Non-linear non-planar parametric responses of an inextensional beam

Abstract

The non-linear integro-differential equations of motion for an inextensional beam are used to investigate the planar and non-planar responses of a fixed-free beam to a principal parametric excitation. The beam is assumed to undergo flexure about two principal axes and torsion. 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. In both cases, the beam possesses a one-to-one internal resonance with one of the natural flexural frequencies in one plane being approximately equal to one of the natural flexural frequencies in the second plane. A combination of the Galerkin procedure and the method of multiple scales is used to construct a first-order uniform expansion for the interaction of the two resonant modes, yielding four first-order non-linear ordinary-differential equations governing the amplitudes and phases of the modes of vibration. The results show that the non-linear inertia terms produce a softening effect and play a significant role in the planar responses of high-frequency modes. On the other hand, the non-linear geometric terms produce a hardening effect and dominate the planar responses of low-frequency modes and non-planar responses for all modes. If the non-linear geometric terms were not included in the governing equations, then non-planar responses would not be predicted. For some range of parameters, Hopf bifurcations exist and the response consists of amplitude- and phase-modulated or chaotic motions.