Vibration control of a string using a scabbard-like actuator

The dynamics of a vibrating string subjected to a constraint at one boundary is investigated in this paper. The constraint is applied by a scabbard that moves a small distance along the mean position of the string. The scabbard is moved instantaneously such that the position and the velocity of the string outside the scabbard is unaffected immediately after application of the constraint, whereas the length of the string covered by the scabbard is brought to rest. The constraint is removed by moving the scabbard back to its original position and the change in energy of the string is investigated for different values of scabbard travel distance and time of application of the constraint. Analytical and numerical simulation results are first provided for the string vibrating in the first mode, and then for a more general case where the string has arbitrary initial conditions. The results show that the energy of the string can increase or decrease depending on the time of application of the constraint for a given distance of travel of the scabbard. This provides the opportunity for active control of vibration of the string through direct physical interaction, using the scabbard as an actuator. A simple feedback control strategy is proposed and numerical simulation results are presented. These results indicate that although removal of the constraint does not change the energy of the string, the effectiveness of the control strategy depends on the time of removal of the constraint.