Non-linear response of buckled beams to 1:1 and 3:1 internal resonances

Samir A. Emam, Ali H. Nayfeh

Research output: Contribution to journalArticlepeer-review

74 Citations (Scopus)


The non-linear response of a buckled beam to a primary resonance of its first vibration mode in the presence of internal resonances is investigated. We consider a one-to-one internal resonance between the first and second vibration modes and a three-to-one internal resonance between the first and third vibration modes. The method of multiple scales is used to directly attack the governing integral-partial-differential equation and associated boundary conditions and obtain four first-order ordinary-differential equations (ODEs) governing modulation of the amplitudes and phase angles of the interacting modes involved via internal resonance. The modulation equations show that the interacting modes are non-linearly coupled. An approximate second-order solution for the response is obtained. The equilibrium solutions of the modulation equations are obtained and their stability is investigated. Frequency-response curves are presented when one of the interacting modes is directly excited by a primary excitation. To investigate the global dynamics of the system, we use the Galerkin procedure and develop a multi-mode reduced-order model that consists of temporal non-linearly coupled ODEs. The reduced-order model is then numerically integrated using long-time integration and a shooting method. Time history, fast Fourier transforms (FFT), and Poincare sections are presented. We show period doubling bifurcations leading to chaos and a chaotically amplitude-modulated response.

Original languageEnglish
Pages (from-to)12-25
Number of pages14
JournalInternational Journal of Non-Linear Mechanics
Publication statusPublished - 2013


  • Buckled beams
  • Frequency-response curves
  • Non-linear dynamics
  • One-to-one internal resonance
  • Three-to-one internal resonance

ASJC Scopus subject areas

  • Mechanics of Materials
  • Mechanical Engineering
  • Applied Mathematics


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