A unified procedure for free transverse vibration of rectangular and annular sectorial plates

  • Siyuan Bao
  • Shuodao WangEmail author


A unified solution procedure applicable for analyzing the free transverse vibration of both rectangular and annular sectorial plates is presented in this study. For the annular sectorial plate, the basic theory is simplified by a variable transformation in the radial direction. The analogies of coordinate system, geometry and potential energy between the two different shapes are drawn and then unified in one framework by introducing the shape parameter. A generalized solving procedure for the two shapes becomes feasible under the unified framework. The solution adopts the spectro-geometric form that has the advantage of describing the geometry of structure by mathematical or design parameters. The assumed displacement field and its derivatives are continuous and smooth in the entire domain, thereby accelerating the convergence. In this study, the admissible functions are formulated in simple trigonometric forms of the mass and stiffness matrices for both rectangular and annular sectorial plates can be obtained, thereby making the method computationally effective, especially for analyzing annular sectorial plates. The generality, accuracy and efficiency of the unified approach for both shapes are fully demonstrated and verified through benchmark examples involving classical and elastic boundary conditions.


Rectangular plate Annular sectorial plate Transverse vibration Spectro-geometric method 



The authors acknowledge the support from the National Natural Science Foundation of China (Grant No. 11202146) and the Qinglan Project of JiangSu Province.


