Keywords: superposition method, vibration, natural frequencies, upperbound, lowerbound, plate, shell.

The paper shows the boundedness of results obtained using Gorman's superposition method for free vibration analysis of plates and shells. It is confirmed numerically that the superposition method gives lowerbounds and upperbounds for plates whose edges are fully clamped and completely free respectively.

The superposition method has been successfully applied for vibration analyses of plates and cylindrical shell [1,2,3]. It is one of the most efficient methods to solve the eigenvalue problems because of its excellent convergence rate [4,5]. However, there are few published results that provide sufficient information for the boundedness of the superposition method. The prediction made by Ilanko [6] is that whether it gives upperbound or lowerbound results depends on the boundary conditions. In cases where the building blocks used in the superposition method is subject to stiffer boundary conditions than those of the original system being modelled, it gives upperbound results. This would be the case where completely free plates are modelled by using the building block whose boundary conditions are slip-shear conditions. On the contrary, it gives lowerbound results when the building blocks are subject to more flexible conditions at the boundaries. This would be the case where fully clamped plates are modelled by using the building blocks whose boundary conditions are simply supported.

The investigation shows, for the first time, the boundedness of the superposition method, which is predicted in the reference [6], is numerically confirmed for those plates. This will be true for open cylindrical shells if exact modes are used for the building blocks. However, it may not be possible to declare the boundedness of the superposition method for the shells where only approximate modes are available for the building blocks. It would be useful to estimate the maximum possible error if using results of the Superposition method together with the bounded results obtained by other methods, for example, the Rayleigh-Ritz method and the finite difference method.

1

D.J. Gorman, "Free vibration analysis of rectangular plates", New York, Elsevier North Holland, 1982.

2

D.J. Gorman, "Vibration analysis of plates by the superposition method", Singapore, World Scientific Publishing, 1999.

3

S.D. Yu, W.L. Cleghorn, R.G. Fenton, "On the accurate analysis of free vibration of open circular cylindrical shells", Journal of Sound and Vibration, 188, 315-336, 1995. doi:10.1006/jsvi.1995.0596

4

Y. Mochida, "Bounded eigenvalues of fully clamped and completely free rectangular plates", ME Thesis, University of Waikato, 2007.

5

Y. Mochida, S. Ilanko, "Bounded natural frequencies of completely free rectangular plates", Journal of Sound and Vibration, 311, 1-8, 2008. doi:10.1016/j.jsv.2007.10.022

6

S. Ilanko, "On the bounds of Gorman's superposition method of free vibration analysis" Journal of Sound and Vibration, 294, 418-420, 2006. doi:10.1016/j.jsv.2005.11.012

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