Quality and Reliability: Metrology, Quality Management, Reliability and Maintenance

Accuracy is closeness of a measurement to the true value; precision is closeness of repeated measurements to one another. A tight cluster of readings far from the true value is precise but not accurate — the signature of a systematic error such as a zero error, a calibration error, an instrument deflection or a temperature different from the 20 °C reference, which can be found and corrected. Random errors scatter readings both ways, from friction, backlash, parallax and operator judgement, and are reduced only by averaging. Gross errors are blunders. Abbe's principle — the measuring axis should be in line with the dimension being measured — removes the error a small angular tilt would otherwise multiply by an offset.

A tolerance is the permitted variation of a size, the difference between its upper and lower limits. In the ISO system the tolerance magnitude is set by an international tolerance (IT) grade, built on the standard tolerance unit i = 0.45 D^(1/3) + 0.001D (i in µm, D in mm, the geometric mean of the diameter step), with IT6 = 10i, IT7 = 16i, IT8 = 25i, IT9 = 40i, IT10 = 64i and IT11 = 100i. For the 18 to 30 mm step, D = √(18 × 30) = 23.24 mm, i = 1.307 µm and IT7 = 20.9 µm, tabulated as 21 µm. The fundamental deviation, denoted by a letter (capital for holes, small for shafts), fixes where the tolerance zone lies relative to the basic size; H holes and h shafts have zero fundamental deviation. The hole-basis system, with H holes and shafts chosen to give the fit, is preferred because holes are made with fixed-size tools such as reamers.

Limit gauges check whether a part lies within its limits without measuring it. Taylor's principle of gauging: the GO gauge checks the maximum-material condition (the smallest hole, the largest shaft) and should check all features at once, being of full form and length; the NO-GO gauge checks the least-material condition and should check one feature at a time, so a single oversize dimension is caught. The gauge itself is made to a tolerance — a common convention takes the gauge-maker's tolerance as 10 % of the work tolerance — and the GO gauge, which rubs every good part, is given a wear allowance that places it slightly inside the work tolerance zone. Interchangeability means any part fits any mating part without selection, which needs the fit to be achieved by the tolerances alone. Selective assembly relaxes that: parts are measured and sorted into size groups, and holes of one group are assembled only with shafts of the matching group, giving a close fit from loose manufacturing tolerances at the cost of full interchangeability and the work of sorting.

• Linear measurement: vernier calipers and micrometers (least count = pitch/number of divisions, e.g. 0.5/50 = 0.01 mm), slip gauges wrung together to build any length, dial indicators and mechanical, optical, electrical and pneumatic comparators, which measure a deviation from a set standard rather than a size.
• Angular measurement: the bevel protractor; the sine bar, on which slip gauges of height h under one roller of a bar of centre distance L set an angle sin θ = h/L (accurate below about 45°, since errors grow as the angle rises); angle gauges; spirit levels; and the autocollimator, which reads tiny angular deviations of a reflecting surface optically.
• Straightness: a straight edge with feeler gauges, a spirit level or an autocollimator stepped along the surface. Flatness: a surface plate with a dial indicator, or an optical flat under monochromatic light, where each interference fringe represents a height change of λ/2.
• Roundness: a roundness tester rotating a stylus about a precise axis, the out-of-roundness judged from a reference circle — least-squares, minimum-zone, minimum-circumscribed or maximum-inscribed; a V-block and dial reading can miss odd-lobed forms. Runout: the total indicator reading as the part turns on its datum axis, combining roundness and eccentricity. Cylindricity: roundness plus straightness of the generators over the whole length, measured on a roundness machine with a vertical traverse or a CMM.
• Optical methods: interferometry (optical flats, laser interferometers for length), the tool-maker's microscope and the profile projector (magnified outlines of threads, gears and small parts), and autocollimators.

