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Figure 3.1: Schematic of the stack blocks and corresponding dimension variables.

3.0 Problem Description

The stack blocks problem is an imaginary assembly used to teach tolerance analysis techniques. It consists of three parts: a base (ground), a sliding block, and a cylinder. The block slides on the base until it contacts the left side wall. The cylinder also contacts the left side wall.

Table 3.1: Manufactured Variables (Independent).

 Variable Name Basic Size Initial Tolerance (+/-) Cylinder Radius B, C 6.620 mm .200 mm Step Width F 3.905 mm .125 mm Step Height G 4.060 mm .150 mm Block Thickness I 6.805 mm .075 mm Step Location J 28.125 mm .350 mm Step Height K 10.675 mm .125 mm

3.1 Design Requirements

Table 3.2: Assembly Variables and Specification Limits.

 Variable Name Basic Size Upper Spec. Limit (USL) Lower Spec. Limit (LSL) Cylin./Ground Contact A 18.7182 mm 19.018 mm 18.418 mm Angle [[phi]]1 74.724[[ring]] -- -- Cylind./Block Contact D 8.6705 mm -- -- Angle [[phi]]2 164.724[[ring]] -- -- Block/Ground Contact E 10.0477 mm -- -- Angle [[phi]]3 105.276[[ring]] -- -- Block/Ground Contact H 2.1894 mm -- -- Angle [[phi]]4 105.276[[ring]] -- -- Block/Ground Contact L 27.2965 mm -- --

Remarks>> Three closed loops are needed to solve for all nine dependent variables.

3.2 Modeling Considerations

• Cylinder radii B and C are not independent dimensions. They must be equivalenced for analysis.

3.3 Design Goal

The goal for this problem is to use the Six-Sigma analysis model with weight-factor allocation to adjust the non-fixed component tolerances until the +/-3[[sigma]] calculated assembly variation corresponds to the specified assembly limits.

3.4 Part Names and DRFs

Figure 3.2: Diagram showing the location of the part DRFs.

Remarks>> Datum reference frames (DRFs) should correspond to locations on the parts to which the component dimensions are referenced.

3.5 Kinematic Joints

Five joints are required to model the stack blocks.

Figure 3.3: Kinematic joint diagram.

Remarks>> Cylindrical slider and edge slider joints both have a rotational and a translational degree of freedom, making ten total dependent variables. However, since joint 1 and joint 2 share the same cylinder center, the rotational degrees of freedom are reduced by one, leaving nine total degrees of freedom. This allows us to solve for this assembly's dependent variables with only three loops.

Table 3.3: Kinematic Joints of the Stack blocks.

 Joint Number Part One Part Two Joint Type 1 Ground Cylinder Cylindrical Slider 2 Cylinder Block Cylindrical Slider 3 Block Ground Edge Slider 4 Ground Block Edge Slider 5 Block Ground Edge Slider

3.6 Network Diagram, Vector Loops, and Design Specifications

The network diagram in figure 3.4 shows that three loops are necessary to describe the stack blocks assembly. A design specification has been applied to the dependent length A (vertical distance from the GROUND DRF to the cylinder center).

Figure 3.4: Network diagram and loop diagram for the stack blocks assembly.

Remarks>> In order to generate loops identical to the ones shown, first manually create loop 3. The remaining two loops can then be created using the automatic loop generation option.

Loop 1 contains a pair of redundant vectors. Redundant vectors occur because each loop that passes though a part must pass through that part's DRF. If CATS does not automatically make those vectors equivalent, it must be done manually or the analysis results will be invalid.

3.7 Geometric Tolerances

Seven geometric tolerances have been applied to the stack blocks assembly.

Figure 3.5: Geometric tolerance diagram.

Remarks>> In general, each joint will have one or two geometric tolerances applied to it. Joint 1 has two geometric tolerances because the roundness of the cylinder and the flatness of the ground both affect the assembly. Joints 4 and 5habe a single geometric tolerance applied because only the flatness of the block affects the assembly variations.

3.8 Sensitivity Matrices

Constraint Sensitivities

A Matrix

 B_C F G I J K X1 1.2635 0.0000 0.0000 0.0000 0.0000 0.0000 Y1 -.96467 0.0000 0.0000 0.0000 0.0000 0.0000 1 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 X2 0.0000 1.0000 0.0000 -.26347 0.0000 0.0000 Y2 0.0000 0.0000 1.0000 .96467 0.0000 0.0000 2 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 X3 0.0000 -1.0000 0.0000 0.0000 1.0000 0.0000 Y3 0.0000 0.0000 -1.0000 0.0000 0.0000 1.0000 3 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

