Piezo Control of Free Vibrations of Damped Beams with Time Delay in the Sensor Feedback

The effect of the time delay between the sensor output and actuation is studied on the displacement and velocity feedback control of a cantilever beam. The system consists of a piezoelectric sensor and actuator pair bonded to an Euler-Bernoulli beam to form an actively controlled laminated structure. The effects of Kelvin-Voigt damping are also included in the problem. The piezoelectric actuators and sensors are taken as patches of partial length which leads to a differential equation formulation with discontinuities. The numerical solution is obtained by replacing the original formulation with an integral equation which is solved by expanding the state function in terms of the eigenfunctions of the freely vibrating beam. This approach leads to an infinite set of linear equations which can be solved with a high degree of accuracy by computing the characteristic equation using a small number of terms. Numerical results are given to analyze the control effectiveness in terms of changes in the natural frequencies for various gains, damping coefficients and time delays.