A screw thread has five elements to check: major diameter (bench micrometer), minor diameter (micrometer with prisms), effective (pitch) diameter, pitch (pitch-measuring machine) and flank angle and form (profile projector, tool-maker's microscope). The effective diameter is measured by the two-wire or three-wire method: wires of diameter d are laid in opposite grooves and the distance M over them is measured, then E = M − d[1 + cosec(θ/2)] + (p/2)cot(θ/2), which for a 60° thread becomes E = M − 3d + 0.866p. The best-size wire touches the flanks exactly at the pitch line, so errors in flank angle do not affect the result: d_best = (p/2)sec(θ/2), which is 0.577p for a 60° thread — 1.443 mm for a 2.5 mm pitch.

Gears are inspected for tooth thickness, profile, pitch, runout and composite error. The gear tooth vernier measures the chordal thickness at the chordal addendum; the base tangent (span) method measures across several teeth with flat-faced micrometer anvils, independent of the addendum and outside diameter; the Parkinson gear tester rolls the gear in tight mesh with a master gear and records centre-distance variation, which is the composite error of all elements together. Profile is checked on an involute tester, and runout with a ball or pin in each tooth space.

Surface roughness is quantified by R_a (the arithmetic mean deviation of the profile from its mean line), R_q (the root-mean-square deviation) and R_z or R_t (peak-to-valley heights). A cut-off length filters out waviness so that only roughness is reported. Contact methods draw a diamond stylus across the surface (the stylus profilometer, with or without a skid as a local datum); they are traceable and robust but slow, can scratch soft surfaces and cannot follow valleys narrower than the stylus tip. Non-contact methods are optical — white-light interferometry, confocal microscopy, laser triangulation and light scattering — fast and harmless to the surface, able to map areas rather than lines, but sensitive to surface reflectivity and slope.

Quality has been defined as fitness for use (Juran), conformance to requirements (Crosby) and, by Taguchi, as the loss a product imposes on society once shipped — a loss that grows with any deviation from target, not only with a deviation outside the specification. The cost of quality has four parts: prevention (training, process design, supplier development), appraisal (inspection, testing, audits), internal failure (scrap, rework, downtime before delivery) and external failure (warranty claims, returns, recalls, lost goodwill). Spending on prevention is what reduces the total, because failure costs usually dwarf the other two.

Process capability compares the specification width with the natural spread of a stable process, ±3σ. Cp = (USL − LSL)/(6σ) measures potential capability, ignoring where the process is centred; Cpk = min[(USL − μ)/(3σ), (μ − LSL)/(3σ)] measures actual capability and equals Cp only when the mean is at the centre of the tolerance. For limits 50.00 ± 0.06 mm and σ = 0.015 mm, Cp = 0.12/0.09 = 1.33; if the mean drifts to 50.02, Cpk = (50.06 − 50.02)/0.045 = 0.89, below 1, and parts will fall outside the upper limit although Cp still looks healthy.

For subgroups of 5 the constants are A2 = 0.577, D3 = 0, D4 = 2.114 and d2 = 2.326, and the process standard deviation is estimated as σ̂ = R̄/d2. With X̿ = 50 and R̄ = 2: X̄-chart limits 50 ± 0.577 × 2 = 48.846 and 51.154; R-chart UCL = 2.114 × 2 = 4.228 and LCL = 0. A negative lower limit on an attribute chart is set to zero. A point outside the limits, or a non-random pattern such as a run of points on one side of the centre line, signals an assignable cause; points scattered within the limits are chance variation, which only a change to the process itself can reduce.

Acceptance sampling decides on a whole lot from a sample. In a single sampling plan (N, n, c) a sample of n is inspected and the lot accepted if the number of defectives is at most c. Its operating characteristic (OC) curve plots the probability of acceptance P_a against the lot's fraction defective p; with Poisson's approximation, λ = np and P_a = Σ_k=0^c e^−λλ^k/k!. For n = 50, c = 1 and p = 0.02, λ = 1 and P_a = e⁻¹(1 + 1) = 0.736. The acceptable quality level (AQL) is a quality the producer wants accepted most of the time; the chance of rejecting it is the producer's risk α. The lot tolerance percent defective (LTPD) is a quality the consumer wants rejected; the chance of accepting it is the consumer's risk β. With rejected lots screened 100 % and defectives replaced, the average outgoing quality is AOQ = P_a p(N − n)/N ≈ P_a p, and its maximum over p is the AOQL. Double and sequential plans reach the same protection with a smaller average sample.

