KEVOS® MAKEOS · Gears
Internal Gear Measurement Between Pins (Even or Odd Teeth, Backlash Allowance, Space Width from a Reading)
What should the measurement between two pins be for my internal spur gear, at the low and high limits of its backlash allowance, and does a reading I have taken fall inside them?
Measurement between two pins (or wires) for an internal spur gear with even or odd teeth by the involute function: the pin centre pressure angle from the space width at the pitch circle, the pin centre radius and the dimension between the pins for the standard space width and for the smallest and largest space-width increases allowed for backlash; the space width a measured reading implies; and the diameter at which the pins touch the flanks, checked against the inside and root diameters.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Gear | ||
| Module | mm | Module of the internal gear, from the drawing. |
| Number of teeth | Number of teeth on the internal gear; the method handles even and odd numbers. | |
| Pressure angle | ° | Standard pressure angle of the gear. |
| Inside (tip) diameter of the internal gear | mm | Inside diameter across the tooth tips of the internal gear, from the drawing or a measurement. |
| Root diameter of the internal gear | mm | Diameter at the bottom of the tooth spaces (the root) of the internal gear. |
| Pins and limits | ||
| Pin diameter | mm | Diameter of the gauge pins; the handbook's Van Keuren standard for internal gears is 1.44 ÷ diametral pitch, which is 1.44 × module in metric. |
| Smallest space-width increase for backlash | mm | Least amount the space width at the pitch circle is opened up for backlash, from the gear drawing. |
| Largest space-width increase for backlash | mm | Greatest amount the space width is opened up for backlash. |
| Measured dimension between the pins | mm | Dimension measured between the inner sides of the two pins with the pins in opposite (even) or nearly opposite (odd) spaces. |
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.
Gear
What this means: Module
- What it is
- Module of the internal gear: pitch diameter ÷ teeth.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.1 mm to 100 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 2 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Scales the pitch diameter and the measurement.
- Get it right
- Take it from the drawing; for an inch gear, module = 25.4 ÷ diametral pitch.
What this means: Number of teeth
- What it is
- Number of teeth on the internal gear.
- Units and format
- Enter a plain number with no unit. The form accepts 7 to 2000. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 60 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Sets the pitch diameter; odd counts give a slightly smaller reading.
- Get it right
- Count carefully: an odd count changes how the pins sit and the formula used.
What this means: Pressure angle
- What it is
- Standard pressure angle of the internal gear teeth.
- Units and format
- Enter a number in °: degrees of angle. The form accepts 10° to 35°. 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
- Changes the base diameter and where the pins sit.
- Get it right
- Common values are 14½°, 20° and 25°; take it from the drawing.
What this means: Inside (tip) diameter of the internal gear
- What it is
- Diameter across the tips of the internal teeth, the smallest diameter of the ring gear.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 1000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 116 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Only used to check that the pins touch the flanks above the tips.
- Get it right
- Measure across the tooth tips or take it from the drawing.
What this means: Root diameter of the internal gear
- What it is
- Diameter at the bottom of the tooth spaces of the internal gear.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 1000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 125 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Only used to check that the pins do not bottom in the spaces.
- Get it right
- Take it from the drawing or the cutter data.
Pins and limits
What this means: Pin diameter
- What it is
- Diameter of the gauge pins or wires placed in two tooth spaces.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.01 mm to 1000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 2.88 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A larger pin sits further out in the space and changes the reading.
- Get it right
- Van Keuren's internal-gear size is 1.44 × module; measure the pins before use.
What this means: Smallest space-width increase for backlash
- What it is
- The least amount the space width at the pitch circle is opened for backlash.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 100 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.04 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 smallest acceptable reading.
- Get it right
- From the drawing's tooth thickness or space width tolerance; zero means no backlash allowance.
What this means: Largest space-width increase for backlash
- What it is
- The greatest amount the space width at the pitch circle is opened for backlash.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 100 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.1 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 largest acceptable reading.
- Get it right
- From the drawing; must be at least the smallest increase.
What this means: Measured dimension between the pins
- What it is
- Dimension measured between the inner sides of the two pins.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 1000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 117.5 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Converted to the space width actually cut and checked against the limits.
- Get it right
- Put the pins in opposite spaces (or as nearly opposite as possible for odd teeth) and take the smallest reading across them.
Method and formulas
- Pitch diameter D = m z; base diameter D_b = D cos φ; standard space width at the pitch circle e₀ = π m ÷ 2.
- Pin centre involute function: inv φ_w = e ÷ D + inv φ − d_p ÷ D_b, with inv x = tan x − x, solved for φ_w.
- Pin centre radius R_w = D_b ÷ (2 cos φ_w); between pins M = 2R_w − d_p for even teeth, M = 2R_w cos(90° ÷ z) − d_p for odd teeth.
- From a reading: R_w from M, cos φ_w = D_b ÷ (2R_w), e = D(inv φ_w − inv φ + d_p ÷ D_b).
- Pins touch the flanks at the diameter 2√(R_b² + (R_b tan φ_w + d_p ÷ 2)²), which must lie between the inside and root diameters.
- Example: 60 teeth of module 2 measure 117.33 mm between 2.88 mm pins for the standard space, and 117.44 to 117.6 mm with the backlash allowance.
Limitations
- Spur internal gears with involute teeth; helical internal gears need a pin size and correction for the helix.
- Pitch and runout errors make readings vary around the gear; measure at several positions.
- The pins must be accurate to well within the tolerance; check their size before use.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Set the inspection limits for an internal (ring) gear
- Turn a pin reading into the space width actually cut
- Check a pin size touches the flanks and not the tooth tips
Who it is for
- Gear inspectors
- Quality engineers
- Machinists cutting internal gears
What you get
- Pin centre pressure angle and radius
- Measurement between pins for the standard space width
- Measurement limits for the backlash allowance
- Space width and increase implied by the reading
- Pin contact diameter against the inside and root diameters
Interpreting the result
- Opening the space width moves the pins outwards, so the dimension between them grows with the backlash allowance.
- For odd teeth the pins are not quite opposite, so the reading is smaller than for an even-tooth gear of the same size.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 2226–2230. Topic reference; no source tables or recommended settings are reproduced.
Learn the Concepts Behind This Tool
These KEVOS® Library readings explain the method this tool uses, with worked examples.
Useful alongside · KEVOS® original book
Measuring and specifying a gear
What it covers: Check tooth thickness with span measurement, identify an unknown gear from its tooth count and diameter, and write the data a gear drawing needs.
KEVOS® tools are screening and decision-support utilities. Verify important financial, engineering, safety, legal or operational decisions using appropriate professional standards and your organisation’s approved data.