Dimensioning and Tolerancing Handbook Episode 2 Part 5 ppsx

Traditional Approaches to Analyzing Mechanical Tolerance Stacks 9-33 9.3.4 Runout Analyzing runout controls in tolerance stacks is similar to analyzing position at RFS. Since runout is always RFS, we can treat the size and location of the feature independently. We analyze total runout the same as circular runout, because the worst-case boundary is the same for both controls. Fig. 9-17 shows a hole that is positioned using runout. Figure 9-18 Concentricity We model the runout tolerance with a nominal dimension equal to zero, and an equal bilateral toler- ance equal to half the runout tolerance. The equation for the Gap in Fig. 9-17 is: Gap = + A/2 + B – C/2 where A = .125 ±.008 B = 0 ±.003 C = .062 ±.005 9.3.5 Concentricity/Symmetry Analyzing concentricity and symmetry controls in tolerance stacks is similar to analyzing position at RFS and runout. Fig. 9-18 is similar to Fig. 9-17, except that a concentricity tolerance is used to control the ∅.062 feature to datum A. Figure 9-17 Circular and total runout 9-34 Chapter Nine The loop diagram for this gap is the same as for runout. The equation for the Gap in Fig. 9-18 is: Gap = + A/2 + B – C/2 where A = .125 ±.008 B = 0 ±.003 C = .062 ±.005 Symmetry is analogous to concentricity, except that it is applied to planar features. A loop diagram for symmetry would be similar to concentricity. 9.3.6 Profile Profile tolerances have a basic dimension locating the true profile. The tolerance is depicted either equal bilaterally, unilaterally, or unequal bilaterally. For equal bilateral tolerance zones, the profile component is entered as a nominal value. The component is equal to the basic dimension, with an equal bilateral tolerance that is half the tolerance in the feature control frame. 9.3.6.1 Profile Tolerancing with an Equal Bilateral Tolerance Zone Fig. 9-19 shows an application of profile tolerancing with an equal bilateral tolerance zone. The equation for the Gap in Fig. 9-19 is: Gap = -A+B where A = 1.255 ±.003 B = 1.755 ±.003 Figure 9-19 Equal bilateral tolerance profile Traditional Approaches to Analyzing Mechanical Tolerance Stacks 9-35 9.3.6.2 Profile Tolerancing with a Unilateral Tolerance Zone Fig. 9-20 shows a figure similar to Fig. 9-19 except the equal bilateral tolerance was changed to a unilateral tolerance zone. The equation for the Gap is the same as Fig. 9-19: Gap = – A + B In this example, however, we need to change the basic dimensions and unilateral tolerances to mean dimensions and equal bilateral tolerances. Therefore, A = 1.258 ±.003 B = 1.758 ±.003 9.3.6.3 Profile Tolerancing with an Unequal Bilateral Tolerance Zone Fig. 9-21 shows a figure similar to Fig. 9-19 except the equal bilateral tolerance was changed to an unequal bilateral tolerance zone. The equation for the Gap is the same as Fig. 9-19: Gap = – A + B Figure 9-20 Unilateral tolerance profile Figure 9-21 Unequal bilateral tolerance profile 9-36 Chapter Nine Figure 9-22 Size datum As we did in Fig. 9-20, we need to change the basic dimensions and unequal bilateral tolerances to mean dimensions and equal bilateral tolerances. Therefore, A = 1.254 ±.003 B = 1.754 ±.003 9.3.6.4 Composite Profile Composite profile is similar to composite position. If a requirement only includes features within the profile, we use the tolerance in the lower segment of the feature control frame. If the requirement includes variations of the profile back to the datum reference frame, we