How to use this calculator
- Choose module or DP. Select the gear pitch system and enter the tooth size.
- Enter tooth counts. Enter pinion and gear tooth counts.
- Set pressure angle and center distance. Use the operating center distance; standard unshifted gears use m*(z1+z2)/2.
- Set addendum coefficients. Use 1.0 for standard full-depth external spur gears unless the drawing says otherwise.
- Read contact ratio. Compare the contact ratio, path of contact and base pitch, then do strength and tolerance checks separately.
How it works
For an external involute spur gear pair, base radii are
r_b = r cos alpha and outside radii are
r_a = r + h_a*m. The working pressure angle follows the
operating center distance:
alpha_w = acos((r_b1 + r_b2) / a)
The path of contact is the useful length of the line of action between the
two addendum circles:
L = sqrt(r_a1^2 - r_b1^2) + sqrt(r_a2^2 - r_b2^2) - a sin(alpha_w)
Base pitch is p_b = pi*m*cos(alpha), so the transverse contact
ratio is:
epsilon_alpha = L / p_b
Use the gear tooth calculator for tooth dimensions, the involute gear calculator for base-circle geometry and DXF export, and the gear mesh force calculator when the geometry needs to feed shaft and bearing load checks.
Worked example
Verified against the live calculator
A standard 20-tooth pinion driving a 40-tooth gear with module 2 mm,
20 degree pressure angle and center distance
a = 2*(20+40)/2 = 60 mm has base pitch
p_b = pi*2*cos(20 deg) = 5.90 mm. The path of contact is about
9.66 mm, so the transverse contact ratio is
epsilon_alpha = 9.66 / 5.90 = 1.64.
Frequently asked questions
What is gear contact ratio?
Contact ratio is the average number of tooth pairs sharing contact during a mesh cycle. A transverse contact ratio above 1 means at least one tooth pair stays engaged; practical spur gears are commonly screened for margin above that.
How do you calculate spur gear contact ratio?
For external involute spur gears, transverse contact ratio is path of contact divided by base pitch: epsilon_alpha = L / p_b. The calculator finds L from the tip radii, base radii, center distance and working pressure angle.
What is a good contact ratio?
As a first-pass screen, values below about 1.2 deserve attention because the mesh has little overlap margin. Final targets depend on load, noise, profile modification, accuracy grade, speed and the governing gear standard.
Does this work for helical gears?
This page reports transverse spur-gear contact ratio only. Helical gears also have overlap contact ratio from helix angle and face width, so total contact ratio needs an added overlap term.
Does this rate gear strength?
No. Contact ratio is geometry. Tooth bending stress, contact stress, pitting, scuffing, backlash, lubrication, face load distribution and manufacturing tolerances need a gear rating method or supplier review.
Method & assumptions
- External involute spur gear pair, transverse contact ratio only.
- Addendum is entered as a coefficient times module; standard full-depth gears use 1.0.
- The entered center distance is the operating center distance; nonstandard profile shifts should be checked against the drawing.
- Helical overlap ratio, profile modification, tip relief, backlash, tooth bending/contact stress, pitting, scuffing and manufacturing tolerances are not modeled.