Change Gear Setups

Having recently been given a Speedway 7x12 mini-lathe for Christmas, I found myself wondering about cutting threads. The instructions provide a table of different thread pitches that can be cut, ranging from 12 to somewhere around fifty, if I recall. But I began wondering what other thread pitches could be cut.

The formula for Threads Per Inch (TPI) set by the change gears of a lathe is:

TPI = L ÷ ((A ÷ B) × (C ÷ D))

where L is the TPI of the lead screw, and A, B, C, and D are the numbers of teeth on the four respective gears of the gear train.

For example, a lathe with a 16 TPI lead screw and 80, 60, 40 and 50 teeth on gears A, B, C, and D respectively, would cut threads at a rate of:

TPI = 16 ÷ ((80 ÷ 60) × (40 ÷ 50))

which works out to 15 TPI. The order of operations is important, so don't forget to do the math inside the parentheses first. Or just use the calculator below.

TPI Calculator
L
A
B
C
D
TPI

There are often multiple ways to accomplish a given TPI. For example, both 80:80:40:40 and 80:40:40:80 will give 16 TPI. Which set is used is mostly a matter of preference and of the convenience of fitting the gears in place.

Note that any two adjacent gears (or A and D) can be replaced with any two gears with the same ratio. This is easiest to see for gears of equal numbers of teeth. For example, 24 TPI can be produced by the combination of 80:80:40:60. As gears A and B have a ratio of 1:1, they can be replaced by any other pair of gears with a ratio of 1:1, such as 20:20:40:60, or 40:40:40:60. In this last example, the middle pair of 40-tooth gears could even be subsequently replaced by 80-tooth gears: 40:80:80:60 also produces 24 TPI.

Furthermore, when gears B and C have the same number of teeth, gear C can be omitted entirely, gear B can be any gear, and gear D slid over on the lead screw shaft so that it is driven directly by gear B (by removing the spacer and placing it in front of gear D).

Of course, this isn't the kind of thing you want to be doing in your head, so I have compiled the following tables. They're intended for use with the 7x10, 7x12, and 7x14 mini-lathes, which are equipped with a 16 TPI lead screw and come with two 80-, 40- and 20-tooth gears, and one 60-, 57-, 55-, 50-, 45-, 30-, and 25-tooth gear. The table will, however, work for any lathe with a 16 TPI lead screw and appropriate gears.

Unfortunately, 25 and 100 TPI are not possible using the standard issue gears, but a 32 tooth gear would solve the problem.

The specifications for the gears used on the Mini-lathe are: module 1, 20° pressure angle, 8mm face width, and a 12mm bore with a 3mm wide by 1.4mm deep keyway. Any gear that meets these specifications can be used on the lathe.

All Integral TPIs

This table gives all 77 integral TPIs possible using the standard issue gears, as well as 25, 75, and 100 TPI, in red, which are only possible with a 32 tooth gear. Also included is the UNC and UNF thread sizes with which the various thread pitches are are used.

I have not tested all these combinations to see if they physically fit on the lathe, but if you run into any problems, the above substitution rules will let you chose gears that fit.

DIAMETER GEAR
TPI UNC UNF A B C D
1 80 20 80 20
2 80 40 80 20
3 80 60 80 20
4 80 40 80 40
5 80 50 80 40
6 80 60 80 40
7 80 Any 35
8 1 60 Any 30
9 7/8 40 30 40 30
10 3/4 80 50 40 40
11 5/8 80 55 40 40
12 9/16 1 40 40 60 45
13 1/2 80 65 40 40
14 7/16 7/8 40 Any 35
15 40 30 40 50
16 3/8 3/4 20 Any 20
18 5/16 9/16 40 Any 45
19 80 57 30 50
20 1/4 7/16-1/2 40 Any 50
21 40 30 20 35
22 40 Any 55
24 #10-12 3/8-5/16 40 Any 60
25 32 Any 50
26 40 Any 65
27 40 30 20 45
28 1/4-#12 20 Any 35
30 40 30 20 50
32 #6-8 #10 30 Any 60
33 40 30 20 55
35 40 50 20 35
36 #8 20 Any 45
38 60 57 20 50
39 80 65 20 60
40 #4-5 #6 20 Any 50
42 20 30 20 35
44 #5 20 Any 55
45 40 50 20 45
48 #4 20 Any 60
50 20 25 20 50
52 20 Any 65
54 20 30 20 45
55 40 55 20 50
56 #2 #3 20 40 20 35
57 40 57 20 50
60 20 30 20 50
63 20 45 20 35
64 #1 #2 20 Any 80
65 40 65 20 50
66 20 40 20 55
70 20 50 20 35
72 #1 20 40 20 45
75 20 50 32 60
76 30 57 20 50
77 20 35 20 55
78 40 65 20 60
80 #0 20 40 20 50
84 20 35 20 60
88 20 40 20 55
90 20 50 20 45
91 20 35 20 65
96 20 40 20 60
99 20 55 20 45
100 20 50 32 80
104 20 40 20 65
108 20 60 20 45
110 20 55 20 50
112 20 35 20 80
114 20 57 20 50
117 20 45 20 65
120 20 60 20 50
128 20 40 20 80
130 20 65 20 50
132 20 60 20 55
143 20 65 20 55
144 20 45 20 80
156 20 65 20 60
160 20 50 20 80
176 20 55 20 80
192 20 60 20 80
208 20 80 20 65
256 Standard Turning 20 80 20 80

Unified National Course Threads and Change Gears

This table gives the TPI for UNC threads of various diameters, and the change gears used to produce the proper TPI.

