Influence of boundary stress singularities on the vibration of clamped and simply-supported sectorial plates with arbitrary radial edge conditions

Abstract

This paper extends previous studies made for sectorial plates having re-entrant (i.e., interior) corners causing stress singularities, to provide accurate frequencies when the circular edge is either clamped or simply-supported. An extensive review of the literature is also given herein spanning nearly the past two decades explaining the free vibration characteristics of sectorial plates. In this work, the classical Ritz method is employed with two sets of admissible functions assumed for the transverse vibratory displacements. These sets include: (1) mathematically complete algebraic–trigonometric polynomials which guarantee convergence to exact frequencies as sufficient terms are retained and (2) corner functions which account for the bending moment singularities at the re-entrant vertex corner of the radial edges having arbitrary edge conditions. Extensive convergence studies summarized herein confirm that the corner functions substantially enhance the convergence and accuracy of non-dimensional frequencies for sectorial plates having either a clamped or hinged circumferential edge and various combinations of clamped, hinged, and free conditions on the radial edges. Accurate (to at least four significant figure) frequencies and normalized contours of the transverse vibratory displacement are presented for the spectra of sector angles [90°, 180° (semi-circular), 270°, 300°, 330°, 350°, 355°, 360° (complete circular)] causing a re-entrant vertex corner of the radial edges. For sector angles of 360°, a clamped–clamped, clamped–hinged, clamped–free, hinged–free or free–free radial crack ensues. One general observation is the substantial reduction in the first six frequencies as the sector angle increases for all plates, except in the first two modes of plates having free–free radial edges.