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SARoboticsTo: NRG TeamFrom: Gale Johnson

Date: Jan 23 2008Subj: Statistical Tolerance Analysis Standard

OverviewA statistical tolerance analysis takes advantage of the fact that each feature on a part has a probability ofbeing made exactly perfect and a probability of being off by the full tolerance allowed. The probability isrepresented by a curve called a normal probability density function. By combining all the probabilities ofthe different parts and dimensions in a design we can determine the probability that a part will have aproblem or fail based on the dimensions and tolerances of the parts.

This document describes the current statistical tolerance analysis method used by many in engineering. Itis intended to be as strait forward as possible. A standardized set of assumptions and actions is listedbelow, along with an example of a tolerance analysis.

A commonly established goal is to have a minimum design sigma of 4.5 for stationary and 6.0 for movinginterfaces. If the assembly stackup involves 4 or more tolerances the statistical (RSS) method is used. Ifless than 4 tolerances are in the tolerance loop the worst case tolerance method should be used. Theworst case stack up should be less than the nominal clearance or margin.

In a statistical (RSS) analysis, each tolerance should account for less than 35% of the total variance. If thevariance of one tolerance exceeds 35% consider reducing the tolerance or using it as a worst case add in.

Steps and Assumptions in an RSS (Statistical) tolerance analysis.

1. Establish a Tolerance Loop, usually with some type of sketch or assembly drawing as reference.Clearly identify the clearance, margin or fit of interest.

2. Enter the Nominal dimensions and tolerances in the loop from the drawings. Note the drawing

number, rev. level and zone location for each tolerance so it can be verified and updated in the future.Columns B-F in the example.

3. Determine areas where a gap or clearance is in the loop but is not the clearance of interest. Thesewould be items such as the gap between a pin or screw and a hole. Usually these gaps are not calledout or controlled on a drawing. Enter the nominal gap as a tolerance (usually 3 sigma) with adescription relating it back to the drawing and features entered in item 2. Columns B-F in the example.

4. Next to the tolerance enter a geometric correction factor or tolerance conversion factor. This

normalizes the tolerance to a linier or radial +- tolerance. This will convert the tolerance to a value thatreflects the movement the tolerance will allow in a particular direction. Column I in the example.

If the tolerance were a true position diameter or a bilateral profile the factor would be .5 indicatingthat the feature can only move +- half the indicated tolerance. If the tolerance is not evenly distributedon both sides of the nominal dimension it must be centered. If the dimension is 1.05 +.15 -.05 , centerthis by entering a dimension of 1.10 +-0.1. Enter the print tolerance (1.05 +.15 -.05) in the descriptionsection to identify the dimension for future reference If the dimension is a Min or Max but the actual

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6. Determine which tolerances should be scaled for rotation about some fixed point such as a clockingand locating pin on a plate. Determine the scaling or multiplication factor for the tolerances and enterthem in the spreadsheet with the affected tolerance. Column J in the example.

7. Determine the type of distribution for each tolerance. The standard assumption is that the parttolerance represents a +- 3 sigma normal distribution. On drawings some dimension are flagged witha CP. This refers to a note stating that this dimension must be held to a particular CPK. A 1.0 CPK = a3 sigma tolerance, a 1.5 CPK = 4.5 sigma tolerance. Column K in the example.

Examples of normal distributions are shown bellow. They represent probabilities of different distribution over the sameclearance area. Examples of distributions with less than 3 sigma would be some gaps, tweaked dimensions and dimensionsthat have been sorted/binned for some reason. A uniform distribution can be modeled as a 1.732 sigma normal distribution.Note: If data is available to determine the actual distribution enter that number. The distribution factor is calculatedas follows. Determine how far the nominal dimension is from the population mean (mean shift = abs( dim-mean)).

Calculate the distribution factor by subtracting the mean shift from the drawing tolerance and then divide by theactual sigma from the population ( dist. Factor = (tol-mean shift)/actual sigma value).

8. The nominal tolerance is then converted to an effective tolerance at the point of interest by multiplyingit by the conversion and scale multipliers. This is done by equations in column M in the example.

9. Determine which tolerances should be added in as worst case. Flag these dimensions by placing a 0in the Tol. used as column or column L in the example. Die-cast dimensions are usually consideredworst case due to historic problems and data. Gaps that are always closed through a purposeful biasshould also be considered worst case.

10. Sum all the effective tolerances that will be used as worst case. This is called the worst case add inand will be used to reduce the nominal clearance or margin. Column N in the example

11. Divide all the remaining effective tolerances by their distributions. This normalizes all the tolerances toa 1 sigma tolerance. If the tolerance is +-0.09 with a distribution of 3 sigma, the 1 sigma tolerance is0.09/3 or 0.03. Column O in the example.

12. Square all the 1 sigma values. This is called the variance of the tolerance. Column P in the example.

13. Sum all the squared 1 sigma values and take the square root. This is the loops 1 sigma tolerance.

14. As a means to evaluate the contribution of each tolerance divide each variance (squared effectivetolerance) by the sum of all the variances (sum of squared effective tolerances). This will give you apercent contribution of each tolerance. In general no single tolerance should account for more than35% of the total variance. If it does consider reducing the tolerance or using it as a worst casetolerance. Column Q in the example.

15. Determine the nominal clearance or fit of interest by summing the nominal dimensions around the

tolerance loop you created.

16. Subtract the worst case add in from the nominal clearance or margin. This is the new adjustedclearance or margin that will be used to calculate the loop design sigma.

17. Divide the new adjusted clearance or margin by the 1 sigma tolerance of the loop. This is the designsigma of the loop. The goal is a minimum of 4.5 sigma for a static clearance and 6 sigma for a moving

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Probability Density Function for Different Distributions

0

2

4

6

8

10

12

14

-0.1 -0.05 0 0.05 0.1

Tolerance Range

%P

robability

Tol. Sigma 6.0

Tol. Sigma 4.5

Tol. Sigma 3.0 (Standard)

Tol. Sigma 1.73 (Uniform)

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New_tol_Study_Template10.xls

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