Detecting vibration frequencies and shapes for timoshenko beam model: A comparison between available eigensolvers

Stochino F.;Cazzani A.;Turco E.

2017-01-01

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Abstract

The Timoshenko beam model [1], which takes into account both rotary inertia and shear deformation, is often considered a more suitable tool for analyzing beam structures than the simpler Euler-Bernoulli one, which disregards such effects. However, the dynamics of Timoshenko beams is still an active area of research, because its spectrum consists of two branches, see, among others, [2], which, above a transition frequency, usually occurring between the first vibration modes, appear to be strictly entangled. This effect creates a challenging tasks for eigenvalue solvers, as it has been already shown for the case of simply-supported Timoshenko beams, where a closed form solution is available for both frequency equations and vibration modes (which come out to be simple sinusoids). Contrarily to what happens for Euler-Bernoulli beams in the case of Timoshenko beams isogeometric elements produce better results than standard finite elements, but do not allow to reproduce exactly the vibration shape beyond some limited part of the spectrum. For instance, by using 5th order NURBS, and 500 degrees of freedom (DOFs), only the first 270 modal shapes are reproduced accurately, while the others present not-acceptable errors, as it has been shown in [3]. In this paper special attention is devoted to the numerical algorithms for solving the eigenvalue problem, using as an example a simply-supported Timoshenko beam under free vibrations conditions. The results obtained using the QZ algorithm, QR algorithm and the more general Lanczos and Arnoldi iteration methods are compared and discussed in order to determine their comparative performances.