AC 2009-138: CALCULATION OF TOLERANCE STACKS USING DIRECT-POSITION APPROACH IN GEOMETRIC DIMENSIONING AND TOLERANCING Cheng Lin, Old Dominion University American Society for Engineering Education, 2009 Page 14.301.1
Calculation of Tolerance Stacks Using Direct-Position Approach in Geometric Dimensioning and Tolerancing Abstract Formulas 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 directly from the extreme positions of the holes specified in an engineering drawing. When compared to other approaches for tolerance stacks, this method can be applied to all three material conditions (Maximum Material Condition, Least Material Condition, and Regardless of Feature Size.) and is easier for students to learn and remember the formulas. A graphical demonstration using position control on two holes in an engineering drawing is applied to explain the approach. 1. Introduction Tolerance stacks are used to describe the problem-solving process in calculating the effects of the accumulated variation that is allowed by specified dimensions and tolerances, which are typically specified on an engineering drawing. Arithmetic tolerance stacks use the worst-case maximum or minimum values of dimensions and tolerances to calculate the maximum and minimum distances between holes or between a hole and the edge of a part 1,2. The application is particularly important in the design stage to maintain a specified minimum solid distances in the part. In addition, stack analysis enables parts to be made precise enough to be assembled interchangeably with the largest possible tolerances permitted by part specification. Several methods are proposed to calculate tolerance stacks using the approaches of graphs, charts, tables, and formulas 3,4,5. However, they are all very complicated for students to learn. A graphical approach called gage method, using the concept of functional gages, seems to provide an effective way for this purpose 2. However, as the functional gages can only be applied to the Maximum Material Condition (MMC), this method can not be applied to other two material conditions: Least Material Condition (LMC) and Regardless of Feature of Size (RFS). In this paper, the direct-position method is proposed to derive the formulas for tolerance stacks. The method not only is easier for students or designers to understand and remember, but can be applied to three material conditions, which are explained in the following session. A graphical example using position control on three material conditions is applied to demonstrate the approach. 2. Three Material Conditions Based on the design and manufacturing needs, geometric tolerances can be specified with different material conditions, which include Maximum Material Condition (MMC), Least Material Condition (LMC), and Regardless of Feature Size (RFS). Characteristics of each material condition are described in the following paragraphs. 2.1. Maximum Material Condition (MMC) To indicate that a geometric tolerance is specified with MMC, a symbol m is added to either a geometric characteristic or a datum. Maximum Material Condition is particularly defined as having the maximum solid volume for a part. Therefore, for internal parts (such as holes or grooves, etc.), MMC is at its minimum feature of size (FOS). For external parts (such as pins or studs, etc.), MMC is at its maximum feature of size. When a geometric characteristic is specified with MMC, the geometric tolerance may have a bonus tolerance when its FOS is approaching to its Least Material Condition (LMC). Figure 1 6 shows a design drawing using MMC Position Tolerance with Datum A as the center axis of the φ0.8 hole. From the table shown in this figure, when the diameter of a part is measured at 1.02, which is the MMC, there is no Page 14.301.2
bonus tolerance and the Position Tolerance remains at 0.05. However, when the diameter is measured at 0.98, which is the LMC, the bonus tolerance is equal to 0.04. The total Position Tolerance in this case increases 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 geometric tolerance or a datum. Least Material Condition is particularly defined as having the least solid volume for a part. Therefore, for internal parts, LMC is at its maximum feature of size. For external parts, LMC is at its minimum feature of size. When a geometric tolerance is specified with LMC, the geometric tolerance may have a bonus tolerance when its FOS is approaching to its MMC. Figure 2 6 shows a design drawing using LMC Position Tolerance with Datum A as the center axis of the φ0.8 hole. From the table shown in this figure, when the diameter of a part is measured at 1.02, which is the MMC, there is a bonus tolerance and the Position Tolerance increases to 0.09. However, when the diameter of a part is measured at 0.98, which is the LMC, there is no bonus tolerance. The Position Tolerance remains at 0.05. LMC is particularly applied 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 concept of RFS has been used prior to the introduction of MMC and LMC principles. Figure 3 6 shows a design drawing using RFS position tolerance. Since there is no modifier added to the position tolerance, according to Rule 2 6, the position tolerance is RFS. From the table shown in this figure, the position tolerance remains the 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, A represents a datum feature; φd 1 and φd 2 represent the diametrical sizes of the two holes respectively; T 1 and T 2 represent the bilateral size tolerances of the two holes respectively; φp 1 and φp 2 represent the RFS