Question 6 of 7: Reliability and robustness of a door lock (15 marks)

Question 6 — Reliability and robustness of a door lock (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.

Given. A residential lockset whose cylinder life is Weibull-distributed with shape $\beta = 2.3$ and characteristic life $\eta = 180\,000$ actuations, used 12 times per day; a latch-engagement clearance specified as $0.80 \pm 0.30$ mm with process standard deviation $\sigma = 0.085$ mm and a measured mean of 0.92 mm; a demonstration test at 90 % confidence; and an electronic module with activation energy $E_a = 0.70$ eV, use temperature 25 °C and stress temperature 85 °C.

Find. The process for improving reliability and robustness, a test programme that would demonstrate it, and the long-term quantitative measures — sample size for a zero-failure demonstration, $B_{10}$ life, capability indices, and the accelerated-test compression factor.

Approach. Distinguish reliability from robustness explicitly, since the question offers them as alternatives; build the improvement process around failure-mode identification and parameter design; then attach the four standard calculations that make each claim measurable.

A. The process for enhancing reliability or robustness

The two words name different things and are improved by different means. Reliability is the probability that the lock performs its function for a stated duration under stated conditions — a statement about time and wear-out. Robustness is insensitivity of performance to variation in manufacture, installation and environment — a statement about variation, and largely time-independent. A lock can be reliable in a laboratory and not robust in the field, which for this product is the usual case, since a residential lock is installed by an amateur into a door that moves seasonally.

The process runs as follows.

1. Define the function and the duty cycle quantitatively. Lock and unlock from both sides; resist forced entry; operate over $-40$ to $+50$ °C in the Canadian climate; survive freezing rain and de-icing salt. Duty is taken as 12 actuations per day, so a 25-year design life is about 110 000 actuations and a 10-year warranty target is about 44 000.
2. Identify the failure modes before designing against them. A design FMEA over the whole product: key-pin and driver-pin wear rounding the shear line, spring relaxation and fatigue, latch-bolt and strike wear, cylinder-cam torsional fatigue, corrosion of the springs and pins, lubricant loss and dust ingress, and — for an electronic cylinder — battery depletion, contact corrosion and firmware fault. Rank by risk priority number and address the top items only.
3. Attack the dominant physics of failure, not the symptom. Pin wear is adhesive and abrasive wear at a sliding contact, so the levers are material pair and hardness (nickel silver against a hardened plug), contact stress (larger radius, better finish), and contaminant exclusion (a sprung dust shutter). Spring relaxation is set and fatigue, so the levers are stress amplitude, alloy and shot peening. Each of these is a design change with a physical mechanism attached, which is what distinguishes engineering from a wish for higher quality.
4. Design for robustness by parameter design before tolerance design. Following Taguchi, first choose nominal values of the control factors so that performance is insensitive to the noise factors — door sag, strike misalignment, temperature, dirt, key wear — and only then tighten tolerances, because tolerance tightening buys robustness with money while parameter design buys it for nothing. For a lock the highest-leverage robustness change is geometric: a lead-in chamfer and a tapered latch bolt that tolerate ±2 mm of strike misalignment, so that a badly hung door does not load the mechanism at all.
5. Design for the installation and for the user, since a large share of field failures are installation defects: fewer parts, keyed features that cannot be assembled wrongly, a template that fixes the backset, and a torque-limited fastener. This is Question 1's ease-of-use argument applied to the installer.
6. Verify by test, then feed the field data back. Reliability growth is achieved by test–analyse–fix cycles, and the programme must plan for several, exactly as in Question 2(C).

B. How this would be tested

Testing has three distinct purposes and needs three distinct tests, which are commonly and expensively confused.

1. Life-cycle endurance to a recognised standard. The industry benchmark is ANSI/BHMA A156.2 for bored and preassembled locksets, in which Grade 1 requires one million cycles, Grade 2 eight hundred thousand and Grade 3 two hundred thousand, together with strength, operational and finish tests; the equivalent Canadian route is CSA certification of the door hardware assembly with the fire-rated variants listed by ULC. The rig is a servo-electric cycler that inserts and rotates the key, throws the bolt and retracts it against a spring-loaded strike, with cycle counting, torque monitoring and periodic teardown for wear measurement. Torque to turn the key, measured every 25 000 cycles, is a far better degradation signal than waiting for a functional failure, because it is continuous rather than binary.

2. A demonstration test that produces a reliability statement. To demonstrate a reliability $R$ at confidence $C$ with zero failures, the required sample size is

$$n = \frac{\ln(1-C)}{\ln R}$$

To show $R = 0.95$ at 90 % confidence, $n = \ln(0.10)/\ln(0.95) = -2.303/-0.0513 = 44.9$, so $\boxed{n = 45}$ units must run the full mission with no failures. To show $R = 0.99$ at the same confidence requires $n = \ln(0.10)/\ln(0.99) = 229.1$, so $\boxed{n = 230}$ units. Quoting the second number is what justifies the entire apparatus of accelerated and degradation testing: 230 locksets each cycled to 110 000 actuations is not a programme any consumer-product budget will carry, so the reliability claim has to be earned by physics and by acceleration rather than by brute-force sampling.

