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PARALLEL STACKED BLOCKS

Figure 7.1: Schematic of the parallel blocks and corresponding dimension variables.

7.0 Problem Description

A stacked blocks assembly is traditionally considered a one-dimensional tolerance problem. In this case, though, we wish to explore the two-dimensional effects of this assembly. The variable of interest is the gap between the vertical surface on the right and the top right corner of the top-most block. If the surfaces of each block were perfectly parallel to each other, then the gap variation would simply be the error in positioning the stack relative to the vertical surface. Since the block surfaces are not perfectly parallel, the stack may lean from side to side.

This assembly will be modeled two ways. First, the tolerance on the thickness of each block will be considered an implied parallelism. In other words, one side of a block may be shorter than the nominal while the opposite side is longer than the nominal. The entire tolerance zone thus becomes an implied parallelism geometric tolerance. As long as both sides stay within the tolerance zone, the part is still considered okay. By using an implied parallelism, a series of rotations is introduced into the assembly. For the second case, a parallelism geometric tolerance will be applied to each block. This effectively reduces the rotation that may be introduced by each block surface.

Table 7.1: Manufactured Variables (Independent).

 Variable Name Basic Size Initial Tolerance (±) A 3.200 in .000 in * B .736 in .005 in C 1.000 in .000 in * D .800 in .005 in E .800 in .005 in F .800 in .005 in G .800 in .005 in H 1.000 in .000 in *

* These variables are for positioning modeling elements. They are not dimensioned lengths, so they do not have tolerances associated with them. They will not appear in the sensitivity matrices.

7.1 Design Requirements

Table 7.2: Assembly Variables and Specification Limits.

 Variable Name Basic Size Upper Spec. Limit (USL) Lower Spec. Limit (LSL) Gap Width .7360 in .7460 in .7260 in

7.2 Modeling Considerations

• For best results, Joints that have either a kinematic or a geometric rotation should be placed at the center point of the expected rotation. For this assembly, since the rotation of the blocks is important, the joints between each block should be located in the center of the mating surfaces. This requires using two redundant vectors, C and H. These two vectors are used only to position the joints in the middle of their mating surfaces, and are not dimensioned lengths. They must either be assigned a zero tolerance in the modeler or equivalenced in the analyzer.

• This assembly does not have a closure constraint, and therefore no kinematic variables. Thus no closed constraint loops are necessary to model it. A single open loop is all that is required.

• When using a gap specification, it is important to place the open loop endpoints at the points between which the variation is desired, even if it means inserting a dummy variable (such as A). CATS will calculate the variation that would occur along a vector that connects the two loop endpoints. If it is neccessary to use a dummy variable, assign it a zero tolerance in order to eliminate it from the sensitivity matrices. This prevents it from biasing the calculated variations.

7.3 Design Goal

The purpose of this chapter is to illustrate the effects of the parallelism constraint on the variation of the Gap. The problem will be analyzed first without the parallelism geometric tolerance (with the thickness tolerance used as an implied parallelism). The problem will then be analyzed with the parallelism geometric tolerance.

7.4 Part Names and DRFs

Figure 7.2: Diagram showing the location of the part DRFs.

7.5 Kinematic Joints

Four rigid joints are used to model this assembly.

Figure 7.3: Kinematic joint diagram.

Table 7.3: Kinematic Joints Of The Parallel Blocks.

 Joint Number Part One Part Two Joint Type 1 Base Block 1 Rigid 2 Block 1 Block 2 Rigid 3 Block 2 Block 3 Rigid 4 Block 3 Block 4 Rigid

Remarks>> Rigid joints are used to avoid introducing any kinematic degrees of freedom into this assembly. This allows us to solve for the Gap variation using a single open loop. If planar joints were used to model the contact between the blocks, a sliding plane degree of freedom would be introduced at each joint, which would make it impossible to solve for the Gap variation.

7.6 Network Diagram, Vector Loops, and Design Specifications

Figure 7.4: Network diagram and loop diagram for the parallel blocks assembly.

Remarks>> The direction of open loops is important when gap and position specifications are used. CATS assumes the first part is fixed in space and the parts "downstream" all rotate relative to it. This arises due to the non-commutative property of matrix multiplication. To generate the correct open loop direction, create the final endpoint (the moving endpoint) first and the starting endpoint (fixed endpoint) second.

7.7 Geometric Tolerances

Figure 7.5: Geometric tolerance diagram.

The bottom of each block is parallel to the top of the block within a .004 in bandwidth. This value will be used for the second analysis.

