Structural Engineering and Mechanics

Techno-Press (테크노프레스)
권호 표지
DOI QR코드

DOI QR Code

Free vibration of circular and annular membranes with varying density by the method of discrete singular convolution

  • Ersoy, Hakan (Akdeniz University, Faculty of Engineering, Mechanical Engineering Department Division of Mechanics) ;
  • Ozpolat, Lutfiye (Akdeniz University, Faculty of Engineering, Civil Engineering Department Division of Mechanics) ;
  • Civalek, Omer (Akdeniz University, Faculty of Engineering, Civil Engineering Department Division of Mechanics)
  • Received : 2008.12.16
  • Accepted : 2009.06.15
  • Published : 2009.07.30
⟨ Previous Next ⟩

Abstract

A numerical method is developed to investigate the effects of some geometric parameters and density variation on frequency characteristics of the circular and annular membranes with varying density. The discrete singular convolution method based on regularized Shannon's delta kernel is applied to obtain the frequency parameter. The obtained results have been compared with the analytical and numerical results of other researchers, which showed well agreement.

Keywords

Acknowledgement

Supported by : Akdeniz University

References

  1. Buchanan, G.R. (2005), "Vibration of circular membranes with linearly varying density along a diameter", J.Sound Vib., 280, 407-414 https://doi.org/10.1016/j.jsv.2004.01.043
  2. Buchanan, G.R. and Peddieson, Jr. J. (1999), "Vibration of circular and annular membranes with variable density", J. Sound Vib., 226(2), 379-382 https://doi.org/10.1006/jsvi.1999.2177
  3. Buchanan, G.R. and Peddieson, Jr. J. (2005), "A finite element in elliptic coordinates with application of membrane vibration", Thin Wall. Struct., 43, 1444-1454 https://doi.org/10.1016/j.tws.2005.04.001
  4. Casperson, L.W. and Nicolet, M.A. (1968), "Vibrations of a circular membrane", Am. J. Phys., 36(8), 669-671 https://doi.org/10.1119/1.1975085
  5. Civalek, Ö. (2007), "Three-dimensional vibration, buckling and bending analyses of thick rectangular plates based on discrete singular convolution method", Int. J. Mech. Sci., 49, 752-765 https://doi.org/10.1016/j.ijmecsci.2006.10.002
  6. Civalek, Ö. (2006), "An efficient method for free vibration analysis of rotating truncated conical shells", Int. J. Press. Vess. Piping, 83, 1-12 https://doi.org/10.1016/j.ijpvp.2005.10.005
  7. Civalek, Ö. (2007), "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
  8. Civalek, Ö. (2007), "Free vibration and buckling analyses of composite plates with straight-sided quadrilateral domain based on DSC approach", Finite Elem. Anal. Des., 43, 1013-1022 https://doi.org/10.1016/j.finel.2007.06.014
  9. Civalek, Ö. (2007), "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
  10. Civalek, Ö. (2007), "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
  11. Gutierrez, R.H., Laura, P.A.A., Bambill, D.V. and Jederlinic, V.A.(1998), "Axisymmetric vibrations of solid circular and annular membranes with continuously varying density", J. Sound Vib., 212(4), 611-622 https://doi.org/10.1006/jsvi.1997.1418
  12. Han, J.B. and Liew, K.M. (1997), "Analysis of moderately thick circular plates using differential quadrature method", J. Eng. Mech., 123(2), 1247-1252 https://doi.org/10.1061/(ASCE)0733-9399(1997)123:12(1247)
  13. Hang, L.T.T., Wang, C.M. and Wu, T.Y. (2005), "Exact vibration results for stepped circular plates with free edges", Int. J. Mech. Sci., 47, 1224-1248 https://doi.org/10.1016/j.ijmecsci.2005.04.002
  14. Ho, S.H. and Chen, C.K. (2000), "Free vibration analysis of non-homogeneous rectangular membranes using a hybrid methods", J. Sound Vib., 233(3), 547-555 https://doi.org/10.1006/jsvi.1999.2808
  15. Houmat, A. (2001), "A sector Fourier p-element for free vibration analysis of sectorial membranes", Comput. Struct., 79, 1147-1152 https://doi.org/10.1016/S0045-7949(01)00013-X
  16. Houmat, A. (2006), "Free vibration analysis of arbitrarily shaped membranes using the trigonometric p-version of the finite element method", Thin Wall. Struct., 44, 943-951 https://doi.org/10.1016/j.tws.2006.08.022
  17. Jabareen, M. and Eisenberger, M. (2001), "Free vibrations of non-homogeneous circular and annular membranes", J. Sound Vib., 240(3), 409-429 https://doi.org/10.1006/jsvi.2000.3249
