A hierarchy of rational Timoshenko dispersion relations

This paper shows that the conventional Timoshenko dispersion relation for bending waves is the (0,1) member of a two-parameter family (m,n) of approximations to the exact dispersion relation obtained from linear elasticity. Higher members of the family are shown to converge rapidly to the exact dispersion relation; in particular, any number of branches can be captured accurately over their entire length, i.e. up to arbitrarily high frequencies and wavenumbers. The theory admits a rigorous accuracy analysis, and demonstrates that the conventional Timoshenko dispersion relation is a completely rational approximation, thus answering some previous doubts about this matter. Lower members of the two-parameter family include the Euler–Bernoulli dispersion relation and a Rayleigh-type inertia correction. Especially useful is the Timoshenko (1,2) dispersion relation, which extends the conventional Timoshenko (0,1) dispersion relation by capturing the first four branches of the exact dispersion relation rather than merely the first two, and is accurate over a wide range. An inertia-corrected Timoshenko dispersion relation is also derived, and is shown to include many previous approximations as special cases.