Machinery Dynamics
Ring spindles run above their critical speed on purpose. The risk is the band they cross getting there, not the speed they settle at.
— rpm
Resonance of the spindle on its bolster
A single lumped mass on a linear spring is the whole model, and a real spindle has several modes — the second and third criticals can sit inside the running range even when the first is safely below it. Bolster stiffness is also not linear: it is a damped flexible mounting whose stiffness varies with amplitude and with the oil, so the critical speed drifts with temperature and wear rather than staying where a datasheet puts it. Unbalance changes continuously through the build as the bobbin fills, and the worst case is rarely the full package. Use this to locate the band to avoid, then confirm with a run-up vibration measurement.
Spinning Spindle Vibration Resonance & Critical Speed — free, with the formula and a worked example, at Textile School.