Planar Vibration of an Elastica Arch: Theory and Experiment

Abstract

This investigation examines the planar, linear vibration of a deep arch that is described by a simply supported elastica. The arch is formed from an elastic rod that buckles nonlinearly under the action of a large, steady end-load. A theoretical model is proposed that governs the planar response of the rod about a generally curved, pre-stressed equilibrium. The model utilizes a geometrically nonlinear rod theory to describe the planar bending and extension of the rod centerline. The equations of motion are linearized about an elastica equilibrium and numerical solutions for free vibration are determined using a variational formulation of the associated eigenvalue problem. Natural frequencies and mode shapes are computed over a large range of centrally and eccentrically applied end-loads. Results from an experimental modal test provide support for the model.