On the vibration of imperfect circular disks

An analytical method is presented for vibration analysis of a rotating circular disk with distributed imperfections such as the nonuniform thickness, density, Young's modulus, Poisson' s ratio and the odd distribution of internal stresses. In this paper, the linear governing equation of the disk is formulated, and then the relation between each imperfection and the split of degenerate modes is derived. In the derivation, the distributed imperfection, which is a periodic function of angular position, is expanded as a Fourier series. The derived simple relation suggests that the zeroth order imperfection causes the shift in the natural frequencies of all modes, whereas the 2m-th order harmonic imperfection results in the split of the natural frequencies, with the equal magnitudes but with different signs, for a pair of original degenerate modes with m nodal diameters.