Tolerance Analysis focuses on dimensional aspects of manufactured physical products and the process of determining appropriate tolerances (read: allowable variations) so that things fit together and work the way they are supposed to. When done properly in conjunction with known manufacturing capabilities, products don't feel sloppy nor inappropriately "tight" (i.e., higher operating efforts) to the customer. The manufacturer also minimizes the no-build scenario and spends less time (and money) in assembly, where workers are trying to force sloppy parts together. Defects are less frequent. There are a wealth of benefits too numerous to list but obvious nonetheless. Let us measure twice and cut once.

In the case of the one-way clutch example, the current MC quality prediction for system outputs provide us with approximately 3- and 6-sigma capabilities (Z-scores). What if a sigma score of three is not good enough? What does the design engineer do to the input standard deviations to comply with a 6 sigma directive?

How do Monte Carlo analysis results differ from those derived via WCA or RSS methodologies? Let us return to the one-way clutch example and provide a practical comparison in terms of a non-linear response. From the previous posts, we recall that there are two system outputs of interest: stop angle and spring gap. These outputs are described mathematically with response equations, as transfer functions of the inputs.

In past blogs, I have waxed eloquent about two traditional methods of performing Tolerance Analysis, the Worst Case Analysis and the Root Sum Squares. With the advent of ever-more-powerful processors and the increasing importance engineering organizations place on transfer functions, the next logical step is to use these resources and predict system variation with Monte Carlo Analysis.

The other RSS equation, that of predicted output mean, has a term dependent on 2^nd derivatives that is initially non-intuitive:

Why is that second term there?

A few posts ago, I explained the nature of transfer functions and response surfaces and how they impact variational studies when non-linearities are concerned. Now that we have the context of the RSS equations in hand, let us examine the behavior of transfer functions more thoroughly.

Root Sum Squares (RSS) approach to Tolerance Analysis has solid a foundation in capturing the effects of variation. In the days of the golden abacus, there were no super-fast processors willing to calculate the multiple output possibilities in a matter of seconds (as can be done with Monte Carlo simulators on our laptops). It has its merits and faults but is generally a good approach to predicting output variation when the responses are fairly linear and input variation approaches normality. That is the case for plenty of Tolerance Analysis dimensional responses so we will utilize this method on our non-linear case of the one-way clutch.

Transfer Functions (or Response Equations) are useful to understand the "wherefores" of your system outputs. The danger with a good many is that they are not accurate. ("All models are wrong, some are useful.") Thankfully, the very nature of Tolerance Analysis variables (dimensions) makes the models considered here concrete and accurate enough. We can tinker with their input values (both nominals and variance) and determine what quality levels may be achieved with our system when judged against spec limits. That is some powerful stuff!

With uncertainty and risk lurking around every corner, it is incumbent on us to account for it in our forward business projections, whether those predictions are financially-based or engineering-centric. For the design engineer, he may be expressing dimensional variance in terms of a tolerance around his nominal dimensions. But what does this mean? Does a simple range between upper and lower values accurately describe the variation?

In my last couple of posts, I provided an introduction into the topic of Tolerance Analysis, relaying its importance in doing upfront homework before making physical products. I demonstrated the WCA method for calculating extreme gap value possibilities. Implicit in the underlying calculations was a transfer function (or mathematical relationship) between the system inputs and the output, between the independent variables and the dependent variable. In order to describe the other two methods of allocating tolerances, it is necessary to define and understand the underlying transfer functions.

As stated in my last post, there are three common approaches to performing Tolerance Analysis. Let us describe the simplest of the three, the Worst Case Analysis (WCA) approach. An engineering-centric term in the Tolerance Analysis world would be Tolerance Stacks, usually meaning in a one-dimensional sense. The explanation begins with probably the most overworked example found in dusty tomes (my apologies in advance).

(I would like acknowledge James Ministrelli, DFSS Master Black Belt and GD&T Guru Extraordinaire, for his help & advice in these posts. Thanks, Jim!)

Tolerance Analysis is the set of activities, the up-front design planning and coordination between many parties (suppliers & customers), that ensure manufactured physical parts fit together the way they are meant to. Knowing that dimensional variation is the enemy, design engineers need to perform Tolerance Analysis before any drill bit is brought to raw metal, before any pellets are dropped in the hopper to mold the first part. Or, as the old carpenter's adage goes: Measure twice, cut once. 'Cause once all the parts are made, it would be unpleasant to find they don't go together. Not a good thing.

The recent and ongoing disaster in the Gulf of Mexico has raised some questions as to the preparedness of Big Oil to respond to an emergency spill.  An examination of their emergency spill plans has garnered criticism but is it justified?  How much effort should companies spend on risk identification and mitigation?