Three-dimensional vibrations of thick, linearly tapered, annular plates

The Ritz method is applied in a three-dimensional (3-D) analysis to obtain accurate frequencies for thick, linearly tapered, annular plates. The method is formulated for annular plates having any combination of free or fixed boundaries at both inner and outer edges. Admissible functions for the three displacement components are chosen as trigonometric functions in the circumferential co-ordinate, and algebraic polynomials in the radial and thickness co-ordinates. Upper bound convergence of the non-dimensional frequencies to the exact values within at least four significant figures is demonstrated. Comparisons of results for annular plates with linearly varying thickness are made with ones obtained by others using 2-D classical thin plate theory. Extensive and accurate (four significant figures) frequencies are presented for completely free, thick, linearly tapered annular plates having ratios of average plate thickness to difference between outer radius (a) and inner radius (b) ratios (h(m)/L) of 0.1 and 0.2 for b/L = 0.2 and 0.5. All 3-D modes are included in the analyses; e.g., flexural, thickness-shear, in-plane stretching, and torsional. Because frequency data given is exact to at least four digits, it is benchmark data against which the results from other methods (e.g., 2-D thick plate theory, finite element methods) and may be compared. Throughout this work, Poisson's ratio v is fixed at 0.3 for numerical calculations.