Quality and inspection

Why it matters

No two manufactured parts are identical. Tool wear, temperature, material batches and operators all add variation. Quality engineering asks two questions: is the process stable, and is it capable of making parts inside the tolerance? Customers, especially in automotive, aerospace and medical work, often require proof in the form of capability indices such as Cpk. As a design engineer you need to read these numbers, because they tell you whether your tolerances can actually be met.

Inspection basics

• Measuring tools: calipers (good to about 0.02 mm with care), micrometers (0.001 to 0.01 mm), height gauges, pin and thread gauges, go/no-go gauges, and coordinate measuring machines (CMMs) for GD&T.
• A gauge must be much better than the tolerance. A common rule of thumb is that measurement uncertainty should be no more than about 10 % of the tolerance (sometimes 25 % is accepted). A caliper is not the right tool for a ±0.005 mm bore.
• Gauge R&R (repeatability and reproducibility) studies check how much of the observed variation comes from the measurement system itself.
• First article inspection checks every dimension on the first part off a new process before production starts.

Statistical process control (SPC)

SPC watches a process over time with control charts. The common X-bar and R chart takes small subgroups (often 5 consecutive parts) at regular intervals and plots each subgroup's mean and range. Control limits (roughly ±3 sigma of the process) come from the data, not from the drawing. A point outside the limits, or patterns such as long runs on one side of the centre line, signal a special cause (a worn tool, a new material lot) to investigate.

Control limits and specification limits are different things. A process can be in control (stable) but not capable (its natural spread is wider than the tolerance), or capable but drifting out of control.

Capability indices

For a feature with lower and upper specification limits LSL and USL, process mean mu and process standard deviation sigma:

Cp  = (USL - LSL) / (6 * sigma)                  potential: spread vs tolerance
Cpk = min(USL - mu, mu - LSL) / (3 * sigma)       actual: also accounts for centring

• Cp ignores where the process is centred; Cpk uses the distance to the nearest limit.
• Cpk is never larger than Cp. They are equal only when the process is perfectly centred.
• Cpk = 1.0 means the nearest limit is 3 sigma away (roughly 0.13 % out of spec on that side if the data are normal). Many customers ask for Cpk >= 1.33 (4 sigma), and often 1.67 for critical features. Check the customer's requirements.

Which sigma? Strictly, Cp and Cpk use the short-term, within-subgroup sigma, often estimated as R-bar / d2 (d2 = 2.326 for subgroups of 5). Using the overall sample standard deviation of all the data gives the performance indices Pp and Ppk. The two are close for a stable process and differ when the process drifts between subgroups. Capability indices also assume the process is stable and the data are roughly normal; check the control chart first.

Worked example 1: by hand

A shaft diameter is specified as 25.00 ± 0.05 mm, so LSL = 24.95 and USL = 25.05. The process mean is 25.02 mm and sigma is 0.01 mm.

Cp  = (25.05 - 24.95) / (6 * 0.01)               = 0.10 / 0.06 = 1.67
Cpk = min(25.05 - 25.02, 25.02 - 24.95) / 0.03   = min(0.03, 0.07) / 0.03 = 1.00

The spread is fine (Cp 1.67), but the process runs 0.02 mm high, so it fails a 1.33 Cpk requirement. Re-centring the process (an offset on the CNC) would fix it without buying a better machine. That is the value of comparing Cp with Cpk.

Worked example 2: from data

The Do tab holds 40 measured diameters in 8 subgroups of 5. Their mean is about 25.013 mm and the average subgroup range is about 0.0184 mm, so:

sigma_within = R-bar / d2 = 0.0184 / 2.326 = 0.0079 mm
Cp  = 0.10 / (6 * 0.0079)              = 2.11
Cpk = (25.05 - 25.013) / (3 * 0.0079)  = 1.57

Here the process is capable and only slightly off centre.

Design takeaways

• Tolerances set the denominator. If the process cannot hold a tolerance with Cpk >= 1.33, either the tolerance, the process, or the design has to change.
• Talk to quality early about which features are critical-to-quality and how they will be measured.
• Tolerance stack-ups add variation from several parts; a statistical (root-sum-square) stack assumes capable, centred processes, so it depends on these numbers.

Common mistakes

• Computing Cpk on an unstable process. The number means nothing if the process is drifting.
• Confusing control limits with specification limits.
• Tiny samples. Capability from 10 parts is very uncertain; studies commonly use at least 25 subgroups or 100 or more parts.
• Measurement system too coarse. If the gauge cannot resolve the variation, the sigma you compute is mostly gauge noise.

Recap

• Inspection needs gauges much better than the tolerance; Gauge R&R checks this.
• SPC control charts show stability; capability indices show fit within the tolerance.
• Cp = (USL − LSL)/6σ; Cpk = nearest-limit distance / 3σ; Cpk ≤ Cp.
• Cp and Cpk use within-subgroup sigma; Pp and Ppk use the overall sigma.

Try it

On the Do tab, finish the Cpk function and compare Cp, Cpk and Ppk for the measured shaft data.