Structural Engineering and Mechanics

Techno-Press (테크노프레스)
권호 표지
DOI QR코드

DOI QR Code

Vibration analysis of plates with curvilinear quadrilateral domains by discrete singular convolution method

  • Civalek, Omer (Akdeniz University, Faculty of Engineering, Civil Engineering Department, Division of Mechanics) ;
  • Ozturk, Baki (Nigde University, Faculty of Engineering, Civil Engineering Department, Division of Mechanics)
  • Received : 2009.07.03
  • Accepted : 2010.06.16
  • Published : 2010.10.20
⟨ Previous Next ⟩

Abstract

A methodology on application of the discrete singular convolution (DSC) technique to the free vibration analysis of thin plates with curvilinear quadrilateral platforms is developed. In the proposed approach, irregular physical domain is transformed into a rectangular domain by using geometric coordinate transformation. The DSC procedures are then applied to discretization of the transformed set of governing equations and boundary conditions. For demonstration of the accuracy and convergence of the method, some numerical examples are provided on plates with different geometry such as elliptic, trapezoidal having straight and parabolic sides, sectorial, annular sectorial, and plates with four curved edges. The results obtained by the DSC method are compared with those obtained by other numerical and analytical methods. The method is suitable for the problem considered due to its generality, simplicity, and potential for further development.

Keywords

References

  1. Bert, C.W. and Malik, M. (1996), "The differential quadrature method for irregular domains and application to plate vibration", Int. J. Mech. Sci., 38(6), 589-606. https://doi.org/10.1016/S0020-7403(96)80003-8
  2. Cheung, Y.K., Tham, L.G. and Li, W.Y. (1988), "Free vibration and static analysis of general plates by spline finite strip method", Comput. Mech., 3, 187-197. https://doi.org/10.1007/BF00297445
  3. Cheung, Y.K. and Cheung, M.S. (1971), "Flexural vibrations of rectangular and other polygonal plates", J. Eng. Mech., 97, 3911-411.
  4. Civalek, $\ddot{O}$. (2004), "Application of differential quadrature (DQ) and harmonic differential quadrature (HDQ) for buckling analysis of thin isotropic plates and elastic columns", Eng. Struct., 26(2), 171-186. https://doi.org/10.1016/j.engstruct.2003.09.005
  5. Civalek, O. (2005), "Geometrically nonlinear dynamic analysis of doubly curved isotropic shells resting on elastic foundation by a combination of harmonic differential quadrature-finite difference methods", Int. J. Pres. Ves. Pip., 82(6), 470-479. https://doi.org/10.1016/j.ijpvp.2004.12.003
  6. Civalek, O.(2006), "An efficient method for free vibration analysis of rotating truncated conical shells", Int. J. Pres. Ves. Pip., 83, 1-12. https://doi.org/10.1016/j.ijpvp.2005.10.005
  7. Civalek, $\ddot{O}$. (2007a), "Nonlinear analysis of thin rectangular plates on Winkler- Pasternak elastic foundations by DSC-HDQ methods", Appl. Math. Model., 31, 606-624. https://doi.org/10.1016/j.apm.2005.11.023
  8. Civalek, $\ddot{O}$. (2007b), "Frequency analysis of isotropic conical shells by discrete singular convolution (DSC)", Struct. Eng. Mech., 25(1), 127-130. https://doi.org/10.12989/sem.2007.25.1.127
  9. Civalek, O.(2007c), "Numerical analysis of free vibrations of laminated composite conical and cylindrical shells: discrete singular convolution (DSC) approach", J. Comput. Appl. Math., 205, 251-271. https://doi.org/10.1016/j.cam.2006.05.001
  10. Civalek, $\ddot{O}$. (2007d), "A parametric study of the free vibration analysis of rotating laminated cylindrical shells using the method of discrete singular convolution", Thin Wall. Struct., 45, 692-698. https://doi.org/10.1016/j.tws.2007.05.004
  11. Geannakakes, G.N. (1990), "Vibration analysis of arbitrarily shaped plates using beam characteristics orthogonal polynomials in semi-analytical finite strip method", J. Sound Vib., 137, 283-303. https://doi.org/10.1016/0022-460X(90)90793-Y
  12. Gordon, W.J. and Hall, C.A. (1973), "Construction of curvilinear co-ordinate systems and application to mesh generation", Int. J. Numer. Meth. Eng., 7, 461-477. https://doi.org/10.1002/nme.1620070405
  13. Gorman, D.J. (1988), "Accurate free vibration analysis of rhombic plates with simply-supported and fullyclamped edge conditions", J. Sound Vib., 125, 281-290. https://doi.org/10.1016/0022-460X(88)90283-0
  14. Han, J.B. and Liew, K.M. (1997), "An eight-node curvilinear differential quadrature formulation for Reissner/ Mindlin plates", Comput. Meth. Appl. Mech. Eng., 141, 265-280. https://doi.org/10.1016/S0045-7825(96)01115-2
  15. Huang, C.S., Leissa, A.W. and McGee, O.G. (1993), "Exact analytical solutions for free vibrations of sectorial plates with simply supported radial edges", J. Appl. Mech., 60, 478-483. https://doi.org/10.1115/1.2900818
  16. Karami, G. and Malekzadeh, P. (2003), "An efficient differential quadrature methodology for free vibration analysis of arbitrary straight-sided quadrilateral thin plates", J. Sound Vib., 263, 415-442. https://doi.org/10.1016/S0022-460X(02)01062-3
  17. Lam, K.Y., Liew, K.M. and Chow, S.T. (1992), "Use of two-dimensional orthogonal polynomials for vibration analysis of circular and elliptical plates", J. Sound Vib., 154, 261-269. https://doi.org/10.1016/0022-460X(92)90580-Q
  18. Lam, K.Y., Liew, K.M. and Chow, S.T. (1990), "Free vibration analysis of isotropic and orthotropic triangular plates", Int. J. Mech. Sci., 32(5), 455-464. https://doi.org/10.1016/0020-7403(90)90172-F
  19. Leissa, A.W. (1969), Vibration of Plates, NASA, SP-160.
  20. Leissa, A.W. (1977), "Recent research in plate vibrations: complicating effects", Shock Vib., 9(11), 21-35. https://doi.org/10.1177/058310247700901106
  21. Leissa, A.W. (1981), "Plate vibration research, 1976-1980: classical theory", Shock Vib., 13(9), 11-12. https://doi.org/10.1177/058310248101300905
  22. Leissa, A.W. (1987), "Recent research in plate vibrations: 1981-1985: Part I. classical theory", Shock Vib., 19(2), 11-18. https://doi.org/10.1177/058310248701900204
  23. Leissa, A.W., McGee, O.G. and Huang, C.S. (1993), "Vibrations of sectorial plates having corner stress singularities", J. Appl. Mech., 60, 131-140.
  24. Li, W.Y., Cheung, Y.K. and Tham, L.G. (1986), "Spline finite strip analysis of general plates", J. Eng. Mech., 112(1), 43-54. https://doi.org/10.1061/(ASCE)0733-9399(1986)112:1(43)
  25. Liew, K.M. and Han, J.B. (1997), "A four-note differential quadrature method for straight-sided quadrilateral Reissner/Mindlin plates", Commun. Numer. Meth. Eng., 13, 73-81. https://doi.org/10.1002/(SICI)1099-0887(199702)13:2<73::AID-CNM32>3.0.CO;2-W
  26. Liew, K.M. (1992), "Frequency solutions for circular plates with internal supports and discontinuous boundaries", Int. J. Mech. Sci., 34(7), 511-520. https://doi.org/10.1016/0020-7403(92)90027-E
  27. Liew, K.M., Xiang, Y. and Kitipornchai, S. (1995), "Benchmark vibration solutions for regular polygonal Mindlin plates", J. Acoust. Soc. Am., 97(5), 2866-2871. https://doi.org/10.1121/1.411852
  28. Liew, K.M. and Lam, K.Y. (1993a), "On the use of 2-D orthogonal polynomials in the Rayleigh-Ritz method for flexural vibration of annular sector plates of arbitrary shape", Int. J. Mech. Sci., 35, 129-139. https://doi.org/10.1016/0020-7403(93)90071-2
  29. Liew, K.M. and Lam, K.Y. (1993b), "Transverse vibration of solid circular plates continuous over multiple concentric annular supports", J. Appl. Mech., 60, 208-210. https://doi.org/10.1115/1.2900749
  30. Liew, K.M., Lim, C.W. and Lim, M.K. (1994), "Transverse vibration of trapezoidal plates of variable thickness: unsymmetric trapezoids", J. Sound Vib., 177(4), 479-501. https://doi.org/10.1006/jsvi.1994.1447
  31. Liew, K.M. and Sumi, Y.K. (1998), "Vibration of plates having orthogonal straight edges with clamped boundaries", J. Eng. Mech., 124, 184-192. https://doi.org/10.1061/(ASCE)0733-9399(1998)124:2(184)
