The Integrated Analysis of Vibrations of Quartz Crystal Plates ‎through Artificial Coupling Factors

Document Type : Research Paper


1 Piezoelectric Device Laboratory, School of Mechanical Engineering and Mechanics, Ningbo University, 818 FengHua Road, Ningbo, 315211 Zhejiang, China

2 Department of Architectural Engineering, Huzhou Vocational & Technical College, 299 Xuefu Road, Huzhou, 313000 Zhejiang, China

3 Department of Architectural Engineering, Ningbo Polytechnic, 1069 Xinda Road, Ningbo, 315800 Zhejiang, China‎


We introduce smaller artificial factors into the elastic constants matrix and manage to make Mindlin first-order plate theoryequations of motions coupled for a uniform and integrated analysis. The energy distributions of the five coupled modes are obtained and all the five vibration modes are identified through the energy calculation. This analytical approach based on artificial couplings of vibration modes suggests that all vibration modes of structural components can be analyzed through the same procedure and computer code if the right elastic constants are modified and the mode identification can be done with the energy method. This is a new technique to study multimode vibrations of structures in a broad frequency range with just one procedure and calculation tool for simplification.


Main Subjects

[1] Parthasarathy, S., Pancholy, M., and Chhapgar, A. F., Piezo-electric Oscillations of Quartz Plates at Even and Half-odd Harmonics, Nature, 171(4344), 1953, 216-217.
[2] Parthasarathy, S. and Singh, H., Amplitude of a Quartz Plate vibrating in Liquids, Nature, 181(4604), 1958, 260-260.
[3] Mindlin, R. D., Yang, J., (Ed.) An Introduction to the Mathematical Theory of Vibrations of Elastic Plates, World Scientific, Hackensack, NJ, 2006.
[4] Mindlin, R. D., Coupled Piezoelectric Vibrations of Quartz Plates, International Journal of Solids and Structures, 10(4), 1974, 453-459.
[5] Lee, P. C. Y., An Accurate Two-Dimensional Theory of Vibrations of Isotropic, elastic plates, Acta Mechanica Solida Sinica, 24(2), 2011, 125-134.
[6] Yong, Y. K., Wang, J., and Imai, T., On The Accuracy of Mindlin Plate Predictions for the Frequency-Temperature Behavior of Resonant Modes in AT- and SC-cut Quartz Plates, IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 46(1), 1999, 1-13.
[7] Wang, J. and Momosaki, E., The Piezoelectrically Forced Vibrations of AT-cut Quartz Strip Resonators, Journal of Applied Physics, 81(4), 1997, 1868-1876.
[8] Wang, J. and Zhao, W., The Determination of the Optimal Length of Crystal Blanks in Quartz Crystal Resonators, IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 52(11), 2005, 2023-2030.
[9] Lee, P. C. Y. and Yong, Y. K., Frequency-temperature Behavior of Thickness Vibrations of Doubly-Rotated Quartz Plates affected by Plate Dimensions and Orientations, Journal of Applied Physics, 60(7), 1986, 2327-2342.
[10] Mindlin, R. D. and Spencer, W. J., Anharmonic, Thickness-Twist Overtones of Thickness-Shear and Flexural Vibrations of Rectangular, AT-cut, Quartz Plates, Journal of The Acoustical Society of America, 42(6), 1967, 1268-1277.
[11] Hu, Y., Cui, Z., Jiang, S., and Yang, J., Thickness-shear Vibration of Circular Crystal Plate in Cylindrical Shell as Pressure Sensor, Applied Mathematics and Mechanics, 27(6), 2006, 749-755.
[12] Zhao, Z., Qian, Z., Wang, B., and Yang, J., Analysis of Thickness-shear and Thickness-twist Modes of AT-cut Quartz Acoustic Wave Resonator and Filter, Applied Mathematics and Mechanics, 36(12), 2015, 1527-1538.
[13] Wang, J., Wang, Y., Hu, W., Zhao, W., Du, J., and Huang, D., Parallel Finite Element Analysis of High Frequency Vibrations of Quartz Crystal Resonators on Linux Cluster, Acta Mechanica Solida Sinica, 21(6), 2008, 549-554.
[14] Yong, Y., Stewart, J. T., Detaint, J., Zarka, A., Capelle, B., and Zheng, Y., Thickness-shear Mode Shapes and Mass-Frequency Influence Surface of a Circular and Electroded AT-cut Quartz Resonator, IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 39(5), 1992, 609-617.
[15] Zargaripoor, A., Daneshmehr, A. R., and Nikkhah Bahrami, M., Study on Free Vibration and Wave Power Reflection in Functionally Graded Rectangular Plates using Wave Propagation Approach, Journal of Applied and Computational Mechanics, 5(1), 2019, 77-90.
[16] Shukla, S. K., Gupta, A., and Sivakugan, N., Analysis of Circular Elastic Plate Resting on Pasternak Foundation by Strain Energy Approach, Geotechnical and Geological Engineering, 29(4), 2011, 613-618.
[17] Lee, E. T. and Eun, H. C., Damage Identification Based on the Proper Orthogonal Mode Energy Curvature, Journal of Vibration and Acoustics, 137(4), 2015.
[18] Berger, J. B., Wadley, H. N. G., and McMeeking, R. M., Mechanical Metamaterials at the Theoretical Limit of Isotropic Elastic Stiffness, Nature, 543(7646), 2017, 533-537.
[19] Huang, Q., Wang, J., Wu, R., Xie, L., and Du, J., Using Kinetic and Strain Energies of Mindlin Plates for Vibration Mode Identification, Piezoelectricity, Acoustic Waves, and Device Applications, Chengdu, Sichuan, China, 471-475, 2017.
[20] Wang, J. and Yang, J., Higher-order Theories of Piezoelectric Plates and Applications, Applied Mechanics Reviews, 53(4), 2000, 87-99.
[21] Wang, J., Yong, Y. K., and Imai, T., Finite Element Analysis of the Piezoelectric Vibrations of Quartz Plate Resonators with Higher-order Plate Theory, International Journal of Solids and Structures, 36(15), 1999, 2303-2319.
[22] Wang, J., Yu, J. D., Yong, Y. K., and Imai, T., A Finite Element Analysis of Frequency–temperature Relations of AT-cut Quartz Crystal Resonators with Higher-order Mindlin Plate Theory, Acta Mechanica, 199(1-4), 2008, 117-130.
[23] Yasir, R. Y., Al-Muslimawi, A. H., and Jassim, B. K., Numerical Simulation of Non-Newtonian Inelastic Flows in Channel based on Artificial Compressibility Method, Journal of Applied and Computational Mechanics, 6(2), 2020, 271-283.
[24] Bechmann, R., Elastic and Piezoelectric Constants of Alpha-Quartz, Physical Review, 110(5), 1958, 1060-1061.
[25] Wang, J., Zhao, W., and Bian, T., A Fast Analysis of Vibrations of Crystal Plates for Resonator Design Applications, 2004 IEEE International Frequency Control Symposium and Exposition, Montreal, Canada, 596-599, 2004.
[26] Wu, R. et al., Free and Forced Vibrations of SC-cut Quartz Crystal Rectangular Plates with the First-order Mindlin Plate Equations, Ultrasonics, 73, 2016, 96-106.