the complete Mill gear loop analysis

The gear loop mesh analysis program is now expanded to work on all three Mill groups in the Plan 27 layout:

Babbage drawing A/093 from July 28, 1841

Each group contains a figure wheel stack (A), fixed and movable long pinions to effect shifts (L and S, respectively), and the necessary connecting pinions (G, J, C, and J). There are two anticipating carriages (F), which mesh to the fixed long pinions of the middle and right groups. The moveable long pinions are meshed with additional pinions (O) meshed to the rack that connects to the Store.

You can see quite a few other wheel stacks in the drawing, which are there to implement multi-precision arithmetic and many other complex features. My simplified version omits them, and includes only the 22 stacks described above.

Still, precisely positioning all the axle centers so that the various loops of gears mesh correctly is a challenge. Babbage's drawings and notes are not clear on whether he had gotten to do that yet.

Each group has a left and right loop of gears, and the rightmost wheel of each group has to mesh to the next group. Also, as Tim Robinson pointed out, the O pinons that link to the rack must mesh the rack perfectly.  The first mesh is "free" since the horizontal position of the rack is arbitrary, but the other O pinions are considerably constrained in order to also mesh.

It's unknown if an analytical solution of a system of interlinked gears like this is possible. My Python Monte-Carlo analysis program, which is now considerably more complex than it was, starts with a tentative placement of all 22 axles, and then systemically perturbs their locations slightly in the search for an arrangement that allows everything to mesh. 

 I use a Solidworks VBA macro to export the coordinates of the initial positions into the analysis program, which looks at over a million possible configurations in just over 10 seconds. It then outputs revised coordinates of the best consistent solution, which the Solidworks macro then uses to change the locations in the existing assembly. It's all a bit kludgy, but it works. Here's the resulting proposed design:

In the end it doesn't take much movement of the axle centers; the maximum coordinate change is 0.17 inches.

For a flavor of what goes on during the analysis, here's the log it produces:

******* analyzing the left group *******

for the starting coordinates of left loop with starting angle of 0.0 degrees for the left wheel

  the angular discrepancy is 0.1478 degrees, or 0.0082 tooth pitches

and for the starting coordinates of the right loop

  the angular discrepancy is 0.2396 degrees, or 0.0106 tooth pitches

doing coordinate search for meshing both loops of the left group...

after 15,625,000,000 possible combinations with 83 left good and 2,785 right good solutions

the best axle locations have a 0.0040 inch maximum coordinate change

for the left loop with initial tooth angle of 0.0000 from the horizontal

  coordinate changes are ['0.0000, 0.0000', '0.0000, 0.0000', '-0.0000, -0.0000', '0.0000, -0.0000', '0.0000, -0.0000', '-0.0000, 0.0000']

  the angular discrepancy is 0.1478 degrees, or 0.0082 tooth pitches

for the right loop with initial tooth angle of 24.6286 from the horizontal

  coordinate changes are ['-0.0000, -0.0000', '-0.0000, 0.0040', '0.0002, 0.0037', '0.0003, 0.0040', '0.0000, -0.0000']

  the angular discrepancy is 0.2405 degrees, or 0.0107 tooth pitches

wheel B moved by delta x 0.0002, delta y 0.0037

******* analyzing the middle group *******

for the starting coordinates of left loop with starting angle of 98.6586374112805 degrees for the left wheel

  the angular discrepancy is 1.1957 degrees, or 0.0664 tooth pitches

and for the starting coordinates of the right loop

  the angular discrepancy is 1.2727 degrees, or 0.0566 tooth pitches

doing coordinate search for meshing both loops of the middle group...

after 15,625,000,000 possible combinations with 17 left good and 305 right good solutions

the best axle locations have a 0.0680 inch maximum coordinate change

for the left loop with initial tooth angle of 98.6586 from the horizontal

  coordinate changes are ['0.0000, 0.0000', '0.0175, 0.0280', '0.0104, -0.0300', '0.0464, -0.0271', '0.0454, -0.0497', '-0.0618, 0.0000']

  the angular discrepancy is 0.2337 degrees, or 0.0130 tooth pitches

for the right loop with initial tooth angle of 140.1315 from the horizontal

  coordinate changes are ['0.0104, -0.0300', '0.0067, 0.0680', '0.0615, 0.0028', '0.0468, -0.0240', '0.0464, -0.0271']

  the angular discrepancy is 0.0392 degrees, or 0.0017 tooth pitches

the rack pinion angular discrepancy is 0.129 degrees. (best was 0.000, worst was 17.701)

wheel C moved by delta x 0.0615, delta y 0.0028

******* analyzing the right group *******

for the starting coordinates of left loop with starting angle of -175.92082189597147 degrees for the left wheel

  the angular discrepancy is 8.5572 degrees, or 0.4754 tooth pitches

doing coordinate search for meshing left loop of the right group...

after 125,000 possible combinations with 9 left good and 0 right good solutions

the best axle locations have a 0.1387 inch maximum coordinate change

for the left loop with initial tooth angle of -175.9208 from the horizontal

  coordinate changes are ['0.0000, 0.0000', '0.0508, 0.0720', '0.0459, 0.0026', '-0.1387, -0.0120', '-0.1289, 0.1324', '-0.1684, 0.0000']

  the angular discrepancy is 0.1954 degrees, or 0.0109 tooth pitches

the rack pinion angular discrepancy is 0.349 degrees. (best was 0.000, worst was 17.835)

final new coordinates:

-12.56476967, -1.15749417,  -12.56476967, -1.15749417,    A

-10.46060713, -2.50749417,  -10.46060713, -2.50749417,  P12

 -8.22047890, -2.71803049,   -8.22047890, -2.71803049,  FP1

 -8.39337477,  0.47729531,   -8.39337477,  0.47729531,  MP1

-10.64309187,  0.44161634,  -10.64309187,  0.44161634,  P11

 -7.99923978,  2.69250583,   -7.99923978,  2.69250583,  RP1

 -5.97059177, -2.69549417,   -5.97063539, -2.69149417,  P14

 -4.05811716, -1.08538832,   -4.05794038, -1.08165015,    B

 -6.15199709,  0.28050583,   -6.15164949,  0.28450583,  P13

 -1.94741317, -2.42513780,   -1.92945858, -2.39297208,  P22

  0.28990470, -2.66369334,    0.30035141, -2.69371680,  FP2

  0.01240110,  0.52425142,    0.05875499,  0.49715004,  MP2

 -2.23531628,  0.62557543,   -2.18973328,  0.57961510,  P21

  0.61337767,  2.69250583,    0.55161702,  2.69250583,  RP2

  2.53961349, -2.62749417,    2.54634436, -2.55949417,  P24

  4.42590683, -0.98679388,    4.48742371, -0.98399078,    C

  2.24340637,  0.23250583,    2.29016363,  0.20850583,  P23

  6.44277449, -2.46403619,    6.55525386, -2.38900982,  P32

  8.68957872, -2.58391412,    8.79702403, -2.58127652,  FP3

  8.52907872,  0.61205830,    8.45190270,  0.60005830,  MP3

  6.28042025,  0.68974420,    6.21315824,  0.82483212,  P31

  9.38595567,  2.69250583,    9.27904439,  2.69250583,  RP3

wrote these new coordinates to c:\temp\inpoints.txt

the analysis took 11 seconds

For anyone interested, I've posted updated source code (including the Solidworks macro) to the github repository (https://github.com/LenShustek/AnalyticalEngine) in the simulations\gear_meshing subdirectory.