DYNAMICS AND ENERGETICS OF A STRING VIBRATING AGAINST AN OBSTACLEPLACED AT ONE BOUNDARY

摘要

In this paper we investigate the vibration of a string with a rigid obstacle placed at one of its boundaries. During vibration, a portion of the string wraps and unwraps around the obstacle. The impact between the string and the obstacle during wrapping is assumed to be inelastic. The length of the string that wraps around the obstacle, is discretized into a finite number of segments for the purpose of analysis. These segments sequentially collide with the obstacle starting from when the string makes first contact with the obstacle till it comes to rest. During wrapping, the energy of the string is dissipated through impact but during unwrapping the energy is conserved. The geometry of the string at any instant of time is determined from the boundary conditions associated with wrapping and unwrapping of the string. A general solution for vibration against convex obstacles of arbitrary geometry was analysed and numerical simulation results are presented for elliptic- and circular-shaped obstacles with different orientations and for different modes of string vibration. The results show that an obstacle at the boundary can be used as a passive mechanism for vibration suppression. The energy dissipated is found to be greater for higher modes of vibration and for obstacle geometries that result in greater length of wrapping.