Study of optimal locations of piezo-patch actuators and sensors on a cantilever beam for maximum frequency gaps

Active vibration control is implemented using multiple piezoelectric actuators and sensors bonded to the top and bottom surfaces of a cantilever beam. The control is exercised using closed-loop displacement feedback. The objective of the study is to determine the optimal locations of patch actuators and sensors such that the frequency gap between higher frequencies of the beam is maximized. Maximizing the frequency gaps is useful in those cases where the excitation frequency can,be placed in between two higher order frequencies to avoid the resonance. In these cases the design requirement is to maximize the difference between the two higher order frequencies such as between the first and second frequencies or between the second and third frequencies, etc. In the present study the frequency gaps between the higher order frequencies are investigated with respect to actuator and sensor locations with a view towards determining the optimal locations for largest frequency gaps. The differential equation governing the vibrations of a beam/piezo patch system is solved using an integral equation approach. The equivalent integral equation formulation of the problem avoids the discontinuities which arise due to partial length piezo patches. The solution is approximated using the eigenfunctions of the freely vibrating structure which leads to a system of algebraic equations. The numerical results are given for various patch combinations and the optimal locations of the actuators and the sensors are determined. It is observed that the optimal locations of the piezo patches depend on the specific frequency gap as well as the patch configurations.