The hypothesis of equal wave speeds for stabilization of Timoshenko beam is not necessary anymore: the time delay cases†

Abstract

AbstractIn the current study, we consider the Bresse–Timoshenko type systems and we prove some stability results for time delay cases into setting of so called simplified Bresse–Timoshenko equations (or truncated version of Bresse–Timoshenko equations) according to contributions of Elishakoff et al. (2010, Advances in Mathematical Modeling and Experimental Methods for Materials and Structures. Solid Mechanics and Its Applications. Springer: Berlin, 249–254.; 2015, Celebrating the Centenary of Timoshenko’s study of effects of shear deformation and rotary inertia. Appl. Mech. Rev.67, 1–11.; 2017, Critical contrasting of three versions of vibrating Bresse-Timoshenko beam with a crack. Int. J. Solids Struct. 109, 143–151.). These equations are free of the so-called ‘second spectrum’ phenomenon, and they have important consequences on stabilization setting. Specifically, following Almeida Júnior and Ramos (2017, On the nature of dissipative Timoshenko systems at light of the second spectrum. Z. Angew. Math. Phys.68, 31.) in a recent contribution that shows that damping effects eliminate the consequences of this spectrum for equal wave propagation velocities, we prove that time delay effects are able of stabilizing the truncated version regardless of any relationship between coefficients of system. It is concluded that dissipative truncated versions of Bresse–Timoshenko equations are advantageous over the classical Bresse–Timoshenko equations in stabilization context.