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.

As stated before, the first derivative of the transfer function with respect to a particular input quantifies how sensitive the output is to that input. However, it is important to recognize that Sensitivity does not equal Sensitivity Contribution. To assign a percentage variation contribution from any one input, one must look towards the RSS output variance (σ_Y²) equation:

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.

Crystal Ball utlizes several powerful functions and features to extract information and descriptive statistics. We are going to review these techniques and present the CB.GetForeStatFN in full detail, including Six Sigma Capability Metrics.

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!)