Vibrations of Ballooning Elastic Strings

Abstract

This paper investigates the nonlinear dynamic response of a linearly elastic string fixed at one boundary and undergoing constant speed circular motion at the other boundary. The response divides into nonlinear steady-state ballooning that is fixed relative to a rotating coordinate system and linearized vibration about the steady state. Single-loop balloons have high tension and purely imaginary eigenvalues. The single-loop vibration frequencies generally decrease with increasing balloon length. Highly extensible strings whirl in first and higher modes with forward whirling modes having lower frequencies. Axially stiff strings exhibit whirling only in higher modes. If the nondimensional string stiffness is larger than 1000, then the inextensible steady-state solutions and the lowest six vibration frequencies match the extensible results to within three percent. One-and-a-half loop balloons are divergently unstable. Long and/or sufficiently extensible strings form low-tension double-loop balloons. Inextensible double balloons are coupled mode flutter unstable. The steady-state balloons, steady-state eyelet tension, and balloon stability are experimentally verified.