Gear tooth vernier: chordal thickness

Gear tooth vernier caliper

It consists of two perpendicular vernier scales. The vertical scale is set to the chordal addendum $h_c$ and acts as a depth stop, resting on the top of the tooth. The horizontal scale measures the chordal thickness $t_c$, the straight line distance between the two flanks at the pitch circle.

        |<-- t_c -->|
     ___|___________|___   <- horizontal vernier jaws
         \    __   /
          \  /  \ /       tooth
    h_c    ||    ||
    (vertical vernier stop on tooth top)

The vertical slide is set to $h_c$ and locked, the instrument is placed over the tooth so that the stop touches the top of the tooth, the jaws are closed on the flanks and the horizontal vernier reading is compared with the calculated chordal thickness. Errors: the measurement depends on the outside diameter of the blank, which must be checked, and the instrument is only accurate to its vernier least count (0.02 mm). Each tooth must be measured separately; several teeth are checked around the gear.

Formulae

For a gear with $z$ teeth and module $m$, the circular thickness at the pitch circle is $\pi m/2$ and

$$t_c = z\,m\sin\!\left(\frac{90^\circ}{z}\right), \qquad h_c = m + \frac{z\,m}{2}\left[1 - \cos\!\left(\frac{90^\circ}{z}\right)\right]$$

Calculation ($m = 5$, $z = 30$)

$$\frac{90^\circ}{z} = 3^\circ$$ $$t_c = 30 \times 5 \times \sin 3^\circ = 150 \times 0.052336 = 7.850\ \text{mm}$$ $$h_c = 5 + \frac{150}{2}\left(1 - \cos 3^\circ\right) = 5 + 75(0.001370) = 5.103\ \text{mm}$$

Answer: chordal thickness = 7.850 mm, chordal addendum = 5.103 mm.