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

Clelia Marchionna

doi:10.3934/cpaa.2013.12.2947

Article Contents

2013, Volume 12, Issue 6: 2947-2971. Doi: 10.3934/cpaa.2013.12.2947

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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

Received: June 30, 2012

Revised: December 31, 2012

Published: October 2013

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Abstract

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.

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Mathematics Subject Classification: Primary: 34C25; Secondary: 35L70.

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