Metrology, Quality and Reliability

Parallel system reliability

A parallel, or redundant, system works as long as at least one of its components works, so adding redundancy raises reliability. If n independent identical components, each with reliability r, are placed in parallel, the system fails only when every component fails. Each component fails with probability 1-r, so the system reliability is R_s = 1-(1-r)^n. For components with different reliabilities, the rule generalizes to R_s = 1-\prod_i (1-R_i).

To find the minimum number of identical components for a target R_s, solve (1-r)^n \le 1-R_s, which gives n \ge \dfrac{\ln(1-R_s)}{\ln(1-r)}. The result is always rounded up to the next integer, because a fraction of a component cannot be installed. For example, with r = 0.5 and a target of 0.99, n \ge 6.64, so 7 components are needed, and 1-0.5^7 = 0.992 confirms this.

A two-engine system that survives if at least one engine runs has reliability 1-(1-R_e)^2 = 2R_e-R_e^2. The expression R_e^2 describes the case where both engines must work, and 2R_e can exceed 1, so it is never a valid reliability.

Mixed networks are reduced step by step. A series block has reliability equal to the product of its members, R_{series} = \prod_i R_i, and a parallel block is replaced by one equivalent reliability using the complement rule above. Reduction proceeds from the innermost blocks outward until a single equivalent value remains. Series reliability can never exceed that of its weakest member, while parallel reliability can never be less than that of its strongest member, and these bounds are useful for checking an answer.

Limits, fits and tolerances

Limits, fits and tolerances describe how a shaft and a hole of basic size N are made to mate. The tolerance of a dimension is the difference between its maximum and minimum limits, and the limit deviations are measured from the basic size. In the hole-basis system, the hole is the reference part whose lower limit equals the basic size, and the shaft is varied to obtain the required fit.

For any pair of mating parts, clearance is hole size minus shaft size. The largest clearance is the largest hole minus the smallest shaft, and the smallest clearance is the smallest hole minus the largest shaft. A negative clearance is an interference. Hence C_{max} = H_{max} - S_{min} and C_{min} = H_{min} - S_{max}.

Fits are classified from these extremes. A clearance fit has C_{min} \ge 0, so the hole is always larger than the shaft. An interference fit has C_{max} \le 0, so the shaft is always larger. A transition fit has C_{min} < 0 < C_{max}, so either outcome may occur. The classification must be made from the actual limits, not from the nominal size or the tolerance alone.

Material conditions are also used in this topic. Maximum material condition (MMC) means the most material: the smallest hole and the largest shaft. Least material condition (LMC) means the largest hole and the smallest shaft. When a hole is taken at its MMC and a shaft at its LMC, the clearance is H_{min} - S_{min}, which must be computed from those specific limits.

Example: a hole 40^{+0.025}_{0} mm and a shaft 40^{-0.010}_{-0.030} mm give C_{min} = 40.000 - 39.990 = 0.010 mm and C_{max} = 40.025 - 39.970 = 0.055 mm, so the fit is a clearance fit.

Surface roughness parameters

Surface roughness is measured from deviations of the real profile about a mean line or datum over a sampling length. The arithmetic average R_a = \dfrac{1}{L}\int_0^L |y|\,dx is the mean of the absolute deviations and is the most common parameter. The root mean square value R_q = \sqrt{\dfrac{1}{n}\sum_{i=1}^{n} y_i^2} squares the deviations before averaging, so for the same profile R_q is never smaller than R_a, and it gives larger weight to high peaks and deep valleys.

The peak-to-valley height R_t is the vertical distance between the highest peak and the lowest valley within the evaluation length, so it is sensitive to single extreme points. For a turned surface produced by a sharp single point tool, the height depends on the feed per revolution and the cutting-edge angles, with the approximation R_t = f/(\tan\kappa_r + \cot\kappa_e) in the angle convention of the tool signature. For a tool with a nose radius r, the approximation R_t \approx f^2/(8r) is used. Feed given in mm/min must first be converted to mm/rev by dividing by the spindle speed.

On engineering drawings, the surface-texture symbol carries roughness limits at fixed positions. When both a maximum and a minimum value are specified, the spread between them is the difference of the two values read from the symbol.

Average roughness can also be estimated from a recorded profile graph. The areas above and below the mean line are summed, and the total area is divided by the evaluation length on the paper, corrected for the vertical magnification. That is, R_a = \dfrac{A}{l_p \, M_v}, where l_p = l \, M_h is the length on the paper and M_h, M_v are the horizontal and vertical magnifications.

