Accurate Nonlinear Equations and a Perturbation Solution for the Free Vibrations of a Circular Elastic Ring

A set of geometrically nonlinear equations is derived for the plane motion of an elastically homogeneous circular ring. For vibrations that are primarily flexural, extensional strain, transverse shearing strain, and rotary inertia terms are shown to be negligible and the equations are reduced to two-coupled cubic equations for the axial force and rotation at a cross section. For free vibrations whose linear part is a single term harmonic in space and time, a perturbation solution shows that the frequency always initially decreases with amplitude (softening nonlinearity). The frequency amplitude relation agrees exactly with recent, independent work by Maewal and Nachbar, but differs substantially for low wave numbers from earlier work of others who used shallow shell-type approximations.