Effect of vibration control on the frequencies of a cantilever beam with non-collocated piezo sensor and actuator

Displacement feedback control of a cantilever beam is studied using non-collocated piezoelectric patch sensors and actuators. The solution to the problem is obtained using two different methods, one analytical and another numerical. The analytical method involves an integral equation formulation of the problem where the eigensolutions of the integral equation are shown to be the eigensolutions of the governing differential equation of motion of the smart beam. This approach eliminates the difficulties associated with discontinuities caused by patch sensors and actuators which introduce Heaviside functions and derivatives of the Heaviside functions into the differential equation formulation. The numerical method of solution uses a finite-element model of the controlled beam with modified beam element mass and stiffness matrices to account for the piezo patches and the control effect. The control circuit consists of a piezoceramic and polyvinylidene fluoride sensor patch and a lead zirconium titanate actuator patch. The mass and stiffness of the piezoceramic actuator patch are taken into account in the mass and stiffness calculations. Numerical examples with non-collocated sensor and actuator patches are presented and the first three natural frequencies are given using the integral equation and the finite-element methods. The results of these methods match very closely which provides a verification of the results.