Spectral analysis of coupled Euler-Bernoulli and Timoshenko beam model

Academic Article


  • AbstractThe 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 \documentclass{article}\pagestyle{empty}\usepackage{amsfonts}\begin{document}$\mathfrak{S}_p$\end{document} 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.
  • Authors

  • Balakrishnan, AV
  • Shubov, Marianna
  • Peterson, CA
  • Status

    Publication Date

  • 2004
  • Has Subject Area


  • completeness
  • matrix differential operator
  • minimality
  • root vectors
  • spectrum
  • Digital Object Identifier (doi)

    Start Page

  • 291
  • End Page

  • 313
  • Volume

  • 84
  • Issue

  • 5