Piezoelectric materials are being used increasingly in vibration and shape control. This work concerns a closed-loop displacement feedback control of a thin rectangular plate reinforced with a sensor patch and an actuator patch. The sensor senses the bending strains of the plate and generates a signal which is amplified and sent to the actuator. The actuator then generates a corresponding signal which causes the plate to bend in the opposite direction. The optimal shapes of the patches (under various constraints) are determined to maximize the minimum vibration frequency.; We present an integral equation approach to convert the partial differential equation into a certain integral equation to which a kernel can be determined explicitly and consequently express the kernel in terms of the shapes of the patches by converting the domain integrals over the patches into the corresponding line integrals over their boundaries using Green's theorem. Then optimizing the shapes of the patches amounts to optimizing the parameterizations of their boundaries with the admissible set composed of all the reasonable parameters.; This method is also valid in optimizing other indices of performance of the vibrating plate and has various applications in elasticity.
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