Structural Engineering and Mechanics

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Vibration analysis of laminated plates with various boundary conditions using extended Kantorovich method

  • Singhatanadgid, Pairod (Department of Mechanical Engineering, Faculty of Engineering, Chulalongkorn University) ;
  • Wetchayanon, Thanawut (Department of Mechanical Engineering, Faculty of Engineering, Chulalongkorn University)
  • Received : 2012.12.12
  • Accepted : 2014.06.29
  • Published : 2014.10.10
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Abstract

In this study, an extended Kantorovich method, employing multi-term displacement functions, is applied to analyze the vibration problem of symmetrically laminated plates with arbitrary boundary conditions. The vibration behaviors of laminated plates are determined based on the variational principle of total energy minimization and the iterative Kantorovich method. The out-of-plane displacement is represented in the form of a series of a sum of products of functions in x and y directions. With a known function in the x or y directions, the formulation for the variation of total potential energy is transformed to a set of governing equations and a set of boundary conditions. The equations and boundary conditions are then numerically solved for the natural frequency and vibration mode shape. The solutions are verified with available solutions from the literature and solutions from the Ritz and finite element analysis. In most cases, the natural frequencies compare very well with the reference solutions. The vibration mode shapes are also very well modeled using the multi-term assumed displacement function in the terms of a power series. With the method used in this study, it is possible to solve the angle-ply plate problem, where the Kantorovich method with single-term displacement function is ineffective.

Keywords

Acknowledgement

Supported by : Chulalongkorn University, Commission on Higher Education

References

  1. Abouhamze, M., Aghdam, M.M. and Alijani, F. (2007), "Bending analysis of symmetrically laminated cylindrical panels using the extended Kantorovich method", Mech. Adv. Mater. Struct., 14(7), 523-530. https://doi.org/10.1080/15376490701585967
  2. Aghdam, M.M. and Falahatgar, S.R. (2003), "Bending analysis of thick laminated plates using extended Kantorovich method", Compos. Struct., 62(3-4), 279-283. https://doi.org/10.1016/j.compstruct.2003.09.026
  3. Aghdam, M.M., Mohammadi, M. and Erfanian, V. (2007), "Bending analysis of thin annular sector plates using extended Kantorovich method", Thin Wall. Struct., 45(12), 983-990. https://doi.org/10.1016/j.tws.2007.07.012
  4. Bercin, A.N. (1996), "Free vibration solution for clamped orthotropic plates using the Kantorovich method", J. Sound. Vib., 196(2), 243-247. https://doi.org/10.1006/jsvi.1996.0479
  5. Chen, X.L., Liu, G.R. and Lim, S.P. (2003), "An element free Galerkin method for the free vibration analysis of composite laminates of complicated shape", Compos. Struct., 59(2), 279-289. https://doi.org/10.1016/S0263-8223(02)00034-X
  6. Chow, S.T., Liew, K.M. and Lam, K.Y. (1992), "Transverse vibration of symmetrically laminated rectangular composite plates", Compos. Struct., 20(4), 213-226. https://doi.org/10.1016/0263-8223(92)90027-A
  7. Dalaei, M. and Kerr, A.D. (1996), "Natural vibration analysis of clamped rectangular orthotropic plates", J. Sound. Vib., 189(3), 399-406. https://doi.org/10.1006/jsvi.1996.0026
  8. Eisenberger, M. and Alexandrov, A. (2003), "Buckling loads of variable thickness thin isotropic plates", Thin Wall. Struct., 41(9), 871-889. https://doi.org/10.1016/S0263-8231(03)00027-2
  9. Kerr, A.D. (1969), "An extended Kantorovich method for the solution of eigenvalue problems", Int. J. Solid. Struct., 5(6), 559-572. https://doi.org/10.1016/0020-7683(69)90028-6
  10. Lee, J.M., Chung, J.H. and Chung, T.Y. (1997), "Free vibration analysis of symmetrically laminated composite rectangular plates", J. Sound. Vib., 199(1), 71-85. https://doi.org/10.1006/jsvi.1996.0653
  11. Leissa, A.W. and Narita, Y. (1989), "Vibration studies for simply supported symmetrically laminated regular plates", Compos. Struct., 12(2), 113-132. https://doi.org/10.1016/0263-8223(89)90085-8
  12. Maheri, M.R. and Adams, R.D. (2003), "Modal vibration damping of anisotropic FRP laminates using the Rayleigh-Ritz energy minimization scheme", J. Sound. Vib., 259(1), 17-29. https://doi.org/10.1006/jsvi.2002.5151
  13. Reddy, J.N. (2003), Mechanics of laminated composite plates and shells, CRC Press, Boca Raton.
  14. Shen, H.S., Chen, Y. and Yang, J. (2003), "Bending and vibration characteristics of a strengthened plate under various boundary conditions", Eng. Struct., 25(9), 1157-1168. https://doi.org/10.1016/S0141-0296(03)00063-4
  15. Shufrin, I. and Eisenberger, M. (2005), "Stability and vibration of shear deformable plates - first order and higher order analyses", Int. J. Solid. Struct., 42(3-4), 1225-1251. https://doi.org/10.1016/j.ijsolstr.2004.06.067
  16. Shufrin, I. and Eisenberger, M. (2006), "Vibration of shear deformable plates with variable thickness - Firstorder and higher-order analyses", J. Sound. Vib., 290(1-2), 465-489. https://doi.org/10.1016/j.jsv.2005.04.003
  17. Shufrin, I., Rabinovitch, O. and Eisenberger, M. (2008a), "Buckling of symmetrically laminated rectangular plates with general boundary conditions - A semi analytical approach", Compos. Struct., 82(4), 521-531. https://doi.org/10.1016/j.compstruct.2007.02.003
  18. Shufrin, I., Rabinovitch, O. and Eisenberger, M. (2008b), "Buckling of laminated plates with general boundary conditions under combined compression, tension, and shear-A semi-analytical solution", Thin Wall. Struct., 46(7-9), 925-938. https://doi.org/10.1016/j.tws.2008.01.040
  19. Ungbhakorn, V. and Singhatanadgid, P. (2006), "Buckling analysis of symmetrically laminated composite plates by the extended Kantorovich method", Compos. Struct., 73(1), 120-128. https://doi.org/10.1016/j.compstruct.2005.02.007
  20. Whitney, J.M. (1987), Structural analysis of laminated anisotropic plates, Technomic, Lancaster, PA.
  21. Yuan, S. and Jin, Y. (1998), "Computation of elastic buckling loads of rectangular thin plates using the extended Kantorovich method", Compos. Struct., 66(6), 861-867. https://doi.org/10.1016/S0045-7949(97)00111-9
  22. Yuan, S., Jin, Y. and Williams, F.W. (1998), "Bending analysis of Mindlin plates by extended Kantorovich method", J. Eng. Mech., 124(12), 1339-1345. https://doi.org/10.1061/(ASCE)0733-9399(1998)124:12(1339)

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