Dimensioning and Tolerancing Handbook Episode 2 Part 8 ppsx

Multi-Dimensional Tolerance Analysis (Automated Method) 13-11 Block Cylinder Frame θ DRF DRF DRF -A- .02 A .04 .02 A .01 Figure 13-14 Applied geometric variations at contact points Component tolerances are set as a result of analyzing tolerance stackup in an assembly and determin- ing how each component dimension contributes to assembly variation. Processes and tooling are se- lected to meet the required component tolerances. Inspection and gaging equipment and procedures are also determined by the resulting component tolerances. Thus, we see that the performance requirements have a pervasive influence on the entire manufacturing enterprise. It is the designer’s task to transform each performance requirement into assembly tolerances and corresponding component tolerances. There are several assembly features that commonly arise in product design. A fairly comprehensive set can be developed by examining geometric dimensioning and tolerancing feature controls and forming a corresponding set for assemblies. Fig. 13-15 shows a basic set that can apply to a wide range of assemblies. Note that when applied to an assembly feature, parallelism applies to two surfaces on two different parts, while GD&T standards only control parallelism between two surfaces on the same part. The same can be said about the other assembly controls, with the exception of position. Position tolerance in GD&T relates assemblies of two parts, while the position tolerance in Fig. 13-15 could involve a whole chain of intermediate parts contributing variation to the position of mating features on the two end parts. An example of the application of assembly tolerance controls is the alignment requirements in a car door assembly. The gap between the edge of the door and the door frame must be uniform and flush (parallel in two planes). The door striker must line up with the door lock mechanism (position). Each assembly feature, such as a gap or parallelism, requires an open loop to describe the variation. You can have any number of open loops in an assembly tolerance model, one per critical feature. Closed loops, on the other hand, are limited to the number of loops required to locate all of the parts in the assembly. It is a unique number determined by the number of parts and joints in the assembly. L = J − P +1 where L is the required number of loops, J is the number of joints, and P is the number of parts. For the example problem: L = 4 − 3 + 1 = 2 which is the number we determined by inspection of the assembly graph. 13-12 Chapter Thirteen The example assembly has a specified gap tolerance between a cylindrical surface and a plane, as shown in Fig. 13-6. The vector loop describing the gap is shown in Fig. 13-16. It begins with vector g, on one side of the gap, proceeds from part-to-part, and ends at the top of the cylinder, on the opposite side of the gap. Note that vector a, at the DRF of the Frame, appears twice in the same loop in opposite directions. It is therefore redundant and both vectors must be eliminated. Vector r also appears twice in the cylinder; however, the two vectors are not in opposite directions, so they must both be included in the loop. Vector g, incidentally, is not a manufactured dimension. It is really a kinematic variable, which adjusts to locate the point on the gap opposite the highest point on the cylinder. It was given zero tolerance, because it does not contribute to the variation of the gap. The steps illustrated above describe a comprehensive system for creating assembly models for tolerance analysis. With just a few basic elements, a wide variety of assemblies may be represented. Next, we will illustrate the steps in performing a variational analysis of an assembly model. 