Free vibration of flexible rotating disks

Perturbation techniques are employed to estimate the free vibration characteristics, i.e., the natural frequencies and mode shapes, of a flexible rotating disk. The normalized disk stiffness ε = 8D/[hb⁴(3 + v)m̄Ω²] is introduced and treated as a small parameter. Then, singular perturbation solutions to the governing eigenvalue problem are derived that are valid up to and including order ε for any given non-zero hub radius. The special case of a zero hub radius is then considered and the corresponding solutions are presented. Next, a regular perturbation formulation valid for the limiting case of a very narrow rotating annulus is developed. The natural frequency/mode shape predictions from the various perturbation formulations are compared with "exact" values obtained from power series solutions of the eigenvalue problem. It is found that the singular perturbation solutions match well with the "exact" values for small stiffnesses, hub radii and nodal circles, while the regular perturbation solution provides excellent accuracy for large hub radii.