Whirling of a base-excited cantilever beam

Abstract

The paper describes the response of a thin elastic cantilever beam excited by a known harmonic displacement at its base. The principal moments of the beam cross-sectional area are equal and the base excitation is in one of the principal directions. If the excitation frequency is slightly less than the resonant frequency of one of the bending modes, planar motions of the beam are unstable and an out-of-plane component of displacement develops. With this motion, the free end of the beam moves in an ellipse while the base motion is planar. Increasing the frequency to a value just above resonance causes the motion to become planar again. This whirling is a result of nonlinear coupling between the in-plane and out-of-plane displacements through the axial inertia. The paper presents the governing equations, develops the frequency–amplitude relations for the whirling response and examines the stability of the response. It is shown all whirling motions are stable and the frequency range for which whirling occurs increases with the level of excitation. Also it is shown for a given level of excitation, the frequency range broadens with increasing mode number.