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The coupled equations of motion for a circular ring or circular ring segment are developed for the case where the ring is rotating about an axis in its plane and subjected to an angular velocity as well as an angular acceleration. Coupling results from bending in and out of the plane of the ring as well as from extension and torsion of the ring. These equations are then applied to special cases to determine the coupled equations of motion for a ring, beam and cable rotating at a constant angular speed. Coupled equations of motion for a non-rotating circular ring or circular ring segment are developed for the cases of extensional motion and inextensional motion. These equations are subsequently linearized and uncoupled for extensional and inextensional motion in the plane of the ring as well as for uncoupled motion out of the plane of the ring. The critical angular speed for lateral dynamic instability is determined for a rotating circular shaft which supports several rotating circular ring segments

Derivation of the coupled equations of motion for a circular ring rotating about an axis in the plane of the ring