SeMA Journal

, Volume 74, Issue 2, pp 165–180 | Cite as

Sinc-Galerkin solution to the clamped plate eigenvalue problem

Article

Abstract

We propose an accurate and computationally efficient numerical technique for solving the biharmonic eigenvalue problem. The technique is based on the sinc-Galerkin approximation method to solve the clamped plate problem. Numerical experiments for plates with various aspect ratios are reported, and comparisons are made with other methods in literature. The calculated results accord well with those published earlier, which proves the accuracy and validity of the proposed method.

Keywords

Sinc functions Sinc-Galerkin Biharmonic problem Eigenvalues 

Mathematics Subject Classification

65N25 31A30 

Notes

Acknowledgments

The authors would like to thank the anonymous reviewer for carefully reading this paper and for his many useful suggestions.

References

  1. 1.
    Bardell, N.S., Dunsdon, J.M., Langley, R.S.: Free vibration of coplanar sandwich panels. Compos. Struct. 38, 463–475 (1997)CrossRefGoogle Scholar
  2. 2.
    Bellman, R., Kashef, B.G., Casti, J.: Differential quadrature: a technique for the rapid solution of non-linear partial differential equations. J. Comput. Phys. 10, 40–52 (1972)CrossRefMATHGoogle Scholar
  3. 3.
    Bellomo, N., Ridolfi, L.: Solution of nonlinear initial -boundary value problems by sinc-collocation-interpolation methods. Comput. Math. Appl. 29, 15–28 (1995)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Belytschko, T., Lu, Y.Y., Gu, L.: Element-free Galerkin methods. Int. J. Numer. Methods Eng. 37, 229–256 (1994)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bialecki, B.: Sinc-collocation methods for two-point boundary value problems. IMA J. Numer. Anal. 11, 357–375 (1991)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Bjørstad, P.E., Tjøstheim, B.P.: High precision solutions of two fourth order eigenvalue problems. Computing 63, 97–107 (1999)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Cabay, S., Jackson, L.W.: A polynomial extrapolation method for finding limits and antilimits of vector sequences. SIAM J. Numer. Anal. 13, 734–752 (1976)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Chakraverty, S., Pradhan, K.K.: Free vibration of exponential functionally graded rectangular plates in thermal environment with general boundary conditions. Aerosp. Sci. Technol. 36, 132–156 (2014)CrossRefGoogle Scholar
  9. 9.
    Chen, W., Zhong, T.X.: A note on the DQ analysis of anisotropic plates. J. Sound Vib. 204, 180–182 (1997)CrossRefGoogle Scholar
  10. 10.
    Cheung, Y.K., Chen, W.J.: Hybrid quadrilateral element based on Mindlin/Reissner plate theory. Comput. Struct. 32, 327–339 (1989)CrossRefMATHGoogle Scholar
  11. 11.
    Chia, C.Y.: Non-linear vibration of anisotropic rectangular plates with non-uniform edge constraints. J. Sound Vib. 101, 539–550 (1985)CrossRefGoogle Scholar
  12. 12.
    Cook, T.: Comparison of finite difference, spectral and sinc biharmonic operators. M.sc thesis, University of Utah (2004)Google Scholar
  13. 13.
    Donning, B.M., Liu, W.K.: Meshless methods for shear-deformable beams and plates. Comput. Methods Appl. Mech. Eng. 152, 47–71 (1998)CrossRefMATHGoogle Scholar
  14. 14.
    El-Gamel, M.: Sinc-collocation method for solving linear and nonlinear system of second-order boundary value problems. Appl. Math. 3, 1627–1633 (2012)CrossRefGoogle Scholar
  15. 15.
    El-Gamel, M.: Numerical solution of Troesch’s problem by sinc-collocation method. Appl. Math. 4, 707–712 (2013)CrossRefGoogle Scholar
  16. 16.
    El-Gamel, M., Behiry, S., Hashish, H.: Numerical method for the solution of special nonlinear fourth-order boundary value problems. Appl. Math. Comp. 145, 717–734 (2003)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    El-Gamel, M., Cannon, J.R., Zayed, A.: Sinc-galerkin method for solving linear sixth order boundary-value problems. Math. Comp. 73, 1325–1343 (2004)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    El-Gamel, M., Mohsen, A., El-Mohsen, A.A.: Sinc-Galerkin method for solving biharmonic problems. Appl. Math. Comp. 247, 386–396 (2014)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Fan, S.C., Cheung, Y.K.: Flexural free vibrations of rectangular plates with complex support conditions. J. Sound Vib. 93, 81–94 (1984)CrossRefGoogle Scholar
  20. 20.
    Geannakakes, G.N.: Natural frequencies of arbitrarily shaped plates using the Rayleigh–Ritz method together with natural co-ordinate regions and normalized characteristic polynomials. J. Sound Vib. 182, 441–478 (1995)CrossRefGoogle Scholar
  21. 21.
    Gavalas, G., El-Raheb, M.: Extension of Rayleigh–Ritz method for eigenvalue problems with discontinuous boundary conditions applied to vibration of rectangular plates. J. Sound Vib. 333, 4007–4016 (2014)CrossRefGoogle Scholar
  22. 22.
    Gorman, D.J.: Free Vibration Analysis of Rectangular Plates. Elsevier, New York (1982)MATHGoogle Scholar
  23. 23.
