Free Vibration of Prestressed Plates and Shallow Shells with Piezoelectric Elements

Abstract

In this paper, the possibility of using piezoelectric elements to create a prestress state in a plate and a shallow shell leading to a change in the natural frequencies is demonstrated numerically and experimentally. Strains in the thin-walled structure are determined using the non-linear relations based on the Reissner–Mindlin theory, which are linearized with respect to the state with a small deviation from the initial equilibrium caused by the inverse piezoeffect. A mathematical formulation of the dynamics problem is based on the variational principle of virtual displacements, taking into account the prestress state. The solution was developed by the finite element method. The validity of the obtained results is confirmed by comparing the natural frequencies and mode shapes of a rectangular plate with a piezoelectric element obtained numerically and experimentally at different values of electrical voltage. A series of calculations has been carried out to analyze the influence of the elasticity modulus, length, thickness, curvature of the shallow shell and voltage on the spectrum of natural frequencies of vibration. It was found that the effectiveness of using this technique decreases with increasing stiffness of the structure as a whole.