Gear measurement

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.

(a) Pitch measurement

Pitch errors cause noise and uneven motion. Two types are measured:

• Single (adjacent) pitch error: the difference between the actual and the theoretical distance between adjacent teeth. A pitch-measuring instrument with a fixed stop on one tooth and a sensitive dial gauge on the next tooth is set to zero with a master tooth; then each tooth is checked in turn.
• Cumulative pitch error: the error in the position of any tooth with respect to the first, measured over several pitches. Also, the base pitch is measured by a base pitch gauge that reads the distance between parallel tangents on the base circle: $p_b = \pi m\cos\alpha$.

(b) Tooth bearing contact test

The gear is coated thinly with Prussian blue (marking compound) and rolled in contact with a master gear under light braking. The contact patch left on the teeth shows the area of bearing. Ideally the contact should be at the centre, covering about 70 to 80 % of the tooth height and length. A contact at the ends or the tip indicates misalignment, taper, wrong centre distance or profile errors. The test is simple and it is used for large gears and final assembly checks.

(c) Composite method

The gear under test is meshed with a master gear on a double-flank roll-testing machine. One gear is mounted on a fixed spindle and the other on a spring-loaded sliding carriage so that the two stay in tight mesh. As they rotate, the centre distance varies with errors of tooth thickness, profile, pitch and runout. The variation is recorded as a chart and read as:

• Total composite error: the total variation of centre distance in one full revolution.
• Tooth-to-tooth composite error: the variation in one pitch rotation.

It is fast and checks the combined effect, so it is common in mass production, but it does not tell which error is present.