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Normal Modes and Localization in Nonlinear Systems

  • Book
  • © 2001

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Editors:
  1. Alexander F. Vakakis

    1. University or Illinois at Urbana/Champaign, Urbana, USA

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Table of contents (15 chapters)

  1. Front Matter

    Pages I-1
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  2. Invariant Manifolds, Nonclassical Normal Modes, and Proper Orthogonal Modes in the Dynamics of the Flexible Spherical Pendulum

    • Ioannis T. Georgiou, Ira B. Schwartz
    Pages 3-31

  3. Normal Vibrations in Near-Conservative Self-Excited and Viscoelastic Nonlinear Systems

    • Yu. V. Mikhlin, B. I. Morgunov
    Pages 33-48

  4. Nonlinear Normal Modes in a System with Nonholonomic Constraints

    • R. H. Rand, D. V. Ramani
    Pages 49-64

  5. Nonlinear Normal Modes of a Parametrically Excited Cantilever Beam

    • Hiroshi Yabuno, Ali H. Nayfeh
    Pages 65-77

  6. Normal Modes and Boundary Layers for a Slender Tensioned Beam on a Nonlinear Foundation

    • F. Pellicano, A. F. Vakakis
    Pages 79-93

  7. The Description of Localized Normal Modes in a Chain of Nonlinear Coupled Oscillators Using Complex Variables

    • L. I. Manevitch
    Pages 95-109

  8. Spatially Localized Models of Extended Systems

    • Ralf W. Wittenberg, Philip Holmes
    Pages 111-132

  9. Mode Localization in Dynamics and Buckling of Linear Imperfect Continuous Structures

    • Angelo Luongo
    Pages 133-156

  10. Dynamics of Relative Phases: Generalised Multibreathers

    • Taehoon Ahn, Robert S. Mackay, Jacques-A. Sepulchre
    Pages 157-182

  11. Nonlinear Modal Analysis of Structural Systems Using Multi-Mode Invariant Manifolds

    • Eric Pesheck, Nicolas Boivin, Christophe Pierre, Steven W. Shaw
    Pages 183-205

  12. Localization in Nonlinear Mistuned Systems with Cyclic Symmetry

    • Walter Sextro, Karl Popp, Tomasz Krzyzynski
    Pages 207-220

  13. Mode Localization Induced by a Nonlinear Control Loop

    • Robert T. M’Closkey, Alex Vakakis, Roman Gutierrez
    Pages 221-236

  14. Transition of Energy to a Nonlinear Localized Mode in a Highly Asymmetric System of Two Oscillators

    • Oleg V. Gendelman
    Pages 237-253

  15. Application of Nonlinear Normal Mode Analysis to the Nonlinear and Coupled Dynamics of a Floating Offshore Platform with Damping

    • Jeffrey M. Falzarano, Robert E. Clague, Ravikiran S. Kota
    Pages 255-274

  16. Performance of Nonlinear Vibration Absorbers for Multi-Degrees-of-Freedom Systems Using Nonlinear Normal Modes

    • Gregory S. Agnes, Daniel J. Inman
    Pages 275-292

  17. Back Matter

    Pages 293-293
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Keywords

About this book

The nonlinear normal modes of a parametrically excited cantilever beam are constructed by directly applying the method of multiple scales to the governing integral-partial differential equation and associated boundary conditions. The effect of the inertia and curvature nonlin­ earities and the parametric excitation on the spatial distribution of the deflection is examined. The results are compared with those obtained by using a single-mode discretization. In the absence of linear viscous and quadratic damping, it is shown that there are nonlinear normal modes, as defined by Rosenberg, even in the presence of a principal parametric excitation. Furthermore, the nonlinear mode shape obtained with the direct approach is compared with that obtained with the discretization approach for some values of the excitation frequency. In the single-mode discretization, the spatial distribution of the deflection is assumed a priori to be given by the linear mode shape ¢n, which is parametrically excited, as Equation (41). Thus, the mode shape is not influenced by the nonlinear curvature and nonlinear damping. On the other hand, in the direct approach, the mode shape is not assumed a priori; the nonlinear effects modify the linear mode shape ¢n. Therefore, in the case of large-amplitude oscillations, the single-mode discretization may yield inaccurate mode shapes. References 1. Vakakis, A. F., Manevitch, L. I., Mikhlin, Y. v., Pilipchuk, V. N., and Zevin A. A., Nonnal Modes and Localization in Nonlinear Systems, Wiley, New York, 1996.

Editors and Affiliations

  • University or Illinois at Urbana/Champaign, Urbana, USA

    Alexander F. Vakakis

Bibliographic Information

  • Book Title: Normal Modes and Localization in Nonlinear Systems

  • Editors: Alexander F. Vakakis

  • DOI: https://doi.org/10.1007/978-94-017-2452-4

  • Publisher: Springer Dordrecht

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer Science+Business Media Dordrecht 2001

  • Hardcover ISBN: 978-0-7923-7010-9Published: 31 January 2002

  • Softcover ISBN: 978-90-481-5715-0Published: 21 January 2011

  • eBook ISBN: 978-94-017-2452-4Published: 29 June 2013

  • Edition Number: 1

  • Number of Pages: VI, 294

  • Additional Information: Reprinted from NONLINEAR DYNAMICS, 25:1-3

  • Topics: Classical Mechanics, Vibration, Dynamical Systems, Control, Ordinary Differential Equations, Partial Differential Equations

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