Minimum-Cost Tolerance Allocation 14-9 14.7 2-D Example: One-way Clutch Assembly The application of tolerance allocation to a 2-D assembly will be demonstrated on the one-way clutch assembly shown in Fig. 14-6. The clutch consists of four different parts: a hub, a ring, four rollers, and four springs. Only a quarter section is shown because of symmetry. During operation, the springs push the rollers into the wedge-shaped space between the ring and the hub. If the hub is turned counterclockwise, the rollers bind, causing the ring to turn with the hub. When the hub is turned clockwise, the rollers slip, so torque is not transmitted to the ring. A common application for the clutch is a lawn mower starter. (Reference 5) c c Vector Loop Ring Hub Roller Spring φ φ b a 2 e 2 Figure 14-6 Clutch assembly with vector loop The contact angle φ between the roller and the ring is critical to the performance of the clutch. Variable b, is the location of contact between the roller and the hub. Both the angle φ and length b are dependent assembly variables. The magnitude of φ and b will vary from one assembly to the next due to the variations of the component dimensions a, c, and e. Dimension a is the width of the hub; c and e/2 are the radii of the roller and ring, respectively. A complex assembly function determines how much each dimension contrib- utes to the variation of angle φ. The nominal contact angle, when all of the independent variables are at their mean values, is 7.0 degrees. For proper performance, the angle must not vary more than ±1.0 degree from nominal. These are the engineering design limits. The objective of variation analysis for the clutch assembly is to determine the variation of the contact angle relative to the design limits. Table 14-5 below shows the nominal value and tolerance for the three independent dimensions that contribute to tolerance stackup in the assembly. Each of the independent variables is assumed to be statistically independent (not correlated with each other) and a normally distributed random variable. The tolerances are assumed to be ±3σ. Table 14-5 Independent dimensions for the clutch assembly Dimension Nominal Tolerance Hub width - a 2.1768 in. .004 in. Roller radius - c .450 in. .0004 in. Ring diameter - e 4.000 in. .0008 in. 14-10 Chapter Fourteen 14.7.1 Vector Loop Model and Assembly Function for the Clutch The vector loop method (Reference 2) uses the assembly drawing as the starting point. Vectors are drawn from part-to-part in the assembly, passing through the points of contact. The vectors represent the independent and dependent dimensions that contribute to tolerance stackup in the assembly. Fig. 14-6 shows the resulting vector loop for a quarter section of the clutch assembly. The vectors pass through the points of contact between the three parts in the assembly. Since the roller is tangent to the ring, both the roller radius c and the ring radius e are collinear. Once the vector loop is defined, the implicit equations for the assembly can easily be extracted. Eqs. (14.4) and (14.5) shows the set of scalar equations for the clutch assembly derived from the vector loop. h x and h y are the sum of vector components in the x and y directions. A third equation, h θ , is the sum of relative angles between consecutive vectors, but it vanishes identically. h x = 0 = b + c sin(φ) - e sin(φ) (14.4) h y = 0 = a + c + c cos(φ) - e cos(φ) (14.5) Eqs. (14.4) and (14.5) may be solved for φ explicitly:       − + = − ce ca 1 cosφ (14.6) The sensitivity matrix [S] can be calculated