Normal Modes and Localization in Nonlinear Systems

  • Alexander F. Vakakis

Table of contents

  1. Front Matter
    Pages I-1
  2. Hiroshi Yabuno, Ali H. Nayfeh
    Pages 65-77
  3. Ralf W. Wittenberg, Philip Holmes
    Pages 111-132
  4. Taehoon Ahn, Robert S. Mackay, Jacques-A. Sepulchre
    Pages 157-182
  5. Eric Pesheck, Nicolas Boivin, Christophe Pierre, Steven W. Shaw
    Pages 183-205
  6. Walter Sextro, Karl Popp, Tomasz Krzyzynski
    Pages 207-220
  7. Robert T. M’Closkey, Alex Vakakis, Roman Gutierrez
    Pages 221-236
  8. Back Matter
    Pages 293-293

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.


beam boundary layer dynamical systems dynamics vibration

Editors and affiliations

  • Alexander F. Vakakis
    • 1
  1. 1.University or Illinois at Urbana/ChampaignUrbanaUSA

Bibliographic information

  • DOI
  • Copyright Information Springer Science+Business Media B.V. 2001
  • Publisher Name Springer, Dordrecht
  • eBook Packages Springer Book Archive
  • Print ISBN 978-90-481-5715-0
  • Online ISBN 978-94-017-2452-4
  • Buy this book on publisher's site