This paper presents an extension of Myklestad’s adaptation of the Holzer method of calculating natural frequencies and mode shapes of systems to the case of a tapered-twisted beam with certain elastic constraints. The details of the solution are so arranged that the bulk of the numerical calculations can be carried out by a technician with only a high-school mathematical background. This extension makes possible the evaluation of the effect of rotation of a beam on a radial line, of certain elastic constraints such as the lashing wires and shrouding used on turbine blades, and of coupling between the torsional and flexural vibrations. In this paper, however, the effects of coupling between the torsional and flexural vibrations will not be considered. The basic differential equation will be solved by the tabular method due to Holzer.