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Acta Mechanica

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24.06.2020 | Original Paper

Issues related to the second spectrum, Ostrogradsky’s energy and the stabilization of Timoshenko–Ehrenfest-type systems

verfasst von: D. S. Almeida Júnior, A. J. A. Ramos, A. Soufyane, M. L. Cardoso, M. L. Santos

Erschienen in: Acta Mechanica | Ausgabe 9/2020

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Abstract

In this paper, we discuss the stabilization properties of a beam model on a Winkler foundation by using Timoshenko–Ehrenfest-type systems, taking into account the influence of the so-called second spectrum. We consider the well-known classical version of the Timoshenko–Ehrenfest beam model as well as the truncated (or simplified) version of the same beam model according to the approach given by Elishakoff (in: Banks-Sills (ed.), Advances in mathematical modelling and experimental methods for materials and structures, solid mechanics and its applications. Springer, Berlin, pp 249–254, 2010). The main novelty of our approach is the concept of applying Ostrogradsky’s energy to both beam models to highlight the physics issues arising in the frequency spectra. Our ideas are an attempt to fill the gap regarding the consequences of the second spectrum in the stabilization scenario for dissipative Timoshenko systems that are partially damped.

Vorheriger Artikel Explicit solution of one boundary value problem of thermoelasticity for a circle with diffusion, microtemperatures, and microconcentrations

Nächster Artikel Variable separation method for solving boundary value problems of isotropic linearly viscoelastic bodies

Elishakoff et al. [27] documents (strongly based on the papers and autobiographical book by Timoshenko) that Stephen Timoshenko was one of the developers of this theory. Specifically, he had a co-author, the physicist Paul Ehrenfest (1880–1933), who collaborated with him on his 1921 and 1922 papers. See also Challamel and Elishakoff [13] for a historical presentation of the beam and plate models in elasticity theory as well as studies of predecessors such as Bresse (i.e., his studies in the nineteenth century), who rigorously derived the set of equations for the curved beam shear in dynamics, which was later (1913; 1916; 1920; 1921; 1922) generalized by Timoshenko and Paul Ehrenfest.

Timoshenko [63] introduced Eqs. (1.3)–(1.4), which take into account the shear deformation and rotary inertia. According to Elishakoff et al. [23, 25], Timoshenko had two predecessors, namely Bresse [12] and Rayleigh [50]. However, Timoshenko did not reference Bresse, though he sometimes referenced Rayleigh. Moreover, Ehrenfest’s name did not appear in his papers dated 1920 and 1921. Koiter [45] did not know these facts when he wrote: “What is generally known as Timoshenko beam theory is a good example of a basic principle in the history of science: a theory which bears someone’s name is most likely due to someone else.” Elishakoff [27] unequivocally proves that the modern theory with the shear coefficient was introduced by Timoshenko and Ehrenfest. It is therefore fair that the theory should be called the Timoshenko–Ehrenfest theory.

Emil Winkler (1835–1888) was the German civil engineer and professor responsible for formulating and solving the problem of an elastic beam on a deformable foundation, which today is known as the Winkler foundation (Fig. 3). It is a beam model on an elastic foundation that assumes a linear force–deflection relationship.

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Titel: Issues related to the second spectrum, Ostrogradsky’s energy and the stabilization of Timoshenko–Ehrenfest-type systems
verfasst von: D. S. Almeida Júnior
A. J. A. Ramos
A. Soufyane
M. L. Cardoso
M. L. Santos
Publikationsdatum: 24.06.2020
Verlag: Springer Vienna
Erschienen in: Acta Mechanica / Ausgabe 9/2020
Print ISSN: 0001-5970
Elektronische ISSN: 1619-6937
DOI: https://doi.org/10.1007/s00707-020-02730-7

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