    Gorman, D.J.: An exact analytical approach to the free vibration analysis of rectangular plates with mixed boundary conditions. J. Sound Vib. 93, 235–247 (1984)CrossRefMATHGoogle Scholar
  24. 24.
    Gorman, D.J., Yu, S.D.: A review of the superposition method for computing free vibration eigenvalues of elastic structures. Comput. Struct. 104, 27–37 (2012)CrossRefGoogle Scholar
  25. 25.
    Grenander, V., Szego, G.: Toeplitz Forms and Their Applications, 2nd edn. Chelsea Publishing Co, Orlando (1985)MATHGoogle Scholar
  26. 26.
    Keer, L.M., Stahl, B.: Eigenvalue problems of rectangular plates with mixed edge conditions. J. Appl. Mech. 39, 513–520 (1972)CrossRefMATHGoogle Scholar
  27. 27.
    Leipholz, H.H.E.: On some developments in direct methods of the calculus of variations. Appl. Mech. Rev. 40, 1379–1392 (1987)CrossRefGoogle Scholar
  28. 28.
    Leissa, A.W.: The free vibration of rectangular plates. J. Sound Vib. 31, 257–293 (1973)CrossRefMATHGoogle Scholar
  29. 29.
    Liew, K.M., Hung, K.C., Lam, K.Y.: On the use of the substructure method for vibration analysis of rectangular plates with discontinuous boundary conditions. J. Sound Vib. 163, 451–462 (1993)CrossRefMATHGoogle Scholar
  30. 30.
    Liew, K.M., Xiang, Y., Kitipornchai, S.: Transverse vibration of thick rectangular plates-I. compressive sets of boundary conditions. Comput. Struct. 49, 1–29 (1993)CrossRefGoogle Scholar
  31. 31.
    Lund, J., Bowers, K.: Sinc Methods for Quadrature and Differential Equations. SIAM, Philadelphia (1992)CrossRefMATHGoogle Scholar
  32. 32.
    McArthur, K., Bowers, K., Lund, J.: The sinc method in multiple space dimensions: model problems. Numer. Math. 56, 789–816 (1990)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Mohsen, A., El-Gamel, M.: A sinc-collocation method for the linear Fredholm integro-differential equations. Z. Angew. Math. Phys. 58, 380–390 (2007)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Mohsen, A., El-Gamel, M.: On the Galerkin and collocation methods for two-point boundary value problems using sinc bases. Comput. Math. Appl. 56, 930–941 (2008)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Mohsen, A., El-Gamel, M.: On the numerical solution of linear and nonlinear Volterra integral and integro-differential equations. Appl. Math. Comput. 217, 3330–3337 (2010)MathSciNetMATHGoogle Scholar
  36. 36.
    Neudecker, H.: A note on Kronecker matrix products and matrix equation systems. SIAM J. Appl. Math. 17, 603–606 (1969)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Nowacki, W.: Free vibrations and buckling of a rectangular plate with discontinuous boundary conditions. Bull. Acad. Pol. Sci. Biol. 3, 159–167 (1955)MATHGoogle Scholar
  38. 38.
    Ota, T., Hamada, M.: Fundamental frequencies of simply supported but partially clamped square plates. Bull. Jpn. Soc. Mech. Eng. 6, 397–403 (1963)CrossRefGoogle Scholar
  39. 39.
    Piskunov, V.H.: Determination of the frequencies of the natural oscillations of rectangular plates with mixed boundary conditions. Prikl. Mekh. 10, 72–76 (1964). (in Ukrainian)Google Scholar
  40. 40.
    Ralph, C., Bowers, K.: The sinc-Galerkin method for fourth-order differential equations. SIAM J. Numer. Anal. 28, 760–788 (1991)MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Shu, C., Richards, B.E.: Application of generalized differential quadrature to solve twodimensional incompressible Navier-Stokes equations. Int. J. Numer. Methods Fluids 15, 791–798 (1992)CrossRefMATHGoogle Scholar
  42. 42.
    Sidi, Avram: Efficient implementation of minimal polynomial and reduced rank extrapolation methods. J. Comput. Appl. Math. 36, 305–337 (1991)MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Stenger, F.: Numerical Methods Based on Sinc and Analytic Functions. Springer, New York (1993)CrossRefMATHGoogle Scholar
  44. 44.
    Timoshenko, S., Woinowsky-Krieger, S.: Theory of Plates and Shells. McGraw-hill, New York (1959)MATHGoogle Scholar
  45. 45.
    Žitňan, P.: Vibration analysis of membranes and plates by a discrete least squares technique. J. Sound Vib. 4, 595–605 (1996)Google Scholar
  46. 46.
    Wieners, C.: Bounds for the N lowest eigenvalues of fourth-order boundary value problems. Computing 59, 29–41 (1997)MathSciNetCrossRefMATHGoogle Scholar
  47. 47.
    Yin, G.: Sinc-collocation method with orthogonalization for singular problem-like Poisson. Math. Comput. 62, 21–40 (1994)CrossRefMATHGoogle Scholar
  48. 48.
    Young, D.: Vibration of rectangular plates by the Ritz method. Trans. Am. Soc. Mech. Eng. J. Appl. Mech. 17, 448–453 (1950)MATHGoogle Scholar

Copyright information

© Sociedad Española de Matemática Aplicada 2016

Authors and Affiliations

  1. 1.Department of Mathematics and Engineering Physics, Faculty of EngineeringMansoura UniversityMansouraEgypt
  2. 2.Department of Engineering Mathematics and Physics, Faculty of EngineeringCairo UniversityGizaEgypt
  3. 3.Centre for Photonics and Smart MaterialsZewail City of Science and TechnologyGizaEgypt

Personalised recommendations