Free flexural vibration of tapered beams

Abstract

Abstract The free flexural vibration behaviour of a range of tapered beams is investigated by making use of the exact solutions of the governing differential equations and then imposing the necessary boundary conditions. This research is prompted by a recently published paper in Computers and Structures which used the Frobenius method of series solution but needed 130 to 300 terms in the series when solving the free vibration problem of tapered beams using the transfer matrix method. This paper investigates the same problem using a direct approach, but with a twofold purpose. The first is to show that an exact solution for the problem is possible by using Bessel functions rather than relying on a series solution which is somehow unnecessary and from a computational standpoint inefficient. The second reason for writing this paper is to broaden the scope of the recently published paper by not focusing on only one type of cross-section, but on a range of cross-sections and yet retaining the exactness of the results. The application of Bessel functions is demonstrated when developing the theory and computing the numerical results. The results presented can be used as an aid to validate the finite element and other approximate methods.