from Eq. (14.6) by differentiation or by finite difference: [ S] =       −− −− =           ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ 21.10469.44043.103 6272.25483.106469.2 e b c b a b eca φφφ The tolerance sensitivities for δφ are in the top row of [S]. Assembly variations accumulate or stackup statistically by root-sum-squares: ( ) ∑ = 2 )( jij xS δδφ = .01159 radians = .664 degrees where δφ is the predicted 3σ variation, δx j is the set of 3σ component variations. By worst case: ∑ = jij xS δδφ = .01691 radians = .9688 degrees where δφ is the predicted extreme variation. 14.8 Allocation by Scaling, Weight Factors Once you have RSS and worst case expressions for the predicted variation δφ, you may begin applying various allocation algorithms to search for a better set of design tolerances. As we try various combina- ( ) ( ) ( ) ( )( )( ) ( )( )( ) ( )( )( ) 222 2 13 2 12 2 11 0008.6272.20004.5483.10004.6469.2 +−+−= ++= eScSaS δδδ ( )( ) ( )( ) ( )( ) 000862722000454831000464692 131211 eScSaS ++= ++= δδδ Minimum-Cost Tolerance Allocation 14-11 tions, we must be careful not to exceed the tolerance range of the selected processes. Table 14-6 shows the selected processes for dimensions a, c, and e and the maximum and minimum tolerances obtainable by each, as extracted from the Appendix for the corresponding nominal size. Table 14-6 Process tolerance limits for the clutch assembly Part Dimension Process Nominal Sensitivity Minimum Maximum (inch) Tolerance Tolerance Hub a Mill 2.1768 -2.6469 .0025 .006 Roller c Lap .9000 -10.548 .00025 .00045 Ring e Grind 4.0000 2.62721 .0005 .0012 14.8.1 Proportional Scaling by Worst Case Since the rollers are vendor-supplied, only tolerances on dimensions a and e may be altered. The propor- tionality factor P is applied to δa and δe, while δφ is set to the maximum tolerance of ±.017453 radians (±1° ). ( ) ( ) ( )( ) ( ) ( ) 0008.6272.20004.5483.10004.6469.2017453. 017453. 131211 PP ePScSaPS xS jij ++= ++= ∑ = δδδ δδφ Solving for P: P = 1.0429 δa = (1.0429)(.004)=.00417 in. δe = (1.0429)(.0008)=.00083 in. 14.8.2 Proportional Scaling by Root-Sum-Squares ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( )( )( ) ( ) ( )( ) 222 2 13 2 12 2 11 2 0008.6272.20004.5483.10004.6469.2017453. 017453. PP ePScSaPS xS jij +−+−= ++= ∑ = δδδ δδφ Solving for P: P = 1.56893 δa = (1.56893)(.004)=.00628 in. δe = (1.56893)(.0008)=.00126 in. Both of these new tolerances exceed the process limits for their respective processes, but by less than .001in each. You could round them off to .006 and .0012. The process limits are not that precise. 14.8.3 Allocation by Weight Factors Grinding the ring is the more costly process of the two. We would like to loosen the tolerance on dimen- sion e. As a first try, let the weight factors be w a = 10, w e = 20. This will change the ratio of the two tolerances and scale them to match the 1.0 degree limit. The original tolerances had a ratio of 5:1. The final ratio will be the product of 1:2 and 5:1, or 2.5:1. The sensitivities do not affect the ratio. 14-12 Chapter Fourteen ( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( )( ) ( )( )( ) ( ) ( )( )( ) 222 2 13 2 12 2 11 2 0008.30/206272.20004.5483.10004.30/106469.2017453. 