Calculation Of Tolerance Stacks Using Direct Position .
AC 2009-138: CALCULATION OF TOLERANCE STACKS USINGDIRECT-POSITION APPROACH IN GEOMETRIC DIMENSIONING ANDTOLERANCINGCheng Lin, Old Dominion UniversityPage 14.301.1 American Society for Engineering Education, 2009
Calculation of Tolerance Stacks Using Direct-Position Approach inGeometric Dimensioning and TolerancingAbstractFormulas for the calculation of position tolerance stacks of Geometric Dimensioning and Tolerancing(GD&T) are presented. This direct-position approach shows that the formulas can be observed directlyfrom the extreme positions of the holes specified in an engineering drawing. When compared to otherapproaches for tolerance stacks, this method can be applied to all three material conditions (MaximumMaterial Condition, Least Material Condition, and Regardless of Feature Size.) and is easier for students tolearn and remember the formulas. A graphical demonstration using position control on two holes in anengineering drawing is applied to explain the approach.1. IntroductionTolerance stacks are used to describe the problem-solving process in calculating the effects of theaccumulated variation that is allowed by specified dimensions and tolerances, which are typically specifiedon an engineering drawing. Arithmetic tolerance stacks use the worst-case maximum or minimum values ofdimensions and tolerances to calculate the maximum and minimum distances between holes or between ahole and the edge of a part1,2. The application is particularly important in the design stage to maintain aspecified minimum solid distances in the part. In addition, stack analysis enables parts to be made preciseenough to be assembled interchangeably with the largest possible tolerances permitted by partspecification.Several methods are proposed to calculate tolerance stacks using the approaches of graphs, charts, tables,and formulas3,4,5. However, they are all very complicated for students to learn. A graphical approach calledgage method, using the concept of functional gages, seems to provide an effective way for this purpose2.However, as the functional gages can only be applied to the Maximum Material Condition (MMC), thismethod can not be applied to other two material conditions: Least Material Condition (LMC) andRegardless of Feature of Size (RFS). In this paper, the direct-position method is proposed to derive theformulas for tolerance stacks. The method not only is easier for students or designers to understand andremember, but can be applied to three material conditions, which are explained in the following session. Agraphical example using position control on three material conditions is applied to demonstrate theapproach.2. Three Material ConditionsBased on the design and manufacturing needs, geometric tolerances can be specified with different materialconditions, which include Maximum Material Condition (MMC), Least Material Condition (LMC), andRegardless of Feature Size (RFS). Characteristics of each material condition are described in the followingparagraphs.2.1. Maximum Material Condition (MMC)Page 14.301.2To indicate that a geometric tolerance is specified with MMC, a symbol m is added to either a geometriccharacteristic or a datum. Maximum Material Condition is particularly defined as having the maximumsolid volume for a part. Therefore, for internal parts (such as holes or grooves, etc.), MMC is at itsminimum feature of size (FOS). For external parts (such as pins or studs, etc.), MMC is at its maximumfeature of size. When a geometric characteristic is specified with MMC, the geometric tolerance may havea bonus tolerance when its FOS is approaching to its Least Material Condition (LMC). Figure 16 shows adesign drawing using MMC Position Tolerance with Datum A as the center axis of the φ0.8 hole. From thetable shown in this figure, when the diameter of a part is measured at 1.02, which is the MMC, there is no
bonus tolerance and the Position Tolerance remains at 0.05. However, when the diameter is measured at0.98, which is the LMC, the bonus tolerance is equal to 0.04. The total Position Tolerance in this caseincreases to 0.09. MMC can be easily found in most GD&T design drawings.Figure 1: A design drawing using MMC position tolerance.2.2. Least Material Condition (LMC)To indicate that a geometric tolerance is specified with LMC, a symbol l is added to either a geometrictolerance or a datum. Least Material Condition is particularly defined as having the least solid volume for apart. Therefore, for internal parts, LMC is at its maximum feature of size. For external parts, LMC is at itsminimum feature of size. When a geometric tolerance is specified with LMC, the geometric