Vibration control of continuous systems using boundary constraint

Vibration suppression in continuous systems, namely strings and membranes, using boundary constraints was investigated. A problem of string vibration, where the string is subjected to a fixed boundary constraint, is studied first. The boundary constraint is in the form of a smooth obstacle and the string is assumed to wrap and unwrap around the obstacle during vibration. Assuming that wrapping of the string around the obstacle results in loss of kinetic energy due to inelastic collision and unwrapping conserves energy, the behavior of the string was investigated for different modes of oscillation and different obstacle shapes. The loss of energy is found to be greater for higher modes of oscillation and for obstacles that induce greater length of wrapping. Next, the vibration of the string subjected to a moving constraint near one boundary is investigated. The constraint is applied by a scabbard, which moves a small distance along the mean position of the string. 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. Unlike the fixed obstacle at the boundary, simulation results show that the energy of the string can increase or decrease depending on the time of application of the constraint. Based of this finding, a semi-active control strategy is proposed for vibration suppression. The control strategy is verified experimentally by substituting the scabbard-like actuator by a pair of solenoids. The experimental results show good match with results obtained through simulations. The control strategy developed for the string is extended to a circular membrane. It is assumed that the membrane is fixed at its outer boundary and a zero displacement circular areal constraint is sequentially applied and removed. To investigate the effect of constraint application and removal on the energy of the membrane, the dynamics of constrained membranes is investigated. For arbitrary size and location of the constraint, the orthogonality of distinct modes is mathematically established and a procedure for accurate computation of the eigenfrequencies and mode shapes is presented. Assuming that the constraint is applied and removed instantaneously at arbitrary time intervals, the change in total energy is investigated for different sizes and locations of the constraint and for different times of application of the constraint. Similar to the vibrating string with the scabbard-like actuator, the results show that energy of the membrane can decrease or increase depending on the time of application of the constraint. Three different semi-active control strategies are presented to the vibration suppression of circular membrane. These control strategies have the potential for use in large space structures where high dimensional tolerances are required.