Non-planar vibrations of a string in the presence of a boundary obstacle

Abstract

We analyze planar and non-planar motions of a string vibrating against a unilateral curved obstacle. Our model incorporates the change in tension due to stretching of the string, which introduces nonlinear coupling between motions in mutually perpendicular directions, as well as the wrapping nonlinearity due to the presence of the obstacle. The system of equations has been discretized by assuming functional form of the displacements which satisfies all the geometrical boundary conditions. This discretized system is then used to investigate the various motions possible both in the absence as well as the presence of the obstacle. In the absence of the obstacle, there are infinitely many planar and two non-planar motions viz. a circular trajectory and a precessing elliptical trajectory for a fixed magnitude of the disturbance. In contrast, the string has only one planar motion when the obstacle is present and two non-planar motions, either an oscillating orbit or a whirling orbit depending on the magnitude of the initial disturbance. To obtain the transition from oscillating to whirling orbits, we perform a stability analysis of the planar motion using Floquet theory. This analysis reveals that there exists a critical amplitude below which the planar motion is neutrally stable and the typical trajectories are ellipses with major and minor radii changing both in magnitude and direction. Beyond the critical amplitude, the planar motion is unstable and we get whirling trajectories which are precessing ellipses again with varying major and minor radii. We further study the effect of changing obstacle parameters on the critical amplitude, and obtain the stability boundaries in the space spanned by the obstacle parameters and the amplitude of the planar vibration. We obtain some interesting values of the obstacle parameters for which small and large amplitude planar motions are stable resulting in oscillating ellipses while motions with intermediate amplitudes are unstable giving precessing ellipses as the non-planar motion. We also find parameters for which the planar motion is always stable and hence, whirling motions are not possible. Finally, we consider non-planar vibrations with inclusion of several modes and observe more complicated non-planar motions due to modal interactions.