Abstract
A p-version, hierarchical finite element for moderately thick isotropic plates is derived and free vibrations are studied. The effects of the rotatory inertia, transverse shear and geometrical non-linearity, due to moderately large displacements, are taken into account. The time domain free vibration equations of motion are obtained by applying the principle of virtual work, and are mapped into the frequency domain by the harmonic balance method. The ensuing frequency domain equations are solved by a predictor–corrector method. The convergence properties of the element, the influence of the plate's thickness and of the width to length ratio on the backbone curves and on the non-linear mode shapes are investigated. The first and higher order modes are analysed and results are compared with published results.
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References
Han, W. and Petyt, M., ‘Geometrically non-linear vibration analysis of thin, rectangular plates using the hierarchical finite element method — I: The fundamental mode of isotropic plates’, Comput. Struct. 63 (1997) 295-308.
Han, W. and Petyt, M., ‘Geometrically non-linear vibration analysis of thin, rectangular plates using the hierarchical finite element method — II: 1st mode of laminated plates and higher modes of isotropic and laminated plates’, Comput. Struct. 63 (1997) 309-318.
Ribeiro, P. and Petyt, M., ‘Nonlinear vibration of plates by the hierarchical finite element and continuation methods’, Int. J. Mech. Sci. 41 (1999) 437-459.
Ribeiro, P. and Petyt, M., ‘Nonlinear free vibration of isotropic plates with internal resonance’, Int. J. Nonlinear Mech. 35 (2000) 263-278.
Ribeiro, P. and Petyt, M., ‘Geometrical non-linear, steady-state, forced, periodic vibration of plates. Part I: Model and convergence studies’, J. Sound Vib. 226 (1999) 955-983.
Ribeiro, P. and Petyt, M., ‘Geometrical non-linear, steady-state, forced, periodic vibration of plates. Part II: Stability study and analysis of multi-modal, multi-frequency response’, J. Sound Vib. 226 (1999) 985-1010.
Mei, C. and Decha-Umphai, K., ‘A finite element method for non-linear forced vibrations of rectangular plates’, AIAA J. 23 (1985) 1104-1110.
Zienkiewicz, O.C. and Taylor, R.L., The Finite Element Method, 4th edn, McGraw-Hill, London, 1988.
Sathyamoorthy, M., ‘Nonlinear vibration analysis of plates: a review and survey of current developments’, Appl. Mech. Rev. 40 (1987) 1553-1561.
Reissner, E., ‘The effect of transverse shear deformation on the bending of elastic plates’, J. Appl. Mech. 12 (1945) 69-77.
Mindlin, R.D., ‘Influence of rotatory inertia and shear on flexural vibrations of isotropic, elastic plates’, J. Appl. Mech. 18 (1951) 31-38.
Petyt, M., Introduction to Finite Element Vibration Analysis, Cambridge University Press, Cambridge, 1990.
Liew, K.M., Wang, C.M., Xiang, X. and Kitipornchai, S., Vibration of Mindlin Plates. Programming the p-Version Ritz Method, Elsevier, Amsterdam, 1998.
Raju, K.K. and Hinton, E., ‘Nonlinear vibrations of thick plates using Mindlin plate elements’, Int. J. Num. Meth. Eng. 15 (1980) 249-257.
Mei, C. and Prasad, C.B., ‘Effects of large deflection and transverse shear on response of rectangular symmetric composite laminates subjected to acoustic excitation’, J. Comp. Mater. 23 (1989) 606-639.
Reddy, J.N., and Chao, W.C., ‘Large deflection and large-amplitude free vibrations of laminated composite-materials plates’, Comput. Struct. 13 (1981) 341-347.
Brockman, R.A., ‘Dynamics of the bilinear Mindlin plate element’, Int. J. Num. Meth. Eng. 24 (1987) 2343-2356.
Chen, W.C. and Liu, W.H., ‘Deflections and free vibrations of laminated plates — Levy-type solutions’, Int. J. Mech. Sci. 32 (1990) 779-793.
Ganapathi, M., Varadan, T.K. and Sarma, S.M., ‘Nonlinear flexural vibrations of laminated orthotropic plates’, Comput. Struct. 39 (1991) 685-688.
Reddy, J.N., ‘Theory and analysis of laminated composite plates’, in: Soares, C.A.M. and Freitas, M.J.M. (eds), Mechanics of Composite Materials and Structures, Kluwer Academic Publishers, Dordrecht, 1999, pp. 1-62.
Nayfeh, A.H. and Balachandra, B., Applied Nonlinear Dynamics: Analytical, Computational, and Experimental Methods, Wiley, New York, 1995.
Bardell, N.S., ‘Free vibration analysis of a flat plate using the hierarchical finite element method’, J. Sound Vib. 151 (1991) 263-289.
Leissa, A.W., Vibration of Plates, NASA Sp-60, 1969.
Raju, K.K., Rao, G.V. and Raju, I.S., ‘Effect of geometric nonlinearity on the free flexural vibrations of moderately thick rectangular plates’, Comput. Struct. 9 (1978) 441-444.
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Ribeiro, P. A Hierarchical Finite Element for Geometrically Non-linear Vibration of Thick Plates. Meccanica 38, 117–132 (2003). https://doi.org/10.1023/A:1022027619946
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DOI: https://doi.org/10.1023/A:1022027619946