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442Chapter 4Tolerance Stack-Up AnalysisTolerance Stack-Up Analysis for a Fixed Fastener Assembly using Geometric Tolerances; Main RulesCalculating Gaps; Working the RouteCalculating Inner and Outer Boundary Means and their Tolerances;Numbers ChartsChapter ObjectivesReaders will learn:1. What a statistical tolerance is.2. How to designate a statistical tolerance on a feature.3. How to be very specific in what a statistical tolerance requires.4. About Populations, Population Parameters, Statistical Target Values, Upper and Lower Specification Limits.5. Symbology for Statistical Process Control formulas.6. About Arithmetic Means, Normal Distribution of Tolerance and the Standard Deviation and Statistical Probability for Tolerance Stack-Up Analysis for Positional Tolerances.t r 23Downloaded From: / on 04/11/2014 Terms of Use: /terms443Chapter Twenty-ThreeFIGURE 23-1 Fixed Fastener Assembly Tolerance Stack-Up AnalysisTolerance Stack-Up Analysis for a Fixed FastenerAssembly using Geometric Tolerances;Main RulesDownloaded From: / on 04/11/2014 Terms of Use: /terms444Tolerance Stack-Up Analysis1) Start at the bottom of the gap and work to the top of the gap. . or Start at the left of the gap and work to the right of the gap.2) Stay on one part until it is exhausted, then jump to another -not back and forth.3) Numbers that lead you left are designated negative (-) . . . &Numbers that lead you right are designated positive (+)Numbers that lead you down are designated negative (-) . . . &Numbers that lead you up are designated positive (+)Tolerance Stack-Up AnalysisMain RulesDownloaded From: / on 04/11/2014 Terms of Use: /terms445Chapter Twenty-ThreeFIGURE 23-2 Calculate Minimum Gap Lower LeftTo calculate the minimum, gap at the lower left of the assembly, we first visualize Part #1 and Part #2 pushed together to create the minimum gap. This will mean the left edge of the slot will merge with the left of the tab. They share the same location in the assembly. Part #1 will choose a route that works from the left edge of the gap to the left edge of the slot where it touches the left edge of the tab. This is to allow us to jump from Part #1 to Part #2. Once on Part #2, we will then proceed to work to the end of the gap. This is demonstrated in the following steps.Calculating Gaps; Working the RouteDownloaded From: / on 04/11/2014 Terms of Use: /terms446Tolerance Stack-Up AnalysisWorking the RoutePART #11) We begin the route at the left of the gap and go left 12mm. This dimension is plus and minus 0.1. Since this step goes left, the 12 is negative.2) We work from the left edge of the part to the right a basic dimension of 95.3 (plus and minus zero) to the center of the slot. Since this step goes right, the 95.3 is a positive number.3) We go to the left (from the center of the slot) to the left edge of the slot. This puts us at the point where Part #1 touches Part #2 so that we can jump to Part #2. To get this dimension, the radius of the slot inner and outer boundaries will have to be calculated. But we know this step is left, therefore a negative number.PART #21) We work from the left edge of the tab to the center of the tab. The step is to the right, therefore a positive number. To get this dimension, the radius of the tabs inner and outer boundaries must be calculated.2) We work to the right (positive) from the center of the tab to the right edge of the part. This is a positive 57.1 basic dimension (plus and minus zero).3) We work from the edge of the part back to the left (negative) a dimension of 136.5 plus and minus 0.7. This brings us to the end of the gap. Since all but two of the numbers in this route are known, we will fill them into the chart. See Figure 23-3.Downloaded From: / on 04/11/2014 Terms of Use: /terms447Chapter Twenty-ThreeFIGURE 23-3 Calculating Minimum Gap Lower LeftDownloaded From: / on 04/11/2014 Terms of Use: /terms448Tolerance Stack-Up AnalysisFIGURE 23-4 Virtual and Resultant Condition Calculations for Tolerance Stack-Up Analysis for Slot and Tab - SummaryNow, we willcomplete the route and calculate the minimum gap.Calculating Inner and Outer Boundary Means and their Tolerances; Numbers ChartsDownloaded From: / on 04/11/2014 Terms of Use: /terms449Chapter Twenty-ThreeFIGURE 23-5 Calculate Minimum Gap Lower LeftFINAL SUMMARY OF STEPSSince we are calculating the minimum gap at the lower left, we will begin the loop at the left side of the gap (on Part #1) and go left (negative) 12mm to the edge of Part #1. Then we will go right (positive) to the center of the slot 95.300. Then we go left (negative) to the edge of the slot 6.095. Now we go right (positive) to the center of the tab 5.985. The loop continues right (positive) to the right edge of Part #2 57.100.The loop reverses and goes left for the last step (negative) to the edge of the gap being calculated 136.500.Downloaded From: / on 04/11/2014 Terms of Use: /terms450Tolerance Stack-Up AnalysisSTEP1Go left from the left edge of the minimum gap to the left edge of Part #1 (-12.000). STEP 2 Go right from the edge of Part #1 to the center of the slot(+95.300).STEP 3Go left to the left edge of the slot (-6.095).STEP 4Go right to the center of the tab (+5.985).STEP 5Go right to the right edge of Part #2 (+57.100).STEP 6Go left to the left edge of Part #2 (-136.500).The negatives are added:-12.000+ -6.095+ -136.500-154.595The positives are added:+95.300+ +5.985+ -+57.100+158.385The negative and positive totals are added: +158.395 + -154.595 +3.790The tolerances are totaled:Tolerances0.100 = Wall0.000 = Basic Dimension of 95.300 0.055 = Slot 0.055 = Tab0.000 = Basic Dimension of 57.100 + 0.700 = Overall Dimension of Part #2 0.910 = Total ToleranceThe total tolerance is subtracted from the 3.79 to get the MIN Gap:3.79 -0.91 2.88 = MIN Gap (at lower left)Downloaded From: / on 04/11/2014 Terms of Use: /terms451Chapter Twenty-ThreeStatistical Tolerancing is applied to features on parts wherein it has been deemed acceptable to forego 100% interchangeability in assemblies. These tolerances are calculated using a predetermined methodology such as Root Sum Square or Monte Carlo or other formulas that attempt to predict how much tolerance is likely to be used during the manufacturing procedure. That “natural tolerance” likely to be consumed by the manufacturing is then compared to the arithmetically available tolerance that has been calculated using formulas such as the fixed and floating fastener assembly formulas discussed in this textbook.The tolerance predicted to be consumed is put over the arithmetically available tolerance for the features in the assembly under consideration to be increased and converted to a percentage. This percentage of the arithmetically available tolerance that is predicted to be used during the manufacture of the features being considered is put under the number one. In other words, one divided by the percentage of the arithmetically calculated tolerance likely to be consumed during manufacturing becomes the percent the individual tolerances will be increased and redistributed back into the assembly.For example, lets say we have an assembly tolerance for the considered features that adds up to 0.91. Then we apply a formula such as the Root Sum Square formula that predicts, in a natural bell curve distribution of parts (a Gaussian Frequency Curve), that we are likely to consume (to within plus or minus 3 sigma) 0.71 of the 0.91.We divide the 0.71 by the 0.91 and find that we have predicted that about 78% of the 0.91 will be consumed during the manufacturing procedure. We then divide the number 1 by 0.78 to get the factor by which we will increase the individual tolerances considered.1 divided by 0.78 is approximately 1.28. Our individual tolerances will each be increased to 128% of their original value. If a Root Sum Square formula was run on the increased tolerances, it would predict (using the same plus or minus 3 sigma procedure as first applied) that of the sum of the increased tolerances, it is likely that only 0.91 (the original arithmetically available tolerance) would be consumed by the assembly.This type of procedure, or one like it, can be used to increase tolerances and, in doing so, decrease the costs associated with production of features using the more tightly toleranced arithmetically calculated tolerances. These ST or Statistical Tolerances should only be allowed to manufacturers who have proven statistically capable in the judgment of the company or department vending the parts out to be produced. An ST tolerance should never be allowed for use by a supplier, who by contrast, is in statistical chaos. Such a supplier may produce entire batches of mating parts that have spanned their larger statistical tolerances and endanger the products quality and its ability to function or assemble properly.Calculating Statistical TolerancingDownloaded From: / on 04/11/2014 Terms of Use: /terms452Tolerance Stack-Up AnalysisFor an example of this methodology, we will use an illustration where, in a fixed fastener two- part assembly, we calculate the MIN GAP for the lower left. To do this, we have to choose the correct route to follow, convert the dimensions to equal bilateral with plus or minus tolerances, then calculate the gap. We used basic dimensions that have tolerances expressed as zero and slots and tabs whose tolerances included both size and position. Still, in the end, we came up with a series of plus and minus tolerances that we used to calculate the minimum gap.6.095 is the radius of the mean dimension of the inner and outer slot boundary averaged. Likewise, the tolerance of plus and minus 0.055 is the tolerance on that slot radius derived from the inner and outer slot boundaries. The mean of the slot boundaries is 2x6.095 or 12.19 with a tolerance of 0.11 (2x0.055). See the following calculations for the slot and the tab.FIGURE 23-6Downloaded From: / on 04/11/2014 Terms of Use: /terms453Chapter Twenty-ThreeFIGURE 23-7 V irtual and Resultant Condition Calculations for Tolerance Stack-Up Analysis for Slot and Tab SummaryNOTE: The numbers used are from the prior calculations from Figure 23-4.Downloaded From: / on 04/11/2014 Terms of Use: /terms454Tolerance Stack-Up AnalysisFIGURE 23-8 Calculated Minimum Gap Lower LeftDownloaded From: / on 04/11/2014 Terms of Use: /terms455Chapter Twenty-ThreeFIGURE 23-9 Calculating the Sum of the SquaresThe arithmetically calculated 100% tolerance allows a minimum gap that is 3.79 - 0.91 = 2.88.The statistically calculated assembly tolerance allows a minimum gap that is 3.79 - 0.71 (rounded off to two decimal places). This 0.71 tolerance is the amount of tolerance likely to be consumed in a natural bell curved distribution of manufactured parts for this assembly. So, if we want to consume 0.91 tolerance, the piece part tolerance should be increased to 128% (rounded off from 127.92184%).Downloaded From: / on 04/11/2014 Terms of Use: /terms456Tolerance Stack-Up AnalysisSo, the 0.055 tolerance for the slot and the tab becomes 1.28 x 0.055 = 0.0704 or a tolerance for each that is 0.070 (rounded off the three decimal places). The