High-frequency vibrations of circular and annular plates with the Mindlin plate theory

  • 45 Accesses


Circular and annular elastic plates have wide applications as essential elements in various engineering structures and products demanding accurate analysis of their vibrations. At higher frequencies, the analysis of vibrations needs appropriate equations, as shown by the Mindlin plate equations for rectangular plates with tailored applications for the analysis of quartz crystal resonators. Naturally, there are equally strong demands for the equations and applications in circular and annular plates with the consideration of higher-order vibration modes. By following the procedure established by Mindlin based on displacement expansion in the thickness coordinate, a set of higher-order equations of vibrations of circular and annular plates are derived and truncated for comparisons with classical and first-order plate equations of circular plates. By utilizing these equations, coupled thickness-shear and flexural vibrations of circular and annular plates are analyzed for vibration frequencies and mode shapes.

This is a preview of subscription content, log in to check access.

Access options

Buy single article

Instant unlimited access to the full article PDF.

US$ 39.95

Price includes VAT for USA

Subscribe to journal

Immediate online access to all issues from 2019. Subscription will auto renew annually.

US$ 199

This is the net price. Taxes to be calculated in checkout.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9


  1. 1.

    Reddy, J.N.: Theory and Analysis of Elastic Plates and Shells. CRC Press, Boca Raton (2006)

  2. 2.

    Reddy, J.N.: Mechanics of Laminated Composite Plates and Shells. CRC Press, Boca Raton (2003)

  3. 3.

    Carrera, E.: Historical review of Zig-Zag theories for multilayered plates and shells. Appl. Mech. Rev. 56, 287–308 (2003)

  4. 4.

    Ravari, M.R.K., Forouzan, M.R.: Frequency equations for the in-plane vibration of orthotropic circular annular plate. Arch. Appl. Mech. 81, 1307–1322 (2011)

  5. 5.

    Filippi, M., Carrera, E., Valvano, S.: Analysis of multilayered structures embedding viscoelastic layers by higher-order, and zig-zag plate elements. Compos. Part B Eng. 154, 77–89 (2018)

  6. 6.

    Fazzolari, F.A., Carrera, E.: Advanced variable kinematics Ritz and Galerkin formulations for accurate buckling and vibration analysis of anisotropic laminated composite plates. Compos. Struct. 94, 50–67 (2011)

  7. 7.

    Reddy, J.N., Wang, C.M., Kitipornchai, S.: Axisymmetric bending of functionally graded circular and annular plates. Eur. J. Mech. A Solid 18, 185–199 (1999)

  8. 8.

    Wang, J., Hashimoto, K.Y.: A two-dimensional theory for the analysis of surface acoustic waves in finite elastic solids. J. Sound. Vib. 295, 838–855 (2006)

  9. 9.

    Tahera, H.R.D., Omidia, M., Zadpoorb, A.A., Nikooyan, A.A.: Free vibration of circular and annular plates with variable thickness and different combinations of boundary conditions. J. Sound Vib. 296, 1084–1092 (2006)

  10. 10.

    Irie, T., Yamada, G., Takagi, K.: Natural frequencies of thick annular plates. J. Appl. Mech. 49, 633–638 (1982)

  11. 11.

    Fadaee, M.: A novel approach for free vibration of circular/annular sector plates using Reddy’s third order shear deformation theory. Meccanica 50, 2325–2351 (2015)

  12. 12.

    Hosseini-Hashemi, S., Es’Haghi, M., Taher, H.R.D., Fadaie, M.: Exact closed-form frequency equations for thick circular plates using a third-order shear deformation theory. J. Sound Vib. 329, 3382–3396 (2010)

  13. 13.

    Haghani, A., Mondali, M., Faghidian, S.A.: Linear and nonlinear flexural analysis of higher-order shear deformation laminated plates with circular delamination. Acta. Mech. 229, 1631–1648 (2018)

  14. 14.

    Thai, H.T., Park, T., Choi, D.H.: An efficient shear deformation theory for vibration of functionally graded plates. Arch. Appl. Mech. 83, 137–149 (2013)

  15. 15.

    Abolghasemi, S., Eipakchi, H.R., Shariati, M.: An analytical procedure to study vibration of rectangular plates under non-uniform in-plane loads based on first-order shear deformation theory. Arch. Appl. Mech. 86, 853867 (2016)

  16. 16.

    Mindlin, R.D., Deresiewicz, H.: Thickness-shear and flexural vibrations of a circular disk. J. Appl. Phys. 25, 1329–1332 (1954)

  17. 17.

    Deresiewicz, H., Mindlin, R.D.: Axially symmetric flexural vibrations of a circular disk. J. Appl. Mech. 22, 86–88 (1955)

  18. 18.

    Deresiewicz, H.: Symmetric flexural vibrations of a clamped circular disc. J. Appl. Mech. 23, 319 (1956)

  19. 19.

