Axial End-Play Stack: Worst-Case Versus Statistical Tolerancing
A gearbox output shaft is located axially by a housing bore, one ball bearing, a spacer, and an end cover. The design requires assembled end play of 0.02 to 0.12 mm so the bearing is neither preloaded nor loose. Drawing dimensions are: housing bore depth 60.00 +/-0.10 mm, bearing width 25.00 +/-0.05 mm, spacer length 20.00 +/-0.06 mm, cover spigot depth 14.90 +/-0.05 mm. All parts are purchased or machined to these tolerances and assembled at random from stock, with no matching or selection. Manufacturing wants to release the drawing as-is because the nominal stack (60.00 - 59.90 = 0.10 mm) is described as "right in the middle of the spec." A prototype built from random stock measured 0.15 mm end play; a second unit showed visible bearing preload. No shims are currently in the bill of materials.
1. Compute the worst-case assembled end-play range with the current tolerances. Is the 0.02-0.12 mm specification achievable? Identify any error in the nominal dimensioning.
Read the worked answer
End play = A - (B + C + D), where A = bore depth, B = bearing width, C = spacer, D = cover spigot.
- Max end play = A_max - (B_min + C_min + D_min) = 60.10 - (24.95 + 19.94 + 14.85) = 60.10 - 59.74 = +0.36 mm
- Min end play = A_min - (B_max + C_max + D_max) = 59.90 - (25.05 + 20.06 + 14.95) = 59.90 - 60.06 = -0.16 mm
So the stack ranges from 0.16 mm of interference (preload) to 0.36 mm of looseness, a spread of 0.52 mm = 2 x (0.10 + 0.05 + 0.06 + 0.05). The spec window is only 0.10 mm wide, so it is not achievable by a factor of about five.
The nominal error: with a 0.10 mm nominal, the assembly sits at the upper limit of a 0.02-0.12 mm spec, not the middle. A properly centered nominal would be 0.07 mm (cover spigot 14.93 mm), which buys 0.05 mm of margin at each end before tolerances are even considered.
2. A colleague proposes replacing the worst-case calculation with a root-sum-square (RSS) stack so the drawing can be released unchanged. Perform the RSS calculation and state whether the result justifies release. Under what conditions is RSS defensible here?
Read the worked answer
Using half-tolerances as 3-sigma limits: w = sqrt(0.10^2 + 0.05^2 + 0.06^2 + 0.05^2) = sqrt(0.0100 + 0.0025 + 0.0036 + 0.0025) = sqrt(0.0186) = 0.136 mm.
Even after re-centering the nominal at 0.07 mm, the +/-3-sigma band is 0.07 +/- 0.136 = -0.066 to +0.206 mm. That still violates the 0.02-0.12 mm window on both sides, so RSS does not rescue the design.
RSS is also the wrong tool here on principle:
- It assumes each dimension is normally distributed and centered on nominal. Bearings are typically ground near mid-tolerance but spacers are often saw-cut and run to the plus side, which shifts the mean.
- The negative end of the distribution is an interference condition (bearing preload, brinelling, heat). There is no sensory warning of a bad build and the failure is progressive. One-sided, invisible failure modes are exactly where the RSS tail probability is least trustworthy and where worst case should govern.
Conclusion: reject. The stack must be fixed structurally, not statistically.
3. Recommend a fix. Rank the candidate actions by cost and effectiveness, and state which tolerance you would change first.
Read the worked answer
Contributions to the 0.26 mm total half-stack: housing bore 0.10 (38%), spacer 0.06 (23%), bearing 0.05 (19%), cover 0.05 (19%).
Options ranked:
- Re-center the nominal to 0.07 mm (cover spigot 14.93). Zero cost, buys 0.05 mm at each end. Do this first regardless.
- Selective shim at assembly. Specify a shim pack in 0.01 mm steps and an assembly instruction to measure and select to 0.06-0.08 mm. Adds ~1 min of assembly labour and one part number, and absorbs all remaining stack variation including supplier drift. Cheapest effective fix.
- Tighten the spacer to +/-0.02 mm (grind instead of saw-cut). Doubles the cost of a cheap part but removes the largest tolerance you actually control.
- Tighten the housing bore depth. Last resort: it is a single-point facing operation in a large casting and tightening it drives real scrap cost on the most expensive part.
Do not tighten the bearing width; you do not control it and it buys little. Recommended release: nominal re-centered, spacer tightened to +/-0.02, and a 0.03-0.10 mm selective shim range specified at assembly.