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

• Published in: Acta mechanica Vol. 231; no. 9; pp. 3565 - 3581
• Main Authors: Almeida Júnior, D. S., Ramos, A. J. A., Soufyane, A., Cardoso, M. L., Santos, M. L.
• Format: Journal Article
• Language: English
• Published: Vienna Springer Vienna 01-09-2020; Springer; Springer Nature B.V
• Subjects: Classical and Continuum Physics; Control; Dynamical Systems; Engineering; Engineering Fluid Dynamics; Engineering Thermodynamics; Frequency spectrum; Heat and Mass Transfer; Original Paper; Solid Mechanics; Stabilization; Theoretical and Applied Mechanics; Vibration
• ISSN: 0001-5970; 1619-6937
• DOI: 10.1007/s00707-020-02730-7

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.