UNC TPI A B C D
1 8 60 Any 30
7/8 9 40 30 40 30
3/4 10 80 50 40 40
5/8 11 80 55 40 40
9/16 12 40 40 60 45
1/2 13 80 65 40 40
7/16 14 40 Any 35
3/8 16 Any 40 20
5/16 18 40 Any 45
1/4 20 40 Any 50
#12 24 40 Any 60
#10 24 40 Any 60
#8 32 30 Any 60
#6 32 30 Any 60
#5 20 20 Any 50
#4 40 20 Any 50
#2 56 20 40 20 35
#1 64 20 Any 80

Unified National Fine Threads and Change Gears

This table gives the TPI for UNF threads of various diameters, and the change gears used to produce the proper TPI.

UNF TPI A B C D
1 12 40 40 60 45
7/8 14 40 Any 35
3/4 16 20 Any 20
9/16 18 40 Any 45
1/2 20 40 Any 50
7/16 20 40 Any 50
3/8 24 40 Any 60
5/16 24 40 Any 60
1/4 28 20 Any 35
#12 28 20 Any 35
#10 32 30 Any 60
#8 36 20 Any 45
#6 40 20 Any 50
#5 44 20 Any 55
#4 48 20 Any 60
#3 56 20 40 20 35
#2 64 20 Any 80
#1 72 20 40 20 45
#0 80 20 40 20 50

This table gives the change gears used to produce metric threads. As the leadscrew of the Mini-lathe is in inches, and not millimeters, it is only able to approximate correct metric threads. This table uses a 21-tooth gear (available from LittleMachineShop.com) to increase accuracy over what is possible with the original gears.
Nominal
Pitch
A B C D Actual
Pitch
Error
1 21 30 45 50 1.000 0.0125%
2 21 30 45 25 2.000 0.0125%
3 21 20 45 25 3.000 0.0125%
4 21 20 60 25 4.001 0.0125%
5 21 20 60 20 5.001 0.0125%
6 65 25 80 55 6.0036 0.0606%
7 57 40 65 21 7.0020 0.0287%
8 57 20 80 45 8.0433 0.5387%
9 55 30 65 21 9.0084 0.0936%
10 55 21 60 25 9.9786 0.2147%
11 65 30 80 25 11.0067 0.0606%
12 57 20 80 30 12.0650 0.5387%
13 57 20 60 21 12.9268 0.5664%
14 57 20 65 21 14.0040 0.0287%
15 80 21 50 20 15.1190 0.7874%
16 80 21 80 30 16.1270 0.7874%
17 80 20 80 30 16.9333 0.3937%
18 57 20 80 20 18.0975 0.5387%
19 60 20 80 20 19.0500 0.2625%
24 80 20 80 21 24.1905 0.7874%

Notes

Let me tell you how I came up with my table - you may find it interesting, and hopefully useful. (All the following shenanigans took place in Excel.)

There are a total of 14 gears. I included a 21-tooth and a 32-tooth gear, and numbered each gear in hexadecimal (0 through F). From a list of all the numbers between 0000 and FFFF (0 through 65535), I then eliminated:

1. all numbers that had more than TWO 0s, 1s, or 2s (representing 20, 40, and 80 tooth gears);
2. all numbers that had more than ONE 3 through D (representing all the gears of which there are only one);
3. all combinations where B>C+D (because B would hit the lead screw shaft).

Then I replaced each digit with the appropriate number of teeth for the gear it represents, and did the math. This left me with something like 5500 pitches. From this list of pitches, I grepped the integral (whole number) pitches, of which there were only 81, though many versions of most.

Finally, I made a list of just the FIRST combination that produced each pitch - which was a mistake, actually, because I had sorted them from greatest to least, so most of the combinations started out with 80 teeth on gear B, which would be impossible to fit (gear C and D have to have a greater number of teeth than gear B to fit, as mentioned). I had to manually go through the list and play musical gears to make them all possible. I'm still not 100% sure they are all possible, as I haven't tested them all.

To extract the metric thread change gear combinations, I calculated the error of each of the 5500 combinations of gears, and then sorted them by error. This placed all the most accurate (lowest error) combinations first, from which I then extracted the first of each pitch.

Then I went and took some asprin.

Credits

Thanks to Mert Baker, Chris Wood (of LittleMachineShop.com), and Frank Hoose (of Mini-lathe.com) for their comments on this web page. I think I asked someone else, too, but I'll be damned if I can remember who ... apologies to whoever it was.