This paper is a study of flexural elastic vibrations of Timoshenko beams with due allowance for the effects of rotary inertia and shear. Two independent formulations are developed, one based on the concepts proposed by Timoshenko and the other on the extended Rayleigh-Ritz energy method. The results obtained from the two formulations are matched in order to determine the correcting coefficients in the simplified formulae which are proposed for the frequency of the first two flexural branches. The proposed formulae are shown to achieve greater accuracy in describing the flexural motions. New accurate solutions are offered for several cross-sectional geometries which enable the accuracy of the available methods to be assessed. Correction coefficients are evaluated for several sections and are compared with previous work.