Suppose you have a standard B&S dividing head
40:1 gear ratio
three hole plates with these hole circles
15,16,17,18,19,20
21,23,27,29,31,33
37,39,41,43,47,49
You want to make a gear with 127 teeth. You calculate that you need to rotate the DH crank by
40/127 = 0.31496...
of a full rotation between each tooth. 127 is prime so the fraction can't be reduced. If you had a plate with 127 holes you could step off by 40 holes between each tooth but you don't have such a plate. Now, with a rotary table, you could make a plate with 127 holes but it would be a massive exercise in tedium to do so.
A procedure called "compound indexing" can supply an answer to this quandry. In this procedure one leaves the hole plate loose on the shaft so it can be rotated. A separate indexing pin, NOT the one on the crank handle, is used to keep the plate from rotating. With the hole plate secured by this extra pin, the crank is rotated some number of holes as would be done in normal, non-compound indexing. Then, the extra pin is released and the hole plate is rotated in the designated direction for the required number of holes. When this is done, the crank, which is fixed to the hole plate by its indexing pin, also rotates. Thus the net rotation of the worm is the algebraic sum of the two separate rotations (crank and hole plate) and the workpiece is rotated 1/40 of that amount.
It should be patently obvious that the hole plates can't be swapped while this tedious process is being done so all the rotations must be accomplished via the holes on the single plate that is mounted. For example, we can't mix 1/15 steps (15 hole circle on plate #1) with 1/49 steps (49 hole circle on plate #3).
Now, you'll note that, in the MH (Machinery's Handbook) table, all the hole plate movements are specified as holes on either the 20 (plate #1), 33 (plate #2) or 49 (plate #3) hole circles. IOW they confine themselves to solutions that only use the outer hole circle on the plate. I'm not certain, but I think this is because it's mechanically difficult to mount the separate indexing pin (the one that indexes the plate) so it can reach the other hole circles. Rich Kuzmack has addressed a method for indexing on all the circles - HSM Jan/Feb 1998 pg. 54.
Some of the solutions in the MH table specify a negative (i.e. opposite to crank rotation direction) indexing of the hole plate. For instance, it indicates that, for 87 divisions, one moves the crank 23 holes on the 29 hole circle and then rotate the hole plate 11 holes on the 33 hole plate IN THE OPPOSITE DIRECTION. Considering the effects of backlash, solutions that require negative rotations are not to be preferred. For both simplicity and accuracy, we would like to always be moving everything in the same direction. Rich has produced a table with all positive crank plate movements in his 1-2/88 article in HSM.
To test my understanding, I wrote a computer program to find all the compound indexing solutions with error less than some input quantity. Running it for this problem, I found that there were acceptable solutions using any of the three plates. The MH solution is the most accurate if one requires indexing on the outer circle of holes. If that condition is relaxed, there is a slightly more accurate solution.
Before discussing the solutions, I need to explain the rather compact output notation used by the program. Here is a typical output line...
1 & 1/16 -1 * ( 2/17) = 0.94485294 [ 3] (-3.0637E-003 %) **
Which says that if we rotate the crank by 1 revolution plus 1 hole on the 16 hole ring and then rotate the plate in the opposite direction (the '-1') by 2 holes in the 17 hole ring we will move the workpiece by approximately [3] times the desired 0.31496063 distance (0.94485294) and the error in doing this will be -3.0637E-3 %. Moving by 3 times the required distance implies that we will need to 'go around' the workpiece three times to complete the desired 127 divisions. (The two asterisks at the end of the line may be ignored. They are used by the program to denote 'best yet found' solution for rapid scanning.)