tolerance mayhave a bonus tolerance when its FOS is approaching to its MMC. Figure 26 shows a design drawing usingLMC Position Tolerance with Datum A as the center axis of the φ0.8 hole. From the table shown in thisfigure, when the diameter of a part is measured at 1.02, which is the MMC, there is a bonus tolerance andthe Position Tolerance increases to 0.09. However, when the diameter of a part is measured at 0.98, whichis the LMC, there is no bonus tolerance. The Position Tolerance remains at 0.05. LMC is particularlyapplied to guarantee a larger minimum thickness than the same drawing using MMC.Figure 2: A design drawing using LMC position tolerance.2.3. Regardless of Feature Size (RFS)Unlike MMC and LMC, Regardless of Feature Size gives no additional geometric tolerance. The conceptof RFS has been used prior to the introduction of MMC and LMC principles. Figure 36 shows a designdrawing using RFS position tolerance. Since there is no modifier added to the position tolerance, accordingto Rule 26, the position tolerance is RFS. From the table shown in this figure, the position tolerance remainsthe same regardless the variations on the feature of sizes.Page 14.301.3
Figure 3: A design drawing using RFS position tolerance.3. Formula for Solid Distance between Two Holes - Regardless of Feature Size (RFS)Figure 4 shows an engineering drawing using RFS in the position-tolerance control. In the drawing, Arepresents a datum feature; φD1 and φD2 represent the diametrical sizes of the two holes respectively; T1and T2 represent the bilateral size tolerances of the two holes respectively; φP1 and φP2 represent the RFSposition tolerances of the two holes respectively; L is the actual location between these two holes and isexpressed in basic dimension 2,7; X represents the solid distance between these two holes. According to thedefinition of RFS2, there is no bonus position tolerance for these two holes.Figure 4: Position Tolerance with RFS.3.1. Formulas for X max – RFSFigure 5 shows the extreme position to determine X max based on the following conditions:a. The two holes are made at their minimum sizes: φ (D1-T1) and φ (D2-T2)b. The centers of the two holes are located at their farthest positions of the position-tolerance zones:Points A and B in Figure 5.Page 14.301.4
Figure 5: Extreme Position for X max – RFS.From Figure 5, X max can be easily determined through the following equation:(1)3.2. Formulas for X min – RFSFigure 6 shows the extreme position to determine X min based on the following conditions:a. The two holes are made at their maximum sizes: φ (D1 T1) and φ (D2 T2)b. The centers of the two holes are located at their closest positions of the position-tolerance zones:Points A and B in Figure 6.Figure 6: Extreme Position for X min – RFS.From Figure 6, X min can be easily determined through the following equation:(2)Equation (2) is similar to Equation (1) except using minus signs on P2/2 and P1/2 in the formulabecause Points A and B are at their closest positions.Page 14.301.5
4. Formula for Solid Distance between Two Holes – Maximum Material Condition (MMC)Figure 7 shows an example of an engineering drawing using MMC in the position-tolerance control. Thedrawing is very similar to Figure 6, except with m added in the feature control frame of the positiontolerances. According to MMC6, a maximum bonus position tolerance of 2T1 is added to φP1 when thediameter of the Hole 1 is made at φ (D1 T1). Similarly, a maximum bonus position tolerance of 2T2Figure 7: Position Tolerance with MMC.can be added to φP2 when Hole 2 is made at φ (D2 T2). No bonus position tolerance will be allowed whenthe diameters of the holes are made at φ (D1-T1) and φ (D2-T2) respectively.4.1. Formulas for X max – MMCFigure 8 shows the extreme condition for X max. Similar to the Session 3.1, it is based on the followingconditions:a. The two holes are made at their minimum sizes: φ (D1-T1) and φ (D2-T2).b. The centers of the two holes are located at their farthest locations of the position-tolerance zones:Points A and B.Figure 8: Extreme Position for X max – MMC.Because there is no bonus position tolerance when the diameters of the holes are made at φ (D1-T1) and φ(D2-T2) respectively, Figure 8 is exactly the same as Figure 5. X max can be easily determined through thefollowing equation:Page 14.301.6
