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Four-hole pattern at MMC: position deviation, bonus, datum shift and composite feature-relating fit; flatness and roundness by least squares and minimum zone

Tool	GD&T Inspection
Source	Shakarji, C. M. (1998). "Least-Squares Fitting Algorithms of the NIST Algorithm Testing System." Journal of Research of the National Institute of Standards and Technology, Vol. 103, No. 6, pp. 633–641; Shakarji, C. M. (2004). "Reference Algorithms for Chebyshev and One-Sided Data Fitting for Coordinate Metrology." CIRP Annals, Vol. 53, No. 1, pp. 439–442; American Society of Mechanical Engineers (2019). Dimensioning and Tolerancing. ASME Y14.5-2018, New York (position, bonus, datum shift, composite and projected-zone definitions).
Level	Professional

Problem

Four Ø0.250 ± 0.005 in holes at true positions (±1.000, ±0.750) in carry a position tolerance Ø0.010 at MMC with a datum referenced at MMB (clearance 0.004 in) and a composite lower segment Ø0.007 at MMC; the axes were measured at the entry and exit of the 0.500 in thick part and the zone is projected 0.750 in. Find each hole's deviation, bonus, allowable and margin, the projected-zone deviation, whether the pattern accepts with the datum shift and in the feature-relating check; then the flatness of a 12-point surface and the roundness of a 24-point profile by least squares and by the minimum zone.

Inputs to Enter

H1 measured (1.0020, 0.7485), size 0.2502, exit (1.0012, 0.7490); H2 (−0.9970, 0.7530), 0.2460, exit (−0.9974, 0.7526); H3 (−1.0040, −0.7455), 0.2510, exit (−1.0032, −0.7462); H4 (1.0065, −0.7520), 0.2472, exit (1.0058, −0.7528) in
Hole 0.250 +0.005/−0.005 in, position Ø0.010 at MMC, datum shift 0.004 in, FRTZF Ø0.007, depth 0.500 in, projected height 0.750 in
Flatness: the 4 × 3 grid of 12 points in the tool's box (tolerance 0.0005 in); roundness: the 24 points on Ø1.000 in (tolerance 0.0005 in)

Open the GD&T Inspection tool (loaded with this example)

Answers

MMC 0.245, LMC 0.255, virtual condition 0.235, resultant condition 0.275 in; ± coordinate equivalent of Ø0.010: ± 0.003536 in
H1: deviation 2√(0.0020² + 0.0015²) = 0.0050 in, bonus 0.0052, allowable 0.0152, margin +0.0102 (projected-height deviation 0.0078, accept); H2 0.0085 / 0.0110; H3 0.0120 / 0.0160; H4 0.0136 against 0.0122, margin −0.0014, REJECT
Datum shift: best translation (0.0019, 0.0005) in within the 0.002 in radius gives a worst margin of +0.0017 in, pattern ACCEPT; composite FRTZF Ø0.007: translation 0.0045 in and rotation −0.167° give a worst margin of +0.00057 in, ACCEPT (−0.0044 in before the fit)
Flatness: least-squares spread 0.000343 in, minimum zone 0.000330 in; roundness: least squares 0.000501 in (reject at 0.0005), minimum zone 0.000451 in (accept), minimum circumscribed 0.000536, maximum inscribed 0.000680 in

Notes

The tool's default inputs are this case; press Submit (the Analysis select switches to the flatness, roundness, orientation, runout and conversion boxes). The data are illustrative, generated for the tool; every number is reproduced by the independent numpy/scipy script References/GDT-Inspection/checks/gdt_check.py (linprog for the minimum zone, brute-force circles, minimax pattern fits), and the same test (test_gdt_inspection.php, 234 checks) fits all 210 non-cone NIST L2 reference pairs to the NIST reference values and reproduces Shakarji's 2004 two-lobed maximum-inscribed circle to 1e-12.