Three-Dimensional Vibrations of Thick Circular and Annular Plates

Finite Element Analysis of Three-Dimensional Vibrations of Thick Circular and Annular Plates

“…It should also be noted that, if we look at the displacements, the mode shown in Figure 3 should not appear for n"0. So and Leissa [3] use a complementary set of functions which replace cos n by sin n , and conversely, to obtain this type of mode. However, either set of displacements will lead to the same system equation with our formulation when n"0, none of the modes is missing, and this might be another advantage of the present method.…”

“…Boundary condition is completely free. The results of a 3-D Ritz method [3] and a 2-D Mindlin theory [9] are also listed. It is observed that very good agreement between the present results and those of the 3-D Ritz method has been achieved.…”

“…This is a displacement "eld that satis"es the 3-D elasticity equations of motion (of circular plates), expressed in cylindrical co-ordinates, when synchronous, sinusoidal motion is assumed. These are also the same displacements used in reference [3] and elsewhere.…”

“…As to 3-D analysis, a 3-D analytical solution based upon series consisting of Bessel functions in the radial co-ordinate and trigonometric functions in the circumferential and thickness co-ordinates was presented by Hutchinson [4] for the vibration analysis of thick, free circular plates. So and Leissa [3] proposed a 3-D Ritz method to solve similar types of problems, made comprehensive comparisons, and presented various vibration modes. However, due to the nature of their approaches, analysis of circular plates with simply supported boundary conditions might be a very tough job for them to do.…”

Three-dimensional free vibration analysis of carbon nanotube reinforced composites annular plates

“…Technically speaking, composites annular plates are widely used in industrial structural elements. There are large number of research papers and reviews on the free vibration analyses of isotropic annular plates [12][13][14][15][16][17][18][19] . A few researchers have analyzed vibration and static behavior of nanocomposites plates [20][21][22][23] .…”