30/2030/10017453. PP ePScSaPS xS jij +−+−= ++= ∑ = δδδ δδφ Solving for P: P = 4.460 δa = (4.460)(10/30)(.004)=.00595 in. δe = (4.460)(20/30)(.0008)=.00238 in. Evaluating the results, we see that δa is within the .006in limit, but δe is well beyond the .0012 inch process limit. Since δa is so close to its limit, we cannot change the weight factors much without causing δa to go out of bounds. After several trials, the best design seemed to be equal weight factors, which is the same as proportional scaling. We will present a plot later that will make it clear why it turned out this way. From the preceding examples, we see that the allocation algorithms work the same for 2-D and 3-D assemblies as for 1-D. We simply insert the tolerance sensitivities into the accumulation formulas and carry them through the calculations as constant factors. 14.9 Allocation by Cost Minimization The minimum cost allocation applies equally well to 2-D and 3-D assemblies. If sensitivities are included in the derivation presented in Section 14.1, Eqs. (14.1) through (14.3) become: Table 14-7 Expressions for minimum cost tolerances in 2-D and 3-D assemblies Worst Case RSS ( ) ( ) ( ) 11 1 11 11 1 1 ++ +         = i i k/k k/ i ii i T SBk SBk T ( ) ( ) ( ) 2/2 1 2/1 2 11 1 2 1 ++ +         = i i kk k i ii i T SBk SBk T ( ) ( ) ( ) ∑ ++ +         + = 11 1 11 11 1 11 1 i i k/k k/ i ii i ASM T SBk SBk S TST ( ) ( ) ( ) ∑ ++ +         + = 2/22 1 2/2 2 11 1 2 2 2 1 2 1 2 1 i i kk k i ii i ASM T SBk SBk S TST Part Dimension Process Nominal Sensitivity B k Minimum Maximum (inch) Tolerance Tolerance Hub a Mill 2.1768 -2.6469 .1018696 .45008 .0025 .006 Roller c Lap .9000 -10.548 .000528 1.130204 .00025 .00045 Ring e Grind 4.0000 2.62721 .0149227 .79093 .0005 .0012 The cost data for computing process cost is shown in Table 14-8: Table 14-8 Process tolerance cost data for the clutch assembly Minimum-Cost Tolerance Allocation 14-13 14.9.1 Minimum Cost Tolerances by Worst Case To perform tolerance allocation using a Worst Case Stackup Model, let T 1 = δa, and T i = δe, then S 1 = S 11 , k 1 = k a , and B 1 = B a , etc. eScSaST ASM δδδ 131211 ++= ( ) ( ) ( ) 1/1 1/1 13 11 131211 ++ +         ++= e k a k e k aa ee a SBk SBk ScSaS δδδ ( ) ( )( )( ) ( )( )( ) ( ) ( ) 790931450081 7909311 627221018696045008 64692014922779093 62722000454831064692017453 ./. )./( da . . . a d. .       ++= The only unknown is δa, which may be found by iteration. δe may then be found once δa is known. Solving for δa and δe: δa =.00198 in. ( )( )( ) ( )( )( ) ( ) ( ) 0030400198 627221018696045008 64692014922779093 790931450081 7909311 de ./. )./( =      = in. The cost corresponding to holding these tolerances would be reduced from C= $5.42 to C= $3.14. Comparing these values to the process limits in Table 14-6, we see that δa is below its lower process limit (.0025< δa <.006), while δe is much larger than the upper process limit (.0005< δe <.0012). If we decrease δe to the upper process limit, δa can be increased until T ASM equals the spec limit. The resulting values and cost are then: δa = .0038 in. δe = .0012 in. C = $4.30 The relationship between the resulting three pairs of tolerances is very clear when they are plotted as shown in Fig. 14-7. Tol e and Tol a are plotted as points in 2-D tolerance space. The feasible region is bounded by a box formed by the upper and lower process limits, which is cut off by the Worst Case limit curve. The original tolerances of (.004, .0008) lie within the feasible region, nearly touching the WC Limit. Extending a line through the original tolerances to the WC Limit yields the proportional scaling results found in section 14.2 (.00417, .00083), which is not much improvement over the original tolerances. The minimum cost tolerances (OptWC) were a significant change, but moved outside the feasible region. The feasible point of lowest cost (Mod WC) resulted at the intersection of the upper limit for Tol e and the WC Limit (.0038, .0012). Tol a Tol e 0 0.001 0.002 0.003 0.004 0.005 0 0.002 0.004 Original Opt WC Mod WC WC Limit Opt WC Original Mod WC WC Limit Feasible Region Figure 14-7 Tolerance allocation results for a Worst Case Model 14-14 Chapter Fourteen This type of plot really clarifies the relationship between the three results. Unfortunately, it is limited to a 2-D graph, so it is only applicable to an assembly with two design tolerances. 