use the tolerance in the upper segment of the feature control frame. Fig. 9-16 shows an example of composite profile tolerancing. Gap 3 is controlled by features within the profile, so we would use the tolerance in the lower segment of the profile feature control frame (∅.008) to calculate the variation for Gap 3. Gap 4, however, includes variations of the profiled features back to the datum reference frame. In this situation, we would use the tolerance in the upper segment of the profile feature control frame (∅.040) to calculate the variation for Gap 4. 9.3.7 Size Datums Fig. 9-22 shows an example of a pattern of features controlled to a secondary datum that is a feature of size. In this example, ASME Y14.5 states that the datum feature applies at its virtual condition, even though it is referenced in its feature control frame at MMC. (Note, this argument also applies for second- ary and tertiary datums invoked at LMC.) In the tolerance stack, this means that we will get an additional “shifting” of the datum that we need to include in the loop diagram. The way we handle this in the loop diagram is the same way we handled features controlled with position at MMC or LMC. We calculate the virtual and resultant conditions, and convert these bound- aries into a nominal value with an equal bilateral tolerance. Traditional Approaches to Analyzing Mechanical Tolerance Stacks 9-37 The value for A in the loop diagram is: • Largest outer boundary = ∅.503 + ∅ .011 = ∅.514 • Smallest inner boundary = ∅.497 – ∅.005 = ∅.492 • Nominal diameter = (∅.514 + ∅ .492)/2 = ∅.503 • Equal bilateral tolerance = ∅.011 An easier way to convert to this radial value is: LMC ±(total size tolerance + tolerance in the feature control frame) = ∅.503 ±(.006+.005) = .503±.011 The value for C in the loop diagram is: • Largest outer boundary = ∅.145 + ∅ .020 = ∅.165 • Smallest inner boundary = ∅139 – ∅.014 = ∅.125 • Nominal diameter = (∅.165 + ∅ .125)/2 = ∅.145 • Equal bilateral tolerance = ∅.020 An easier way to convert to this radial value is: LMC ±(total size tolerance + tolerance in the feature control frame) = ∅.145 ±(.006+.014) = .145 ±.020 The equation for the Gap in Fig. 9-22 is: Gap = – A/2 + B/2 – C/2 where A = .503 ±.011 B = .750 ±0 C = .145 ±.020 9.4 Abbreviations Variable Definition a i sensitivity factor that defines the direction and magnitude for the ith dimension. In a one-dimensional stackup, this value is usually +1 or -1. Sometimes, in a one-dimensional stackup, this value may be +.5 or 5 if a radius is the contributing factor for a diameter callout on a drawing. a j sensitivity factor for the jth, fixed component in the stackup a k sensitivity factor for the kth, variable component in the stackup C f correction factor used in the MRSS equation C f,resized correction factor used in the MRSS equation, using resized tolerances i x f ∂ ∂ partial derivative of function y with respect to x i d g the mean value at the gap. If d g is positive, the mean “gap” has clearance, and if d g is negative, the mean “gap” has interference d i the mean value of the ith dimension in the loop diagram 9-38 Chapter Nine D i dimension associated with i th random variable x i F wc resize factor that is multiplied by the original tolerances to achieve a desired assembly performance using the Worst Case Model F mrss resize factor that is multiplied by the original tolerances to achieve a desired assembly performance using the MRSS Model F rss resize factor that is multiplied by the