position tolerances of the two holes respectively; L is the actual location between these two holes and is expressed in basic dimension 2,7 ; X represents the solid distance between these two holes. According to the definition of RFS 2, there is no bonus position tolerance for these two holes. Figure 4: Position Tolerance with RFS. 3.1. Formulas for X max RFS Figure 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 (D 1 -T 1 ) and φ (D 2 -T 2 ) 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 RFS Figure 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 (D 1 +T 1 ) and φ (D 2 +T 2 ) 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: Equation (2) is similar to Equation (1) except using minus signs on P 2 /2 and P 1 /2 in the formula because Points A and B are at their closest positions. (2) 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. The drawing is very similar to Figure 6, except with m added in the feature control frame of the position tolerances. According to MMC 6, a maximum bonus position tolerance of 2T 1 is added to φp 1 when the diameter of the Hole 1 is made at φ (D 1 +T 1 ). Similarly, a maximum bonus position tolerance of 2T 2 Figure 7: Position Tolerance with MMC. can be added to φp 2 when Hole 2 is made at φ (D 2 +T 2 ). No bonus position tolerance will be allowed when the diameters of the holes are made at φ (D 1 -T 1 ) and φ (D 2 -T 2 ) respectively. 4.1. Formulas for X max MMC Figure 8 shows the extreme condition for X max. Similar to the Session 3.1, it is based on the following conditions: a. The two holes are made at their minimum sizes: φ (D 1 -T 1 ) and φ (D 2 -T 2 ). 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 (D 1 -T 1 ) and φ (D 2 -T 2 ) respectively, Figure 8 is exactly the same as Figure 5. X max can be easily determined through the following equation: Page 14.301.6
(3) 4.2. Formulas for X min MMC Figure 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 (D 1 +T 1 ) and φ (D 2 +T 2 ). Because the holes are made at their Least Material Condition (LMC), bonus tolerances of φ2t 1 and φ2t 2 are added to their respective original position tolerances φp 1 and φp 2. 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. The drawing is very similar to Figure 4, except m is replaced with l in the feature control frame of the position tolerances. According to Foster 7, a maximum bonus position tolerance of 2T 1 can be added to φp 1 when the Hole 1 is made at φ (D 1 -T 1 ), and a maximum bonus position tolerance of 2T 2 can be added to φp 2 when the Hole 2 is made at φ (D 2 T 2 ). No bonus position tolerance will be allowed when the holes are made at φ (D 1 +T 1 ) and φ (D 2 +T 2 ) respectively. Page 14.301.7
Figure 10: Position Tolerance with LMC. 5.1. Formulas for X max LMC The extreme condition of the two holes for X max is shown in Figure 11. Similar to the Session 3.1, it is based on the following conditions: a. The two holes are at their minimum sizes: φ (D 1 -T 1 ) and φ (D 2 -T 2 ). 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 LMC The extreme position of the two holes for X min is shown in Figure 12, which is based on the following conditions: c. The two holes are at their maximum sizes: φ (D 1 +T 1 ) and φ (D 2 +T 2 ). Because the holes are made at 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 in Figure 12. Figure 12: Extreme Position for X min LMC. Page 14.301.8
It can be seen that the drawing is exactly the same as Figure 6. Therefore the equation for X min is exactly the same as Equation (2). (6) 6. Examples Figure 13 shows an MMC design drawing. Based on the information, Equations (3) and (4) can be applied to 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 applied to calculate X max and X min : (9) (10) From 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. Page 14.301.9
Figure 14: An LMC design drawing. 7. Conclusions This method presents an alternate approach to calculate the tolerance stacks using the direct-position method. When compared all the derived equations, maximum value of X min can be obtained when the drawing is specified with RFS or LMC. Minimum value of X min, which is undesirable, can be found in the drawing when specified in MMC. From the teaching experience of GD&T, students prefer to use this approach because the formulas can be derived directly from the observing of the graphic expression. In addition, the approach can also be applied to derive the formulas for other geometric characteristic symbols such as parallelism, perpendicularity, concentricity, runout, and profile tolerances. Students also become more understanding of the total tolerance zone in the position control of GD&T when applying to three material conditions. References 1. Scholz, F., Tolerance Stack Analysis, Methods Research and Technology, Boeing Information and Support Service, 1995 2. Krulikowski, A., Fundamentals of Geometric Dimensioning and Tolerancing, Delma, 1998 3. Ngoi, B., Applying the coordinate Tolerance System to Tolerance Stack Analysis Involving Position Tolerance, International Journal Manufacturing Technology Vol. 15; PP. 404-408, 1999 4. Ngoi, 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, 2000 5. Yen, D., Graph-based set-up planning and tolerance decomposition for computer-aided fixture design, International Journal of Production Research, v 38, n 1, Jan 10, p 21-37, 2000 6. Lin, C., and Verma, A., Clarifications of Rule 2 in Teaching Geometric Dimensioning and Tolerancing, ASEE Annual Conference, Session 1147, June 2007 7. Foster, L., Geo-Metrics III, Prentice Hall, 1994 Page 14.301.10