The nonlinear normal modes of a geometrically nonlinear multi-span beam consisting of n segments, coupled by means of torsional stiffeners are examined. Assuming that the stiffeners possess large torsional stiffness, the beam displacements are decomposed into static and flexible components. It is shown that the static components are much smaller in magnitude than the flexible ones. A Galerkin approximation is subsequently employed to discretize the problem, whereby the computation of the nonlinear normal modes of the multi-span beam is reduced to the study of the periodic solutions of a set of weakly coupled, weakly nonlinear ordinary differential equations. Numerous stable and unstable, localized and non-localized nonlinear normal modes of the multi-span beam are detected. Assemblies consisting of n = 2, 3, and 4 beam segments are examined, and are found to possess stable, strongly localized nonlinear normal modes. These are free synchronous oscillations during which only one segment of the assembly vibrates with finite amplitude. As the number of periodic segments increases, the structure of the nonlinear normal modes becomes increasingly more complicated. In the multi-span beams examined, nonlinear mode localization is generated through two distinct mechanisms: through Pitchfork or Saddle-node mode bifurcations, or as the limit of a continuous mode branch when a coupling parameter tends to zero.
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October 1996
Research Papers
Localized and Non-Localized Nonlinear Normal Modes in a Multi-Span Beam With Geometric Nonlinearities
J. Aubrecht,
J. Aubrecht
Department of Mechanical and Industrial Engineering, University of Illinois at Urbana—Champaign, Urbana, IL 61801
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A. F. Vakakis
A. F. Vakakis
Department of Mechanical and Industrial Engineering, University of Illinois at Urbana—Champaign, Urbana, IL 61801
Search for other works by this author on:
J. Aubrecht
Department of Mechanical and Industrial Engineering, University of Illinois at Urbana—Champaign, Urbana, IL 61801
A. F. Vakakis
Department of Mechanical and Industrial Engineering, University of Illinois at Urbana—Champaign, Urbana, IL 61801
J. Vib. Acoust. Oct 1996, 118(4): 533-542 (10 pages)
Published Online: October 1, 1996
Article history
Received:
November 1, 1993
Revised:
January 1, 1995
Online:
February 26, 2008
Citation
Aubrecht, J., and Vakakis, A. F. (October 1, 1996). "Localized and Non-Localized Nonlinear Normal Modes in a Multi-Span Beam With Geometric Nonlinearities." ASME. J. Vib. Acoust. October 1996; 118(4): 533–542. https://doi.org/10.1115/1.2888332
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