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Error Term Propagation

When hand-building prototypes, one has the luxury of mixing and matching parts to get everything to fit. Sandpaper and glue have also been known to be in the vicinity. But, since the advent of the assembly line, production parts need to fit the same way every time. How do you determine whether your display is going to line up with the enclosure so that all of the text is readable, the mounting screws line up, and your product looks good? Time to sharpen your pencil.

In a perfect world, every dimension on a part would match the drawing exactly and there would be no variation between parts. In the real world every manufacturing process has some amount of uncertainty, a statistical account of the variation from part to part.  To ensure that all of the parts will fit together nicely even with the part-to-part variation, we create a calculation often called a “tolerance stack-up”.

Input

The first input is specified by the manufacturers of off-the-shelf parts. These are fixed inputs to the tolerance calc. “As built” parts may or may not be tighter than the specified tolerance, but it probably wouldn’t be prudent to depend on that. Unfortunately, most manufacturers will not adjust the given tolerances unless you’re buying big, big quantities, so we treat these values as fixed.

The second input is based on the process – these are slightly adjustable input tolerances.  For example, most machinists should be able to meet a standard tolerance for machining steel or aluminum of 0.005”. However, you can specify better tolerances if necessary, for the right price. Plastic part tolerances can vary widely based on the molding process, part geometry, and material characteristics, but a molder should be able to give guidance.

Stack it up

Once we have all of the inputs, we create a table to determine the overall tolerance. For medical devices, we often consider absolute tolerances, adding everything up as stated to account for 100% of the variation. As a less-conservative alternative, the Root Sum of the Squares method accounts for a large portion of the variation by accounting for independent variables differently than dependent variables. As the number of variables increases, the probability that all of those variables measure to their extreme tolerance values independently gets smaller. Thus, the population that is accounted for gets larger, but it never reaches 100%.

Summing tolerances together is as simple as adding the values and adding the tolerances to determine the total value. However, subtraction, multiplication, and division are a little more complicated. This document explains the methods (and a few examples) for accumulating absolute tolerances for some of these cases.

Output

The output of the tolerance calculation should ideally be a dimension, such as the width of the display window for the enclosure. For example, if you’ve added up all of the tolerances to be 0.017” and the width of the display is 3.886” ± 0.0197”, the display window should be 3.886” + 0.0197″ + 0.017″ = 3.923”. These are the inspection dimensions the quality control department will be checking against before a new fabricated or off-the-shelf part makes it into the assembly.

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