Explain how 'Centrifugal Expansion' led to the failure of an external ring in a high-speed coupling.

In this failure mode, the shaft's rotational speed exceeded the 'Clinging Speed' of the ring. As the speed increased, the centrifugal force $F_c = m r imes ext{omega}^2$ overcame the elastic 'cling' force of the ring. Once the ring expanded and lost contact with the groove bottom, it began to spin independently of the shaft. This friction caused localized heating, reducing the material's yield strength and causing further expansion. Eventually, the ring 'popped' out of the groove due to the lack of radial constraint. The fix involves using a 'Self-Locking' ring design or increasing the ring's cross-section to increase its mass-to-stiffness ratio favorably.