OpenSpecEngineering

Finds the basic geometry of a pair of standard spur gears: pitch, base and outside diameters, center distance, ratio and contact ratio. It also warns when the teeth interfere or the contact ratio is too low to run smoothly. Choose a standard module or diametral pitch, the two tooth counts and the pressure angle. Designers use it to lay out a gear pair before checking tooth strength and ordering gears.

Equations

d = m·N     a = (d₁ + d₂)/2     d_b = d·cos φ     d_a = d + 2m

m_c = [√(r_a1² − r_b1²) + √(r_a2² − r_b2²) − a·sin φ] / (π·m·cos φ)

m is the module, the pitch diameter per tooth in mm. A diametral pitch P (teeth per inch of pitch diameter) is converted with m = 25.4/P. N₁ and N₂ are the pinion and gear tooth counts, d the pitch diameter, a the center distance, φ the pressure angle, d_b the base diameter the involute is generated from, and d_a the outside (tip) diameter. r_a and r_b are the tip and base radii. Full-depth teeth have an addendum of one module [1], [3]. The contact ratio m_c is the length of the path of contact divided by the base pitch: the average number of tooth pairs in mesh [1]. The gear ratio is N₂/N₁.

The page flags interference when either tip circle reaches past the point where the line of action touches the other gear’s base circle [1]. The modules are the ISO 54 series [2]; the diametral pitches are those in general use [1].

Limits and common mistakes

• The page covers standard, full-depth spur gears without profile shift. Helical gears, stub teeth and shifted profiles have other diameters and center distances.
• It gives geometry only, not tooth strength or wear. Check bending and contact stress before choosing a face width.
• With 20° teeth a pinion of fewer than about 18 teeth can interfere with a large gear. Use more teeth or a 25° pressure angle rather than undercut teeth.
• Keep the contact ratio at 1.2 or more for smooth, quiet running.

Worked example

A conveyor drive needs a 2.5:1 speed reduction with spur gears of module 2.5. Try an 18-tooth pinion on a 45-tooth gear and find the center distance for the gearbox housing.

1. Choose Tooth Size m = 2.5 mm. The module fills in as 2.5 mm.
2. Enter Pinion Teeth 18 and Gear Teeth 45.
3. Choose Pressure Angle 20° and press Solve. The Solution shows: Pinion Pitch Diameter (d1): 45 mm
   Gear Pitch Diameter (d2): 112.5 mm
   Center Distance: 78.75 mm
   Gear Ratio (N2/N1): 2.5
   Pinion Base Diameter (db1 = d1·cos φ): 42.2862 mm
   Gear Base Diameter (db2 = d2·cos φ): 105.715 mm
   Pinion Outside Diameter (d1 + 2m): 50 mm
   Gear Outside Diameter (d2 + 2m): 117.5 mm
   Contact Ratio (mc): 1.63281

Check: the center distance is m·(N₁ + N₂)/2 = 2.5 × 63/2 = 78.75 mm. No note appears, so the teeth don’t interfere, and a contact ratio of 1.63 is typical of 20° spur gears.