  18. Kang, S.W. and Lee, J.M. (2000), "Application of free vibration analysis of membranes using non-dimensional dynamic influence function", J. Sound Vib., 234, 455-470 https://doi.org/10.1006/jsvi.1999.2872
  19. Kang, S.W., Lee, J.M. and Kang, Y.J. (1999), "Vibration analysis of arbitrarily shaped membranes using nondimensional dynamic influence function", J. Sound Vib., 221, 117-132 https://doi.org/10.1006/jsvi.1998.2009
  20. Laura, P.A.A., Bambill, D.V. and Gutierrez, R.H. (1997), A note on transverse vibrations of circular, annular, composite membranes”, J. Sound Vib., 205(5), 692-697 https://doi.org/10.1006/jsvi.1996.0839
  21. Laura, P.A.A., Rossi, R.E. and Gutierrez, R.H. (1997), "The fundamental frequency of non-homogeneous rectangular membranes", J. Sound Vib., 204(2), 373-376 https://doi.org/10.1006/jsvi.1996.0931
  22. Leung, A.Y.T., Zhu, B., Zheng J. and Yang, H. (2003), "A trapezoidal Fourier p-element for membrane vibrations", Thin Wall. Struct., 41, 479-491 https://doi.org/10.1016/S0263-8231(02)00117-9
  23. Liew, K.M. and Liu, F.-L. (2000), "Differential quadrature method for vibration analysis of shear deformable annular sector plates", J. Sound Vib., 230(2), 335-356 https://doi.org/10.1006/jsvi.1999.2623
  24. Liew, K.M. and Yang, B. (2000), "Elasticity solution for free vibrations of annular plates from three-dimensional analysis", Int. J. Solids Struct., 37, 7689-7702 https://doi.org/10.1016/S0020-7683(99)00306-6
  25. Liew, K.M., Han, J.-B. and Xiao, Z.M. (1997), "Vibration analysis of circular Mindlin plates using the differential quadrature method", J. Sound Vib., 205(5), 617-630 https://doi.org/10.1006/jsvi.1997.1035
  26. Lim, C.W., Li, Z.R. and Wei, G.W. (2005), "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
  27. Lim, C.W., Li, Z.R., Xiang, Y., Wei, G.W. and Wang, C.M. (2005), "On the missing modes when using the exact frequency relationship between Kirchhoff and Mindlin plates", Adv. Vib. Eng., 4, 221-248
  28. 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
  29. Masad, J.A. (1996), "Free vibrations of a non-homogeneous rectangular membrane", J. Sound Vib., 195, 674-678 https://doi.org/10.1006/jsvi.1996.0454
  30. Mei, C. (1969), "Free vibrations of circular membranes under arbitrary tension by the finite element method", J.Acoust. Soc. Am., 46(3), 693-700 https://doi.org/10.1121/1.1911750
  31. Oden, J.T. and Sato, T. (1697), "Finite strains and displacements of elastic membranes by the finite element method", Int. J. Solids Struct., 3, 471-488 https://doi.org/10.1016/0020-7683(67)90002-9
  32. Pronsato, M.E., Laura, P.A.A. and Juan, A. (1999), "Transverse vibrations of a rectangular membrane with discontinuously varying density", J. Sound Vib., 222(2), 341-344 https://doi.org/10.1006/jsvi.1998.2021
  33. Shu, C. (1996), "Free vibration analysis of composite laminated conical shells by generalized differential quadrature", J. Sound Vib., 194, 587-604 https://doi.org/10.1006/jsvi.1996.0379
  34. Shu, C. (1999), "Application of differential quadrature method to simulate natural convection in a concentricannulus", Int. J. Numer. Meth. Fluids, 30, 977-933 https://doi.org/10.1002/(SICI)1097-0363(19990830)30:8<977::AID-FLD873>3.0.CO;2-J
  35. Shu, C. and Du, H. (1997), "A generalized approach for implementing general boundary conditions in the GDQ free vibration analysis of plates", Int. J. Solids Struct., 34, 837-846 https://doi.org/10.1016/S0020-7683(96)00056-X
  36. Shu, C. and Richards, B.E. (1992), "Application of generalized differential quadrature to solve two-dimensional incompressible navier-stokes equations", Int. J. Numer. Meth. Fluids, 15, 791-798 https://doi.org/10.1002/fld.1650150704
  37. Shu, C. and Xue, H. (1997), "Explicit computations of weighting coefficients in the harmonic differential quadrature", J. Sound Vib., 204(3), 549-555 https://doi.org/10.1006/jsvi.1996.0894
  38. Shu, C. and Xue, H. (1998), "Comparison of two approaches for implementing stream function boundary conditions in DQ simulation of natural convection in a square cavity", Int. J. Heat Fluid Flow, 19, 59-68 https://doi.org/10.1016/S0142-727X(97)10010-8
  39. Shu, C., Chen, W. and Du, H. (2000), "Free vibration analysis of curvilinear quadrilateral plates by the differential quadrature method", J. Comp. Phy., 163, 452-466 https://doi.org/10.1006/jcph.2000.6576
