BIZOMAT MAKEOS · Gears
Spur Gear Contact Diameters, Contact Ratio and LPSTC
Where does contact start on each gear, what is the contact ratio, and where is the lowest point of single tooth contact?
For two spur gears at standard centre distance: the contact (start of active profile) diameter on each gear, the contact ratio by two independent formulas, the lowest point of single tooth contact (LPSTC) on each gear for contact stress checks, and the maximum hob tip radius for the tooth system.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Module | mm | Module (25.4 ÷ diametral pitch). |
| Teeth on gear 1 (pinion) | Number of teeth on the first gear. | |
| Teeth on gear 2 | Number of teeth on the second gear. | |
| Pressure angle | ° | Pressure angle. |
| Outside diameter of gear 1 (0 = standard) | mm | Actual outside diameter of gear 1; 0 uses the standard m (z + 2). |
| Outside diameter of gear 2 (0 = standard) | mm | Actual outside diameter of gear 2; 0 uses the standard value. |
| Dedendum constant b (dedendum = b × module) | Dedendum as a multiple of module for the tooth system (1.25 for standard full depth). |
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 the gear pair: pitch diameter divided by number of teeth. Both gears must share it to mesh. It scales every diameter the tool reports.
- 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 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- All diameters scale in proportion to the module; the contact ratio does not change.
- Get it right
- For inch gears convert diametral pitch to module: module = 25.4 ÷ diametral pitch.
What this means: Teeth on gear 1 (pinion)
- What it is
- The number of teeth on the first gear, normally the pinion (the smaller gear 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 20 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Small tooth counts at low pressure angles can undercut or interfere; the tool reports interference rather than a result.
What this means: Teeth on gear 2
- What it is
- The number of teeth on the second gear, which meshes with gear 1 at the standard centre distance.
- Units and format
- Enter a plain number with no unit. The form accepts 6 to 2000. 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.
- Get it right
- Either gear can be the larger; keep each outside diameter with its own gear.
What this means: Pressure angle
- What it is
- The pressure angle of the tooth system: the angle between the line of action and the tangent to the pitch circles, shared by both gears.
- 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.
- Get it right
- Use the tooth system’s pressure angle from the drawing or cutter; both gears must have the same one.
What this means: Outside diameter of gear 1 (0 = standard)
- What it is
- The actual outside (tip) diameter of gear 1 if it differs from standard, for example after topping or with a long-addendum design.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 100000 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
- A larger tip diameter on gear 1 starts contact lower on gear 2 and raises the contact ratio.
- Get it right
- Leave it at 0 to use the standard tip diameter, module × (teeth + 2).
What this means: Outside diameter of gear 2 (0 = standard)
- What it is
- The actual outside (tip) diameter of gear 2 if it differs from standard.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 100000 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
- A larger tip diameter on gear 2 starts contact lower on gear 1 and raises the contact ratio.
- Get it right
- Leave it at 0 to use the standard tip diameter; measure a topped gear across the tips.
What this means: Dedendum constant b (dedendum = b × module)
- What it is
- The dedendum of the tooth system as a multiple of the module, the depth of the tooth below the pitch circle. Used only for the maximum hob tip radius.
- Units and format
- Enter a plain number with no unit. The form accepts 1 to 2. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1.25 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Take it from the tooth system standard you cut to; it does not affect the contact diameters or contact ratio.
Method and formulas
- With P = 1 ÷ m: cos A = N cos φ ÷ (D₀ P); tan b = tan φ − (N ÷ n)(tan A − tan φ); contact diameter d_c = n cos φ ÷ (P cos b), and likewise for the other gear.
- Contact ratio m_f = N (tan A − tan B) ÷ 2π = n (tan a − tan b) ÷ 2π (both are reported as a check).
- LPSTC: tan c = tan a − 2π ÷ n; d_L = n cos φ ÷ (P cos c).
- Maximum hob tip radius = m (0.785398 cos φ − b sin φ) ÷ (1 − sin φ).
- Example: 20 and 40 teeth, module 1, contact begins at 18.878 mm on the pinion with a contact ratio of 1.6352.
Limitations
- Standard centre distance and standard tooth thickness; operating centre distance changes the result.
- A contact diameter at or below the base circle indicates interference.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Specify the form diameter a ground or hobbed profile must reach
- Find the critical diameter for Hertz stress
- Check contact ratio after changing outside diameters
Who it is for
- Gear designers
- Gear inspectors
- Machinists
What you get
- Contact diameter on each gear
- Contact ratio
- LPSTC diameters and maximum hob tip radius
Interpreting the result
- Keep the true involute form diameter below the contact diameter, and the contact ratio above about 1.4 for power gears.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 2155–2157. Topic reference; no source tables or recommended settings are reproduced.
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Open the Spur Gear Contact Diameters, Contact Ratio and LPSTC calculator