13.5 Steps in Analyzing an Assembly Tolerance Model In a 2-D or 3-D assembly, component dimensions can contribute to assembly variation in more than one direction. The magnitude of the component contributions to the variation in a critical assembly feature is determined by the product of the process variation and the tolerance sensitivity, summed by worst case Figure 13-15 Assembly tolerance controls Concentricity & Runout A A u±du Assembly Gap Part 1 Part 2 A -A- Parallelism u ± du Assembly Length Assembly Angle φ ± d φ θ± d θ A Perpendicularity & Angularity -A- A B Position -A- Multi-Dimensional Tolerance Analysis (Automated Method) 13-13 Gap Block Frame θ DRF DRF DRF a f r Cylinder a r g U 2 Loop 3 Figure 13-16 Open loop describing critical assembly gap or Root Sum Squared (RSS). If the assembly is in production, actual process capability data may be used to predict assembly variation. If production has not yet begun, the process variation is approximated by substituting the specified tolerances for the dimensions, as described earlier. The tolerance sensitivities may be obtained numerically from an explicit assembly function, as illus- trated in Chapter 12. An alternative procedure will be demonstrated, which does not require the derivation of an explicit assembly function. It is a systematic method, which may be applied to any vector loop assembly model. Step 1. Generate assembly equations from vector loops The first step in an analysis is to generate the assembly equations from the vector loops. Three scalar equations describe each closed vector loop. They are derived by summing the vector components in the x and y directions, and summing the vector rotations as you trace the loop. For closed loops, the compo- nents sum to zero. For open, they sum to a nonzero gap or angle. The equations describing the stacked block assembly are shown below. For Closed Loops 1 and 2, h x , h y , and h θ are the sums of the x, y, and rotation components, respectively. See Eqs. (13.1) and (13.2). Both loops start at the lower left corner, with vector a. For Open Loop 3, only one scalar equation (Eq. (13.6)) is needed, since the gap has only a vertical component. Open loops start at one side of the gap and end at the opposite side. Closed Loop 1 h x = a cos(0) + U 2 cos(90) + R cos(90 + φ 3 ) + e cos(90 + φ 3 − 180) + U 3 cos(θ) + c cos(−90)+ b cos(−180) = 0 h y = a sin(0) + U 2 sin(90) + R sin(90 + φ 3 ) + e sin(90 + φ 3 − 180) + U 3 sin(θ) (13.1) + c sin(−90) + b sin(−180) = 0 h θ = 0 + 90 + φ 3 – 180 + 90 − θ − 90 – 90 +180 = 0 13-14 Chapter Thirteen Closed Loop 2 h x = a cos(0) + U 1 cos(90) + r cos(0) + r cos(− φ 1 ) + R cos(− φ 1 + 180) + e cos(− φ 1 − φ 2 ) + U 3 cos(θ) + c cos(– 90) + b cos(– 180) = 0 h y = a sin(0) + U 1 sin(90) + r sin(0) + r sin(− φ 1 ) + R sin(− φ 1 + 180) + e sin(− φ 1 − φ 2 ) + U 3 sin(θ) + c sin(– 90) + b sin(− 180) = 0 (13.2) h θ = 0 + 90 – 90 – φ 1 + 180 – φ 2 – 180 + 90 – θ – 90 – 90 + 180 = 0 Open Loop 3 Gap = r sin(– 90) + r sin(180) + U 1 sin(– 90) + f sin(90) + g sin(0) (13.3) The loop equations relate the assembly variables: U 1 , U 2 , U 3 , φ 1 , φ 2 , φ 3 , and Gap to the component dimensions: a, b, c, e, f, g, r, R, and θ. We are concerned with the effect of small changes in the component variables on the variation in the assembly variables. Note the uniformity of the equations. All h x components are in terms of the cosine of the angle the vector makes with the x-axis. All h y are in terms of the sine. In fact, just replace the cosines in the h x equation with sines to get the h y equation. The loop equations always have this