Boundary Control of Beams

Abstract

In this chapter, distributed vibration control of a laminated elastic beam is studied. It is assumed that two thin layers of a mono—axially oriented piezoelectric material, i.e., d₃₁ only and d₃₂ = 0, are respectively bonded on the top and bottom surfaces of the elastic beam. One layer serves as a distributed sensor and the other as a distributed actuator. The effective axis of the piezoelectric layers is aligned with the x—axis to ensure the maximum piezoelectric effects in sensor and actuator applications. It is intended to use the distributed sensor signal as a feedback reference in a closed—loop feedback control system. Two control algorithms, namely a displacement feedback and a velocity feedback are implemented and their control effectiveness evaluated. In the displacement feedback, the distributed sensor signal is amplified and fed back to the distributed piezoelectric actuator. (Note that the sensor signal is proportional to strains which can be ultimately expressed in terms of displacements as presented in Chapter 4. Thus, the conventional “displacement” feedback is used. In general, the dominating vibration component contributes to a higher strain level and consequently to a higher percentage of the total output signal.) In the velocity feedback, this sensor output is differentiated, amplified, and then fed back into the distributed actuator. (Note that the signal used in the velocity feedback is actually a strain/unit-time. Since the strain is ultimately expressed in terms of displacements, the time derivative of the displacement is the velocity.