(3)4.2. Formulas for X min – MMCFigure 9 shows the extreme condition for X min, which is based on the following conditions:a. The two holes are made at their maximum sizes: φ (D1 T1) and φ (D2 T2). Because the holes aremade at their Least Material Condition (LMC), bonus tolerances of φ2T1 and φ2T2 are added totheir respective original position tolerances φP1 and φP2.b. The centers of the two holes are at their closest locations, which are indicated as Points A and B.Figure 9: Extreme Position for X min – MMC.From Figure 9, X min can be easily determined through the following equation:(4)5. Formula for Solid Distance between Two Holes – Least Material Condition (LMC)Figure 10 shows an example of an engineering drawing using LMC in the position-tolerance control. Thedrawing is very similar to Figure 4, except m is replaced with l in the feature control frame of the positiontolerances. According to Foster7, a maximum bonus position tolerance of 2T1 can be added to φP1 when theHole 1 is made at φ (D1 -T1), and a maximum bonus position tolerance of 2T2 can be added to φP2 when theHole 2 is made at φ (D2 – T2). No bonus position tolerance will be allowed when the holes are made at φ(D1 T1) and φ (D2 T2) respectively.Page 14.301.7
Figure 10: Position Tolerance with LMC.5.1. Formulas for X max – LMCThe extreme condition of the two holes for X max is shown in Figure 11. Similar to the Session 3.1, it isbased on the following conditions:a. The two holes are at their minimum sizes: φ (D1-T1) and φ (D2-T2).b. The centers of the two holes are at their farthest locations: Points A and B in Figure 11.Figure 11: Extreme Position for X max – LMC.From Figure 11, X max can be easily expressed by the following equation:(5)5.2. Formulas for X min – LMCThe extreme position of the two holes for X min is shown in Figure 12, which is based on the followingconditions:c. The two holes are at their maximum sizes: φ (D1 T1) and φ (D2 T2). Because the holes are madeat their Least Material Condition (LMC), no bonus position tolerance for these two holes.d. The centers of the two holes are at their closest locations, which are located at Points A and B inFigure 12.Page 14.301.8Figure 12: Extreme Position for X min – LMC.
It can be seen that the drawing is exactly the same as Figure 6. Therefore the equation for X min is exactlythe same as Equation (2).(6)6. ExamplesFigure 13 shows an MMC design drawing. Based on the information, Equations (3) and (4) can be appliedto calculate X max and X min:(7)(8)Figure 13: An MMC design drawing.Figure 14 shows an LMC design drawing. Based on the information, Equations (5) and (6) can be appliedto calculate X max and X min:(9)(10)Page 14.301.9From the results shown in these two cases, both values in LMC are larger than the values in MMC.Therefore, it is better to use LMC in the design when using the same dimensioning to seek for larger X min.
Figure 14: An LMC design drawing.7. ConclusionsThis method presents an alternate approach to calculate the tolerance stacks using the direct-positionmethod. When compared all the derived equations, maximum value of Xmin can be obtained when thedrawing is specified with RFS or LMC. Minimum value of Xmin, which is undesirable, can be found in thedrawing when specified in MMC. From the teaching experience of GD&T, students prefer to use thisapproach because the formulas can be derived directly from the observing of the graphic expression. Inaddition, the approach can also be applied to derive the formulas for other geometric characteristic symbolssuch as parallelism, perpendicularity, concentricity, runout, and profile tolerances. Students also becomemore understanding of the total tolerance zone in the position control of GD&T when applying to threematerial conditions.References1.2.3.4.5.6.7.Scholz, F., “Tolerance Stack Analysis, Methods Research and Technology”, Boeing Informationand Support Service, 1995Krulikowski, A., “Fundamentals of Geometric Dimensioning and Tolerancing”, Delma, 1998Ngoi, B., “ Applying the coordinate Tolerance System to Tolerance Stack Analysis InvolvingPosition Tolerance”, International Journal Manufacturing Technology Vol. 15; PP. 404-408, 1999Ngoi, B., “Nexus method for stack analysis of geometric dimensioning and tolerancing (GDT)problems”, International Journal of Production Research, v 38, n 1, Jan 10, p 21-37, 2000Yen, D., “Graph-based set-up planning and tolerance decomposition for computer-aided fixturedesign”, International Journal of Production Research, v 38, n 1, Jan 10, p 21-37, 2000Lin, C., and Verma, A., “Clarifications of Rule 2 in Teaching Geometric Dimensioning andTolerancing”, ASEE Annual Conference, Session 1147, June 2007Foster, L., “Geo-Metrics III”, Prentice Hall, 1994Page 14.301.10
2.2. Least Material Condition (LMC) To indicate that a geometric tolerance is specified with LMC, a symbol l is added to either a geometric tolerance or a datum. Least Material Condition is p articularly defined as having the least solid volum e for a part. Therefore, f
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