Nonlinear Responses of Buckled Beams to Subharmonic-Resonance Excitations

Samir A. Emam, Ali H. Nayfeh

Research output: Contribution to journalArticlepeer-review

77 Citations (Scopus)


We investigated theoretically and experimentally the nonlinear response of a clamped-clamped buckled beam to a subharmonic resonance of order one-half of its first vibration mode. We used a multi-mode Galerkin discretization to reduce the governing nonlinear partial-differential equation in space and time into a set of nonlinearly coupled ordinary-differential equations in time only. We solved the discretized equations using the method of multiple scales to obtain a second-order approximate solution, including the modulation equations governing its amplitude and phase, the effective nonlinearity, and the effective forcing. To investigate the large-amplitude dynamics, we numerically integrated the discretized equations using a shooting method to compute periodic orbits and used Floquet theory to investigate their stability and bifurcations. We obtained interesting dynamics, such as phase-locked and quasiperiodic motions, resulting from a Hopf bifurcation, snapthrough motions, and a sequence of period-doubling bifurcations leading to chaos. Some of these nonlinear phenomena, such as Hopf bifurcation, cannot be predicted using a single-mode Galerkin discretization, We carried out an experiment and obtained results in good qualitative agreement with the theoretical results.

Original languageEnglish
Pages (from-to)105-122
Number of pages18
JournalNonlinear Dynamics
Issue number2
Publication statusPublished - Jan 2004
Externally publishedYes


  • Buckled beams
  • Galerkin discretization
  • Nonlinear dynamics
  • Subharmonic resonance

ASJC Scopus subject areas

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


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