In-Plane Vibrations of a Thin Rotating Disk
A model for the in-plane oscillations of a thin rotating disk has been derived using a nonlinear strain measure to calculate the disk energy. This accounts for the stiffening of the disk due to the radial expansion resulting from its rotation. The corresponding nondimensionalized natural frequencies are seen to depend only on the nondimensionalized rotation speed and have been calculated. The radially expanded disk configuration is linearly stable over the range of rotation speeds studied here. The sine and cosine modes for all nodal diameters couple to each other at all non-zero rotation speeds and the strength of this coupling increases with rotation speed. This coupling causes the reported frequencies of the stationary disk to split. The zero, one and two nodal diameter in-plane modes do not have a critical speed corresponding to the vanishing of the backward travelling wave frequency. The use of a linear strain measure in earlier work incorrectly predicts instability of the rotating equilibrium and the existence of critical speeds in these modes.