Active Control of a Beam for Generating Points of Zero Displacements and Zero Slopes

Piezoelectric transducers have been used extensively as the distributed actuators and sensors in active control of structural vibrations. Piezoelectric actuator/sensors are distributively bonded on or embedded in the host structure and have the inherent advantage of integrating over their surface area, which leads to potentially more robust implementations as compared to implementations that use shaker/accelerometers. For this reason piezoelectric actuator/sensors have attracted more and more attention in recent years. In this paper, a theoretical analysis is presented of the active control of a vibrating beam using collocated triangular and rectangular piezoelectric actuator/sensor pairs. The aim of this study is to generate points of zero displacements and zero slopes at any designated position. So the control systems impose a virtual clamped boundary condition at the control position on the beam, in which both displacement and slope are driven to zero. Two independent single-input single-output (SISO) control systems similar to direct velocity feedback (DVFB) are implemented, i.e. for the rectangular pair the voltage signal measured by a triangular piezoelectric sensor is electronically multiplied by a fixed gain and fed directly back to a collocated piezoelectric actuator. The triangular and rectangular piezoelectric actuator/ sensor pairs positioned at one end of the beam are used to measure and control the displacement and slope of the structure respectively. The active control systems are unconditionally stable for any type of primary disturbance acting on the structure due to the collocated actuator/sensors. It should be noted that the presented control strategy is different to DVFB. In DVFB, when the control gain is increased, the vibration energy of the beam is initially reduced at resonance frequencies because of the active damping effect. However this effect does not continue. When large control gains are implemented, the overall kinetic energy of the beam is increased to the same or even higher values than those of the beam without control systems because the vibration of the beam is rearranged into a new set of lightly damped resonance frequencies. Imposing a virtual clamped boundary condition at the control position is clearly more complicated than DVFB, because in addition to the zero displacement constraints, the zero slope constraints must also be satisfied. The proposed control system allows for certain points of the structure to remain stationary without using any rigid supports. Furthermore, such control systems have the potential to create a region of nearly zero vibration for any ‘excitation’ frequency. This means that no progressive waves or reflected waves exist in the designated region, thus significantly reducing the vibration level in that region of the beam. The control systems impose a virtual clamped boundary condition at the control position on the beam in which the displacement and slope are driven to zero. As a result, the vibration of the actively controlled beam can be described in terms of two beams clamped at the control position. A numerical analysis is then performed to verify the proposed control system. It is found that the new resonance frequencies and mode shapes seen in the simulations are consistent with the natural frequencies and natural modes of the controlled beam derived analytically. The capability of the proposed method for generating a zero-vibration region is also numerically demonstrated.