The steady-state forced-vibration problem, for a beam or rotating shaft on damped, flexible end supports, is completely solved by the procedure described. The analysis is an extension of a Holzer-type iteration method applied to beams by Myklestad and others. Resonant frequencies of any mode, forced amplitudes, shears, and moments at any point on the beam, and phase effects introduced by the damping, are all evaluated. The damped problem is handled conveniently using complex notation with its vector interpretation. Vector diagrams, which give a physical idea of the vibration, are shown for the two numerical examples of the paper.