Measurement of Screw Threads

(a) Errors in threads

1. Pitch errors: progressive (pitch increases or decreases uniformly), periodic (repeats every revolution due to eccentric lead screw), drunken thread (pitch varies along the helix), irregular.
2. Flank angle errors: the thread angle is incorrect, or one flank is different from the other.
3. Major and minor diameter errors: wrong depth and root form.
4. Effective (pitch) diameter errors: the diameter at which the thread and gap widths are equal is wrong.

Effect: the bolt (external) and nut (internal) cannot assemble because the pitch error or angle error makes the bolt act as if it has a larger effective diameter. Contact then occurs only on a few points on flanks, so the load carrying area drops, the thread wears fast and can strip. The effective diameter of the bolt must be reduced (or that of the nut increased) to compensate the errors, which wastes tolerance.

(b) Numerical

A pitch error $\delta P$ forces the flanks of the threads to ride over each other; to assemble it, the effective diameter of the bolt must be reduced by an equal amount, therefore the virtual effective diameter is increased by

$$\delta E = \delta P\,\cot\frac{\alpha}{2}$$

For $\alpha = 60^\circ$: $\cot 30^\circ = 1.732$

$$\delta E = 1.732 \times 0.03 = 0.052\ \text{mm}$$

Virtual effective diameter $= 18.350 + 0.052 = 18.402$ mm.

For Whitworth ($\alpha = 55^\circ$): $\cot 27.5^\circ = 1.921$

$$\delta E = 1.921 \times 0.03 = 0.0576\ \text{mm}, \quad E_{virtual} = 18.350 + 0.058 = 18.408\ \text{mm}$$

Answer: increase = 0.052 mm (metric), virtual effective diameter = 18.402 mm; for Whitworth 0.058 mm (18.408 mm).

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).

A toolmaker's microscope has a base with a table carrying the work, which can be moved in two perpendicular directions by micrometer heads (least count 0.01 mm or 0.005 mm), a microscope with a cross-hair eyepiece that rotates and has an angle scale, and a light source below the table that throws a sharp shadow of the thread profile.

Major and minor diameters

The thread is held on its axis on the table (between centres). The cross-wire is set on the edge of the crest on one side and the reading of the micrometer head is taken; the table is moved until the cross-wire touches the opposite crest, and the second reading is taken. The difference is the major diameter. The same is done at the roots to get the minor diameter.

Pitch

The cross-wire is set to a definite point (a flank) of a thread and the micrometer reading is noted. The table is moved parallel to the axis until the same point of the next thread (or after $n$ threads) comes in the cross-wire; the difference of readings is the pitch (or $n$ pitches, better for accuracy). Pitch error follows from comparison with the nominal pitch. The table must move parallel to the axis; the head is tilted by the helix angle to bring the flank into focus.

Thread angle

The eyepiece is turned until its cross-wire lies along one flank and the angle scale is read; then it is turned until it lies along the other flank and the scale is read again. The difference of the two readings gives the included angle (for a 60 degree thread, 60 degrees). The half-angles of each flank are measured separately to check symmetry.

Limitations: the microscope only measures the projected profile, so errors arise if the helix angle is not compensated for, and the contrast depends on lighting.