Involute tooth profile and its properties

Law of gearing

For a constant angular velocity ratio, the common normal at the point of contact of the two teeth must always pass through a fixed point on the line of centres, called the pitch point $P$. Then

$$\frac{\omega_1}{\omega_2} = \frac{O_2P}{O_1P} = \text{constant}$$

(Proof: along the common normal the velocities of the contact point on both bodies must have equal components. This gives $\omega_1 \cdot O_1N_1 = \omega_2 \cdot O_2N_2$, where $O_1N_1$, $O_2N_2$ are perpendiculars from the centres to the common normal. Similar triangles give $O_1N_1/O_2N_2 = O_1P/O_2P$.)

Generation of involute

An involute is the curve traced by a point on a taut string as it is unwound from a circle, called the base circle. A point on a straight line that rolls without slipping on the base circle also traces the involute.

        T      string (tangent)
        |\
        | \
        |  * P  (point on involute)
        |  /
   -----|-/-----
  /     T0      \
 |   base circle  |
  \              /
   --------------

The string length $TP$ equals the arc $T_0T$ on the base circle that has been unwound.

Involute satisfies the law of gearing

Take two base circles of radii $r_{b1}$, $r_{b2}$ and a common tangent $N_1N_2$ to both (the string). When one circle drives the other, the contact point of the two involutes always lies on this tangent. So the common normal at the contact point is always $N_1N_2$, a fixed straight line crossing the line of centres at the same pitch point. Hence the velocity ratio is constant:

$$\frac{\omega_1}{\omega_2} = \frac{r_{b2}}{r_{b1}}$$

This line $N_1N_2$ is the line of action; the angle it makes with the common tangent of the pitch circles is the pressure angle $\phi$.

Properties of involute

1. The normal to an involute at any point is tangent to the base circle.
2. The radius of curvature at any point equals the length of the tangent from that point to the base circle.
3. The involute starts on the base circle and exists only outside it.
4. Path of contact is a straight line, so the pressure angle and the direction of the tooth force remain constant.
5. $r_b = r\cos\phi$ (base circle radius from pitch circle radius).

Advantages

• Constant velocity ratio is kept even if the centre distance changes slightly (the base circles do not change).
• Constant pressure angle gives smooth transmission and steady bearing loads.
• Simple to manufacture: a rack cutter has straight sides, so one cutter serves all gears of the same module and pressure angle.
• Interchangeability of gears of the same module and pressure angle.