Vibration and stability of a long heavy elastica on rigid foundation

Abstract

A long heavy elastica resting on a horizontal rigid foundation and under the action of a pair of equal and opposite compressive force tends to buckle away from the foundation. The deformation may be symmetric or asymmetric. Self-contact may also occur. Previous investigation demonstrated through experimental observation that the symmetric non-self-contact deformation becomes unstable when a symmetry breaking bifurcation occurs. After the symmetry-breaking bifurcation, the elastica branches to a non-self-contact asymmetric deformation when the end shortening continues to increase. In this paper, we present a vibration method which is capable of predicting the stability of an elastica with self-contact. The main difficulty of the vibration analysis is the variation of contact points between the elastica and the foundation, and variation of the self-contact point during vibration. After transforming the governing equations from the original Lagrangian description to an Eulerian one, the equations are linearized near the neighborhood of the static deformation. From the calculated natural frequencies, one can determine the stability of the long heavy elastica.