On the resonant nonlinear traveling waves in a thin rotating ring

Large amplitude, traveling wave motion of an inextensible, linearly elastic, rotating ring is analyzed. Equations governing the planar dynamics of a thin rod, curved in its undeformed state and moving in a horizontal reference frame which rotates about a fixed axis, are obtained via Hamilton's extended principle. The equations are specialized to study the behavior of a rotating circular ring and approximate solutions are obtained near resonance utilizing a perturbation analysis. Undamped free and viscously damped forced traveling wave motion is considered. The motion is found to consist of a forward and a backward traveling wave which may be coupled due to the non-linear terms present in the equations of motion