14.9.2 Minimum Cost Tolerances by RSS Repeating the minimum cost tolerance allocation using the RSS Stackup Model: ( ) ( ) ( ) 2 13 2 12 2 11 2 eScSaST ASM δδδ ++= ( ) ( )( )( ) ( )( )( ) )790932()450082(2 )790932(2 2 222 62722101869645008 64692014922779093 62722 )0004)(548310()64692()017453( ./. ./ da . a d. .       + += Solving for δa by iteration and δe as before: δa = .00409 in. ( )( )( ) ( )( )( ) ( ) ( ) ( ) 790932450082 7909321 00409 62722101869645008 64692014922779093 ./. )./( . de       = = .00495 in. The cost corresponding to holding these tolerances would be reduced from C= $5.42 to C= $2.20. Comparing these values to the process limits in Table 14-6, we see that δa is now safely within its process limits (.0025< δa <.006), while δe is still much larger than the upper process limit (.0005< δe <.0012). If we again decrease δe to the upper process limit as before, δa can be increased until it equals the upper process limit. The resulting values and cost are then: δa = .006 in. δe = .0012 in. C = $4.07 The plot in Fig. 14-8 shows the three pairs of tolerances. The box containing the feasible region is entirely within the RSS Limit curve. The original tolerances of (.004, .0008) lie near the center of the feasible region. Extending a line through the original tolerances to the RSS Limit yields the proportional scaling results found in section 14.2 (.00628, .00126), both of which lie just outside the feasible region. The Tol a Tol e 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0 0.002 0.004 0.006 Original Opt RSS Mod RSS RSS Limit Opt RSS Mod RSS RSS Limit Original Feasible Region Figure 14-8 Tolerance allocation results for the RSS Model ( ) ( ) ( ) ( ) ( ) ( ) 2/22 2/2 13 112 13 2 12 2 11 ++ +         ++= e k a k e k aa ee a SBk SBk ScSaS δδδ Minimum-Cost Tolerance Allocation 14-15 minimum cost tolerances (OptRSS) were a significant change, but moved far outside the feasible region. The feasible point of lowest cost (ModRSS) resulted at the upper limit corner of the feasible region (.006, .0012). Comparing Figs. 14-7 and 14-8, we see that the RSS Limit curve intersects the horizontal and vertical axes at values greater than .006 inch, while the WC Limit curve intersects near .005 inch tolerance. The intersections are found by letting Tol a or Tol e go to zero in the equation for T ASM and solving for the remaining tolerance. The RSS and WC Limit curves do not converge to the same point because the fixed tolerance δc is subtracted from T ASM differently for WC than RSS. 14.10 Tolerance Allocation with Process Selection Examining Fig. 14-7 further, the feasible region appears very small. There is not much room for tolerance design. The optimization preferred to drive Tol e to a much larger value. One way to enlarge the feasible region is to select an alternate process for dimension e. Instead of grinding, suppose we consider turning. The process limits change to (.002< δe <.008), with B e = .118048 k e = 45747. Table 14-9 shows the revised data. Table 14-9 Revised process tolerance cost data for the clutch