Advanced Search
Article Contents
Article Contents

Free vibrations in space of the single mode for the Kirchhoff string

Abstract Related Papers Cited by
  • We study a single mode for the Kirchhoff string vibrating in space. In 3D a single mode is generally almost periodic in contrast to the 2D periodic case. In order to show a complete geometrical description of a single mode we prove some monotonicity properties of the almost periods of the solution, with respect to the mechanical energy and the momentum. As a consequence of these properties, we observe that a planar single mode in 3D is always unstable, while it is known that a single mode in 2D is stable (under a suitable definition of stability), if the energy is small.
    Mathematics Subject Classification: Primary: 34C25; Secondary: 35L70.


    \begin{equation} \\ \end{equation}
  • [1]

    M. Abramowitz and I. A. Stegun, "Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables," Chapters 16, 17, New York: Dover, National bureau of standards, 1964.


    L. P. Bonorino, E. H. M. Brietzke, J. P. Lukaszczyk and C. A. Taschetto, Properties of the period function for some Hamiltonian systems and homogeneous solutions of a semilinear elliptic equation, J. Differential Equations, 214 (2005), 156-175.doi: 10.1016/j.jde.2004.08.007.


    G. F. Carrier, On the non-linear vibration problem of an elastic string, Q. Appl. Math., 3 (1945), 157-165.


    T. Cazenave and F. B. Weissler, Unstable simple modes of the nonlinear string, Q. Appl. Math., 54 (1996), 287-305.


    C. Chicone, The monotonicity of the period function for planar Hamiltonian vector fields, J. Differential Equations, 69 (1987), 310-321.doi: 10.1016/0022-0396(87)90122-7.


    C. Chicone, "Ordinary Differential Equations with Applications," Springer-Verlag, New York, 2006.


    A. Cima, A. Gasull and F. Mañosas, Period function for a class of Hamiltonian systems, J. Differential Equations, 168 (2000), 180-199.doi: 10.1006/jdeq.2000.3912.


    W. R. Dean, Note on the evaluation of an elliptic integral of the third kind, J. London Math. Soc., 18 (1943), 130-132.doi: 10.1112/jlms/s1-18.3.130.


    R. W. DickeyStability of periodic solutions of the non linear string, Q. Appl. Math., 38 (1980/81), 253-259.


    G. Gallavotti, "The Elements of Mechanics," Springer-Verlag, New York, 1983. Also available from: Ipparco Editore, 2007. http://ipparco.roma1.infn.it/pagine/deposito/2007/elements.pdf.


    M. Ghisi and M. Gobbino, Stability of simple modes of the Kirchhoff equation, Nonlinearity, 14 (2001), 1197-1220.doi: 10.1088/0951-7715/14/5/314.

  • 加载中

Article Metrics

HTML views() PDF downloads(80) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint