Skip to main content
SpringerLink
Log in

On the use of GDQ for vibration characteristic of non-homogeneous orthotropic rectangular plates of bilinearly varying thickness

  • Published:
Acta Mechanica Aims and scope Submit manuscript

Abstract

The generalized differential quadrature (GDQ) method is used to analyze the effect of non-homogeneity, orthotropy, and thickness variation on the vibration characteristics of rectangular plates on the basis of Kirchhoff’s plate theory. The non-homogeneity of the plate material is assumed to arise due to the exponential variations in Young’s moduli, shear modulus, and density of the plate material with the in-plane coordinates. The thickness of the plate is taken as the Cartesian product of linear variations along two concurrent edges of the plate. In the GDQ method, the derivative of a function with respect to a space variable at a given grid point is approximated as a weighted linear sum of the function values at all of the grid points in the computational domain of that variable. Numerical results have been obtained for four different combinations of boundary conditions at the edges, namely: (1) fully clamped; (2) two opposite edges are clamped and the other two are simply supported; (3) two opposite edges are clamped and the other two are free; and (4) two opposite edges are simply supported and the other two are free. The effect of various plate parameters on the natural frequencies is studied for the first three modes of vibration. Three-dimensional mode shapes for a specified plate are plotted. A comparison of results obtained by the present method with those available in the literature is presented.

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

Similar content being viewed by others

References

  1. Bert C.W., Malik M.: Free vibration analysis of tapered rectangular plates by differential quadrature method: a semi-analytical approach. J. Sound Vib. 190, 41–63 (1996)

    Article  Google Scholar 

  2. Lal R., Gupta U.S., Gupta U.S.: Quintic splines in the study of transverse vibrations of non-uniform orthotropic rectangular plates. J. Sound Vib. 207(1), 1–13 (1997)

    Article  MATH  Google Scholar 

  3. Bambill D.V., Rossit C.A., Laura P.A.A., Rossi R.E.: Transverse vibrations of an orthotropic rectangular plate of linearly varying thickness and with a free edge. J. Sound Vib. 235, 530–538 (2000)

    Article  Google Scholar 

  4. Ashour A.S.: A semi-analytical solution of the flexural vibration of orthotropic plates of variable thickness. J. Sound Vib. 240, 431–445 (2001)

    Article  MATH  Google Scholar 

  5. Zenkour A.M.: Elastic behavior of an orthotropic beam/one-dimensional plate of uniform and variable thickness. J. Eng. Math. 44(4), 331–344 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  6. Civalek Ö.: Fundamental frequency of isotropic and orthotropic rectangular plates with linearly varying thickness by discrete singular convolution method. Appl. Math. Model. 33, 3825–3835 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  7. Mozafaria H., Ayob A.: Effect of thickness variation on the mechanical buckling load in plates made of functionally graded materials. Proc. Technol. 1, 496–504 (2012)

    Article  Google Scholar 

  8. Malhotra S.K., Ganesan N., Veluswami M.A.: Vibration of orthotropic square plates having variable thickness (parabolic variation). J. Sound Vib. 119, 184–188 (1987)

    Article  Google Scholar 

  9. Grigoenko A.Ya., Efimova T.L.: Spline-approximation method applied to solve natural-vibration problems for rectangular plates of varying thickness. Int. Appl. Mech. 41(10), 1161–1169 (2005)

    Article  Google Scholar 

  10. Xu Y., Zhou D., Liu K.: Three-dimensional thermo-elastic analysis of rectangular plates with variable thickness subjected to thermo-mechanical loads. J. Thermal Stress. 33, 1136–1155 (2010)

    Article  Google Scholar 

  11. Grigorenko Y.M., Grigorenko A.Y., Efimova T.L.: Spline-based investigation of natural vibrations of orthotropic rectangular plates of variable thickness within classical and refined theories. J. Mech. Mater. Struct. 3(5), 929–952 (2008)

    Article  Google Scholar 

  12. Lal R.: Transverse vibration of orthotropic nonuniform rectangular plates with continuously varying density. Indian J. Pure Appl. Math. 34(4), 587–606 (2003)

    MATH  Google Scholar 

  13. Liu M.F., Chang T.P., Wang Y.H.: free vibration analysis of orthotropic rectangular plates with tapered varying thickness and Winkler Spring Foundation. Mech. Des. Struct. Mach. 39, 320–333 (2011)

    Article  Google Scholar 

  14. Pachenari Z., Attarnejad R.: Free vibration of tapered mindlin plates using basic displacement functions. Arab. J. Sci. Eng. 39, 4433–4449 (2014)

    Article  MathSciNet  Google Scholar 

  15. Grossi R.O., Laura P.A.A.: Transverse vibrations of orthotropic rectangular plates with thickness varying in two directions and with edges elastically restrained against rotation. Fibre Sci. Technol. 14, 311–317 (1980–1981)

  16. Tomar J.S., Gupta A.K.: Effect of thermal gradient on frequencies of an orthotropic rectangular plate whose thickness varies in two directions. J. Sound Vib. 98(2), 257–262 (1985)

    Article  MATH  Google Scholar 

  17. Huang M., Ma X.Q., Sakiyama T., Matuda H., Morita C.: Free vibration analysis of orthotropic rectangular plates with variable thickness and general boundary conditions. J. Sound Vib. 288, 931–955 (2005)

    Article  Google Scholar 

  18. Xu Y.P., Zhou D.: Three-dimensional elasticity solution of functionally graded rectangular plates with variable thickness. Compos. Struct. 91, 56–65 (2009)

    Article  Google Scholar 

  19. Bahmyari E., Ranji A.R.: Free vibration analysis of orthotropic plates with variable thickness resting on non-uniform elastic foundation by element free Galerkin method. J. Mech. Sci. Technol. 26(9), 2685–2694 (2012)

    Article  Google Scholar 

  20. Gupta A.K., Kumar S.: Thermal effect on vibration of orthotropic rectangular plate with thickness variation as linearly and parabolically in x- and y-directions respectively. J. Math. Sci. Appl. 1(2), 32–35 (2013)

    Google Scholar 

  21. Lal R., Lal R.: Transverse vibrations of non-homogeneous orthotropic rectangular plates of variable thickness: a spline technique. J. Sound Vib. 306, 203–214 (2007)

    Article  MATH  Google Scholar 

  22. Lal R., Saini R.: Buckling and vibration of non-homogeneous rectangular plates subjected to linearly varying in-plane force. Shock Vib. 20, 1–16 (2013)

    Article  Google Scholar 

  23. Lal R., Kumar Y.: Characteristic orthogonal polynomials in the study of transverse vibrations of nonhomogeneous rectangular orthotropic plates of bilinearly varying thickness. Meccanica 47, 175–193 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  24. Singh B., Saxena V.: Transverse vibration of rectangular plates with bidirectional thickness variation. J. Sound Vib. 198(1), 51–65 (1996)

    Article  MathSciNet  Google Scholar 

  25. Shu C.: Differential Quadrature and Its Application in Engineering. Springer, London (2000)

    Book  MATH  Google Scholar 

  26. Chakraverty S., Petyt M.: Free vibration analysis of elliptic and circular plates having rectangular orthotropy. Struct. Eng. Mech. 7(1), 53–67 (1999)

    Article  Google Scholar 

  27. Lal R., Kumar Y.: Boundary characteristic orthogonal polynomials in the study of transverse vibrations of nonhomogeneous rectangular plates with bilinear thickness variation. Shock Vib. 19, 349–364 (2012)

    Article  Google Scholar 

  28. Liew K.M., Lim M.K.: Transverse vibration of trapezoidal plates of variable thickness: symmetric trapezoids. J. Sound Vib. 165(1), 45–67 (1993)

    Article  MATH  Google Scholar 

  29. Eftekhari S.A., Jafari A.A.: Accurate variational approach for free vibration of variable thickness thin and thick plates with edges elastically restrained against translation and rotation. Int. J. Mech. Sci. 68, 35–46 (2013)

    Article  Google Scholar 

  30. Marangoni R.D., Cook L.M., Basavanhally N.: Upper and lower bounds to the natural frequencies of vibration of clamped rectangular orthotropic plates. Int. J. Solids Struct. 14, 611–623 (1978)

    Article  MATH  Google Scholar 

  31. Xing Y.F., Liu B.: New exact solution for free vibrations of thin orthotropic rectangular plates. Compos. Struct. 89(4), 567–574 (2009)

    Article  MathSciNet  Google Scholar 

  32. Jafari A.A., Eftekhari S.A.: An efficient mixed methodology for free vibration and buckling analysis of orthotropic rectangular plates. Appl. Math. Comput. 218, 2670–2692 (2011)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, 247667, India

    Roshan Lal & Renu Saini

Authors
  1. Roshan Lal
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Renu Saini
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Renu Saini.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lal, R., Saini, R. On the use of GDQ for vibration characteristic of non-homogeneous orthotropic rectangular plates of bilinearly varying thickness. Acta Mech 226, 1605–1620 (2015). https://doi.org/10.1007/s00707-014-1272-4

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00707-014-1272-4

Keywords

Search

Navigation