Six sigma aims for a process whose specification limits lie six standard deviations from the mean. Allowing for the conventional long-term drift of the mean by 1.5σ, that corresponds to about 3.4 defects per million opportunities. Projects follow DMAIC — define, measure, analyse, improve, control — for existing processes and DMADV for new designs, run by trained Green and Black Belts. Total quality management makes quality everyone's responsibility: customer focus, continuous improvement (kaizen) through the Deming plan-do-check-act cycle, employee involvement through quality circles, decisions based on data using the seven QC tools (check sheet, histogram, Pareto chart, cause-and-effect diagram, scatter diagram, control chart and stratification), and long-term supplier partnership. Just-in-time inventory management produces and delivers only what is needed, when it is needed, in the quantity needed: small lots made possible by set-up reduction, pull signals by kanban (number of kanbans N = D L(1 + α)/C, with demand rate D, lead time L, safety factor α and container size C), and inventory treated as waste that hides problems.

ISO 9001:2015 specifies requirements for a quality management system — not for any product — built on the process approach, risk-based thinking and the PDCA cycle, and on seven quality-management principles: customer focus, leadership, engagement of people, process approach, improvement, evidence-based decision making and relationship management. Like other current ISO management-system standards it follows a common high-level structure of ten clauses, the requirements sitting in clauses 4 to 10 (context, leadership, planning, support, operation, performance evaluation, improvement). ISO 14001:2015 sets requirements for an environmental management system: identifying environmental aspects and impacts, meeting compliance obligations, taking a life-cycle perspective and improving environmental performance. Organisations are certified by accredited third-party bodies after an audit, and re-audited to keep the certificate. Because both standards share one structure they are often run as a single integrated management system, and that is where sustainable manufacturing meets quality management: defects and rework are wasted material and energy, so lean and six-sigma waste reduction, life-cycle assessment, design for remanufacture and recycling, and cleaner processes serve the quality and the environmental objectives at once — the triple bottom line of economic, environmental and social performance.

Reliability R(t) is the probability that an item performs its function without failure for a time t under stated conditions. The hazard rate h(t) = f(t)/R(t) follows the bathtub curve: decreasing in infant mortality (manufacturing defects, removed by burn-in), roughly constant during useful life (random failures), and increasing in wear-out. With a constant failure rate λ the exponential model applies: R(t) = e^−λt, and the mean time between failures is MTBF = ∫R dt = 1/λ — so an item reaches its MTBF with only e⁻¹ = 36.8 % probability. The Weibull distribution R(t) = exp[−(t/η)^β] covers all three regions through its shape parameter: β < 1 decreasing, β = 1 constant (exponential), β > 1 increasing hazard. Repair times are often modelled as exponential or lognormal. Maintainability M(t) is the probability that repair is completed within t; for an exponential repair rate μ, M(t) = 1 − e^−μt and the mean time to repair MTTR = 1/μ. Inherent availability A = MTBF/(MTBF + MTTR) is the long-run fraction of time the item is up: MTBF 190 h and MTTR 10 h give 0.95.

Maintenance is breakdown (run to failure — acceptable only where failure is cheap and safe), preventive (time- or usage-based overhaul and part replacement, worthwhile only for items with an increasing hazard rate — replacing a constant-hazard item early buys nothing), and predictive or condition-based (vibration analysis, thermography, oil and wear-debris analysis, ultrasonic and motor-current monitoring, acting when the measured condition deteriorates). Replacement: for an item that deteriorates, with maintenance costs rising each year, replace it at the end of the year that minimises the average annual cost of ownership (capital cost less resale plus cumulative maintenance, divided by years) — equivalently, just before next year's maintenance would exceed that average. For items that fail suddenly, such as lamps, compare individual replacement on failure with group replacement of all at fixed intervals, which is cheaper per item. Total productive maintenance involves operators in routine care (autonomous maintenance) alongside planned maintenance, and measures the result by overall equipment effectiveness, OEE = availability × performance rate × quality rate, attacking the six big losses: breakdowns, set-up and adjustment, minor stops and idling, reduced speed, defects and rework, and start-up losses.