Feb 2016, Vol 9: #2
 
This month, Skip reviews the use of Designed Experiments (DoE) for tolerance studies during product design. Are you up for a challenge? Read on! -Carol
Using Designed Experiments to Set Tolerances during Critical Parameter Development and Management
Tolerance analysis comes in two forms: 1) Analysis of tolerance ranges using mathematical equations from 1st Principles enhanced by use of Monte Carlo Simulations, and 2) analysis of tolerance ranges using physical prototypes under the control of Designed Experimentation (DoEs). In this newsletter we will focus on the use of DoEs to help define tolerances whether they be unilateral (USL or LSL) or bilateral (USL and LSL).
DoEs have many uses during Critical Parameter Development. To be clear, we are trying to discover the differences between important factors and parameters (well behaved, easily controlled) vs. those that have problems as defined by the Big 7 Critical Parameter metrics discussed in numerous past newsletters. In general, DoEs are applied sequentially during iterative studies including:
  1. Screening studies to determine main effects that stand out statistically when compared to random sources of common cause variation.
  2. Modelingstudies to determine interactions and the coefficients associated with linear and non-linear behavior as well as significant interactions. Here the result is a usable math model [Y = f(Xs)] that describes the important factors and parameters and their relationships that govern the ideal function of the design. If these factors and parameters are well behaved in light of the Big 7 metrics then they are deemed important and ECO (Easy, Common & Old). If certain factors and parameters, particularly anti-synergistic interaction relationships between them, are present and troublesome, then these are a subset of Critical Parameters that need special control plans to force them to behave in light of our tolerances.
  3. Optimizationstudies to determine the set points that adjust the mean of our output Y to reside on a desired target. This is used primarily to force Cpk to equal Cp by forcing the difference between the mean of Y and the target to equal zero.
  4. Robustnessstudies to determine the set points for controllable factors and parameters (Xs) that minimize the standard deviation (s) of the output response Y. This is done when realistic and stressful sources of variation (noises) are purposefully induced and changed in conjunction (interaction) with purposeful changes of the Xs. Uniquely useful interactions can occur between the controllable Xs and the induced noises that enable the discovery of set points of the Xs that leave the function of the design insensitive to the noises perturbations. We now have Xs that govern the mean of Y and separate Xs that govern the standard deviation of Y in the presence of stressful noises!
  5. Tolerancestudies finish developmental learning prior to freezing the design in preparation for verification and validation tests, ongoing reliability growth tests and transfer into Production and the Supply Chain management process. Tolerances in this environment are usually associated with ΔY as a function of ΔXs that govern variation ranges in functional responses that cannot be analytically modeled and simulated using Monte Carlo studies on a computer - thus done on proto type hardware. Tolerance DoEs can be run under nominal (non-stressed) conditions and when done in conjunction with the ability to adjust the mean back on target using the Xs that were discovered in Optimization of the Mean studies, result in the ability to run Cp studies. When the tolerance DoEs are run under realistically stressful noise conditions, even with the mean adjustment parameters available for tuning the mean, the results often characterize Cpk performance because the noise may result in the mean still being driven off target to some degree by the stressful noises. In modern software packages like JMP and Minitab these are called Pp and Ppk studies!
DoE Architectures for Tolerance Studies
Tolerance analysis using DoEs is classified as yet another form of Screening study. We are not after a math model in this case, but rather a Variance model from ANOVA as driven by purposeful excursions of the Xs away from their robust nominal set points. As I said, this can be done under no-noise (nominal) conditions or stressful noise conditions. Think of it this way - the tolerances were exercised and defined under the best of operational conditions but it is unknown if they are effective in controlling variation of Y when the real noise attacks the function in the customer's use environment. Is it risky to not know this information? Should the time be taken to define "robust" tolerances? Sit down with your peers, technical leaders and project manager and discuss the risk of being under-informed.
DoE Options for tolerance analysis are pretty straight forward. See Chapters 13-15 in my text Tolerance Design for complete instructions. You have two choices:
  1. Fractional Factorial DoEs that are set at 2 levels. Remember you are forming a hypothesis about proposed USLs and LSLs to find out if they work in light of your Cp and Cpk goals/constraints.

a) Study the extremes of your proposed LSL and USL conditions (considered Worst Case Tolerance conditions from all Xs being studies in the DoE - an unlikely condition!)

 

b) Study proportions of the extremes to be more realistic about the probability of all your Xs moving away from their target values at the same time (forms of Root Sum of Squares (RSS) approach):

      • Approx. 1/3rd of your proposed USL and LSL values. Usually this is close to equaling 1 std. deviation of the expected range of the variation of your Xs out toward their estimated USL or LSL.
      • Square Root of 3/2 of your 1 std. deviation values for your Xs (see Ch. 15 of Tolerance Design for a complete explanation of this unusual setting recommendation!)

2. Fractional Factorial DoEs that are set at 3 levels. These are typically Placket-Burman designs that allow you to include your nominal value in between your LSL, USL or RSS portions of them.

 
"Robust" versions of these options will call for inducing the compounded noises outside of the tolerance DoE. These two compounded noise extremes are typically run as replicates, thus enabling the mean and std. deviation of Y to be computed and used in Cp / Cpk assessment of performance at your tolerance extremes.
Taguchi's L18 Array can be used to include the compounded noises in the 1st column to induce noise inside of the DoE runs. This still requires at least 2 replicates of the DoE to get the data necessary to compute the mean and std. deviation of each row of the DoE.
If you have a strong, anti-synergistic interaction between two Xs, you will typically have to tightly control the variation of one of them while allowing more generous variability in the other. This is a case where for example: Y = C0+C1X1 + C2X2 + C3 X1*X2 is real and the X1*X2 interaction is anti-synergistic. Either X1 or X2 is going to have to be tightly toleranced and controlled rigorously out in the supply chain. This is Critical Parameter identification and management! We would prefer our designs not possess such hyper-sensitivity but alas it can happen to you!
When using orthogonal, Fractional Factorial screening experiments to assess tolerance sensitivities; you still use ANOVA to define the factor main effects on Y. Here the variation induced in Y comes from the excursions away from the X's nominal value: either out to1 sigma, square root of 3/2 sigma. or the full distance out to your proposed USL and LSL values under study. The key is to be realistic in setting your proposed spec. limits by having discussions with your suppliers of materials and parts. Also, if your design is adjustable by Service Engineers, be sure you understand their adjustment capabilities in the field or in a Service Center.
Using DoEs to explore tolerance limits or proportions of them is profoundly useful for complex designs that are not capable of being assessed using computer-aided tolerance studies. Often, development teams will say their functional relationships are a mysterious "black box" of un-model-able physics or chemistry under the current state-of-the-art. Empirical exploration of X set point limits only requires a well-controlled, modular or adjustable proto type that has capable measurement systems for both Xs and Ys so we trust the data coming out of the DoE based upon precise settings of the Xs.
Happy Tolerancing!
Is there a topic you'd like us to write about? Have a question? We appreciate your feedback and suggestions! Simply "reply-to" this email. Thank you!
Sincerely,
 
Carol Biesemeyer
Business Manager and Newsletter Editor
Product Development Systems & Solutions Inc.
About PDSS Inc.
Product Development Systems & Solutions (PDSS) Inc.  is a professional services firm dedicated to assisting companies that design and manufacture complex products.  We help our clients accelerate their organic growth and achieve sustainable competitive advantage through functional excellence in product development and product line management.
  
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