What does tolerance stack-up mean?

Dimensional and geometric tolerances play a fundamental role in the detailed design phase of a product.

Indeed, it is necessary that each designer is able to indicate the correct tolerances in the drawing to guarantee:

• the design intent, i.e. the functionality of the component in operation
• the assemblability
• the interchangeability
• the cost target: the use of tight tolerances often implies the use of more expensive processes
• the design robustness by reducing the intrinsic variability introduced by the production processes

However, also in the manufacturing process the component will go through several transformation steps, which involves the creation of semi-finished products at each stage of the process.

Therefore, it is essential that the tolerances indicated on the stage drawings are suitable to guarantee two main requirements:

• that the tolerances requested in the final drawing are satisfied
• that the estimated stock allowance is sufficient

These are not obvious requirements, as very often it is not possible to manufacture all the final dimensions directly in the machining process, but these dimensions will be the result of the algebraic sum of 2 or more dimensions, which will then form a sort of “stack of tolerances”.

There are several techniques for carrying out a tolerance analysis. In this post we will focus mainly on how to perform a tolerance stack up on linear dimensions, using the “worst case” technique.

The Worst-Case Analysis

As mentioned, there are several ways to perform an appropriate tolerance analysis.

One way is, for example, to consider the standard deviation of the various processes involved. However, these values are not always available and they may vary over time depending on various factors, such as wear, temperature, etc.

A simpler technique is to consider the so-called “worst case” scenario: in this case, the lower and upper tolerances that concur in the final dimension are considered.

It is clear that this condition (i.e. the worst case) is unlikely to occur, therefore it can be quite restrictive in terms of tolerance values to be indicated in the semi-finished products, consequently increasing processing times and costs.

However, this method has some advantages:

• It’s simple: the calculation does not require particular mathematical notions, so it can be taught and understood at all levels
• It’s practical: this is fundamental feature in an extremely concrete and practical context such as manufacturing
• It’s reliable: if the tolerances are respected, it guarantees 100% fulfillment of the drawing requirements. If not fully respected, there is still a high probability that the pieces are still compliant, as statistically it is very difficult for the tolerance chain to work in the worst case

A practical example of tolerance stack-up

Now let’s imagine to manufacture an item in different machining stages and then having to turn this item on the spindle. In this case, there will be different 0-points and some features will have to be finished in 2 or more steps. What tolerances and how much stock will we have to indicate in our stage drawings?

See the example in Figure 1: the starting points are indicated with a dot, the machined surface with an arrow. The values next to the dots and arrows indicate the stock.

Assuming that the final dimension of the groove is 2 +/- 0.25 mm, are we mathematically sure that the piece conforms to the design?

A simple method is as follows:

• The last machined surface of the groove (0 mm) is identified
• Go to the other extreme of the quota and identify the stock (0.5 mm)
• Move up until you find a quota associated with the same stock
• Go to the other extreme of the quota and identify the stock
• Continue in this way until you reach the other surface of the groove with 0 stock

The tolerance stack-up is unique, which means that there is only one way to calculate the worst case value of a feature.

The final tolerance could be +/- 0.3, which is > +/- 0.25. This means that there is no absolute certainty that, fulfilling the tolerances indicate in the stage drawings, the piece will be conform.

And what about the nominal value? The calculation represents the verification that the right tolerances have been taken into consideration. Following the direction of the arrow, it results:

|4,5 – 11.5 + 5| = 2

If it turned out to be a different value, it would mean that we didn’t consider the right chain of tolerances.

Tolerance stack-up and Industry 4.0

What would happen if for some reason the semi-finished items did not fully comply with the tolerances indicated in the stage drawings at any intermediate stage of the process?

One approach could be not to intervene on the downstream process, considering the occurrence of the “worst case” is statistically unlikely. On the other hand, another approach could be to recalculate the tolerance stack up and tightening some downstream tolerances to ensure that the final value indicated is fulfilled.

In this sense, some of the Industry 4.0 technologies could help us, in particular the Internet of Things, Big Data Analytics, Cloud Computing and IT Systems Integration.

In a 4.0 scenario, each machine would be able to measure the dimensional values of the semi-finished items, send the data to a centralized system capable of processing the tolerance stack-up, then send back the information to the downstream process with the new tolerances to be fulfilled.

In this case, if tighter tolerances are required, the system could decide autonomously to change machine, tool or machining parameter, according to the manufacturing capability of the company.

In this way, an extremely flexible and reconfigurable manufacturing systems would be created.

Where to start?

Performing a tolerance stack up during the manufacturing design process effectively means to design a robust system capable of producing items according to the design intent. Therefore, this approach makes it possible to reduce the defects ratio.

Finally, thanks to the Internet of Things, Big Data Analytics, Cloud Computing and IT Systems integration it is possible to implement a flexible and autonomous production system, able to self-adapt the process in real time according to the real dimensions of the semi-finished items.

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