tolerance for the wall becomes 1.28 x 0.100 = 0.128. The tolerance for the overall dimension becomes:1.28x 0.7 = 0.896.This is the answer to the problem of how much the tolerances can be increased. Increase all tolerances in the route of numbers to 128% of their original values.Since, statistically, the tolerance originally assigned would not be fully consumed, the minimum gap calculation given originally as 3.79 - 0.91 = 2.88 MIN GAP becomes a statistical probability within 3 sigma of consuming only 3.79 - 0.71 = 3.08 MIN GAP.Given the newly assigned statistically calculated tolerances, with each piece part given a statistically calculated tolerance, we have a mathematical possibility of a MIN GAP that is 3.79 minus the sum of the statistical tolerances. They are: 0.070 (slot) + 0.070 (tab) + 0.128 (wall) + 0.896 (overall dimension) = 70 (slot) + 0.070 (tab) + 0.128 (wall) + 0.896 (overall dimension) 1.164And it leaves a MIN GAP of 3.790 - 1.164 = 2.626. So, arithmetically we might have a MIN GAP that is 2.626, but this is highly unlikely. The slot and tab statistical tolerances of 0.07 when squared become 0.0049 each. The wall statistical tolerance of 0.128 when squared becomes 0.016384. The overall dimension statistical tolerance of 0.898 when squared becomes 0.806404. When added, these tolerances are:0.0049 + 0.0049 + 0.016384 + 0.8064040.832588 Sum of Squares of Tolerance The square root of 0.832588 = 0.91 (rounded off) Square Root of Sum of SquaresSo, we have again shown by the RSS (Root Sum Square) formula that even though the statistical assembly tolerance (1.164) is greater (to 128%) than originally calculated arithmetically as 0.91, the amount the tolerance likely to be consumed is still only 0.91.This statistical approach assumes a zero mean shift for all the dimensions being used (and that the process capability equals 1). It is based on manufacturing processes that are in statistical control, not in statistical chaos. Those not employing Statistical Process Control in the manufacture of the workpieces should not use the RSS tolerancing methodology described here.Downloaded From: / on 04/11/2014 Terms of Use: /terms457Chapter Twenty-ThreeAlso, the RSS model assumes that parts produced for the assembly have been mixed and the components picked at random. The logic of the RSS Model is interesting. It basically allows more tolerances for those manufacturers that need it leastthose using SPC controls. It calculates that the chances of producing a part that spans its larger Statistical Tolerance (ST) are so small that if it does happen, the randomly selected mating parts will make up for the potential problem by not spanning their tolerances. And, in fact, it presupposes that the mating part will be produced so much better than its tolerance extremes as to allow the parts to assemble well. If this is a false assumption, unacceptable functional conditions may arise, such as interference of material. In general, the RSS method is not used if there are less than four dimensions in the stack-up analysis.The 100% tolerancing method scares many professionals when they see that a line fit possibly exists between mating features. This happens when the inner boundaries of holes (or slots) and the outer boundaries of shafts (or tabs) that are mating features are the same value. Professionals calculating the worst mating conditions of such features can see the line fit possibility and sometimes are uncomfortable with this. If that makes them uncomfortable, then allowing more tolerance using the RSS calculations, and consequently a greater possibility of interference, should make them even more unsettled.Back in 1968, a man named A. Bender wrote a paper for SAE (Society of Automotive Engineers) entitled: Statistical Tolerancing as it Relates to Quality Control and the Designer. In this paper, he suggested a safety factor be added to the RSS formula. Instead of just taking the square root of the sum of the squares of the individual features tolerances, he suggested a factor of 1.5 be multiplied by the answer of the standard RSS solution or, in other words, 1.5 times the square root of the sum of the squares of the individual features tolerances. This means this factor estimates one is more likely to use more of the original tolerance than the regular RSS formula calculates is likely consumed. This was so that the additional tolerance given to the piece parts in the assembly was not quite so risky. It is known that, in most cases, producing features at their worst-case condition is unlikely, but it is also known that it happens. Some studies have shown that the RSS methodology does not accurately reflect the reality of what is produced, so to have a cushion would be wise.In years since, many authors and statisticians have suggested other safety or correction factors. They are often based on studies they have done that include probable or measured repeatability and/or accuracy rates of particular manufacturing processes used on specific products. These correction factors often range from 1.4 to 1.8, although one of the most common remains the 1.5 suggested by Bender all those years ago. Before trying an arbitrary correction factor, it is wise to look to your company to see if they have established which, if any, of these factors have been approved.To practice any of the methods shown in this unit, pick examples from earlier units in this book and calculate the statistical tolerances with the RSS method with or without correctional factors such as 1.5.Downloaded From: / on 04/11/2014 Terms of Use: /terms4