Question 3 of 7: Factors that limit a design, and how to overcome them

Question 3: Factors that limit a design, and how to overcome them (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 — Five common factors which limit a design

The five below are chosen because they bind on almost every product, they arise from different sources, and each admits a different class of response. They are listed with the reason each one is a limit rather than a preference.

Three further limits deserve mention even though they fall outside the five: schedule, because a market window closes whether or not the design is finished; freedom to operate, because a competitor's patent can make an otherwise excellent solution unusable; and the supply base, because a component that only one vendor makes is a design constraint dressed as a purchase order.

Part B — What an engineer can do about each

1. Cost. Replace cost estimating with target costing: fix the allowable cost from the market price and treat it as a demand, then allocate it across functions and design to the allocation. Attack the structure of the cost rather than trimming it — move cost from the variable term into the fixed term where volume justifies it, using the two-term model $c(n) = T/n + u$, and reduce assembly content, since labour is bought in whole operators. For a product with a target retail price of CAD 349, a 42 per cent retail margin and a required 35 per cent manufacturer gross margin, the allowable factory cost is

Every subsequent decision is then measured against a number that exists before the design does, which is the only way the constraint actually binds.

2. Process capability. The first move is to find out whether the problem is capital or setting, because the two cost three orders of magnitude apart. Capability is measured by comparing the tolerance band with the process spread, and by comparing it again with the spread and the offset.

Given. A dimension with a bilateral tolerance of $T = 0.090$ mm, a process standard deviation of $\sigma = 0.0125$ mm, and a mean offset from the target of 0.015 mm; the customer requires $C_{pk} \geq 1.33$. Find. The current capability, the defect rate, and the cheapest route to the requirement.

1. Measure the potential capability. $C_p$ compares the tolerance band with six standard deviations of process spread and ignores where the process is centred: $$C_p = \frac{T}{6\sigma} = \frac{0.090}{6(0.0125)} = \boxed{1.200}$$
2. Measure the actual capability. $C_{pk}$ takes the nearer specification limit, so it charges the process for being off-centre: $$C_{pk} = \frac{\min(USL-\mu,\ \mu-LSL)}{3\sigma} = \frac{0.045-0.015}{3(0.0125)} = \boxed{0.800}$$
3. Convert the gap into parts. The upper limit is $z_U = 2.40$ standard deviations away and the lower is $z_L = 4.80$, so the expected defective fraction is $\Phi(-2.40) + \Phi(-4.80) = \boxed{8\,198\ \text{ppm}}$, almost all of it at one end.
4. Diagnose before spending. A high $C_p$ with a low $C_{pk}$ says the equipment can hold the tolerance and is simply aimed wrongly, which is a setup and control problem — an adjustment and a control chart, not a machine. Centring alone lifts $C_{pk}$ to 1.200, which is a very large gain for almost no money.
5. Close the remainder by choosing between two levers. Reaching 1.33 from a centred process needs either less variation, $\sigma \leq T/(6 \times 1.33) = 0.0113$ mm, or more tolerance, $T \geq 6(1.33)(0.0125) = 0.0998 \approx 0.100$ mm. The second is usually a design decision available for free: widening the band by 10 micrometres, or moving the tolerance onto a cheaper feature by re-choosing the datum scheme, buys the same capability as a new machine.

3. Material limits. Change what is being asked before changing the material. Derive the material index that the loading case actually rewards — for a beam of free section it is $E^{1/2}/\rho$ and for a panel of free thickness it is $E^{1/3}/\rho$ — and note that using the wrong idealisation flatters the metals enough to reorder a shortlist. Then exploit shape, which is free: a thin-walled tube at the same second moment of area as a solid section carries a small fraction of its mass, and shape efficiency is usually a larger lever than substitution. Where the property space is genuinely empty, use a hybrid — a sandwich, a coating, a reinforcement — to occupy a point no monolithic material reaches.

4. Codes and standards. Treat compliance as a screening step in Ashby's sense: it eliminates candidates outright and must therefore run before any ranking, not after. Obtain the actual test method rather than the clause summary, and run it on an early prototype, because designing to the clause and designing to the test are different activities and only the test issues certificates. Engage the certification body during design rather than after it, and prefer designs that meet the intent by construction — double insulation, an inherently stable base geometry — over designs that meet it by a margin that manufacturing variation can erode.

5. The human user. The population is data, so the design must either adjust or accept exclusion, and the engineer's job is to state which. Provide adjustability over a range derived from the relevant percentiles, quote the accommodation fraction rather than the range, and cap actuation forces at an accessible-control limit such as the 22.2 N of CSA B651 so that the adjustment is usable by the people who most need it. Where the parameter route is provably closed — where the required dimension lies outside what the architecture can reach — change the concept rather than quietly relaxing the target, and record the limit.