Parametric Resonances of Columns with Damping

  • Yoshihiko SugiyamaEmail author
  • Mikael A. Langthjem
  • Kazuo Katayama
Part of the Solid Mechanics and Its Applications book series (SMIA, volume 255)


This chapter discusses the mathematical aspects of parametric resonances of columns with damping, dealing generally with dynamical systems with damping. In addition to the first-order approximation for the resonance boundaries of Mathieu-Hill equations, an approach to the second-order approximation is suggested.


  1. 1.
    Arscott, F. M. (1964). Periodic differential equations. Oxford: Pergamon Press.zbMATHGoogle Scholar
  2. 2.
    Magnus, W., & Winkler, S. (1966). Hill’s equation. New York: Wiley.zbMATHGoogle Scholar
  3. 3.
    Bolotin, V. V. (1964). The Dynamic Stability of Elastic Systems. San Francisco: Holden-Day, Inc.Google Scholar
  4. 4.
    Nayfeh, A. H., & Mook, D. T. (1995). Nonlinear oscillations. New York: Wiley.CrossRefGoogle Scholar
  5. 5.
    Seyranian A. P. & Mailybaev, A. A. (2003). Multiparameter Stability Theory with Mechanical Applications. Singapore: World Scientific Publishing.Google Scholar
  6. 6.
    Hsu, C. S. (1963). On the parametric excitation of a dynamical systems having multiple degrees of freedom. Journal of Applied Mechanics, 30, 367–372.CrossRefGoogle Scholar
  7. 7.
    Struble, R. A., & Fletcher, J. E. (1962). General perturbation solution of the Mathieu equation. Journal of the Society for Industrial and Applied Mathematics, 10, 314–328.CrossRefGoogle Scholar
  8. 8.
    Weidenhammer, F. (1951). Der eingespannte, axial-pulsierend belaste Stab als stabilitätsproblem. Ingenieur-Archiv, 19, 162–191.Google Scholar
  9. 9.
    Schmidt, G., & Weidenhammer, F. (1961). Instabilitäten gedämpten rheolinearer Schwingungen. Mathematischen Nachrichten, 23, 301–318.CrossRefGoogle Scholar
  10. 10.
    Yamamoto, T. & Saito, A. (1967). On the oscillations of summed and differential types under parametric excitation (vibratory systems with damping), Transactions of the Japan Society of Mechanical Engineers, 33, 905–914 (in Japanese).CrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Yoshihiko Sugiyama
    • 1
    Email author
  • Mikael A. Langthjem
    • 2
  • Kazuo Katayama
    • 3
  1. 1.Small Spacecraft Systems Research Center, Osaka Prefecture UniversitySakaiJapan
  2. 2.Department of Mechanical Systems EngineeringYamagata UniversityYonezawaJapan
  3. 3.Daicel CorporationTatsunoJapan

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