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.
Keywords:
averaging, Kapitza’s pendulum, Whitney’s pendulum, forced oscillations, averaging on an infinite interval.
\Bibitem{Pol20}
\by Ivan Yu. Polekhin
\paper The Method of Averaging for the Kapitza – Whitney Pendulum
\jour Regul. Chaotic Dyn.
\yr 2020
\vol 25
\issue 4
\pages 401--410
\mathnet{http://mi.mathnet.ru/rcd1073}
\crossref{https://doi.org/10.1134/S1560354720040073}
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https://www.mathnet.ru/eng/rcd1073
https://www.mathnet.ru/eng/rcd/v25/i4/p401
This publication is cited in the following 6 articles:
ALYOUSEF HAIFA A., SALAS ALVARO H, ALOTAIBI B.M., EL-TANTAWY S. A., “On the Analytical Approximations to Kapitza'S Pendulum Oscillator Using Novel Techniques”, Rom. J. Phys., 70:1-2 (2025), 103
D. D. Kulminskiy, M. V. Malyshev, “Experimental Study of the Accuracy of a Pendulum Clock with a Vibrating Pivot Point”, Rus. J. Nonlin. Dyn., 20:4 (2024), 553–563
Ivan Yu. Polekhin, “A Topological–Analytical Method for Proving Averaging Theorems on an Infinite Time Interval in a Degenerate Case”, Proc. Steklov Inst. Math., 322 (2023), 188–197
Ivan Polekhin, “Asymptotically stable non-falling solutions of the Kapitza–Whitney pendulum”, Meccanica, 58 (2023), 1205–1215
Ivan Yu. Polekhin, “The Spherical Kapitza – Whitney Pendulum”, Regul. Chaotic Dyn., 27:1 (2022), 65–76
Ivan Polekhin, 2020 International Conference Nonlinearity, Information and Robotics (NIR), 2020, 1
Citation in format AMSBIB
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