I analysed the chain sag and the current parobola approximation. see my analysis here and I conclude the parabola equation is good. And my catenary calculations needed successive approximation, which is coomputing intensive.
On my 11.5 ft wide top beam, I actually got around 1/64th of an inch on my maslowCNC with the parabola approximation, chain stretch (8.1m/m/N) and chain tension calculations among parameter improvements for gross results. Then a residual error correction grid for details.
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