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Free transverse vibration of shear deformable super-elliptical plates

  • Altekin, Murat (Department of Civil Engineering, Yildiz Technical University,)
  • Received : 2016.05.23
  • Accepted : 2017.01.19
  • Published : 2017.04.25

Abstract

Free transverse vibration of shear deformable super-elliptical plates with uniform thickness was studied based on Mindlin plate theory using finite element method. Quadrilateral isoparametric elements were used in the paper. Sensitivity analysis was made to determine the influence of the thickness, the aspect ratio, and the shape of the plate on the natural frequency. Accuracy of the results computed in the current study was validated by comparing them with the solutions available in the literature. The results reveal that the frequencies of clamped super-elliptical plates lie in the range bounded by elliptical and rectangular plates irrespective of the aspect ratio, and furthermore, the frequency decreases if the super-elliptical power increases. A similar trend was observed for simply supported plates with high aspect ratio. The free vibration response for the first and the second symmetric-antisymmetric (SA) modes were found to be different for high aspect ratio. The results reveal that using insufficient number of degrees of freedom results in finding a totally different relation between the super-elliptical power and the frequency.

Keywords

References

  1. Abdelbari, S., Fekrar, A., Heireche, H., Saidi, H., Tounsi, A. and Bedia, E.A.A. (2016), "An efficient and simple shear deformation theory for free vibration of functionally graded rectangular plates on Winkler- Pasternak elastic foundations", Wind Struct., 22(3), 329-348. https://doi.org/10.12989/was.2016.22.3.329
  2. Altay, G. and Dokmeci, M.C. (2006), "A polar theory for vibrations of thin elastic shells", Int. J. Solid. Struct., 43(9), 2578-2601. https://doi.org/10.1016/j.ijsolstr.2005.06.027
  3. Altekin, M. and Altay, G. (2008), "Static analysis of point-supported super-elliptical plates", Arch. Appl. Mech., 78(4), 259-266. https://doi.org/10.1007/s00419-007-0154-9
  4. Altekin, M. (2008), "Free linear vibration and buckling of super-elliptical plates resting on symmetrically distributed point-supports on the diagonals", Thin-Wall. Struct., 46(10), 1066-1086.
  5. Altekin, M. (2009), "Free vibration of orthotropic super-elliptical plates on intermediate supports", Nuclear Eng. Des., 239(6), 981-999. https://doi.org/10.1016/j.nucengdes.2009.02.007
  6. Altekin, M. (2010a), "Bending of orthotropic super-elliptical plates on intermediate point supports", Ocean Eng., 37(11-12), 1048-1060. https://doi.org/10.1016/j.oceaneng.2010.03.015
  7. Altekin, M. (2010b), "Free in-plane vibration of super-elliptical plates", Shock Vib., 18(3), 471-484. https://doi.org/10.1155/2011/613521
  8. Altekin, M. (2014), "Large deflection analysis of point supported super-elliptical plates", Struct. Eng. Mech., 51(2), 333-347. https://doi.org/10.12989/sem.2014.51.2.333
  9. Bathe, K.J. (1996), Finite Element Procedures, Prentice Hall, New Jersey, USA.
  10. Bayer, I., Guven, U. and Altay, G. (2002), "A parametric study on vibrating clamped elliptical plates with variable thickness", J. Sound Vib., 254(1), 179-188. https://doi.org/10.1006/jsvi.2001.4099
  11. 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
  12. Bhat, R.B., Singh, J. and Mundkur, G. (1993), "Plate characteristic functions and natural frequencies of vibration of plates by iterative reduction of partial differential equation", J. Vib. Acoust., 115(2), 177-181. https://doi.org/10.1115/1.2930328
  13. Ceribasi, S. and Altay, G. (2009), "Free vibration of super elliptical plates with constant and variable thickness by Ritz method", J. Sound Vib., 319(1-2), 668-680. https://doi.org/10.1016/j.jsv.2008.06.010
  14. Ceribasi, S. (2012), "Static and dynamic analysis of thin uniformly loaded super elliptical FGM plates", Mech. Adv. Mater. Struct., 19(5), 325-335.
