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Civil-Comp Proceedings
ISSN 1759-3433
CCP: 93
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Paper 189

Natural Frequencies of Tapered Cylindrical Shells by Wittrick-Williams Algorithm

N. El-Kaabazi and D. Kennedy

Cardiff School of Engineering, Cardiff University, United Kingdom

Full Bibliographic Reference for this paper
N. El-Kaabazi, D. Kennedy, "Natural Frequencies of Tapered Cylindrical Shells by Wittrick-Williams Algorithm", in , (Editors), "Proceedings of the Tenth International Conference on Computational Structures Technology", Civil-Comp Press, Stirlingshire, UK, Paper 189, 2010. doi:10.4203/ccp.93.189
Keywords: vibration, shells, dynamic stiffness method, boundary value problem, Wittrick-Williams algorithm, transcendental eigenvalue problem.

Shells are structures designed to cover wide areas without internal supports. This type of structure can survive extreme loads, and has been used since antiquity. Modern applications include aircraft, spacecraft, pressure vessels, storage tanks and cooling towers.

In this paper, dynamic stiffness equations are formulated for tapered cylindrical shells, under the assumptions of Donnell, Timoshenko and Flügge theories. These are transformed to governing ordinary differential equations of order eight, which are solved numerically using the boundary-value solver COLSYS [1], enabling the assembly of the dynamic stiffness matrix. Natural frequencies are found by solving the resulting transcendental eigenvalue problem using the Wittrick-Williams algorithm [2].

This algorithm requires knowledge of the number of clamped end natural frequencies exceeded at any trial value of the eigenparameter. The paper presents and validates a new method, based on the recently discovered "member stiffness determinant" [3], for dividing the shell into a sufficient number of smaller elements, so as to ensure that all their clamped end natural frequencies exceed the highest frequency of interest.

Good agreement is obtained with previously published natural frequencies for constant and variable thickness cylindrical shells. A parametric study is performed, using the Donnell theory, to examine the natural frequencies of shells with linear and quadratic, asymmetrically and symmetrically tapered axial variation of thickness. The study considers the effects of varying the ratios of length to radius and radius to thickness, and covers clamped-clamped, simply supported and cantilevered end conditions.

Increasing the amount of thickness variation decreases the natural frequencies in the case of asymmetric taper, but increases them in cases of symmetric taper. Natural frequencies decrease as the length to radius and radius to thickness ratios increase. For cantilevered end conditions, the natural frequencies increase if material is concentrated at the clamped end of the shell.

Solution times appear to be competitive, and there is no loss of accuracy for higher natural frequencies because an exact formulation is used. The method could be further developed to find the natural frequencies and critical buckling loads of orthotropic and anisotropic shells, and shells with varying radius, e.g. conical and spherical shells, and domes.

U. Ascher, J. Christiansen, R.D. Russell, "Collocation software for boundary-value ODEs", ACM Transactions on Mathematical Software, 7(2), 209-222, 1981. doi:10.1145/355945.355950
W.H. Wittrick, F.W. Williams, "A general algorithm for computing natural frequencies of elastic structures", Quarterly Journal of Mechanics and Applied Mathematics, 24(3), 263-284, 1971. doi:10.1093/qjmam/24.3.263
F.W. Williams, D. Kennedy, M.S. Djoudi, "The member stiffness determinant and its uses for the transcendental eigenproblems of structural engineering and other disciplines", Proceedings of The Royal Society A, 459(2032), 1001-1019, 2003. doi:10.1098/rspa.2002.1074

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