Eigenfrequencies of Tapered Beams with Intermediate Point Supports

Abstract

This paper studies the vibrational characteristics of multi-span beams with continuously varying rectangular cross-section of depth and breadth proportional to xs and xt respectively where s and t may be given arbitrary real numbers for a truncated beam and arbitrary positive numbers for a sharp ended beam and x is the coordinate along the centreline of the beam, measured from the sharp end of the beam. The Bernoulli-Euler theory of bending is used to describe the dynamic deflection of the beam. A new set of admissible functions are developed from the static solutions of the tapered beam with intermediate point supports under a Taylor series of loads. The unknown coefficients in the static beam functions are uniquely determined by the boundary conditions and the zero displacement conditions at the locations of the point supports. The eigenfrequency equation is derived by the Rayleigh-Ritz method. The numerical results are tabulated for both truncated and sharp ended beams with one and two intermediate point supports and compared with other existing values in the literature. Good agreement is observed. It is shown that the first few eigenfrequencies can be obtained with high accuracy by the present models and only a small number of terms of the static beam functions has been used.