assembly Part Dimension Process Nominal Sensitivity B k Minimum Maximum (inch) Tolerance Tolerance Hub a Mill 2.1768 -2.6469 .1018696 .45008 .0025 .006 Roller c Lap .9000 -10.548 .000528 1.130204 .00025 .00045 Ring e Turn 4.0000 2.62721 .118048 .45747 .002 .008 Milling and turning are processes with nearly the same precision. Thus, B e and B a are nearly equal as are k e and k a . The resulting RSS allocated tolerances and cost are: δa =.00434 in. δe = .00474 in. C = $2.54 The new optimization results are shown in Fig. 14-9. The feasible region is clearly much larger and the minimum cost point (Mod Proc) is on the RSS Limit curve on the region boundary. The new optimum point has also changed from the previous result (Opt RSS) because of the change in B e and k e for the new process. The resulting WC allocated tolerances and cost are: δa = .00240 in. δe = .00262 in. C = $3.33 Tol a Tol e 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0 0.002 0.004 0.006 Original Opt RSS Mod RSS Mod Proc RSS Limit Opt RSS Mod RSS Mod Proc RSS Limit Original Feasible Region Figure 14-9 Tolerance allocation results for the modified RSS Model 14-16 Chapter Fourteen The modified optimization results are shown in Fig. 14-10. The feasible region is the smallest yet due to the tight Worst Case (WC) Limit. The minimum cost point (Mod Proc) is on the WC Limit curve on the region boundary. Tol a Tol e 0 0.001 0.002 0.003 0.004 0.005 0 0.002 0.004 Original Opt WC Mod WC Mod Proc WC Limit Opt WC Original Mod WC Mod Proc WC Limit Feasible Region Figure 14-10 Tolerance allocation results for the modified WC Model Cost reductions can be achieved by comparing cost functions for alternate processes. If cost-versus- tolerance data are available for a full range of processes, process selection can even be automated. A very systematic and efficient search technique, which automates this task, has been published. (Reference 4) It compares several methods for including process selection in tolerance allocation and gives a detailed description of the one found to be most efficient. 14.11 Summary The results of WC and RSS cost allocation of tolerances are summarized in the two bar charts, Figs. 14-11 and 14-12. The changes in magnitude of the tolerances are readily apparent. Costs have been added for comparison. WC Cost Allocation Results 0 0.002 0.004 Original Opt WC Mod WC Mod Proc a c e $5.42 $3.14 $4.30 $3.33 Tolerance Figure 14-11 Tolerance allocation results for the WC Model Minimum-Cost Tolerance Allocation 14-17 Summarizing, the original tolerances for both WC and RSS were safely within tolerance constraints, but the costs were high. Optimization reduced the cost dramatically; however, the resulting tolerances exceeded the recommended process limits. The modified WC and RSS tolerances were adjusted to con- form to the process limits, resulting in a moderate decrease in cost, about 20%. Finally, the effect of changing processes was illustrated, which resulted in a cost reduction near the first optimization. Only the allocated tolerances remained in the new feasible region. A designer would probably not attempt all of these cases in a real design problem. He would be wise to rely on the RSS solution, possibly trying WC analysis for a case or two for comparison. Note that the clutch assembly only had three dimensions contributing to the tolerance stack. If there had been six or eight, the difference between WC and RSS would have been much more significant. It should be noted that tolerances specified at the process limit may not be desirable. If the process is not well controlled, it may be difficult to hold it at the limit. In such cases, the designer may want to back off from the limits to allow for process uncertainties. 