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In the article ‘step plan tolerance analysis‘, the steps to achieve a good tolerance analysis are briefly summarized. In the articles that followed, the method of adding tolerances is discussed. Much attention was paid to the statistics of dimensional variations of manufactured parts and how to deal with them. Tolerance analysis with Cpk requirements was also discussed. An important and often difficult part was covered very briefly: setting up the tolerance chain.

## Drafting the tolerance chain

Establishing the tolerance chain is often seen as one of the most difficult parts of tolerance analysis. Especially with complex, three-dimensional structures, it can be difficult to see and describe the actual ‘path’ of the tolerance chain. And you need that very chain to name all the parts that are important in the analysis. Mechanical (or software) adjustments/calibration in the assembly process can also complicate the analysis.

Using an example, setting up the chain is discussed and tips & tricks are given.

## Without a ‘critical dimension’ no tolerance chain

Without a critical dimension, you can’t establish a tolerance chain. But what exactly is this critical dimension? In short, it is a (or the) specification. And it is the specification of which you want to determine the influence of tolerances. Step 1 in your tolerance analysis is to determine that specification/critical dimension.

The specification can be anything. Sometimes the specification is accuracy: think of the imaging chip in a (digital) camera relative to the lens (or lens system). Other times it is a question of whether something fits or not. Or the question of whether a carriage or (linear) motor has sufficient range. Or, for example, in an assembly with rotating parts, whether the run-out is not too large.

Sometimes there is only one critical dimension in the assembly, but often there are several. Each critical dimension has its own tolerance chain. And its own tolerance analysis.

## Different machine states

Step 2 in your analysis is to describe in which (machine) state (or condition) the specification applies. Sometimes there is a different specification for the condition before an adjustment and after an adjustment. Or there are different use states (pressure, temperature, motion, forces, et cetera). For example, consider a robot whose position is calibrated at low speed, but will operate at high speed.

## Make a simplified sketch of the assembly

Step 3. Once you have an unambiguous definition of the critical dimension, make a simple sketch of the structure. If you start from your assembly drawing in CAD, it may well be so complex that you can’t see the forest for the trees. Ideally, you make your sketch with something like PowerPoint so you can explain the construction to any interested party. What matters is the essence of the assembly.

Once you have that sketch, then you (hopefully) have all the parts that belong in your chain. Then the puzzling begins to find the corresponding chain. That can sometimes be quite complicated. Keep the goal in mind: you want to create a proper (correct) tolerance chain and you’d prefer not to take that long. How you get the chain is not actually important. Any method is permissible.

Here is a simple sketch of a construction where the rubber gasket should be mounted with a prescribed pressure. In this (first) design, the nut is tightened until the nut rests against the larger diameter part of the bolt.

Now you can draw the critical dimension in the sketch. Since the pressure on the gasket is prescribed, that pressure is your specification and therefore your critical dimension. Your analysis is then of course about assessing the influence of part tolerances on gasket pressure. The construction chosen provides a nominal indentation and tolerances cause a variation on that indentation. Via the properties of the gasket material, you can then calculate the pressure.

The critical dimension K in your sketch then looks like this:

## There is only one dimension per part in the chain

Now that you have the critical dimension, step 4 is to create the chain, the “route” through the assembly. This is not too difficult in this example. Each part always has two interfaces in your tolerance chain. And between those two interfaces you can draw a dimension line (hopefully it’s already there) with a corresponding tolerance.

Often you start at either end of the critical dimension/specification and figure out which dimension in the part affects your critical dimension (specification). That dimension doesn’t necessarily have to be on a drawing (but hopefully it does). And that dimension must interface with a subsequent part in your tolerance chain.

In this example, it will be clear that the following parts have an influence:

• thickness of the (unstressed) gasket;
• thickness of washer;
• length of bolt;
• thickness of the housing.

## Tolerance chain is always closed

A correct tolerance chain is always closed. That is, there is an unambiguous route from the ‘one side’ of the specification to the ‘other side’ of the specification. If multiple routes are possible, then the structure is over-constrained or something is wrong in your chain.

If there is a gap in your tolerance chain, then one or more parts are missing. Sometimes there may appear to be a gap in your chain because the parts are not in contact. But then the chain may still be closed because there is a mechanism that determines the dimension of that gap. Consider, for example, the air gap of an air bearing.

In the following figure, you see the overall chain drawn. The part dimensions that are important are coded:

• P = thickness of the (unstressed) gasket;
• R = thickness of the washer;
• B = length of the bolt;
• H = thickness of the housing.