Modeling and Related Results for Current-actuated Piezoelectric Beams by including Magnetic Effects∗

Piezo-electric material can be controlled with current as the electrical variable, instead of voltage. The main purpose of this paper is to derive the governing equations for a currentcontrolled piezo-electric beam and to investigate stabilizability. Besides the consideration of current control, there are several new aspects to the model here. Most significantly, magnetic effects are included. For the electromagnetic part of the model, electrical potential and magnetic vector potential are chosen to be quadratic-through thickness to include the induced effects of the electromagnetic field. Two sets of decoupled system of partial differential equations are obtained; one for stretching motion and another one for bending motion. Hamilton’s principle is used to derive a boundary value problem that models a single piezo-electric beam actuated by a charge (or current) source at the electrodes. Current or charge controllers at the electrodes can only control the stretching motion. Attention is therefore focused on control of the stretching equations in this paper. It is shown that the Lagrangian of the beam is invariant under certain transformations. A Coulomb-type gauge condition which is widely used in the electromagnetic theory is used here. This gauge condition decouples the electrical potential equation from the equations of the magnetic potential. A semigroup approach is used to prove that the Cauchy problem is well-posed. Unlike the voltage or charge actuation, a bounded control operator in the natural energy space is obtained in the current actuation case. The paper concludes with analysis of stabilizability and comparison with other actuation approaches and models.