Non-linear non-planar oscillations of a cantilever beam under lateral base excitations

Abstract

The non-planar responses of a cantilevered beam subject to lateral harmonic baseexcitation are investigated using two non-linear coupled integrodifferential equations of motion. The equations contain cubic non-linearities due to curvature and inertia. Two uniform beams with rectangular cross sections are considered: one has an aspect ratio near unity, and the other has an aspect ratio near 6.27. A combination of the Galerkin procedure and the method of multiple scales is used to construct a first-order uniform expansion for the case of a one-to-one internal resonance and a primary resonance. The results show that the non-linear geometric terms are important for the responses of low-frequency modes because they produce hardening spring effects. On the other hand, the non-linear inertia terms dominate the responses of high-frequency modes. We also obtain quantitative results for non-planar motions and investigate their dynamic behavior. For different range of parameters, the non-planar motions can be steady whirling motions, whirling motions of the beating type, or chaotic motions. Furthermore, we investigate the effects of damping.