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.
Highlights
Strings are the simplest continuous systems which exhibit vibrations
The real complication arises in situations where either the transverse motions are large enough that the assumption of small vibrations no longer holds or the supports are not ideal giving rise to non-trivial boundary conditions
We have studied the non-planar motions of string vibrating against a smooth unilateral obstacle
Summary
Strings are the simplest continuous systems which exhibit vibrations. The applications of strings are quite diverse, for example, in cranes to lift loads and transmit power, in ropeways to suspend heavy weights, in elevators as a support system, in a classical Atwood machine as a classroom demonstration to verify laws of motion, etc. The real complication arises in situations where either the transverse motions are large enough that the assumption of small vibrations no longer holds or the supports are not ideal giving rise to non-trivial boundary conditions. Under these situations, we can not get exact solutions to the string vibrations problem. The peculiar feature of the Indian stringed musical instruments is the presence of a finite-sized curved bridge. Sundry theoretical studies have been reported in the literature[4,5,6] All these studies have been restricted to planar motions of vibrating string. We analyze the dynamics of a string, explore the possibility of different types of motion and numerically investigate the results
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