Question 3 of 7: Specifying Tolerances, and Design for Manufacture and Assembly

Question 3: Specifying Tolerances, and Design for Manufacture and Assembly (15 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Part A — Three questions for the manufacturing engineer

A tolerance is a commercial decision disguised as a number. It should be set by the requirement at the top and by process capability at the bottom, and the manufacturing engineer holds the second of those. The three questions below are chosen because each one changes the tolerance that should be written on the drawing, and because between them they cover capability, cost and measurement.

Question 1: What is the short-term standard deviation of your process on this feature, on the machine and material we will actually use? This is the question that converts an opinion into a number. If the process holds $\sigma = 0.012$ mm, then a tolerance can be quoted at whatever capability the programme requires, and if it does not, no amount of drawing authority will make it. The qualifiers matter: capability measured on a different machine, a different cavity or a different resin lot is not evidence about this part.

Given. A bilateral tolerance of ±0.05 mm on a moulded terminal position, a process standard deviation of 0.012 mm, and an observed mean offset of 0.020 mm toward the upper limit. Find. $C_p$, $C_{pk}$, the expected defect rate, and the tolerance that would be needed to hold $C_p = 1.33$.

1. Compute the potential capability from the spread alone. With a tolerance band $T = 0.10$ mm, $$C_p=\frac{T}{6\sigma}=\frac{0.10}{6\times 0.012}=1.389$$ so $\boxed{C_p = 1.389}$. On spread alone the process is comfortably capable, which is exactly the reassurance that $C_p$ is not entitled to give.
2. Compute the achieved capability including the setting error. The nearer limit governs, $$C_{pk}=\frac{\text{USL}-\mu}{3\sigma}=\frac{0.050-0.020}{3\times 0.012}=0.833$$ giving $\boxed{C_{pk}=0.833}$. The corresponding tail areas are at $z=2.50$ and $z=5.83$, so the expected defect rate is about 6,210 parts per million — roughly one part in 161, against a target of fewer than 64 ppm. The gap between the two indices is the whole diagnosis: the process spread is fine and the process is mis-centred, so the corrective action is a setting adjustment, not a capital replacement.
3. Find the tolerance that a capability target would require. For $C_p = 1.33$ at the same spread, $T = 1.33\times 6\times 0.012 = 0.0958$ mm, i.e. about ±0.048 mm. The current ±0.05 mm is therefore the right order of magnitude and it is the centring that must be fixed — a conclusion no one could reach from the drawing alone.

Question 2: What does each tolerance band cost you, and where are the step changes in that cost? Manufacturing cost against tolerance is not smooth; it steps where the process, the fixture, the number of operations or the inspection method changes. Asking for the cost curve, and specifically for the step locations, tells the designer whether tightening from ±0.10 to ±0.05 mm is nearly free or triggers a secondary machining operation and a hundred-per-cent inspection regime. This is the same discontinuity argument as the takt-time step in Question 1, and it is where most avoidable cost is created.

Question 3: How will this feature be located, held and measured in production, and is my datum scheme the one you would actually use? A tolerance is meaningless without the datum reference frame it is measured from, and a datum scheme chosen for design convenience is frequently unusable on the shop floor — the surface may be a moulding gate, a non-functional flash region, or inaccessible in the fixture. Establishing whether the measurement is capable is part of this question: gauge repeatability and reproducibility should consume well under 10 per cent of the tolerance band, and on a ±0.05 mm feature that is a real constraint on the gauge. A tolerance that cannot be measured repeatably cannot be enforced, and it will be settled by argument rather than by data.

Part B — The main concepts behind DFMA, and how to use them

DFMA is the deliberate use of manufacturing and assembly knowledge to shape the design while the design is still cheap to change. It rests on four ideas.

Idea 1: most of the cost is committed long before it is spent. By the end of concept and embodiment design, typically 70 to 80 per cent of the eventual product cost is locked in, while only a few per cent has actually been disbursed. Detail-stage cost reduction therefore works on the small fraction that remains free. DFMA exists to move the manufacturing conversation to the left, into the region where it can still change the answer.

Idea 2: eliminate parts before optimising them. The Boothroyd questions — relative motion, fundamentally different material, necessary separation for assembly or service — identify parts that exist only through habit. Eliminating a part removes its piece price, its handling and insertion time, its tooling, its inventory, its inspection, its documentation and its failure mode simultaneously. No amount of optimising a screw is worth as much as not needing the screw.

Idea 3: make the remaining assembly operations easy and error-proof. The standard measures are: assemble along a single vertical axis so gravity assists and no re-orientation fixture is needed; provide generous lead-ins and chamfers; make parts either clearly symmetric so orientation does not matter, or clearly asymmetric so the wrong orientation is physically impossible; avoid parts that tangle or nest; and provide unobstructed access and clear vision at every insertion.

Idea 4: design for the specific process that will make the part, not for processes in general. Uniform wall thickness, draft, generous radii and moulded-in features for injection moulding; nesting efficiency and bend-relief for sheet metal; access, tool run-out and standard tooling for machining. Each rule is a statement about the physics of a particular process, and it is worth marks to say which process each rule belongs to.

Used together on a real product, these give the sequence demonstrated in Question 1: score the existing design to establish a baseline; identify the parts that fail the three questions; consolidate them into multifunctional mouldings; convert the remaining fasteners to integral snap-fits where the load path permits; re-orient the assembly onto one axis; re-score; and then, and only then, cost the surviving alternatives through the $c(n)=T/n+u$ model to check that the consolidated part’s tooling is affordable at the planned volume. The hair-dryer redesign in Question 1 followed exactly this sequence and returned a part-count reduction from 32 to 17, an assembly-time reduction from 196 to 92 s, and a DFA index improvement from 0.122 to 0.261.

Two cautions belong in a complete answer. Part consolidation can create a single part that is expensive to tool, difficult to mould and impossible to service, so the DFA recommendation must always be tested against the cost model and against the service strategy. And DFMA is only effective if manufacturing engineers are in the room during concept and embodiment; run as a review at the end of detail design, it degrades into a list of changes that are all too expensive to make.