  32. Liew, K.M., Lam, K.Y. and Chow, S.T. (1990), "Free vibration analysis of rectangular plates using orthogonal plate function", Comp. Struct., 34(1), 79-85. https://doi.org/10.1016/0045-7949(90)90302-I
  33. Liew, K.M. and Lam, K.Y. (1990), "Application of two-dimensional orthogonal plate function to flexural vibration of skew plates", J. Sound Vib., 139(2), 241-252. https://doi.org/10.1016/0022-460X(90)90885-4
  34. Liew, K.M. and Lam, K.Y. (1991), "A Rayleigh-Ritz approach to transverse vibration of isotropic and anisotropic trapezoidal plates using orthogonal plate functions", Int. J. Solids Struct., 27(2), 189-203. https://doi.org/10.1016/0020-7683(91)90228-8
  35. Liew, K.M. (1993a), "On the use of pb-2 Rayleigh-Ritz method for free flexural vibration of triangular plates with curved internal supports", J. Sound Vib., 165(2), 329-340. https://doi.org/10.1006/jsvi.1993.1260
  36. Liew, K.M. (1993b), "Treatments of over-restrained boundaries for doubly connected plates of arbitrary shape in vibration analysis", Int. J. Solids Struct., 30(3), 337-347. https://doi.org/10.1016/0020-7683(93)90170-C
  37. Liew, K.M. and Lim, M.K. (1993), "Transverse vibration of trapezoidal plates of variable thickness: symmetric trapezoids", J. Sound Vib., 165(1), 45-67. https://doi.org/10.1006/jsvi.1993.1242
  38. Lim, C.W., Li, Z.R., Xiang, Y., Wei, G.W. and Wang, C.M. (2005a), "On the missing modes when using the exact frequency relationship between Kirchhoff and Mindlin plates", Adv. Vib. Eng., 4, 221-248.
  39. Lim, C.W., Li, Z.R. and Wei, G.W. (2005b), "DSC-Ritz method for high-mode frequency analysis of thick shallow shells", Int. J. Numer. Meth. Eng., 62, 205-232. https://doi.org/10.1002/nme.1179
  40. Liu, F.L. and Liew, K.M. (1999), "Free vibration analysis of Mindlin sector plates: numerical solutions by differential quadrature method", Comput. Meth. Appl. Mech. Eng., 177, 77-92. https://doi.org/10.1016/S0045-7825(98)00376-4
  41. Malik, M. and Bert, C.W. (2000), "Vibration analysis of plates with curvilinear quadrilateral planforms by DQM using blending functions", J. Sound Vib., 230(4), 949-954. https://doi.org/10.1006/jsvi.1999.2584
  42. Ng, C.H.W., Zhao, Y.B. and Wei, G.W. (2004), "Comparison of discrete singular convolution and generalized differential quadrature for the vibration analysis of rectangular plates", Comput. Meth. Appl. Mech. Eng., 193, 2483-2506. https://doi.org/10.1016/j.cma.2004.01.013
  43. Shu, C., Chen, W. and Du, H. (2000), "Free vibration analysis of curvilinear quadrilateral plates by the differential quadrature method", J. Comput. Phys., 163, 452-466. https://doi.org/10.1006/jcph.2000.6576
  44. Singh, B. and Chakraverty, S. (1992), "On the use of orthogonal polynomials in the Rayleigh-Ritz method for study of transverse vibration of elliptic plates", Comp. Struct., 43, 439-443. https://doi.org/10.1016/0045-7949(92)90277-7
  45. Wang, C.M., Xiang, Y., Watanabe, E. and Utsunomiya, T. (2004), "Mode shapes and stress-resultants of circular Mindlin plates with free edges", J. Sound Vib., 276(3-5), 511-525. https://doi.org/10.1016/j.jsv.2003.08.010
  46. Wang, C.M. (1994), "Natural frequencies formula for simply supported Mindlin plates", J. Vib. Acoust., 116(4), 536-540. https://doi.org/10.1115/1.2930460
  47. Wang, C.M., Xiang, Y., Utsunomiya, T. and Watanabe, E. (2001), "Evaluation of modal stress resultants in freely vibrating plates", Int. J. Solids Struct., 38(36-37), 6525-6558. https://doi.org/10.1016/S0020-7683(01)00040-3
  48. Wang, G. and Cheng-Tzu, T.H. (1994), "Static and dynamic analysis of arbitrary quadrilateral flexural plates by B3-spline functions", Int. J. Solids Struct., 31, 657-667. https://doi.org/10.1016/0020-7683(94)90144-9
  49. Wang, X., Striz, A.G. and Bert, C.W. (1994), "Buckling and vibration analysis of skew plates by the differential quadrature method", AIAA J., 32(4), 886-889. https://doi.org/10.2514/3.12071
  50. Wei, G.W., Kouri, D.J. and Hoffman, D.K. (1998), "Wavelets and distributed approximating functionals", Comput. phys. Commun., 112, 1-6. https://doi.org/10.1016/S0010-4655(98)00051-4
  51. Wei, G.W. (1999), "Discrete singular convolution for the solution of the Fokker-Planck equations", J. Chem. Phys., 110, 8930-8942. https://doi.org/10.1063/1.478812
  52. Wei, G.W. (2001a), "A new algorithm for solving some mechanical problems", Comput. Meth. Appl. Mech. Eng., 190, 2017-2030. https://doi.org/10.1016/S0045-7825(00)00219-X
  53. Wei, G.W. (2001b), "Vibration analysis by discrete singular convolution", J. Sound Vib., 244, 535-553. https://doi.org/10.1006/jsvi.2000.3507
  54. Wei, G.W. (2001c), "Discrete singular convolution for beam analysis", Eng. Struct., 23, 1045-1053. https://doi.org/10.1016/S0141-0296(01)00016-5
  55. Wei, G.W., Zhao, Y.B. and Xiang, Y. (2001), "The determination of natural frequencies of rectangular plates with mixed boundary conditions by discrete singular convolution", Int. J. Mech. Sci., 43, 1731-1746. https://doi.org/10.1016/S0020-7403(01)00021-2
  56. Wei, G.W. and Gu, Y. (2002), "Conjugate filter approach for solving Burgers' equation", J. Comput. Appl. Math., 149, 439-456. https://doi.org/10.1016/S0377-0427(02)00488-0
  57. Wei, G.W., Zhao, Y.B. and Xiang, Y. (2002a), "Discrete singular convolution and its application to the analysis of plates with internal supports. Part 1: Theory and algorithm", Int. J. Numer. Meth. Eng., 55, 913-946. https://doi.org/10.1002/nme.526
  58. Wei, G.W., Zhao, Y.B. and Xiang, Y. (2002b), "A novel approach for the analysis of high-frequency vibrations", J. Sound Vib., 257(2), 207-246. https://doi.org/10.1006/jsvi.2002.5055
  59. Wu, W.X., Shu, C. and Wang, C.M. (2006), "Computation of modal stress resultants for completely free vibrating plates by LSFD method", J. Sound Vib., 297, 704-726. https://doi.org/10.1016/j.jsv.2006.04.019
  60. Xiang, Y., Liew, K.M. and Kitipornchai, S. (1993), "Transverse vibration of thick annular sector plates", J. Eng. Mech., 119, 1579-1599. https://doi.org/10.1061/(ASCE)0733-9399(1993)119:8(1579)
  61. Zhao, Y.B., Wei, G.W. and Xiang, Y. (2002), "Discrete singular convolution for the prediction of high frequency vibration of plates", Int. J. Solids Struct., 39, 65-88. https://doi.org/10.1016/S0020-7683(01)00183-4
  62. Zhao, Y.B. and Wei, G.W. (2002), "DSC analysis of rectangular plates with non-uniform boundary conditions", J. Sound Vib., 255(2), 203-228. https://doi.org/10.1006/jsvi.2001.4150
  63. Zhao, S., Wei, G.W. and Xiang, Y. (2005), "DSC analysis of free-edged beams by an iteratively matched boundary method", J. Sound Vib., 284, 487-493. https://doi.org/10.1016/j.jsv.2004.08.037

