Exact expressions for numerical evaluation of high order modes of vibration in uniform Euler-Bernoulli beams

In the literature, expressions can be found for the mode shape functions of uniform Euler-Bernoulli beams. However, these expressions permit the evaluation of only the first 12 modes or so due to numerical issues associated with the evaluation of near field waves that increase exponentially with frequency. To overcome this problem, approximate expressions have been developed that avoid the evaluation of these waves. This article presents in tabular form alternative, exact expressions of beam mode shape functions that are numerically stable. These formulations have not been reported in the literature before and they are of special interest for researchers working in the area of structural acoustics and vibration. Beams with combinations of the classical boundary conditions of clamped, free, pinned, and sliding are considered.