Non-planar motions of a string vibrating against a smooth unilateral obstacle

Abstract

Earlier theoretical studies on string vibrations against smooth unilateral obstacle assumed planar motions. The present paper analyzes the non planar motions of a string vibrating against a smooth unilateral curved obstacle. In particular, we study the effect of obstacle in determining different types of non planar motions of a string. The tension is assumed to be variable along the length due to the stretching of the string, which introduces nonlinear coupling between the perpendicular modes. The system of equations has been discretized by assuming functional form of the modes which satisfies all the geometrical boundary conditions. The mathematical model has been numerically investigated for different initial conditions and chosen value of obstacle parameters. The trajectory of a typical point on the string is oscillating ellipse and it transforms into rotating ellipses with time varying major and minor axis for larger amplitudes.