One note about critical speed. The equation for critical speed is derived in several theoretical papers. But I have never seen an experimental confirmation. It's something we need to verify one day, probably with some very simple tops made for this purpose.
Totally agree. For one thing, still bothered by the differences in the formulas for critical total angular speed about the spin axis (w3), and critical spin rate about that axis (s).
Would love to confirm that the simpler — and much more commonly cited — formula for w3 is adequate for our purposes. [Note that w3 = s + p cos(a) where p is the precession rate about the vertical, and a is the tilt angle measured from the vertical. In a sleeping top (a = 0), w3 = s. We usually don't measure p.]
But testing these formulas won't be easy. Critical speed is just the lower limit of stability against small perturbations in tilt angle. In theory, a top shouldn't fall at or above critical speed, but in a quiet environment, the top might stay up in a metastable state for some time thereafter. This would be especially true of a perfectly balanced top with a worn tip or on a deeply concave support.
HOWEVER, even if the formulas turn out to be poor at predicting first-scrape or topple speeds, I'm already quite comfortable with them as
qualitative guides for spin time maximization. I've made thousands of LEGO tops, and I always make them spin as long as possible without compromising higher design priorities.
The simple w3 formula has never led me astray.