3. Robustness testing against deliberately imposed noise. This is where the lock is tested as it will be installed, not as it was designed. A designed experiment (an $L_9$ or $L_{18}$ orthogonal array) with control factors — chamfer angle, spring rate, clearance, lubricant — crossed against a noise array of strike misalignment ($-2$, 0, $+2$ mm), temperature ($-30$, $+20$, $+50$ °C), dust loading and key wear. The response is the signal-to-noise ratio of the operating torque, and the configuration selected is the one with the flattest response, not the lowest mean. Alongside this run environmental tests: salt spray to ASTM B117, thermal cycling and freeze–thaw with a wet cylinder, and dust ingress.

Accelerated testing for the electronic module. Where a smart cylinder is included, temperature-accelerated life testing uses the Arrhenius factor

$$AF = \exp\left[\frac{E_a}{k}\left(\frac{1}{T_u}-\frac{1}{T_s}\right)\right] = \exp\left[\frac{0.70}{8.617\times10^{-5}}\left(\frac{1}{298.15}-\frac{1}{358.15}\right)\right] = \boxed{96.0}$$

so a three-year field life of $3 \times 8760 = 26\,280$ hours is represented by $26\,280/96.0 = 274$ chamber hours — under twelve days. This is the only reason a reliability claim on an electronic product is affordable, and it is valid only if the acceleration does not introduce a failure mechanism that would not occur in the field, which must be confirmed by teardown.

C. Measuring and quantifying reliability and robustness over the long term

Reliability is quantified with a life distribution, not with a single number. Fitting the cycle-test failures to a two-parameter Weibull, $R(t) = \exp[-(t/\eta)^{\beta}]$, the shape parameter itself is diagnostic: $\beta < 1$ indicates infant mortality (an assembly or supplier problem), $\beta \approx 1$ a random external cause, and $\beta > 1$ wear-out, which is what a lock should show. With $\beta = 2.3$ and $\eta = 180\,000$ cycles the design measures are

$$B_{10} = \eta\left[-\ln(0.9)\right]^{1/\beta} = 180\,000\,(0.10536)^{1/2.3} = \boxed{67\,660\ \text{cycles}}$$

which at 12 actuations per day is $67\,660/(12\times365) = 15.4$ years before the first tenth of the population has failed, with the characteristic life $\eta$ itself reached at 41.1 years and $R(\eta) = 0.368$ by definition. Reliability at specific milestones follows directly: $R(100\,000) = 0.772$ and $R(200\,000) = 0.280$. Against the ANSI/BHMA Grade 1 demand of one million cycles this design is nowhere near Grade 1 and is a solid Grade 3 product, which is the correct and useful conclusion to draw from the numbers rather than a marketing claim.

Robustness is quantified with capability indices and with transmitted variation. For the latch-engagement clearance specified at $0.80 \pm 0.30$ mm,

$$C_p = \frac{USL-LSL}{6\sigma} = \frac{1.10-0.50}{6(0.085)} = 1.176$$

but with the process running at a mean of 0.92 mm rather than 0.80,

$$C_{pk} = \frac{\min(USL-\mu,\ \mu-LSL)}{3\sigma} = \frac{1.10-0.92}{3(0.085)} = \boxed{0.706}$$

The gap between $C_p = 1.176$ and $C_{pk} = 0.706$ is the whole diagnosis: the process has ample spread capability and is simply mis-centred, so the corrective action is a setting adjustment costing nothing, not a capital investment in a more precise process. Recentring at 0.80 mm alone restores $C_{pk}$ to 1.176. Reporting $C_p$ without $C_{pk}$ conceals exactly this, and reporting $C_{pk}$ without $C_p$ conceals whether the fix is cheap or expensive.

Long-term measurement then combines four streams: warranty and field-return rates converted to failures per unit-year and plotted as a Weibull or a Crow–AMSAA reliability-growth curve across production lots; ongoing reliability testing of a small sample drawn from production each month, which catches supplier drift that the original qualification cannot; statistical process control on the key characteristics with $C_{pk}$ tracked as a time series rather than as a one-time certificate; and a returned-parts teardown programme, because a failure rate tells you how often and only a teardown tells you why. The quantities reported to management should be $B_{10}$ and $B_{1}$ life, field failure rate per unit-year against the warranty accrual, $C_{pk}$ on the critical characteristics, and the Weibull $\beta$ trend — a $\beta$ falling toward 1 in the field data is the earliest available warning of a new supplier or assembly problem.