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This paper is concerned with the dynamics of hyperelastic solids and structures. We seek for a smart control actuation that produces a desired (prescribed) displacement field in the presence of transient imposed forces. In the literature, this problem is denoted as displacement tracking, or also as shape morphing problem. One talks about shape control, when the displacements to be tracked do vanish. In the present paper, it is assumed that the control actuation is provided by imposed eigenstrains, e.g., by the electric field in piezoelectric actuators, or by thermal actuators, or via analogous physical effects, such as magneto-striction or pre-stress. Structures with a controlled eigenstrain-type actuation belong to the class of smart structures. The action of the eigenstrains can be conveniently characterized by actuation stresses. Our theoretical derivations are performed in the framework of the theory of small incremental dynamic deformations superimposed upon a statically pre-deformed configuration of a hyperelastic solid or structure. We particularly ask for a distribution of incremental actuation stresses, such that the incremental displacements follow exactly a prescribed trajectory field, despite the imposed incremental forces are present. An exact solution of this problem is presented under the assumption that the actuation stresses can be tailored freely and applied everywhere within the body. Extending a Neumann-type solution strategy, it is shown that the actuation stresses due to the distributed control eigenstrains must satisfy certain quasi-static equilibrium conditions, where auxiliary body-forces and auxiliary surface tractions are to be taken into account. The latter auxiliary loading can be directly computed from the imposed forces and from the desired displacement field to be tracked. Hence, despite the problem is a dynamic one, a straightforward computation of proper actuator distributions can be obtained in the framework of quasi-static equilibrium conditions. Necessary conditions for the functioning of this concept are presented. Particularly, it must be required that the intermediate configuration is infinitesimally superstable. Previous results of our group for the case of shape control and displacement tracking in linear elastic structures are included as special cases. The high potential of the solution is demonstrated via Finite Element computations for an irregularly shaped four-corner plate in a state of plain strain.

SUMMARY This paper is concerned with active control of torsional vibrations in laminated rods by piezoelectric shear sensors and actuators. A piezoelectric layer can be used as a sensor by utilizing the direct piezoelectric effect, measuring either the charge on short-circuited electrodes or the voltage on open electrodes. On the other hand, the converse piezoelectric effect enables actuation by applying an electric potential difference to the electrodes of a piezoelectric layer. The sensor and actuator equations are formulated in the framework of an extended Saint-Venant torsion theory considering additional cross-sectional warping due to piezoelectric eigenstrains. A solution of the shape control problem for torsional vibrations is presented, i.e. the necessary distribution of actuation strains in order to completely compensate vibrations caused by external excitations. For the case, in which the external excitations are not known exactly, a feedback control solution is presented using one piezoelectric layer as a sensor and a second one as an actuator. For the examples of a rectangular and a circular cross-section, the theoretical results are validated by three-dimensional finite element computations, showing a very good coincidence. Copyright © 2011 John Wiley & Sons, Ltd.

In this paper, the piezoelectric compensation of torsional vibrations in rods caused by external excitations is studied. As an illustrative example, a laminated rod containing piezoelectric shear actuators is assumed to be fixed at the one end, and the other end is subjected to a torsional couple; additionally, a distributed torsional couple per unit length is acting. In such a system, cross-sectional warping is known to be present. The consideration of piezoelectric eigenstrains requires an extension of Saint Venant’s theory of torsion, which is achieved by introducing an additional warping function. Using D’Alembert’s principle, the boundary value problems for Saint Venant’s warping function, the additional warping function and the torsional angle are obtained. From the latter boundary value problems, the distribution of piezoelectric actuation is derived in order to completely compensate the external excitations, i.e. an analytical solution of the corresponding shape control problem is obtained. Finally, the results are verified by means of three-dimensional finite element computations.

In many applications, nonlinear beams undergoing bending, axial and shear deformation are important structural elements. In the present paper, a shear deformable beam finite element is presented for such applications. Since displacements and displacement gradients are chosen as the nodal degrees of freedom, an equivalent displacement and rotation interpolation is retrieved. The definition of strain energy is based on Reissner’s nonlinear rod theory with special strain measures for axial strain, shear strain and bending strain. Furthermore, a thickness deformation is introduced by adding an according term to the virtual work of internal forces. This underlying formulation is extended for piezo-electric actuation. The obtained beam finite elements are applied to a two-link robot with two flexible arms with tip masses. Distributed and concentrated masses cause flexural vibrations, which are compensated by means of piezo-electric actuators attached to the arms. A numerical example of a highly flexible robot with piezo-electric actuation and feedforward control is presented to show the applicability of the finite element.