original tolerances to achieve a desired assembly performance using the RSS Model g m minimum value at the (assembly) gap. This value is zero if no interference or clearance is allowed. µ y mean of random variable y n number of independent variables (dimensions) in the equation (stackup) p number of independent, fixed dimensions in the stackup q number of independent, variable dimensions in the stackup r the total number of measurements in the population of interest σ y standard deviation of function y t i equal bilateral tolerance of the ith component in the stackup T i tolerance associated with ith random variable x i t jf equal bilateral tolerance of the jth, fixed component in the stackup t kv equal bilateral tolerance of the kth, variable component in the stackup t kv,wc,resized equal bilateral tolerance of the kth, variable component in the stackup after resizing, using the Worst Case Model t kv,rss,resized equal bilateral tolerance of the kth, variable component in the stackup after resizing, using the RSS Model t kv,mrss,resized equal bilateral tolerance of the kth, variable component in the stackup after resizing, using the MRSS Model t mrss expected assembly gap variation (equal bilateral) using the MRSS Model t mrss,resized the expected variation (equal bilateral) using the MRSS Model and resized tolerances t rss the expected variation (equal bilateral) using the RSS Model t rss,resized the expected variation (equal bilateral) using the RSS Model and resized tolerances t wc maximum expected variation (equal bilateral) using the Worst Case Model t wc,resized maximum expected variation (equal bilateral) using the Worst Case Model and resized tolerances USL i upper specification limit of the ith dimension x i ith independent variable y function consisting of n independent variables (x 1 ,…,x n ) Z i standard normal transform of ith dimension Z y standard normal transform of y Traditional Approaches to Analyzing Mechanical Tolerance Stacks 9-39 9.5 Terminology MMC = Maximum Material Condition: The condition in which a feature of size contains the maximum amount of material within the stated limits of size. LMC = Least Material Condition: The condition in which a feature of size contains the least amount of material within the stated limits of size. VC = Virtual Condition: A constant boundary generated by the collective effects of a size feature’s specified MMC or LMC material condition and the geometric tolerance for that material condition. RC = Resultant Condition: The variable boundary generated by the collective effects of a size feature’s specified MMC or LMC material condition, the geometric tolerance for that material condition, the size tolerance, and the additional geometric tolerance derived from the feature’s departure from its specified material condition. 9.6 References 1. Bender, A. May 1968. Statistical Tolerancing as it Relates to Quality Control and the Designer. Society of Automotive Engineers, SAE paper No. 680490. 2. Braun, Chuck, Chris Cuba, and Richard Johnson. 1992. Managing Tolerance Accumulation in Mechanical Assemblies. Texas Instruments Technical Journal. May-June: 79-86. 3. Drake, Paul and Dale Van Wyk. 1995. Classical Mechanical Tolerancing (Part I of II). Texas Instruments Technical Journal. Jan Feb: 39-46. 4. Gilson, J. 1951. A New Approach to Engineering Tolerances. New York, NY: Industrial Press. 5. Gladman, C.A. 1980. Applying Probability in Tolerance Technology: Trans. Inst. Eng. Australia. Mechanical Engineering ME5(2): 82. 