C29, 70K40

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Polekhin, I.Y. The Method of Averaging for the Kapitza – Whitney Pendulum. Regul. Chaot. Dyn. 25, 401–410 (2020). https://doi.org/10.1134/S1560354720040073

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