Double Legendre polynomial quadrature-free method for axisymmetric vibration of functionally graded piezoelectric circular plates

A double Legendre polynomial quadrature-free method was presented to analyze axisymmetric free vibration problem for axially polarized Functionally Graded Piezoelectric circular plates. The present analytical method directly incorporated the boundary conditions into the vibration control equation. At the same time, the mechanical displacements and the electrical potential were defined as the form of double Legendre polynomials series. Then, the solution of eigenvalue was used to obtain the resonant/anti-resonant frequencies. In view of including lots of time-consuming integration calculations, the quadrature-free process was deduced to improve the computational efficiency. The valid of the presented method was verified by the comparison with the existing literature and the commercial software COMSOL. Furthermore, the effects of the material gradient index, radius-thickness ratio on the frequencies, and electro-mechanical coupling efficient were studied. The results showed that the frequency-radius product tends to be stable with the increase of the radius-thickness ratio, and the increase of gradient index led to the increase of the frequency-radius product.