A Simple Frequency Expression for the In-Plane Vibrations of Rotating Rings

An approximate but accurate expression for determining the reference in-plane flexural frequencies of rotating rings is presented. The ring reference frequencies are with respect to a coordinate system that rotates with the ring at the mean speed of the forward and backward traveling waves analogous to the standing wave approach for rotating disks. The formula is based on adapting the correction factor for shear, centrifugal stiffening, and centrifugal mass of prismatic bars to be applied to Hoppe’s “Classical” formula for stationary, slender, inextensible rings. The result is a closed-form analytical solution applicable to arbitrary shaped rotating rings with distributed loading. The formula has been verified for a wide range of ring radial depths and nodal diameters by comparison to elaborate NASTRAN solid finite element models that account for shear, centrifugal stiffening, and centrifugal mass effects. Excellent correlations, within 1 percent, for all test cases were achieved. Forward and backward traveling-wave frequencies are obtained by using the reference frequency expression into Bryan’s “classical” formula for thin rotating rings. Centrifugal mass contribution was shown to be particularly important for thin rings vibrating at low nodal diameters.