Volume 29, 2023
|Number of page(s)||24|
|Published online||27 February 2023|
Do shape memory alloy cables restrain the vibrations of girder bridges? A mathematical point of view
Univ. Polytechnique Hauts-de-France, INSA Hauts-de-France, CERAMATHS-Laboratoire de Matériaux Céramiques et de Mathématiques, 59313 Valenciennes, France
* Corresponding author: firstname.lastname@example.org
Accepted: 3 October 2022
We study the energy decay of a damped Euler-Bernoulli beam which is subject to a point-wise feedback force representing a Shape Memory Alloy (SMA) cable. The problem we consider is that of the paper by [Liu et al. Int. J. Struct. Stab. Dyn. 17 (2017) 1750076] but, for simplicity, our mod-elization does not take into account the additional stiffness term they considered. An explicit expression is given for the resolvent of the underlying operator as well as its eigenvalues and eigenfunctions. We show the exponential decay of the energy. The fastest decay rate is given by the supremum of the real part of the spectrum of the infinitesimal generator of the underlying semigroup since we prove the existence of a Riesz basis. To the question ”Do Shape Memory Alloy cables restrain the vibrations of girder bridges?”, the experiments in [Liu et al. Int. J. Struct. Stab. Dyn. 17 (2017) 1750076] answer positively. Our study does not allow to give a definite answer yet. The only presence of these cables may not be enough. Some physical parameters have to be chosen carefully.
Mathematics Subject Classification: 74K10 / 35B40 / 35Q74 / 34L10 / 34L15 / 34L20 / 35B35 / 35E15 / 93D23
Key words: Euler-Bernoulli beam / pointwise stabilization / resolvent operator / eigenvalues and their localization / eigenfunctions / Riesz basis / exponential stability / rate of decay
© The authors. Published by EDP Sciences, SMAI 2023
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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