Exact and Numerically Stable Expressions for Euler-Bernoulli and Timoshenko Beam Modes

In this work we present a general procedure for deriving exact, analytical, and numerically stable expressions for the characteristic equations and the eigenmodes of the Timoshenko and the Euler-Bernoulli beam models. This work generalizes the approach recently described in Gonçalves et al. (2018), which allows the numerical stabilization for the case of the Timoshenko beam model. Our results enable the reliable computation of the eigenvalues and the eigenmodes of both beam models for any number of modes. In addition to presenting the necessary details for stabilizing the solutions to the eigenvalue problem for the two beam models, we also tabulate the results for a large number of the common boundary conditions so that one can compare the predictions of all those models. Therefore, another contribution of our work is the presentation of both the conventional as well as the novel, numerically stabilized results for both the Euler-Bernoulli and the Timoshenko beam models in one manuscript with consistent notation, and for the most common boundary conditions. The code for the stabilized Timoshenko expressions as well as for the finite element verification are made available through a Mendeley Data repository linked to this manuscript.