Linear and angular measurement, comparators and gauges

Why it matters

Tolerances on a drawing mean nothing unless the shop can measure or gauge them. Bore diameters in cylinder blocks, valve-seat angles, crankpin sizes and gear tooth thicknesses are checked with slip gauges, micrometers, comparators, sine bars and limit gauges thousands of times a day. Choosing an instrument with enough resolution, using it correctly and designing go/no-go gauges are core inspection skills.

Key ideas

Measurement terms.

• Accuracy – closeness of a reading to the true value. Precision (repeatability) – closeness of repeated readings to each other.
• Resolution (least count) – the smallest change an instrument can show. Sensitivity/magnification – output change per unit input change.
• A common rule: the instrument's least count should be about one-tenth of the tolerance being checked.

Errors. Systematic errors (calibration, zero error, temperature, Abbe offset) can be corrected; random errors are reduced by repeated readings. Standard reference temperature is 20 °C – a 100 mm steel part grows by about 1.2 µm per °C. Abbe's principle: the measured length should lie on the line of the scale; an offset multiplies any angular play into a length error (callipers violate it, micrometers obey it). Parallax and excessive measuring force are common reading errors.

Standards. Line standards (scales) carry the length between engraved lines; end standards (slip gauges, length bars) between flat faces. Slip (gauge) blocks are hardened, lapped steel or ceramic blocks in sets; they are wrung together to build any length to about 1 µm. Use the fewest blocks, starting from the smallest decimal place, and add protector slips at the ends for wear.

Linear instruments.

• Vernier calliper – least count = one main-scale division − one vernier division (commonly 0.02 mm).
• Outside micrometer – a precision screw; least count = pitch / thimble divisions (0.5 mm/50 = 0.01 mm); vernier micrometers read 0.001 mm.
• Vernier height gauge on a surface plate; bore gauges, inside micrometers and telescoping gauges for holes.
• Coordinate measuring machines (CMMs) for 3-D geometry.

Comparators measure the deviation of a part from a master (slip gauges) rather than its absolute size, with high magnification:

• mechanical – dial indicator, lever types;
• optical – lever plus mirror and light beam;
• pneumatic – back-pressure or flow change of air escaping between a jet and the work; non-contact, ideal for bores and multi-diameter gauging;
• electrical/electronic – LVDT, which gives a voltage proportional to plunger displacement. Dial indicators also check run-out, flatness and alignment.

Angular measurement.

• Universal bevel protractor – vernier reading to 5 minutes of arc.
• Sine bar – a hardened bar with two rollers a fixed centre distance L apart (100, 200 or 250 mm); one roller is raised on slip gauges of height h so that sin θ = h/L. A dial indicator run along the work shows whether it is parallel to the bar. Accuracy falls rapidly beyond about 45° (error ∝ tan θ), so larger angles are measured from the complementary angle.
• Sine centres for conical parts; angle gauges combined by addition and subtraction.
• Spirit levels and clinometers for small inclinations and alignment.
• The autocollimator (optical instrument, reads seconds of arc) for straightness, flatness and squareness.

Limit gauges. Gauges check whether a part lies within limits; they do not give its size.

• Plug gauge for holes: GO end = lower limit of the hole, NO-GO end = upper limit.
• Ring or snap gauge for shafts: GO = upper limit of the shaft, NO-GO = lower limit.
• Taylor's principle: the GO gauge checks the maximum material condition and should check all related features (full form, full length); the NO-GO gauge checks the least material condition and should check only one feature at a time (point contact), so out-of-roundness or bending is detected.
• Gauge tolerance (manufacturing tolerance on the gauge) is commonly taken as about 10 % of the work tolerance, and a wear allowance (commonly about 10 % of the gauge tolerance) is added to the GO gauge only, because only the GO side wears in normal use. Conventions on where the gauge zones lie differ between textbooks and standards (e.g. IS 3455) – follow the one stated in the question.

Formulas

LC (vernier) = s / n Least count; s = value of one main-scale division (mm), n = number of vernier divisions spanning (n − 1) main divisions.

LC (micrometer) = p / N p = screw pitch (mm), N = thimble divisions.

sin θ = h / L Sine bar; h = slip-gauge height (mm), L = roller centre distance (mm).

dθ = dh / (L·cos θ) Angular error (rad) caused by an error dh in the slip-gauge pile; grows with tan θ in relative terms.

ΔL = L·α·ΔT Thermal error (mm); α = coefficient of expansion (1/°C; about 11.5 × 10⁻⁶ for steel), ΔT = departure from 20 °C.

T_g = 0.1·T_w W = 0.1·T_g Common rule for gauge tolerance and wear allowance; T_w = work tolerance.

Worked examples

Example 1 (standard) – least counts and a slip-gauge pile.

1. Vernier: 50 vernier divisions equal 49 main divisions of 1 mm ⇒ LC = s/n = 1/50 = 0.02 mm.
2. Micrometer: pitch 0.5 mm, 50 thimble divisions ⇒ LC = 0.5/50 = 0.01 mm.
3. Building 41.125 mm from a typical 87-piece metric set: take 1.005 (clears the third decimal), then 1.12 (clears the second), then 9.0 and 30: 1.005 + 1.12 + 9.0 + 30 = 41.125 mm with four blocks.

Example 2 (GATE level) – sine bar and gauge design. (a) A 200 mm sine bar is set to check a 30° taper. Slip-gauge height h = 200 × sin 30° = 100.000 mm. If the pile is in error by 2 µm, dθ = 0.002/(200 × cos 30°) = 1.155 × 10⁻⁵ rad = 2.4 seconds of arc. (b) Design a plug gauge for a 25 H7 hole (25.000–25.021 mm), with gauge tolerance = 10 % of work tolerance, wear allowance = 10 % of gauge tolerance, and gauge zones inside the work tolerance.

1. Work tolerance = 0.021 mm ⇒ T_g = 0.0021 mm; W = 0.00021 mm.
2. GO end: starts at the lower limit + wear allowance = 25.00021 mm; GO size 25.0002 to 25.0023 mm.
3. NO-GO end: at the upper limit, within the zone: 25.0189 to 25.0210 mm.

Example 3 – thermal error. A 100 mm steel gauge used at 28 °C: ΔL = 100 × 11.5 × 10⁻⁶ × 8 = 0.0092 mm – nearly half the tolerance of an IT7 hole of that size.

Common mistakes

• Interchanging the GO and NO-GO limits for holes and shafts (GO always corresponds to maximum material).
• Adding wear allowance to the NO-GO gauge.
• Using a sine bar beyond about 45°, where small height errors cause large angle errors.
• Reading a comparator deviation as an absolute size without adding the master's size.
• Building slip-gauge piles starting with the largest block; start from the last decimal place.
• Ignoring temperature – measurements are referred to 20 °C.

For GATE ME

Expect least-count calculations, slip-gauge selection, sine-bar heights and errors, limit-gauge design (GO/NO-GO sizes with gauge tolerance and wear allowance), Taylor's principle, and matching instruments to applications (comparator types, autocollimator, bevel protractor, LVDT). Practise one complete gauge-design problem with the convention stated.

Quick check

1. Least count of a micrometer with 1 mm pitch and 100 divisions?
2. Slip-gauge height for a 100 mm sine bar at 30°?
3. Which end of a plug gauge is sized at the hole's lower limit?
4. Which comparator is non-contact and well suited to bores?
5. To which gauge is wear allowance added?

Answers: 1. 0.01 mm; 2. 50 mm; 3. The GO end; 4. Pneumatic comparator; 5. The GO gauge.