B Matrix

 A D E H L q1 q2 q3 q4 X1 0.0000 -.96467 0.0000 0.0000 0.0000 -18.718 10.048 0.0000 0.0000 Y1 1.0000 -.26347 -1.0000 0.0000 0.0000 6.6200 0.0000 0.0000 0.0000 1 0.0000 0.0000 0.0000 0.0000 0.0000 -1.0000 1.0000 0.0000 0.0000 X2 0.0000 0.0000 0.0000 -.96467 0.0000 0.0000 10.048 4.0600 0.0000 Y2 0.0000 0.0000 -1.0000 -.26347 0.0000 0.0000 0.0000 -3.9050 0.0000 2 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 1.0000 0.0000 X3 0.0000 0.0000 0.0000 .96467 -.96467 0.0000 0.0000 -4.0600 10.675 Y3 0.0000 0.0000 0.0000 .26347 -.26347 0.0000 0.0000 3.9050 -28.125 3 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -1.0000 1.0000

F Matrix

 1 2 3 4 5 6 7 X1 1 1 0.26347 0.26347 -1 0 0 Y1 0 0 -0.96467 -0.96467 0 0 0 1 0 0 0 0 0 0 0 X2 0 0 0 0 -1 -0.26347 0 Y2 0 0 0 0 0 0.96467 0 2 0 0 0 0 0 0 0 X3 0 0 0 0 0 0.26347 -0.26347 Y3 0 0 0 0 0 -0.96467 0.96467 3 0 0 0 0 0 0 0

Tolerance Sensitivities

-B-1A Matrix

 B_C F G I J K A 1.3097 -.20262 .74186 1.0366 -.07050 .25814 D 1.3097 .09432 -.34534 -3.536E-17 -.09432 .34534 E 0.0000 -.29694 1.0872 1.0366 .02382 -.08720 H 0.0000 .97149 .23849 -.27312 .06514 -.23849 L 0.0000 .00682 -.02498 -.27312 1.0298 .02498 q1 0.0000 -.01049 .03842 3.934E-18 .01049 -.03842 q2 0.0000 -.01049 .03842 3.934E-18 .01049 -.03842 q3 0.0000 .01049 -.03842 -3.934E-18 -.01049 .03842 q4 0.0000 .01049 -.03842 -3.934E-18 -.01049 .03842

-B-1F Matrix

 1 2 3 4 5 6 7 A 0.27312 0.27312 1.0366 1.0366 -3.89e-16 0.76903 0.26759 D 1.0366 1.0366 0.27312 0.27312 -1.0366 -0.35799 0.35799 E 0 0 0 0 0.27312 1.127 -0.0904 H 0 0 0 0 -1.0366 -0.0259 -0.24722 L 0 0 0 0 -1.0366 -0.0259 -0.24722 q1 0 0 0 0 1.49e-17 0.03983 -0.03983 q2 0 0 0 0 1.49e-17 0.03983 -0.03983 q3 0 0 0 0 -1.49e-17 -0.03983 0.03983 q4 0 0 0 0 -1.49e-17 -0.03983 0.03983

3.9 Predicted Assembly Variation

Table 3.4: Independent Variable Tolerances and Control Factors

 Dim.Name +/- Tol. Std.Dev. Cp Dk Cpk Sk Wt. Factor Tol. Basic Fixed B-C 0.2 mm .0667 1 0.25 0.75 0 1 1 No F 0.125 mm .0417 1 0.25 0.75 0 2 2 No G 0.15 mm .0500 1 0.25 0.75 0 2 2 No I 0.075 mm .0250 1 0.25 0.75 0 0 0 Yes J 0.35 mm .1167 1 0.25 0.75 0 4 4 No K 0.125 mm .0417 1 0.25 0.75 0 3 3 No

Table 3.5: Kinematic Assembly Variables (No Geometric Tolerances)

 Variable Name Degree of Freedom +/- Assembly Variation (ZASM = 3.000) Worst Case RSS Case 6-SIG Case A translation (mm) .53325 .29889 .39852 D translation (mm) .40172 .27275 .36367 E translation (mm) .29718 .18495 .24660 H translation (mm) .23030 .13362 .17816 L translation (mm) .38864 .36105 .48140 q1 rotation ([[ring]]) .89099 .48446 .64594 q2 rotation ([[ring]]) .89099 .48446 .64594 q3 rotation ([[ring]]) .89099 .48446 .64594 q4 rotation ([[ring]]) .89099 .48446 .64594

Table 3.6: Geometric Tolerances

 Feat. Joint Part Name Feature Type Tolerance Band Char. Length 1 1 Ground Flatness .080 mm N/A 2 1 Cylinder Circularity .020 mm N/A 3 2 Cylinder Circularity .020 mm N/A 4 2 Block Flatness .050 mm N/A 5 3 Ground Flatness .080 mm N/A 6 4 Block Flatness .050 mm N/A 7 5 Block Flatness .050 mm N/A

Table 3.7: Kinematic Assembly Variables (Geometric Tolerances Included)

 Variable Name Degree of Freedom +/- Assembly Variation (ZASM = 3.000) Worst Case RSS Case 6-SIG Case A translation (mm) .60910 .30109 .40018 D translation (mm) .52248 .27956 .36880 E translation (mm) .33854 .18741 .24845 H translation (mm) .27859 .14004 .18303 L translation (mm) .43693 .36347 .48322 q1 rotation ([[ring]]) 1.0051 .49113 .65096 q2 rotation ([[ring]]) 1.0051 .49113 .65096 q3 rotation ([[ring]]) 1.0051 .49113 .65096 q4 rotation ([[ring]]) 1.0051 .49113 .65096

Remarks>> As is apparent from the geometry of the stack blocks, all angular variations are identical.

