Analytic and Finite Element Solutions for Active Displacement Feedback Control using PZT Patches

An analytical solution to the equation of motion of a beam controlled with piezoceramic (PZT lead zirconate titanate) sensor and actuator patches is proposed. The contribution of the mass and stiffness of the piezoceramic patches to the piezo structure are taken into account. The equation of motion for the controlled structure includes Heaviside functions and derivatives of the Heaviside function due to finite patch lengths making the equation of motion difficult to solve using conventional methods. In the present study, an integral equation is introduced where the eigensolutions of the integral equation are eigensolutions of the differential equation of motion for the controlled beam. A finite element model of the controlled beam is also formulated. The model contains modified beam element mass and stiffness matrices to account for the piezo patches and control effect. Two case studies are presented and the first three natural frequencies and mode shapes are found using the integral equation and finite element solutions. The results from the integral equation solution match very closely the results from the finite element solution.