    Iyengar, K.T.S.R., Raman, P.V.: Free vibration of circular plates of arbitrary thickness. J. Acoust. Soc. Am. 64, 1088–1092 (1978)

  20. 20.

    Iyengar, K.T.S.R., Raman, P.V.: Axisymmetric free vibration of thick annular plates. J. Acoust. Soc. Am. 68, 1748–1749 (1980)

  21. 21.

    Celep, Z.: Free vibration of some circular plates of arbitrary thickness. J. Sound Vib. 70, 379–388 (1980)

  22. 22.

    Xiang, Y., Liew, K.M., Kitipornchai, S.: Vibration of circular and annular Mindlin plates with internal ring stiffeners. J. Acoust. Soc. Am. 110, 3696–3705 (1996)

  23. 23.

    He, H.J., Yang, J.S., Jiang, Q.: Thickness-shear and thickness-twist vibrations of circular AT-cut quartz resonators. Acta Mech. Solida Sin. 26, 245–254 (2013)

  24. 24.

    Liu, B., Xing, Y.F., Wang, W., Yu, W.D.: Thickness-shear vibration analysis of circular quartz crystal plates by a differential quadrature hierarchical finite element method. Compos. Struct. 131, 1073–1080 (2015)

  25. 25.

    Zhu, F., Wang, B., Dai, X.Y., Qian, Z.H., Iren, K., Vladimir, K., Huang, B.: Vibration optimization of an infinite circular AT-cut quartz resonator with ring electrodes. Appl. Math. Model. 72, 217–229 (2019)

  26. 26.

    Mindlin, R.D.: Influence of rotatory inertia and shear on flexural vibrations of isotropic, elastic plates. J. Appl. Mech. 18, 31–38 (1951)

  27. 27.

    Mindlin, R.D.: Forced thick-shear and flexural vibrations of piezoelectric crystal plates. J. Appl. Phys. 23, 83–88 (1952)

  28. 28.

    Tiersten, H.F., Mindlin, R.D.: Forced vibrations of piezoelectric crystal plates. Q. Appl. Math. 20, 107–119 (1962)

  29. 29.

    Mindlin, R.D.: High frequency vibrations of piezoelectric crystal plates. Int. J. Solids Struct. 8, 895–906 (1972)

  30. 30.

    Mindlin, R.D., (edited by Yang, J.S.): An Introduction to the Mathematical Theory of Vibrations of Elastic Plates. World Scientific, New Jersey (2007)

  31. 31.

    Senjanovic, I., Hadzic, N., Vladimir, N., Cho, D.S.: Natural vibrations of thick circular plate based on the modified Mindlin theory. Arch. Mech. 66, 389–409 (2014)

  32. 32.

    Wang, W.J., Wang, J., Chen, G.J., Ma, T.F., Du, J.K.: Mindlin plate equations for the thickness-shear vibrations of circular elastic plates. In: Proceedings of 2012 Symposium on Piezoelectricity, Acoustic Waves, and Device Applications, pp. 357–360 (2012)

  33. 33.

    Chen, H., Wang, J., Ma, T.F., Du, J.K.: An analysis of free vibrations of isotropic, circular plates with the Mindlin plate theory. In: Proceedings of 2014 Symposium on Piezoelectricity, Acoustic Waves, and Device Applications, pp. 397–400 (2014)

  34. 34.

    Wang, J., Chen, H., Ma, T.F., Du, J.K., Yi, L.J., Yong, Y.K.: Analysis of thickness-shear vibrations of an annular plate with the Mindlin plate equations. In: Proceedings of IEEE International Frequency Control Symposium on European Frequency and Time Forum (2015)

  35. 35.

    Wang, J., Yang, J.S.: Higher-order theories of piezoelectric plates and applications. Appl. Mech. Rev. 53, 87–99 (2000)

  36. 36.

    Liew, K.M., Wang, C.M., Xiang, Y., Kitipornchai, S.: Vibration of Mindlin Plates: Programming the p-version Ritz Method. Elsevier, Oxford (1998)

  37. 37.

    Lee, P.C.Y., Yu, J.D., Li, X.P., Shih, W.H.: Piezoelectric ceramic disks with thickness-graded materials properties. IEEE Trans. Ultrason. Ferroelectr. Freq. Control 46, 205–215 (1999)

Download references


This research is supported in part by the National Natural Science Foundation of China (Grant Nos. 11372145 and 11672142). Additional funding is from the National Key R&D Program of China (Grant No. 2017YFB1102900). It was also supported by the Research Project Foundation of Zhejiang Educational Department (Grant No. Y201636501). The authors also received financial support from the K. C. Wong Magna Fund established and administered by Ningbo University.

Author information

Correspondence to Ji Wang.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Chen, H., Wu, R., Xie, L. et al. High-frequency vibrations of circular and annular plates with the Mindlin plate theory. Arch Appl Mech (2020) doi:10.1007/s00419-019-01654-6

Download citation


  • Plate
  • Mindlin
  • Vibration
  • Circular
  • Frequency