  15. Chakraverty, S. (2009), Vibration of Plates, CRC Press, Boca Raton, USA.
  16. Chen, C.C., Lim, C.W., Kitipornchai, S. and Liew, K.M. (1999), "Vibration of symmetrically laminated thick super elliptical plates", J. Sound Vib., 220(4), 659-682. https://doi.org/10.1006/jsvi.1998.1957
  17. Chen, C.C. and Kitipornchai, S. (2000), "Free vibration of symmetrically laminated thick perforated plates", J. Sound Vib., 230(1), 111-132. https://doi.org/10.1006/jsvi.1999.2612
  18. Civalek, O. and Ulker, M. (2004), "Harmonic differential quadrature (HDQ) for axisymmetric bending analysis of thin isotropic circular plates", Struct. Eng. Mech., 17(1), 1-14. https://doi.org/10.12989/sem.2004.17.1.001
  19. Civalek, O. (2007), "Three-dimensional vibration, buckling and bending analyses of thick rectangular plates based on discrete singular convolution method", Int. J. Mech. Sci., 49(6), 752-765. https://doi.org/10.1016/j.ijmecsci.2006.10.002
  20. Civalek, O. and Ersoy, H. (2009), "Free vibration and bending analysis of circular Mindlin plates using singular convolution method", Commun. Numer. Meth. Eng., 25(8), 907-922. https://doi.org/10.1002/cnm.1138
  21. Civalek, O. and Ozturk, B. (2010), "Vibration analysis of plates with curvilinear quadrilateral domains by discrete singular convolution method", Struct. Eng. Mech., 36(3), 279-299. https://doi.org/10.12989/sem.2010.36.3.279
  22. DeCapua, N.J. and Sun, B.C. (1972), "Transverse vibration of a class of orthotropic plates", J. Appl. Mech., 39(2), 613-615. https://doi.org/10.1115/1.3422735
  23. Duran, R.G., Hervella-Nieto, L., Liberman, E., Rodriguez, R. and Solomin, J. (1999), "Approximation of the vibration modes of a plate by Reissner-Mindlin equations", Math. Comput., 68(228), 1447-1463. https://doi.org/10.1090/S0025-5718-99-01094-7
  24. Eftekhari, S.A. and Jafari, A.A. (2013), "A simple and accurate Ritz formulation for free vibration of thick rectangular and skew plates with general boundary conditions", Acta Mechanica, 224(1), 193-209. https://doi.org/10.1007/s00707-012-0737-6
  25. El-Sayad, M.A. and Ghazy, S.S.A. (2012), "Rayleigh-Ritz method for free vibration of mindlin trapezoidal plates", Can. J. Sci. Eng. Math., 3(4), 159-166.
  26. Farag, A.M., Mohamed, W.F., Ata, A.A. and Burhamy, B.M. (2013), "Computational initial value method for vibration analysis of symmetrically laminated composite plate", Int. Scholar. Scientific Res. Innovat., 7(1), 545-554.
  27. Geannakakes, G.N. (1995), "Natural frequencies of arbitrarily shaped plates using the Rayleigh-Ritz method together with natural co-ordinate regions and normalized characteristic orthogonal polynomials", J. Sound Vib., 182(3), 441-478. https://doi.org/10.1006/jsvi.1995.0210
  28. Gutierrez, R.H., Laura, P.A.A. and Rossit, C.A. (2000), "Fundamental frequency of transverse vibration of a clamped rectangular orthotropic plate with a free-edge hole", J. Sound Vib., 235(4), 697-701. https://doi.org/10.1006/jsvi.2000.2910
  29. Han, J.B. and Liew, K.M. (1997), "Analysis of moderately thick circular plates using differential quadrature method", J. Eng. Mech., 123(12), 1247-1252. https://doi.org/10.1061/(ASCE)0733-9399(1997)123:12(1247)
  30. Han, J.B. and Liew, K.M. (1999), "Axisymmetric free vibration of thick annular plates", Int. J. Mech. Sci., 41(9), 1089-1109. https://doi.org/10.1016/S0020-7403(98)00057-5
  31. Hashemi, S.H. and Arsanjani, M. (2005), "Exact characteristic equations for some of classical boundary conditions of vibrating moderately thick rectangular plates", Int. J. Solid. Struct., 42(3-4), 819-583. https://doi.org/10.1016/j.ijsolstr.2004.06.063
  32. Hasheminejad, S.M., Keshvari, M.M. and Ashory, M.R. (2014), "Dynamic stability of super elliptical plates resting on elastic foundations under periodic in-plane loads", J. Eng. Mech., 140(1), 172-181. https://doi.org/10.1061/(ASCE)EM.1943-7889.0000630
  33. Hughes, T.J.R., Taylor, R.L. and Kanoknukulchai, W. (1977), "A simple and efficient finite element for plate bending", Int. J. Numer. Method. Eng., 11(10), 1529-1543. https://doi.org/10.1002/nme.1620111005
  34. Irie, T., Yamada, G. and Aomura, S. (1980), "Natural frequencies of Mindlin circular plates", J. Appl. Mech., 47(3), 652-655. https://doi.org/10.1115/1.3153748
  35. Iyengar, K.T.S.R. and Raman, P.V. (1978), "Free vibration of circular plates of arbitrary thickness", J. Acoust. Soc. Am., 64(4), 1088-1092. https://doi.org/10.1121/1.382068
  36. Jazi, S.R. and Farhatnia, F. (2012), "Buckling analysis of functionally graded super elliptical plate using pb- 2 Ritz method", Adv. Mater. Res., 383-390, 5387-5391.
