Rotating Beam with Tip Mass Analyzed by a Variational Method

Abstract

Hamilton's principle is used in the determination of a variational principle, which is stated in forms making it suitable for the case of: (1) moderately large displacements of a beam which is initially stressfree, or (2) small displacements of a beam having initial stresses. In the latter form, the principle is applied to the problem of determining the effect of tip mass and rotational speed on the natural frequencies of a rotating beam carrying a concentrated tip mass. The variational principle stated here allows for an independent selection of displacement and moment coordinate functions in the manner suggested by Reissner. It thus imposes fewer constraints than would be imposed by the Rayleigh-Ritz method of assuming displacement coordinate functions only. The effect of tip mass is shown to be much greater on the frequency of the second mode than on the fundamental mode.