Numerical evaluation of high-order modes of vibration in uniform Euler–Bernoulli beams

Many vibration text books give expressions for the mode shape functions of uniform Euler–Bernoulli beams. However, the common forms of these expressions permit the evaluation of only the first 12 modes or so due to numerical issues. This article presents alternative and approximate forms for the evaluation of beam mode shape functions that are numerically stable. Although the approximations allow numerical evaluation of the mode shapes for all modes of vibration, the penalty is that some errors occur in the calculations for low-order modes, and these errors are quantified. Beams with combinations of the classical boundary conditions of clamped, free, pinned, and sliding are covered.