Natural frequencies of thick, complete, circular rings with an elliptical or circular cross-section from a three-dimensional theory |
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Authors: | Jae-Hoon Kang Arthur W. Leissa |
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Affiliation: | (1) Department of Architectural Engineering, Chung-Ang University, Seoul, South Korea;(2) Department of Mechanical Engineering, Colorado State University, Fort Collins, CO, USA |
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Abstract: | A three-dimensional (3D) method of analysis is presented for determining the free vibration frequencies and mode shapes of thick, complete (circumferentially closed), circular rings with an elliptical or circular cross-section. Displacement components ur, uθ, and uz in the radial, circumferential, and axial directions, respectively, are taken to be periodic in θ and in time, and algebraic polynomials in the r and z directions. Potential (strain) and kinetic energies of the circular rings are formulated, and upper-bound values of the frequencies are obtained by minimizing the frequencies. As the degree of the polynomials is increased, frequencies converge to the exact values. Convergence to four-digit exactitude is demonstrated for the first five frequencies of the rings. Novel numerical results are presented for the circular rings having an elliptical cross-section based upon 3D theory. Comparisons are also made between the frequencies from the present 3D Ritz method and ones obtained from thin and thick ring theories, experiments, and other 3D methods. |
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Keywords: | Three-dimensional vibration analysis Circular ring Thick ring Elliptical cross-section Ritz method |
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