The free vibration of a type of tapered beams

Abstract

This paper studies the vibrational characteristics of tapered beams with continuously varying rectangular cross-section of depth and breadth proportional to x s and x t , respectively, where both s and t are arbitrary real numbers for a truncated beam and arbitrary positive numbers for a sharp ended beam and x is the axial co-ordinate measured from the sharp end of the beam. The Bernoulli–Euler theory of bending is used to describe the motion of the beam. A new set of beam functions are developed as the admissible functions, which are the complete solution of the tapered beam under an arbitrary static load expanded into a Taylor series. The eigenfrequency equation is obtained by the Rayleigh–Ritz method. The accuracy is assured from the convergency and comparison studies. The effect of the location of the Taylor series expanding point on the convergency is discussed. The analysis shows that the present approach is convergent for arbitrary truncation factor by taking the midpoint of the beam as the expanding point of the Taylor series. Numerical results are tabulated for three different tapered beams with various boundary conditions and truncation factors. It is shown that the eigenfrequencies can be obtained with high accuracy by using only a small number of terms of the static beam functions.