  40. Wang, C.M. and Lee, K.H. (1996), "Deflection and stress-resultants of axisymmetric Mindlin plates in terms of corresponding Kirchhoff solutions", Int. J. Mech. Sci., 38(11), 1179-1185 https://doi.org/10.1016/0020-7403(96)00019-7
  41. Wang, C.M., Xiang, Y., Watanabe, E. and Usunomiya, 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
  42. Wang, X. and Wang, Y. (2004), "Re-analysis of free vibration of annular plates by the new version of differential quadrature method", J. Sound Vib., 278(3), 685-689 https://doi.org/10.1016/j.jsv.2003.12.008
  43. 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
  44. Wei, G.W. (2001), "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
  45. Wei, G.W. (2001), "Discrete singular convolution for beam analysis", Eng. Struct., 23, 1045-1053 https://doi.org/10.1016/S0141-0296(01)00016-5
  46. Wei, G.W. (2001), "Vibration analysis by discrete singular convolution", J. Sound Vib. 244, 535-553 https://doi.org/10.1006/jsvi.2000.3507
  47. Wei, G.W., Zhao Y.B. and Xiang, Y. (2002), "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
  48. Wei, G.W., Zhao Y.B. and Xiang, Y. (2002), "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
  49. Willatzen, M. (2002), "Exact power series solutions for axisymmetric vibrations of circular and annular membranes with continuously varying density in the general case", J. Sound Vib., 258(5), 981-986 https://doi.org/10.1006/jsvi.2002.5155
  50. Wu, W.X., Shu, C. and Wang, C.M. (2007), "Vibration analysis of arbitrarily shaped membranes using local radial basis function-based differential quadrature method", J. Sound Vib., 306, 252-270 https://doi.org/10.1016/j.jsv.2007.05.015
  51. Xiang, Y. (2002), "Exact vibration solutions for circular Mindlin plates with multiple concentric ring supports", Int. J. Solids Struct., 39, 6081-6102 https://doi.org/10.1016/S0020-7683(02)00494-8
  52. Xiang, Y. (2003), "Vibration of circular Mindlin plates with concentric elastic ring supports", Int. J. Mech. Sci., 45(3), 497-517 https://doi.org/10.1016/S0020-7403(03)00059-6
  53. Xiang, Y. (2003), "Vibration of circular Mindlin plates with concentric elastic ring supports", Int. J. Mech. Sci.,45, 497-517 https://doi.org/10.1016/S0020-7403(03)00059-6
  54. Xiang, Y. and Zhang, L. (2005), "Free vibration analysis of stepped circular Mindlin plates", J. Sound Vib., 280,633-655 https://doi.org/10.1016/j.jsv.2003.12.017
  55. Xiang, Y., Liew, K.M. and Kitipornchai, S. (1993), "Transverse vibration of thick annular sector plates", J. Eng.Mech., 119, 1579-1597 https://doi.org/10.1061/(ASCE)0733-9399(1993)119:8(1579)
  56. Xiang, Y., Zhao, Y.B. and Wei, G.W. (2002), "Discrete singular convolution and its application to the analysis of plates with internal supports. Part 2: Applications", Int. J. Numer. Meth. Eng., 55, 947-971 https://doi.org/10.1002/nme.527
  57. 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
  58. Zhao, Y.B., Wei, G.W. and Xiang, Y. (2002), "Plate vibration under irregular internal supports", Int. J. SolidsStruct., 39, 1361-1383 https://doi.org/10.1016/S0020-7683(01)00241-4

Cited by

  1. Free vibration analysis of composite, circular annular membranes using wave propagation approach vol.39, pp.16, 2015, https://doi.org/10.1016/j.apm.2015.03.057
  2. Generalized Differential Quadrature Finite Element Method for vibration analysis of arbitrarily shaped membranes vol.79, 2014, https://doi.org/10.1016/j.ijmecsci.2013.12.008
  3. Strong Formulation IsoGeometric Analysis for the vibration of thin membranes of general shape vol.120, 2017, https://doi.org/10.1016/j.ijmecsci.2016.10.033
  4. Power series solution of circular membrane under uniformly distributed loads: investigation into Hencky transformation vol.45, pp.5, 2009, https://doi.org/10.12989/sem.2013.45.5.631
  5. Closed-form solution of axisymmetric deformation of prestressed Föppl-Hencky membrane under constrained deflecting vol.69, pp.6, 2009, https://doi.org/10.12989/sem.2019.69.6.693
  6. A Review on the Discrete Singular Convolution Algorithm and Its Applications in Structural Mechanics and Engineering vol.27, pp.5, 2009, https://doi.org/10.1007/s11831-019-09365-5