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An Introduction to Parametric Resonance

  • Jonatan Peña-RamírezEmail author
  • Rob H. B. Fey
  • Henk Nijmeijer
Chapter

Abstract

In many engineering, physical, electrical, chemical, and biological systems, oscillatory behavior of the dynamic system due to periodic excitation is of great interest. Two kinds of oscillatory responses can be distinguished: forced oscillations and parametric oscillations. Forced oscillations appear when the dynamical system is excited by a periodic input. If the frequency of an external excitation is close to the natural frequency of the system, then the system will experience resonance, i.e. oscillations with a large amplitude.

Keywords

Parametric Resonance Primary System Roll Resonance Secondary System Parametric Roll 
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.

References

  1. 1.
    Amabili, M., Paidoussis, M. P.: Review of studies on geometrically nonlinear vibrations and dynamics of circular cylindrical shells and panels, with and without fluid-structure interaction. Applied Mechanics Reviews, 56(4):349–381, (2003).CrossRefGoogle Scholar
  2. 2.
    Armstrong, E. H.: Some recent developments of regenerative circuits. Proceedings of the Institute of Radio Engineers, 10:244–260, (1922).Google Scholar
  3. 3.
    Arscott, F.: Periodic differential equations. Pergamon Press. New York, (1964).zbMATHGoogle Scholar
  4. 4.
    Ataka, H.: On superregeneration of an ultra-short-wave receiver. Proceedings of the Institute of Radio Engineers, 23(8):841–884, (1935).Google Scholar
  5. 5.
    Barrow, W. L., Smith, D. B., Baumann, F. W.: A further study of oscillatory circuits having periodically varying parameters. Journal of The Franklin Institute, 221(3):403–416, (1936).CrossRefGoogle Scholar
  6. 6.
    Biran, A.: Ship hydrostatics and stability. Butterworth-Heinemann. Oxford, (2003).Google Scholar
  7. 7.
    Broer, H. W., Levi, M.: Geometrical aspects of stability theory for Hill’s equations. Archive for rational mechanics and analysis, 131(3):225–240, (1995).CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Brown, D. K.: The way of a ship in the midst of the sea: the life and work of William Froude. Periscope Publishing Ltd. ISBN-10: 1904381405, (2006).Google Scholar
  9. 9.
    France, W. N., Levadou, M., Trakle, T. W., Paulling, J. R., Michel, R. K., Moore, C.: An investigation of head-sea parametric rolling and its influence on container lashing systems, Marine Technology, 40(1):1–19, (2003).Google Scholar
  10. 10.
    Galeazzi, R.: Autonomous supervision and control of parametric roll resonance. Ph.D. Thesis, Kongens Lyngby, Denmark, (2009).Google Scholar
  11. 11.
    Holden, C., Galeazzi, R., Rodríguez, C., Perez, T., Fossen, T. I., Blanke, M., Neves, M. A. S.: Nonlinear container ship model for the study of parametric roll resonance. Modeling, identification and control, 28(4):87–103, (2007).CrossRefGoogle Scholar
  12. 12.
    Ivanov, M. I.: Free tides in two-dimensional uniform depth basins. Fluid Dynamics, 39(5): 779–789, (2004).CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Jeffreys, H.: The free oscillation of water in an elliptical lake. Proceedings of the London Mathematical Society (second edition), 23:455–476, (1923).Google Scholar
  14. 14.
    Lamb, H.: Hydrodynamics. Sixth edition. Cambridge University Press. Cambridge, United Kingdom (1945).Google Scholar
  15. 15.
    Markevich, N. I., Sel’kov, E. E.: Parametric resonance and amplification in excitable membranes. The Hodgkin-Huxley model. Journal of Theoretical Biology, 140(1):27–38, (1989).Google Scholar
  16. 16.
    Mathieu, É.: Mémoire sur le mouvement vibratoire d’une membrane de forme elliptique. Journal of Mathematics, 13:137–203, (1868).Google Scholar
  17. 17.
    McLachlan, N. W.: Theory and Applications of Mathieu Functions. Oxford Press, London (1951).Google Scholar
  18. 18.
    Mestrom, R. M. C., Fey, R. H. B., van Beek, J. T. M., Phan, K. L., Nijmeijer, H.: Modelling the dynamics of a MEMS resonator: simulations and experiments. Sensors and Actuators A: Physical, 142(1):306–315, (2008).CrossRefGoogle Scholar
  19. 19.
    Nayfeh, A. H., Mook, D. T.: Nonlinear Oscillations. John Wiley %26 Sons. New York, (1979).Google Scholar
  20. 20.
    Piccardo, G., Tubino, F.: Parametric resonance of flexible footbridges under crowd-induced lateral excitation. Journal of Sound and Vibration, 311:353–371, (2008).CrossRefGoogle Scholar
  21. 21.
    Rhoads, J. F., Shaw, S. W., Turner, K. L.: Nonlinear dynamics and its applications in micro- and nanoresonators. Journal of Dynamic Systems, Measurement, and Control, 132(034001):1–14, (2010).Google Scholar
  22. 22.
    Ruby, L.: Applications of the Mathieu equation. American Journal of Physics, 64(1):39–44, (1996).CrossRefMathSciNetGoogle Scholar
  23. 23.
    Sahu, S. K. and Datta, P. K.: Research advances in the dynamic stability behavior of plates and shells: 1987–2005 Part I: Conservative Systems. Applied Mechanics Reviews, 60:65–75, (2007).CrossRefGoogle Scholar
  24. 24.
    Seyranian, A. P.: The swing: parametric resonance. Journal of Applied Mathematics and Mechanics, 68(5):757–764, (2004).CrossRefMathSciNetGoogle Scholar
  25. 25.
    Taylor, G. I.: Tidal oscillations in gulfs and rectangular basins. Proceedings of the London Mathematical Society, 20:148–181, (1920).CrossRefGoogle Scholar
  26. 26.
    Tondl, A., Ruijgrok, T., Verhulst, F., Nabergoj, R.: Autoparametric resonance in mechanical systems. Cambridge University Press, Cambridge, United Kingdom (2000).zbMATHGoogle Scholar
  27. 27.
    van der Pol B, Strutt, M. J. O.: On the stability of the solutions of Mathieu’s equation. Philosophical Magazine, Series 7, 5(27):18–38, (1928).Google Scholar
  28. 28.
    Whittaker, E. T., Watson, G. N.: A Course of Modern Analysis: an introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions. Cambridge University Press, 3rd edition, Cambridge, United Kingdom (1920).Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Jonatan Peña-Ramírez
    • 1
    Email author
  • Rob H. B. Fey
    • 1
  • Henk Nijmeijer
    • 1
  1. 1.Department of Mechanical EngineeringEindhoven University of TechnologyEindhovenThe Netherlands

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