Calculation Of Tolerance Stacks Using Direct Position ...

1 AC 2009-138: Calculation OF Tolerance Stacks Using . Direct - Position APPROACH IN GEOMETRIC DIMENSIONING AND. TOLERANCING. Cheng Lin, Old Dominion University Page American Society for Engineering Education, 2009. 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.

2 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 part1,2. 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 formulas3,4,5. 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 purpose2.

3 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. Maximum Material Condition (MMC).

4 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 16 shows a Page design drawing Using MMC Position Tolerance with Datum A as the center axis of the hole. From the table shown in this figure, when the diameter of a part is measured at , which is the MMC, there is no bonus Tolerance and the Position Tolerance remains at However, when the diameter is measured at , which is the LMC, the bonus Tolerance is equal to The total Position Tolerance in this case increases to MMC can be easily found in most GD&T design drawings.

5 Figure 1: A design drawing Using MMC Position Tolerance . 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 26 shows a design drawing Using LMC Position Tolerance with Datum A as the center axis of the hole. From the table shown in this figure, when the diameter of a part is measured at , which is the MMC, there is a bonus Tolerance and the Position Tolerance increases to However, when the diameter of a part is measured at , which is the LMC, there is no bonus Tolerance . The Position Tolerance remains at LMC is particularly applied to guarantee a larger minimum thickness than the same drawing Using MMC.

6 Figure 2: A design drawing Using LMC Position Tolerance . 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 36 shows a design drawing Using RFS Position Tolerance . Since there is no modifier added to the Position Tolerance , according to Rule 26, 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 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; D1 and D2 represent the diametrical sizes of the two holes respectively; T1. and T2 represent the bilateral size tolerances of the two holes respectively; P1 and P2 represent the RFS.

7 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 RFS2, there is no bonus Position Tolerance for these two holes. Figure 4: Position Tolerance with RFS. 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: (D1-T1) and (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 Figure 5: Extreme Position for X max RFS. From Figure 5, X max can be easily determined through the following equation: (1). 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: (D1+T1) and (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.

8 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 formula because Points A and B are at their closest positions. Page 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 MMC6, a maximum bonus Position Tolerance of 2T1 is added to P1 when the diameter of the Hole 1 is made at (D1+T1). Similarly, a maximum bonus Position Tolerance of 2T2. Figure 7: Position Tolerance with MMC. can be added to P2 when Hole 2 is made at (D2+T2). No bonus Position Tolerance will be allowed when the diameters of the holes are made at (D1-T1) and (D2-T2) respectively. Formulas for X max MMC.

9 Figure 8 shows the extreme condition for X max. Similar to the Session , it is based on the following conditions: a. The two holes are made at their minimum sizes: (D1-T1) and (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 (D1-T1) and . (D2-T2) respectively, Figure 8 is exactly the same as Figure 5. X max can be easily determined through the following equation: Page (3). 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: (D1+T1) and (D2+T2). Because the holes are made at their Least Material Condition (LMC), bonus tolerances of 2T1 and 2T2 are added to their respective original Position tolerances P1 and P2. b. The centers of the two holes are at their closest locations, which are indicated as Points A and B.

10 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 Foster7, a maximum bonus Position Tolerance of 2T1 can be added to P1 when the Hole 1 is made at (D1 -T1), and a maximum bonus Position Tolerance of 2T2 can be added to P2 when the Hole 2 is made at (D2 T2). No bonus Position Tolerance will be allowed when the holes are made at . (D1+T1) and (D2+T2) respectively. Page Figure 10: Position Tolerance with LMC. Formulas for X max LMC. The extreme condition of the two holes for X max is shown in Figure 11. Similar to the Session , it is based on the following conditions: a. The two holes are at their minimum sizes: (D1-T1) and (D2-T2).