Cited by

  1. New analytic solutions for free vibration of rectangular thick plates with an edge free vol.131-132, 2017, https://doi.org/10.1016/j.ijmecsci.2017.07.002
  2. On new analytic free vibration solutions of rectangular thin cantilever plates in the symplectic space vol.53, 2018, https://doi.org/10.1016/j.apm.2017.09.011
  3. New benchmark solutions for free vibration of clamped rectangular thick plates and their variants vol.78, 2018, https://doi.org/10.1016/j.aml.2017.11.006
  4. Strong Formulation Finite Element Method Based on Differential Quadrature: A Survey vol.67, pp.2, 2010, https://doi.org/10.1115/1.4028859
  5. Free transverse vibration of shear deformable super-elliptical plates vol.24, pp.4, 2017, https://doi.org/10.12989/was.2017.24.4.307
  6. On the Convergence of Laminated Composite Plates of Arbitrary Shape through Finite Element Models vol.2, pp.1, 2010, https://doi.org/10.3390/jcs2010016
  7. A Review on the Discrete Singular Convolution Algorithm and Its Applications in Structural Mechanics and Engineering vol.27, pp.5, 2010, https://doi.org/10.1007/s11831-019-09365-5
  8. Approximate analysis of quadrilateral slabs having various cases of boundary conditions and aspect ratios vol.24, pp.9, 2021, https://doi.org/10.1177/1369433220982099