Output Annihilation and Optimal H 2—Control of Plate Vibrations by Piezoelectric Actuation

This paper is concerned with the piezoelectrically induced actuation and control of plates. Flexural vibrations of linear elastic laminated plates with layers made of ceramic or polymeric piezoelectric materials are considered, see (Rao et al, 1994). In the present formulation, the mechanical modelling of such a type of smart or intelligent plates is related to the fundamental duality between load-stresses, induced by imposed forces, and self-stresses due to eigenstrains (Reisner, 1931). The distributed actuating effect of piezoelectric layers is considered as a distribution of eigenstrains counteracting the load-stresses. Assuming perfect bond between the piezoelectric layers and their substrate, this eigenstrain problem is formulated within the context of the classical lamination theory of thin plates (Jones, 1975). A convolution integral is derived for the piezoelectrically induced vibrations by using dynamic bending moments due to single forces as kernel functions. This Maysel-type integral (Irschik et al, 1993), (Irschik et al, 1995) is accompanied by an analogous integral statement for the load-induced vibrations in the case of simply supported rectangular plates. Comparing these two formulations, an annihilation problem is solved. It is shown, what kind of distribution of the piezoelectric actuators is able to annihilate precisely a dynamic plate deflection generated by a prescribed distribution of imposed forces. Thus, the problem of distributed actuation by piezoelectric effects is concisely connected to the control by distributed forces. For the analogous problem of bending of beams see (Irschik et al, 1994).