Calculation of the Optimum Controls of Transient Vibrations of Smart Beams and Plates

This paper addresses the optimization problem of finding the control voltage applied to a single piezoelectric patch actuator or several such actuators which brings the transient displacements and velocities of all points of a beam or rectangular plate to zero with the least control effort. To solve the problem an efficient functional-analytic approach known as the method of l-moments is used. Making use of the Fourier series solutions of the vibration problem of smart beams and plates explicit equations are derived which allow the calculation of open-loop optimal controls for beams and plates. The approach is illustrated by plots of the calculated optimal controls for beams and plates as well as numerical simulations of the response in the presence of control. These results provide valuable information about the best control possibilities that can be achieved when using piezoelectric patch actuators to control transient vibrations of beams and plates.