  37. Kim, C.S. (2003), "Natural frequencies of orthotropic elliptical and circular plates", J. Sound Vib., 259(3), 733-745. https://doi.org/10.1006/jsvi.2002.5278
  38. Krishnamoorthy, C.S. (1994), Finite Element Analysis: Theory and Programming, (Second Edition), Tata McGraw-Hill Publishing Company Limited, New Delhi.
  39. Kutlu, A. and Omurtag, M.H. (2012), "Large deflection bending analysis of elliptic plates on orthotropic elastic foundation with mixed finite element method", Int. J. Mech. Sci., 65(1), 64-74. https://doi.org/10.1016/j.ijmecsci.2012.09.004
  40. Kutlu, A., Ugurlu, B., Omurtag, M.H. and Ergin, A. (2012), "Dynamic response of Mindlin plates resting on arbitrarily orthotropic Pasternak foundation and partially in contact with fluid", Ocean Eng., 42, 112-125. https://doi.org/10.1016/j.oceaneng.2012.01.010
  41. Lam, K.Y. and Hung, K.C. (1990), "Orthogonal polynomials and sub-sectioning method for vibration of plates", Comput. Struct., 34(6), 827-834. https://doi.org/10.1016/0045-7949(90)90353-4
  42. 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(2), 261-269. https://doi.org/10.1016/0022-460X(92)90580-Q
  43. Leissa, A.W. (1973), "The free vibration of rectangular plates", J. Sound Vib., 31(3), 257-293. https://doi.org/10.1016/S0022-460X(73)80371-2
  44. Leissa, A. (1993), Vibration of Plates, Acoustical Society of America, Columbus, USA.
  45. Liew, K.M., Lam, K.Y. and Chow, S.T. (1990), "Free vibration analysis of rectangular plates using orthogonal plate function", Comput. Struct., 34(1), 79-85. https://doi.org/10.1016/0045-7949(90)90302-I
  46. Liew, K.M., Xiang, Y. and Kitipornchai, S. (1993a), "Transverse vibration of thick rectangular plates-I. Comprehensive sets of boundary conditions", Comput. Struct., 49(1), 1-29. https://doi.org/10.1016/0045-7949(93)90122-T
  47. Liew, K.M., Hung, K.C. and Lim, K.M. (1993b), "A continuum three-dimensional vibration analysis of thick rectangular plates", Int. J. Solid. Struct., 30(24), 3357-3379. https://doi.org/10.1016/0020-7683(93)90089-P
  48. Liew, K.M., Xiang, Y. and Kitipornchai, S. (1995), "Research on thick plate vibration: a literature survey", J. Sound Vib., 180(1), 163-176. https://doi.org/10.1006/jsvi.1995.0072
  49. Liew, K.M. (1996), "Solving the vibration of thick symmetric laminates by Reissner/Mindlin plate theory and the p-Ritz method", J. Sound Vib., 198(3), 343-360. https://doi.org/10.1006/jsvi.1996.0574
  50. Liew, K.M., Wang, C.M., Xiang, Y. and Kitipornchai, S. (1998a), Vibration of Mindlin Plates, Elsevier Science Ltd., Oxford, UK.