Table 3.8: Six-Sigma Percent Contributions To A (No Geometric Tolerances)

 Variable Name Variance Six-Sigma B_C 1.355e-2 76.81 G 2.446e-3 13.86 I 1.194e-3 6.77 K 2.057e-4 1.17 other 2.470e-4 1.40

Remarks>> The roller radius is the dominant cause of variation in A.

Table 3.9: Six-Sigma Percent Rejects

 Spec. Name Spec. Type Nominal Dimension (+/-) Computed Variation With Geometric Tolerances Without Geometric Tolerances Dep. Length 18.718 .40018 Z Rej. Z Rej. ZASM = 3.000 USL 19.018 Upper Tail 2.25 12256 2.26 11962 (Rejects in PPM) LSL 18.418 -2.25 12256 -2.26 11962

Remarks>>The Z-scores are significantly below the specified 3[[sigma]] limits. The component tolerances must be tightened to meet the specifications.

3.10 Tolerance And Nominal Allocation

Weight Factor Tolerance Allocation

Weight Factor Tolerance Allocation adjusts dimension tolerances according to the user-assigned weight factors. The weight factors that are large compared to the others receive a greater portion of the unused variation when there is a positive variance pool (assembly variation is smaller than the specified assembly limits) and are reduced the least when there is a negative variance pool (assembly variation exceeds the specified assembly limits). The user should assign larger weight factors to the tolerances he wants to become (or remain) as large as possible.

Table 3.10: Six-Sigma Weight Factor Tolerance Allocation (No Geometric Tolerances).

 Assembly Specs. Nom. USL LSL +/- ZASM Dep. Length (mm) 18.7182 19.0182 18.4182 3.000
 Dimension Name Specified Values Allocated Values Nom. +/-Tol. Nom. +/-Tol. STDEV % Cont. B_C (mm) 6.6200 .20000 6.6200 .11293 .05019 43.21 F(mm) 3.9050 .12500 3.9050 .14116 .06274 1.62 G (mm) 4.0600 .15000 4.0600 .16939 .07529 31.19 I (mm) 6.8050 .07500 6.8050 .07500 .03333 11.94 * J (mm) 28.1250 .35000 28.1250 .79050 .35133 6.14 K (mm) 10.6750 .12500 10.6750 .21174 .09411 5.90 Assem. Total Nom. +/-Var. Nom. +/-Var. STDEV 100.00 A (mm) 18.7182 .39852 18.7182 .30000 .10000 Min./Max. 18.3197 19.1168 18.4182 19.0182 * Fixed Nom./Tol.
 Before Optimization After Optimization Rejects Z PPM Z PPM Upper Tail 2.26 11962.3 3.00 1350.0 Lower Tail -2.26 11962.3 -3.00 1350.0 Total Rejects 23924.7 Total Rejects 2699.9

Nominal Allocation

Nominal Allocation is used to estimate the dimension nominals that will center the assembly variable or match the upper or lower assembly variation with the upper or lower specification limit. The larger the weight factor assigned to a dimension, the more that nominal may be adjusted.

Table 3.11: Six-Sigma Nominal Allocation (No Geometric Tolerances).

Lower Specification Limit Justified.

 Assembly Specs. Nom. USL LSL +/- ZASM Dep. Length (mm) 18.7182 19.0182 18.4182 3.000
 Dimension Name Specified Values Allocated Values Nom. +/-Tol. Nom. +/-Tol. STDEV % Cont. B_C (mm) 6.6200 .20000 6.6263 .20000 .08889 76.81 F(mm) 3.9050 .12500 3.8240 .12500 .05556 0.72 G (mm) 4.0600 .15000 4.0821 .15000 .06667 13.86 I (mm) 6.8050 .07500 6.8050 .07500 .03333 6.77 * J (mm) 28.1250 .35000 27.6592 .35000 .15556 0.68 K (mm) 10.6750 .12500 10.7704 .12500 .05556 1.17 Assem. Total Nom. +/-Var. Nom. +/-Var. STDEV 100.00 A (mm) 18.7182 .39852 18.8168 .39852 .17712 Min./Max. 18.3192 19.1168 18.4182 19.2153 * Fixed Nom./Tol.
 Before Optimization After Optimization Rejects Z PPM Z PPM Upper Tail 2.26 11962.3 1.52 64674.9 Lower Tail -2.26 11962.3 -3.00 1350.0 Total Rejects 23924.7 Total Rejects 66024.9

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