Stability Regions for the Damped Mathieu Equation

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Previous article Next article Stability Regions for the Damped Mathieu EquationJames H. Taylor and Kumpati S. NarendraJames H. Taylor and Kumpati S. Narendrahttps://doi.org/10.1137/0117033PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] W. J. Cunningham, Introduction to nonlinear analysis, McGraw-Hill Electrical and Electronic Engineering Series, McGraw-Hill Book Co., Inc., New York, 1958ix+349 MR0093611 0088.06702 Google Scholar[2] N. W. McLachlan, Theory and Application of Mathieu Functions, Clarendon Press, Oxford, 1951 Google Scholar[3] K. S. Narendra and , R. M. Goldwyn, A geometrical criterion for the stability of certain nonlinear nonautonomous systems, IEEE Trans. Circuit Theory, CT-11 (1964), 406–407 CrossrefGoogle Scholar[4] I. W. Sandberg, A frequency-domain condition for the stability of feedback systems containing a single time-varying nonlinear element, Bell System Tech. 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Remote Control, 22 (1961), 857–875 MR0133563 0107.29601 Google Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails Nano- and microparticle Nonlinear Damping Identification in quadrupole trapInternational Journal of Non-Linear Mechanics, Vol. 147 | 1 Dec 2022 Cross Ref Design of a metastructure for vibration isolation with quasi-zero-stiffness characteristics using bistable curved beamNonlinear Dynamics, Vol. 108, No. 3 | 9 March 2022 Cross Ref Numerical simulation of Faraday waves in a rectangular tank and damping mechanism of internal bafflesJournal of Fluids and Structures, Vol. 109 | 1 Feb 2022 Cross Ref Stability of rotatory solitary states in Kuramoto networks with inertiaPhysical Review E, Vol. 105, No. 2 | 18 February 2022 Cross Ref A Floquet-Based Analysis of Parametric Excitation Through the Damping CoefficientJournal of Vibration and Acoustics, Vol. 143, No. 4 | 10 November 2020 Cross Ref Experimental and numerical study of nonlinear modal characteristics of Faraday wavesOcean Engineering, Vol. 221 | 1 Feb 2021 Cross Ref Approximate Floquet Analysis of Parametrically Excited Multi-Degree-of-Freedom Systems With Application to Wind TurbinesJournal of Vibration and Acoustics, Vol. 141, No. 1 | 24 July 2018 Cross Ref Stochastic stability of the non-homogeneous damped Mathieu equationNonlinear Dynamics, Vol. 86, No. 3 | 12 August 2016 Cross Ref Floquet-Based Analysis of General Responses of the Mathieu EquationJournal of Vibration and Acoustics, Vol. 138, No. 4 | 25 May 2016 Cross Ref Periodic Solutions, Eigenvalue Curves, and Degeneracy of the Fractional Mathieu EquationJournal of Physics: Conference Series, Vol. 698 | 7 April 2016 Cross Ref Continuum modeling of thermal transport in superlattices and layered materials for new energy matierlasActa Physica Sinica, Vol. 65, No. 11 | 1 Jan 2016 Cross Ref Parametric Resonance Cancellation Via Reshaping Stability Regions: Numerical and Experimental ResultsIEEE Transactions on Control Systems Technology, Vol. 22, No. 2 | 1 Mar 2014 Cross Ref Fractional Ince equation with a Riemann–Liouville fractional derivativeApplied Mathematics and Computation, Vol. 219, No. 22 | 1 Jul 2013 Cross Ref Vibrational control of Mathieu's equation2013 IEEE/ASME International Conference on Advanced Intelligent Mechatronics | 1 Jul 2013 Cross Ref Transition curves in a parametrically excited pendulum with a force of elliptic typeProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 468, No. 2148 | 19 September 2012 Cross Ref Dynamic stability of parametrically-excited linear resonant beams under periodic axial forceChinese Physics B, Vol. 21, No. 11 | 8 November 2012 Cross Ref Statistical Properties of Fluctuation of Charged Fine Particles in an AC TrapProcedia IUTAM, Vol. 5 | 1 Jan 2012 Cross Ref Parametric instability induced scalar gravitational waves from a model pulsating neutron starPhysics Letters B, Vol. 705, No. 1-2 | 1 Nov 2011 Cross Ref Chaos-Induced Diffusion in a Nonlinear Dissipative Mathieu Equation for a Charged Fine Particle in an AC TrapJournal of the Physical Society of Japan, Vol. 80, No. 4 | 15 Apr 2011 Cross Ref The Lyapunov Stability for the Linear and Nonlinear Damped Oscillator with Time-Periodic ParametersAbstract and Applied Analysis, Vol. 2010 | 1 Jan 2010 Cross Ref Particle clustering in periodically forced straining flowsJournal of Fluid Mechanics, Vol. 624 | 10 April 2009 Cross Ref Stability chart of parametric vibrating systems using energy-rate methodInternational Journal of Non-Linear Mechanics, Vol. 39, No. 8 | 1 Oct 2004 Cross Ref Hopf Bifurcation in PD Controlled Pendulum or ManipulatorJournal of Dynamic Systems, Measurement, and Control, Vol. 124, No. 2 | 10 May 2002 Cross Ref Designing against parametric instability in following seasOcean Engineering, Vol. 27, No. 6 | 1 Jun 2000 Cross Ref Stability of dynamical systems - An overviewJournal of Guidance, Control, and Dynamics, Vol. 13, No. 3 | 1 May 1990 Cross Ref Some recent results on the stability of linear time varying systemsSadhana, Vol. 13, No. 3 | 1 Nov 1988 Cross Ref On the stability of the damped mathieu equationMechanics Research Communications, Vol. 15, No. 6 | 1 Nov 1988 Cross Ref Stability of Time Varying Linear SystemsSpace Dynamics and Celestial Mechanics | 1 Jan 1986 Cross Ref Historical PerspectiveAnalysis of Periodically Time-Varying Systems | 1 Jan 1983 Cross Ref The Mathieu EquationAnalysis of Periodically Time-Varying Systems | 1 Jan 1983 Cross Ref Floquet exponent for instability intervals of Hill's equationCommunications on Pure and Applied Mathematics, Vol. 32, No. 1 | 1 Jan 1979 Cross Ref Stability Diagram Approximation for the Lossy Mathieu EquationJ. A. RichardsSIAM Journal on Applied Mathematics, Vol. 30, No. 2 | 28 July 2006AbstractPDF (583 KB)A Technique for Determining Stability Regions for the Damped Mathieu EquationH. Gunderson, H. Rigas, and F. S. VanVleckSIAM Journal on Applied Mathematics, Vol. 26, No. 2 | 12 July 2006AbstractPDF (420 KB)The Corduneanu-Popov Approach to the Stability of Nonlinear Time-Varying SystemsJames H. Taylor and Kumpati S. NarendraSIAM Journal on Applied Mathematics, Vol. 18, No. 2 | 1 August 2006AbstractPDF (1025 KB) Volume 17, Issue 2| 1969SIAM Journal on Applied Mathematics223-493 History Submitted:16 May 1968Published online:12 July 2006 InformationCopyright © 1969 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0117033Article page range:pp. 343-352ISSN (print):0036-1399ISSN (online):1095-712XPublisher:Society for Industrial and Applied Mathematics