How is the maximum allowable rotational speed $N_{max}$ calculated for an external spiral retaining ring to prevent liftoff?

For external rings, centrifugal force causes the ring to expand, potentially losing its grip on the groove. The maximum RPM is calculated as $N_{max} = \sqrt{\frac{4.48 \cdot 10^{11} E I}{w \rho R_m^3 (R_o - R_i)}}$, where $w$ is the weight of the material per unit length and $\rho$ is the density. If the application speed exceeds $N_{max}$, the ring must be designed with a 'Self-Locking' feature. This involves a tab-and-slot mechanism that mechanically prevents the ring from expanding beyond the groove diameter, allowing it to withstand speeds that would otherwise cause a standard spiral ring to fail.