An exact,closed form,solution for the flexural vibration of a thin annular plate having a parabolic thickness variation

An exact, closed form, solution is obtained for the transverse vibrations, with nodal diameters and circles, of a thin annular plate having a parabolic thickness variation. Representative numerical values for the frequency parameter and typical mode shapes are presented for three different combinations of simple boundary conditions. The corresponding exact solution for an aeolotropic annular plate of the same geometry is also presented. Aside from possible design applications, these exact, closed form, data can be used as test cases for assessing the accuracy of various approximate methods of solution. The analysis involves only the powers of the radius and is simpler than that for the constant thickness solution which involves Bessel functions.