Analysis of vibration spectrum of a Timoshenko beam with boundary damping by the wave method

Abstract The Timoshenko beam is a model of a thick beam and is important in structural engineering. Recently, there has been considerable interest in the analysis and control of the vibration of the Timoshenko beam. The understanding of the spectrum of vibration is fundamental in order to accomplish such research. The classical separation of variables approach leads to complicated transcendental equations which are difficult to analyze. In this paper, we develop the wave propagation method to tackle this problem. Asymptotic estimates for the eigenfrequencies are obtained for a Timoshenko beam with dissipative conditions. Such estimates show that the Timoshenko beam has two branches in its spectrum corresponding to the propagation of waves with two velocities for the displacement and angular rotation variables. The numerical computations using a high accuracy Legendre-tau spectral method.