Piezoelectric transducers are well-established devices in vibration control, energy harvesting, sensing and structural health monitoring. In this contribution the main focus is laid on vibration control, in particular on shape control. This notion describes a special displacement tracking technique where one intends to completely annihilate (or strongly reduce) structural deflections. An excellent literature overview on shape control is presented by Irschik [
7]. Shape control has been first introduced by Hafka and Adelman [
8] who derived an analytical computation of the temperature field of a supporting structure to reduce distortions of space structures from their original shape. Vibrations of rotary wings were attenuated in Nitzsche and Breitbach [
9], where smart devices were used to construct geometric modal filters that are able to control some critical modes. Austin et al. [
10] designed adaptive wings, which included actuators to minimize the aerodynamic performance. Agrawal and Treanor [
11] minimized a quadratic cost function of an unloaded shear-rigid cantilever, which contains the error between desired and achieved static deflection, to obtain the best locations for piezoceramics actuators. Shirazi et al. [
12] designed a robust controller for tracking the tip deflection of a piezoelectric cantilever. A reduced order model was assumed for the cantilever via Galerkin’s method and the effects of the uncontrolled modes on the overall performance were studied. A planar hexagonal reflector was controlled by 30 PZT-actuators in Song et al. [
13]. The proposed shape control method was also experimentally verified and considered model errors and uncertainties. The TLBO-method (teacher-learning based optimization) was implemented in the software code of COMSOL Multiphysics, which was used to model a piezoelectric bimorph, to nearly avoid structural deflections, see Sumit et al. [
14]. In a further study [
15] several optimization techniques were compared (simulated annealing (SA), genetic algorithm (GA), particle swarm optimization (PSO) and TLBO) for the same benchmark problem, finding that the SA method gave better values of the objective function and converged faster. Locations and actuation voltages of a cantilever plate are optimized by genetic algorithm by Wang et al. [
16]. The deflections of laminated composite hybrid plate under thermo-electro-mechanical loads were investigated in Gohari et al. [
17]. Assuming Kirchhoff kinematics, the plate was then controlled based on a double integral multivariable transformation method. Results were compared and verified against those available in the literature. Bendine and Wankhade [
18] used a first order shear deformation theory and computed the required voltage via an adopted genetic algorithm to maintain a desired shape of the beam varying both loading and boundary conditions. If a piezoelectric layer or a multilayer beam with attached piezoelectric transducers is modeled within the framework of Bernoulli-Euler one achieves total elimination of the transverse deflection if the quasi-static bending moment of the smart control devices is equal, but sign-reversed to the quasi-static moment caused by the external forces, see Irschik et al. [
19]. This result was experimentally verified by Nader [
20]: relative deflections of a support-excited beam with attached piezoelectric patches are attenuated if the patches are properly placed and voltage-actuated. Irschik and Pichler et al. [
21] showed that the distribution of the actuating stress has to be equal to the statically admissible stress to avoid vibrations of linear elastic structures. It is interesting to note that for statically indeterminate beams no deformation will occur if the electrodes of the piezoelectric layers are properly shaped in case of electrical actuation. Similarly, one may measure no voltage signal in case of sensing: this holds e.g. for a clamped-clamped beam with a constant distribution of the piezoelectric layer, see Hubbard and Burke [
22]. These distributions are nil-potent shape functions, see Irschik et al. [
23,
24] and [
21], i.e. no matter how the voltage control signal is chosen, deformations will not occur. For thicker beams certain subsections are controlled by Krommer [
25]. In [
26] Krommer and Irschik consider both shear and extension actuation mechanisms and analytically showed that perfect annihilation of vibrations is possible. The role of the electrical boundary conditions at the vertical faces is studied by Krommer and Irschik [
27], where a weak form of the charge equation of electrostatics is solved to find a solution for the electric potential distribution. In a previous work of them, they investigated if either the assumption of a vanishing electric displacement field or electric field in axial direction (i.e.
\(D_x = 0\) or
\(E_x = 0\) holds) gives a better correlation to finite element results, Krommer and Irschik [
28]. For Reissner-Mindlin plate considering piezo- and pyroelectricity, the direct piezoelectric and the pyrelectric effects are incorporated in terms of effective stiffness parameters, see Krommer and Irschik [
39]. It was found that the eigenfrequencies are higher for vanishing in-plane components of the electric displacement field than those for vanishing in-plane components of the electric field. If the electrodes cannot be considered as perfect in a sense that the equipotential area condition is fulfilled, one uses the notion resistive or moderately conductive electrodes, see Buchberger and Schoeftner [
29] and Schoeftner et al. [
30,
31]. If the electrode resistivity varies along the beam length in a certain manner the bending vibrations may be also attenuated, see [
32]. Instead of properly tuning the electrode resistivity, it is much easier to attach patches at certain locations onto the elastic substrate which are then connected via resistances causing a desired voltage drop. Hence the theoretical framework and the experimental realization for this kind of vibration control technique is demonstrated in [
33]. For monofrequent harmonic excitations shape control is also possible if the width of the layers is proportional to the quasi-static bending moment distribution and if the attached inductive electric circuit is driven in resonance. This principle is similar to a perfectly tuned vibration absorber connecting the mechanical and the electrical domain. Boley’s iterative method is extended by Schoeftner and Benjeddou [
38] in when the compatibility equations and the charge equation of electrostatics are solved simultaneously for each layer. As representative example a simply-supported piezoelectric bimorph is investigated with sinusoidal voltage and distributed loads. It is shown that the error between analytical results and the solution from two-dimensional plane stress results decreases with each iteration.
This contribution presents results for shape control of laminated piezoelectric beams by considering shear rigidity. The necessary electric voltage actuation is calculated by minimizing the mean square error of the deflection. As suggested by Krommer and Irschik in [
26] results are compared to two-dimensional finite element results under plane stress assumptions. From a theoretical point of view concerning piezoelectric modeling and control aspects, the method presented here is a particular case of the results in [
26] neglecting the shear piezoelectric mode
\(\tilde{e}_{15}\) and the in-plane electric displacement and electric field, i.e.
\(D_x \approx E_x \approx 0\). The equivalent single layer theory for piezoelectric composites is assumed because the elastic moduli of the layered beam are of the same order of magnitude. First the differential equations of a piezoelectric beam are derived based on Timoshenko’s kinematic assumption and the constitutive relations for PZT-5A. The indirect and the direct piezoelectric effect are considered by approximately solving the charge equation of electrostatics to find relations between electric displacement, electric field and the displacement field. In a next step one computes the electric field from the constitutive relations. Consequently, the electric potentials of the piezoelectric layers are obtained by integration. Then the shear force and the bending moment are calculated. The latter has not only contributions from the mechanical degrees of freedom, but also from the electric field. Inserting into Newton’s law one finds the beam differential equations. In order to solve the shape control problem, a performance criterion is set up: the goal is to calculate the voltage actuation in order to minimize the square of the residual deflection. Finally several examples are presented in order to verify the proposed shape control method with finite element calculations, where the focus is laid on thick or moderately thick beams, i.e. the thickness-to-length ratios are
\(\lambda = 1/5\) and
\(\lambda = 1/10\). Comparing the results to two-dimensional finite element results in MATLAB one observes that the consideration of the shear influence becomes more and more important if the thickness of the elastic core, which is usually a non-piezoelectric material, is much larger than the thickness of the piezoelectric layers.