A Perturbation Method for Finite Element Modeling of Piezoelectric Vibrations in Quartz Plate Resonators

Y. K. Yong, Zhen Zhang

Research output: Contribution to journalArticlepeer-review

19 Scopus citations


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 the 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. 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, previously derived by Lee, Syngellakis, and Hou [1], were employed in the formulation of the finite element matrix equation. The equations may be used for modeling third harmonic overtone of thickness shear vibrations. Results from the perturbation method for SC-Cut quartz plates compared well with the direct solution of the piezoelectric finite element equations. This method will result in significant savings in computer memory and computational time. Resonance and anti-resonance frequencies of a certain mode could be calculated easily by using the same eigen-pair 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 steadystate response to an electrical excitation is calculated. The accuracy of the perturbation method is dependent on the magnitude of piezoelectric coupling constant in the mode of vibration. For example, an investigation on the first few modes of vibration in a cube made of lithium niobate, which has high piezoelectric coupling constants, yielded divergent results for the perturbation method. Hence, the method should not be used for materials with high piezoelectric coupling constants.

Original languageEnglish (US)
Pages (from-to)551-562
Number of pages12
JournalIEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control
Issue number5
StatePublished - Sep 1993

All Science Journal Classification (ASJC) codes

  • Instrumentation
  • Acoustics and Ultrasonics
  • Electrical and Electronic Engineering


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