Manual machining
Dividing head calculator
Turns = 40 / N; 7 divisions is 5 5/7 turns, or 5 full turns plus 15 holes on the 21-hole circle.
Technical review pending. Formulas and values are cited. This notice comes down after a machinist review.
Crank turns per division
turns
turns = R / N = 40 / 7 = 5 5/7 = 5.7143 turns
5 full turns plus 15 holes on the 21-hole circle (Brown and Sharpe, plate 2). Check: 5 x 9° + (15 / 21) x 9° = 51.4286° of spindle. Angle per division 360 / 7 = 51.4286° (51° 25' 43").
| Plate | Circle | Holes to move | Full turns |
|---|---|---|---|
| Brown and Sharpe, plate 2 | 21 holes | 15 | 5 |
| Brown and Sharpe, plate 3 | 49 holes | 35 | 5 |
Source: Machinery's Handbook (dividing heads; simple, angular and differential indexing); US Army TC 9-524, Fundamentals of Machine Tools, chapter 8 (index plates); US Navy NAVEDTRA 14345, Machinery Repairman, chapter 7 (plain, angular and differential indexing).
The count is only as good as the plate. Check the first division with an indicator or by counting the full set before cutting.
| Plate | Hole circles |
|---|---|
| Brown and Sharpe, plate 1 | 15, 16, 17, 18, 19, 20 |
| Brown and Sharpe, plate 2 | 21, 23, 27, 29, 31, 33 |
| Brown and Sharpe, plate 3 | 37, 39, 41, 43, 47, 49 |
Source: US Army TC 9-524, chapter 8; US Navy NAVEDTRA 14345, chapter 7
| Plate | Hole circles |
|---|---|
| Cincinnati standard plate, side 1 | 24, 25, 28, 30, 34, 37, 38, 39, 41, 42, 43 |
| Cincinnati standard plate, side 2 | 46, 47, 49, 51, 53, 54, 57, 58, 59, 62, 66 |
Plates vary by head and by age; count the holes on your plate before trusting a table. Counts neither set can index by simple indexing on a 40:1 head (2 to 100 divisions): 61, 63, 67, 69, 71, 73, 77, 79, 81, 83, 87, 89, 91, 93, 97, 99; use differential indexing or a high-number plate. The Brown and Sharpe set also misses 51, 53, 57, 59, 96.
Source: US Army TC 9-524, chapter 8; US Navy NAVEDTRA 14345, chapter 7
How it works
turns = R / N; turns = angle x R / 360; holes = (p / q) x circle
| Symbol | Meaning | Unit |
|---|---|---|
| R | Worm ratio, crank turns per spindle revolution (40 on a standard dividing head) | turns/rev |
| N | Number of divisions | count |
| turns | Crank turns per division | turns |
| p / q | The fraction of a turn left after the full turns, reduced | turn |
| circle | Holes in the hole circle; it has to be a multiple of q | holes |
| holes | Spaces to move past the start hole | holes |
| angle | Spindle angle to index; one crank turn is 360 / R (9° on 40:1, 4° on 90:1) | ° |
| A | Differential: an approximate count near N that the plates index | count |
Differential indexing (40:1 head)
turns = 40 / A; gear ratio = (A - N) x 40 / A
| Symbol | Meaning | Unit |
|---|---|---|
| A | Approximate count the plates index, near N | count |
| N | Divisions wanted | count |
| gear ratio | Gear on the spindle : driven gear on the plate side. A > N: one or three idlers; A < N: none or two | ratio |
When no circle comes out even for an angle, the calculator lists the nearest hole count on each circle with its error: error = (holes / circle - fraction) x (360 / R) x 3,600 arc-seconds. The best three show in the table.
Worked example: 7 divisions on a 40:1 head
Index 7 equal divisions with the Brown and Sharpe plates.
- Turns = 40 / 7 = 5.7143 = 5 5/7 turns.
- 5/7 needs a circle divisible by 7. Plate 2 has 21: 5/7 x 21 = 15 holes; plate 3 has 49: 35 holes. Cincinnati: 28 (20 holes), 42 (30 holes), 49 (35 holes).