Partial listing of program output:
Number of divisions = 127
Approximation accuracy = 0.005000 %
Required increment = 40/127 = 0.31496063
PLATE #1 (15,16,17,18,19,20 holes)
0 & 1/16 +1 * (15/17) = 0.94485294 [ 3] (-3.0637E-003 %) **
1 & 1/16 -1 * ( 2/17) = 0.94485294 [ 3] (-3.0637E-003 %) **
0 & 15/17 +1 * ( 1/16) = 0.94485294 [ 3] (-3.0637E-003 %) **
0 & 5/16 +1 * (18/19) = 1.25986842 [ 4] (+2.0559E-003 %) **
0 & 18/19 +1 * ( 5/16) = 1.25986842 [ 4] (+2.0559E-003 %) **
1 & 17/19 +1 * (10/16) = 2.51973684 [ 8] (+2.0559E-003 %) **
3 & 8/18 +1 * (13/20) = 4.09444444 [13] (-1.0684E-003 %) **
3 & 17/18 +1 * ( 3/20) = 4.09444444 [13] (-1.0684E-003 %) **
4 & 8/18 -1 * ( 7/20) = 4.09444444 [13] (-1.0684E-003 %) **
4 & 17/18 -1 * (17/20) = 4.09444444 [13] (-1.0684E-003 %) **
4 & 3/20 -1 * ( 1/18) = 4.09444444 [13] (-1.0684E-003 %) **
4 & 13/20 -1 * (10/18) = 4.09444444 [13] (-1.0684E-003 %) **
3 & 15/17 +1 * (16/19) = 4.72445820 [15] (+1.0320E-003 %) **
3 & 16/19 +1 * (15/17) = 4.72445820 [15] (+1.0320E-003 %) **
4 & 14/15 +1 * ( 8/19) = 5.35438596 [17] (+1.0320E-003 %) **
5 & 14/15 -1 * (11/19) = 5.35438596 [17] (+1.0320E-003 %) **
5 & 13/15 +1 * ( 2/17) = 5.98431373 [19] (+1.0320E-003 %) **
5 & 13/19 +1 * ( 6/20) = 5.98421053 [19] (-6.9252E-004 %) **
6 & 13/19 -1 * (14/20) = 5.98421053 [19] (-6.9252E-004 %) **
5 & 6/20 +1 * (13/19) = 5.98421053 [19] (-6.9252E-004 %) **
6 & 6/20 -1 * ( 6/19) = 5.98421053 [19] (-6.9252E-004 %) **
PLATE #2 (21,23,27,29,31,33 holes)
2 & 10/21 +1 * ( 1/23) = 2.51966874 [ 8] (-6.4700E-004 %) **
3 & 10/21 -1 * (22/23) = 2.51966874 [ 8] (-6.4700E-004 %) **
2 & 27/29 +1 * (28/33) = 3.77951933 [12] (-2.1769E-004 %) **
3 & 27/29 -1 * ( 5/33) = 3.77951933 [12] (-2.1769E-004 %) **
2 & 28/33 +1 * (27/29) = 3.77951933 [12] (-2.1769E-004 %) **
3 & 28/33 -1 * ( 2/29) = 3.77951933 [12] (-2.1769E-004 %) **
5 & 14/23 +1 * ( 2/33) = 5.66930171 [18] (+1.8299E-004 %) **
6 & 14/23 -1 * (31/33) = 5.66930171 [18] (+1.8299E-004 %) **
5 & 2/33 +1 * (14/23) = 5.66930171 [18] (+1.8299E-004 %) **
6 & 2/33 -1 * ( 9/23) = 5.66930171 [18] (+1.8299E-004 %) **
6 & 4/29 +1 * ( 5/31) = 6.29922136 [20] (+1.3904E-004 %) **
7 & 4/29 -1 * (26/31) = 6.29922136 [20] (+1.3904E-004 %) **
6 & 5/31 +1 * ( 4/29) = 6.29922136 [20] (+1.3904E-004 %) **
7 & 5/31 -1 * (25/29) = 6.29922136 [20] (+1.3904E-004 %) **
PLATE #3 (37,39,41,43,47,49)
The Machinery's Handbook recommendation:
2 & 23/39 +1 * (12/49) = 2.83464155 [ 9] (-1.4536E-004 %)
3 & 29/41 +1 * (33/47) = 4.40944473 [14] (-9.2668E-005 %) **
MH's preoccupation with the outer row of holes leads to downright errors in their table. For 51, divisions, MH shows:
8 & 41/47 -1 * (12/49) = 8.62744247 [11] (-9.8686E-005 %)
Rich converts this to a more usable positive hole plate rotation in his HSM article:
7 & 41/47 +1 * (37/49) = 8.62744247 [11] (-9.8686E-005 %)
but there are several NEAR ZERO error solutions on plate #1, e.g.,
1 & 5/15 +1 * ( 4/17) = 1.56862745 [ 2] (+0.0000E+000 %) **
1 & 4/17 +1 * ( 6/18) = 1.56862745 [ 2] (+0.0000E+000 %) **
2 & 10/15 +1 * ( 8/17) = 3.13725490 [ 4] (+0.0000E+000 %) **
Sorry for the extra long post but I wanted to put Rich's excellent work in context and make everyone aware that even MH has to be used with forbearance. Hopefully, this treatise will make compound indexing a bit more understandable.
