Effective diameter by three-wire method

Method

Three wires of equal diameter $d$ are placed in the thread grooves, two on one side and one on the opposite side. The distance $M$ over the wires is measured with a micrometer (or a comparator with a fixed anvil). The effective diameter is then calculated from $M$, the wire diameter and the thread form.

     wire
    __O__          M measured across
   /     \       the outer surfaces of
  /       \      the wires with a
 /   thread \    micrometer
    O   O

For a thread of included angle $\alpha$ and pitch $P$ (wire touching the flanks at the pitch line):

$$M = E + d\left(1 + \operatorname{cosec}\frac{\alpha}{2}\right) - \frac{P}{2}\cot\frac{\alpha}{2}$$

For 60 degrees this becomes $M = E + 3d - 0.866P$.

Best wire size

The wire should touch the flanks at the pitch (effective) diameter, where the gap width is $P/2$, so that an error in the thread angle has the least effect on the result. The contact point is $P/4$ from the centre line of the groove, and the radius of the wire is $d/2$, so

$$\frac{P}{4} = \frac{d}{2}\cos\frac{\alpha}{2} \;\Rightarrow\; d_{best} = \frac{P}{2\cos(\alpha/2)}$$

For 60 degrees, $d_{best} = 0.5774\,P$.

Numerical

$d = 0.5774 \times 2.5 = 1.443$ mm

$$E = M - 3d + 0.866P = 20.563 - 3(1.443) + 0.866(2.5) = 20.563 - 4.330 + 2.165 = 18.398\ \text{mm}$$

Error $= 18.398 - 18.376 = +0.022$ mm, the thread is slightly oversize in effective diameter.

Answer: effective diameter = 18.398 mm (+0.022 mm over nominal).