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 piezocer amic actuators. The optimal control law is derived by using a maximum princ iple developed for one-dimensional structures where the control function ap pears in the boundary conditions in the form of a moment. The objective fun ction is specified as a weighted quadratic functional of the displacement a nd velocity which is to be minimized at a specified terminal time using con tinuous piezoelectric actuators. The expenditure of control force is includ ed in the objective functional as a penalty term. The explicit solution of the problem is developed for cantilever beams using eigenfunction expansion s of the state and adjoint variables. The effectiveness of the proposed con trol mechanism is assessed by plotting the displacement and velocity agains t time. It is shown that both quantities are damped out substantially as co mpared to an uncontrolled beam and this reduction depends on the magnitude of the control moment. The capabilities of piezo actuation are also investi gated by means of control moment versus piezo and beam thickness graphs whi ch indicate the required minimum level of voltage to be applied on piezo ma terials 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, p iezo and beam thicknesses. (C) 2000 Elsevier Science Ltd. All rights reserv ed.