Accurate Finite Difference Calculation Of The Natural Frequencies Of Euler-Bernoulli Beams Using Richardson Extrapolation

Abstract

A method is presented to obtain accurate estimations of the natural frequencies of nonuniform Euler-Bernoulli beams (the method can deal easily with all possible combinations of classical boundary conditions, including rotating and supporting end springs). The method is based on a repeated finite difference calculation with different step sizes and the use of the Richardson extrapolation (also known as the deferred approach to the limit) on the calculated eigenvalues. Implemented in double precision arithmetic in a PC/286/87 personal computer, this scheme is shown to achieve about ten correct digits in the first five natural frequencies of uniform beams (six correct digits for modes as high as 20) and to agree satisfactorily with exact and approximate natural frequencies of several types of nonuniform beams reported by other authors. The finite difference approximation to the Euler-Bernoulli equation is solved by the Rayleigh quotient iteration method (a variant of the inverse power method) so that mode shapes can be calculated along with the natural frequencies.