Gear Tooth Strength Calculation and Equation

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The following is general guidelines for determining or estimating your required gear tooth strength.

Wilfred Lewis, in 1892, presented his expression for tooth beam strength which is now reknowned as the classic Lewis equation. As a static beam resisting a fixed load in position and magnitude, this equation is usually adequate. However, it does not take into account the dynamics of meshing teeth. In that regard, later investigators have modified and improved the original Lewis equation.

When a gear train system is transmitting power and motion, it is safe to assume that all of the load is being carried by one tooth.This is most correct because as the load approaches the end of the tooth, where the bending force would be the greatest, a second tooth comes into mesh to share the load. Simple results can be obtained from the Lewis bending strength equation.

Eq. 1
Wt = ( S · F · Y ) / Dp

Eq. 2
DP = t / pd


W = Maximum transmitted load (lbs, N)
S = Maximum bending tooth stress taken as 1/3 of the tensile strength (psi, N/mm2)
F = Face width of gear (in, mm)
DP = Diametral Pitch, (in, mm).
Y = Lewis Factor (See Lewis Factor for Gears) (no units)
t = number of teeth, #
pd = pitch circle of gear teeth (in, mm)

The conversion formula for the module (m) from the diametral pitch (DP) is as follows:

m = 25.4 /DP

The Lewis factor is dimensionless and independent of tooth size, and a function only of shape.

Open Gear Tooth Strength Calculator

Spur Gear Tooth Profile

The maximum bending tooth stress (S) is valid for well lubricated, low shock applications. For high shock, poorly lubricated applications, the safe stress could be as low as .025S. If your design calls for an harsh environment for your gear application, you might want to lower S to assure a reasonable amount of gear life.


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