International Journal of Theoretical Physics

, Volume 36, Issue 8, pp 1921–1940 | Cite as

Parametric excitation of subharmonic oscillations

  • A. M. Elnaggar
  • A. A. Alhanadwah


Subharmonic oscillations of order one-half for a single-degree-of-freedom system with quadratic, cubic, and quartic nonlinearities under parametric excitation are investigated. Two approximate methods (multiple scales and generalized synchronization) are used for comparison. The modulation equations (reduced equations) of the amplitudes and the phases are obtained. Steady-state solutions (periodic solutions) and their stability are determined. Numerical solutions are carried out, and graphical representations of the results are presented and discussed. The results obtained by the two methods are in excellent agreement.


Periodic Solution Multiple Scale Parametric Excitation Generalize Synchronization Jump Phenomenon 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Balbi, J. H. (1973). Gènèralisation de la mèthode de la moyenne.InternationalJournal of Non-Linear Mechanics,8, 313–324.MATHMathSciNetCrossRefGoogle Scholar
  2. Elnaggar, A. M. (1985). Existence and determination of superharmonic synchronizations as solution of a quasi-linear physical system.Indian Journal of Pure and Applied Mathematics,16(2), 139–142.MATHMathSciNetGoogle Scholar
  3. Elnaggar, A. M., and El-Bassiouny, A. F. (1992). Response of self-excited three-degree-of-freedom systems to multifrequency excitations,International Journal of Theoretical Physics,31, 1531–1548.MathSciNetCrossRefGoogle Scholar
  4. Elnaggar, A. M., and El-Bassiouny, A. F. (1995). Principal parametric resonances of two-degree-of-freedom systems with quadratic and cubic non-linearities.Bulletin of the faculty of Science, Assiut University,24(2-C), 25–40.MathSciNetGoogle Scholar
  5. Elnaggar, A. M., and El-Basyouny, A. F. (1993). Harmonic, subharmonic, superharmonic, simultaneous subsuperharmonic and combination resonances of self-excited two coupled second order systems to multifrequency excitation.Acta Mechanica Sinica,9(1), 61–71.MATHGoogle Scholar
  6. Elnaggar, A. M., El-Diriny, M. A. (1995). Response of two internal resonant oscillators subjected to combined quadratic parametric and external excitation.Bulletin of the Faculty of Science, Assiut University,24(2-C), 85–102.MathSciNetGoogle Scholar
  7. Elnaggar, A. M., and Hamd-Allah, G. M. (1982). Analytical studies for even subharmonic synchronization of a weakly nonlinear conservative physical system,Kyungpook Mathematical Journal,22(2), 203–213.MathSciNetGoogle Scholar
  8. Evan-Iwanowski, R. M. (1976).Resonance Oscillations in Mechanical Systems, Elsevier, New York.Google Scholar
  9. Gerald, C. F. (1980).Applied Numerical Analysis. Addison-Wesley, Reading, Massachusetts.Google Scholar
  10. Haag, J. (1962).Oscillatory, Motions, Wadsworth.Google Scholar
  11. Ibrahim, R. A. (1985).Parametric Random Vibration Wiley-Interscience, New York.Google Scholar
  12. Nayfeh, A. H., and Mook, D. T. (1979).Nonlinear Oscillations, Wiley-Interscience, New York.Google Scholar
  13. Schmidt, G., and Tondl, A. (1986).Nonlinear Vibrations, Akademie-Verlag, Berlin.Google Scholar
  14. Zavodney, L. D. (1987). A theoretical and experimental investigation of parametrically excited nonlinear mechanical systems, Ph. D. Dissertation, Virgnia Polytechnic Institute and State University.Google Scholar
  15. Zavodney, L. D., and Nayfeh, A. H. (1988). The response of a single-degree-of-freedom system with quadratic and cubic non-linearities to a fundamental parametric resonance.Journal of Sound and Vibration,120(1), 63–93.MathSciNetCrossRefADSGoogle Scholar
  16. Zavodney, L. D., Nayfeh, A. H., and Sanchez, N. E. (1989). The response of a single-degree-of-freedom system with quadratic and cubic non-linearities to a principal parametric resonance,Journal of Sound and Vibration,129(3), 417–442.MathSciNetCrossRefADSGoogle Scholar

Copyright information

© Plenum Publishing Corporation 1997

Authors and Affiliations

  • A. M. Elnaggar
    • 1
  • A. A. Alhanadwah
    • 1
  1. 1.Department of Mathematics, Faculty of Science, BenhaZagazig UniversityEgypt

Personalised recommendations