On New Symplectic Approach for Exact Free Vibration Solutions of Moderately Thick Rectangular Plates with Two Opposite Edges Simply Supported

Year 2009, Volume: 1 Issue: 3, 13 - 28, 01.09.2009

R. Li Y. Zhong

Abstract

The purpose of this paper is to report the effective application of a new symplectic approach for exact free vibration solutions of moderately thick rectangular plates. By way of a simple but rigorous derivation, the governing differential equations for free vibration of the plates are transferred into Hamilton canonical equations. The whole state variables are then separated. Using the method of eigenfunction expansion in the symplectic geometry, the free vibration analysis of moderately thick rectangular plates with two opposite edges simply supported is performed and exact vibration solutions are obtained. The method eliminates the need to pre-determine any trial functions hence more reasonable than other available methods. Comprehensive numerical results are presented to validate the approach proposed here by comparison with those established in the open literature

Keywords

Moderately thick plate, free vibration, exact solution, symplectic approach

References

• [1] A.W. Leissa, Vibration of plates (NASA SP 160), Office of Technology Utilization, Washington, D.C., 1969.
• [2] D.J. Gorman, Free vibration analysis of rectangular plates, Elsevier, New York, 1982.
• [3] K.M. Liew, Y. Xiang, S. Kitipornchai, Research on thick plate vibration: a literature survey. Journal of Sound and Vibration 180 (1995) 163-176.
• [4] Institute of Mechanics, Chinese Academy of Science, Flexure, stability and vibration of sandwich plates and shells, Science Press, Beijing, 1977 (in Chinese).
• [5] C.W. Bert, J.N. Reddy, W.C. Chao, V.S. Reddy, Vibration of thick rectangular plates of bimodulus composite materials. J.Applied Mech. 48 (1981) 371-376.
• [6] Y.V.K.S. Rao, G. Singh, Vibration of corner supported thick composite plates. Journal of Sound and Vibration 111 (1986) 510-514.
• [7] C.N. Chang, F.K. Chiang, Vibration analysis of a thick plate with an interior cut-out by a finite element method. Journal of Sound and Vibration 125 (1988) 477-486.
• [8] J.N. Reddy, A.A. Khdeir, Buckling and vibration of laminated composite plates using various plate theories. American Institute of Aeronautics and Astronautics Journal 27 (1989) 1808-1817.
• [9] G. Aksu, S.A. Al-Kaabi, Free vibration analysis of Mindlin plates with linearly varying thickness. Journal of Sound and Vibration 119 (1987) 189-205.
• [10] G. Aksu, M.B. Felemban, Frequency analysis of corner point supported Mindlin plates by a finite difference energy method. Journal of Sound and Vibration 158 (1992) 531-544.
• [11] C.M. Wang, W.X. Wu, C. Shu, T. Utsunomiya, LSFD method for accurate vibration modes and modal stress-resultants of freely vibrating plates that model VLFS. Computers and Structures 84 (2006) 2329-2339.
• [12] D.J. Dawe, O.L. Roufaeil, Rayleigh_Ritz vibration analysis of Mindlin plates. Journal of Sound and Vibration 69 (1980) 345-359.
• [13] S. Wang, D.J. Dawe, Vibration of shear-deformable rectangular plates using a spline-function Rayleigh-Ritz approach. International Journal for Numerical Methods in Engineering 36 (1993) 695-711.
• [14] K.M. Liew, Y. Xiang, S. Kitipornchai, Transverse vibration of thick rectangular plates—I. Comprehensive sets of transverse boundary conditions. Computers and Structures 49 (1993) 1-29.
• [15] L.W. Chen, J.L. Doong, Large amplitude vibration of an initially stressed moderately thick plate. Journal of Sound and Vibration 89 (1983) 499-508.
• [16] J.A. Bowlus, A.N. Palazotto, J.M. Whitney, Vibration of symmetrically laminated rectangular plates considering deformation and rotatory inertia. American Institute of Aeronautics Journal 25 (1987) 1500-1511.
• [17] P.R. Benson, E. Hinton, A thick finite strip solution for static, free vibration and stability problems. Int. J. for Num. Meth. Eng. 10 (1976) 665-678.
