The multi-field coupled vibration analysis of AT-cut quartz crystal resonators with parallelism error

Mengjie Li, Nian Li, Peng Li, Dianzi Liu, Iren E. Kuznetsova, Zhenghua Qian, Tingfeng Ma

Research output: Contribution to journalArticlepeer-review


During the fabrication of quartz crystal resonators (QCRs), parallelism error is inevitably generated, which is rarely investigated. In order to reveal the influence of parallelism error on the working performance of QCRs, the coupled vibration of a non-parallel AT-cut quartz crystal plate with electrodes is systematically studied from the views of theoretical analysis and numerical simulations. The two-dimensional thermal incremental field equations are solved for the free vibration analysis via the coefficient-formed partial differential equation module of the COMSOL Multiphysics software, from which the frequency spectra, frequency–temperature curves, and mode shapes are discussed in detail. Additionally, the piezoelectric module is utilized to obtain the admittance response under different conditions. It is demonstrated that the parallelism error reduces the resonant frequency. Additionally, symmetry broken by the non-parallelism increases the probability of activity dip and is harmful to QCR’s thermal stability. However, if the top and bottom surfaces incline synchronously in the same direction, the influence of parallelism error is tiny. The conclusions achieved are helpful for the QCR design, and the methodology presented can also be applied to other wave devices.
Original languageEnglish
Pages (from-to)349-360
Number of pages12
JournalActa Mechanica Solida Sinica
Issue number2
Early online date21 Mar 2023
Publication statusPublished - Apr 2023


  • Admittance
  • Mode coupling
  • Parallelism error
  • Quartz crystal resonators
  • Resonant frequency

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