14.12 References 1. Chase, K. W. and A. R. Parkinson. 1991. A Survey of Research in the Application of Tolerance Analysis to the Design of Mechanical Assemblies: Research in Engineering Design. 3(1):23-37. 2. Chase, K. W., J. Gao and S. P. Magleby. 1995. General 2-D Tolerance Analysis of Mechanical Assemblies with Small Kinematic Adjustments. Journal of Design and Manufacturing. 5(4): 263-274. 3. Chase, K.W. and W.H. Greenwood. 1988. Design Issues in Mechanical Tolerance Analysis. Manufacturing Review. March, 50-59. 4. Chase, K. W., W. H. Greenwood, B. G. Loosli and L. F. Hauglund. 1989. Least Cost Tolerance Allocation for Mechanical Assemblies with Automated Process Selection. Manufacturing Review. December, 49-59. 5. Fortini, E.T. 1967. Dimensioning for Interchangeable Manufacture. New York, New York: Industrial Press. 6. Greenwood, W.H. and K.W. Chase. 1987. A New Tolerance Analysis Method for Designers and Manufacturers. Journal of Engineering for Industry, Transactions of ASME. 109(2):112-116. 7. Hansen, Bertrand L. 1963. Quality Control: Theory and Applications. Paramus, New Jersey: Prentice-Hall. 8. Jamieson, Archibald. 1982. Introduction to Quality Control. Paramus, New Jersey: Reston Publishing. 9. Pennington, Ralph H. 1970. Introductory Computer Methods and Numerical Analysis. 2nd ed. Old Tappan, New Jersey: MacMillan. 10. Speckhart, F.H. 1972. Calculation of Tolerance Based on a Minimum Cost Approach. Journal of Engineering for Industry, Transactions of ASME. 94(2):447-453. 11. Spotts, M.F. 1973.Allocation of Tolerances to Minimize Cost of Assembly. Journal of Engineering for Industry, Transactions of the ASME. 95(3):762-764. RSS Cost Allocation Results 0 0.002 0.004 0.006 Original Opt RSS Mod RSS Mod Proc a c e $5.42 $2.20 $4.07 $2.54 Tolerance Figure 14-12 Tolerance allocation results for the RSS Model [...]... 13.600 -20 .99 9 Ream 0.000-0. 599 0.600-0 .99 9 1.000-1. 499 1.500 -2. 799 2. 800-4. 499 4.500-7. 799 Turn / bore / shape 0.000-0. 599 0.600-0 .99 9 1.000-1. 499 1.500 -2. 799 2. 800-4. 499 4.500-7. 799 7.800-13. 599 13.600 -20 .99 9 Mill 0.000-0. 599 0.600-0 .99 9 1.000-1. 499 1.500 -2. 799 2. 800-4. 499 4.500-7. 799 7.800-13. 599 13.600 -20 .99 9 Drill 0.000-0. 599 0.600-0 .99 9 1.000-1. 499 1.500 -2. 799 2. 800-4. 499 B k Min Tol Max Tol 0.001 893 78... 14A -2 Cost-tolerance functions for metal removal processes Size Range A Lap / Hone 0.000-0. 599 0.600-0 .99 9 1.000-1. 499 1.500 -2. 799 2. 800-4. 499 4.500-7. 799 7.800-13. 599 13.600 -20 .99 9 Grind / Diamond turn 0.000-0. 599 0.600-0 .99 9 1.000-1. 499 1.500 -2. 799 2. 800-4. 499 4.500-7. 799 7.800-13. 599 13.600 -20 .99 9 Broach 0.000-0. 599 0.600-0 .99 9 1.000-1. 499 1.500 -2. 799 2. 800-4. 499 4.500-7. 799 7.800-13. 599 13.600 -20 .99 9... 0.000 528 16 0.0 022 0173 0.00033 1 29 0.00 026 156 0.000381 19 0.000 59 824 0.00 427 422 0 .95 08781 1.13 020 36 0 .98 08618 1 .25 90 875 1. 326 92 9 7 1.3073 528 1 .27 16314 1. 022 1757 0.00 02 0.00 025 0.0003 0.0004 0.0005 0.0006 0.0007 0.0008 0.0004 0.00045 0.0005 0.0006 0.0008 0.001 0.00 12 0.0015 0. 024 84363 0.01 525 616 0. 020 50 72 0.0133561 0.014 92 2 68 0. 024 67047 0.051 199 44 0.0831 790 8 0.6465 727 0. 722 198 9 0.70 390 47 0.7 827 624 0. 790 9 32. .. 