Center distance
-Standard, unshifted gearsContact ratio
-Aim for 1.2 or moreTangential tooth force
-Enter power and speedGeometry
| Pinion | Gear | |
|---|---|---|
| Module / diametral pitch | - | - |
| Pitch diameter d | - | - |
| Outside diameter da | - | - |
| Root diameter df | - | - |
| Base diameter db | - | - |
| Whole depth | - | - |
| Circular pitch | - | - |
Mesh and loads
| Gear ratio z2/z1 | - |
| Gear speed n2 | - |
| Pitch line velocity | - |
| Pinion torque | - |
| Radial force Wr | - |
| Normal force Wn | - |
| Minimum teeth without undercut | - |
Checks
How the calculator works
The gears are full-depth involute spur gears with no profile shift, cut to the standard rack of ISO 53 and ANSI/AGMA: addendum ha = 1·m, dedendum hf = 1.25·m (a clearance of 0.25·m). In US units the module is m = 25.4 / P.
- Diameters. Pitch d = m·z; outside da = d + 2m; root df = d − 2.5m; base db = d·cosα.
- Center distance. a = (d1 + d2)/2 = m(z1 + z2)/2.
- Contact ratio. εα = [√(ra1² − rb1²) + √(ra2² − rb2²) − a·sinα] / (π·m·cosα).
- Undercut. The theoretical minimum is zmin = 2 / sin²α. A pinion below it needs profile shift, which is outside this calculator.
- Loads. T = P/ω, Wt = 2T/d1, Wr = Wt·tanα, Wn = Wt/cosα. Pitch line velocity v = π·d1·n1.
Friction and efficiency losses are ignored; the forces are for the transmitted power at the pitch circle. Power is taken as the transmitted tooth power, not the motor rating.
Worked example: module 2, 20 and 60 teeth
Module 2 mm, 20° pressure angle, z1 = 20, z2 = 60, 5 kW at 1500 rpm. Pitch diameters 40 mm and 120 mm, so the center distance is 80 mm. Outside diameters 44 mm and 124 mm, root diameters 35 mm and 115 mm, base diameters 37.59 mm and 112.76 mm. The contact ratio is (11.44 + 25.79 − 27.36)/5.904 = 1.67. The speed ratio is 3, so the gear runs at 500 rpm. Torque on the pinion is 5000/157.08 = 31.83 N·m, so Wt = 2·31.83/0.040 = 1592 N, Wr = 579 N, Wn = 1694 N, and the pitch line velocity is 3.14 m/s.
Questions
How do you calculate spur gear pitch diameter?
The pitch diameter is d = m × z in metric (module m in mm, z teeth) or d = z / P in US units (P = diametral pitch, teeth per inch). Example: a 20-tooth gear of module 2 has a pitch diameter of 40 mm; a 24-tooth gear of 8 diametral pitch has 3.000 in.
How do you find the center distance of two spur gears?
For standard (unmodified) gears the center distance is the sum of the two pitch radii: a = m (z1 + z2) / 2. A 20-tooth pinion meshing with a 60-tooth gear at module 2 sits at 80 mm. Profile shifted gears use a different distance, which this calculator does not cover.
What is the contact ratio of a gear pair and what should it be?
The contact ratio is the average number of tooth pairs sharing the load. It is the length of the path of contact divided by the base pitch. Values above about 1.2 are the practical minimum; 1.4 to 2.0 is typical for spur gears. Below 1.0 the mesh is broken and the gears cannot run smoothly.
What is the minimum number of teeth for a spur gear without undercut?
For a full-depth tooth (addendum = 1 module) the theoretical minimum is 2 / sin²α, which is 17.1 teeth at 20 degrees, so 18 teeth is safe and 17 is the usual limit. At 14.5 degrees it is about 32 teeth and at 25 degrees about 11.2. Fewer teeth need profile shift or a stub tooth.
How do you get the tooth force from power and speed?
Torque is T = P / ω with ω = 2π n / 60. The tangential force on the pitch circle is Wt = 2 T / d, the radial (separating) force is Wr = Wt tan α, and the normal force is Wn = Wt / cos α. Example: 5 kW at 1500 rpm on a 40 mm pitch diameter gives T = 31.8 N·m and Wt = 1592 N.
Does this calculator check gear tooth strength?
No. It gives geometry, contact ratio, velocity and tooth forces. Bending and surface-durability checks under ISO 6336 or AGMA 2101 need load factors, material grade and quality that depend on your application, so use those standards or a gear specialist for a rating.
Engineering aid for guidance. Geometry of standard full-depth involute spur gears and static tooth forces only; not a strength rating (ISO 6336, AGMA 2101), profile shift, backlash, helical or internal gears. Verify critical designs; you remain responsible for the design. Nothing you enter leaves your browser except through a share link you copy yourself.