Keywords: displacement tracking, shape control, eigenstrain, smart structures, piezoelectric actuation.

In the classical framework, structural mechanics deals with the computation of displacements and stresses, which are produced by a given set of loadings, such as imposed forces, imposed boundary conditions and imposed initial conditions. The solution of such direct problems has been brought to a high art, particularly in the linear theory of elastic structures. In the present paper, we treat a problem, which is inverse to the classical direct framework of structural mechanics, and which we call the displacement tracking problem: we seek for a control actuation, which, when being subjected together with some imposed transient force loading, produces a desired (prescribed) displacement field in a linear elastic structure. In our study, special emphasis is made on upon smart structures, i.e. on a control actuation by eigenstrains induced via piezoelectric or thermal actuators, or via analogous effects, such as magnetostriction or pre-stress.

The strategy followed in the present study is to seek for analytical solutions of the displacement tracking problem, assuming that no constraints are present with respect to the spatial and time-wise distribution of the control actuation, the latter being assumed to be distributed throughout the structure. A feed-forward control situation is treated, in which both, the structural parameters as well as the imposed loadings are known as functions of space and time. The corresponding analytical solutions for distributed actuators afterwards serve as a starting point for the design of sub-optimal discrete actuator nets. Our analytical solution, in which we derive in the framework of the linear theory of elastodynamics, can be summarized as follows: 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 loading and from the desired displacement field to be tracked. Hence, despite the fact that we are dealing with a dynamic problem, a straightforward computation of proper actuator distributions can be performed in the framework of quasi-static equilibrium conditions only.

After having presented a proof of the above stated analytical solution, we give a short review of the previous contributions of our group, mainly concerning structural shape control, i.e. the case of zero-displacement tracking [1,2,3]. We then present an exemplary numerical validation of our analytical solution for displacement tracking by means of finite element computations. The presented analytical solution finally is used as a suitable starting point for the actuator design, in the case where restrictions with respect to the spatial applicability and the intensity of actuation have to be taken into account. Here, we report on an experimental verification in the laboratory, using a framed structure, which is equipped with a net of piezoelectric actuator patches.

1

H. Irschik, U. Pichler, "An extension of Neumann´s method for shape control of force-induced elastic vibrations by eigenstrains", International Journal of Solids and Structures, 41, 871-884, 2004. doi:10.1016/j.ijsolstr.2003.09.023

2

M. Krommer, H. Irschik, "Sensor and Actuator Design for Displacement Control of Continuous Systems", Smart Structures and Systems 3, 147-172, 2007.

3

Ch. Zehetner, H. Irschik, "Displacement compensation of beam vibrations caused by rigid-body motions", Smart Materials and Structures, 14, 862-868, 2005. doi:10.1088/0964-1726/14/4/046

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