Subharmonic parametric resonance of simply supported buckled beams

Samir A. Emam, Mostafa M. Abdalla

Research output: Contribution to journalArticlepeer-review

20 Citations (Scopus)


This study presents the nonlinear vibrations of simply supported beams that are subjected to a subharmonic resonance excitation of order two. The applied axial force has a static component to buckle the beam and a dynamic component for parametric excitation. The equation governing the nonlinear transverse deformations is an integral partial differential equation. The static counter part of the governing equation represents the nonlinear buckling problem that is solved for the critical buckling loads and exact buckled configurations. A reduced-order model based on the Galerkin’s discretization is obtained where a multi-mode discretization is assumed. The significance of the number of modes retained in the discretization is examined. It is found out that for simply supported beams, the second and higher modes have no contribution on either the static or the dynamic response of the beam. To study the nonlinear dynamics of the problem, a shooting method is used to locate periodic orbits and long-time numerical integration using fifth-order Runge–Kutta method is used to find the fast Fourier transforms of the response. A forward sweep, where the amplitude of the force is increased, while the frequency is kept fixed, is used to drive the beam. We find local attractors coexisting with snapthrough motion. Nonlinear instabilities such as period-doubling leading to chaos, symmetry-breaking, cyclic-fold, period-demultiplying, and jumping are reported. A bifurcation diagram showing all possible attractors near and far from the equilibrium positions is presented. This diagram is a helpful tool to design beams for a desired response especially for energy harvesting applications where high-amplitude vibration is needed.

Original languageEnglish
Pages (from-to)1443-1456
Number of pages14
JournalNonlinear Dynamics
Issue number2
Publication statusPublished - Jan 2014


  • Bifurcation
  • Buckled beams
  • Nonlinear
  • Parametric resonance
  • Subharmonic

ASJC Scopus subject areas

  • Mechanical Engineering
  • Aerospace Engineering
  • Ocean Engineering
  • Applied Mathematics
  • Electrical and Electronic Engineering
  • Control and Systems Engineering


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