Free Transverse Vibration of Mindlin Annular and Circular Plate with General Boundary Conditions

  • Qingjun Hao
  • Zhaobo ChenEmail author
  • Wenjie Zhai
Conference paper
Part of the Mechanisms and Machine Science book series (Mechan. Machine Science, volume 75)


In this paper, free vibration of Mindlin annular plate with arbitrary boundary conditions is investigated by an accurate solution method. The effects of the restrained stiffness of the springs on modal properties are incorporated in the current framework. The displacement components can be expressed as Fourier cosine series plus auxiliary polynomial functions, which aim to eliminate the discontinuities of displacement and its derivatives at both ends and to enhance the convergence of the results effectively. The current solution method can be used to general boundary conditions without changing the calculate method, which is different from the most existing studies. Numerical examples were presented and discussed for several different geometric and material properties as well as distinct boundary conditions.


Mindlin annular plate Fourier–Ritz method General boundary conditions Free vibration 



The authors would like to thank the anonymous reviewers for their very valuable comments.

Conflict of Interest

None declared.


  1. 1.
    Reissner, E.: The effect of transverse shear deformation on the bending of elastic plates. J. Appl. Mech. 12(2), 69–72 (1945)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Mindlin, R.D.: Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates. J. Appl. Mech. 18(1), 31–38 (1951)zbMATHGoogle Scholar
  3. 3.
    Reddy, J.N.: A simple higher-order theory for laminated composite plates. J. Appl. Mech. 51, 745–752 (1984)CrossRefGoogle Scholar
  4. 4.
    Librescu, L.: On the theory of anisotropic elastic shells and plates. Int. J. Solids Struct. 3(1), 53–68 (1967)CrossRefGoogle Scholar
  5. 5.
    Soni, S.R., Amba Rao, C.L.: On radially symmetric vibrations of orthotropic non uniform disks including shear deformation. J. Sound Vib. 42, 57–63 (1975)Google Scholar
  6. 6.
    Chakraverty, S., Petyt, M.: Natural frequencies for free vibration of nonhomogeneous elliptic and circular plates using two-dimensional orthogonal polynomials. Appl. Math. Model. 21(7), 399–417 (1997)CrossRefGoogle Scholar
  7. 7.
    Chakraverty, S., Bhat, R.B., Stiharu, I.: Free vibration of annular elliptic plates using boundary characteristic orthogonal polynomials as shape functions in the Rayleigh–Ritz method. J. Sound Vib. 241(3), 524–539 (2001)CrossRefGoogle Scholar
  8. 8.
    Wu, T.Y., Liu, G.R.: Free vibration analysis of circular plates with variable thickness by the generalized differential quadrature rule. Int. J. Solids Struct. 38, 7967–7980 (2001)CrossRefGoogle Scholar
  9. 9.
    Wu, T.Y., Wang, Y.Y., Liu, G.R.: Free vibration analysis of circular plates using generalized differential quadrature rule. Comput. Methods Appl. Mech. Eng. 191, 5365–5380 (2002)CrossRefGoogle Scholar
  10. 10.
    Civalek, O., Gurses, M.: Free vibration of annular Mindlin plates with free inner edge via discrete singular convolution method. Arab. J. Sci. Eng. 34 (2009)Google Scholar
  11. 11.
    Shi, X., Shi, D., Qin, Z., Wang, Q.: In-plane vibration analysis of annular plates with arbitrary boundary conditions. Sci. World J. 2014 (2014)Google Scholar
  12. 12.
    Shi, X., Shi, D., Li, W.L., et al.: A unified method for free vibration analysis of circular, annular and sector plates with arbitrary boundary conditions. J. Vib. Control 22(2), 442–456 (2016)CrossRefGoogle Scholar
  13. 13.
    Hao, Q., Zhai, W., Chen, Z.: Free vibration of connected double-beam system with general boundary conditions by a modified Fourier–Ritz method. Arch. Appl. Mech. 88(5), 741–754 (2018)CrossRefGoogle Scholar
  14. 14.
    Chen, J.T., Chen, I.L., Chen, K.H., Lee, Y.T., Yeh, Y.T.: A meshless method for the free vibration analysis of circular and rectangular clamped plates using radial basis function. Eng. Anal. Boundary Elem. 28, 535–545 (2004)CrossRefGoogle Scholar
  15. 15.
    Zhou, Z., et al.: Natural vibration of circular and annular thin plates by Hamiltonian approach. J. Sound Vib. 330(5), 1005–1017 (2011)CrossRefGoogle Scholar
  16. 16.
    Vera, S.A., Sanchez, M.D., Laura, P.A.A., et al.: Transverse vibrations of circular, annular plates with several combinations of boundary conditions. J. Sound Vib. 213, 757–762 (1998)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.School of Mechatronics EngineeringHarbin Institute of TechnologyHarbinPeople’s Republic of China

Personalised recommendations