KEYWORDS: Control systems, Systems modeling, Fourier transforms, Smart materials, Electronic components, System integration, Commercial off the shelf technology, Modeling, Silicon, Ceramics
In this work we obtain some boundary observability inequalities with Dirichlet boundary conditions on displacement vector and an excat controllability of a coupled electro-mechanical system with coefficients depending on both space and time variables. Let us mention that some results on modeling of such coupled systems have been derived by Senouci-Bereksi and Hoffmann for piezoelectric plates and electrical networks and junctions and Senouci-Bereksi and Lenczner for piezoelectric shells coupled to some active electronic networks under different configurations. Some results on observability and exact controllability for general elastic systems with constant coefficients not depending in time, have been obtained by Alabau and Komornik. To our knowledge this work seems to be new and very interesting for researchers working on smart materials and systems investigations.
In this paper, we propose a new approach to the modeling of specific electronic networks and junctions. This approach is based on some results obtained by Chechkin, Jikov, Lukkassen and Piatniski (C4) in the theory of periodic wire networks and junctions, which are based on singular measures on these structures. Using this approach, we generalize some results obtained by Canon and Lenczner (C2) and Lenczner (L2) concerning electronic circuits models. As application, the model of piezoelectric thin plate, obtained by Hoffmann and Botkin (H1, H4) is used, where the piezoelectric patches are chosen as junction domains.
In this paper, we first formulate the equations of the electrical network. This network is composed of resistors, current sources, voltage sources and voltage to voltage amplifiers. It has an arbitrary shape. It is connected to piezoelectrical patches which are distributed transducers on the elastic thin shell. A 2D Reisner-Mindlin model of piezoelectrical thin shell is obtained under coupled to electronic network. A second model is derived under co- localization assumption. The two models are written under the form of a global variational formulation. Sufficient conditions on the network are stated in order to insure existence and uniqueness of the solution of the coupling. They are based on a graph interpretation of abstract conditions stated in the framework of functional analysis. Finally, numerical simulations of the Reisner-Mindlin shell in vibrations are presented for a particular choice of electric circuit. This last is designed in order to act as a stabilizer of the shell vibrations.
In this paper we state the homogenized model of a periodic electro-piezo-mechanical thin shells. The electrical network is including resistive devices, tension sources, current sources and voltage to voltage amplifiers. It is distributed evenly on the surface of the shell. The model derivation is based on a two-scale convergence which is based on the two- scale transform introduced by T. Arbogast, J. Douglas and U. Hornung. The present work is based on some results of two- scale convergence of gradients which leads to the same results than the usual two-scale convergence introduced by G. Allaire. We apply these results and those introduced in the framework of M. Lenczner and G. Senouci-Bereksi for the electrical network, to derive the two-dimensional electro-piezo- mechanical thin shell model applying Reisner-Mindlin kinematic.
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