Dynamic Green´s function method applied to vibrations of piezoelectric shells

In order to efficiently calculate thermoelastic deformations and also Green´s functions Maysel´s formula is a proper tool in many applications. It can be embedded within multiple field analyses when considering the isothermal background. By generalizing Maysel´s formula, it becomes possible to include both the piezoelectric effects and inertia Either some or each of the layers of a composite shell made of piezoelectric materials behave as distributed actuators so that the shell becomes an intelligent or smart structure. To extinguish vibrations say of a shell, an electrical field with a proper control must be applied. The construction of dynamic Green´s functions makes the mechanical part of the control problem readily available. A version of that multiple field analysis is presented with the best auxiliary problem constructed in the background. In such a manner, all kinds of symmetries present in the actual coupled problem are preserved, and, for the sake of simplifying the auxiliary problem, constraints on the shell are relaxed. The three-dimensional formulation is at first specialized for the case of thin layered shells of revolution, and applied to an infinite circular cylindrical shell with two ring supports, the piezoelectric influence function being derived. Solutions of the (relaxed) auxiliary problem are more readily available in the frequency space. As a consequence, the time convolution in the dynamic version of Maysel´s formula is replaced by a Fourier integral, which is eventually subjected to the fast Fourier transformation