KEVOS® MAKEOS · Threads
Thread Measurement over Wires by the Involute Helicoid Formula (Multi-Start Threads and Worms, Pitch Diameter from a Reading)
For a multi-start thread or worm with a large lead angle, what is the exact measurement over three wires by the involute helicoid formula, how far is the simplified Buckingham formula out, and what pitch diameter does my reading over wires give?
Measurement over wires by the Buckingham exact involute helicoid formula the handbook gives for threads and worms where the lead angle matters: lead L = P S, tan B = L ÷ π E, the normal or axial half-angle (tan An = tan A cos B), tan F = tan An ÷ sin B, Rb = (E ÷ 2) cos F, Ta = T ÷ tan B, tan Hb = cos F tan H, inv G = Ta ÷ E + inv F + W ÷ (2 Rb cos Hb) − π ÷ S and M = 2 Rb ÷ cos G + W for any wire that touches the flanks; compared with the simplified formula M = E + W (1 + sin An) with its own wire W = T cos B ÷ cos An, and the pitch diameter worked back from a measured reading.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Pitch diameter E | mm | Pitch diameter (basic, maximum or minimum) for which the reading is wanted. |
| Pitch P (axial) | mm | Axial pitch of the thread, from one thread to the next. |
| Number of starts S | starts | Number of starts (threads) on the screw or worm; the lead is the pitch times the starts. |
| Thread angle given in the | choice | Whether the half-angle you enter is in the normal plane (the half-angle of a milling cutter) or in the axial plane. |
| Half the included thread angle | ° | Half the included thread angle: 14.5° for Acme, 30° for 60° threads, 20° for a 40° worm. |
| Wire size | choice | Use the wire diameter you enter, or the wire the simplified formula requires. |
| Wire diameter W | mm | Diameter of the wires or pins, measured precisely. |
| Reading and accuracy | ||
| Measured reading over wires (0 if none) | mm | Your measured reading over the wires, to work back to the pitch diameter; 0 to skip. |
| Accuracy you need on M | mm | How close the reading must be; decides whether the simplified formula is good enough. |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
What this means: Pitch diameter E
- What it is
- Pitch diameter for which you want the reading over wires.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 10000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 15.875 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The reading rises almost one for one with it.
- Get it right
- Use the basic, maximum or minimum pitch diameter from the thread limits.
What this means: Pitch P (axial)
- What it is
- Axial pitch of the thread, from one thread to the next.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 1000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 3.6576 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Sets the thread width T = P ÷ 2 and the lead.
- Get it right
- For a multi-start thread the pitch is the lead ÷ the starts.
What this means: Number of starts S
- What it is
- Number of starts (threads) on the screw or worm.
- Units and format
- Enter a number in starts: a number of thread starts. The form accepts 1 starts to 100 starts. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 6 starts as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Raises the lead and the lead angle.
- Get it right
- Count the thread ends at the end of the screw.
What this means: Thread angle given in the
- What it is
- Plane in which the thread half-angle is given.
- Units and format
- Choose one of 2 options from the list: Normal plane (cutter half-angle An); Axial plane (half-angle A).
- Example
- The form opens with “Normal plane (cutter half-angle An)” selected as a worked example. Choose the option that matches your case.
- Effect on the result
- Changes which half-angle is converted by the lead angle.
- Get it right
- Normal plane for a thread milled with a cutter of that angle; axial plane for a lathe-cut thread.
What this means: Half the included thread angle
- What it is
- Angle between one thread flank and a line square to the axis, half the included angle.
- Units and format
- Enter a number in °: degrees of angle. The form accepts 5° to 45°. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 20° as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A smaller angle makes the lead angle matter more.
- Get it right
- 14.5° for Acme, 30° for 60° threads, 20° for a 40° worm.
What this means: Wire size
- What it is
- Whether to use your wire size or the size the simplified formula needs.
- Units and format
- Choose one of 2 options from the list: Wire diameter I enter; Wire of the simplified formula (T cos B ÷ cos An).
- Example
- The form opens with “Wire diameter I enter” selected as a worked example. Choose the option that matches your case.
- Effect on the result
- Changes the reading over wires.
- Get it right
- The exact formula works with any wire that touches the flanks.
What this means: Wire diameter W
- What it is
- Diameter of the measuring wires or pins.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 100 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1.781302 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The reading rises with the wire size.
- Get it right
- Measure the wires precisely; any error goes straight into the reading.
Reading and accuracy
What this means: Measured reading over wires (0 if none)
- What it is
- Reading over the wires you measured.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 10000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Gives the pitch diameter of the thread you measured.
- Get it right
- Measure with the recommended contact pressure; 0 to skip.
What this means: Accuracy you need on M
- What it is
- How close the reading over wires must be.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 10 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.0025 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Decides whether the simplified formula is good enough.
- Get it right
- For gauges, a few micrometres; for general work, more.
Method and formulas
- L = P S; tan B = L ÷ π E; T = P ÷ 2; tan An = tan A cos B (or tan A = tan An ÷ cos B).
- Simplified (3): W = T cos B ÷ cos An; M = E + W (1 + sin An).
- Exact (4): tan F = tan An ÷ sin B; Rb = (E ÷ 2) cos F; Ta = T ÷ tan B; H = 90° − B; tan Hb = cos F tan H.
- inv G = Ta ÷ E + inv F + W ÷ (2 Rb cos Hb) − π ÷ S; M = 2 Rb ÷ cos G + W.
- From a reading: the pitch diameter E for which formula (4) gives the measured M, found by successive trial.
- Example: the exact reading over 1.7813 mm wires is 18.252 mm; the simplified formula gives 18.266 mm.
Limitations
- The formula is exact for involute helicoid threads and a very close approximation for milled threads of intermediate profile.
- The wire must touch both flanks; very small or large wires may touch the crest or root instead.
- Contact pressure, wire accuracy and wire wear all affect the reading; measure the wires precisely.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Find the reading over wires for a worm or multi-start thread with a large lead angle
- Decide whether the simplified Buckingham formula is accurate enough
- Work out the pitch diameter from a reading over wires
Who it is for
- Inspectors and metrologists
- Machinists
- Gauge makers
What you get
- Lead angle and normal and axial half-angles
- Wire size of the simplified formula
- Measurement M by the exact involute helicoid formula
- Measurement by the simplified formula and the difference
- Pitch diameter from your reading
Interpreting the result
- For single-start threads with small lead angles the simplified formula is close; for worms and multi-start threads the difference grows with the lead angle.
- Under some conditions a wire may touch one flank at two points; balls of the same diameter can be used instead.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 1997–2001. Topic reference; no source tables or recommended settings are reproduced.
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