Electrical analogs for plate equations and their applications in mechanical vibration suppression by P.Z.T. actuators
Before the beginning of digital-computers era, a lot of research was carried out in order to find electric circuits the governing equations of which were analogous to the ones of mechanical systems. The mentioned circuits were called ectro-mechanical analogs. They were used as analogical computers for the simulation and the design of mechanical systems. The actual technological development of piezoelectric actuators, which are devices able to efficiently transduce energy between the electrical and mechanical form, induced us to consider again those electro-mechanical analogs in order to create coupled piezo-electro-mechanical systems. Our idea is that the coupling between electro-mechanical phenomena is maximum when the propagation of both electrical and mechanical waves are governed by similar equations. Let us remark that because of the propagating mechanical wave-speed is much lower than the light-speed for every material, it is not possible to search for an efficient electro-mechanical coupling inside a piezoelectric continuum. Consequently circuits able to support the propagation of electric-potential waves have been considered. In this work, the equations for the elastica and for the plate are considered and their circuital analogs are derived using their finite-difference approximation. Afterwards, the coupling between the two structures is modelled considering piezoelectric actuators uniformly distributed on the mechanical system and connected to the nodes of the electric circuit. Then the electro-mechanical coupled equations are derived, and an analytical solution is found for a particular case. Finally some numerical simulations showing the efficiently energy exchange is presented.