Two-to-one internal resonances in nonlinear two degree of freedom system with parametric and external excitations

A. F. El-Bassiouny, M. M. Kamel, Ayman Abdel-Khalik

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12 Citations (Scopus)


A system of two degrees of freedom with quadratic and cubic nonlinearities and subjected to parametric and external excitations is studied and solved. The method of multiple scale perturbation technique (MSPT) is used to analyze the response of this system. Four ordinary differential equations are derived to describe the modulation of the amplitudes and phases of the two modes of vibrations for the principal parametric resonances. The steady-state solutions and their stability are determined and representative numerical results are included. The theoretical resonance cases of this system have been obtained from the first approximation differential equations and some of them are confirmed by applying well-known numerical technique. The stability of the obtained numerical solution is studied using phase-plane method. The effects of different parameters on the vibration of this system are investigated.

Original languageEnglish
Pages (from-to)45-56
Number of pages12
JournalMathematics and Computers in Simulation
Issue number1
Publication statusPublished - 17 Apr 2003
Externally publishedYes



  • Homogenous
  • Multiple scale perturbation technique
  • Resonance

ASJC Scopus subject areas

  • Information Systems and Management
  • Control and Systems Engineering
  • Applied Mathematics
  • Computational Mathematics
  • Modelling and Simulation

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