Planar free vibrations of off-axis piezoelectric laminates

The frequencies of free vibration and the through-thickness mode shapes for off-axis piezoelectric laminates in cylindrical bending are computed using an exact solution and an approximate discrete-layer theory. In the exact solution, the in-plane displacement and electrostatic potential functions for the case of simple support are selected to reduce the equations of motion and the charge equation into a system of ordinary differential equations. The unknown field distributions are found to be a function of eight unknown constants per layer and the resonant frequency. These are calculated using an iterative procedure. The discrete-layer approach uses one-dimensional approximation functions to represent the through-thickness behavior of the fields combined with analytic approximations in the axial direction. Example problems are studied for several geometric, material, and support conditions. For the case of simple support, excellent agreement is found between the two formulations.