Theory of the Vibrating String as an Angular Motion Sensor

Abstract

The important deviations from ideal “gyroscopic” behavior of the plane of a vibrating string, driven by longitudinal end motion, are obtained theoretically. The amplitude of lateral string oscillation is obtained, and perturbation methods are applied to determine conditions under which these oscillations begin and/or continue. Elastic asymmetry is shown to cause the string elements to move out of the nominal plane of vibration into thin elliptical orbits, balanced in a “field” which deviates from linearity because of tension variation. A general expression for orbital precession in a nearly-linear central field is derived and applied to string orbits to obtain the differential equation of motion of the nominal string plane. Finally, motion of the string plane caused by asymmetrical damping is derived.