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• W2100020546 abstract "Previous article Next article Stabilization of the Inverted Linearized Pendulum by High Frequency VibrationsMark Levi and Warren WeckesserMark Levi and Warren Weckesserhttps://doi.org/10.1137/1037044PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstractIn this note we give a simple geometrical picture that explains why an inverted pendulum is stabilized by high frequency vibrations.[1] A. Stephenson, On a new type of dynamical stability, Manchester Memoirs, 52 (1908), 1–10 39.0768.02 Google Scholar[2] J. A. Blackburn, , H. J. T. Smith and , N. Gronbech-Jensen, Stability and Hopf bifurcations in an inverted pendulum, Amer. J. Phys., 60 (1992), 903–908 93h:70014 CrossrefISIGoogle Scholar[3] B. van der Pol and , M. J. O. Strutt, On the stability of the solutions of Mathieu's equation, The London, Edinburgh and Dublin Phil. Mag. 7th series, 5 (1928), Google Scholar[4] P. L. Kapitza, D. ter Haar, Dynamical stability of a pendulum when its point of suspension vibrates, and Pendulum with a vibrating suspensionCollected Papers of P. L. Kapitza, Vol. 2, Pergamon Press Ltd., London, 1965, 714–737 Google Scholar[5] Mark Levi, Stability of the inverted pendulum—a topological explanation, SIAM Rev., 30 (1988), 639–644 10.1137/1030140 90c:58093 0667.34050 LinkISIGoogle Scholar[6] J. J. Stoker, Nonlinear Vibrations in Mechanical and Electrical Systems, Interscience Publishers, Inc., New York, N.Y., 1950xix+273 11,666a 0035.39603 Google Scholar[7] I. Z. Shtokalo, Linear differential equations with variable coefficients. (Criteria of stability and unstability of their solutions.), Translated from Russian, Hindustan Publishing Corpn., Delhi, 1961, 1961xii+100 27:5979 Google Scholar[8] V. I. Arnold, Ordinary differential equations, The M.I.T. Press, Cambridge, Mass.-London, 1973ix+280 50:13679 0296.34001 Google Scholar[9] V. A. Yakubovich and , V. M. Starzhinskii, Linear differential equations with periodic coefficients, John Wiley and Sons, New York, 1975 Google Scholar[10] D. J. Acheson, A pendulum theorem, Proc. Roy. Soc. Lond. A, 443 (1993), 239–245 0784.70019 CrossrefISIGoogle Scholar[11] D. J. Acheson and , T. Mullin, Upside-down pendulums, Nature, 366 (1993), 215–216 10.1038/366215b0 CrossrefISIGoogle ScholarKeywordsstabilityhyperbolic and elliptic rotationsapplications of linear algebra Previous article Next article FiguresRelatedReferencesCited byDetails A note on parametric resonance induced by a singular parameter modulationInternational Journal of Non-Linear Mechanics, Vol. 139 Cross Ref The pendulum under vibrations revisited20 January 2021 | Nonlinearity, Vol. 34, No. 1 Cross Ref Dynamic Response of an Inverted Pendulum System in Water under Parametric Excitations for Energy Harvesting: A Conceptual Approach7 October 2020 | Energies, Vol. 13, No. 19 Cross Ref On the existence and stabilization of an upper unstable limit cycle of the damped forced pendulumJournal of Computational and Applied Mathematics, Vol. 371 Cross Ref Reverse rotation of soft ferromagnetic ball in rotating magnetic fieldJournal of Magnetism and Magnetic Materials, Vol. 476 Cross Ref On the existence of periodic motions of the excited inverted pendulum by elementary methods25 May 2018 | Acta Mathematica Hungarica, Vol. 155, No. 2 Cross Ref Introduction Cross Ref On stabilizability of the upper equilibrium of the asymmetrically excited inverted pendulum1 January 2018 | Electronic Journal of Qualitative Theory of Differential Equations, No. 45 Cross Ref Schrödinger’s equation and “bike tracks” — A connectionJournal of Geometry and Physics, Vol. 115 Cross Ref Dynamics and vibrational control of an underwater inverted pendulum Cross Ref An extension of the Levi-Weckesser method to the stabilization of the inverted pendulum under gravity13 December 2013 | Meccanica, Vol. 49, No. 5 Cross Ref Scattering and Localization Properties of Highly Oscillatory Potentials23 May 2013 | Communications on Pure and Applied Mathematics, Vol. 67, No. 1 Cross Ref Vibrational control of Mathieu's equation Cross Ref Resonances in the heavy symmetrical top with vibrating pivot19 February 2013 | The European Physical Journal Plus, Vol. 128, No. 2 Cross Ref Stabilization of the Controlled Inverted Pendulum by a Control with Delay Cross Ref On the periodic symmetric electrostatic forcing of a microcantilever Cross Ref Lateral stability of a periodically forced electrostatic comb drive Cross Ref Difference Equations as Difference Analogues of Differential Equations Cross Ref Stabilization by vertical vibrationsMathematical Methods in the Applied Sciences, Vol. 32, No. 9 Cross Ref KAM dynamics and stabilization of a particle sliding over a periodically driven curveApplied Mathematics Letters, Vol. 20, No. 6 Cross Ref Geodesics on vibrating surfaces and curvature of the normal family3 October 2005 | Nonlinearity, Vol. 18, No. 6 Cross Ref Some disconjugacy criteria for differential equations with oscillatory coefficientsMathematische Nachrichten, Vol. 278, No. 12-13 Cross Ref GEOMETRY OF VIBRATIONAL STABILIZATION AND SOME APPLICATIONS20 November 2011 | International Journal of Bifurcation and Chaos, Vol. 15, No. 09 Cross Ref The Parametrically Forced Pendulum: A Case Study in 1 1/2 Degree of FreedomJournal of Dynamics and Differential Equations, Vol. 16, No. 4 Cross Ref Analysis of a Rotating Pendulum With a Mass Free to Move Radially1 September 1998 | Journal of Applied Mechanics, Vol. 65, No. 3 Cross Ref Local and global instabilities of spatially developing flows: cautionary examples8 July 1997 | Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, Vol. 453, No. 1962 Cross Ref Moving averages for periodic delay differential and difference equations Cross Ref Equilibria and stability of an n-pendulum forced by rapid oscillations Cross Ref Bifurcations and stability of the vertically forced n-pendulum as n→∞ Cross Ref Volume 37, Issue 2| 1995SIAM Review History Submitted:31 January 1994Accepted:06 June 1994Published online:17 February 2012 InformationCopyright © 1995 © Society for Industrial and Applied MathematicsKeywordsstabilityhyperbolic and elliptic rotationsapplications of linear algebraMSC codes34A2634D1070J25PDF Download Article & Publication DataArticle DOI:10.1137/1037044Article page range:pp. 219-223ISSN (print):0036-1445ISSN (online):1095-7200Publisher:Society for Industrial and Applied Mathematics" @default.
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