摘要

The problem of parametric vibrations of elastic three-layer shells composed of middle orthotropic dielectric or metal layer and two piezoelectric layers is studied. On the basis of the mechanical Kirchoff - Love hypothesis and adequate assumptions for an electrical field the constitutive equations for forces and moments are obtained for varying electrode positions, type of polarization and electrical boundary conditions. It is shown how nonlinear and linearized equations describing the parametric vibrations of the arbitrary shaped shells can be obtained if the constitutive equations, universal equations of motion, kinematic equations and boundary conditions are used. The linearized equations describe a region of dynamic instability (RDI). On the boundary of RDI the harmonic motion occurs. This gives an opportunity to reduce the problem of investigations of the principal RDI to solving the eigenvalue problems and the problem of static stability. Method of finite elements is developed to solve these problems. The problem of parametric vibrations of a three-layer cylindrical piezoelectric panel is considered in detail. The analytical solution of the problem is obtained for the case of simply supported edges. Correlation of an analytical and finite-element solutions demonstrates high accuracy of the first. The problem of parametric vibrations under harmonic mechanical load is solved for the open-circuited and short-circuited conditions. The essential influence of the electric boundary conditions on the size of RDI that can be used for control of the parametric vibrations of the shells is shown. The finite-element solution of the problem of parametric vibrations of cylindrical piezoelectric panel with clamped edges is obtained. The numerical results point to essential influence of mechanical boundary conditions on the size and position of RDI.