What is 'Clinging Speed' and how is it calculated for a spiral retaining ring on a rotating shaft?

Clinging speed is the rotational speed at which centrifugal force causes an external retaining ring to expand and lose its grip on the groove bottom. If the ring loses contact, it can vibrate, cause wear, or even fly off the shaft. The formula for the maximum allowable speed $N$ is $N = _x000c_rac{70.4}{D_m} imes ext{sqrt}_x000c_rac{E imes I imes ext{clinging force}}{w imes R^3}$ where $w$ is the weight of the ring per inch. For high-speed applications (e.g., turbochargers), we design 'Self-Locking' rings. These have a small tab that locks into a notch on the adjacent turn, preventing the ring from expanding radially even at speeds exceeding 50,000 RPM.