Gear Bending Analysis

Lewis Equation

Bending stress for gears is given by the Lewis equation.

Approximating the tooth as a rectangular cantilever beam, recall that

$$\sigma =\frac{Mc}{I}$$

where $c=\frac{t}{2}$ and $I=\frac{b{h}^3}{12}=\frac{F{t}^3}{12}$.

Thus, the Lewis equation is given as:

$$\sigma =\frac{W_{t}l\frac{t}{2}}{F{t}^3/12}=\frac{6W_{t}l}{F{t}^2}$$

By similar triangles, we have

$$\frac{t/2}{x}=\frac{l}{t/2}⟶2x=\frac{t^2}{2l}⟶\frac{6l}{t^2}=\frac{3}{2x}$$

With this, we can get the Lewis equation as

$$\begin{array}{rl}\sigma & =\frac{W_t}{F}\cdot \frac{6}{t^2}\\ & =\frac{W_t}{F}\cdot \frac{3}{2x}\\ & =\frac{W_t}{F\cdot \frac{2x}{3}}\\ & =\frac{W_{t}P}{F\frac{2x}{3}P}\\ & =\frac{W_{t}P}{FY}\end{array}$$

where $Y=\frac{2xP}{3}$ is a form factor. $Y$ can be found from a table:

To account for dynamic effects, a dynamic factor is used:

$$\begin{array}{rl}\sigma & =K_v\frac{W_{t}P}{FY}[\text{psi}]\\ & =K_v\frac{W_t\frac{1}{m}}{FY}[\text{MPa}]\end{array}$$

where $K_v={\left(\frac{a+V^b}{a}\right)}^c$, with $V$ being the pitch line (linear) velocity.

$a$, $b$ and $c$ can be found from a table:

Allowable Stress

Allowable stress ${\sigma }_{\text{all}}$ can be obtained from:

• Yield strength $S_y$ for moderate operating conditions
• Fatigue strength (or endurance limit $S_e$) for infinite life bending
• Safety factor is applied to compute ${\sigma }_{\text{all}}$:

$${\sigma }_{\text{all}}=\frac{S_e}{n_s}$$

Example