- Check: 360° / 7 = 51.4286°; 5 x 9° + (15 / 21) x 9° = 45° + 6.4286° = 51.4286°.
Result: 5 full turns plus 15 holes on the 21-hole circle (plate 2).
Worked example: 23° on a 40:1 head
Index the spindle 23° with the Brown and Sharpe plates.
- Turns = 23 / 9 = 2.5556 = 2 5/9 turns.
- 5/9 needs a circle divisible by 9: plate 1 has 18: 10 holes; plate 2 has 27: 15 holes; Cincinnati side 2 has 54: 30 holes.
- Check: 2 x 9° + (10 / 18) x 9° = 18° + 5° = 23°.
- 23° 30' = 23.5°; 23.5 / 9 = 2 11/18 turns: 2 turns plus 11 holes on 18, or 33 holes on 54.
- An angle that does not come out even: 10.25° = 1 5/36 turns, and no Brown and Sharpe circle divides by 36. Nearest: 1 turn plus 6 holes on 43 (10.2558°, +20.9 arc-seconds), then 4 holes on 29 (-31.0 arc-seconds).
Result: 2 full turns plus 10 holes on the 18-hole circle.
Worked example: 57 divisions by differential indexing
The Brown and Sharpe plates cannot index 57 directly (no 57-hole circle). Pick A = 60.
- Crank 40 / 60 = 2/3 turn = 14 holes on the 21-hole circle.
- Gear ratio = (60 - 57) x 40 / 60 = 3 x 2/3 = 2/1: for example a 48-tooth gear on the spindle and a 24-tooth driven gear.
- A is greater than N: one or three idlers in a simple train.
- Check: 40 / 57 - 40 / 60 = 0.701754 - 0.666667 = 0.035088 turn = 2/57; the plate supplies that much per division through the gears.
Result: 14 holes on 21 each division, 2:1 gearing. The nearest counts the plates index on each side of 57 are 56 and 58.
Short checks
- 36 divisions: 40 / 36 = 1 1/9 turns = 1 turn plus 2 holes on 18 or 3 holes on 27 (Cincinnati: 6 holes on 54).
- 6 divisions (hex): 6 2/3 turns = 6 turns plus 12 holes on 18.
- 90:1 rotary table, 7 divisions: 90 / 7 = 12 6/7 turns, on a circle divisible by 7 on the table's plate (enter it as a custom plate); 4° per handwheel turn.
Shop notes
- Set the sector arms to span the holes to move, then swing them after each index. Count spaces moved, not the hole the pin starts in.
- Turn the crank in one direction only; if you overshoot, back up past the hole and come forward again to keep backlash out.
- Lock the spindle before each cut and release the spindle lock before indexing.
- For gear cutting, the involute cutter number and blank size come from the spur gear calculator; this page gives the indexing.
FAQ
How many turns for 7 divisions on a dividing head?
5 5/7 turns on a 40:1 head: 5 full turns plus 15 holes on the 21-hole circle, or 35 holes on the 49-hole circle.
How do you index an angle on a dividing head?
Divide the angle by 9° per crank turn on a 40:1 head. 23° is 2 5/9 turns: 2 turns plus 10 holes on the 18-hole circle.
What hole circles come on Brown and Sharpe plates?
Plate 1: 15, 16, 17, 18, 19, 20; plate 2: 21, 23, 27, 29, 31, 33; plate 3: 37, 39, 41, 43, 47, 49 holes.
Why can't the plates index 57 divisions?
40 / 57 needs a 57-hole circle, and the Brown and Sharpe plates have none. Differential indexing at A = 60 does it: 14 holes on 21 with a 2:1 gear ratio from the spindle. The Cincinnati standard plate has a 57-hole circle: 40 holes on it.
How many turns on a 90:1 rotary table for 5 divisions?
90 / 5 = 18 full turns; each handwheel turn is 4°.
How many turns for 36 divisions?
1 1/9 turns: 1 turn plus 2 holes on the 18-hole circle.
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