ANALYSIS OF VIBRATION SPECTRUM OF A TIMOSHENKO BEAM WITH BOUNDARY DAMPING BY THE WAVE METHOD

The Timoshenko beam is a model of a thick beam and is important in str uctural 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 appro ach leads to complicated transcendental equations which are difficult to analyze. In this paper, we develop the wave propagation method to t ackle this problem. Asymptotic estimates for the eigenfrequencies are obtained for a Timoshenko beam with dissipative conditions. Such estim ates show that the Timoshenko beam has two branches in its spectrum co rresponding to the propagation of waves with two velocities for the di splacement and angular rotation variables. The asymptotic estimates ar e confirmed by numerical computations using a high accuracy Legendre-t au spectral method.