Three-dimensional vibration of a buckled elastica supported by spherical hinges

In this paper we study the effect of extensibility on the vibration characteristics of a spatial buckled rod (elastica) under edge thrust and supported by spherical hinges at the ends. The nonlinear equations of motion are written within the framework of director theory. The elastica in question admits only plane deformations. There are three types of vibration modes, in-plane, symmetric out-of-plane, and anti-symmetric (twisting) out-of-plane. Most of the natural frequencies decrease as the end shortening increases, except the first in-plane mode without a nodal point. This mode is inadmissible in an inextensible elastica. This may be considered a flaw in the inextensible elastica model when dynamic behavior is concerned. In the limit case when the static deformation is small, a small-deformation theory taking into account axial extensibility is formulated and compared with the elastica model. The natural frequency of the first in-plane mode derived from small-deformation theory agrees very well with the one calculated from the extensible elastica model in the post-buckling range. However, all others are found to be independent of end shortening. This obviously unreasonable result is due to the limitation of small-deformation theory.