Acceptance sampling OC curve

The operating characteristic (OC) curve of an acceptance sampling plan plots the probability P_a of accepting a lot against its fraction defective p. An effective plan accepts good lots with high probability and rejects bad lots with high probability, and the steepness of the curve shows how well the plan discriminates between them.

Two risks are read from the curve. The producer's risk \alpha is the probability of rejecting a lot whose quality is at the acceptable quality level p_0, so \alpha = 1 - P_a(p_0); it is the Type I error. The consumer's risk \beta is the probability of accepting a lot at the lot tolerance percent defective p_1, so \beta = P_a(p_1); it is the Type II error. Rejecting a good lot is the producer's risk, while accepting a bad lot is the consumer's risk, and the two terms must not be interchanged.

For a single sampling plan with sample size n and acceptance number c, P_a = \sum_{d=0}^{c}\binom{n}{d}p^d(1-p)^{n-d}. When c = 0, this reduces to P_a = (1-p)^n. This function decreases from 1 at p = 0 and is convex when n is small, and larger values of n or c make the curve steeper around the design point.

For a given plan the two risks trade off against each other. Making the acceptance criterion stricter lowers the consumer's risk but raises the producer's risk, and reducing the sample size flattens the curve, which increases the consumer's risk.

An ideal OC curve is a vertical step at p_0: every lot with p \le p_0 is accepted and every lot above p_0 is rejected. For this ideal curve \alpha = 0 and \beta = 0, but a real finite sample cannot achieve it.

Attribute control charts

Attribute control charts monitor counts or proportions of nonconforming items instead of measured values. A defective item is a unit that fails the specification, whereas a defect is a single imperfection, and one item may contain several defects. The correct chart depends on what is counted and on whether the inspection unit has a constant size.

The p-chart plots the fraction defective of samples of size n, with centre line \bar{p} and limits \bar{p} \pm 3\sqrt{\bar{p}(1-\bar{p})/n}. The limits change with n when the sample size varies. The c-chart plots the number of nonconformities in a constant-size inspection unit, with centre line \bar{c} and limits \bar{c} \pm 3\sqrt{\bar{c}}, because defect counts follow a Poisson distribution. A negative lower control limit is set to zero.

Example: if the average number of defects per constant-size unit is 9, then \bar{c} = 9, the upper limit is 9 + 3\times 3 = 18, and the lower limit is 9 - 9 = 0.

In short, the p-chart is used when each unit is classified as conforming or not, and the c-chart is used when the number of imperfections per unit is recorded with a fixed unit size.

Exponential reliability function

For an exponential time-to-failure distribution, the failure rate \lambda is constant and the reliability is R(t) = e^{-\lambda t}. The mean time to failure is \text{MTTF} = 1/\lambda. Because \lambda does not change with age, the probability of failure in the next short interval is the same at any time, which is the memoryless property.

Failure rates are often given in unusual units, such as failures per 4000 hours. Convert \lambda to a consistent time unit before forming the product \lambda t; for example, a rate of 10^{-6} per hour corresponds to 1 failure per million hours.

For a series system of components that are all essential, the system failure rate is the sum of the component rates, \lambda_s = \sum_i \lambda_i, so the system MTTF is the reciprocal of this sum. For example, four components each with a rate of 5\times10^{-5} per hour in series give \lambda_s = 2\times10^{-4} per hour and an MTTF of 5000 hours.

Six sigma defect level

Six sigma is a quality target in which a process produces no more than 3.4 defects per million opportunities, a level that includes the conventional 1.5 sigma shift. The defect level is expressed as \text{DPMO} = \dfrac{\text{number of defects}}{\text{units} \times \text{opportunities per unit}} \times 10^{6}. The denominator counts opportunities, not units, so a unit with many critical-to-quality features offers several chances for a defect.

Example: if 30 defects are found in 4000 units, each with 5 critical features, then \text{DPMO} = \dfrac{30}{4000\times 5}\times 10^{6} = 1500, which is far above the six sigma level of 3.4. Products with different batch sizes can be compared only after each defect count is divided by its own total number of opportunities.

Control chart Type-I error

A Type I error in a control chart is a false alarm, in which a point signals an out-of-control condition although the process is in control. For a standard three-sigma Shewhart chart with normal data, the probability of such a signal for one point is about 0.0027.

When several decision rules are used together, each rule has its own false-alarm probability \alpha_i. If the rules are independent, the probability that none of them signals falsely is \prod_i (1-\alpha_i), so the overall Type I error is \alpha = 1 - \prod_i (1-\alpha_i). It is not the sum of the individual probabilities, although the sum is a useful upper bound.