Static and dynamic shape control of slender beams by piezoelectric actuation and resistive electrodes

In the present paper a concept of static and dynamic shape control for slender piezoelastic beams is addressed. The notion shape control means to find a weighting sequence for the piezoelectric control units, such that force-induced lateral deflections are completely annihilated along the beam axis. In our case this spatial control intensity for the piezoelectric transducers is realized with so-called resistive electrodes: by prescribing only the maximum and minimum voltage level, any desirable voltage distribution along the beam axis may be obtained. For the derivation of the shape control method an extended version of the Bernoulli–Euler beam theory is used, which unifies the kinematics of a slender piezoelectric beam and the Telegrapher’s equations for moderately conductive electrodes. The outcome is an actuator equation for the displacement, which is fourth order in space and second order in time, and a sensor equation, which is second order in space and first order in time (diffusion equation for the voltage). The developed shape control method is verified by several numerical examples in the static case as well as in the dynamic frequency regime for a clamped-free beam. We show that the presented method is perfectly suited for static loads, being able to cancel force-induced structural deformations along the beam axis, whereas the deflections in the higher frequency regime can be nearly canceled out. The method is efficient, as long as a non-dimensional constant, involving the resistance, the capacitance per unit length, the length of the electrodes and the excitation frequency, is small.