6. Greenwood, W.H., and K. W. Chase. May 1987. A New Tolerance Analysis Method for Designers and Manufacturers. Transactions of the ASME Journal of Engineering for Industry. 109. 112-116. 7. Hines, William, and Douglas Montgomery.1990. Probability and Statistics in Engineering and Management Sciences. New York, New York: John Wiley and Sons. 8. Kennedy, John B., and Adam M. Neville. 1976. Basic Statistical Methods for Engineers and Scientists. New York, NY: Harper and Row. 9. The American Society of Mechanical Engineers. 1995. ASME Y14.5M-1994, Dimensioning and Tolerancing. New York, NY: The American Society of Mechanical Engineers. 10. Van Wyk, Dale and Paul Drake. 1995. Mechanical Tolerancing for Six Sigma (Part II). Texas Instruments Technical Journal. Jan-Feb: 47-54. 10-1 Statistical Background and Concepts Ron Randall Ron Randall & Associates, Inc. Dallas, Texas Ron Randall is an independent consultant specializing in applying the principles of Six Sigma quality. Since the 1980s, Ron has applied Statistical Process Control and Design of Experiments principles to engineering and manufacturing at Texas Instruments Defense Systems and Electronics Group. While at Texas Instruments, he served as chairman of the Statistical Process Control Council, a Six Sigma Cham- pion, Six Sigma Master Black Belt, and a Senior Member of the Technical Staff. His graduate work has been in engineering and statistics with study at SMU, the University of Tennessee at Knoxville, and NYU’s Stern School of Business under Dr. W. Edwards Deming. Ron is a Registered Professional Engi- neer in Texas, a senior member of the American Society for Quality, and a Certified Quality Engineer. Ron served two terms on the Board of Examiners for the Malcolm Baldrige National Quality Award. 10.1 Introduction Statistics do a fine job of enumerating what has already occurred. Industry’s most urgent needs are to estimate what will happen in the future. Will the product be profitable? How often will defects occur? The job of statistics is to help estimate the future based on the past. When designing any part or system, it is necessary to estimate and account for the variation that is likely to occur in the parts, materials, and product features. Statistics can help estimate or model the most likely outcome, and how much variation there is likely to be in that outcome. From these models, esti- mates of manufacturability and product performance can be made long before production. Knowledge of the probabilities of defects prior to production is important to the financial success of the product. Changes to the design or manufacturing processes that are completed prior to production are far less costly than changes made during production or changes made after the product is fielded. Statistics can help estimate these probabilities. Chapter 10 10-2 Chapter Ten 10.2 Shape, Locations, and Spread Historical data or data from a designed experiment when displayed in a histogram will: • Have a shape • Have a location relative to some important values such as the average or a specification limit • Have a spread of values across a range. For example, Fig. 10-1 contains full indicator movement (FIM) runout values of 1,000 steel shafts, measured in thousandths of an inch (mils). Ideally, these 1,000 shafts would all be the same, but the histogram begins to reveal some information about these shafts and the processes that made them. The thousand data points are displayed in a histogram