Tolerance analysis

Tolerance analysis is a calculation that checks how the tolerances of individual part dimensions combine and affect a resulting functional dimension of an assembly, such as a clearance, an interference, the gap between covers or the coaxiality of holes. In practice it is often called a tolerance stack-up. European engineering literature also speaks of dimensional chains: the individual dimensions form a closed loop, and the resulting dimension is its closing link.

The simplest approach is the worst-case method (also called the min/max method). The tolerance of the closing dimension is the sum of the tolerances of all links in the chain, so the assembly will work with any combination of dimensions within tolerance. Five dimensions with a tolerance of ±0.1 mm give a result of ±0.5 mm. In long chains, this leads to very tight and expensive part tolerances.

The statistical RSS method (root sum square) relies on the fact that all deviations will practically never sit at their limits at the same time. For the same example it gives approximately ±0.22 mm (0.1 × √5). It does, however, assume independent, approximately normally distributed and centered manufacturing processes, and it accepts a small share of assemblies out of tolerance. Monte Carlo simulation randomly generates thousands of virtual assemblies and can handle nonlinear relationships, asymmetric distributions and geometric tolerances in 3D.

Simple linear chains are commonly calculated in a spreadsheet, more complex 3D problems in specialized software. The inverse task is tolerance allocation: starting from the accuracy required of the resulting dimension, the tolerances of the individual parts are set with regard to manufacturing capability and cost.

Run a tolerance analysis wherever function depends on a dimension made up of several parts: axial play of bearings and shafts, the stroke of a mechanism, gap and flush between visible covers, the alignment of holes for a bolt passing through several parts, the position of a connector relative to its opening in the enclosure, or seal compression. It pays to do it before ordering molds and production parts, because changes later cost far more.

The result also supports cost decisions, since it shows which tolerances can be relaxed and which are truly critical. If the required accuracy cannot be achieved economically, the options are an adjustment feature (shims, a set screw, slotted holes) or selective assembly.

A common mistake is leaving out contributors that do not appear in the dimensions: bolt clearance in holes, geometric tolerances (flatness, perpendicularity, position), coating thickness after surface finishing, or compression of flexible parts. In assemblies of different materials, such as aluminum and steel, thermal expansion over the operating temperature range also matters.

Do not use the statistical method without knowing the actual processes. For small batches, off-center processes or safety-critical dimensions, the worst-case method is the safer choice, or RSS with a correction factor. Feed the results back into the drawings: specify the tolerances of critical dimensions explicitly and verify that the supplier's process holds them consistently (a process capability of at least Cpk 1.33 is often required).