Interference and minimum number of teeth

Interference

Involute action exists only outside the base circle. If the tip of the driven gear tooth goes below the base circle of the pinion (that is, beyond the tangent point $N$ of the line of action), the tip digs into the flank of the pinion below its base circle. This mismatch is called interference. It causes undercutting, weakens the tooth and gives noise and wear.

   pinion base circle
         ____
        /    \----- N (tangent point)
       |  O   |   \
        \____/     \  line of action
                    \ K  <- gear tip beyond N:
                           interference

Minimum teeth with a rack

Let pinion pitch radius be $R = mz/2$ and the rack addendum $= a_w m$ ($a_w = 1$ for full depth). The rack tip should not go beyond the point $N$ (the tangent point on the pinion base circle). The distance from the pitch point $P$ to the rack's tip along the line of action is $\dfrac{a_w m}{\sin\phi}$. The distance $PN = R\sin\phi$. For no interference:

$$\frac{a_w m}{\sin\phi} \le R\sin\phi = \frac{mz}{2}\sin\phi$$ $$z_{min} = \frac{2a_w}{\sin^2\phi}$$

Values for full depth ($a_w = 1$), $\sin^2 20^\circ = 0.1170$

Pinion meshing with a gear (ratio $G = 3$)

Interference is avoided when the gear addendum circle does not pass the pinion's base-circle tangent point $N_1$:

$$(R_G + a_wm)^2 \le R_G^2\cos^2\phi + (R + R_G)^2\sin^2\phi$$

with $R_G = GR$. Solving the resulting quadratic for the pinion pitch radius $R$:

$$z_{min} = \frac{2a_w\left[G + \sqrt{G^2 + (2G+1)\sin^2\phi}\right]}{(2G+1)\sin^2\phi}$$

For $G = 3$, $\phi = 20^\circ$, $a_w = 1$: $z_{min} = 14.98$, so a pinion of at least 15 teeth is needed (the value rises to 17.1 as $G \to \infty$, which is the rack case, and drops to 12.32 for $G = 1$).

Methods to avoid interference

• Use more teeth on the pinion than $z_{min}$.
• Use stub teeth (shorter addendum) or larger pressure angle.
• Use profile (addendum) correction, that is, a long-addendum pinion and short-addendum gear (X-gears).
• Increase the centre distance (gives a larger working pressure angle).

Answer: $z_{min} = 2a_w/\sin^2\phi = 31.90$ (14.5 deg), 17.10 (20 deg), 11.20 (25 deg); for $G = 3$, 20 deg: 14.98, take 15 teeth.