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Spectral analysis of coupled Euler-Bernoulli and Timoshenko beam model.
- Source :
-
ZAMM -- Journal of Applied Mathematics & Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik . May2004, Vol. 84 Issue 5, p291-313. 23p. - Publication Year :
- 2004
-
Abstract
- The present paper is devoted to the asymptotic and spectral analysis of a system of coupled Euler-Bernoulli and Timoshenko beams. The model is governed by a system of two coupled differential equations and a two parameter family of boundary conditions modelling the action of the self-straining actuators. The above equations of motion form a coupled linear hyperbolic system, which is equivalent to a single operator evolution equation in the energy space. That equation defines a semigroup of bounded operators. This is a dynamics generator of the semigroup which is our main object of interest in the present paper. We prove that for each set of boundary parameters, the dynamics generator has a compact inverse and this inverse operator belongs to class <UEQN>\documentclass{article}\pagestyle{empty}\usepackage{amsfonts}\begin{document}$\mathfrak{S}_p$\end{document}</UEQN> of compact operators with p > 1. We also show that if both boundary parameters are not purely imaginary numbers, then the dynamics generator is a nonselfadjoint operator in the energy space. However, its inverse operator is a finite-rank perturbation of a selfadjoint operator. The latter fact is crucial for the proof of the fact that the root vectors of the dynamics generator form a complete and minimal set in the energy space. We will use the spectral results in our forthcoming papers to prove that the dynamics generator of the system is a Riesz spectral operator in the sense of Dunford and to use the latter fact for the solution of several boundary and distributed controllability problems via the spectral decomposition method. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 00442267
- Volume :
- 84
- Issue :
- 5
- Database :
- Academic Search Index
- Journal :
- ZAMM -- Journal of Applied Mathematics & Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik
- Publication Type :
- Academic Journal
- Accession number :
- 13508633
- Full Text :
- https://doi.org/10.1002/zamm.200310097