In these cases, there is far more concern about the Extremes than middle values. ... Extreme of both directions at a given Probability for each Measure of Interest.
What is purpose of Variation Accumulation (Tolerance Analysis)? There are two major Reasons for Variation Accumulation: 1. Determine if assemblies will fit. 2. Determine the extent of Variation likely in production. Other reasons for Variation Accumulation: 1. Learning what are leading Contributors, and how much. 2. Learning how a particular change will affect Results. In these cases, there is far more concern about the Extremes than middle values. The only exception might be how the effect of a change is determined. This can be more accurately determined with a New Analysis. Especially, if Analyses do not take very long. First Results should not be sacrificed for accuracy of the last. To find relevant Results, the shape of the Curve is not important. Mathematicians may want to know as a curiosity. All Middle Results “work,” so there is no Engineering curiosity. Resulting two goals of Variation Accumulation software: 1. Extreme of both directions at a given Probability for each Measure of Interest. 2. Contributors in Order of Significance for each direction, and Portion of Contribution. This information only requires Extremes, not Middle Values. Level of quality Since accurate Extremes are necessary, the following information is needed: 1. Total number of Parts are going to be made. 2. How many of these Assemblies might be bad, or not fit. From these two Numbers, a reasonable number of Standard Deviations can be selected from the Z Table (𝑁𝜎). This result is not likely to be three sigma. This will be used for all Contributors. This method is not assuming Normal Distribution, because technically it can’t be Normal for a Finite Number of Parts. But, Z Table is a reasonable Standard. Inputs will generally follow the Normal curve because of the Central Limit Theorem and because the Variation has multiple causes. Quality level is determined at this Point to best accommodate nonlinearity. Total Effect (𝑇𝑖 ) on the Measurement of Interest should be determined with individual Contributor (𝐶𝑖 ) at (𝑁𝜎). Standard Deviation should be based on Variation, instead of Tolerance, or results will not reflect reality. Unfortunately, this information is hard to get. Total Effect will be in a Measurement of Interest Frame of Reference, rather than a Contribution Frame of Reference. The Transform Function causes Skew, and associated Mean Shift, if there is nonlinearity. These Mean Shifts are exactly additive and appears as interactions, if there is more than one Mean Shift. Total Effect should be determined in both directions, because these Values will be different due to any nonlinearity. This is the maximum extent that the Measurement of Interest can change each way, with Extremes of this particular Contribution. These Values (both directions) will be used for both of the following Results. Incidentally, they will be exactly Contributor times Average Extreme Sensitivity (𝑇𝑖 = 𝐶𝑖 𝑆𝑖 ). Measurement of Interest Extremes (𝑀) Total Effects are statistically added together to equal Measure of Interest Extreme, because they cannot occur simultaneously at same Probability (𝑀2 = ∑ 𝑇𝑖2 ). It is based on the Formula (𝜎𝑇2 = 𝜎𝐴2 + 𝜎𝐵2 + 2þ𝜎𝐴 𝜎𝐵 ). This is true for all Distributions, Skewed or not. The þ = 0, because Dimensions are independent, so last term goes away. All 𝜎 2 are multiplied be the same N, so N can be included in each term. Therefore, the First Formula is derived with easy, Verifiable Assumptions. Combine all Positive Changes and combine all Negative Changes. This will result in two different amounts, due to any nonlinearity. These answers will be at the previously accepted Probability of Failure. They will be as correct as possible for nonlinear Problems. It will be more accurate than any Commercially Available Software.
Contribution Portions Contributions will be a statistical Portion (𝐶𝑜𝑛𝑡𝑟𝑖𝑏𝑢𝑡𝑖𝑜𝑛 = 𝑇𝑖2 ⁄𝑀2 ). Since (𝑇𝑖2 ) is a subset of(∑ 𝑇𝑖2 ), then (𝐶𝑜𝑛𝑡𝑟𝑖𝑏𝑢𝑡𝑖𝑜𝑛 = 𝑇𝑖2 ⁄∑ 𝑇𝑖2 𝑜𝑟 𝑇𝑖2 ⁄𝑀2 ). Combine all Positive Changes and combine all Negative Changes. This will result in two different Contribution Pareto Charts, due to nonlinearity. This Answer will be at the previously accepted Probability of Failure. They will be as correct as possible for nonlinear Problems. It will be more accurate than any Commercially Available Software.
