On the stability of the upside–down pendulum with damping

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Restricted accessMoreSectionsView PDF ToolsAdd to favoritesDownload CitationsTrack Citations ShareShare onFacebookTwitterLinked InRedditEmail Cite this article Bartuccelli M. V., Gentile G. and Georgiou K. V. 2002On the stability of the upside–down pendulum with dampingProc. R. Soc. Lond. A.458255–269http://doi.org/10.1098/rspa.2001.0859SectionRestricted accessOn the stability of the upside–down pendulum with damping M. V. Bartuccelli M. V. Bartuccelli Department of Mathematics and Statistics, University of Surrey, Guildford GU2 7XH, UK Google Scholar Find this author on PubMed Search for more papers by this author , G. Gentile G. Gentile Dipartimento di Mathematica, Università di Roma Tre, Rome I–00146, Italy Google Scholar Find this author on PubMed Search for more papers by this author and K. V. Georgiou K. V. Georgiou Department of Mathematics and Statistics, University of Surrey, Guildford GU2 7XH, UK Google Scholar Find this author on PubMed Search for more papers by this author M. V. Bartuccelli M. V. Bartuccelli Department of Mathematics and Statistics, University of Surrey, Guildford GU2 7XH, UK Google Scholar Find this author on PubMed Search for more papers by this author , G. Gentile G. Gentile Dipartimento di Mathematica, Università di Roma Tre, Rome I–00146, Italy Google Scholar Find this author on PubMed Search for more papers by this author and K. V. Georgiou K. V. Georgiou Department of Mathematics and Statistics, University of Surrey, Guildford GU2 7XH, UK Google Scholar Find this author on PubMed Search for more papers by this author Published:19 December 2001https://doi.org/10.1098/rspa.2001.0859AbstractA rigorous analysis is presented in order to show that, in the presence of friction, the upward equilibrium position of the vertically driven pendulum, with a small nonvanishing damping term, becomes asymptotically stable when the period of the forcing is below an appropriate threshold value. As a by–product we obtain an analytic expression of the solution for initial data close enough to the equilibrium position. Next Article VIEW FULL TEXT DOWNLOAD PDF FiguresRelatedReferencesDetailsCited by Bhadra N and Banerjee S (2020) Dynamics of a system of coupled inverted pendula with vertical forcing, Chaos, Solitons & Fractals, 10.1016/j.chaos.2020.110358, 141, (110358), Online publication date: 1-Dec-2020. Wright J, Bartuccelli M and Gentile G (2017) Comparisons between the pendulum with varying length and the pendulum with oscillating support, Journal of Mathematical Analysis and Applications, 10.1016/j.jmaa.2016.12.076, 449:2, (1684-1707), Online publication date: 1-May-2017. Bartuccelli M, Gentile G and Wright J (2016) Stable dynamics in forced systems with sufficiently high/low forcing frequency, Chaos: An Interdisciplinary Journal of Nonlinear Science, 10.1063/1.4960614, 26:8, (083108), Online publication date: 1-Aug-2016. Singh R, Tayal V and Singh H (2016) A review on Cubli and non linear control strategy 2016 IEEE 1st International Conference on Power Electronics, Intelligent Control and Energy Systems (ICPEICES), 10.1109/ICPEICES.2016.7853425, 978-1-4673-8587-9, (1-5) Ghose Choudhury A and Guha P (2014) Damped equations of Mathieu type, Applied Mathematics and Computation, 10.1016/j.amc.2013.11.106, 229, (85-93), Online publication date: 1-Feb-2014. Wright J, Bartuccelli M and Gentile G (2014) The effects of time-dependent dissipation on the basins of attraction for the pendulum with oscillating support, Nonlinear Dynamics, 10.1007/s11071-014-1386-1, 77:4, (1377-1409), Online publication date: 1-Sep-2014. Leine R (2012) Non-smooth stability analysis of the parametrically excited impact oscillator, International Journal of Non-Linear Mechanics, 10.1016/j.ijnonlinmec.2012.06.010, 47:9, (1020-1032), Online publication date: 1-Nov-2012. Gajamohan M, Merz M, Thommen I and D'Andrea R (2012) The Cubli: A cube that can jump up and balance 2012 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS 2012), 10.1109/IROS.2012.6385896, 978-1-4673-1736-8, (3722-3727) Bartuccelli M, Deane J and Gentile G (2008) Frequency locking in an injection-locked frequency divider equation, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 465:2101, (283-306), Online publication date: 8-Jan-2009. Champneys A and Fraser W (2004) Resonance Tongue Interaction in the Parametrically Excited Column, SIAM Journal on Applied Mathematics, 10.1137/S0036139902418274, 65:1, (267-298), Online publication date: 1-Jan-2004. Bartuccelli M, Gentile G and Georgiou K (2001) On the dynamics of a vertically driven damped planar pendulum, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 10.1098/rspa.2001.0841, 457:2016, (3007-3022), Online publication date: 8-Dec-2001. This Issue08 February 2002Volume 458Issue 2018 Article InformationDOI:https://doi.org/10.1098/rspa.2001.0859Published by:Royal SocietyPrint ISSN:1364-5021Online ISSN:1471-2946History: Published online19/12/2001Published in print08/02/2002 License: Citations and impact KeywordsMathieu's equationbasins of attractionperturbation theorystability