Manual machining
Manual machining calculators for the lathe and mill
Taper per foot = 12 x (D - d) / l, and tailstock offset = (D - d) x L / (2 x l); the tools below cover the rest of the manual lathe and mill math.
Technical review pending. Formulas and values are cited. This notice comes down after a machinist review.
Manual lathe and mill formulas
TPF = 12 x (D - d) / l; offset = (D - d) x L / (2 x l); compound = included angle / 2; turns = 40 / N; compound travel = h x 1.1490 at 29.5°
| Symbol | Meaning | Unit |
|---|---|---|
| TPF | Taper per foot | in/ft |
| D, d | Large and small end diameters | in |
| L | Overall work length between centers (offset formula) | in |
| l | Taper length | in |
| compound | Compound setting for a short steep taper | ° |
| N | Number of divisions on a 40:1 dividing head | |
| turns | Crank turns per division | turns |
| h | Radial thread depth | in or mm |
Source: Machinery's Handbook (tapers; dividing heads; thread cutting); National Bureau of Standards, 1959 notice defining the inch as 25.4 mm; Logan Engineering, Cutting Metric Threads (127-tooth and 37/47 transposing gears)
Worked example: 36 divisions on a dividing head
A 40:1 dividing head, 36 equal divisions.
- Turns per division: 40 / 36 = 1.1111 turns.
- As a fraction: 1 4/36 = 1 1/9 turns.
- One full turn of the crank, plus 1/9 of a turn counted on a hole circle divisible by 9.
Result: 1 1/9 turns of the crank per division.
Manual machining tools
- Taper per foot, taper angle, Morse and 7/24 tapers come from the taper calculator.
- Offset for turning a taper between centers comes from the tailstock offset calculator.
- Dial moves for a diameter and compound travel for threading come from the cross slide and compound infeed calculator.
- Spacer thickness and pocket depth for cutting vise jaws are in the soft jaws guide.
- RPM for a belt or gear setting comes from the spindle speed calculator.
- Thread depth and the number of passes are in the threading formulas guide.
- A finished thread gets checked over wires with the 3-wire thread measuring calculator.
- The gage block stack for an angle comes from the sine bar calculator.
- Dovetails are measured over pins with the dovetail calculator.
- Hex and square across corners, and the flats for a dividing head job, come from the across corners calculator.
- Hole coordinates for the DRO come from the bolt circle calculator.
- Center holes for work between centers are sized in the center drill chart.
- Dividing head turns and hole circles for any number of divisions. Coming
- Rotary axis feed rate. Coming
- Shop power guides for running machines are coming.
How to choose a taper setup
- Long, shallow tapers between centers: tailstock offset.
- Steeper and short tapers: a taper attachment or the compound.
- Short, steep tapers: set the compound to half the included angle.
On a home-shop machine
- Tool-maker charts assume rigid production machines. On a light mill or benchtop lathe, start at the low end of the range and work up.
- Change gears for metric threads on an inch lathe use a 127-tooth transposing gear, because 127 = 5 x 25.4: paired with a 100-tooth gear it converts inch lead to millimeters with no ratio error. The lathe manual gives the gear train for each pitch.
- If a 127-tooth gear does not fit, a 37 and 47 tooth pair comes close: 37 / 47 = 0.787234 against 100 / 127 = 0.787402, about 0.02% short (0.0021 in over 10 in of thread).
FAQ
How do you turn a taper on a manual lathe?
Use tailstock offset for long, shallow tapers between centers. For short, steep tapers, set the compound to half the included angle.
How many turns on a 40:1 dividing head for 36 divisions?
40 / 36 = 1 1/9 turns (1.1111): one full turn plus 1/9 turn, counted on a hole circle divisible by 9.
Why does a 127-tooth gear cut metric threads on an inch lathe?
An inch is defined as 25.4 mm, and 25.4 = 127 / 5. A 127-tooth gear paired with a 100-tooth gear in the train converts the inch lead screw to millimeter pitches with no ratio error. The lathe manual gives the full gear train for each pitch.
What if a 127-tooth gear does not fit the lathe?
A 37 and 47 tooth pair stands in for 100 / 127: 37 / 47 = 0.787234 against 0.787402, about 0.02% short, roughly 0.0021 in over 10 in of thread.
How far do you feed the compound for a 1/2-13 thread?
0.0542 in at 29.5°: 0.0472 in radial depth x 1.1490.
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