Nonlinear effects in a vibrating string (wire)

Abstract

Many unusual effects are observed in vibrating strings or wires as a consequence of tensional changes and longitudinal motion which yield coupled nonlinear equations for transverse motion in orthogonal directions. For a wire driven sinusoidally in a transverse direction, notable effects observed at driving frequencies near a natural resonance include the generation of motion perpendicular to the plane of the driving force, sudden jumps in amplitude and phase, whirling motion, and hysteresis [R. J. Hanson, J. M. Anderson, and H. K. Macomber, J. Acoust. Soc. Am. 96, 1549–1556 (1994)]. Further work demonstrates the generation of second, third, and higher harmonics with amplitudes sometimes comparable to the first. These harmonics, present in both the driving and perpendicular directions, contribute to a very complex total motion. All of the effects are very sensitive to asymmetries which produce slight differences in free vibration frequencies in two orthogonal directions. For the large amplitude free vibration of a plucked string, there are also significant consequences of the nonlinear effect of the tension changes. These effects for a plucked string will be related to several musical instruments. Phenomena for both the driven and plucked cases will be qualitatively demonstrated with video displays of the signals from optical motion detectors.