Application of active piezoelectric patches in controlling the dynamic response of a thin rectangular plate under a moving mass

The governing differential equation of motion for an undamped thin rectangular plate with a number of bonded piezoelectric patches on its surface and arbitrary boundary conditions is derived using Hamilton’s principle. A moving mass traveling on an arbitrary trajectory acts as an external excitation for the system. The effect of the moving mass inertia is considered using all the out-of-plane translational acceleration components. The method of eigenfunction expansion is used to transform the equation of motion into a number of coupled ordinary differential equations. A classical closed-loop optimal control algorithm is employed to suppress the dynamic response of the system, determining the required voltage of each piezoactuator at any time interval. In a numerical example for a simply supported square plate under two different loading paths, the effect of the mass velocity and mass weight of the moving load on the dynamic behavior of the uncontrolled system is investigated. The results show that, depending on the path of the moving mass, the inertia effect is very important, causing different behaviors of the system. In addition, the number of vibrational modes involved in determining the dynamic response of the system is crucial. The inertia effect is more important for an orbiting mass loading case compared to the case in which the moving mass is traversing the plate on a straight line. A number of equally spaced piezo patches are used on the lower surface of the plate to control the displacement of the center point of the plate. The implemented control mechanism proves to be very efficient in suppressing the near resonant dynamic response of the system, requiring fairly low levels of voltage for each patch. Increasing the area of the employed piezo patches would reduce the required maximum voltage for controlling the response of the system.