BIZOMAT MAKEOS · Gears
Gear Pair for a Given Centre Distance (Profile Shift, Tips, Contact Ratio)
What profile shifts make this spur gear pair fit my centre distance, what tip diameters does that give, and is the contact ratio and undercut acceptable?
Design of an external spur gear pair to an off-standard centre distance: working pressure angle from the centre distance, the sum of profile shift coefficients from the involute function, its split between pinion and gear, tip diameters with or without tip shortening to keep the standard clearance, the contact ratio with those tips (as the contact ratio tool), the undercut limit for each gear, and the working pressure angle check against the profile shift tool.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Module | mm | Module of both gears. |
| Pinion teeth | Teeth on the pinion. | |
| Gear teeth | Teeth on the gear. | |
| Pressure angle | ° | Pressure angle of the tooling. |
| Centre distance required | mm | Centre distance the housing fixes. |
| Share of the total shift on the pinion | % | How much of the total profile shift goes on the pinion (more on the pinion helps strength and avoids undercut). |
| Tip shortening | choice | Whether the tip diameters are reduced so the clearance stays at the standard value. |
| Smallest contact ratio wanted | Least transverse contact ratio you accept. |
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: Module
- What it is
- The module of both gears, pitch diameter ÷ number of teeth, which must be the same on pinion and gear to mesh.
- 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 2 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The standard centre distance is proportional to the module.
What this means: Pinion teeth
- What it is
- The number of teeth on the pinion, the smaller gear of the pair.
- Units and format
- Enter a plain number with no unit. The form accepts 6 to 300. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 14 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Small tooth numbers risk undercut; the tool checks it against the pinion’s shift.
What this means: Gear teeth
- What it is
- The number of teeth on the gear, the larger wheel of the pair.
- Units and format
- Enter a plain number with no unit. The form accepts 6 to 1000. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 40 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
What this means: Pressure angle
- What it is
- The standard pressure angle of the cutting tool used for both gears, before any profile shift.
- 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
- This is the tool’s pressure angle, not the working pressure angle the tool calculates.
What this means: Centre distance required
- What it is
- The centre distance between the gear shafts that the housing fixes, which the pair must run at. Profile shift makes up the difference from the standard centre distance.
- 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 55 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 distance than standard increases the total profile shift needed.
- Get it right
- If it is too small for these gears, the tool stops; check the module and tooth numbers.
What this means: Share of the total shift on the pinion
- 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 70% as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Type the percentage itself, for example 70, not 0.7.
What this means: Tip shortening
- What it is
- Whether the tip diameters are reduced so the clearance at the tooth roots stays at the standard value after profile shift.
- Units and format
- Choose one of 2 options from the list: Shorten tips to keep standard clearance; No tip shortening.
- Example
- The form opens with “Shorten tips to keep standard clearance” selected as a worked example. Choose the option that matches your case.
- Get it right
- Without shortening, positively shifted pairs can have less root clearance than standard.
What this means: Smallest contact ratio wanted
- What it is
- The least transverse contact ratio you accept, the average number of tooth pairs in mesh, from your design basis.
- Units and format
- Enter a plain number with no unit. The form accepts 1 to 2.5. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1.2 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Method and formulas
- Standard centre distance a = m (z₁ + z₂) ÷ 2; working pressure angle cos α_w = a cos α ÷ a_w.
- Σx = (inv α_w − inv α)(z₁ + z₂) ÷ (2 tan α); split by the share (as the profile shift tool in reverse).
- Centre distance factor y = (a_w − a) ÷ m; tip shortening k = Σx − y; tip radius r_a = m (z ÷ 2 + 1 + x − k).
- Contact ratio ε = [√(r_a1² − r_b1²) + √(r_a2² − r_b2²) − a_w sin α_w] ÷ (π m cos α) (as the contact ratio tool); no undercut if x ≥ 1 − z sin²α ÷ 2.
- Example: 55 mm needs Σx = 0.53289 (pinion 0.37302, gear 0.15987) at 22.69° working angle; contact ratio 1.4127.
Limitations
- External spur gears of standard full-depth proportions.
- Tooth tip thickness (pointing) at large shifts is not checked: verify it on the drawing.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Fit a gear pair into an existing housing
- Avoid undercut on a small pinion
- Check the contact ratio after shifting
Who it is for
- Gear engineers
- Design engineers
- Machinists
What you get
- Working pressure angle and total profile shift
- Shift, tip diameter and undercut check for each gear
- Contact ratio
Interpreting the result
- Putting most of the positive shift on the pinion strengthens it and keeps it clear of undercut.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 2141–2160. Topic reference; no source tables or recommended settings are reproduced.
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Open the Gear Pair for a Given Centre Distance (Profile Shift, Tips, Contact Ratio) calculator