Vibration damping in beams via piezo actuation using optimal boundary control

Open-loop optimal control theory is formulated and applied to damp out the vibrations of a beam where the control action is implemented using piezoceramic actuators. The optimal control law is derived by using a maximum principle developed for one-dimensional structures where the control function appears in the boundary conditions in the form of a moment. The objective function is speci®ed as a weighted quadratic functional of the displacement and velocity which is to be minimized at a speci®ed terminal time using continuous piezoelectric actuators. The expenditure of control force is included in the objective functional as a penalty term. The explicit solution of the problem is developed for cantilever beams using eigenfunction expansions of the state and adjoint variables. The e€ectiveness of the proposed control mechanism is assessed by plotting the displacement and velocity against time. It is shown that both quantities are damped out substantially as compared to an uncontrolled beam and this reduction depends on the magnitude of the control moment. The capabilities of piezo actuation are also investigated by means of control moment versus piezo and beam thickness graphs which indicate the required minimum level of voltage to be applied on piezo materials in relation to geometric dimensions of the combined active/passive structure. The graphs show the magnitude of the control moment which can be achieved using piezoceramics in terms of problem inputs such as voltage, piezo and beam thicknesses