in Fig. 10-1. A histogram displays the frequency (how often) a range of values is present. The histogram has a shape, its location is concentrated between the values 0.000 and 0.005, and is spread out between the values 0 and 0.030. The range that occurs most often is 0.000 to 0.002, but there are many shafts that are larger than this. Statistics can help quantify the histogram. With knowledge of the type of distribution (shape), the mean of the sample (location), and the standard deviation of the sample (spread), one can estimate the chance that a shaft will exceed a certain value like a specification. We will come back to this example later. 3020100 400 300 200 100 0 x(FIM).001 Frequency 10.3 Some Important Distributions Data that is measured on a continuous scale like inches, ohms, pounds, volts, etc. is referred to as vari- ables data. Data that is classified by pass or fail, heads or tails, is called attributes data. Variables data may be more expensive to gather than attributes data, but is much more powerful in its ability to make estimates about the future. 10.3.1 The Normal Distribution The normal distribution is a mathematical model. All mathematical models are wrong, in that there is always some error. Some models are useful. This is one of them. Karl Frederick Gauss described this distribution in the eighteenth century. Gauss found that repeated measurements of the same astronomical quantity produced a pattern like the curve in Fig. 10-2. This pattern has since been found to occur almost everywhere in life. Heights, weights, IQs, shoe sizes, Figure 10-1 Histogram of runout (FIM) data [...]... 2. 2 2. 3 2. 4 2. 5 0 5. 0000E-01 0 ###### 0.01 6.0365E-03 7.9762E-03 1.0444E- 02 1. 355 3E- 02 1.7 429 E- 02 2 .22 16E- 02 2.8067E- 02 3 .51 48E- 02 4.3633E- 02 5. 3699E- 02 6 .5 52 2 E- 02 7. 927 0E- 02 9 .50 98E- 02 1.1314E-01 1.3 350 E-01 1 .5 6 25 E-01 1.8141E-01 2. 0897E-01 2. 3885E-01 2. 7093E-01 3. 050 3E-01 3.4090E-01 3.7 828 E-01 4.1683E-01 4 .5 620 E-01 4.9601E-01 0. 02 5. 8677E-03 7.7602E-03 1.0170E- 02 1. 320 9E- 02 1.7003E- 02 2.1692E- 02 2.7 429 E- 02. .. 7.1 428 E-03 9.3867E-03 1 .22 25E- 02 1 .57 78E- 02 2.0182E- 02 2 .55 88E- 02 3 .21 57 E- 02 4.0 059 E- 02 4.9471E- 02 6. 057 1E- 02 7.3 52 9 E- 02 8. 850 8E- 02 1. 056 5E-01 1 . 25 07E-01 1.4686E-01 1.7106E-01 1.9766E-01 2. 2663E-01 2. 57 85E-01 2. 9116E-01 3 .26 36E-01 3.6317E-01 4.0 129 E-01 4.4038E-01 4.8006E-01 0.06 5 .23 35E-03 6.9468E-03 9.1375E-03 1.1911E- 02 1 .53 86E- 02 1.9699E- 02 2.4998E- 02 3.1443E- 02 3. 920 4E- 02 4.8 457 E- 02 5. 9380E- 02 7 .21 45E- 02. .. 2. 1692E- 02 2.7 429 E- 02 3.4380E- 02 4 .27 16E- 02 5 .26 16E- 02 6. 4 25 5E- 02 7.7804E- 02 9.3417E- 02 1.1 123 E-01 1.3136E-01 1 .53 86E-01 1.7879E-01 2. 0611E-01 2. 357 6E-01 2. 6763E-01 3.0 153 E-01 3.3 724 E-01 3.7448E-01 4. 129 4E-01 4. 52 2 4E-01 4. 920 2E-01 0.03 5. 7030E-03 7 .54 94E-03 9.9031E-03 1 .28 74E- 02 1. 658 6E- 02 2.1178E- 02 2.6804E- 02 3.3 6 25 E- 02 4.1815E- 02 5. 155 1E- 02 6.3008E- 02 7.6 358 E- 02 9.1 759 E- 02 1.0935E-01 1 .29 24E-01 1 .51 51E-01... 9.4803E-07 1 .54 12E-06 2. 4837E-06 3.9675E-06 6 .28 17E-06 9. 856 8E-06 1 .5 327 E- 05 2. 3617E- 05 3.6 057 E- 05 5. 454 5E- 05 8.1 753 E- 05 1 .21 40E-04 1.7860E-04 2. 6032E-04 3. 759 0E-04 5. 3776E-04 7. 621 7E-04 1.0702E-03 1.4889E-03 2. 0 52 2 E-03 2. 8 027 E-03 3.7 924 E-03 0.08 1.9915E-07 3. 323 4E-07 5. 5003E-07 9. 026 3E-07 1.4686E-06 2. 3689E-06 3.7875E-06 6.0 020 E-06 9. 426 4E-06 1.4671E- 05 2. 2 627 E- 05 3. 