  1. 1.
    Reddy, J.N.: Theory and Analysis of Elastic Plates and Shells. CRC Press, Boca Raton (2007)Google Scholar
  2. 2.
    Xing, Y.F., Liu, B.: New exact solutions for free vibrations of rectangular thin plates by symplectic dual method. Acta Mech. Sin. 25, 265–70 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Li, R., Wang, P., Xue, R., Guo, X.: New analytic solutions for free vibration of rectangular thick plates with an edge free. Int. J. Mech. Sci. 131–132, 179–190 (2017)CrossRefGoogle Scholar
  4. 4.
    Li, R., Wang, P., Tian, Y., Wang, B., Li, G.: A unified analytic solution approach to static bending and free vibration problems of rectangular thin plates. Sci. Rep. 5 (2015).
  5. 5.
    Gorman, D.J.: Vibration Analysis of Plates by the Superposition Method. World Scientific, Singapore (1999)CrossRefzbMATHGoogle Scholar
  6. 6.
    Dozio, L., Ricciardi, M.: Free vibration analysis of ribbed plates by a combined analytical-numerical method. J. Sound Vib. 319, 681–97 (2009)CrossRefGoogle Scholar
  7. 7.
    Bert, C.W., Wang, X., Striz, A.G.: Differential quadrature for static and free vibration analyses of anisotropic plates. Int. J. Solids Struct. 30, 1737–44 (1993)CrossRefzbMATHGoogle Scholar
  8. 8.
    Du, H., Lim, M.K., Lin, R.M.: Application of differential quadrature to vibration analysis. J. Sound Vib. 181, 279–93 (1995)CrossRefzbMATHGoogle Scholar
  9. 9.
    Wei, G.W.: Vibration analysis by discrete singular convolution. J. Sound Vib. 244, 535–53 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Baltacıoglu, A.K., Akgoz, B., Civalek, O.: Nonlinear static response of laminated composite plates by discrete singular convolution method. Compos. Struct. 93, 153–161 (2010)CrossRefGoogle Scholar
  11. 11.
    Gürses, M., Civalek, O., Korkmaz, A., Ersoy, H.: Free vibration analysis of symmetric laminated skew plates by discrete singular convolution technique based on first-order shear deformation theory. Int. J. Numer. Methods Eng. 79(3), 290–313 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Baltacıoglu, A.K., Civalek, O., Akgoz, B., Demir, F.: Large deflection analysis of laminated composite plates resting on nonlinear elastic foundations by the methsod of discrete singular convolution. Int. J. Press. Vessels Pip. 88, 290–300 (2011)CrossRefGoogle Scholar
  13. 13.
    Belytschko, T.: Meshless methods: an overview and recent developments. Comput. Methods Appl. Mech. Eng. 139(2), 3–47 (1996)CrossRefzbMATHGoogle Scholar
  14. 14.
    Li, S., Liu, W.: Meshfree and particle method and their applications. Appl. Mech. Rev. 55(1), 1–34 (2002)CrossRefGoogle Scholar
  15. 15.
    Warburton, G.B.: The vibration of rectangular plates. Proc. Inst. Mech. Eng. Ser. A 168, 371–84 (1954)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Young, D.: Vibration of rectangular plates by the Ritz method. J. Appl. Mech. 17, 448–53 (1950)zbMATHGoogle Scholar
  17. 17.
    Leissa, A.W.: The historical bases of the Rayleigh and Ritz methods. J. Sound Vib. 287, 961–78 (2005)CrossRefGoogle Scholar
  18. 18.
    Bassily, S.F., Dickinson, S.M.: On the use of beam functions for problems of plates involving free edges. J. Appl. Mech. 42, 858–64 (1975)CrossRefzbMATHGoogle Scholar
  19. 19.
    Goncalves, P.J.P., Brennan, M.J., Elliott, S.J.: Numerical evaluation of high-order modes of vibration in uniform Euler–Bernoulli beams. J. Sound Vib. 301, 1035–1039 (2007)CrossRefzbMATHGoogle Scholar
  20. 20.
    Dickinson, S.M., Di Blasio, A.: On the use of orthogonal polynomials in the Rayleigh–Ritz method for the study of the flexural vibration and buckling of isotropic and orthotropic rectangular plates. J. Sound Vib. 108, 51–62 (1986)CrossRefzbMATHGoogle Scholar
  21. 21.
    Kim, C.S., Young, P.G., Dickinson, S.M.: On the flexural vibration of rectangular plates approached by using simple polynomials in the Rayleigh–Ritz method. J. Sound Vib. 143, 379–94 (1990)CrossRefGoogle Scholar
  22. 22.
    Zhou, D.: Natural frequencies of elastically restrained rectangular plates using a set of static beam functions in the Rayleigh–Ritz method. Comput. Struct. 57, 731–5 (1995)CrossRefzbMATHGoogle Scholar
  23. 23.
    Cheung, Y.K., Zhou, D.: Vibrations of rectangular plates with elastic intermediate line-supports and edge constraints. Thin-Walled Struct. 37, 305–31 (2000)CrossRefGoogle Scholar
  24. 24.
    Zhou, D.: Vibrations of point-supported rectangular plates with variable thickness using a set of static tapered beam functions. Int. J. Mech. Sci. 44, 149–64 (2002)CrossRefzbMATHGoogle Scholar
  25. 25.
    Dozio, L.: On the use of the trigonometric Ritz method for general vibration analysis of rectangular plates. Thin-Walled Struct. 49(1), 129–144 (2011). CrossRefGoogle Scholar
  26. 26.
    Li, W.L.: Free vibration of beams with general boundary conditions. J. Sound Vib. 237(4), 709–725 (2000)CrossRefGoogle Scholar
  27. 27.
    Li, W.L., Daniels, M.: A Fourier series method for the vibrations of elastically restrained plate arbitrarily loaded with springs and masses. J. Sound Vib. 252, 768–781 (2002)CrossRefGoogle Scholar
  28. 28.
    Li, W.L., Zhang, X., Du, J., Liu, Z.: An exact series solution for the transverse vibration of rectangular plates with general elastic boundary supports. J. Sound Vib. 321, 254–69 (2009)CrossRefGoogle Scholar
