Bevel Gears

Bevel Gears

96% axial + rolling friction

Some Properties

• Based on rolling cones
• Intersecting axes
• Can turn any corner angle
• Low power
• Straight bevels are “spur”
• Spiral bevels are “helical”
• Axial, radial, and tangential forces

Types

• Straight
• Spiral
• Hypoid (Kind of a hollow circular gear)

• There are two cones, one for the pinion and one for the gear
• Each one has a corresponding pitch cone angle
• $\gamma$ is the pitch cone angle of the pinion
• $\Gamma$ is the pitch cone angle of the gear
• $\gamma +\Gamma =90°$ (usually)
• $\tan \Gamma =\frac{\frac{dG}{2}}{\frac{dP}{2}}\tan \gamma =\frac{d\frac{p}{2}}{d\frac{g}{2}}$
• $\tan \Gamma \cdot \tan \gamma =1$

Using diameters

You can use $d_p$ and $d_G$ to find $m$ (module), and $P$ (diametral pitch)
$\tan \Gamma =\frac{d_p}{d_G}=\frac{N_P}{N_G}=\frac{1}{\tan \gamma }$

Straight Bevel Gears

The Pinion

• $W_{np}$ is the normal reaction force on the pinion (P), along the line of action at contact point
  • $W_{np}=\frac{W_{tp}}{\cos \varphi }$
• $W_{tp}$ is the tangential force on the pinion (P), along the line of torque transfer
  • $P=\omega T=\left(2\pi n\right)\left(\frac{D_p}{2}{W}_{tp}\right)=\pi n{D}_p{W}_{tp}$
  • THIS IS APPLICABLE TO ALL GEARS
• $W_{xp}$ is the axial force on the pinion
• $\varphi$ is the pressure angle of the pinion and is the angle between the line of action and the torque transfer line
• $W_{rp}$ is the radial for on the pinion
  • $W_{rp}=W_{np}\sin \varphi \cos \gamma =W_{tp}\tan \varphi \cos \gamma$

The Gear

• $W_{rg}=W_{xp}$
• $W_{xg}=W_{rp}$
• $W_{tg}=W_{tp}$

General Procedure

• Load distribution has components in all directions $W_r,W_t,W_x$ which are applied at the pitch radii $\left(R_p,R_G\right)$
• Given $W_t$ for either the pinion or the gear, we can use $W_{rP}=W_{tp}\tan \varphi \cos \gamma$ and $W_{xp}=W_{tp}\tan \varphi \sin \gamma$ first to get radial and axial stuff
• The pinion is perpendicular to the gear so the radial direction for the pinion is the axial direction for the gear as noted above.