Measurement of screw threads and gears

Why it matters

Threads and gears are the most common precision features in machines, and their fit depends on several interacting dimensions rather than one diameter. A bolt with the correct major diameter can still fail to assemble if its effective diameter or flank angle is wrong, and a gear with the right outside diameter can still be noisy if its tooth thickness or profile is off. This topic gives the standard shop-floor and laboratory methods to check them.

Key ideas

Screw thread elements.

• Major diameter: the largest diameter (crest of an external thread). Minor (root) diameter: the smallest.
• Effective (pitch) diameter: the diameter of the imaginary coaxial cylinder that cuts the thread so that the widths of thread and space along it are equal (each p/2). It decides the fit between nut and bolt, so it is the most important thread dimension.
• Pitch p: axial distance between corresponding points on adjacent threads. Lead: axial advance in one turn; lead = number of starts × pitch.
• Flank angle: half the included thread angle (30° for the 60° ISO metric thread, 27.5° for the 55° Whitworth thread).
• Thread form errors: pitch errors (progressive, periodic, drunken) and flank-angle errors effectively change the effective diameter; this extra amount is called the virtual effective diameter.

Measuring thread elements.

• Major diameter: bench micrometer or floating carriage micrometer against a setting cylinder.
• Minor diameter: micrometer with small V-shaped prisms that sit in the root.
• Effective diameter: thread micrometer (cone and V anvils), the two-wire method or the three-wire method. Three precision wires of equal diameter are placed in the grooves (one side two wires, the other one) and the distance over the wires M is measured with a micrometer.
• Best-size wire: the wire that touches the flanks exactly at the effective diameter. It makes the result independent of small flank-angle errors.
• Pitch: pitch-measuring machine; thread angle and form: optical projector or toolmaker's microscope.

Gear elements (involute spur gear). Module m = d/N (mm), pitch circle diameter d = mN, addendum = m (full-depth standard teeth), dedendum = 1.25 m, pressure angle typically 20°. Tooth thickness on the pitch circle (arc) = πm/2.

Measuring gear elements.

• Tooth thickness with a gear tooth vernier calliper: one vernier is set to the chordal addendum (height from the tip to the pitch-circle chord); the other measures the chordal thickness. Requires an accurate outside diameter because the depth is set from the tip.
• Base tangent (span) method: a disc micrometer measures across S teeth; the anvils touch the involutes tangentially at points on a line tangent to the base circle, so the result does not depend on the outside diameter. Ideal for production.
• Constant chord method: measures at the point where a rack tooth touches the gear; the chord is the same for all tooth numbers of a given module.
• Composite tests: the Parkinson gear tester rolls the gear against a master and records centre-distance variation, giving a combined check of profile, pitch, run-out and tooth thickness.
• Other elements: run-out with a ball or pin in each space and a dial indicator; profile with an involute tester; pitch with a pitch-measuring instrument; backlash with a dial indicator on a mating pair.

Formulas

Lead = n_s · p

• n_s: number of starts; p: pitch (mm).

d_best = p / (2·cos(θ/2))

• θ: included thread angle. For ISO metric (θ = 60°): d_best = 0.5774·p.

E = M − d·(1 + cosec(θ/2)) + (p/2)·cot(θ/2) (three-wire method)

• E: effective diameter (mm); M: measurement over the wires (mm); d: wire diameter (mm). For θ = 60°: E = M − 3d + 0.8660·p.

E = d_major − 0.6495·p (basic effective diameter of an ISO metric external thread)

w = N·m·sin(90°/N) h = m·[1 + (N/2)·(1 − cos(90°/N))]

• w: chordal tooth thickness at the pitch circle (mm); h: chordal addendum (mm); N: number of teeth; m: module (mm).

W = m·cos φ·[π·(S − 0.5) + N·inv φ], inv φ = tan φ − φ (rad)

• W: base tangent length over S teeth (mm); φ: pressure angle. A suitable S ≈ N·φ°/180 + 0.5, rounded.

Worked examples

Example 1 (standard): best wire and effective diameter of an M20 × 2.5 thread.

Given: ISO metric thread, θ = 60°, p = 2.5 mm; measured over three best-size wires M = 20.54 mm.

1. d_best = 0.5774 × 2.5 = 1.443 mm.
2. E = M − 3d + 0.8660·p = 20.54 − 3 × 1.443 + 0.8660 × 2.5.
3. E = 20.54 − 4.329 + 2.165 = 18.376 mm.
4. Basic E = 20 − 0.6495 × 2.5 = 18.376 mm.

Effective diameter = 18.376 mm, equal to the basic value.

Example 2 (GATE level): gear tooth vernier settings.

Given: spur gear, m = 5 mm, N = 30 teeth, standard addendum.

1. Half-angle per tooth: 90°/N = 90/30 = 3°.
2. w = N·m·sin 3° = 30 × 5 × 0.052336 = 7.850 mm.
3. h = m·[1 + (N/2)(1 − cos 3°)] = 5 × [1 + 15 × (1 − 0.998630)] = 5 × 1.02056 = 5.103 mm.

Set the depth vernier to 5.103 mm; the chordal thickness should read 7.850 mm (arc thickness πm/2 = 7.854 mm, so the chord is very slightly smaller).

Common mistakes

• Taking the effective diameter as the mean of major and minor diameters. It is not; use the thread-form relation or a wire measurement.
• Confusing pitch and lead on multi-start threads.
• Using a wire that is not the best size and then ignoring the flank-angle error it introduces.
• Forgetting cosec and cot are of the half angle θ/2 in the wire formula.
• Setting the gear tooth vernier from a nominal tip diameter without checking the actual outside diameter; any error shifts the measured chord.
• Using degrees instead of radians in inv φ.

For GATE PI

Expect numericals on best-size wire diameter, effective diameter from a two- or three-wire measurement, chordal thickness and chordal addendum for a gear tooth vernier, base tangent length, and simple module/pitch-circle/lead calculations. Conceptual questions cover which method measures which element and the advantage of the base tangent and Parkinson tests.

Quick check

1. A three-start thread has pitch 1.5 mm. What is its lead?
2. What is the best-size wire for a metric thread of pitch 3 mm?
3. Which gear measurement is independent of the outside diameter?
4. What is the pitch circle diameter of a gear with module 4 mm and 25 teeth?

Answers: 1. 4.5 mm. 2. 0.5774 × 3 = 1.732 mm. 3. The base tangent (span) method. 4. 100 mm.