Exact Analytical Solutions for the Vibrations of Sectorial Plates With Simply-Supported Radial Edges

[+] Author and Article Information
C. S. Huang, O. G. McGee

Department of Civil Engineering, Ohio State University, Columbus, OH 43210

A. W. Leissa

Department of Engineering Mechanics, Ohio State University, Columbus, OH 43210

J. Appl. Mech 60(2), 478-483 (Jun 01, 1993) (6 pages) doi:10.1115/1.2900818 History: Received October 21, 1991; Revised August 07, 1992; Online March 31, 2008


The first known exact analytical solutions are derived for the free vibrations of sectorial thin plates having their radial edges simply supported, with arbitrary conditions along their circular edges. This requires satisfying: (1) the differential equation of motion, (2) boundary conditions along the radial and circular edges, and (3) proper regularity conditions at the vertex of the radial edges. The solution to the differential equation involves ordinary and modified Bessel functions of the first and second kinds, of non-integer order, and four constants of integration. Utilizing a careful limiting process, the regularity conditions are invoked to develop two equations of constraint among the four constants for sector angles exceeding 180 deg (re-entrant corners). Moment singularities for re-entrant corners are shown to be the same as the ones determined by Williams (1952) for statically loaded sectorial plates. Frequency determinants and equations are generated for circular boundaries which are clamped, simply-supported, or free. Nondimensional frequency parameters are presented for all three types of configurations for sector angles of 195, 210, 270, 330, and 360 deg (i.e., re-entrant corners).

Copyright © 1993 by The American Society of Mechanical Engineers
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