Distributed sensing and control of a simply supported plate

A mathematical model of the transverse vibration of a plate, with collocated piezoelectric actuators/sensors, to be used for free and forced vibration control purposes is presented. For the analysis of plate flexural vibration, the “Finite Element”, and “Galerkin” procedures are described. Hamilton's Principle is used to derive the equations of the motion of a Mindlin plate model, which takes into account the transverse shear effects. The coupled differential equations of the motion are solved using the finite element method. Test results show that the application of this theory along with an appropriate choice of basis functions allows the analysis of a wide range of thicknesses. The formulated approach avoids shear locking and provides excellent accuracy and convergence characteristics. The performance of the plate with/without attached layers of piezoelectric material is compared using a finite element model. The comparison between the Galerkin formulation and the finite element method shows the accuracy of each method within the desired plate thickness range. The eigenfunctions of a Poisson-Kirchoff plate model have been used as basis functions in the Galerkin formulation. The state transition matrix of a Poisson-Kirchoff plate model is computed.; To ensure the highest energy efficiency, controllability, and observability of the system a process of optimization is described. Separation of variables, a double Fourier expansion, coupled with Navier's method are used in the Poisson-Kirchoff plate model to find the optimal location of the sensors/actuators.; A quadratic control objective is defined as a measure of system performance. The control objective is composed of those error variables that are important to the design and they are used to approximate the high order plant system by a lower order model. To guarantee that the error in the reduced model is smaller than the desired error, a minimum required number of modes in the plate model is specified. The application of the formulated techniques is illustrated via numerical example.; The energy based control strategy “Linear Coupling Control (LCC)” is used for controlling free and forced vibrations in a simply supported plate. The control strategy is implemented by coupling a virtual second order linear system (controller) to an oscillatory plant and creating an energy exchange between the two systems. The energy transfer phenomenon between two oscillators is maximized by coupling the appropriate states of the plant and the controller. Once energy is transferred from the plant to the controller it is dissipated via linear damping. To provide a basis for comparing the results of the proposed control algorithm and the conventional control methods, linear quadratic optimal control was implemented as a control algorithm for the plate. The experimental verification of these approaches to vibration suppression is presented for the cases where it is undesirable to add actuators with significant mass and stiffness to the structure. The virtual controller is implemented on a personal computer. The experimental results have shown the controller's potential to eliminate free and forced vibration. In free vibration the controller reduces the oscillation time to one fourth. For forced vibration, the energy coupling control law provides an over 80% reduction of the steady state amplitude.