BIZOMAT MAKEOS · Gears
Gear Tooth Measurement for Backlash (Span and Over Pins)
What span measurement and measurement over pins should this gear show once the teeth are thinned for the backlash I want?
Inspection dimensions for an external involute gear cut thin for backlash: the tooth thickness reduction for this gear’s share of the normal backlash range, the number of teeth spanned and the base tangent length (span) at nominal and at both backlash limits (as the span measurement tool), the face width the anvils need on a helical gear, and for spur gears the measurement over pins at nominal and at both limits (as the gear over pins tool).
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Normal module | mm | Normal module. |
| Number of teeth | Number of teeth. | |
| Normal pressure angle | ° | Normal pressure angle. |
| Helix angle (0 = spur) | ° | Helix angle; 0 for a spur gear (over-pins values are given for spur gears). |
| Profile shift coefficient | Profile shift coefficient of this gear. | |
| Face width | mm | Face width (for the span anvils on a helical gear). |
| Smallest normal backlash wanted (pair) | mm | Smallest normal backlash the pair must have. |
| Largest normal backlash wanted (pair) | mm | Largest normal backlash the pair may have. |
| Share of the backlash taken by this gear | % | How much of the backlash comes from thinning this gear (50% for equal sharing). |
| Pin diameter (0 = 1.728 × module) | mm | Pin or ball diameter for the over-pins check. |
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: Normal module
- What it is
- The normal module of the gear: the tooth size measured square to the tooth, as on the drawing or the hob. For a spur gear it equals the ordinary module, pitch diameter ÷ number of teeth.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.3 mm to 50 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 3 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The nominal span is proportional to the module.
- Get it right
- On a helical gear use the normal module, not the transverse module.
What this means: Number of teeth
- What it is
- The number of teeth on the gear being inspected. It sets how many teeth the span is measured over and whether the over-pins measurement uses the even or odd tooth formula.
- Units and format
- Enter a plain number with no unit. The form accepts 6 to 500. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 30 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
What this means: Normal pressure angle
- What it is
- The pressure angle of the tooth profile in the normal section, as made by the cutting tool and shown on the gear drawing.
- Units and format
- Enter a number in °: degrees of angle. The form accepts 14.5° to 30°. 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.
- Get it right
- Use the normal pressure angle for helical gears, not the transverse value.
What this means: Helix angle (0 = spur)
- What it is
- The helix angle of the teeth at the pitch diameter. Zero means a spur gear, for which the over-pins measurement is also calculated.
- Units and format
- Enter a number in °: degrees of angle. The form accepts 0° to 45°. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0° as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Enter exactly 0 for a spur gear; any other value treats it as helical and drops the over-pins results.
What this means: Profile shift coefficient
- What it is
- The profile shift coefficient of this gear: how far the cutting tool was moved away from (positive) or towards (negative) the gear centre, as a multiple of the module. Take it from the gear drawing.
- Units and format
- Enter a plain number with no unit. The form accepts -1 to 1.5. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- This is the coefficient for this gear only, not the sum for the pair.
What this means: Face width
- What it is
- The axial width of the toothed face. On a helical gear the micrometer anvils must fit across the face at the span, so the tool checks the face is wide enough.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 2000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 30 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- It only matters for helical gears; a spur gear always passes this check.
What this means: Smallest normal backlash wanted (pair)
- What it is
- The least normal backlash the gear pair must have when assembled, measured square to the tooth flanks. With its share it sets the least tooth thinning, and so the largest span.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 5 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.08 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- This is backlash for the pair, not for this gear alone; the share field splits it.
What this means: Largest normal backlash wanted (pair)
- What it is
- The most normal backlash the gear pair may have when assembled, measured square to the tooth flanks. With its share it sets the most tooth thinning, and so the smallest span.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 5 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.16 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- It must not be smaller than the smallest backlash, or the tool stops.
What this means: Share of the backlash taken by this gear
- What it is
- Units and format
- Enter a number in %: percent: the number is read as a percentage, so type 8 for 8 %. The form accepts 0% to 100%. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 50% as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The span limits fall below nominal in direct proportion to this share.
- Get it right
- Type the percentage itself, for example 50 for an equal split, not 0.5.
What this means: Pin diameter (0 = 1.728 × module)
- What it is
- The diameter of the pins or balls placed in opposite tooth spaces for the over-pins measurement on a spur gear. Use the actual size of the pins you hold.
- 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 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Enter 0 to use 1.728 × module; a pin too small for the tooth space stops the calculation.
Method and formulas
- Normal backlash j_n = (Δs₁ + Δs₂) cos α_n, so this gear’s thinning Δs = share × j_n ÷ cos α_n; the span falls by share × j_n.
- Span W = m_n cos α_n [π(k − 0.5) + z inv α_t] + 2x m_n sin α_n with k from the usual rule (as the span measurement tool); anvils need a face width of W sin β cos α_n.
- Over pins (spur): solve inv φ = s ÷ d + inv α + d_p ÷ d_b − π ÷ z with the thinned tooth thickness s; M = d_b ÷ cos φ + d_p (even teeth) (as the gear over pins tool).
- Example: span over 4 teeth should read 32.178–32.218 mm (nominal 32.258 mm).
Limitations
- External gears; internal gears need a different method.
- Over-pins values are for spur gears; helical gears need the helical over-balls formula.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Set the inspection limits for a gear drawing
- Convert a backlash requirement into a span measurement
- Check a gear on the shop floor with a micrometer
Who it is for
- Gear engineers
- Inspectors
- Machinists
What you get
- Tooth thinning for the backlash limits
- Teeth spanned and span measurement limits
- Measurement over pins limits (spur gears)
Interpreting the result
- The span is easier to measure on the machine and does not depend on the outside diameter.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 2236–2248. Topic reference; no source tables or recommended settings are reproduced.
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Open the Gear Tooth Measurement for Backlash (Span and Over Pins) calculator