0.0 324 526 1 0.046 821 58 0.0 420 49 92 0.048 096 84 0.06 92 9 088 0.0 92 0 390 7 0.6000163 0.5654 92 0.6 021 191 0.6 021 191 0.5654 92 0.540 92 5 4 0.0005 0.0006 0.0008 0.001 0.00 12 0.0015 0.00 12 0.0015 0.0 02 0.0 025 0.003 0.004 0.0 720 1641 0.08 596 95 02 0.10 123 3386 0.118003 02 0.11804756 0. 125 76137 0.1 599 7103 0.15300611 0.46 822 793 0.457471 42 0.44 723 008 0.43 898 69 0.457471 42 0.46536684 0.43 898 69 0.46 822 793 0.0008 0.001 0.00 12 0.0015... 0.0 02 0.0 025 0.003 0.004 0.003 0.004 0.005 0.006 0.008 0.01 0.0 12 0.015 0.08 623 08 0.108788 12 0. 095 44417 0.1018 695 8 0.14 399 071 0. 1 29 7 620 9 0.1 391 6564 0.17114563 0. 425 91 73 0.4044547 0.4431 399 0.4500 798 0.4044547 0.4431 399 0.4500 798 0. 425 91 73 0.00 12 0.0015 0.0 02 0.0 025 0.003 0.004 0.005 0.006 0.003 0.004 0.005 0.006 0.008 0.01 0.0 12 0.015 0.00301435 0.00085 791 0.00318631 0.00644133 0.0 022 3316 1. 095 5 124 ... 0.741 3 29 1 0.6548 091 0.6017646 0.00 02 0.00 025 0.0003 0.0004 0.0005 0.0006 0.0007 0.0008 0.0005 0.0006 0.0008 0.001 0.00 12 0.0015 0.0 02 0.0 025 0.04385 52 0.04670538 0.040713 62 0.048 524 0.0637 591 0.0 92 2 92 3 0.144046 0.171785 0.5486 19 0.5 523 0115 0.58686634 0.5 797 61 0.5 596 08 0. 521 758 0.4 695 7 0.4 590 7 0.00 025 0.0003 0.0004 0.0005 0.0006 0.0007 0.0008 0.001 0.0008 0.001 0.00 12 0.0015 0.0 02 0.0 025 0.003 0.004 0.0 324 526 1... 0.001 0.0 02 0.003 0.004 0 Ream 4 3 2 1 0 0 Figure 14A -2 Plot of fitted cost versus tolerance functions 14 -22 Chapter Fourteen B k Lap / Hone 0.006 1.5 0.004 1 0.0 02 0.5 0 0 0 2 4 6 8 10 12 14 16 18 0 2 4 6 8 10 12 14 16 18 0 2 4 6 8 10 12 14 16 18 0 2 4 6 8 10 12 14 16 18 Grind / Diamond turn 0.1 1.5 1 0.05 0.5 0 0 0 2 4 6 8 10 12 14 16 18 Broach 1.5 0 .2 0.15 0.1 0.05 0 1 0.5 0 0 2 4 6 8 10 12 14 16... 1.3801 824 1. 190 6 627 1. 095 5 124 1.3801 824 0.003 0.004 0.005 0.006 0.008 0.005 0.006 0.008 0.01 0.0 12 Minimum-Cost Tolerance Allocation Lap / Hone 14 -21 Turn / bore / shape 8 3 6 2 4 1 2 0 0 0 0.0005 0.001 0.0015 Grind / Diamond turn 0 0.005 0.01 0.015 0.005 0.01 0.015 0.004 0.008 0.0 12 Mill 8 2 6 1.5 4 1 2 0.5 0 0 0 0.0005 0.001 0.0015 0.0 02 0.0 025 Broach 0 Drill 2 6 1.5 4 1 2 0.5 0 0 0 0.001 0.0 02 0.003... 0.05 0.5 0 0 0 1 2 3 4 5 6 0 2 Figure 14A-3 Plot of coefficients versus size for cost-tolerance functions 4 6 Minimum-Cost Tolerance Allocation B Turn / bore / shape 14 -23 k 0 .2 0.15 0.1 0.05 0 1.5 1 0.5 0 0 2 4 6 0 2 4 6 8 10 12 14 16 18 0 8 10 12 14 16 18 2 4 6 8 10 12 14 16 18 Mill 0 .2 0.15 0.1 0.05 0 1.5 1 0.5 0 0 2 4 6 8 10 12 14 16 18 Drill 0.01 1.5 1 0.005 0.5 0 0 0 1 2 3 4 0 1 2 3 Figure 14A-3... 14A -2 and plotted in Figs 14A -2 and 14A-3 Turn 2. 5 2 Size 4: Data Cost Size 4: Fitted 1.5 Size 5: Data Size 5: Fitted 1 Size 6: Data Size 6: Fitted 0.5 0 0 0.0 02 0.004 0.006 0.008 0.01 0.0 12 Tolerance Figure 14A-1 Plot of cost-versus-tolerance for fitted and raw data for the turning process Table 14A-1 Relative cost of obtaining various tolerance levels Minimum-Cost Tolerance Allocation 14- 19 14 -20 . Model: ( ) ( ) ( ) 2 13 2 12 2 11 2 eScSaST ASM δδδ ++= ( ) ( )( )( ) ( )( )( ) ) 790 9 32( )4500 82( 2 ) 790 9 32( 2 2 222 627 221 018 696 45008 646 92 0 14 92 2 7 790 93 627 22 )0004)(548310()646 92 ( )017453( ./. ./ da . 0.0005 0.00 12 0.600-0 .99 9 0.046 821 58 0.5654 92 0.0006 0.0015 1.000-1. 499 0.0 420 49 92 0.6 021 191 0.0008 0.0 02 1.500 -2. 799 0.048 096 84 0.6 021 191 0.001 0.0 025 2. 800-4. 499 0.06 92 9 088 0.5654 92 0.00 12 0.003 4.500-7. 799 . turn 0.000-0. 599 0. 024 84363 0.6465 727 0.00 02 0.0005 0.600-0 .99 9 0.01 525 616 0. 722 198 9 0.00 025 0.0006 1.000-1. 499 0. 020 50 72 0.70 390 47 0.0003 0.0008 1.500 -2. 799 0.0133561 0.7 827 624 0.0004 0.001 2. 800-4. 499