form. This makes the equations very easy to derive. In a CAD implementation, equation generation may be automated. The h θ equations are the sum of relative rotations from one vector to the next as you proceed around the loop. Counterclockwise rotations are positive. Fig. 13-17 traces the relative rotations for Loop 1. A final rotation of 180 is added to bring the rotations to closure. While the arguments of the sines and cosines in the h x and h y equations represent the absolute angle from the x-axis, the angles are expressed as the sum of relative rotations up to that point in the loop. Using relative rotations is critical to the correct assembly model behavior. It allows rotational variations to propagate correctly through the assembly. −θ φ 3 R e c b U 3 U2 a +90° -90° -90° +90° x-axis -180° +180° hθ = 0 + 90 + φ 3 – 180 + 90 – θ – 90 – 90 +180 = 0 Relative rotations Loop 1 Figure 13-17 Relative rotations for Loop 1 Multi-Dimensional Tolerance Analysis (Automated Method) 13-15 A shortcut was used for the arguments for vectors U 2 , c, and b. The sum of relative rotations was replaced with their known absolute directions. The sum of relative angles for U 2 is (− θ 1 − θ 2 + 90), but it must align with the angled plane of the frame (θ ). Similarly, vectors b and c will always be vertical and horizontal, respectively, regardless of the preceding rotational variations in the loop. Replacing the angles for U, C, and b is equivalent to solving the h θ equation for θ and substituting in the arguments to eliminate some of the angle variables. If you try it both ways, you will see that you get the same results for the predicted variations. The results are also independent of the starting point of the loop. We could have started with any vector in the loop. Step 2. Calculate derivatives and form matrix equations The loop equations are nonlinear and implicit. They contain products and trigonometric functions of the variables. To solve for the assembly variables in this system of equations would require a nonlinear equation solver. Fortunately, we are only interested in the change in assembly variables for small changes in the components. This is readily accomplished by linearizing the equations by a first-order Taylor’s series expansion. Eq. (13.4) shows the linearized equations for Loop 1. 3 3 2 2 1 1 3 3 2 2 1 1 3 3 2 2 1 1 3 3 2 2 1 1 3 3 2 2 1 1 3 3 2 2 1 1 U U h U U h U U hhhh h R R h r r h e e h c c h b b h a a h h U U h U U h U U hhhh h R R h r r h e e h c c h b b h a a h h U U h U U h U U hhhh h R R h r r h e e h c c h b b h a a h h zzzzzz zzzzzzz z yyyyyy yyyyyyy y xxxxxx xxxxxxx x δδδδφ φ δφ φ δφ φ δθ θ δδδδδδδ δδδδφ φ δφ φ δφ φ δθ θ δδδδδδδ δδδδφ φ δφ φ δφ φ δθ θ δδδδδδδ ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ = (13.4) where δa represents a small change in dimension a, and so on. Note that the terms have been rearranged, grouping the component variables a, b, c, e, r, R, and θ together and assembly variables U 1 , U 2 , U 3 , φ 1 , φ 2 , and φ 3 together. The Loop 2 and Loop 3 equations may be expressed similarly. Performing the partial differentiation of the respective h x , h y , and h θ equations yields the coefficients of the linear system of equations. The partials are easy to perform because there are only sines and cosines to deal with. Eq. (13.5) shows the partials of the Loop 1 h x equation. 13-16 Chapter Thirteen Component Variables Assembly Variables (??U ? h )f( R h r h )f( e h )( c h )( b h )( a h x x x x x x x sin 90cos 0 270cos 90cos 180cos 0cos 3 3 3 −= ∂ ∂ += ∂ ∂ = ∂ ∂ += ∂ ∂ −= ∂ ∂ −= ∂ ∂ = ∂ ∂ )?