Straight bevel gear geometry and formative teeth

Definitions

• Pitch cone angle ($\gamma$): the angle between the gear axis and the pitch cone element.
• Cone distance ($A_0$): the length of a pitch cone element from the apex to the large-end pitch circle.
• Back cone: a cone whose elements are perpendicular to the pitch cone elements at the large end of the tooth. When developed (laid flat), it gives a spur gear.
• Formative (virtual) number of teeth: the number of teeth on the spur gear whose pitch radius equals the back cone distance.

Tredgold's approximation

The tooth profile of a bevel gear is taken as that of a spur gear lying on the back cone surface. The back cone radius is $r_v = \dfrac{d}{2\cos\gamma}$, so the virtual spur gear has pitch diameter $d/\cos\gamma$ and

$$z' = \frac{z}{\cos\gamma}$$

This $z'$ is used for tooth strength design and to check interference ($z'$ need not be an integer).

       apex
        /|\
       / | \   A0 = cone distance
      /  |  \
     /   |   \
    /_(gamma)_\
    gear pitch dia d

Shafts at 90 degrees

For shaft angle $\Sigma = 90^\circ$:

$$\tan\gamma_1 = \frac{z_1}{z_2}, \qquad \gamma_2 = 90^\circ - \gamma_1$$

Calculation

$$\gamma_1 = \tan^{-1}\frac{20}{40} = 26.57^\circ, \qquad \gamma_2 = 90^\circ - 26.57^\circ = 63.43^\circ$$

Pitch diameters (large end):

$$d_1 = mz_1 = 6 \times 20 = 120\ \text{mm}, \qquad d_2 = mz_2 = 6 \times 40 = 240\ \text{mm}$$

Cone distance:

$$A_0 = \sqrt{\left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2} = \sqrt{60^2 + 120^2} = 134.16\ \text{mm}$$

Face width limit (to avoid weak tooth tips and cutting problems): $b \le A_0/3$ or $10m$, whichever is less. Here $10m = 60$ mm, so

$$b_{max} = \frac{134.16}{3} = 44.72\ \text{mm}$$

Formative numbers of teeth:

$$z_1' = \frac{20}{\cos 26.57^\circ} = 22.36, \qquad z_2' = \frac{40}{\cos 63.43^\circ} = 89.44$$

Tooth proportions (large end): addendum $= m = 6$ mm, dedendum $= 1.2m = 7.2$ mm.

Answer: $\gamma_1 = 26.57^\circ$, $\gamma_2 = 63.43^\circ$, $d_1 = 120$ mm, $d_2 = 240$ mm, $A_0 = 134.16$ mm, $b_{max} = 44.72$ mm, $z_1' = 22.36$, $z_2' = 89.44$.