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A closed-form solution for natural frequencies of thin-walled cylinders with clamped edges


This paper presents an approximate closed-form solution for the free-vibration problem of thin-walled clamped-clamped cylinders. The used indefinite equations of motion are classic. They derive from Reissner’s version of Love’s theory, properly modified with Donnell’s assumptions, but an innovative approach has been used to find the equations of natural frequencies, based on a solving technique similar to Rayleigh's method, on the Hamilton's principle and on a proper constructions of the eigenfuctions. Thanks to the used approach, given the geometric and mechanical characteristics of the cylinder, the model provides the natural frequencies via a sequence of explicit algebraic equations; no complex numerical resolution, no iterative computation, no convergence analysis is needed. The predictability of the model was checked both against FEM analysis results and versus experimental and numerical data of literature. These comparisons showed that the maximum error respect to the exact solutions is less than 10% for all the comparable mode shapes and less than 5%, on the safe side, respect to the experimental data for the lowest natural frequency. There are no other models in the literature which are both accurate and easy to use. The accurate models require complex numerical techniques while the analytical models are not accurate enough. Therefore the advantage of this novel model respect to the others consists in a best balance between simplicity and accuracy; it is an ideal tool for engineers who design such shells structures.