( U h )( U h U h )f(e)f(R f h f h f h x x x x x x cos 90cos 0 270sin90sin 0 0 3 2 1 33 3 2 1 = ∂ ∂ = ∂ ∂ = ∂ ∂ +−+−= ∂ ∂ = ∂ ∂ = ∂ ∂ (13.5) Each partial is evaluated at the nominal value of all dimensions. The nominal component dimensions are known from the engineering drawings or CAD model. The nominal assembly values may be obtained by querying the CAD model. The partial derivatives above are not the tolerance sensitivities we seek, but they can be used to obtain them. Step 3. Solve for assembly tolerance sensitivities The linearized loop equations may be written in matrix form and solved for the tolerance sensitivities by matrix algebra. The six closed loop scalar equations can be expressed in matrix form as follows: [A]{δX} + [B]{δU} = {0} where: [A] is the matrix of partial derivatives with respect to the component variables, [B] is the matrix of partial derivatives with respect to the assembly variables, {δX} is the vector of small variations in the component dimensions, and {δU} is the vector of corresponding closed loop assembly variations. We can solve for the closed loop assembly variations in terms of the component variations by matrix algebra: {δU} = −[B −1 A]{δX} (13.6) The matrix [B -1 A] is the matrix of tolerance sensitivities for the closed loop assembly variables. Performing the inverse of the matrix [B] and multiplying [B -1 A] may be carried out using a spreadsheet or other math utility program on a desktop computer or programmable calculator. Multi-Dimensional Tolerance Analysis (Automated Method) 13-17 For the example assembly, the resulting matrices and vectors for the closed loop solution are: { }                       = δθ δ δ δ δ δ δ δ R r e c b a X { }                       = 3 2 1 3 2 1 δφ δφ δφ δ δ δ δ U U U U [ ]                                   ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ = ? h R h r h e h c h b h a h ? h R h r h e h c h b h a h ? h R h r h e h c h b h a h ? h R h r h e h c h b h a h ? h R h r h e h c h b h a h ? h R h r h e h c h b h a h ??????? yyyyyyy xxxxxxx ??????? yyyyyyy xxxxxxx A                     − +−−−−− −+−−+−−− − ++− −++− = 1000000 )cos()180sin()sin()sin(100 )sin()180cos()cos(1)cos(011 1000000 )cos()90sin(0)270sin(100 )sin()90cos(0)270cos(011 31121 31121 333 333 θφφφφ θφφφφ θφφ θφφ U U U U                     − −−− −−− − −− −−− = 1000000 6144.156907.6907.9563.100 7738.47232.7232.12924.011 1000000 6144.159563.09563.100 7738.42924.02924.011 13-18 Chapter Thirteen [ ]                                       ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ = 321321 321321 321321 321321 321321 321321 f h f h f h U h U h U h f h f h f h U h U h U h f h f h f h U h U h U h f h f h f h U h U h U h f h f h f h U h U h U h f h f h f h U h U h U h ?????? yyyyyy xxxxxx ?????? yyyyyy xxxxxx B ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )                                   −− −−−           −−− −− −− −−           −− − −       +++ +−+− = 011000 0cos cos 180cos cos )sin(0)90sin( 0sin sin 180sin sin )cos(0)90cos( 100000 00)sin()90sin(0 )270cos(90cos )270sin(90sin 00)cos()90cos(0 21 21 1 1 21 21 1 1 33 33 φφ φφ φ φ θ φφ φφ φ φ θ θ φφ φφ θ e e R r e e R r eR eR                     −− − −− = 011000 00804.166144.529237.01 05968.528764.3195631.00 100000 3856.40029237.10 3446.140095631.00 [ ]                     −− −− − − −− = − 000100 5384.100483.6923.00483. 5384.200483.6923.00483. 0001500457.1 000013057. 