Vibrations of Tapered Timoshenko Beams in Terms of Static Timoshenko Beam Functions

In this paper, the free vibrations of a wide range of tapered Timoshenko beams are investigated. The cross section of the beam varies continuously and the variation is described by a power function of the coordinate along the neutral axis of the beam. The static Timoshenko beam functions, which are the complete solutions of a tapered Timoshenko beam under a Taylor series of static load, are developed, respectively, as the basis functions of the flexural displacement and the angle of rotation due to bending. The Rayleigh-Ritz method is applied to derive the eigenfrequency equation of the tapered Timoshenko beam. Unlike conventional basis functions which are independent of the cross-sectional variation of the beam, these static Timoshenko beam functions vary in accordance with the cross-sectional variation of the beam so that higher accuracy and more rapid convergence have been obtained. Some numerical results are presented for both truncated and sharp-ended Timoshenko beams. On the basis of convergence study and comparison with available results in the literature it is shown that the first few eigenfrequencies can be given with quite good accuracy by using a small number of terms of the static Timoshenko beam functions. Finally, some valuable results are presented systematically.