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

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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

Abstract
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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 and 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.
[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.
[3]	G. F. Carrier, On the non-linear vibration problem of an elastic string, Q. Appl. Math., 3 (1945), 157-165.
[4]	T. Cazenave and F. B. Weissler, Unstable simple modes of the nonlinear string, Q. Appl. Math., 54 (1996), 287-305.
[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.
[6]	C. Chicone, "Ordinary Differential Equations with Applications," Springer-Verlag, New York, 2006.
[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.
[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.
[9]	R. W. Dickey, Stability of periodic solutions of the non linear string,, Q. Appl. Math., 38 (): 253.
[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.
[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.

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.
[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.
[3]	G. F. Carrier, On the non-linear vibration problem of an elastic string, Q. Appl. Math., 3 (1945), 157-165.
[4]	T. Cazenave and F. B. Weissler, Unstable simple modes of the nonlinear string, Q. Appl. Math., 54 (1996), 287-305.
[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.
[6]	C. Chicone, "Ordinary Differential Equations with Applications," Springer-Verlag, New York, 2006.
[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.
[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.
[9]	R. W. Dickey, Stability of periodic solutions of the non linear string,, Q. Appl. Math., 38 (): 253.
[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.
[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.

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