Finite element analysis of the piezoelectric vibrations of quartz plate resonators with higher!order plate theory

A _nite element formulation of the piezoelectric vibrations of quartz resonators based on Mindlin plate theory is derived[ The higher!order plate theory is employed for the development of a collection of suc! cessively higher!order plate elements which can be e}ective for a broad frequency range including the fundamental and overtone modes of thickness!shear vibrations[ The presence of electrodes is also considered for their mechanical e}ects[ The mechanical displacements and electric potential are combined into a generalized displacement _eld\ and the subsequent derivations are carried out with all the generalized equations[ Through the standard _nite element procedure\ the vibration frequency\ the vibration mode shapes and the electric potential distribution are obtained[ The frequency spectra are compared with some well!known experimental results with good agreement[ Our previous experience with _nite element analysis of high!frequency quartz plate vibrations leads us to believe that memory and computing time will always remain as key issues despite the advances in computers[ Hence\ the use of sparse matrix techniques\ e.cient eigenvalue solvers\ and other reduction procedures are explored[ Þ 0888 Elsevier Science Ltd[ All rights reserved[