making gear loops mesh

 My design is based on a simplification of Babbage's Plan 27 from 1841. There are, like for all of his plans, many details unspecified, and many hidden subtleties. 

One such subtlety relates to the ability of loops of gears to mesh correctly. Consider the following section of the Mill, from drawing A/093 dated 28 July 1841:

The circles represent the pitch circles of the gears and are tangent in the drawing to enforce the required separation between the axle centers.

• ''A and 'A are stacks of 8" diameter figure wheels with 80 teeth. Two occupy one "cage" at each decimal position, so each stack holds two interleaved independent numbers.
• ''L are fixed long pinions, and ''S are movable long pinions, which together serve as shifters to multiply or divide numbers by 10. They are made of concentric locked gears of different pitch.
• ''G, ''J, 'G, and 'J are linking pinions that can be moved into place vertically to link the long pinions with either of the two digits in a cage

When not in motion, gears are locked so that the figure wheels are at rotated into a quantized position that represents a decimal digit.

Note that there are two loops of 5 gears each: 

• ''A to ''J to '' to ''S to ''G and back to ''A
• ''L to 'J to 'A to 'G to ''S and back to ''L

All the gears around each loop should mesh perfectly. It makes no sense to expect them to turn as drawn, because with 5 gears the direction of rotation of the last gear is wrong. But if the linking pinions J and G connect to different figure wheels in the A cages, they will turn. That is used frequently to "give off" one number in a stack to the other number in the same stack.

The perfect mesh is important so that the position for the lock is the same for both figure wheels in a cage. But because the last gear turns in the wrong direction, the "perfect mesh" only holds for the quantized and locked position of the figure wheels. 

There is considerable freedom to move the axle centers while still obeying the required separation, but requiring the teeth to mesh around the loops imposes an additional constraint. The question is: do the axle positions shown above allow a perfect mesh around both loop?

To answer that question I used a CAD program to trace the pitch circles and identify their centers.

Here each circle is marked with the tooth pitch (DP = diametral pitch, the number of teeth per inch of diameter), the number of teeth, and the diameter. The sizes were measured from the drawing using the ruler that the Science Museum kindly included at the top of the scan, and knowing that this drawing was made 1/2 actual size.

Now I could overlay each pitch circle with the appropriate gear and see if they mesh.

With a random starting rotation for ''A, they don't -- see the interface between ''G and ''A, and between 'G and ''S. But that's not surprising because the Babbage drawing doesn't show where the quantized locked position is.

Tim Robinson suspects there is a way to use geometry/trigonometry to define and solve mathematical equations that produce solutions to problems like this. Maybe. I'm not smart enough to do that. Anyway, it will be difficult because some of the problems have many possible solutions.

So with the help of Google Gemini, I wrote a Python program that propagates the angular position of a loop of meshed gears given their geometry, and computes the angular discrepancy from a perfect mesh of the last gear back to the first.  It was complicated by the concentric locked gears with different diametral pitches in the long pinions, but the trick was to allow each gear to have a different pitch when driven than when driving.

I then used that program to analyze every possible starting rotation of the ''A gear in .01 degree increments, and searched for the one out of 450 which had the smallest final angular discrepancy for the left gear loop. The answer was that some tooth vertex should be 0.20 degrees from the horizontal, which creates a final angular error of only .0035 degrees. And indeed that makes the left gear loop mesh correctly.

But not, unfortunately the right loop, which still has an angular error of 6.6 degrees at the interface between 'G and ''S. There are no more degrees of freedom, so there is no way, given the axle positions as measured from Babbage's drawing, to make both loops mesh simultaneously. 

So what change in axle positions would make it work? To find that out I used a Monte Carlo simulation, where I perturbed the axle centers by tiny amounts while keeping all the necessary axle-to-axle distances constant, and searched for the smallest angular deviation from a perfect mesh for both loops simultaneously. After trying 312,500,000 combinations of movements, of which 680,000 were geometrically feasible, there were 898 that had less than .01 degree of error in both loops, and the best of those had a maximum axle center coordinate change of only .02". And indeed making those small changes to the axle positions makes both loops mesh almost perfectly.

What's amazing is how little the axle positions had to change -- at most .02", which is less than the measurement (or scanning) accuracy of the Babbage drawings. That doesn't bode well for fabrication, because that kind of accuracy is hard to achieve. Hopefully small errors will be corrected by the beveled edges of the locks and the gear teeth as the locks are inserted and the rotational positions are quantized.

My proposed figure wheel and pinion sizes are different, for a whole variety of reasons I can explain later. But the same technique worked to create axle positions that allow both gear loops to mesh perfectly.

For anyone interested in my gear analysis source code, I've posted it to the github repository (https://github.com/LenShustek/AnalyticalEngine) in the simulations\gear_meshing subdirectory.