  29. 29.
    Shi, X.J., Shi, D.Y., Li, W.L., Wang, Q.S.: Free transverse vibrations of orthotropic thin rectangular plates with arbitrary elastic edge supports. J. Vibroeng. 16(1), 389–398 (2014)Google Scholar
  30. 30.
    Shi, D.Y., Wang, Q.S., Shi, X.J., Pang, F.Z.: A series solution for the inplane vibration analysis of orthotropic rectangular plates with non-uniform elastic boundary constraints and internal line supports. Arch. Appl. Mech. 85(1), 51–73 (2015)CrossRefGoogle Scholar
  31. 31.
    Leissa, A.W.: Vibration of Plates. U. S. Government Printing Office, Washington DC (1969)Google Scholar
  32. 32.
    Aghdam, M.M., Mohammadi, M., Erfanian, V.: Bending analysis of thin annular sector plates using extended Kantorovich method. Thin-Walled Struct. 45(12), 983–990 (2007)CrossRefGoogle Scholar
  33. 33.
    Irie, T., Yamada, G., Ito, F.: Free vibration of polar-orthotropic sector plates. J. Sound Vib. 67(1), 89–100 (1979)CrossRefzbMATHGoogle Scholar
  34. 34.
    Wang, X., Wang, Y.: Free vibration analyses of thin sector plates by the new version of differential quadrature method. Comput. Methods Appl. Mech. Eng. 193(36–38), 3957–3971 (2004)CrossRefzbMATHGoogle Scholar
  35. 35.
    Civalek, Ö., Ülker, M.: Harmonic differential quadrature (HDQ) for axisymmetric bending analysis of thin isotropic circular plates. Struct. Eng. Mech. 17(1), 1–14 (2004)CrossRefzbMATHGoogle Scholar
  36. 36.
    Mizusawa, T., Kito, H., Kajita, T.: Vibration of annular sector mindlin plates by the spline strip method. Comput. Struct. 53(5), 1205–1215 (1994)CrossRefzbMATHGoogle Scholar
  37. 37.
    Srinivasan, R.S., Thiruvenkatachari, V.: Free vibration of annular sector plates by an integral equation technique. J. Sound Vib. 89(3), 425–432 (1983)CrossRefzbMATHGoogle Scholar
  38. 38.
    Zhou, D., Lo, S.H., Cheung, Y.K.: 3-D vibration analysis of annular sector plates using the Chebyshev–Ritz method. J. Sound Vib. 320(1–2), 421–437 (2009)CrossRefGoogle Scholar
  39. 39.
    Xiang, Y., Liew, K.M., Kitipornchai, S.: Transverse vibration of thick annular sector plates. J. Eng. Mech. 119(8), 1579–1599 (1993)CrossRefGoogle Scholar
  40. 40.
    Shi, D.Y., Shi, X.J., Li, W.L.: Vibration analysis of annular sector plates under different boundary conditions. Shock Vib. 2014, 1–11 (2014). MathSciNetGoogle Scholar
  41. 41.
    Shi, X., Li, C., Wang, F., Shi, D.: Three-dimensional free vibration analysis of annular sector plates with arbitrary boundary conditions. Arch. Appl. Mech. 87, 1781–1796 (2017)CrossRefGoogle Scholar
  42. 42.
    Zhao, Y.K., Shi, D.Y., Meng, H.: A unified spectro-geometric-Ritz solution for free vibration analysis of conical–cylindrical–spherical shell combination with arbitrary boundary conditions. Arch. Appl. Mech. 87(6), 961–988 (2017)CrossRefGoogle Scholar
  43. 43.
    Jin, G., Te, Y., Me, X., Chen, Y., Su, X., Xie, X.: A unified approach for the vibration analysis of moderately thick composite laminated cylindrical shells with arbitrary boundary conditions. Int. J. Mech. Sci. 75, 357–376 (2013)CrossRefGoogle Scholar
  44. 44.
    Wang, Q., Shi, D., Liang, Q., Ahad, F.: A unified solution for free inplane vibration of orthotropic circular, annular and sector plates with general boundary conditions. Appl. Math. Model. 40(21–22), 9228–9253 (2016)MathSciNetCrossRefGoogle Scholar
  45. 45.
    Civalek, O., Korkmaz, A., Demir, C.: Discrete singular convolution approach for buckling analysis of rectangular Kirchhoff plates subjected to compressive loads on two-opposite edges. Adv. Eng. Softw. 41(4), 557–560 (2010)CrossRefzbMATHGoogle Scholar
  46. 46.
    Civalek, O.: Analysis of thick rectangular plates with symmetric cross-ply laminates based on first-order shear deformation theory. J. Compos. Mater. 42(26), 2853–2867 (2008)CrossRefGoogle Scholar
  47. 47.
    Talebitooti, M.: Three-dimensional free vibration analysis of rotating laminated conical shells: layerwise differential quadrature (LW-DQ) method. Arch. Appl. Mech. 83, 765–781 (2013)CrossRefzbMATHGoogle Scholar
  48. 48.
    Civalek, O.: Nonlinear dynamic response of laminated plates resting on nonlinear elastic foundations by the discrete singular convolution-differential quadrature coupled approaches. Compos. Part B 50, 171–179 (2013)CrossRefGoogle Scholar
  49. 49.
    Civalek, O.: The determination of frequencies of laminated conical shells via the discrete singular convolution method. J. Mech. Mater. Struct. 1, 163–182 (2006)CrossRefGoogle Scholar
  50. 50.
    Li, W.L.: Vibration analysis of rectangular plates with general elastic boundary supports. J. Sound Vib. 273, 619–635 (2004)CrossRefGoogle Scholar
  51. 51.
    Mirtalaie, S.H., Hajabasi, M.A.: Free vibration analysis of functionally graded thin annular sector plates using the differential quadrature method. Proc. Inst. Mech. Eng. C J. Mech. Eng. Sci. 225(3), 568–583 (2011)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Civil EngineeringSuzhou University of Science and TechnologySuzhouChina
  2. 2.School of Mechanical and Aerospace EngineeringOklahoma State UniversityStillwaterUSA

Personalised recommendations