Skip to main content
SpringerLink
Log in

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

  • Original
  • Published:
Archive of Applied Mechanics Aims and scope Submit manuscript

Abstract

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.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  1. Reddy, J.N.: Theory and Analysis of Elastic Plates and Shells. CRC Press, Boca Raton (2007)

    Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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). https://doi.org/10.1038/srep17054

  5. Gorman, D.J.: Vibration Analysis of Plates by the Superposition Method. World Scientific, Singapore (1999)

    Book  MATH  Google Scholar 

  6. Dozio, L., Ricciardi, M.: Free vibration analysis of ribbed plates by a combined analytical-numerical method. J. Sound Vib. 319, 681–97 (2009)

    Article  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  8. Du, H., Lim, M.K., Lin, R.M.: Application of differential quadrature to vibration analysis. J. Sound Vib. 181, 279–93 (1995)

    Article  MATH  Google Scholar 

  9. Wei, G.W.: Vibration analysis by discrete singular convolution. J. Sound Vib. 244, 535–53 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  13. Belytschko, T.: Meshless methods: an overview and recent developments. Comput. Methods Appl. Mech. Eng. 139(2), 3–47 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  14. Li, S., Liu, W.: Meshfree and particle method and their applications. Appl. Mech. Rev. 55(1), 1–34 (2002)

    Article  Google Scholar 

  15. Warburton, G.B.: The vibration of rectangular plates. Proc. Inst. Mech. Eng. Ser. A 168, 371–84 (1954)

    Article  MathSciNet  Google Scholar 

  16. Young, D.: Vibration of rectangular plates by the Ritz method. J. Appl. Mech. 17, 448–53 (1950)

    MATH  Google Scholar 

  17. Leissa, A.W.: The historical bases of the Rayleigh and Ritz methods. J. Sound Vib. 287, 961–78 (2005)

    Article  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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). https://doi.org/10.1016/j.tws.2010.08.014

    Article  Google Scholar 

  26. Li, W.L.: Free vibration of beams with general boundary conditions. J. Sound Vib. 237(4), 709–725 (2000)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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. 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)

    Article  Google Scholar 

  31. Leissa, A.W.: Vibration of Plates. U. S. Government Printing Office, Washington DC (1969)

    Google Scholar 

  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)

    Article  Google Scholar 

  33. Irie, T., Yamada, G., Ito, F.: Free vibration of polar-orthotropic sector plates. J. Sound Vib. 67(1), 89–100 (1979)

    Article  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  39. Xiang, Y., Liew, K.M., Kitipornchai, S.: Transverse vibration of thick annular sector plates. J. Eng. Mech. 119(8), 1579–1599 (1993)

    Article  Google Scholar 

  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). https://doi.org/10.1155/2014/517946

    Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  50. Li, W.L.: Vibration analysis of rectangular plates with general elastic boundary supports. J. Sound Vib. 273, 619–635 (2004)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

Download references

Acknowledgements

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

Author information

Authors and Affiliations

  1. School of Civil Engineering, Suzhou University of Science and Technology, Suzhou, 215011, Jiang Su, China

    Siyuan Bao

  2. School of Mechanical and Aerospace Engineering, Oklahoma State University, Stillwater, OK, 74078, USA

    Shuodao Wang

Authors
  1. Siyuan Bao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shuodao Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shuodao 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

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bao, S., Wang, S. A unified procedure for free transverse vibration of rectangular and annular sectorial plates. Arch Appl Mech 89, 1485–1499 (2019). https://doi.org/10.1007/s00419-019-01519-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00419-019-01519-y

Keywords

Search

Navigation