Shaft batch acceptance with a biased micrometer and a marginal gauge study
A 60-piece batch of 30.000 mm plus or minus 0.015 mm steel shafts has been turned and is waiting for final inspection before dispatch. The inspector uses a 25-50 mm digital micrometer whose calibration certificate expired three months ago. Checked against a certified 30 mm gauge block at 20 degrees Celsius, the micrometer reads 30.006 mm. The shop floor is at 31 degrees Celsius and the parts have not been soaked; the micrometer and the gauge block were kept in the inspection room at 20 degrees Celsius. A gauge repeatability and reproducibility study run last month on this same feature gave GRR = 0.008 mm. Five shafts read 30.018, 30.012, 30.021, 29.990 and 30.006 mm. The customer rejects any piece outside the limits and charges a 15 JD per piece handling fee for a rejected batch. The delivery truck leaves in four hours. Coefficient of thermal expansion for steel is 11.5 x 10 to the power -6 per degree Celsius.
1. Correct the five readings for bias and temperature, then give an accept, reject or re-measure decision for each shaft with the reasoning.
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Bias correction. The micrometer reads 0.006 mm high against a certified 30 mm block, so true size = reading - 0.006 mm. The certificate has expired, which is exactly why an ad hoc check block comparison is being used; treat the 0.006 mm as a known systematic error, not a random one.
Temperature correction. The part is at 31 C, the micrometer at 20 C, so the part is measured about 11 C above reference. Expansion = 30 x 11.5 x 10^-6 x 11 = 0.0038 mm, about 0.004 mm. The shaft reads approximately 0.004 mm larger than its 20 C size, so its true 20 C size is about reading - 0.004 mm. Note the direction: the part is bigger, not the gauge.
Limits are 29.985 to 30.015 mm.
- 30.018 - 0.006 - 0.004 = 30.008 mm: accept.
- 30.012 - 0.006 - 0.004 = 30.002 mm: accept.
- 30.021 - 0.006 - 0.004 = 30.011 mm: accept, but it started as an apparent reject (30.021 is over the limit). Without the corrections a good shaft would have been scrapped.
- 29.990 - 0.006 - 0.004 = 29.980 mm: reject, and it is 0.005 mm below the lower limit even before corrections - a real reject.
- 30.006 - 0.006 - 0.004 = 29.996 mm: accept.
The lesson is that the bias and the thermal offset are the same order of magnitude as the tolerance band, so uncorrected readings are meaningless in this job. The remaining 55 pieces must be re-measured with the same correction, not accepted on the strength of five.
2. Judge whether the measurement system is fit to release the batch, and state the decision rule you would apply.
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Gauge capability. GRR as a percentage of tolerance = 0.008 / 0.030 = 26.7 percent. The usual bands are: under 10 percent acceptable, 10-30 percent marginal and conditionally acceptable, over 30 percent unacceptable. At 26.7 percent this system sits at the top of the marginal band.
What that means in practice: the measurement uncertainty is roughly a quarter of the tolerance, so a part reading within about 0.004 mm of either limit cannot be confidently classified. Those parts must be re-measured with a better method - a comparator set to the gauge block, a bore or snap gauge, or a second inspector - rather than guessed at.
Decision rule (guard band). Accept only if the corrected reading is inside the limits by at least the measurement uncertainty; reject only if it is outside the limits by at least that amount; everything in between goes to re-measurement or to the customer as a concession with data attached.
System problems to record:
- Expired calibration - the micrometer should have been withdrawn from service, and the fact that a check block found 0.006 mm of bias suggests the drift is real.
- No temperature soak - parts must be at 20 C or the readings corrected, and the correction must be written on the inspection record.
- A GRR at 26.7 percent is a warning to schedule a proper gauge study and, if this feature is critical, to invest in a comparator.
Given four hours to dispatch, the workable decision is: sort the batch with corrected readings and a guard band, ship only parts with clear margin, quarantine the borderline pieces, and raise a non-conformance note on the calibration lapse the same day.
3. Set out what must appear on the inspection record and how the batch decision should be communicated to the customer, so that the workshop's position is defensible.
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Inspection record contents:
- Part number, drawing revision, batch or heat number, quantity inspected and sampling basis (100 percent in this case).
- Nominal and limits used, stated numerically (30.000 / 29.985 / 30.015 mm).
- Instrument identity and its calibration status - including the fact that the certificate had expired, which must not be concealed; a record that hides it is worthless as evidence.
- The bias correction applied (0.006 mm) and how it was established, with the check-block reading.
- Temperature at measurement and the correction applied (about 0.004 mm).
- Raw and corrected readings per piece, with the accept, reject or re-measure decision against the guard band.
- Name and signature of the inspector and the date.
Communication to the customer: send a short, factual deviation note before the truck leaves: total quantity, number of pieces clearly conforming, number rejected and why, the guard-band rule used, and the measurement uncertainty of the method. Offer the borderline pieces for a joint re-measurement or as a concession. Do not present a batch as fully conforming when the measurement system cannot support that claim - if the customer's incoming inspection later finds pieces outside limits, a false declaration turns a small sorting cost into a credibility and liability problem. The 15 JD per-piece handling fee is a strong argument for sorting now, at the workshop, where the operation is cheap.