Finding the Exact 8:1 Gear Ratio
The starting point of this project was figuring out how to achieve an exact 8:1 gear ratio with the capstan drives. In my capstan drive video, I designed all the drives to have an 8:1 gear ratio—but when measured, none of them actually did. The commenters on that video pretty accurately diagnosed my mistake. I had made the outer diameter of the big drum 8 times larger than the outer diameter of the small drum, but this doesn't account for the diameter of the rope. It's the same kind of oversight that led to The SAT Question Everyone Got Wrong in 1982. Instead of using the outer diameters, I should have used the effective diameters—the diameters that touch the centerline of the rope wrapped around the drums. The effective diameter of a drum is the same as the pitch diameter of a gear.
In theory, using the effective diameters should result in a perfect 8:1 gear ratio—but in practice, it's a bit more complicated. Since the rope is thin and flexible, measuring its exact diameter is tricky. It gets even harder when the rope is wrapped around the drum and put under tension, which slightly compresses it and reduces its diameter. While approximating the rope’s diameter can get you close to an 8:1 ratio, achieving true accuracy requires a numerical approach of calculation. First, I used the estimated rope diameter to build two capstan drives: one with a gear ratio slightly below 8:1 and one slightly above it. The measured gear ratios were 7.912:1 and 8.213:1, respectively. The only thing I changed between the two drives, to achieve the different gear ratios, was the outer diameters of the big drums. With the two known ratios and their corresponding outer drum diameters, I used linear interpolation to calculate the outer drum diameter that would produce an exact 8:1 ratio. I then built a third capstan drive using the interpolated value. The measured gear ratio on that drive was 8.000619:1, which confirmed the accuracy of the method. The CAD files for the 8:1 capstan drive test stand are available on my Patreon Shop and through my Patreon Builder Tier.
In theory, knowing the big drum diameter for the exact 8:1 drive should allow me to back-calculate the rope’s precise diameter, which I could then use to design other capstan drives with exact gear ratios. I have yet to confirm that this works.