Exact analytical solutions for free vibrations of thick sectorial plates with simply supported radial edges

Abstract

The first known exact analytical solutions are derived for the free vibrations of thick (Mindlin) sectorial plates having simply supported radial edges and arbitrary conditions along the circular edge. The general solutions to the Mindlin differential equations of motion contain noninteger order ordinary and modified Bessel functions of the first and second kinds, and six arbitrary constants of integration. By exercising a careful limiting process, three regularity conditions at the vertex of the radial edges are invoked to yield three equations of constraint among the six constants for sector angles exceeding 180° (re-entrant corners). Three additional linearly independent equations among the six constants are obtained by satisfying the three boundary conditions along the circular edge. Frequency determinant equations are derived for Mindlin sectorial plates with circular boundaries which are clamped, simply supported, or free. Nondimensional frequency parameters are presented for over a wide range of salient and re-entrant sector angles (30° ⩽ α ⩽ 360°), and thickness-to-radius ratios of 0.1, 0.2 and 0.4. Frequency results obtained for Mindlin sectorial plates are compared to those determined for classically thin sectorial plates, and the results are found to be considerably different than those derived from thin plate theory, particularly for the fundamental frequencies of plates having sector angles slightly greater than 180° when the circular boundary is free. The frequencies for 360° sectorial plates (i.e. circular plates having a hinged crack) are compared with those for complete circular ones.