Selecting Gears

Failure Modes of Gears (Spurs)

• Gears can fail in two ways:
  • Failure by bending
  • Pitting failure (surface / contact wear)

For both cases we need to satisfy $\sigma \le {\sigma }_{all}$

Lewis Bending Equation

Here, we approximate a tooth as a cantilever beam…
From these definitions from Stress Concentration

• $\sigma =\frac{Mc}{I}$
• $c=\frac{t}{2}$
• $I=\frac{b{h}^3}{12}$
• Where $M=W_{t}l$ due to $W_t$
• In this case our b is sometimes F which is the thickness of a tooth and our h is sometimes t which is the gear tooth width

We can derive the Lewis bending question

• $\sigma =\frac{W_t}{F\left(\frac{2}{3}x\right)}\left(\frac{P}{P}\right)=\frac{W_{t}P}{F\left(\frac{2}{3}xP\right)}=\frac{W_{t}P}{FY}$
• Where $Y=\frac{2}{3}xP$ and is non-dimensional (has no units)
• Most of these parameter other than $W_t$ come from tables
• This Lewis equation assumes that your bending moment is static

We want to use this equation with a dynamic factor for dynamic loading conditions at the gear:

• $K_v$
• $\sigma =K_v\frac{W_{t}P}{FY}$ where $K_v={\left(\frac{a+V^b}{a}\right)}^c$ and $V=\omega R$ (tangential velocity)
  • Note that here $K_v$ is the “cyclic factor“

Our bending strength ${\sigma }_{all}$ can be derived from Failure Theories, specifically using:

• $S_y$ for moderate operating conditions
• $S_e$ for infinite life bending

Pitting Failure

• The Hertz contact stress equation affects surface durability
• ${\sigma }_c=-C_p{\left[\frac{K_v{W}^t}{F\cos \varphi }\left(\frac{1}{r_1}+\frac{1}{r_2}\right)\right]}^{\frac{1}{2}}$ where ${\sigma }_{c.\text{all}}=S_e$
• Here, $r_1=\frac{d_p\sin \varphi }{2},r_2=\frac{d_G\sin \varphi }{2}$ which are the radii lines of action for the gear and pinion
• And $C_p-{\left[\frac{1}{\pi \frac{\left(1-{\nu }_p^2\right)}{E_p}+\left(\frac{1-{\nu }_G^2}{E_G}\right)}\right]}^{\frac{1}{2}}$ is the elastic coefficient
• $\nu$ 1 and 2 here are poisson ratios given in a table
• $E$ is the elastic modulus of the gear and pinion
• Here, ${\sigma }_C$ is compared to ${\sigma }_{c,all}$ which is parameterized by $H_B$ or the Brinell Hardness which comes from a table given something like an ASTM number
• Note that the contact stresses here are always in compression, therefore are (-)