Vibration of a string wrapping and unwrapping around an obstacle

The problem of a string vibrating against a smooth obstacle is investigated in this paper. The obstacle is located at one of the boundaries and the string is assumed to wrap and unwrap around the obstacle during vibration. The wrapping of the obstacle is modeled by a series of perfectly inelastic collisions between the obstacle and adjacent segments of the string and unwrapping is assumed to be energy conserving. The geometry of the string is determined iteratively starting from an initial configuration where the string is vibrating in a single mode and is not in contact with the obstacle. The obstacle can be regarded as a passive mechanism for vibration suppression in which the energy lost during each cycle of oscillation depends on the energy content of the string at the beginning of the cycle. Numerical simulation results are provided for the string vibrating in different modes for circular- and elliptic-shaped obstacles. The loss of energy is found to be greater for higher modes of oscillation and for obstacles that induce greater length of wrapping.