  51. Liew, K.M., Kitipornchai, S. and Lim, C.W. (1998b), "Free vibration analysis of thick superelliptical plates", J. Eng. Mech., 124(2), 137-145. https://doi.org/10.1061/(ASCE)0733-9399(1998)124:2(137)
  52. Liew, K.M. and Theo, T.M. (1999), "Three-dimensional vibration analysis of rectangular plates based on differential quadrature method", J. Sound Vib., 220(4), 577-599. https://doi.org/10.1006/jsvi.1998.1927
  53. Liew, K.M. and Feng, Z.C. (2001), "Three-dimensional free vibration analysis of perforated superelliptical plates via the p-Ritz method", Int. J. Mech. Sci., 43(11), 2613-2630. https://doi.org/10.1016/S0020-7403(01)00051-0
  54. Liew, K.M., Wang, J., Ng, T.Y. and Tan, M.J. (2004), "Free vibration and buckling analysis of sheardeformable plates based on FSDT meshfree method", J. Sound Vib., 276(3-5), 997-1017. https://doi.org/10.1016/j.jsv.2003.08.026
  55. Lim, C.W. and Liew, K.M. (1995), "Vibrations of perforated plates with rounded corners", J. Eng. Mech., 121(2), 203-213. https://doi.org/10.1061/(ASCE)0733-9399(1995)121:2(203)
  56. Lim, C.W., Kitipornchai, S. and Liew, K.M. (1998), "A free-vibration analysis of doubly connected superelliptical laminated composite plates", Compos. Sci. Technol., 58(3-4), 435-445. https://doi.org/10.1016/S0266-3538(97)00167-X
  57. Lin, C.C. and Tseng, C.S. (1998), "Free vibration of polar orthotropic laminated circular and annular plates", J. Sound Vib., 209(5), 797-810. https://doi.org/10.1006/jsvi.1997.1293
  58. Ma, Y.Q. and Ang, K.K. (2006), "Free vibration of Mindlin plates based on the relative displacement plate element", Finite Element. Anal. Des., 42(11), 1021-1028. https://doi.org/10.1016/j.finel.2006.03.001
  59. Nallim, L.G. and Grossi, R.O. (2008), "Natural frequencies of symmetrically laminated elliptical and circular plates", Int. J. Mech. Sci., 50(7), 1153-1167. https://doi.org/10.1016/j.ijmecsci.2008.04.005
  60. Narita, Y. (1984), "Note on vibrations of point supported rectangular plates", J. Sound Vib., 93(4), 593-597. https://doi.org/10.1016/0022-460X(84)90428-0
  61. Narita, Y. (1986), "Free vibration analysis of orthotropic elliptical plates resting on arbitrarily distributed point supports", J. Sound Vib., 108(1), 1-10. https://doi.org/10.1016/S0022-460X(86)80306-6
  62. Rao, S.S. and Prasad, A.S. (1975), "Vibrations of annular plates including the effects of rotatory inertia and transverse shear deformation", J. Sound Vib., 42(3), 305-324. https://doi.org/10.1016/0022-460X(75)90247-3
  63. Reddy, J.N. (1993), An Introduction to the Finite Element Method, (Second Edition), McGraw-Hill International editions, Singapore.
  64. Senjanovic, I., Vladimir, N. and Tomic, M. (2013), "An advanced theory of moderately thick plate vibrations", J. Sound Vib., 332(7), 1868-1880. https://doi.org/10.1016/j.jsv.2012.11.022
  65. Senjanovic, I., Hadzic, N., Vladimir, N. and Cho, D.-S. (2014), "Natural vibrations of thick circular plate based on the modified Mindlin theory", Arch. Mech., 66(6), 389-409.
