Extreme Reduction - 11 million to one gearing

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Buy at https://www.shapeways.com/product/EQJQZEVWU/extreme-reduction-11-million-to-one-gearing Copyright (c) 2015, M. Oskar van Deventer. Extreme Reduction is an ultra-compact gearing system with an extreme reduction. The gear ratio is 11.373.076 : 1, that is, over eleven million to one!!! That extreme reduction is achieved in three stages.
-Stage 1: Two sets of planetary gears with a reduction of 24 : 1 and 25 : 1, respectively.
-Stage 2: Two subtractions, 26×26-25×27=1 and 25×25-24×26=1, respectively.
-Stage 3: Another subtraction, namely (25×26+1×25)-(24×27+1×26)=1.

The gear ratio of this mechanism scales to the 5th power with the number of gear teeth. That is, for ×2 times the number of teeth, you get ×32 times the gear ratio. Here are some exaple gear ratios:
-Teeth = 4, gear ratio = 2.185
-Teeth = 9, gear ratio = 87.220
-Teeth = 16, gear ratio = 1.322.209
-Teeth = 25, gear ratio = 11.373.076
-Teeth = 36, gear ratio = 67.320.649
-Teeth = 49, gear ratio = 305.880.100
-Teeth = 64, gear ratio = 1.141.624.705
The number of teeth of a gear needs to be an integer square plus or minus one because of the subtracting mechanism.

Whereas high reduction radios are typically achieved with worm gears, a planetary gear system like this is much more compact. The gears have a limited number of teeth, which makes the system very robust. Also unlike worm gears, this planetary gears system is coaxial, which makes it easier to place.

The application of this type of extreme reduction gears is unclear. A patient person could use it to move a heavy train locomotive with a dental dril.

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