• [18] D.J. Dawe, Finite strip models for vibration of Mindlin plates. Journal of Sound and Vibration 59 (1978) 441-452.
• [19] O.L. Roufaeil, D.J. Dawe, Vibration analyses of rectangular plates by the finite strip method. Computers and Structures 12 (1980) 833-842.
• [20] M.S. Cheung, M.Y.T. Chan, Static and dynamic analysis of thin and thick sectorial plates by the finite strip method. Comp.Struct. 14 (1981) 79-88.
• [21] T. Mizusawa, Vibration of rectangular Mindlin plates with tapered thickness by the spline strip method. Computers and Structures 46 (1993) 451-463.
• [22] T. Mikami, J. Yoshimura, Application of the collocation method to vibration analysis of rectangular Mindlin plates. Comp. Struct. 18 (1984) 425-431.
• [23] P. Malekzadeh, G. Karami, M. Farid, A semi-analytical DQEM for free vibration analysis of thick plates with two opposite edges simply supported. Computer Methods in Applied Mechanics and Engineering 193 (2004) 4781-4796.
• [24] Ö. Civalek, Three-dimensional vibration, buckling and bending analyses of thick rectangular plates based on discrete singular convolution method. International Journal of Mechanical Sciences 49 (2007) 752–765.
• [25] W.X. Zhong, F.W. Williams, Physical interpretation of the symplectic orthogonality of the eigensolutions of a Hamiltonian or symplectic matrix. Computational Mechanics in Structural Engineering 49 (1993) 749-750.
• [26] W.X. Zhong, X.X. Zhong, Method of separation of variables and Hamiltonian system. Numerical Methods for Partial Differential Equations 9 (1993) 63-75.
• [27] W.X. Zhong, A new systematic methodology for theory of elasticity, Dalian University of Technology Press, Dalian, 1995 (in Chinese).
• [28] W.X. Zhong, W.A. Yao, New solution system for plate bending. Computational Mechanics in Structural Engineering 31 (1999) 17-30.
• [29] W.A. Yao, W.X. Zhong, Symplectic elasticity, Higher Education Press, Beijing, 2002 (in Chinese).
• [30] W.A. Yao, W.X. Zhong, Symplectic solution system for Reissner plate bending. Applied Mathematics and Mechanics (English Edition) 25 (2004) 178-185.
• [31] C.W. Lim, S. Cui, W.A. Yao, On new symplectic elasticity approach for exact bending solutions of rectangular thin plates with two opposite sides simply supported. International Journal of Solids and Structures 44 (2007) 5396-5411.
• [32] Y. Zhong, R. Li, Y. Liu, B. Tian, On new symplectic approach for exact bending solutions of moderately thick rectangular plates with two opposite edges simply supported. International Journal of Solids and Structures 46 (2009) 2506-2513.
• [33] Y. Zhong, R. Li, Exact bending analysis of fully clamped rectangular thin plates subjected to arbitrary loads by new symplectic approach. Mechanics Research Communications (2009) doi: 10.1016/j.mechrescom.2009.04.001
• [34] S. Bao, Z. Deng, A general solution of free vibration for rectangular thin plates in Hamilton systems. Journal of Dynamics and Control 3 (2005) 10-16 (in Chinese).
• [35] C.W. Lim, C.F. Lü, Y. Xiang, W. Yao, On new symplectic elasticity approach for exact free vibration solutions of rectangular Kirchhoff plates. International Journal of Engineering Science 47 (2009) 131-140.
• [36] Y. Xing, B. Liu, New exact solutions for free vibrations of rectangular thin plates by symplectic dual method. Acta Mechanica Sinica 25 (2009) 265–270.
• [37] G. Zou, An exact symplectic geometry solution for the static and dynamic analysis of Reissner plates, Comp. Meth. App. Mech. Eng 156 (1998) 171-178.
• [38] E. Reissner, On the theory of bending of elastic plates. Journal of Mathematics and Physics 23 (1944) 184-191.
• [39] E. Reissner, The effect of transverse shear deformation on the bending of elastic plates. Journal of Applied Mechanics 12 (1945) A69-A77.
• [40] E. Reissner, On bending of elastic plates. Quarterly of Applied Mathematics 5 (1947) 55-68.
• [41] H.C. Hu, Variational principle in elasticity and its applications, Science Press, Beijing, 1981 (in Chinese).