9901.3810470.16337.1007413. B 1 Multi-Dimensional Tolerance Analysis (Automated Method) 13-19 {δU} = -[B -1 A]{δX} (13.7)                                           − −− −−− −−−− −− =                       δθ δ δ δ δ δ δ δφ δφ δφ δ δ δ R r e c b a U U U 1000000 .8461.0208.08320000 1.8461.0208.08320000 10.0080.30570.305701.04571.0457 17.07391.045711.04571.3057.3057 11.28251.23112.4948851.04571.3057.3057 3 2 1 3 2 1 Estimates for variation of the assembly performance requirements are obtained by linearizing the open loop equations by a procedure similar to the closed loop equations. In general, there will be a system of nonlinear scalar equations which may be linearized by Taylor’s series expansion. Grouping terms as before, we can express the linearized equations in matrix form: {δV} = [C]{δX} + [E]{δU} (13.8) where {δV} is the vector of variations in the assembly performance requirements, [C] is the matrix of partial derivatives with respect to the component variables, [E] is the matrix of partial derivatives with respect to the assembly variables, {δX} is the vector of small variations in the component dimensions, and {δU} is the vector of corresponding closed loop assembly variations. We can solve for the open loop assembly variations in terms of the component variations by matrix algebra, by substituting the results of the closed loop solution. Substituting for {δU}: {δV} = [C]{δX} − [E][B −1 A]{δX} = [C−Ε B −1 A]{δX} The matrix [C−E B -1 A] is the matrix of tolerance sensitivities for the open loop assembly variables. The B -1 A terms come from the closed loop constraints on the assembly. The B -1 A terms represent the effect of small internal kinematic adjustments occurring at assembly time in response to dimensional variations. The internal adjustments affect the {δV} as well as the {δU}. It is important to note that you cannot simply solve for the values of {δU} in Eq. (13.6) and substitute them directly into Eq. (13.8), as though {δU} were just another component variation. If you do, you are treating {δU} as though it is independent of {δX}. But {δU} depends on {δX} through the closed loop constraints. You must evaluate the full matrix [C−E B -1 A] to obtain the tolerance sensitivities. Allowing the B -1 A terms to interact with C and E is necessary to determine the effect of the kinematic adjustments on {δV}. Treating them separately is similar to taking the absolute value of each term, then summing for Worst Case, rather than summing like terms before taking the absolute value. The same is true for RSS analysis. It is similar to squaring each term, then summing, rather than summing like terms before squaring. For the example assembly, the equation for {δV} reduces to a single scalar equation for the Gap variable. [...]... |S 18| δf + |S19|δg = |– 30573| 0.3 + |.30573| 0.3 + |– 1| 0.3 +|− 1.04569| 0.3 + |– 3.4949| 0.1+ |1 .23 11| 0.3 + | −11 . 28 25| 0.01745 + |1| 0.5 + |0| 0 = ± 2. 2 129 mm RSS: δGap = [(S 11δa )2 + (S 12 b )2 + (S 13δc )2 + (S 14δe )2 + (S 15δr )2 + (S 16δR )2 + (S 17δθ) + (S 18 f )2 + (S 19δg )2] 5 = [(−.30573 ⋅ 0.3 )2 + (.30573 ⋅ 0.3 )2 +(− 1⋅ 0.3 )2 + (− 1.04569 ⋅ 0.3 )2 + (−3.4949 ⋅ 0.1 )2 + (1 .23 11 ⋅ 0.3 )2 + (− 11 . 28 25... Inserting values into the equation yields: )( 0 720 2 ) 1 / ( 1.46 82 3 ) T (1 4 389 9) / (1.46 82 3 ) + B )( 15997 )   1 / (1 46537) 1 / (1 46 82 3 )  ( 46537 )( 125 76 )  (1 4 389 9) / (1.46537) +  (.46 82 3 )( 0 720 2 )  (1 4 389 9) / (1.46 82 3 )  TB  TB ( 4 389 9 )(.15997 )  ( 4 389 9 )( 15997 )       (.46 82 3 015 − 0015 − 0 025 − 0 025 = T B +   ( 4 389 9 The values of k and B for each nominal dimension were obtained... 0.00015 0.00 02 0.0003 0.0005 0.00 08 0.00 12 0.0 02 0.003 0.005 0.600 0.999 0.00015 0.00 025 0.0004 0.0006 0.001 0.0015 0.0 025 0.004 0.006 1.000 1.499 0.00 02 0.0003 0.0005 0.00 08 0.00 12 0.0 02 0.003 0.005 0.0 08 1.500 2. 799 0.00 025 0.0004 0.0006 0.001 0.0015 0.0 025 0.004 0.006 0.010 2. 80 0 4.499 0.0003 0.0005 0.00 08 0.00 12 0.0 02 0.003 0.005 0.0 08 0.0 12 4.500 7.799 0.0004 0.0006 0.001 0.0015 0.0 025 0.004 0.006... 