A fully numerical and consistent method using the three-dimensional theory of elasticity is presented in this paper to study the free vibrations of an axially symmetric solid. The solid is defined in the cylindrical coordinates by a quadrilateral cross section in the plane bounded by four straight and/or curved edges. The cross section is then mapped using the natural coordinates (ξ,η) to simplify the mathematics of the problem. The displacement fields are expressed in terms of the product of two simple algebraic polynomials in ξ and η, respectively. Boundary conditions are enforced in the later part of the solution by simply controlling coefficients of the polynomials. The procedure setup in this paper is such that it was possible to investigate the free axisymmetric and asymmetric vibrations of a wide range of problems, namely; circular disks, cylinders, cones, and spheres with considerable success. The numerical cases include circular disks of uniform as well as varying thickness, conical/cylindrical shells and finally a spherical shell of uniform thickness. Convergence study is also done to examine the accuracy of the results rendered by the present method. The results are compared with the finite element method using the eight-node isoparametric element for the solids of revolution and published data by other researchers.
Vibration of Thick Circular Disks and Shells of Revolution
Contributed by the Applied Mechanics Division of THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS for publication in the ASME JOURNAL OF APPLIED MECHANICS. Manuscript received by the Applied Mechanics Division, June 24, 1999; final revision, Aug. 29, 2002. Associate Editor: V. K. Kinra. Discussion on the paper should be addressed to the Editor, Prof. Robert M. McMeeking, Chair, Department of Mechanics and Environmental Engineering, University of California–Santa Barbara, Santa Barbara, CA 93106-5070, and will be accepted until four months after final publication in the paper itself in the ASME JOURNAL OF APPLIED MECHANICS.
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Singh , A. V., and Subramaniam , L. (March 27, 2003). "Vibration of Thick Circular Disks and Shells of Revolution ." ASME. J. Appl. Mech. March 2003; 70(2): 292–298. https://doi.org/10.1115/1.1544542
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