  66. Singh, B. and Chakraverty, S. (1992), "On the use of orthogonal polynomials in the Rayleigh-Ritz method for the study of transverse vibration of elliptic plates", Comput. Struct., 43(3), 439-443. https://doi.org/10.1016/0045-7949(92)90277-7
  67. Szilard, R. (1974), Theory and Analysis of Plates, Prentice Hall, Englewood Cliffs, USA.
  68. Szilard, R. (2004), Theories and Applications of Plate Analysis, John Wiley & Sons Inc., USA.
  69. Tang, H.W., Yang, Y.T. and Chen, C.K. (2012), "Application of new double side approach method to the solution of super-elliptical plate problems", Acta Mechanica, 223(4), 745-753. https://doi.org/10.1007/s00707-011-0592-x
  70. Thai, H.T. and Choi, D.H. (2013), "Analytical solutions of refined plate theory for bending, buckling and vibration analyses of thick plates", Appl. Math. Model., 37(18-19), 8310-8323. https://doi.org/10.1016/j.apm.2013.03.038
  71. 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
  72. Wang, C.M. and Wang, L. (1994), "Vibration and buckling of super elliptical plates", J. Sound Vib., 171(3), 301-314. https://doi.org/10.1006/jsvi.1994.1122
  73. Wang, C.M., Xiang, Y. and Kitipornchai, S. (1995), "Vibration frequencies for elliptical and semi-elliptical Mindlin plates", Struct. Eng. Mech., 3(1), 35-48. https://doi.org/10.12989/sem.1995.3.1.035
  74. Wang, C.Y. (2015a), "Vibrations of completely free rounded rectangular plates", J. Vibration Acoust., 137(2), doi:10.1115/1.4029159.
  75. Wang, C.Y. (2015b), "Vibrations of completely free rounded regular polygonal plates", Int. J. Acoust. Vib., 20(2), 107-112.
  76. Wang, X., Yang, J. and Xiao, J. (1995a), "On free vibration analysis of circular annular plates with nonuniform thickness by the differential quadrature method", J. Sound Vib., 184(3), 547-551. https://doi.org/10.1006/jsvi.1995.0332
  77. Wu, T.Y. and Liu, G.R. (2001), "Free vibration analysis of circular plates with variable thickness by the generalized differential quadrature rule", Int. J. Solid. Struct., 38(44-45), 7967-7980. https://doi.org/10.1016/S0020-7683(01)00077-4
  78. Wu, L. and Liu, J. (2005), "Free vibration analysis of arbitrary shaped thick plates by differential cubature method", Int. J. Mech. Sci., 47(1), 63-81. https://doi.org/10.1016/j.ijmecsci.2004.12.003
  79. Zhang, D.G. (2013), "Non-linear bending analysis of super-elliptical thin plates", Int. J. Non-Linear Mech., 55, 180-185. https://doi.org/10.1016/j.ijnonlinmec.2013.06.006
  80. Zhang, D.G. and Zhou, H.M. (2014), "Nonlinear symmetric free vibration analysis of super elliptical isotropic thin plates", CMC: Computers, Materials & Continua, 40(1), 21-34.
  81. Zhong, H. and Yu, T. (2007), "Flexural vibration analysis of an eccentric annular Mindlin plate", Arch. Appl. Mech., 77(4), 185-195. https://doi.org/10.1007/s00419-006-0083-z
  82. Zhou, D., Cheung, Y.K., Au, F.T.K. and Lo, S.H. (2002), "Three-dimensional vibration analysis of thick rectangular plates using Chebyshev polynomial and Ritz method", Int. J. Solid. Struct., 39(26), 6339- 6353. https://doi.org/10.1016/S0020-7683(02)00460-2
  83. Zhou, D., Au, F.T.K., Cheung, Y.K. and Lo, S.H. (2003), "Three-dimensional vibration analysis of circular and annular plates via the Chebyshev-Ritz method", Int. J. Solid. Struct., 40(12), 3089-3105. https://doi.org/10.1016/S0020-7683(03)00114-8
  84. Zhou, D., Lo, S.H., Cheung, Y.K. and Au, F.T.K. (2004), "3-D vibration analysis of generalized super elliptical plates using Chebyshev-Ritz method", Int. J. Solid. Struct., 41(16-17), 4697-4712. https://doi.org/10.1016/j.ijsolstr.2004.02.045
  85. Zhou, D., Au, F.T.K., Cheung, Y.K. and Lo, S.H. (2006), "Effect of built-in edges on 3-D vibrational characteristics of thick circular plates", Int. J. Solid. Struct., 43(7-8), 1960-1978. https://doi.org/10.1016/j.ijsolstr.2005.05.007
  86. http://people.sc.fsu.edu/-jpeterson/FEMbook.pdf

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