457 7E- 05 5 .23 55 E- 05 7. 854 3E- 05 1.1674E-04... 1.0 656 E-07 6 .27 33E-08 3.6642E-08 2. 124 0E-08 1 .22 21E-08 6.9804E-09 3. 959 2E-09 2. 2303E-09 1 .24 81E-09 6.9395E-10 3.8348E-10 2. 1065E-10 1. 150 6E-10 6 . 25 02E-11 3.3775E-11 1.8160E-11 9.7185E- 12 5. 1775E- 12 2.7466E- 12 1. 451 2E- 12 7.6389E-13 4.0068E-13 2. 0948E-13 1.0919E-13 5 .2 5. 3 5. 4 5. 5 5. 6 5. 7 5. 8 5. 9 6 6.1 6 .2 6.3 6.4 6 .5 6.6 6.7 6.8 6.9 7 7.1 7 .2 7.3 7.4 7 .5 0 1.7 958 E-07 5. 1 ###### 0.01 1. 022 8E-13 1.9 629 E-13... 5. 9380E- 02 7 .21 45E- 02 8.6915E- 02 1.0383E-01 1 .23 02E-01 1.4 457 E-01 1.6 853 E-01 1.9489E-01 2. 2363E-01 2. 54 63E-01 2. 8774E-01 3 .22 76E-01 3 .59 42E-01 3.9743E-01 4.3644E-01 4.7608E-01 0.07 5. 0848E-03 6. 755 6E-03 8.8940E-03 1.1604E- 02 1 .50 04E- 02 1. 922 6E- 02 2.4419E- 02 3.0742E- 02 3.8364E- 02 4.7460E- 02 5. 820 7E- 02 7.0781E- 02 8 .53 43E- 02 1. 020 4E-01 1 .21 00E-01 1. 423 1E-01 1.6602E-01 1. 921 5E-01 2. 2065E-01 2. 51 43E-01 2. 8434E-01... 1. 328 0E-13 2. 54 56E-13 4.8643E-13 9 .26 42E-13 1. 758 0E- 12 3. 323 4E- 12 6 . 25 70E- 12 1.1 729 E-11 2. 1887E-11 4.0646E-11 7 .51 00E-11 1.3803E-10 2. 52 2 8E-10 4 .58 45E-10 8 .28 11E-10 1.4865E-09 2. 651 2E-09 4.6968E-09 8 .26 36E-09 1.4436E-08 2. 50 35E-08 4.3091E-08 7.3602E-08 1 .24 73E-07 0.08 6.4700E-14 1 .24 41E-13 2. 3 855 E-13 4 .56 00E-13 8.6875E-13 1.6492E- 12 3.1189E- 12 5. 8745E- 12 1.1017E-11 2. 056 8E-11 3. 821 4E-11 7.0645E-11 1 .29 91E-10... 1. 022 8E-13 1.9 629 E-13 3. 755 8E-13 7.1 627 E-13 1.3612E- 12 2 .57 73E- 12 4.8604E- 12 9. 127 2E- 12 1.7063E-11 3.1 750 E-11 5. 8784E-11 1.0 827 E-10 1.9834E-10 3.6 128 E-10 6 .54 17E-10 1.1773E-09 2. 1 051 E-09 3.7395E-09 6 .59 76E-09 1. 155 9E-08 2. 0104E-08 3.4709E-08 5. 9469E-08 1.0110E-07 1.7 051 E-07 0. 02 9 .58 13E-14 1.8393E-13 3. 52 0 3E-13 6.7 159 E-13 1 .27 68E- 12 2.4183E- 12 4 .5 6 25 E- 12 8 .57 15E- 12 1.6032E-11 2. 9845E-11 5. 52 8 5E-11 1.0188E-10... 7.1976E-06 1. 126 3E- 05 1.7466E- 05 2. 6839E- 05 4.0864E- 05 6.1646E- 05 9 .21 38E- 05 1.3644E-04 2. 0017E-04 2. 9094E-04 4.1894E-04 5. 9766E-04 8.4471E-04 1.1 828 E-03 1.6410E-03 2. 255 6E-03 3.0718E-03 4.1 452 E-03 0. 05 2. 324 2E-07 3.8691E-07 6.3872E-07 1.0 455 E-06 1.6967E-06 2. 729 5E-06 4.3 52 5 E-06 6.8790E-06 1.0774E- 05 1.6 723 E- 05 2. 5 721 E- 05 3.9198E- 05 5.9187E- 05 8. 854 6E- 05 1.3 124 E-04 1. 927 2E-04 2. 8038E-04 4.0411E-04 5. 7704E-04... 7.0645E-11 1 .29 91E-10 2. 3 758 E-10 4.3199E-10 7.8078E-10 1.4 024 E-09 2. 5 029 E-09 4.4371E-09 7.8 121 E-09 1.3 657 E-08 2. 3702E-08 4.0 827 E-08 6.9790E-08 1.1837E-07 0.09 6. 059 6E-14 1.1 655 E-13 2. 2 355 E-13 4 .27 45E-13 8.1465E-13 1 .54 71E- 12 2. 926 9E- 12 5. 5 151 E- 12 1.0348E-11 1.9 327 E-11 3 .5 927 E-11 6.6 450 E-11 1 .22 26E-10 2. 2372E-10 4.0702E-10 7.3611E-10 1. 323 0E-09 2. 3 627 E-09 4.1915E-09 7.3848E-09 1 .29 19E-08 2. 2438E-08 3.8680E-08 . control frame) = ∅.1 45 ±(.006+.014) = .1 45 ±. 020 The equation for the Gap in Fig. 9 -22 is: Gap = – A /2 + B /2 – C /2 where A = .50 3 ±.011 B = . 750 ±0 C = .1 45 ±. 020 9.4 Abbreviations Variable Definition a i sensitivity. the 50 samples (Fig. 10-4). Here we begin to see a central tendency between 10.0 and 10 .5 and a gradual decline in frequency as we move away from the center. 12. 5 12. 011 .51 1.010 .51 0.09 .59 .08 .58 .0 15 10 5 0 Frequency Normal,. data. P-Value: 0.843 A-Squared: 0 .21 7 Anderson-Darling Normality Test N: 1000 StDev: 0.964749 Average: 0. 0 25 13 35 3 2 1 0 - 1 - 2 - 3 .999 .99 . 95 .80 .50 . 2 0 . 05 .01 .001 Probability y=lnx Figure