The classical Euler beam equation is modified by introducing a hysteresis damping term which is proportional to the strain but which is in phase with the strain velocity. General solutions are obtained for the cantilever, clamped, and pin‐supported beams whose supports are moved by a prescribed function of time. For sinusoidal vibration of the support, the analysis of a cantilever beam is compared with experiment and with the corresponding solution in which the damping is proportional to the rate of strain as formulated by Sezawa. The hysteresis type damping gives results closer to the experimental values than Sezawa's method. The shear stress and displacement in a cantilever beam whose support falls on an elastic platform are investigated. The analysis is applicable to clamped and pin‐supported beams by substituting the appropriate resonant frequencies and eigenfunctions. Simple formulas for dynamic load factors at the resonant frequencies are found by comparing the maximum displacement or shear stress of the beam under sinusoidal vibration of its supports with the corresponding displacement or shear stress for a static loading under a centrifugal acceleration of the same amplitude as the sinusoidal acceleration. Rice's theory of random noise is applied to the response of the beam on a support vibrating in a random manner.