0.01745 )2 + (1 ⋅ 0.5 )2 + (0 ⋅ 0 )2 ].5 = ± 0 .86 75 mm By forming similar expressions, we may obtain estimates for all the assembly variables (Table 13-1) Table 13-1 Estimated variation in open and closed loop assembly features Assembly Variable U1 U2 U3 φ1 2 φ3 Gap Mean or Nominal 59.0 026 mm 41.47 08 mm 16. 327 9 mm 43. 683 8° 29 .31 62 17.0000° 5.9974 mm WC ±δU 1.6 129 mm 1.5 089 mm 0. 985 5 mm 2. 68 1. 68 1.00° 2. 2 129 ... 1.00 0 720 2 46 82 3 0 02 001736 00637 E 1.00 125 76 46537 006 0 024 98 007 92 F 1.00 0 720 2 46 82 3 0 02 001736 00637 * * 0 025 * 0 025 * 0 025 * 024 5(WC) 0150(WC) 0150(RSS) G Assembly Variation 0111(RSS) Assembly Cost $11.07 $8. 06 Acceptance Fraction $9.34 1.000 9973 “True Cost” $11.07 $8. 08 *Fixed tolerances Min Cost Allocation Results Original Tol $9.34 B D Min Cost: WC E $11.07 F Min Cost: RSS $8. 06 0.000 0.0 02 0.004... coupling of b and c with the other variables The calculated variations are shown in Table 13-5 Table 13-5 Variation results for modified nominal geometry Assembly Variable U1 U2 U3 φ1 2 φ3 Gap Mean or Nominal 59.0453 mm 41.5135 mm 26 . 784 8 mm 43. 683 8° 29 .31 62 17° 5.9547 mm WC ±δ U 1.6497 mm 1.9 088 mm 0.9909 mm 2. 80 ° 1 .80 ° 1.00° 2. 1497 mm RSS ± δU 0.7659 mm 0 .84 01 mm 0.49 08 mm 1.97° 1. 08 1.00° 0 .89 80 mm The... 12| δb + |S 13|δc + |S14|δe + |S15|δr + |S16|δR + |S17|δθ = |.3057| 0.3 + |−.3057| 0.3 + |1| 0.3 + |1.0457| 0.3 + |2. 4949| 0.1 + |−1 .23 11| 0.3 + |11 . 28 25| 0.01745 = ± 1.6 129 mm Multi-Dimensional Tolerance Analysis (Automated Method) 13 -21 RSS: δU1 = [(S 11δa )2 + (S 12 b )2 + (S 13δc )2 + (S 14δe )2 + (S 15δr )2 + (S 16δR )2 + (S 17δθ )2] .5 = [(.3057 ⋅ 0.3 )2 + (−.3057⋅ 0.3 )2 + (1 ⋅ 0.3 )2 + (1.0457 ⋅ 0.3 )2. .. the individual expressions to obtain the corresponding values of TD, TE, and TF, and the predicted cost T B = 0 025  ( 46 82 3 )(.0 720 2 )  TD = TF =    (.4 389 9 )(.15997 )   ( 46537 )( 125 76 TE =   ( 4 389 9 )( 15997  1 / (1 46 82 3 ) ( 1.4 389 9) / ( 1.46 82 3 ) = 0017 TB )  1 / (1 46537) ( 1.4 389 9) / ( 1.46537)  TB )  = 0 025 C = AB + BB (TB )kB + AD + BD (TD )k D + AE + BE (TE ) kE + AF + BF (TF... 0015* * * 0 08 003 0 12 B 8. 000 Process Tolerance Limits Min Tol Max Tol C 5093 0 025 * * * D 400 0 02 0005 00 12 E 7.711 006 0 025 010 F 400 0 02 0005 00 12 0 025 * * * G 5093 * Fixed tolerances The average clearance is the vector sum of the average part dimensions in the loop: Required Clearance = 020 ± 015 Average Clearance =–A+B–C+D–E+F–G = – 0505 + 8. 000 – 5093 + 400 – 7.711 + 400 – 5093 = 020 The worst... (∑ (A λ= k i Bi 2Ti kj j (k i + 2 ) + Bj / Tj ))+ λ ∂∂T (∑T 2 j ) 2 − T asm = 0 (i=1,…n) (i=1,…n) i (i=1,…n) Eliminating λ by expressing it in terms of T1 (arbitrarily selected): k B  Ti =  i i  k B   1 1 1 / ( ki + 2 ) (k1 + 2 ) / ( ki + 2 ) T1 (14.1) Substituting for each of the Ti in the assembly tolerance sum: 2 / (ki +2 )  k i Bi  ( )( )   + T 12 k1 +2 / ki + 2 (14 .2) k B   1 1 . 0.7659 mm U 2 41.5135 mm 1.9 088 mm 0 .84 01 mm U 3 26 . 784 8 mm 0.9909 mm 0.49 08 mm φ 1 43. 683 8° 2. 80 ° 1.97° φ 2 29.31 62 1 .80 ° 1. 08 φ 3 17° 1.00° 1.00° Gap 5.9547 mm 2. 1497 mm 0 .89 80 mm 13 -26 Chapter. (13.7)                                           − −− −−− −−−− −− =                       δθ δ δ δ δ δ δ δφ δφ δφ δ δ δ R r e c b a U U U 1000000 .84 61. 020 8. 08 320 000 1 .84 61. 020 8. 08 320 000 10.0 080 .30570.305701.04571.0457 17.07391.045711.04571.3057.3057 11 . 28 251 .23 1 12. 49 488 51.04571.3057.3057 3 2 1 3 2 1 Estimates for variation. or Nominal ±δ U ±δ U U 1 59.0 026 mm 1.6 129 mm 0.6653 mm U 2 41.47 08 mm 1.5 089 mm 0.6344 mm U 3 16. 327 9 mm 0. 985 5 mm 0.4941 mm φ 1 43. 683 8° 2. 68 1.94° φ 2 29.31 62 1. 68 1.04° φ 3 17.0000° 1.00° 1.00° Gap 5.9974 mm 2. 2 129