A Direct Solution for the Transverse Vibration of Euler-Bernoulli Wedge and Cone Beams

Abstract

A number of publications have appeared on the transverse vibration of "complete" and truncated wedge and cone beams. The mode shape equations are linear differential equations with regular singularity. The natural frequencies of "complete" beams are found in several publications. For truncated beams several numerical solutions have been proposed and the natural frequencies documented (in most cases) for cantilever wedge and cone beams clamped at the large end. Analytical solutions of the mode shape equations have been derived based on Bessel functions. Only in one publication are the natural frequencies tabulated, for nine combinations of clamped, pinned and free boundary conditions, but the accuracy of the frequencies is limited to two figures after the decimal point for the fundamental frequency, and the higher mode frequencies are rounded to integers. The results are not suitable for judging the quality of the various approximate methods. In this paper a direct solution of the mode shape equation is presented. The sliding boundary condition is also considered and the first three dimensionless natural frequencies computed for 16 combinations of the boundary conditions are tabulated for values of truncation factors from 0·05 to 0·7. The tabulated results are benchmarks, and frequencies for truncation factors not in the tables may be interpolated to within 0·1% accuracy.