Steady State Vibration of Damped Timoshenko Beams

A method for the analysis of the steady state vibration of damped Timoshenko beams is presented. The usual four differential equations of the Timoshenko Theory are increased to eight differential equations with the addition of damping in the form of complex modulii. These equations are solved by numerical integration and, therefore, no restrictions need be placed on the longitudinal variation of the beam cross-section or the forcing function. The solution gives the beam response at all values of the forcing frequency and polar-plots are used for the determination of the resonant frequencies. To illustrate the wide applicability of the method the behavior of three beams, typical of those encountered in practice, is studied. The results of these examples show the importance of considering shear deformation when determining resonant frequencies and the great effect that the damping coefficient has on the magnitude of the response at resonance.