5.3 Datum Calculation Walkthrough

The Part and the Drawing

Consider a plate with:

• Datum A — the bottom planar face (primary, RMB).
• Datum B — a datum hole, nominal diameter Ø12.0 ± 0.1, referenced at MMB in the feature control frame.
• Datum C — a datum target hole (tertiary), constraining the remaining rotational degree of freedom.

A controlled bore, Ø20.0 ± 0.2, has the feature control frame:

⌖|Position|Ø0.4(M)|A|B(M)|C

The bore is dimensioned 80 mm from A and 50 mm from B in the DRF.

Step 1 — Compute the Datum B Virtual Condition (MMB Boundary)

For a hole referenced at MMB, the virtual condition (the MMB boundary) is:

VC_B = MMC_B − (geometric tolerance on B, if any)

The MMC of a hole is its smallest permitted size: Ø11.9. Datum B has no自身的 position tolerance (it is a datum), so:

VC_B = Ø11.9 − 0 = Ø11.9 mm

The MMB simulator for B is therefore a fixed pin of Ø11.9 mm. This is the boundary the part must clear on the simulator.

Step 2 — Compute the Datum Shift Available from B at MMB

Suppose the actual mating size of datum hole B is Ø12.05 mm (measured). The datum shift available is the clearance between the actual hole and the virtual-condition pin:

Datum shift_B = actual_mating_size_B − VC_B = 12.05 − 11.9 = Ø0.15 mm of total diameter shift

Expressed as a radial shift (the amount the part can move in any direction in the plane perpendicular to B's axis):

Radial shift_B = 0.15 / 2 = 0.075 mm

This is the maximum the entire DRF (and therefore the controlled bore's tolerance zone) can translate relative to the fixed pin. The shift is a property of the DRF, not a bonus to the bore's own tolerance.

Step 3 — Compute the Bore's Own Tolerance and Bonus

The bore's MMC is its smallest size: Ø19.8. The position tolerance is Ø0.4 at MMC, so the bore's virtual condition is:

VC_bore = MMC_bore + tol = 19.8 + 0.4 = Ø20.2 mm (for an internal feature: VC = MMC + tol)

Because the tolerance is specified at (M) (MMC), the bore receives a bonus tolerance when it is produced larger than MMC. Suppose the bore's actual mating size is Ø20.10:

Bonus_bore = actual − MMC = 20.10 − 19.8 = Ø0.30

The total allowable position tolerance for this bore is:

Total tol_bore = 0.4 (geometric) + 0.30 (bonus) = Ø0.70 mm

The bore's axis may lie within a Ø0.70 mm cylinder centered at the true position (80, 50) in the DRF.

Step 4 — Measure the Bore's Actual Position

The CMM, aligned to the DRF established by A|B|C, reports the bore's axis center at (80.04, 50.06) — i.e., 0.04 mm off in the A-direction and 0.06 mm off in the B-direction. The radial position error is:

Pos error = √(0.04² + 0.06²) = √(0.0016 + 0.0036) = √0.0052 ≈ 0.0721 mm

The diametral position error is 2 × 0.0721 ≈ 0.144 mm, well within the bore's own Ø0.70 tolerance zone. The bore would pass on its own merits without datum shift.

Step 5 — When Datum Shift Saves the Part

Now suppose the bore had been measured at (80.10, 50.25) — a larger error. The radial error:

Pos error = √(0.10² + 0.25²) = √(0.01 + 0.0625) = √0.0725 ≈ 0.2693 mm → Ø0.539 mm

This exceeds the bore's own Ø0.70 zone? No—Ø0.539 < Ø0.70, still passes. Let's push it further: a bore at (80.15, 0.50.40) gives √(0.15² + 0.40²) = √(0.0225 + 0.16) = √0.1825 ≈ 0.4272 → Ø0.854 mm. This exceeds Ø0.70, so on its own the bore fails.

But datum shift is available: the DRF can be translated by up to Ø0.15 mm diametral (0.075 radial) to bring the bore within its zone. We need the bore's center within Ø0.70 of true position after shifting the DRF by up to 0.075 radial. The shifted requirement:

Required shift = (Ø0.854 / 2) − (Ø0.70 / 2) = 0.427 − 0.35 = 0.077 mm radial

Available shift is 0.075 mm radial. The required shift (0.077) just exceeds the available (0.075). The part fails — but only by a hair, and only because we accounted for datum shift. Without considering datum shift, the part fails by Ø0.854 − Ø0.70 = Ø0.154; with datum shift, it fails by Ø0.004 of effective shift. This is the Senior-level distinction: datum shift nearly rescued the part, and the inspector must report both numbers.

Step 6 — Reporting and the Senior Trap

A complete Senior report for this bore states:

1. The bore's actual mating size (Ø20.10) and its bonus (Ø0.30) → total geometric tolerance Ø0.70.
2. The bore's measured position error (Ø0.854) on its own → fail.
3. The datum shift available from B at MMB (Ø0.15 diametral / 0.075 radial).
4. The position error after applying the maximum beneficial shift (Ø0.854 − Ø0.15 = Ø0.704) → still fail, because Ø0.704 > Ø0.70.
5. Conclusion: Reject. The part fails the position requirement even when datum shift is fully exploited.

The Senior trap is to either (a) ignore datum shift and reject by Ø0.154 when the part may actually pass, or (b) conflate datum shift with the bore's bonus tolerance and pass the part by Ø0.154 when it should fail. The two are separate: bonus comes from the bore's size, shift comes from the datum's size. Compute and report each independently, then combine only when evaluating final acceptance.

Summary Table

This is the capstone calculation for the datum chapter: it uses the primary planar datum (A), the secondary feature-of-size datum at MMB (B), the tertiary (C), the controlled feature's own MMC bonus, and the datum shift — all in one acceptance decision.