The superposition method developed by Gorman has been successfully applied in the analysis of out-of-plane vibrations of plates [1] as well as vibrations of cylindrical shells [2]. Recently, the superposition method was also applied for free vibration analysis of doubly curved thin shallow shells of rectangular planform with any possible combination of edges simply-supported and clamped out-of-plane, and shear-diaphragm and constrained in-plane [3]. However, the previous studies so far do not include free vibration analysis of doubly curved shells with a free edge using the Superposition Method.

Previous studies of shells were limited to building blocks, with only the simply-supported or clamped and shear-diaphragm or constrained edge conditions. The steady state solutions of these building blocks could be expressed as series of functions using Galerkin's Method. The building blocks resulted in sine series for out-of-plane displacements, and sine/cosine series for in-plane displacements. However, it has not been possible to express the solutions of building blocks where a free edge condition is involved in series of functions that exactly satisfies the prescribed boundary conditions because mixed derivatives appear in the formulation of the free edge condition. A remedy for this situation is to employ additional building blocks corresponding to slip-shear out-of-plane conditions and tangentially free, normally constrained in-plane conditions. These building blocks result in cosine series for out-of-plane displacement and cosine or sine series for in-plane displacements. The building blocks are superimposed to complement each other and satisfy the free edge conditions, which is no net forces or moments at the edge.

In computation, false roots for the natural frequency parameters were also gathered in addition to the wanted natural frequencies. By carefully inspecting the determinant versus frequency parameter plot, some of those unwanted false roots are eliminated, however, at the time of writing this paper, excluding all of the false roots still remains a challenge. However, the method is to be expected to work, because the series used are complete and satisfy the governing partial differential equations and being complete, offer the potential to satisfy the edge conditions.

1

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

2

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

3

Y. Mochida, S. Ilanko, M. Duke, Y. Narita, "Free vibration analysis of doubly curved shallow shells using the Superposition-Galerkin method", Journal of Sound and Vibration, 331, 1413-1425, 2012. doi:10.1016/j.jsv.2011.10.031

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