Exact In-Plane Natural Frequencies in Purely Radial Modes and Corresponding Critical Speeds of Functionally Graded (FG) Non-Uniform Disks
This paper analytically tackles with the axisymmetric in-plane purely radial free vibrations of hollow continuously hyperbolically varying thickness disks made of functionally power-law graded materials having identical inhomogeneity indexes for both elasticity modulus and the material density. The equation of motion which is in the form of a linear second-order homogeneous Bessel’s ordinary differential equation with constant coefficients is derived by using the linear elasticity theory under plane stress assumption. For practical use, three boundary conditions are chosen as a stress-free annulus, and disks mounted on a rigid shaft at the inner surface with/without rigid casing at the outer surface. For those constraints, characteristic free vibration equations are offered in closed forms. The present frequencies are validated with the results for uniform isotropic and homogeneous disks in the open literature. The influences of boundary conditions, the disk profile parameters, the inhomogeneity indexes, and the aspect ratios (inner radius/outer radius) on the pure radial frequencies are all investigated. Results are reported in both graphical and tabular forms. It was mainly observed that the variation of the vibrational parameters have much more influence over the fundamental frequencies which strictly correspond to the critical rotational speeds of rotating disks. The extensive literature survey showed that there is no such reported analytical solutions to the problem in question. In this regard, the present analytical solutions deserve to be appreciated although such an additional restriction for the inhomogeneity indexes of both the elasticity modulus and the density have been used in the formulation.
In-Plane Free Vibration, Critical Rotational Speeds, Variable Thickness Disk, Functionally Graded, Axisymmetric
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