Integral Equation Approach for Beams with Multi-Patch Piezo Sensors and Actuators

An integral equation approach is developed for the solution of an adaptive beam problem with the beam undergoing free vibrations. The beam is controlled by a closed-loop system consisting of multiple patches of sensors and actuators which are bonded to the bottom and top surfaces of the beam. The coupling between sensors and actuators can be chosen arbitrarily and the control is exercised by displacement feedback. The integral equation governing the vibrations of the beam/piezo-patch system is derived by converting the corresponding differential equation, which is non-standard as a result of the discontinuities caused by the piezo patches. The mathematical formulation involves Heaviside and distribution functions in a differential setting, while the integral equation avoids these difficulties and is expressed in terms of a smooth kernel which is developed using a Green's function approach based on suitable patch functions. The numerical results are obtained for various locations of patches, gain factors and coupling configurations, and the first three eigenfrequencies and eigenfunctions of the beam/piezo system are given in table and graph forms.