7.8 Sensitivity Matrices

Constraint Sensitivities

C Matrix

 B D E F G X -1.0000 .00000 .00000 .00000 .00000 Y .00000 1.0000 1.0000 1.0000 1.0000 q .00000 .00000 .00000 .00000 .00000

G Matrix

 a1 a2 a3 a4 X 3.2000 -2.4000 -1.6000 -.80000 Y -1.0000 1.0000 1.0000 1.0000 q -1.0000 1.0000 1.0000 1.0000

Remarks>> There are no closed loops and no kinematic assembly variables, so A Matrix, B Matrix, F Matrix, and D Matrix are all zero. By applying zero tolerance to the A, C, and H vectors, they are eliminated from all the sensitivity matrices.

Tolerance Sensitivities

C-B-1A Matrix

 B D E F G X -1.0000 .00000 .00000 .00000 .00000 Y .00000 1.0000 1.0000 1.0000 1.0000 q .00000 .00000 .00000 .00000 .00000

G-B-1F Matrix

 a1 a2 a3 a4 X 3.2000 -2.4000 -1.6000 -.80000 Y -1.0000 1.0000 1.0000 1.0000 q -1.0000 1.0000 1.0000 1.0000

7.9 Resultant Tolerances Before Optimization

Table 7.4: Independent Variable Tolerances and Control Factors

 Dim. Name ± Tol. Std. Dev. Cp Dk Cpk Sk Wt. Factor Tol. Basic Fixed B .005 in .0015 1 0.25 0.75 0 1 1 No D .005 in .0015 1 0.25 0.75 0 1 1 No E .005 in .0015 1 0.25 0.75 0 1 1 No F .005 in .0015 1 0.25 0.75 0 1 1 No G .005 in .0015 1 0.25 0.75 0 1 1 No

Table 7.5: Geometric Tolerances (Case 1)

 Feat. Joint Part Name Feature Type Tolerance Band Char. Length b1 1 Block 1 Parallelism .010 in 2.0 in b2 2 Block 2 Parallelism .010 in 2.0 in b3 3 Block 3 Parallelism .010 in 2.0 in b4 4 Block 4 Parallelism .010 in 2.0 in

Remarks>> These geometric tolerances represent case 1--the tolerance on the thickness of the blocks acts as an implied parallelism.

Table 7.6: RSS Percent Rejects (Case 1)

 Spec. Name Spec. Type Nominal Dimension (±) Computed Variation With Geometric Tolerances Without Geometric Tolerances Gap Gap .736 .02247 Z Rej. Z Rej. ZASM = 3.000 USL .746 Upper Tail 1.33 9.1e4 6.00 9.9e-4 (Rejects in PPM) LSL .726 Lower Tail -1.33 9.1e4 -6.00 9.9e-4

Remarks>> With no geometric tolerances applied, the gap variation is simply equal to the tolerance of B.

Table 7.7: RSS Percent Contributions To Gap (Geometric Tolerances Included)

 Variable Name Variance Statistical RSS a1 2.844e-5 50.69 a2 1.600e-5 28.51 a3 7.111e-6 12.67 B 2.778e-6 4.95 a4 1.778e-6 3.17 Other .00000 0.00

Table 7.8: Geometric Tolerances (Case 2)

 Feat. Joint Part Name Feature Type Tolerance Band Char. Length b1 1 Block 1 Parallelism .004 in 2.0 in b2 2 Block 2 Parallelism .004 in 2.0 in b3 3 Block 3 Parallelism .004 in 2.0 in b4 4 Block 4 Parallelism .004 in 2.0 in

Remarks>> These geometric tolerances represent case 2--the parallelism on the blocks is a specified parallelism.

Table 7.9: RSS Percent Rejects (Case 2)

 Spec. Name Spec. Type Nominal Dimension (±) Computed Variation With Geometric Tolerances Without Geometric Tolerances Gap Gap .736 .01009 Z Rej. Z Rej. ZASM = 3.000 USL .746 Upper Tail 2.97 1473 6.00 9.9e-4 (Rejects in PPM) LSL .726 Lower Tail -2.97 1473 -6.00 9.9e-4

Table 7.10: RSS Percent Contributions To Gap (Geometric Tolerances Included)

 Variable Name Variance Statistical RSS a1 4.551e-6 40.24 B 2.778e-6 24.56 a2 2.560e-6 22.63 a3 1.138e-6 10.06 a4 2.844e-7 2.51 Other .00000 0.00

Remarks>> By specifying a parallelism bandwidth, the potential variation in the Gap was more than cut in half, and the rejects were reduced by nearly 180,000 PPM.

7.10 Nominals And Tolerances After Optimization

No tolerance or nominal allocation will be performed on the parallel blocks assembly. The variations that contribute the most to the Gap variation are geometric tolerances, which can't be automatically adjusted by the CATS analyzer. If the user wishes, they can be adjusted manually through the input screens and the analysis repeated.

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