Steam-turbine buckets are normally covered at the tips by a band. The band is applied in segments, each segment elastically joining together a given number of buckets. In studying vibrational behavior the buckets should be treated as banded groups rather than as individual cantilevers. A method of analysis, suitable for digital computation, is presented which gives all of the natural frequencies and mode shapes of a banded group of buckets subject to the usual limitations of the elementary beam theory. Axial and torsional motion as well as tangential is considered. Also a procedure is included for evaluating vibration amplitude and stress at resonance. Damping is assumed to be small and the energy input to the banded group from a prescribed form of nozzle stimulus is equated to the energy dissipated in damping to determine vibration-stress levels. Sample calculations have been made on the I.B.M. Electronic Data Processing Machine, Type 704, and the results of these calculations are given in a companion paper (1).