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November  2013, 12(6): 2947-2971. doi: 10.3934/cpaa.2013.12.2947

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

Clelia Marchionna 1,

1. 

Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy

Received  July 2012 Revised  January 2013 Published  May 2013

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.
Keywords: Kirchhoff string, monotonicity of the almost periods, single modes, stability..
Mathematics Subject Classification: Primary: 34C25; Secondary: 35L7.
Citation: Clelia Marchionna. Free vibrations in space of the single mode for the Kirchhoff string. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2947-2971. doi: 10.3934/cpaa.2013.12.2947
References:
[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.  Google Scholar

[2]

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.  Google Scholar

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

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

[5]

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.  Google Scholar

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

[7]

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.  Google Scholar

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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.  Google Scholar

[9]

R. W. Dickey, Stability of periodic solutions of the non linear string,, Q. Appl. Math., 38 (): 253.   Google Scholar

[10]

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.  Google Scholar

[11]

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.  Google Scholar

show all references

References:
[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.  Google Scholar

[2]

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.  Google Scholar

[3]

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

[4]

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

[5]

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.  Google Scholar

[6]

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

[7]

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.  Google Scholar

[8]

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.  Google Scholar

[9]

R. W. Dickey, Stability of periodic solutions of the non linear string,, Q. Appl. Math., 38 (): 253.   Google Scholar

[10]

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.  Google Scholar

[11]

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.  Google Scholar

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