Static and dynamic shape control of laminated beams with resistively interconnected piezoelectric patches

In this contribution an extended Bernoulli-Euler theory for a laminated beam hosting several resistively interconnected piezoelectric patches is presented. Based on this theory it is shown how to calculate the optimal reference voltage level and to design the attached resistive network, which connects the electrodes of the piezoelectric layers, if one intends to follow a certain trajectory of the lateral deformation (shape control) of a tip-loaded beam. If one tries to annihilate the harmonic vibrations along the beam axis, we show that the design criterion derived in the static case yields that vibrations are attenuated as long as an electrical time constant is small compared to the time constant of the harmonic excitation. This parameter involves the number of the piezoelectric patches, the piezoelectric capacitance and the resistances of the electric circuit. The proposed shape control method is also verified by a three-dimensional finite element calculation in ANSYS.