A perturbation method for finite element modeling of piezoelectric vibrations in quartz plate resonators

Abstract

When the piezoelectric stiffening matrix is added to the mechanical stiffness matrix of a finite element model, its sparse matrix structure is destroyed. A direct consequence of this loss in sparseness is a significant rise in memory and computational time requirements for the model. For weakly coupled piezoelectric materials, the matrix sparseness can be retained by a perturbation method which separates the mechanical eigenvalue solution from its piezoelectric effects. Using this approach, a perturbation and finite element scheme for weakly coupled piezoelectric vibrations in quartz plate resonators has been developed. Finite-element matrix equations were derived specifically for third-overtone thickness-shear, SC-cut quartz plate resonators with electrode platings. High-frequency piezoelectric plate equations were employed in the formulation of the finite element matrix equation. Results from the perturbation method compared well with the direct solution of the piezoelectric finite element equations. This method will result in significant savings in the computer memory and computational time. Resonance and antiresonance frequencies of a certain mode could be calculated easily by using the same eigenpair from the purely mechanical stiffness matrix. Numerical results for straight crested waves in a third overtone SC-cut quartz strip with and without electrodes are presented. The steady-state response to an electrical excitation is calculated.

Parallel finite element analysis of high frequency vibrations of quartz crystal resonators on Linux cluster

“…A detailed introduction of the finite element implementation of the Mindlin plate theory has been given before in a series of papers [3,4,[10][11][12]14] . We are concentrating on the parallel computing with sparse matrix handling function in this paper.…”

“…Earlier finite element analysis of quartz crystal resonators have been done with the Mindlin plate equations, which are developed for the analysis of the high frequency vibrations of quartz crystal plates, by Lee, Zee and Brebbia [1] . Following this pioneer work, extensive research with two-dimensional piezoelectric plate equations has been done by Yong and his collaborators [2][3][4][5][6][7] and the three-dimensional piezoelectricity by other researchers [8][9][10][11][12][13][14] . The two-dimensional equations, or the Mindlin plate theory [15] , have been the first choice because of the reduced problem size in the finite element implementation.…”

3C-5 The Analysis of Thickness-Shear Vibrations of the Third-Order Overtone Mode of Quartz Crystal Plates with Mindlin Plate Theory

“…This means we need to continue our efforts on improving the higher-order equations and making necessary validation through detailed comparison of the dispersion relations. For the vibrations in the vicinity of the fundamental thickness-shear vibrations, both analytical solutions with straight-crested waves [5][6][7] and finite element solutions [9][10][11][12][13] have been obtained with Mindlin plate theory. There are obvious advantages in utilizing the Mindlin plate theory for finite element analysis because the resulted linear systems can be much smaller.…”

Finite element and experimental results of transverse vibration for GT-cut and SC-cut quartz plates with fixed and free boundary conditions and multiple methods of excitation

“…This has led to rapid growth in the study of SC-cut quartz. Yong and Zhang [11] used a perturbation method for the modeling of vibrations of SC-cut quartz plates. Yong and Zhang [12] used two-dimensional piezoelectric plate equations to provide an excellent approximation of vibration behavior of the third overtone of thickness-shear in square and circular SC-cut quartz with partial electrode.…”